Richard L. Liboff
Kinetic Theory
Classical, Quantum,
and Relativistic Descriptions
Third Edition
With 112 Illustrations
Springer

Graduate Texts in Contemporary Physics
S.T. Ali, J.P. Antoine, and J.P. Gazeau: Coherent States, Wavelets and
Their Generalizations
A. Auerbach: Interacting Electrons and Quantum Magnetism
T.S. Chow: Mesoscopic Physics of Complex Materials
B. Felsager: Geometry, Particles, and Fields
P. Di Francesco, P. Mathieu, and D. Senechal: Conformal Field Theories
A. Gonis and W.H. Butler: Multiple Scattering in Solids
K. Gottfried and T.-M. Yan: Quantum Mechanics: Fundamentals, 2nd Edition
K.T. Hecht: Quantum Mechanics
J.H. Hinken: Superconductor Electronics: Fundamentals and
Microwave Applications
J. Hladik: Spinors in Physics
Yu.M. Ivanchenko and A.A. Lisyansky: Physics of Critical Fluctuations
M. Kaku: Introduction to Superstrings and M-Theory, 2nd Edition
M. Kaku: Strings, Conformal Fields, and M-Theory, 2nd Edition
H.V. Klapdor (ed.): Neutrinos
R.L. Liboff (ed): Kinetic Theory? Classical, Quantum, and Relativistic
Descriptions, 3rd Edition
J.W. Lynn (ed.): High-Temperature Superconductivity
HJ. Metcalf and P. van der Straten: Laser Cooling and Trapping
R.N. Mohapatra: Unification and Supersymmetry: The Frontiers of
Quark-Lepton Physics, 3rd Edition
R.G. Newton: Quantum Physics: A Text for Graduate Students
H. Oberhummer: Nuclei in the Cosmos
G.DJ. Phillies: Elementary Lectures in Statistical Mechanics
R.E. Prange and S.M. Girvin (eds.): The Quantum Hall Effect
S.R.A. Salinas: Introduction to Statistical Physics
B.M. Smirnov: Clusters and Small Particles: In Gases and Plasmas
(continued after index)

Richard L. Liboff
School of Electrical Engineering
Cornell University
Ithaca, NY 14853
richard@ee.cornell.edu
Series Editors
R. Stephen Berry Joseph L. Birman Mark P. Silverman
Department of Chemistry Department of Physics Department of Physics
University of Chicago City College of CUNY Trinity College
Chicago, IL 60637 New York, NY 10031 Hartford, CT 06106
USA USA USA
H. Eugene Stanley Mikhail Voloshin
Center For Polymer Studies Theoretical Physics Institute
Physics Department Tate Laboratory of Physics 424
Boston University The University of Minnesota
Boston, MA 02215 Minneapolis, MN 55455
USA USA
Library of Congress Cataloging-in-Publication Data
Liboff, Richard L., 1931-
Kinetic theory: classical, quantum, and relativistic descriptions / Richard L. Liboff
3rd ed.
p. cm.—(Graduate texts in contemporary physics)
Includes bibliographical references and index.
ISBN 0-387-95551-8 (alk. paper)
1. Kinetic theory of matter. I. Title. II. Series.
QC174.9L54 2003
530.13'6—dc21 2002026658
ISBN 0-387-95551-8 Printed on acid-free paper.
The second edition of this book was published © 1998 by John Wiley & Sons.
© 2003 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010,
USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection
with any form of information storage and retrieval, electronic adaptation, computer software, or by
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The use in this publication of trade names, trademarks, service marks, and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not they are
subject to proprietary rights.
Printed in the United States of America.
987654321 SPIN 10887268
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Springer-Verlag New York Berlin Heidelberg
A member of BertelsmannSpringer Science+Business Media GmbH

To the Memory of
Harold Grad
Teacher and Friend

I learned much from my teachers, more
from my colleagues and most of all from
my students
Rabbi Judah ha-Nasi (ca. 135-C.E.)
Babylonian Talmud (Makkot, 10a)

Ludwig Boltzmann (University of Vienna, courtesy AIP Emilio Segre Visual
Archives)

Preface to the Third Edition
Since the first edition of this work, kinetic theory has maintained its position as
a cornerstone of a number of disciplines in science and mathematics. In physics,
such is the case for quantum and relativistic kinetic theory. Quantum kinetic
theory finds application in the transport of particles and radiation through
material media, as well as the non-stationary quantum-many-body problem.
Relativistic kinetic theory is relevant to controlled thermonuclear fusion and to
a number of problems in astrophysics. In applied mathematics, kinetic theory
relates to the phenomena of localization, percolation, and hopping, relevant to
transport properties in porous media. Classical kinetic theory is the foundation
of fluid dynamics and thus is important to aerospace, mechanical, and chemical
engineering. Important to the study of transport in metals is the Lorentz-
Legendre expansion, which in this new edition appears in an appendix. A new
section in Chapter 1 was included in this new edition that addresses constants of
motion and symmetry. A number of small but important revisions were likewise
made in this new edition. A more complete description of the contents of the
text follows.
The text comprises seven chapters. In Chapter 1, the transformation theory
of classical mechanics is developed for the purpose of deriving Liouville's the-
theorem and the Liouville equation. Four distinct interpretations of the solution
to this equation are presented. The fourth interpretation addresses Gibbs's no-
notion of a distribution function that is the connecting link between the Liouville
equation and experimental observation. The notion of a Markov process is
discussed, and the central-limit theorem is derived and applied to the random
walk problem.
In Chapter 2, the very significant BBKGY hierarchy is obtained from the
Liouville equation, and the first two equations of this sequence are applied in
the derivation of conservation of energy for a gas of interacting particles. In

x Preface to the Third Edition
nondimensionalizing this sequence, parameters emerge that differentiate be-
between weakly and strongly coupled fluids. Correlation functions are introduced
through the Mayer expansions. Examining a weakly coupled fluid composed
of particles interacting under long-range interaction leads to the Vlasov equa-
equation and the closely allied concept of a self-consistent solution. Prigogine's
diagrammatic technique and related operator formalism for examining the
Liouville equation are described. The Bogoliubov ansatz concerning the equi-
equilibration of a gas, as well as the Klimontovich formulation of kinetic theory,
are also included in this chapter.
The Boltzmann equation is derived in Chapter 3 and applied to the derivation
of fluid dynamic equations and the H theorem. Poincare's recurrence theorem
is proved and is discussed relative to Boltzmann's H theorem. Transport coeffi-
coefficients are defined, and the Chapman-Enskog expansion is developed. Results
of this technique of solution to the Boltzmann equation are compared with
experimental data and are found to be in good agreement for various molec-
molecular samples. Grad's method of solution of the Boltzmann equation involving
expansion in tensor Hermite polynomials is described. The chapter continues
with a derivation of the Druy vesteyn distribution relevant to a current carrying
plasma in a dc electric field. In the last section of the chapter, the topic of irre-
versibility is revisited. Ergodic and mixing flows are discussed. Action-angle
variables are introduced, and the notions of classical degeneracy and resonant
domains in phase space are described in relation to the chaotic behavior of
classical systems. A statement of the closely allied KAM theorem is also given.
In the first half of Chapter 4, the Vlasov equation is applied to linear
wave theory for a two-component plasma composed of electrons and heavy
ions. Landau damping and the Nyquist criterion for wave instabilities are
described. The chapter continues with derivations of other important ki-
kinetic equations: Krook-Bhatnager-Gross (KBG), Fokker-Planck, Landau, and
Balescu-Lenard equations. A table is included describing the interrelation of
the classical kinetic equations discussed in the text. The chapter concludes with
a description of the widely used Monte Carlo numerical analysis in kinetic
theory.
Quantum kinetic theory is developed in Chapter 5. A brief review of ba-
basic principles leads to a description of the density matrix, the Pauli equation,
and the closely related Wigner distribution. Various equivalent forms of the
Wigner-Moyal equation are derived. A quantum modified KBG equation is
applied to photon transport and electron propagation in solids. Thomas-Fermi
screening and the Mott transition are also discussed. The Uehling-Uhlenbeck
quantum modified Boltzmann equation is developed and applied to a Fermi
liquid. The chapter continues with an overview of classical and quantum hier-
hierarchies of equations connecting reduced distributions. A table of hierarchies
is included where the reader is easily able to view distinctions among these
sets of equations. The Kubo formula, described previously in Chapter 3, is
revisited and applied to the derivation of a quantum expression for electrical

Preface to the Third Edition xi
conductivity. The chapter concludes with an introduction to Green's function
analysis and related diagrammatic representations.
Chapter 6 addresses relativistic kinetic theory. The discussion begins with el-
elementary concepts, including a statement of Hamilton's equations in covariant
form. Stemming from a covariant distribution function in four space, together
with Maxwell's equations in covariant form, a relativistic Vlasov equation
is derived for a plasma in an electromagnetic field. An important compo-
component of this chapter is the derivation and compilation of a table of Lorentz
invariants in kinetic theory. The chapter continues with a derivation of the
relativistic Maxwellian and concludes with a brief description of relativity in
non-Cartesian coordinates.
Chapter 7 has been added in this new edition, and addresses kinetic and
thermal properties of metals and amorphous media. The first component
of the chapter begins with a review of the notion of thermopower, and the
Wiedermann-Franz law is derived. The discussion continues with a formula-
formulation for electrical and thermal conductivity in metals (encountered previously
in Chapter 5), stemming from the quantum Boltzmann equation, in which
Bloch's classic low-temperature T5 dependence of metallic resistivity and
canonical high-temperature linear T dependence are derived. In addition, the
formalism yields a residual resistivity at 0 K. The chapter continues with a
discussion of properties of amorphous media and related processes of local-
localization, hopping, and percolation. Bloch waves and the notion of extended
states are reviewed. Anderson's parameter of the ratio of the spread-of-states
to the band width is introduced. Localization occurs at some critical value of
this parameter. At smaller values of the parameter, energies of localized and
extended states are separated at the "mobility edge." Transition of the Fermi
energy from the domain of extended states to the domain of localized states
represents the Mott metal-insulator transition. Mechanisms of electrical con-
conduction are discussed in three temperature intervals in which the notions of
thermally assisted and variable-range hopping emerge. The chapter continues
with the concepts of bond and site percolation. A number of percolation scal-
scaling laws are discussed. The chapter concludes with a review of localization
in second quantization. Throughout the chapter, many discussions related to
material science are included.
Each chapter is preceded by a brief introductory statement of the subject
matter contained in the chapter. Problems appear at the end of each chapter,
many of which carry solutions. A number of problems include self-contained
descriptions of closely allied topics. In such cases, these are listed in the chapter
table of contents under the heading, Topical Problems. In addition to references
cited in the text, a comprehensive list of references is included in Appendix E.
Assorted mathematical formulas are included in Appendices A and B, includ-
including a list of properties of Laguerre and Hermite polynomials (B4). Appendix
D, addressing the Lorentz-Legendre expansion in kinetic theory, is new to this
edition.

xii Preface to the Third Edition
Stemming from the observation that science and society are inextricably
entwined, a time chart is included (Appendix E) listing early contributors to
science and technology of the classical Greek and Roman eras. The reader will
note that a central figure in this display is the Greek philosopher, Democritus,
who, at about 400 BCE, was the first to propose an atomic theory of matter.
Readers of my earlier work [Introduction to the Theory of Kinetic Equations,
Wiley, New York A969)] will recall that it, too, included a time chart describing
contributions to dynamics from the fifteenth to the nineteenth centuries. The
appendix on Mathematical Formulas has been expanded in this new edition to
include a list of properties of Laguene and Hermite polynomials (Appendix
B4).
Many individuals have contributed to the development of this work. I remain
indebted to these kind colleagues and would like here to express my sincere
gratitude for their encouragement, support, and constructive criticism: Sidney
Leibovich, Terrence Fine, Robert Pay, Christof Litwin, Kenneth Gardner, Neal
Maresca, K. C. Liu, Danny Heffernan, Edwin Dorchek, Philip Bagwell, Ronald
Kline, Steve Seidman, S. Ramakrishna, G. George, Timir Datta, William Mor-
rell, Wayne Scales, Daniel Koury, Erich Kunhardt, Marvin Silver, Hercules
Neves, James Hartle, Kenneth Andrews, Clifford Pollock, Veit Elser, Chuntong
Ying, Michael Parker, Jack Freed, Richard Zallen, Abner Shimony, Philip
Holmes, Lloyd Hillman, Arthur Ruoff, L. Pearce Williams, Lloyd Motz, John
Guckenheimer, Isaac Rabinowitz, Gregory Schenter, and Ilya Prigogine.
Some of these individual are former students. It is due to my association
with these gifted and talented colleagues that the talmudic inscription for this
work is motivated.
Peace,
Richard L. Liboff

Contents
Preface to the Third Edition ix
1 The Liouville Equation 1
1.1 Elements of Classical Mechanics 2
1.1.1 Generalized Coordinates and the Lagrangian ... 2
1.1.2 Hamilton's Equations 4
1.1.3 Constants of the Motion 6
1.1.4 T-Space 7
1.1.5 Dynamic Reversibility 9
1.1.6 Equation of Motion for Dynamical Variables . . . 10
1.2 Canonical Transformations 11
1.2.1 Generating Functions 11
1.2.2 Generating Other Transformations 14
1.2.3 Canonical Invariants 15
1.2.4 Group Property of Canonical Transformations . . 15
1.2.5 Constants of Motion and Symmetry 16
1.3 Liouville Theorem 17
1.3.1 Proof 17
1.3.2 Geometric Significance 18
1.3.3 Action Generates the Motion 19
1.4 Liouville Equation 20
1.4.1 The Ensemble: Density in Phase Space 20
1.4.2 First Interpretation of D(q, p, t) 20
1.4.3 Most General Solution: Second Interpretation of
D(q,p,t) 22
1.4.4 Incompressible Ensemble 23
1.4.5 Method of Characteristics 24

xiv Contents
1.4.6 Solutions to the Initial-Value Problem 25
1.4.7 Liouville Operator 27
1.5 Eigenfunction Expansions and the Resolvent 29
1.5.1 Liouville Equation Integrating Factor 29
1.5.2 Example: The Ideal Gas 30
1.5.3 Free-Particle Propagator 32
1.5.4 The Resolvent 33
1.6 Distribution Functions 35
1.6.1 Third Interpretation of D(q, p, t) 35
1.6.2 Joint-Probability Distribution 36
1.6.3 Reduced Distributions 37
1.6.4 Conditional Distribution 37
1.6.5 s-Tuple Distribution 38
1.6.6 Symmetric Properties of Distributions 39
1.7 Markov Process 41
1.7.1 Two-Time Distributions 41
1.7.2 Chapman-Kolmogorov Equation 41
1.7.3 Homogeneous Processes in Time 43
1.7.4 Master Equation 43
1.7.5 Application to Random Walk 44
1.8 Central-Limit Theorem 47
1.8.1 Random Variables and the Characteristic Function 47
1.8.2 Expectation, Variance, and the Characteristic
Function 47
1.8.3 Sums of Random Variables 49
1.8.4 Application to Random Walk 50
1.8.5 Large n Limit: Central-Limit Theorem 54
1.8.6 Random Walk in Large n Limit 55
1.8.7 Poisson and Gaussian Distributions 57
1.8.8 Covariance and Autocorrelation Function .... 59
Problems 60
2 Analyses of the Liouville Equation 77
2.1 BBKGY Hierarchy 78
2.1.1 Liouville, Kinetic Energy, and Remainder
Operators 78
2.1.2 Reduction of the Liouville Equation 80
2.1.3 Further Symmetry Reductions 81
2.1.4 Conservation of Energy from BYi and BY2 ... 83
2.2 Correlation Expansions: The Vlasov Limit 86
2.2.1 Nondimensionalization 86
2.2.2 Correlation Functions 89
2.2.3 The Vlasov Limit 90
2.2.4 The Vlasov Equation: Self-Consistent Solution . 92
2.2.5 Debye Distance and the Vlasov Limit 95

Contents xv
2.2.6 Radial Distribution Function 96
2.3 Diagrams: Prigogine Analysis 97
2.3.1 Perturbation Liouville Operator 98
2.3.2 Generalized Fourier Series 99
2.3.3 Interpretation of a/Coefficients 99
2.3.4 Equations of Motion for a(k) Coefficients 102
2.3.5 Selection Rules for Matrix Elements 103
2.3.6 Properties of Diagrams 104
2.3.7 Long-Time Diagrams: The Boltzmann Equation . 107
2.3.8 Reduction of Time Integrals 108
2.3.9 Application of Diagrams to Plasmas 113
2.4 Bogoliubov Hypothesis 115
2.4.1 Time and Length Intervals 115
2.4.2 The Three Temporal Stages 115
2.4.3 Bogoliubov Distributions 117
2.4.4 Density Expansion 117
2.4.5 Construction of F2@) 119
2.4.6 Derivation of the Boltzmann Equation 121
2.5 Klimontovich Picture 124
2.5.1 Phase Densities 124
2.5.2 Phase Density Averages 125
2.5.3 Relation to Correlation Functions 126
2.5.4 Equation of Motion 127
2.6 Grad's Analysis 127
2.6.1 Liouville Equation Revisited 127
2.6.2 Truncated Distributions 128
2.6.3 Grad's First and Second Equations 129
2.6.4 The Boltzmann Equation 130
Problems 132
3 The Boltzmann Equation, Fluid Dynamics, and Irreversibility 137
3.1 Scattering Concepts 138
3.1.1 Separation of the Hamiltonian 138
3.1.2 Scattering Angle 140
3.1.3 Cross Section 143
3.1.4 Kinematics 146
3.2 The Boltzmann Equation 149
3.2.1 Collisional Parameters and Derivation 149
3.2.2 Multicomponent Gas 153
3.2.3 Representation of Collision Integral for Rigid
Spheres 153
3.3 Fluid Dynamic Equations and the Boltzmann TL Theorem 155
3.3.1 Collisional Invariants 155
3.3.2 Macroscopic Variables and Conservative Equations 157

xvi Contents
3.3.3 Conservation Equations and the Boltzmann
Equation 159
3.3.4 Temperance: Variance of the Velocity Distribution 161
3.3.5 Irreversibility 162
3.3.6 Poincare Recurrence Theorem 163
3.3.7 Boltzmann and Gibbs Entropies 164
3.3.8 Boltzmann's TL Theorem 166
3.3.9 Statistical Balance 167
3.3.10 The Maxwellian 168
3.3.11 The Barometer Formula 170
3.3.12 Central-Limit Theorem Revisited 172
3.4 Transport Coefficients 174
3.4.1 Response to Gradient Perturbations 174
3.4.2 Elementary Mean-Free-Path Estimates 178
3.4.3 Diffusion and Random Walk 186
3.4.4 Autocorrelation Functions, Transport Coefficients,
and Kubo Formula 188
3.5 The Chapman-Enskog Expansion 194
3.5.1 Collision Frequency 194
3.5.2 The Expansion 195
3.5.3 Second-Order Solution 199
3.5.4 Thermal Conductivity and Stress Tensor 202
3.5.5 Sonine Polynomials 203
3.5.6 Application to Rigid Spheres 207
3.5.7 Diffusion and Electrical Conductivity 208
3.5.8 Expressions of £2(/'q) for Inverse Power Interaction
Forces 209
3.5.9 Interaction Models and Experimental Values ... 211
3.5.10 The Method of Moments 213
3.6 The Linear Boltzmann Collision Operator 217
3.6.1 Symmetry of the Kernel 217
3.6.2 Negative Eigenvalues 218
3.6.3 Comparison of Boltzmann and Liouville Operators 219
3.6.4 Hard and Soft Potentials 220
3.6.5 Maxwell Molecule Spectrum 220
3.6.6 Further Spectral Properties 222
3.7 The Druyvesteyn Distribution 225
3.7.1 Basic Parameters and Starting Equations 225
3.7.2 Legendre Polynomial Expansion 227
3.7.3 Reduction of 7o(/o \ F) 232
3.7.4 Evaluation of 70 and/i Integrals 233
3.7.5 Druyvesteyn Equation 236
3.7.6 Normalization, Velocity Shift, and Electrical
Conductivity 237

Contents xvii
3.8 Further Remarks on Irreversibility 240
3.8.1 ErgodicFlow 240
3.8.2 Mixing Flow and Coarse Graining 243
3.8.3 Action-Angle Variables 245
3.8.4 Hamilton-Jacobi Equation 248
3.8.5 Conditionally Periodic Motion and Classical
Degeneracy 249
3.8.6 Bruns's Theorem 253
3.8.7 Anharmonic Oscillator 253
3.8.8 Resonant Domains and the KAM Theorem .... 255
Problems 258
4 Assorted Kinetic Equations with Applications to Plasmas and
Neutral Fluids 278
4.1 Application of the Vlasov Equation to a Plasma 279
4.1.1 Debye Potential and Dielectric Constant 279
4.1.2 Waves, Instabilities, and Damping 285
4.1.3 Landau Damping 288
4.1.4 Nyquist Criterion 292
4.2 Further Kinetic Equations of Plasmas and Neutral Fluids . 294
4.2.1 Krook-Bhatnager-Gross Equation 295
4.2.2 KBG Analysis of Shock Waves 298
4.2.3 The Fokker-Planck Equation 301
4.2.4 The Landau Equation 307
4.2.5 The Balescu-Lenard Equation 309
4.2.6 Convergent Kinetic Equation 315
4.2.7 Fokker-Planck Equation Revisited 317
4.3 Monte Carlo Analysis in Kinetic Theory 319
4.3.1 Master Equation 319
4.3.2 Equivalence of Master and Integral Equations . . 320
4.3.3 Interpretation of Terms 321
4.3.4 Application of Random Numbers 323
4.3.5 Program for Evaluation of Distribution Function . 324
Problems 325
5 Elements of Quantum Kinetic Theory 329
5.1 Basic Principles 330
5.1.1 The Wave Function and Its Properties 330
5.1.2 Commutators and Measurement 332
5.1.3 Representations 334
5.1.4 Coordinate and Momentum Representations . . . 335
5.1.5 Superposition Principle 337
5.1.6 Statistics and the Pauli Principle 339
5.1.7 Heisenberg Picture 339
5.2 The Density Matrix 341

xviii Contents
5.2.1 The Density Operator 342
5.2.2 The Pauli Equation 346
5.2.3 The Wigner Distribution 351
5.2.4 Weyl Correspondence 354
5.2.5 Wigner-Moyal Equation 357
5.2.6 Homogeneous Limit: Pauli Equation Revisited . . 359
5.3 Application of the KBG Equation to Quantum Systems . 363
5.3.1 Equilibrium Distributions 363
5.3.2 Photon Kinetic Equation 366
5.3.3 Electron Transport in Metals 369
5.3.4 Thomas-Fermi Screening 376
5.3.5 Mott Transition 378
5.3.6 Relaxation Time for Charge-Carrier Phonon
Interaction 379
5.4 Quantum Modifications of the Boltzmann Equation . . . 385
5.4.1 Quasi-Classical Boltzmann Equation 385
5.4.2 Kinetic Theory for Excitations in a Fermi Liquid 387
5.4.3 H Theorem for Quasi-Classical Distribution . . . 392
5.5 Overview of Classical and Quantum Hierarchies 394
5.5.1 Second Quantization and Fock Space 394
5.5.2 Classical and Quantum Distribution Functions . . 395
5.5.3 Equation^ of Motion 399
5.5.4 Generalized Hierarchies 400
5.6 Kubo Formula Revisited 403
5.6.1 Charge Density and Current 403
5.6.2 Identifications for Kubo Formula 404
5.6.3 Electrical Conductivity 405
5.6.4 Reduction to Drude Conductivity 407
5.7 Elements of the Green's Function Formalism 408
5.7.1 Schrodinger Equation Green's Function 408
5.7.2 The s-Body Green's Function 410
5.7.3 Averages and the Green's Function 410
5.7.4 The Quasi-Free Particle 413
5.7.5 One-Body Green's Function 414
5.7.6 Retarded and Advanced Green's Functions . . . . 415
5.7.7 Coupled Green's Function Equations 415
5.7.8 Diagrams and Expansion Techniques 416
5.8 Spectral Function for Electron-Phonon Interactions ... 421
5.8.1 Hamiltonian 421
5.8.2 Green's Function Equations of Motion 423
5.8.3 The Spectral Function 425
5.8.4 Lorentzian Form 426
5.8.5 Lifetime and Energy of a Quasi-Free Particle . . 428
Problems 429

Contents xix
6 Relativistic Kinetic Theory 449
6.1 Preliminaries 450
6.1.1 Postulates 450
6.1.2 Events, World Lines, and the Light Cone 450
6.1.3 Four-Vectors 451
6.1.4 Lorentz Transformation 452
6.1.5 Length Contraction, Time Dilation, and Proper
Time 454
6.1.6 Covariance, Hamiltonian, and Hamilton's
Equations 454
6.1.7 Criterion for Relativistic Analysis 455
6.2 Covariant Kinetic Formulation 456
6.2.1 Distribution Function 456
6.2.2 One-Particle Liouville Equation 457
6.2.3 Covariant Electrodynamics 458
6.2.4 Vlasov Equation 459
6.2.5 Covariant Drude Formulation of Ohm's Law . . . 460
6.2.6 Lorentz Invariants in Kinetic Theory 461
6.2.7 Relativistic Electron Gas and Darwin Lagrangian 464
6.3 The Relativistic Maxwellian 467
6.3.1 Normalization 467
6.3.2 The Nonrelativistic Domain 469
6.4 Non-Cartesian Coordinates 470
6.4.1 Covariant and Contravariant Vectors 470
6.4.2 Metric Tensor 471
6.4.3 Lagrange's Equations 472
6.4.4 The Christoffel Symbol 473
6.4.5 Liouville Equation 474
Problems 474
7 Kinetic Properties of Metals and Amorphous Media 480
7.1 Metallic Electrical and Thermal Conduction 482
7.1.1 Background 482
7.1.2 Thermopower 484
7.1.3 Electron-Phonon Scattering Matrix Elements . . 487
7.1.4 Quantum Boltzmann Equation 493
7.1.5 Perturbation Distribution 496
7.1.6 Electrical Resistivity 498
7.1.7 Scale Parameters of p0 503
7.1.8 Electron Distribution Function 504
7.1.9 Thermal Conductivity 504
7.2 Amorphous Media 505
7.2.1 Background 505
7.2.2 Localization 508
7.2.3 Conduction Mechanisms and Hopping 511

xx Contents
7.2.4 Percolation Phenomena 514
7.2.5 Pair-Connectedness Function and Scaling .... 517
7.2.6 Localization in Second Quantization 520
Problems 524
Location of Key Equations 527
List of Symbols 529
Appendices
A Vector Formulas and Tensor Notation 537
A.I Definitions 537
A.2 Vector Formulas and Tensor Equivalents 539
B Mathematical Formulas 541
B.I Tensor Integrals and Unit Vector Products 541
B.2 Exponential Integral, F Function, £ Function, and Error
Function 542
B.2.1 Exponential Integral 542
B.3 Other Useful Integrals 545
B.4 Hermite and Laguerre Polynomials and the Hypergeometric
Function 546
C Physical Constants 550
D Lorentz-Legendre Expansion 552
E Additional References 554
E.I Early Works 554
E.2 Recent Contributions to Kinetic Theory and Allied Topics 558
F Science and Society in the Classical Greek and Roman Eras 563
Index 565

CHAPTER 1
The Liouville Equation
Introduction
This chapter begins with a review of the basic principles and equations of
classical mechanics. These lead naturally to the Liouville theorem, which to-
together with the notions of ensemble and phase space, give rise to the Liouville
equation. This equation is perhaps the most significant relation in all of clas-
classical kinetic theory, and in the remainder of the chapter various techniques
of solution to this equation, as well as various interpretations of its solution,
are presented. Thus, for example, it is found that the Liouville equation is an
equation of motion for the TV -particle joint-probability distribution, where TV
denotes the number of particles in the system. Integration of this distribution
over subdomains in phase space gives reduced distributions whose equations
of motion (BBKGY Eqs.) are developed in Chapter 2.
A discussion is included of the Chapman-Kolmogorov equation and the
closely allied master equation, which is applied to the random-walk problem.
The master equation emerges again in Chapter 5 in analysis of the quantum
mechanical density matrix.
The chapter concludes with a brief description of probability theory and a
derivation of the central-limit theorem. This theorem is applied to obtain the
long-time displacement of the random walk and yields the Gaussian distribu-
distribution. This distribution is fundamental to diffusion theory and is encountered
again in Chapter 3. The central-limit theorem is likewise again encountered in
Chapter 3 in a derivation of the distribution of center-of-mass momenta of a
stationary gas of molecules. The discrete Poisson distribution is obtained from
random-walk results and applied to small shot noise.

2 1. The Liouville Equation
Classical mechanics and probability theory are the foundations of kinetic
theory. Consequently, a firm grasp of the basic principles and relations of these
formalisms will lead to a richer working knowledge of kinetic theory.
1.1 Elements of Classical Mechanics
1.1.1 Generalized Coordinates and the Lagrangian
The number of degrees of freedom a given system has is equal to the min-
minimum number of independent parameters necessary to uniquely determine
the location and orientation of the system in physical space. Such indepen-
independent parameters are called generalized coordinates. Thus, for example, a rigid
dumbbell molecule free to move in three-space has five degrees of freedom:
three to locate the center of mass of the molecule and two (angles) to determine
the orientation of the molecule. A fluid of TV such molecules has 5N degrees
of freedom. A rigid triangular molecule has six degrees of freedom: five to
locate an edge of the triangle and an additional angle to fix the orientation of
the plane of the molecule about this axis.
For a system with TV degrees of freedom, generalized coordinates are labeled
#i, #2> • • •, Qn- These parameters may be taken to comprise the components
of an TV -dimensional vector,
= (q\,q2, -",qN) A.1)
The corresponding generalized velocities also comprise components of an
TV-dimensional vector, and we write
q = (<7i,<?2, ...,qN) A.2)
The composite vector (q, q) determines the state of the system.
Let the system at hand exist in a conservative force field with kinetic energy
T(q, q) and potential V(q). The Lagrangian of the system is then given by
L(q,q) = T(q,q)-V(q) A.3)
The dynamics of the system may be formulated in terms of Hamilton's prin-
principle. This principle states that the motion of the system between two fixed
points, (q, t)\ and (q, rJ> renders the action integral:
S= f L(q,q,t)dt A.4)
an extremum.1 That is,
r2
8 / L(q,q,t)dt =0 A.5)
More precisely, a minimum.

1.1 Elements of Classical Mechanics
Q2--
fi t2
FIGURE 1.1. End points qx and q2 are held fixed in the variation A.5).
where 8 denotes a variation about the motion of the system.2 See Fig. 1.1
Lagrange's equations
The dynamical principle A.5) may be employed to obtain a differential
equation for L(q,q,t). That is, let q(t) be the motion that renders S an
extremum.
Effecting the variation A.5) and Taylor-series expanding the integrand gives
= / L(q+Sq,q+Sq,t)dt - L(q,q,t)dt
a
8L
—
8q
SL
—
8q
f
- / L(q,q,t)dt A.6)
Note that
d (dL
d (dL
Inserting this expression into the preceding equation gives
3L
Sq
1 J\
d SdLX
Sqdt = 0
Jq dt \dqj_
Since the end points of the trajectory are held fixed in the variation, the first
bracketed "surface terms" vanish. Insofar as the remaining integral must vanish
for any arbitrary, infinitesimal variation 8q, it is necessary that
A.7)
A = 1,2,..., N)
Furthermore, the variation is a virtual displacement, that is, on that occurs in
zero time.

4 1. The Liouville Equation
Here we have returned to the component representation of (q.q). Equa-
Equations A.7) are called Lagrange's equations.
1.1.2 Hamilton's Equations
When written in the form
d (dL\ dL
-r \-rr =— A.8)
dt \dqij dqt
Lagrange's equations A.7) reveal the following important observation: If the
coordinate qt is missing in the Lagrangian, the quantity dL/dqt is constant in
time. In this event, we say that the coordinate q\ is cyclic or ignorable.
To better exhibit this symmetry principle of mechanics, we effect the
following transformation:
(qi,qi) -> (quPi) A.9)
where
dL
P/ = tt- A.10)
The variable /?/, so defined, is called the canonical momentum or momentum
conjugate to the coordinate q\.
The transformation A.9) is accomplished through a Legendre transforma-
transformation (familiar to thermodynamics). This transformation carries the Lagrangian
L(q, q) to the Hamiltonian H(q, p) by the following recipe:
L
Equations of motion in terms of the Hamiltonian are obtained by forming
the differential of the latter equation.
~d(dL)a
LU,r
dL
dqrHl
dL
dqrHl
dL
dqt qi
dL "
dt
With Lagrange's equations A.7) and the definition A.10), we obtain
r) T
dH(q, p, t) = \\qi dp{ - p{ dq{) - — dt
*—r dt
Expanding the left side of this equation and identifying terms gives
A.12)
together with dH/dt = -dL/dt. Equations A.12) are called Hamilton's
equations.

1.1 Elements of Classical Mechanics
The Hamiltonian and the energy
An important theorem concerning the Hamiltonian is as follows. Consider a
system of particles with radii vectors {r/}. Consider that the transformation to
the set of generalized coordinates,
i\ =ri(qu...,qN) A.13)
is independent of time and that the Lagrangian of the system is not an explicit
function of time. Then, with Lagrange's equations A.7), we obtain
dL ^—v dL . dL
It =
dL
)
d (dL
It follows that
d
7l
We may conclude that for such systems H is constant in time. This constant
may further be identified with the energy, E (see Problem 1.32).
H
constant
A.15)
Electrodynamic and relativistic Hamiltonian
Two important Hamiltonians that emerge in physics are as follows. The
Hamiltonian of a particle of charge e and mass m that moves in an elec-
electromagnetic field with vector potential A and scalar potential O is given by
(see Problem A.22)
A.16)
e
p — m\ + -A
c
The electric £ and magnetic field B are related to the fields through the relations
A.17)
c dt
Hamilton's equations applied to the preceding Hamiltonian returns the Lorentz
force law
A.18)

6 1. The Liouville Equation
The second example addresses a relativistic particle with rest mass m. It is
given by3'4
H = y/p2c2 + m2c4 + V(r) A.19)
The relativistic momentum has the form
i = ^ A.19a)
1 — pz c
In the preceding expressions, c is the speed of light.
A transformation of the potentials O and A in which the fields £, B
are invariant is called a 'gauge transformation.' Consider, for example, the
transformation
AA A + A.19b)
c dt
Substitution of these expressions into A.17) indicates that the fields £, B remain
the same. (See Problem 1.22).
1.1.3 Constants of the Motion
If a dynamical function
W = W(qu >>.,
remains constant as the motion of the system unfolds in time, W is a constant
of the motion or an integral of the system.
How many such constants are there? For a system with TV degrees of freedom,
integration of Hamilton's equations A.12) gives
dH /" dH
Pi = ► Pi(t) - pi@) - —dt
8q k 8(h A.20)
r dH
/ —
Jo dpi
qi(O qi@)+ / dt
opi
Thus we find that the state of the system is specified by 2N constants,
{^/(O), pi@)}.5 This conclusions may also be inferred geometrically.
In 2N + 1-dimensional space, 2N surfaces define a curve. For example, in
three-space, the two surfaces
f(x,y,z) = Cx
g(x,y,z) = C2
3 An alternative expression for H written in covariant form is given in Chapter 6.
4The Hamiltonian of an aggregate of TV interacting particles is given by D.27).
5 Although initial values are constant, subsequent motion may grow chaotic. These
topics are further discussed in Section 3.8

1.1 Elements of Classical Mechanics
r-space
FIGURE 1.2. For a harmonic oscillator, the state of the system lies on an ellipse in
P-space.
define a curve. Only one of three variables x, y, z is independent. A curve is a
one-dimensional locus. Thus the intersection of the two surfaces written above
may be written in parametric form as
x=x(t), y = y(O, z = z(t)
Before proceeding further with this geometric construction, we define F-space.
1.1.4 F-Space
For a system with TV degrees of freedom. F-space is a 2N dimensional Carte-
Cartesian space whose axes are the {qt, pt} variables. Thus the state of the system at
any given instant (q\, ..., qN, p\,..., pN) is a single point in 27V-dimensional
F-space.
Let the system be a single harmonic oscillator for which F-space is two-
dimensional. The orbit of the oscillator may be written
q — a cos cot, p = b sin cot
from which we find
1
Thus the state of the system (q, p) lies on an ellipse in F-space. See Fig. 1.2.
This example illustrates that time is suppressed in F-space. All we know
from the curve shown in Fig. 1.2 is that the particle has q, p values that lie
somewhere on the ellipse. We do not know what these values are at a specific
time.

8 1. The Liouville Equation
System trajectory
/V-dimensional r-space
FIGURE 1.3. The system trajectory is a curve in f-space. At any instant of time, the
curve intersects F-space at a single point (q, p).
FIGURE 1.4. Dynamical trajectory of a harmonic oscillator in three-dimensional
B+1) f-space. Projection onto F-space is an ellipse.
Time is exhibited explicitly in 2N + 1 f-space. This Cartesian space is
comprised of F-space and an additional orthogonal time axis. The system
trajectory or dysfunctional orbit,
[<?i@, • • •, <?/v@; Pi(O, • • • > Pn(O]
is a curve in f-space. See Fig. 1.3. The system trajectory for the harmonic
oscillator is shown in Fig. 1.4.
A constant of the motion
Wl(q,p,t) = Cl
is constant on the system trajectory. But W = C\ is a surface in f-space (that
is, a 27V-dimensional locus). The intersection of 27V such independent surfaces

1.1 Elements of Classical Mechanics
/
Projected
path in
-space
Po
The
dynamical
path
The plane
Po=P (The "energy surface")
P
FIGURE 1.5. Dynamical path of a free particle in three-dimensional f-space.
(W\, W2, • • •, WN) determines a curve (one-dimensional locus) in F-space.6
This curve may be written
Pi = Pi(ri),
t(r]), Z
A.21)
which is the system trajectory. So again we find that the dynamical orbit is deter-
determined by 2N constants of the motion. In practice, we are often more interested
in the projected motion in F-space. This motion, we recall, is determined by
2N — 1 constants. See also Section 3.8.6.
These concepts are well illustrated by the simple example of a free particle
moving in one dimension. Since this system has one degree of freedom, it has
only two constants of motion:
= X — Vt
= P
Intersection of these two surfaces in three-dimensional f-space reveals the
orbit
_L_ ,
X — Xq -| 1
m
See Fig. 1.5. These topics are returned to in Section 3.8.
7.7.5 Dynamic Reversibility
Consider a system whose Hamiltonian is invariant under time reversal. That
is, H(t) — H(—t). Suppose {g@, p(t)} is a trajectory for this system. Then
another solution is {q(-f), -p(-t)}. To establish this property first, set t' — -t
6Such constants that intersect the energy surface are called isolating integrals.
Constants such as /(//), where / is any smooth function, are evidently constant on
the energy surface and therefore have no intersection with it.

10 1. The Liouville Equation
P^
Po
-Po
-Pi
<7o
(a)
(b)
FIGURE 1.6. (a) A motion and (b) dynamically reversed motion.
in Hamilton's equations A.12). There results
dp
dt'
dH
dq
It'
dH
~dp~
So if we set p' — —/?, q' = q, Hamilton's equations A.12) are returned.
It follows that, if {g@> p@} is a solution to Hamilton's equations, then
{q(—t), —p(—t)} is also a solution. This situation is depicted in Fig. 1.6.
1.1.6 Equation of Motion for Dynamical Variables
Let u(q, p, t) denote a dynamical variable. Then we may write
du
It
du
du
dpi
du
dt
With Hamilton's equations, the preceding equation may be rewritten
du
du dH dH du \ du
dqi dpi dqi dpi
dt
A.22)
A.23)
Let us introduce the following Poisson bracket notation relevant to two
dynamical variables A and B:
[A, B]
dB dA
dpi
A.24)

1.2 Canonical Transformations
11
Then A.23) may be written
= [u, H] +
A.25)
This is the equation of motion for the dynamical variable u(q,p,t). The
following properties of Poisson brackets are easily verified:
A.26a)
A.26b)
A.26c)
A.26d)
A.26e)
[A,B] = -
[A, c] — 0 (c is constant)
[A + B,F] = A[B,F]
[AB,F] = A[B,F] + [A,F]B
[A, [B, F]] + [B[F, A]] + [F, [A, B]] = 0
d
Jt
[A,B]
,B
dt
[A,B(A)]=0
A,
8B
IF
A.26f)
A.26g)
The relation A.26e) is called Jacob's identity. In A.26g), the function B(A) is
assumed to have a Taylor series expansion.
In the event that a dynamical variable u is not explicitly dependent on time,
then du/dt = 0 and A.25) becomes
du
It
[u,H]
A.27)
If, further, u — u(H), then with the property A.26e) we may write
du
dt
0
A.28)
So we may conclude that any dynamical variable that is function only of the
Hamiltonian is a constant of the motion.
With A.25), Hamilton's equations A.12) may be rewritten
q = [q, H]
P = [p, H]
Note also the fundamental Poisson bracket relations:
[<?/, qr] = [pi, Pr] - 0, [qi, pi>]
A.29a)
A.29b)
A.30)
1.2 Canonical Transformations
1.2.1 Generating Functions
Consider the transformation of variables
q, p -> q',p'
B.1)

12 1. The Liouville Equation
If these two sets of variables are to describe the dynamics of the same system,
then when
8 I L(q,q)dt = 0 B.2)
we must also find
8 Lf(q\q')dt=0 B.3)
When this agreement is obeyed, the transformation B.1) is said to be canonical.
The variational equality B.3) will follow from B.2) provided a function
G\(q, q\ t) exists such that
rr -n T'r ' "\ , dGi(q,q\t)
L(q, q) = L(q ,q)-\ B.4)
at
Substituting this relation into B.2) gives
8 f Ldt = 0 = 8 f L\q\ q')dt + &{Gx[q, q\q), t]t2 - Gx[q, q{q\
= &JL'(q',q')dt B.5)
We may conclude that the existence of a function G\(q,q',t) that satisfies B.4)
guarantees that the transformation B.1) is canonical.
The specifics of this transformation are obtained by casting B.4) in terms
of the Hamiltonian of the system (deleting summational indexes):
- H(p, q) = l^pq -H(q,p)+ — B.6)
Transposing terms gives
J2 p'dq' + (/// ~H)dt= dGM, <]', 0 B.7)
With the left side of B.7) written for the expansion of dG(q, qr, t), we may
conclude that
dGx(q,q') , dGx(q,q')
aq dq'
B.8)
Here we have assumed that G\ is not explicitly dependent on time.
Otherwise, B.7) further implies that
l- = H'-H B.9)
dt
Equations B.8) indicate the manner in which G\(q,q') "generates" the
transformation B.1). For G\ non-time dependent, B.7) gives
B.10)

1.2 Canonical Transformations 13
From the development leading to this equation, we conclude that if the dif-
differences of differential forms on the left side of B.10) are equal to a total
differential, then the transformation is canonical.
We wish to examine more carefully the nature of the generating function
G\(q, q')> For a system with TV degrees of freedom, B.8) comprises IN equa-
equations. Thus consider that B.8) refers specifically to the components (qs, ps)
and q's, p's- The left equation of B.8) is an implicit relation for obtaining
q' = q's(q, p). Such inversion is possible if and only if7
det
Once having obtained q's — qrs(q, p), the right equation of B.8) gives p's —
pfs(q, p), thereby completing the transformation B.1).
As an example of a continuous function of q, q' that does not generate a
canonical transformation, consider
where f(q) and h{q') are arbitrary continuous functions. This function does
not satisfy B.11) and does not give a canonical transformation. Thus, when it
is stated that the relations B.10) guarantees the transformation to be canonical,
it is tacitly assumed that G\(q, q') obeys B.11).
Exchange transformation
An instructive example of the relation B.8) is given by the generating function
G\(q,q') = -qq'
There results
Save for a sign reversal, the roles of p and q are reversed. Accordingly, this
transformation is called the exchange transformation.
Hamiltonian criterion
In the preceding description it was stated that, for a transformation to be canon-
canonical, Hamilton's principle must be obeyed in both coordinate frames [that is,
B.2) and B.3)]. Equivalently, we say that the transformation (q, p) -> (q\ p')
is canonical if and only if there exists a function H'(q\ p') such that the
equations of motion in the new frame maintain the Hamiltonian form A.12):
dH' , dH'
7For further discussion, see E. C. G. Sudarshan and N. Mukunda, Classical
Dynamics: A Modern Perspective, Wiley, New York A974).

14 1. The Liouville Equation
7.2.2 Generating Other Transformations
It is possible through Legendre transformations of B.10) to effect generating
functions that are functions of parts of "new" and "old" variables other than
q, q'. Thus, for example, rewriting B.10) as
' = dGx{q,q')
gives
It is evident from the differentials on the left side of this equation that the
variables of the functions on the right are q, p''. Thus we may write
pdq
'dp' = dG2(q, pf)
B.13)
so that
B.14)
where again, G2 obeys B.11) with respect to variables q, p'.
An instructive example of this transformation is given by the ordinary
transformation of coordinates:
This transformation may be effected by the generating function
B.15)
With B.14) we find
t = Ml)
Pk
The second of these three equations returns B.15), whereas the third completes
the total canonical transformation:
We return to the generating relations B.14) in our derivation of Liouville's
theorem (Section 1.3).

1.2 Canonical Transformations 15
1.2.3 Canonical Invariants
A canonical invariant is a dynamical quantity that remains invariant under a
canonical transformation. An important example of a canonical invariant is the
Poisson brackets of two dynamical functions A, B. That is, if
then
[A, B]qp -^>[A\ B%,j = [A, B]q,p B.16)
A simple proof of this statement can be constructed if we keep in mind that
a canonical transformation is a change in variables related to a given system.
Consider that B is the Hamiltonian of the system at hand. Then by A.27) (for
time-independent B)
dA _
dt
But the time rate of change of A can only depend on the properties of motion
of the system and not on a particular choice of variables. So dA/dt is the same
in all coordinate frames and B.16) follows.
1.2.4 Group Property of Canonical Transformations
Let the canonical transformation
(q, p)—l->(q',p') B.17)
be associated with the generating function G(q, qr). That is,
Similarly, let
(q',p')-^+(q",p") B.19)
be related to the generating function K(q\ q") so that
p'dq' - ^ p"dq" = dK(q\ q") B.20)
Adding dGtodK gives
pdq - V^ p"dq" = d<&(q,q") B.21)
where we have set
JO = d[G + K]
The relation B.21) implies that the transformation
It
B.22)

16 1. The Liouville Equation
is also canonical. Since the transformation C3 is obtained by first effecting C\
and then effecting C2, we may say that C3 is the product of C\ and C2. This is
written as
C3 = Ci ® C2 B.23)
Thus we find that canonical transformations obey the group property, that is,
the product of any two canonical transformations is itself canonical.
1.2.5 Constants of Motion and Symmetry
Constants of motion of a system may be associated with symmetries of a
system. Consider a system described by the coordinates: (q\, qi,--, qn) and
suppose that dH/dqj = 0. With A.2) we conclude that Pj = constant. It
follows that if a system is symmetric with respect to displacements of a given
coordinate, the momentum conjugate to that displacement is constant. As noted
in Problem 1.2, such coordinates are called cyclic. Conservation principles
follow from this symmetry rule.
For example, consider an aggregate of TV particles. The coordinates of the
system are (R, q\, q'2 • • -q'3N), where {q[} are particle Cartesian coordinates
relative to the center-of-mass and R, P are coordinates and momenta of the
center-of-mass, respectively. The Hamiltonian for this system is obtained from
the kinetic energy form given in Problem 1.7, namely,
H = {&+E p?ilm>+E v ^ B-24>
where M is mass of the center-of-mass and V(r/y.) is interparticle potential.
Thus, with homogeneity of space, H is independent of R and one may conclude
that the momentum of the center-of-mass is constant.
In the study of conservation of angular momentum we write the angular
momentum of the system as [see (PI.9)]
where Jcm = R x P is the angular momentum of the center-of-mass and j[ is
the angular momentum of the /th particle relative to the center-of-mass. We note
that rotation of all particles in the system about a given origin that is not coin-
coincident with the center-of-mass, is equivalent to rotation of the center-of-mass
about this same origin. As space is isotropic if follows that the Hamiltonian of
the system is invariant to this rotation. The related conservation rule may be
obtained in the study of infinitesimal variation. Thus, consider the change in
angular momentum of the center-of-mass due to infinitesimal displacement of
the center-of-mass in the time interval, 8t,
8JCM = R x MRSt oc R x 8R = R x R80 B.26)

1.3 Liouville Theorem
17
where 86 is the angle swept out by 5R. The vectors (8Jcm> R, RSO) form an
orthogonal triad. Again, as space is isotropic, system dynamics are invariant
to changes in 0. With B.26), we conclude that the angular momentum of the
center-of-mass is constant. (See also Problem 1.2)
Invariance of a Hamiltonian to a given variable may be examined through
Hamilton's equations A.12) or the Poisson-bracket equations A.25). For the
latter case, if H is independent of a variable u that is not explicitly time
dependent, then
du
—
dt
= [u,H]=0
and u — constant [see A.27).] A similar situation occurs in quantum mechan-
mechanics. If u and H commute, (the Poisson brackets become the commutator) then
H is independent of u. In this event if u is not an explicit function of time,
then the expectation of u is constant. (See Section 5.1.)
1.3 Liouville Theorem
1.3.1 Proof
The Liouville theorem states that the Jacobian of a canonical transformation is
unity. For a system with TV degrees of freedom, we write (in various notations)
J
q,p
',/>')
, p)
dq\_ dq^
dqx dqx
dq\_ dq^_
dpN dpN
dp\
dqx
dp
dp'
N
dq}
dp'
N
N
dp
N
= 1
C.1)
This is a 2N x 2N determinant. To prove the Liouville theorem, we note the
following:
1. Jacobians may be treated like fractions so that
[d(q,p')] I
[3(9,
C2)
2. When the same quantities appear in both partial derivatives, the Jacobian
reduces to one in fewer variables with repeated variables taken as constant. It
follows that C.2) may be rewritten
j —
dq'/dq\v> Jn

18 1. The Liouville Equation
The subscript p' on the "numerator" term Jn denotes that all TV of the p'
variables are held fixed in q differentiation. The ik element of Jn is
Since (q, p) —> (q'p') is a canonical transformation, we may employ the
generating equations B.14) to obtain
riK _ - -2
Jn ~
dp[dqk
In a similar manner, the denominator term may be written
So we find
jik jki
Jn = Jd
and we may conclude that Jn differs from Jd by an interchange of rows and
columns, which as may be recalled, has no effect on a determinant. Thus C.3)
gives
J - t = l
which establishes Liouville's theorem.
1.3.2 Geometric Significance
This theorem has an important geometrical consequence. Under an arbitrary
transformation (q, p) —> (q\ //), a volume integral in F-space transforms as
follows:
C.5)
\q' p' J
ft ft'
In this expression, Q denotes volume in phase space.
If the transformation is canonical, then C.5) becomes
ff dqdp= ff dq' dp' C.6)
We may conclude that volume elements in F-space are canonical invariants.
See Fig. 1.7.

1.3 Liouville Theorem
19
t
FIGURE 1.7. Volume elements in F-space are canonical invariants. Q = Qf.
1.3.3 Action Generates the Motion
We now come to an important relation that is a central point in connecting
dynamics to kinetic theory.
Consider the action integral corresponding to motion in the interval from t
t+T
/
L(q,q)dx
C.7)
Differentiating, we obtain
dS _
It ~
Here we have labeled
p' = p{t + T), q' s q(t + T)
Assuming that H is time independent, C.8) gives
C.8)
C.9)
C.10)
Three important conclusions are evident from this relation:
1. The action S is a generating function for the actual physical motion in
time.
2. The differential motion in time is a canonical transformation.
3. Because of the group property of canonical transformations, B.23), the
extended motion in time is a canonical transformation in time.
These properties will come into play in our first derivation of the Liouville
equation to follow.

20 1. The Liouville Equation
t
P
F-space
FIGURE 1.8. The ensemble comprises M points in F-space.
1.4 Liouville Equation
1.4.1 The Ensemble: Density in Phase Space
As mentioned previously, the state of a system is a single point in F-
space. As time evolves, the system point moves on the system trajectory:
[qi(t),..., qN(t)\ pi@,..., PaKOL
Now imagine a large number of independent replicas of the same system.
Such an abstract collection of identical systems is called an ensemble. Suppose
there are M systems in the ensemble. Then at t = 0 the state of the ensemble
is J\f points in F-space. See Fig. 1.8.
1.4.2 First Interpretation of D(q, /?, t)
Let us introduce the function D(q, p, t), which is defined as follows. The
product
D(q, p, t)dqdp = D(q, p, t)dQ D.1a)
represents the number of system points in the phase volume d£2 about the point
(q, p) at the time t. We may write
^ D.1b)
ail
An important property of the ensemble is that trajectories of the ensemble
never cross in F-space. This follows from the fact that for a system with TV
degrees of freedom the system trajectory [given by A.2)] is uniquely specified
by 2N pieces of data: fo@), p@)]. See Fig. 1.9.
Consider that, at a given instant of time, ensemble points in a differential
volume of F-space are contained within a continuously closed surface. As
time evolves, with the property that trajectories do not cross, we conclude that
points interior to the surface remain interior and that surface points remain
surface points. See Fig. 1.10.

1.4 Liouville Equation 21
t
r-space
FIGURE 1.9. An impossible situation in F-space. Only one system trajectory passes
through (q0, Po)-
t + dt
FIGURE 1.10. Interior points remain interior as time evolves. dM = dM'.
If dM represents the number of enclosed system points, then we may
conclude that in the evolution of the system
dM -* dM' = dM
D.2)
Let dQ denote the volume occupied by these ensemble points. As established
by C.10), the motion of these points comprises a canonical transformation.
Consequently, with the Liouville theorem C.6), we may write
D.3)
dQ ->
Combining these relations, we find
dM (dM
dQ
Recalling the definition D.1),
dQ
dN_
dQ
D.4)
D =
dM
~dQ

22 1. The Liouville Equation
we may conclude that D(q,p,t) is constant on a system trajectory or,
equivalently, that
dD
-T- = 0 D.5)
dt
on a system trajectory. This is the Liouville equation. It may be rewritten in
Poisson bracket form. We have found previously A.25) that if u(q, p, t) is any
dynamical variable, then
du du
— = [w, //] + — D.6)
dt dt
With D.5) we may then write the Liouville equation in the form
D.7)
[D9H]=0
1.4.3 Most General Solution: Second Interpretation of D(q, p, t)
We wish to establish the following important theorem: g(q, p, t) is a solution
of the Liouville equation if and only if g is a constant of the motion.
1. Let g be a constant of the motion. Then,
dt
But
-r = — + [g, H]
dt dt
Therefore,
and g is a solution to the Liouville equation.
2. Let g be a solution to the Liouville equation. Then
But
so that
dt
and we may conclude that g is a constant of the motion.

1.4 Liouville Equation 23
We may conclude that the most general solution to the Liouville equation
is an arbitrary function of all the constants of the motion. We may write this
most general solution in the form
D.8)
Thus, knowledge of the most general solution of the Liouville equation is
equivalent to knowledge of all the constants of the motion:
g\ =
gi =
D.9)
glN = g2N(q, P,t)
Inverting these equations gives the dynamical orbits:
> 0
P\ = P\(g\, ..-,
PN —
Thus we may conclude that knowledge of the most general solution to the
Liouville equation is equivalent to knowledge of all the orbits of the system.
This is our second interpretation of D(q, p, t).
1.4.4 Incompressible Ensemble
A second derivation of the Liouville equation stems from a fluid-dynamic
interpretation of ensemble flow in F-space.
Since system points in an ensemble are neither created nor destroyed, the
rate of change of the number of system points in the volume Q, fQ DdQ
is equal to the net flux of points that pass through the closed surface S that
bounds Q. Let u denote the velocity of system points in the neighborhood of
the element of surface dS, where, in general,
u = (<?i,<72, ...,<7yv;pi, P2, -.., Pn) D.10)
Then the net flux out of the volume through the closed surface S is <§> u • D dS.
We conclude that
—
I DdQ = - <f£u • DdS = - f V • uDdQ
s

24 1. The Liouville Equation
d
Jn
D + V • (uD)
0
dt
Passing to the limit Q —> 0, together with the mean-value theorem, gives the
equation
hV-uD^O D.11)
dt }
Consider the term V • u. With D.10) and Hamilton's equation A.12), we find
dpi
v-u = 2 I -—i-
d2H d2H
1=0 D.12)
Thus the fluid of system points is incompressible. Combining the latter two
equations returns the Liouville equation, which now appears as
dD
— +u-VD = 0 D.13)
dt
Note that
Hi dP
d
D.14)
dpi dqt dqi
which renders D.13) in the Poisson-bracket form D.7).
1.4.5 Method of Characteristics
The property D.8) that the most general solution to the Liouville equation is an
arbitrary function of the 2N constants of motion may be seen in another way.
In 2N + 1 dimensional f-space, the gradient of D(q, p, t) has components
VD = (—, , ..., ) D.15)
\ dt oq\ oPn /
In the notation, the Liouville equation D.7) appears as
V-VD = 0 D.16)
where
V=(l,—,...,-— D.16a)
V dp\ dqNJ
The relation D.16) says that the gradient of D is normal to the vector V in
f-space. This will be the case if D is a function of orbits that are tangent at

1.4 Liouville Equation 25
every point to the vector V. With D.16a), we see that such orbits are given by
dt_ = dqx = dq2 = = dpN
1 ~ dH/dpi dH/dp2 '" dH/dqN
These are evidently Hamilton's equations A.12), whose solutions in the present
context are called characteristic curves.
In general, the gradient of a function is normal to surfaces on which the
function is constant. If D is a function of all the constants of the motion, then
all orbits lie on a surface of constant D. Tangents to this surface therefore have
components in F-space given by D.17).8
So, once again, we find that solution to the Liouville equation is an arbitrary
function of the orbits of the system or, equivalently, the constants of motion of
the system.
1.4.6 Solutions to the Initial-Value Problem
In what follows, we present four approaches to solving the initial-value problem
for the Liouville equation.
Taylor series expansion: Case 1
Let the initial distribution be
D(q,p,0) = D0(q,p) D.18)
Expanding D(q, p, t) about t = 0 at fixed values of (q, p) gives
3D
o
132D
D.19)
o
Employing the Liouville equation D.7), we find
3D
dt
d2D
[H,D]
3D
dt2 dt
= [H, [H, D]]
Inserting these relations into the expansion D.19) gives the solution
D(q, p, At) - {1 + At[H, ] + ^-[H, [H, ]] + • • •}Do(<7, P) D-20)
This solution may be viewed geometrically as the evolution of D in time at a
point in the hyper (q, p) F-plane in f-space. See Fig. 1.11.
8The method of characteristics play a key role in the analysis of the wave equation.
For further discussion, see R. Courant and K. O. Friedrichs, Supersonic Flow and
Shock Waves, vol. I, Wiley-Interscience, New York A956).

26 1. The Liouville Equation
q.p
FIGURE 1.11. The series solution D.20) gives D(q, p, AT) along a line of constant
(q, p) in f-space.
Solution from trajectories: Case 2.
In this method of solution, it is assumed that the orbits
q
P = Po + p@
D.21)
as well as the initial distribution D.15) are known. The functions q(t) and p(t)
vanish at t — 0, so
4@) - q0, p@) - po
Solution in this case is derived based on the property that D(q, p,t) is constant
along system trajectories. Thus, we write
D.22)
D(q, p, t) = Do[<7 - 9@, P - P@]
This function has the following properties:
1. At t = 0,
D.22a)
which is the correct initial value.
2. For values of g, p on the system trajectory,
q ~ q(t) =
D.22b)
It follows that on the system trajectory in f-space
D(q, p, t) = £>o(9o> Po) = constant
D.22c)

1.4 Liouville Equation 27
q(t).p(t)
*(Qo,
Po)
FIGURE 1.12. Graphical display of the solution D.22) with D(q, p, t) constant
along a system trajectory.
We may conclude that D.22) is the solution to the Liouville equation corre-
corresponding to the initial value D.18). This solution may be viewed geometrically
as depicted in Fig. 1.12.
7.4.7 Liouville Operator
The Liouville operator emerges in writing the Liouville equation in the form
of the Schroedinger equation:
dD
= i[H,D] = AD D.23)
dt
Thus
\ dq dp dp dq
D.23a)
Properties of A.
We wish to show that A is Hermitian in the space
an element of C2n if and only if the norm of ty,
> A function ^/(q, p) is
dq dp < oo
An operator A is Hermitian providing
A = A
D.24)
where AT is the Hermitian adjoint of A. The quantity Af may be defined in
terms of its matrix elements in £2n>
A*/ =
dq dp
D.25)

28 1. The Liouville Equation
where Uk and w/ are elements of a set that spans £2^- The matrix representation
of D.24) is given by
A*/ - (A,*)* D.26)
or, equivalently,
1 u*kkuidqdp — I utA*u*kdqdp D.26a)
We consider an TV-body Hamiltonian
D-27)
where <$> is potential of interaction between particles. With D.23a) and D.27),
we obtain
r ^^dHdui duidH\
Aw=i / «ij^(_-l--l_)^dp D.28)
J *^ \d d d d )
duidH
dp
Writing
3
op dp dp
in the first term in D.28) and
K dq dq K dq
in the second term and dropping surface terms gives9
dutdH
". r (dHduk dukdH\ ■
1 I u, y I I dq dp
. J ' ^\dq dq dq dp ) H Vm
= Ajk D.29)
We may conclude that A is a Hermitian operator. An immediate consequence
of this property is that (see Problem 1.15):
^
1. Eigenvalues of A are real.
^^ ^
2. Eigenfunctions of A are orthogonal.
These properties will be employed in Section 1.5 in construction of an
eigenfunction expansion for D(q, p, t).
'Here we are assuming that dH/dp is independent of q.

1.5 Eigenfunction Expansions and the Resolvent 29
Two points should now be made concerning the preceding derivation. First,
note that the property D.24) is quite general. That is, in its derivation noth-
nothing was said of the particulars of the interaction between molecules. We
merely assumed two-body conservative interactions. The second point is some-
what more pragmatic. Thus, if eigenvalues of A are real, then eigenvalues of
—/A = [//, ] are purely imaginary. With the structure of the Liouville
equation D.7), it follows that there are solutions to this equation that oscil-
oscillate in time. The reader may witness a common phenomenon that stems from
this property, sound propagation. Sound waves carry pressure-density distur-
disturbances. The manner in which these fluid-dynamic variables are related to the
Liouville equation is fully discussed in Chapter 2.
1.5 Eigenfunction Expansions and the Resolvent
7.5.7 Liouville Equation Integrating Factor
With the Liouville equation written as D.20), our third (case 3) solution to the
initial-value problem may be obtained as follows. Multiplying D.20) through
by the integrating factor (assuming A to be time-dependent)
exp i i
i f dtf
Jo
permits the equation to be rewritten
d
expi
Integration gives the solution
i f dt'
Jo
A D(t)
0
E.1)
For short time intervals we may write
L
At
dt'A 2^ At A
Expanding the exponential in E.1) about At — 0 then
D(q,p,t) =
1 -iAtA + -(-iAtAJ-\
With D.20) this expansion may be written
D(q,p,t)= J' ■ - ■"- — (A°
gives
D{q,p,Q) E.2)
[H, [H, ]] + •■•
■D(q,p,0)
E.3)
This relation was previously obtained D.20) through Taylor-series expansion
of D(a. n n.

30 1. The Liouville Equation
For longer time intervals, we revert to eigenproperties of A discussed
in the preceding section. Again, we consider the initial-value problem with
D(q, p, 0) as given by D.22a). Furthermore, it is assumed that the (orthogo-
nal) eigenfunctions and (real) eigenvalues of A are known. Here we are taking
A to be time independent. Thus we may write
E.4)
Assuming that the set {\j/n} spans the Hilbert space containing D(q, p, 0), we
may write
Vn
By virtue of the orthogonality of the functions i/rn, the coefficients Dn are given
by
n = (fn\Do(q,p)) E.6)
Substituting the series E.5) into E.1), we obtain
Vn
which gives the solution10
E.7)
This completes our third method of solution for D(q, p, t).
1.5.2 Example: The Ideal Gas
As an example of the application of E.7), we will construct the solution to
the Liouville equation for a collection of TV noninteracting molecules (that is,
an ideal gas). The gas is confined to a cubical box of edge length L and of
sufficiently large size. The Hamiltonian is purely kinetic and is given by
N 2
^ 0<jc<o<L E.8)
s
2m s
where x^l) denotes any Cartesian component of xs. Let us call the A operator
^
for this case Ao. Thus
Ao = — i > • = — i} \s E.9)
10
Here we recall that, with f(x) denoting any continuous function, f{A)\f/n =

1.5 Eigenfunction Expansions and the Resolvent 31
Here we have reverted to velocity \s = ps/m. Eigenfunctions of Ao satisfy
the eigenvalue equation
E.10)
where (k) represents a sequence of wave-vectors:
(k) = (k!.k2, ...,k,v) E.10a)
With E.9), E.10) becomes
Substituting the trial solution
A exp I / ^k, • x, I E.12)
into E.11) gives the eigenvalues
N
E.13)
s=\
Assuming that V\k) satisfies periodic boundary conditions gives (see Prob-
Problem 1.17)
2tt
k, = — n, E.14)
where the components of the vector ns are integers. The constant A in E.12)
is fixed by normalization. We obtain
exp I / y ks -x, I E.15)
L3N/2
s
Substituting these eigenfunctions in the expansion E.7) gives the explicit form:
E.16)
(k)
where x^ and p^ represent 37V-dimensional vectors. To employ the initial
data D.22a), we examine E.16) at the time t — 0.
A)() E.17)
(k)
With the orthogonality of the eigenfunctions ^(k), E.17) gives
\5 -x,
E.18)

32 1. The Liouville Equation
Having obtained the coefficients D(k>, the solution E.16) is complete. Substi-
Substituting E.13) for o>(k) and E.15) for i/r^) into E.16) permits the solution to be
more explicitly written:
E.19)
Thus we find that our eigenfunction solution to the initial-value problem for
the Liouville equation is a function of the 2 x CN) constants of the motion:
(Pi, P2, . . . , Ptf, X] - Vif, X2 - V2f, . . . , XN - \Nt)
1.5.3 Free-Particle Propagator
Again consider the free-particle Hamiltonian E.8) and related operator Ao
given by E.9). The formal solution to the initial-value problem is given by E.1),
which in the present case we write as
D(xi, x2,..., vl5..., vyv, 0 = e~itA°D(xu ..., \N, 0) E.20)
AT
d
dxs
-it Ao = -
E.20a)
s=\
As particle variables are mutually independent, the factors in exp(—it Ao) com-
commute. Thus we may examine the evolution of the coordinates and momenta of
particle 1.
-t\
d
Expanding the exponential gives
d
exp
— t\\
d
dx\
1
2
dx
= D(x,,-rv,,v,,0) E.21)
Repeating this construction for all A^ particles indicates that the value of
D(xN, \N, t) is the value it had t seconds earlier on a free-particle trajectory.
Note also that
e-itk°f(xN, p", 0) = f(xN - vNt, p", 0) = f(xN, pN, t) E.22)
so that exp(—itA0) propagates variables backward in time.11 For this reason,
exp(—it Ao) is called the free-particle propagator.
ii
This operator occurs again in Section 2.4, where we label exp(—it A) = A_,.

1.5 Eigenfunction Expansions and the Resolvent 33
1.5.4 The Resolvent
Our last technique of solution (case 4) to the initial-value problem for the
Liouville equation stems from the Laplace transform defined by
I
00
-st
D(s) = / dte~stD(t) E.23)
To obtain an equation for D(s), we operate on the Liouville equation D.20) as
follows
dte~st (i — = AD
dt
Substituting the expansion
= d_
dt dt
into the preceding equation gives
, / ( /
,• / ( —De~st I dt + is / e~s'Ddt = A / e~s'Ddt
Jo \dt ) Jo Jo
iDe
00
-st
+isD(s) =
o
With Re s > 0, we obtain
- /£>@) = (A - is)D(s) E.24)
or, equivalently,
D(s) = -i(A-isylD@)
E.25)
D(s) = -iRD@)
where
R = (A-il
represents the resolvent operator.
The inverse of E.23) is written12
/
00
ts
dse'sD{s)
y-ioo
where the line s — y in the complex s-plane lies to the right of all the
singularities of R. See Fig. 1.13.
As noted previously, since A is Hermitian, its eigenvalues are real. In this
event we may conclude that the singularities of R occur for s purely imaginary.
12The Laplace transform comes into play again in discussion of Landau damping
(Section 4.1.3).

34 1. The Liouville Equation
Im s
t
Res
FIGURE 1.13. Path of integration in inverse Laplace transform E.22) lies to the right
of all singularities of R.
FIGURE 1.14. Closing the path of the inverse Laplace transform for singularities of
R corresponding to s pure imaginary and finite in number.
This permits us to close the path of E.22) as shown in Fig. 1.14. Here we are
assuming that R has a finite number of singularities and that R is a bounded
operator for all nonreal s.13
13The operator R is bounded in C1N, if for any element \jr of C2N with finite norm,
, there exists a finite constant M such that
< m

1.6 Distribution Functions
35
7
FIGURE 1.15. System points lie in the energy shell in F-space.
1.6 Distribution Functions
1.6.1 Third Interpretation of D(q, p, t)
To this point in our discourse, we have discovered two interpretations of
D(q, p, t): A) A density of system points in F-space, and B) a function whose
general form for a given system implies a complete description of the state of
the system in time.
In this section we come to yet another interpretation of D(q, p, t). We will
find that, apart from a multiplicative constant, it is the joint probability density
for the system.
The energy shell
Consider an isolated system with TV degrees of freedom with Hamiltonian
H(q, p) = E — constant
The system lies on this surface in F-space. Quantum mechanics prescribes an
uncertainty to E, so we consider a small spread of energies about the mean
value of E.H This generates an energy shell in F-space. System points of
the ensemble lie in the energy shell, which we take to be of volume Q. See
Fig. 1.15.
As previously noted (Section 1.4), we call the number of system points in the
ensemble AT and D(q, p, t) the ensemble density function. Since an individual
member of the ensemble is at a given point q, p in F-space, and since such
points may be identified with the classical state of the system, we may write
f
Jn
= D
number of occupied states in Q F.1)
14Thus, for example, the smallest volume in six-dimensional phase space for which
the state of a single particle can be defined is h3, where h is Planck's constant. This
concept is further discussed in Section 5.3.2.

36 1. The Liouville Equation
It follows that in the volume AQ, C £2,
AJ\f —I D dq dp = number of occupied states in A £2 F.1a)
Jaq
Dividing these two expressions, we find
AJ\f ratio of number of states occupied in A £2 to the
—7- = F.2)
N total number of occupied states of the ensemble
We may equate F.2) to the to the probability that an unspecified state is
occupied in AQ}5 We label this entity /AQ /# dq dp (each member of the
ensemble is comprised of TV particles) so that
/
Ja
/ f /
aq Jq & dq dp
Taking AQ, to be infinitesimal F.3) gives
Ddqdp
H/ F.3)
fN dq dp = F.4)
Jq d dq dp
We may conclude that
fN(q, P, t) = CD(q, p, t) F.5)
where C is a constant. It follows that fN(q, /?, t) satisfies the Liouville equation
F.6)
1.6.2 Joint-Probability Distribution
What does it mean to say that fNdQis the probability that a state is occupied
in the volume dQ? We have been writing (q, p) for (xl9 x2, ..., xN\ pl5 ...,
P/v). So if the state (q, p) is occupied, then particle 1 is in the state Xj, pi,
particle 2 is in the x2,p2, and so on. It follows that fu(q, p, t)dq dp is the
probability that particle 1 is in the volume dx\ dp\ about the point Xi, pi and
particle 2 is in the volume dx2d^2 about the point x2, p2, and so on, at the
time t. Therefore, we may label fN the N-body joint-probability density for
the TV-body system. Thus, whereas D(q, p, t) is relevant to an ensemble of
system points, /#(#, /?, t) addresses a single system.
Note that with F.4), fN enjoys the proper probability-density normalization
f fNdqdp = \ F.7)
JQ
The integration is over the entire space accessible to the system. That is, the
energy shell.
15This interpretation is due to J. W. Gibbs. See Collected Works of J. Willard
Gibbs, Dover, New York A960).

1.6 Distribution Functions 37
If G(q, p, t) is any dynamical variable, then with fN a probability density
we may write the following for the expectation or average of G.
(G)=ffNGdqdp F.8)
The experimental meaning of (G) is given by
N
1
(G) = lim -VGi F.9)
In this expression, G/ represents the observed value of G in the /th experimental
run, each run performed under identical circumstances.16
The last equations say that if we solve the Liouville equation for fN and
with F.8) calculate (G), the same value will be found if we perform the
experimental evaluation F.9).
1.6.3 Reduced Distributions
Let us introduce the notation
d\ = dx\ dp\, dl =
relevant, respectively, to particle 1,2,
Consider a system of TV identical particles. We direct our attention to the sub-
subsystem comprised of s < N particles. The probability of finding this subsystem
in the phase volume d\ dl • • • ds about the state A, 2,..., s) is
/s(l,..., s)d\ - - - ds
To construct fs from fN, we must integrate out information on the state of
particles s + 1,..., N from fN(l, 2,..., N). That is,
F.10)
This relation is returned to in Chapter 2 in the derivation of the equations of
motion for the reduced distributions.
1.6.4 Conditional Distribution
In addition to the joint-probability distribution fN(\, 2, ..., N, t), we also
encounter conditional distributions. Thus, for example, the product
h\N\l | 2, 3, ..., N)d\
16Phase averages, as relevant to the ergodic theorem, are further discussed in
Section 3.8.1.

38 1. The Liouville Equation
represents the probability that particle 1 is in the phase volume d\ about the
phase point 1, granted that particles 2,..., TV are in the volume dl... dN
about the point 2,..., N. If particles are statistically independent, then
h\N\l |2,3,...,A0 = /iO) F.11)
More generally, Bayes's formula relates h{N) to fN as follows:
17
, ..., N)
F.12a)
F.12b)
and so forth. These distributions are returned to later in our discussion of a
Markov process.
1.6.5 s-Tuple Distribution
The s-tuple distribution, Fs(l,..., s), is defined as follows: The product
Fs(l,..., s)d\ • • • ds
represents the probable number of s -tuples of particles such that one of the
particles is in the phase volume d\ about the point 1, another in dl about 2,
and so on, at a given time. Thus, for example, F\ A) d\ is the probable number
of particles in the phase element d\ about 1 at a specific time. The product
F2dld2 is the probable number of pairs of particles such that in each pair one
particle is in the state d\ about 1 and the other is in dl about 2 at a specific
time.
The relation between the joint-probability s-particle distribution function fs
and the s-tuple distribution function Fs is as follows: The function fs relates to
the s -particle state of a specific group of s particles. The function Fs refers to the
same s-particle state but is independent of the specific particles occupying this
state and independent of the manner in which these particles occupy this state.
Thus the function /2O, 2) is a property of the configuration where particle 1
occupies the state 1, and particle 2, the state 2. The function F2(l, 2) is a
measure of the number of pairs of particles that occupy these states. To find
the relation between Fs and fs, we first note that the number of ways of choosing
s particles from the total number TV is given by
AM
s\(N -s)\
17
Explicit time dependence is tacitly assumed in all distributions.

1.6 Distribution Functions 39
Assuming identical particles, each such choice gives the same value for fs.
Thus
)f F.13)
Here we have used a bar over Fs to denote that a given s-tuple state has only
been counted once in F.13). The number of ways of distributing each subgroup
of s particles while still obtaining the same s-tuple state is s\ This gives the
desired relation
)/ fs F.14)
\j (N-s)\
Thus, for example,
^i =#/i F.14a)
F2 = N(N - l)/2 F.14b)
Normalization is given by
Fsd\---ds=s\[ / fsdl'-ds= rKj ' F.15)
\sJJ (N-s)\
The average kinetic and potential energies of a collection of particles are best
written in terms of the Fs functions. Thus, for example,
1 f p2
Ek = - / ^i(x, p, 0—dxdp
\3 m F.16)
£<d = - / F2(x, x\p, pr, 0*(x, xf)dxdpdxf dp'
where O(x, xr) is the two-particle interaction potential. These relations are
returned to in Chapter 2 in our proof of the conservation of total kinetic and
potential energy, EK and E&, respectively, from the equations of motion for
and F2.
1.6.6 Symmetric Properties of Distributions
As the s-tuple distribution Fs does not address the state of individual particles,
it must be a symmetric function of its arguments. Thus, in the relations F.14)
and following, it is assumed that fs is properly symmetrized. Suppose fs is not a
symmetric function of its arguments; then a symmetric function is constructed
as follows:
Y\ f(l F.17)
C 1
P(\ s)
where the sum is over the permutations P of the numbers, 1 ... s and fs is the
given asymmetric distribution.

40 1. The Liouville Equation
Example
Consider the example that one molecule in an ideal gas of N identical molecules
is moving with velocity v° and is at the origin at time t — 0. The remaining
molecules are stationary at given positions, (xSJ,..., x^). Let us construct a
symmetric distribution fN that describes this state of the system. From this
distribution, let us then obtain f\ and the number density n(x, t).
The symmetric distribution is given by
- E
- v°0
x 5(v2) • • • 8(\N)8(x2 - x°2) • • • 8(xN - x°N) F.18)
The permutation operator permutes the phase variables (zj, ..., z#), where
Z/ = (x/, V/). Note, in particular, that F.18) may be written in determinant
form. Setting
Dn
/ - v?r)
F.19)
we find
D12
D31
D
N2
^23 ^
33
D
IN
... d
F.20)
The symbol 0 reminds us that only positive signs are employed in the
expansion of the determinant. To obtain f\, we must evaluate
All terms in F.20) integrate to unity except for terms in the first column. The
minor of each such term is of dimension (N — 1). There results
/id)
N\
(N -
\k
k=\
8(yl)8(xl -x°2)
F.21)
rO

1.7 Markov Process 41
This result is closely akin to a superposition state in quantum mechanics where
the particle in question has equal probability of being in one of a number of
states.18
The macroscopic variable number density, n(x, t), is such that n(x, t) dx is
the number of molecules in the volume dx about x at time t. It is obtained from
/i(l) as follows.19
' MVdYx F.22)
Returning to our example, inserting fx A) as given by F.21) into the preceding
formula gives
n
w(x, 0 = 8(x - v?0 + ^ S(x - x?) F.23)
i=2
The number density is peaked along the trajectory x = \®t and at the fixed
sites x?. Furthermore, note that
w(x, t)dx = N F.24)
which is the correct normalization for the number density.
1.7 Markov Process
1.7.1 Two-Time Distributions
Let us introduce the phase vector
z = A,2, ...,
In terms of this variable, the two-time, or conditional, distribution functions
may be written Y[(z>f I Zo, h)- The product
t I zo,to)dz G.1)
represents the probability of finding the system in the state dz about the point
z at time t, granted that it was in the state z0 at t0.
1.7.2 Chapman-Kolmogorov Equation
The two-time distribution has three fundamental properties:
18The superposition principle is discussed in Section 5.1.4.
19Macroscopic variables are more fully described in Chapter 3.

42 1. The Liouville Equation
t'
Time
FIGURE 1.16. Summing over intermediate states in the CK equation.
1. The system undergoes a transition to some state in the interval (t — t0).
/
«/V2allz
G.2)
2. The system does not change its state in zero time.
, to) = S(z -
G.3)
3. The Y[ function obeys the Chapman-Kolmogorov (CK) equation:
Y\(z, t | zo, *o) - J Y\(z, t I z\ t') Y\(z\ t' I zo, k) dzf G.4)
The integral in G.4) represents a sum over intermediate states z\ as depicted
in Fig. 1.16. Note that the integral in G.4) is independent of the arbitrary
intermediate time, f.
Careful examination of G.4) indicates that it contains an approximation.
That is, the integrand function should more accurately be written
\\(z,t | z\f';zo,
G.5)
That is, this term should include the information that the system was in the
state zo at t0. The approximate form of the function G.5) as written in G.4)
is appropriate to systems with "short memories". A process so characterized
is called a Markov process. In this sense property 3 is not an exact relation,
as opposed to the precise properties 1 and 2. Note further that in the situation
depicted in Fig. 1.16 it is uncertain which orbit passes from zo to z. In the
event that Y\(z I Zo) is equivalent to the joint-probability function F.4), which
satisfies the Liouville equation D.7), then as we have found previously there
is only one orbit through zo- We may conclude that the CK equation is more
relevant to systems governed by probabilistic laws, such as occur in random
processes and quantum systems. Thus, the equivalent differential equation

1.7 Markov Process 43
to F.6), derived below for discrete processes [see G.20)], is first applied to
the random-walk problem. It is returned to in Chapter 4 in the derivation of
the Fokker-Planck equation and in Chapter 5, where it is applied to quantum
processes.
1.7.3 Homogeneous Processes in Time
For systems whose behavior is homogenous in time, processes can only depend
on time intervals, not on initial or final times. In such cases we write
I z\\\h-tx\) G.6)
With |*2 — t\ | = At and setting t0 = 0, the CK equation G.4) becomes
Y\(z \zo\t + At) = j Y\(z | z'\ At) Y\(z' Uo; 0 dz' G.7)
1.7A Master Equation
Our aim at this point is to obtain a differential equation for \\ appropriate to
the case where phase variables go over to a countable set:
{Z} -►{/}, / = 0,1,2,... G.8)
The integral equation G.7) then becomes
Y\(l | Z0;r + At) = J2Y\(l I j\At)Y\(j | Z0;0 G.9)
Conditions 1 and 2 become
v/
Y\(l \ lo\O) = 8llo G.11)
For/ ^ j, we note that
Y\(l I j\ At) -> 0 as Ar -> 0 G.12)
so we may write
Y\(l | j; A*) = wjiAt + ..., l^j G.13)
From the property
Y\(l I /; Ar) -> 1 as Ar -> 0 G.14)
we may write
J"J(Z | /; At) = 1 — (probability that there is a transition out of I in At)
G.15)

44 1. The Liouville Equation
Equivalently,
G.16)
Equations G.13) and G.16) may be combined to give (note that the summation
now includes all V values)
Y\(l I j\ At) - Wji At + Sij I
\
1 - At
G.17)
v
Let us check this equation. For j —U
Y\(l I'; ao = wu At + l - At
ww = 1 - At
ww
v/
which is G.17).
Substituting the key result G.17) into the CK equation G.9) gives
Y\(l \lo;
E
Wji At + Stj
— At
w
Summing the 5/y factor gives
Y\(l | /0, t
]~[o*l'o;0
G.18)
V/'
Changing the dummy variable V to j in the second sum gives
Y\(l |/0;
= E
j
w
/0;0
G.19)
Passing to the limit At —> 0, we obtain
G.20)
which is a canonical form of the master equation. Its meaning is best revealed
again through a diagram. See Fig. 1.17. The wjt coefficient in G.20) represents
the probable rate at which transitions from j to I occur. The master equation in
a form closely allied to G.20) comes into play in a number of instances in our
discussion of application of kinetic theory to quantum systems in Chapter 5.
7.7.5 Application to Random Walk
We wish to apply the master equation G.20) to the random-walk problem in
one dimension. For this problem, time is replaced by the total number of steps,

Gain
transition
to-
1.7 Markov Process 45
Loss transition
V
FIGURE 1.17. Geometrical description of terms in the master equation G.20).
n
/
FIGURE 1.18. The random walk in one dimension (at n = 10, / = 2).
n = 11 At, with An = 1. Furthermore, we set
P(Z,/i)s]~[(Z + Zo|Zo;/iAO
which denotes the probability that the particle has moved through the displace-
displacement I in n steps. (See Fig. 1.18.) Setting ibji = WjiAt and with reference
to G.19), we write
00
, n - 1) -
n - 1)] G.21)
j=-oo
The transition probability wjn has the value
1
G.22)
Inserting this form into G.21) and summing over j gives
_ 1
~ 2
, n) -
- 1, „ - 1)] -

46 1. The Liouville Equation
There results
P(Z, n) = -[P(l + 1, ft — 1) + P(l — 1, w — 1)] G.23)
2
This equation states that step n follows step n — \ and that the displacement I
comes from I ± 1 with equal probability. We wish to establish the following
key result for the random-walk problem:
(I2) oc n G.24)
To this end, we define the moments:
G.25)
Stemming from the property P(l, 0) = 5/0 it is readily shown that
Afi(/i) = O G.26)
For M2, with G.23), we write
1 ^ 9 9
M2(w) = - 2^[/2P(/ - 1, n - 1) + 12PA + 1, w - 1)] G.27)
which may be rewritten
M2(n) = \
+ (Z + 1JP(Z + 1, /i - 1) - A + 2Z)P(Z + 1, /i - 1)]
Setting k = I — !,£ = / + ! allows the preceding to be written
M2(n)
We note
and
+
that
k2 P(k n — 1"! -I-
/ |_rC 1 \K i Ti
k
-—>
y 2 k P (k yi 1
k
k
2kP(k,
l)-2/
) = 2M
n-\)
n -
— ^
-i)
j
— 1
i
+ P(/t, n
-\) + P
) = 0
-1)]
'(ln-
1)]

1.8 Central-Limit Theorem 47
There results
* G.28)
Af2(/i) = Af2(/i-l) + l
Since M2@) = 0, we find
M2{n) = /i G.29)
which was to be shown. This relation, or equivalently G.24), plays an important
role in the theory of diffusion and will be returned to in our discussion of
transport coefficients (Chapter 3).
1.8 Central-Limit Theorem
1.8.1 Random Variables and the Characteristic Function
In the preceding example we did not obtain an expression for the probability
P(l, n) relevant to the random-walk problem. That is, we did not obtain a
solution to the discrete equation G.23).
This result is readily obtained from elementary notions of probability theory,
whereas the long-time behavior of P(Z, n) may be found with aid of the central-
limit theorem. To obtain these results, we first describe some basic language
of probability theory.
A random variable (rv) is a mapping, §, from a sample space to the real
line. The sample space is a space of objects or outcomes of experiments. Thus,
for example, in a coin flip the sample space consists of two results, which we
may label H and T. The mapping § maps H and T onto, say, the values 1,
2. For a sample space comprised of elements of the alphabet, § maps A, B,
C, ... onto, say, the numbers 1, 2, 3, .... See Fig. 1.19. These are examples
of discrete rv's, in which case § maps the sample space onto integers. For a
sample space comprised of a continuum of elements, § maps the sample space
onto the continuous real line.
We may associate a probability with such mappings, which we write P(§),
or P(§ = x) (where x are elements of the real line) or, more simply, P(x).
Thus
00 x.00
^ P(x) = 1 or / dxP(x) = 1 (8.1)
-oo J-oo
18.2 Expectation, Variance, and the Characteristic Function
The expectation or average of § is written
? (8.2a)

48 1. The Liouville Equation
(a) Coin Flip
Sample space
Real line
(b) Choosing letters from the alphabet.
Sample space
1 2 3... Real line
FIGURE 1.19. Examples of sample spaces.
whereas the variance is written
Vx
(8.2b)
is given by
(8.3)
Thus we see that the characteristic function is the Fourier transform of P(x).
Inverting (8.3) gives
The characteristic function <j>(a) of the probability function
<Ka) = £{eia) = J2 P(x)eiax
P(x)
i r
— /
da<j){a)e
—iax
(8.4)
From (8.3), we find
da
a=0
(8.5)
da2
a=0
Consider the expression
jax
In <p@) = In
= In 1 = 0
(8.6)

1.8 Central-Limit Theorem
49
Product sample space:
S=S'®S"
Product real space:
X ® .It
Real line
FIGURE 1.20. The random variable £ maps pairs of elements from the sample spaces
Sf and S" onto the real line.
Continuing in this manner, we obtain the series expansion
In4>(a) = 0 + i£(!-)a - -D&a2 + • •
(8.7)
This expansion will come into play in the derivation of the central-limit theorem
in Section 8.5.
1.8.3 Sums of Random Variables
Let £' have the probability distribution P'(x) and £" the distribution P"(x).
Let | = I' + £". Then § is a valid rv because it maps pairs of elements from
a sample space S onto the real line. See Fig. 1.20. Let us obtain P(§ = x). If
' = k, then § = Jk, providing f=x- jfc, [§' + §"] = § = x]. It follows that
= x - k) (8.8)
where Pjoini represents a joint-probability distribution. If §' and §" are
statistically independent, then
(8.9)
Consider the characteristic function
(8.10)
' and §" are statistically independent, then (8.10) reduces to
cj){a) = £{eia^ )£(eia^ ) = <f>'(a)(/)"(a) (8.11)
So the characteristic function of a sum of statistically independent rv's is equal
to the product of characteristic functions related respectively to individual rv's.

50 1. The Liouville Equation
More generally, we may write, for n statistically independent rv's.
n n
(j){CL) — I I 0r(tf), %n — y.%r
r=l r=l
The related probability that measurement finds the sum §„ = x is given by (8.4).
Thus, for example, consider the following problem involving three slot ma-
machines. The first can show the numbers 1, 3, 5, and 7, the second the numbers
1, 2, and 3, and the third the numbers 0, 8, and 9. What is the probability that
the sum of readings is 4 in any single turn of the machines? To answer this
question, we note first that the respective probability distributions for the three
machines are P'(x) = |, P"(x) = |, and P"\x) = |. Thus
\e + ^ + e5ia + elia)
= J2 P"(x)eiax = -(eia + e2ia + e3ia)
4>"\a) = J2 P'"(x)eiax = -(e&ia + e9ia + eOia)
X
The answer to our problem is then given by
- x — /
3 2n J_n
PD) = - x - x - x — / 2e0iada =
4 3 3 2 J 18
We turn next to application of the notion of a summed rv to the random-walk
problem discussed previously in Section 7.4.
1.8.4 Application to Random Walk
Each step in the random walk is considered an independent event. The rv of
the rth step, £r, maps the sample space (L, R) onto the numbers —1, +1. The
probability distribution of §r is
(8.13)
It follows that in any single step
= 1 (8.14)
Vx
Here we are assuming that the random walk is biased so that p and q need
not be equal. For an unbiased random walk, p = q = \. The characteristic
function for this process is

1.8 Central-Limit Theorem 51
so that
(8.16)
The probability that the net displacement to the right is Z after n total steps
is written P(Z, n) [as in G.23)]. The displacement rv after n steps, £, is given
by (8.12). As £r are statistically independent rv's, and all (j)r{a) are equal,
with (8.12) we write
<Ka) = [4>r(a)]n = [peia + qe~ia]n (8.17)
Recalling (8.4), we write
1 C71
/
da<j){a)e-ial (8.18)
From the binomial expansion, we may write
w
Inserting this result into (8.18) indicates that the integrand contains the factor
expz<z(n — 2m — I) so that integration gives a nonzero result only when m =
(n - Z)/2. There results
, /I) - p(n+/)/2 (n-0/2 (g
[(l/2)( + l)]![(l/2)(/)]!^ H
The meaning of the p and q factors are as follows. The net displacement to
the right is Z. Remaining steps are n — L Of these, (n — Z)/2 must be to the
right and (n — Z)/2 must be to the left. The total number of steps to right is
I + (n — 1I2 — (n + /)/2). For an unbiased random walk, p = q = \. In
general, (8.20) is called the binomial distribution.
Simple substitution indicates that the distribution (8.20) satisfies G.23)
rewritten for a biased random walk:
P(Z, n) = qP(Z + 1, n - Z) + pP(Z - 1, w + 1) G.23a)
See Problems 1.26 and 1.27.
Poisson distribution
An important distribution relevant to many applications may be obtained
directly from (8.20). Let
n +Z

52 1. The Liouville Equation
denote the total number of steps to the right. Then (8.20) gives
n) = nl V(l - pT~r (8.21)
r\(n — r)\
We may interpret this form to give the probability that an event with probability
p occurs r times in n trials.
Let us consider the limit n —> oo, p —> 0, with np = X <£ n, constant. The
meaning of this assumption is best illustrated by viewing the given process as
one in time so that n is the total number of subintervals (say, seconds) in a total
time interval. The probability p of an event occurring in one of the subintervals
will decrease if the number of these subintervals is increased. It is evident,
however, that in this process, the average, X — np — (r), remains fixed.
With the preceding assumption, we write
ln(l - p) 2^ -p
1 - p = e
Thus we obtain
Furthermore,
(n-r)\
-(r-1)]
nr
Inserting these results into (8.21) gives
(8.22)
r\
This distribution gives the probability of finding a total number of events r in
X/p trials, where p is the probability of an event.
The distribution (8.22) is called the Poisson distribution. As it stems
from (8.20), it is relevant to spontaneous events with no memory, like rain-
raindrops on a roof or particles emitted by a radioactive sample or electrons by a
hot cathode.
Note that (8.22) is properly normalized. That is,
y p(r) = e-k y
r r
Furthermore,
which agrees with our previous interpretation of X. In addition, the variance,
D(r) = <(r - (r)J) = X

1.8 Central-Limit Theorem 53
so that
D(r) = S(r) = X (8.23)
in a Poisson distribution.
Small shot noise
As an elementary example of these results, consider that a hot cathode emits
electrons to a collecting plate in an elementary dc circuit. We label
(/) = average emission current
(Q) = r (/) = total charge collected by anode in time r
Thus the average number of electrons collected in time r (which we identify
with (r» is
e
where e is electronic charge. If we assume electrons are emitted in a Poisson
distribution, then with (8.23) we obtain20
(8.24)
Under ordinary circumstances,
SI
«1
and 81 is not readily observable. However, with careful observation, Hull and
Williams in 192521 were able to employ (8.24) to infer the value of electronic
charge e. They found
I^ = 1.585 xl<T19C
which is seen to differ from present-day values,
e= 1.602 x 1(T19C
by only 1%.
Noise stemming from fluctuations in the number of particles (or charges, as in
J resent case) is termed shot noise. Noise stemming from fluctuations in velocities
is called Johnson noise (see Section 3.4.4).
21 A. W. Hull and N. W. Williams, Phys. Rev. 25, 147 A925).

54 1. The Liouville Equation
1.8.5 Large n Limit: Central-Limit Theorem
The central-limit theorem addresses a sum of n statistically independent rv's,
{§,.}, which all have the same probability distributions, P(§r), and same ex-
expectations, £(£r), and same variances, D(%r). Furthermore, it is assumed
that D(%r) < oo for all r. The theorem addresses the asymptotic domain
n —> oo relevant to the displacement rv and related characteristic function
given by (8.12). We find that
a=0
Likewise,
"«■>=-
It follows that if all £(£r) are equal and all D(£r) are equal, as in the stated
assumptions above, then
£(§„) - /if(§r) (8.25a)
(8.25b)
where %r denotes any of the rv's of the process. The result (8.25a) reflects the
fact that §„ is a sum of n statistically independent rv's. To understand (8.25b),
we recall that £>(£„) [see (8.2b)] is a measure of the deviation from the mean
of £„, which, we expect, should also grow with n. From the definition of the
characteristic function, we write
— [
27T J-n
~ial
= /) = P(l, n)=— [ dae~ial(f)(a) (8.26)
2 J
With our previous result (8.7), the preceding integral becomes
-ial
n)= — / dae~ial exp
2n /
1
2
.2
(8.27)
We have found [see (8.25)] that £>(£„) increases with n under the stated assump-
assumptions. Thus, as n —> oo, only the a ^ 0 values contribute to the integral (8.27).
In this event, the limits of integration in (8.27) may be replaced by (—oo, +oo)
without incurring gross error in the result. Setting

1.8 Central-Limit Theorem 55
permits (8.27) to be rewritten
da exp | —
D
u
■
(8.28)
In
m2/2D
V2ttD
where X is the dummy variable as implied. Thus we obtain the key result of
the central-limit theorem:
exp(~[^~g5J] ) (8.29a)
With (8.25), this result may be written
1
exp <
2nD(l=)
(8.29b)
Here we have written § for any of the n statistically independent rv's,
The central-limit theorem is employed in the following to obtain the asymp-
asymptotic displacement in the random-walk problem. It comes into play again in
Chapter 3 in a derivation of the distribution for center-of-mass momentum of
a fluid.
1.8.6 Random Walk in Large n Limit
The result (8.29) is the widely employed relation of the central-limit theorem.
It gives the probability that the rv, £, has the value I after n steps (with n ^> 1)
of the given process.
In applying this result to the random-walk process, we recall (8.16)
) - P -
r) - 4pq
Combining these results with (8.25) and substituting into the central-limit
formula (8.29), we find
1
exp <
Snpq
(8.30)
For an unbiased random walk, we set p = q = \ and (8.30) becomes
1
B7mI/2
exp -
In
(8.31)
The continuum limit
For large n, we may assume that the displacement / grows large compared to
th unit step interval, which we label 8. In this limit, it is consistent to introduce

56 1. The Liouville Equation
a continuous displacement variable x in place of the discrete variable I so that
x — 18. Let Ax be an interval that contains many 8 intervals, but sufficiently
small so that P(l,n) does not change appreciably over Ax. Then the probability
that the net displacement lies in the interval (x, x + Ax) at the nth step is
x—v Ax
P(x,n)Ax = J2 P(^n)=P(l9n)—
V/eA*
where the integer Ax/8 is the number of intervals in the displacement Ax and
therefore represents the number of terms in the partial sum. Substituting (8.31)
into this equation gives
1 x \
) (8-32)
8Bnnylz \ )
Associating n with the time interval of the walk, n = t, and setting 82 = A
gives the probability distribution
The function P(x, t) at any instant of time is a well-known distribution. That
is, with
cr2 = At
(8.33) becomes
P(x) = —L=e-x2/2a2 (8.34)
\f2na
which we recognize to be the Gaussian distribution. Note in particular that
P(x) as given by (8.34) maintains normalization.
/»OO
/ P(x)dx =
J — 00
Furthermore,
D(x) = (x2) - (xJ = (x2)
'00
.2
r
(x2) = / xzP(x)dx = az (8.35)
J — 00
so that we may associate a2 with the variance of the Gaussian (and a with
standard deviation). Note that the preceding result returns the fundamental
finding G.24), now written as
(jc2) oc t (8.36)
This result, as well as the Gaussian distribution (8.34), will be encountered
again in our discussion of diffusion (see Section 3.4.3).

1.8 Central-Limit Theorem 57
Stemming from the central-limit theorem, the Gaussian distribution (8.34),
as well as the Poisson distribution (8.22), is relevant to independent events
with no memory.
Asymptotic values
Let us consider application of the Gaussian (8.34) to the problem of evaluating
the probability that observation finds x in the domain \x\ < Xo, where xo is
some constant value. This probability is given by
P(\x\ < x0) = 2 / P(x)dx = — / e~y2 dy = erf —°— (8.37)
Jo V7* Jo (TV 2
where erf is written for the error function. Thus, at Xo = cr, we find
P(\x\ < a) = erf -7= 2^ 0.68 (8.38)
V2
There is — 70% probability of finding |jc | < o.
For large z, we note (see Appendix B, Section B.2)
erfz-1-=(l
/ \ 2z2
Thus, in the large x0 domain, we find
> x0) = 1 - erf -^ - J — 1 - -—— + • • •) (8.39)
V2 Vn xo/cr \ (xo/crJ )
This result will come into play in Section 3.3.12 in application of the central-
limit theorem to a fluid.
1.8.7 Poisson and Gaussian Distributions
The Poisson distribution (8.22) is relevant to discrete events with r = 0, 1, 2,
•.., whereas the Gaussian (8.34) is a continuous function ofx. Furthermore, the
Poisson distribution gives the main value (r) = X > 0, whereas the Gaussian
gives (x)• = 0. See Fig. 1.21.
We wish to illustrate a limiting transformation in which the Poisson dis-
distribution goes to the Gaussian distribution with unit variance, a — 1. The
transformation is
r — X
x = —— (8.40)
v A.
•
in the limit that X becomes large. Note that in this limit the new variable x
grows continuous. Two approximations come into play in this transformation:
The first is Stirling's approximation for large r,
r\ ^ y/lmr ( - )

58 1. The Liouville Equation
t
P{r)
n
JH.
Poisson
distribution.
The parameter
r is discrete.
P(x)
Gaussian
distribution
<x> = 0
FIGURE 1.21. Collapse of the Poisson to Gaussian distribution under the transfor-
transformation x = (r — X)/\fk, X —> oo.
or, equivalently,
lnr! ~ - lnB7rr) + r(lnr —
The second approximation in the expansion
x2
ln(l +x) = x 1
2
forx <^C 1.
We recall the Poisson distribution (8.22).
P(r) =
r\
We wish to transform this distribution under (8.40):
r — jcVa H- k
Let us call the new distribution P(x) so that
P{r)dr = P{x)dx
P(x) =
ax
(8.41a)
(8.41b)
(8.42)
(8.43)

1.8 Central-Limit Theorem 59
r\
with r = r(x), as given by (8.42). Taking the In of (8.43) gives
\nP(x)= (r+ -jlnA.-A.-lnr! (8.44)
With (8.42) and (8.41), we write
x x
In r = In A H — 1
2A
and
.2 -l
lnr! = \nV2n + (r + - )
lnA
X X'
— r
2A
Substituting these results into (8.44) and neglecting terms of O(;t/\/A) gives
e~x2/2
e
P(x) = -7= (8.45)
V27T
which agrees with the Gaussian (8.34) for a — 1.
Covariance and Autocorrelation Function
Another important function entering in probability theory is the covariance of
two rv's written cov(§r, §"). It is defined as
More concisely, we may write
<cr - <r>xr-
(8.46)
Note in particular that if £' and £" are statistically independent or uncorrelated
then
cov(r, D = 0 (8.47)
The correlation function of two rv's is given by the dimensionless ratio
, „ = covq, n 8 48
VDcr)D(r')
where D(|) is the variance (8.2b).
The random variable of a stochastic process is a function of time. Thus, for
example, in a stochastic process one writes
2
2

60 1. The Liouville Equation
H
FIGURE 1.22. The coordinates for Problem 1.1.
The autocorrelation function of a stochastic process is the image of the
covariance and we write
(8.50)
If (§@) = 0, we may set t2 — t\-\-r and the preceding becomes
covK(r,), £(r2)] - <§(*,)£(*, + r)) - <£@£(* + r)) (8.51)
In the limit of no correlation between the distinct stochastic processes,
(8.52)
which vanishes by previous assumption.
The autocorrelation function is returned to in Section 3.4.4 concerning
analysis of transport coefficients and in Section 4.2.5 in derivation of the
Balescu-Lenard equation. In the quantum domain it may be reformulated in
terms of the Green's function. See Problem 5.42.
Problems
1.1. A weightless rigid rod with a frictionless movable bead on it is constrained to
rotate in a fixed plane with one end fixed. The plane of rotation is parallel to
the gravity field. The bead has mass m.
(a) Choose a set of generalized coordinates and write down the Lagrangian
for the system.
(b) What are Lagrange's equations for this system?
Answer
(a) Generalized coordinates are r, 0 (see Fig. 1.22).
(b) The r-equation
. dL
dr
= mrO — mg cos 6
dL\ d
=
dt \dr ) dt

Problems 61
This equation gives
d
(mr) = mrO2 — rag cos
dt
The first term on the right side represents centripetal force, whereas the
second term is the gravity force.
The 0 equation:
dL . dL 7.
— = mgr smO, —- = mr 0
do 6 do
d
—(mr 0) = mgrsinO
dt
That is, the rate of change of angular momentum is equal to the applied
torque.
1.2. (a) What is the Hamiltonian for a free particle expressed in spherical
coordinates?
(b) What are the constant momenta of the particle in this representation?
(c) What are the constant momenta in Cartesian coordinates?
Answer
(a) The Hamiltonian is given by
2ra 2rar2 2rar2 sin2 0
which may be rewritten
p2 I2
2ra 2rar2
where L denotes the total angular momentum of the particle.
(b) Since 0 is a cyclic coordinate, p0 is constant. With
dH
we find
p0 = rar20 sin2 0 = constant
Two remaining constant are H and L2.
(c) In Cartesian coordinates
2 i 2 i 2
„ Px + Pv + Pz
ti =■
2ra
Since jc, y, z are cyclic,
p = constant
which again comprises three independent constants.

62 1. The Liouville Equation
1.3. Show that the Lorentz force law A.18) follows from the electrodynamic
Hamiltonian A.16).
1.4. An inductively coupled double-circuit network has the Lagrangian
2 2
Hi
^ i = l L
In this expression, / is current, V is voltage, L is inductance, C is capacitance,
and Mjk is mutual inductance.
(a) What are Lagrange's equations for this system?
(b) What is the Hamiltonian for this system?
1.5. Establish (a) Jacobi's identity A.26e), and (b) the relation A.26g).
Answer (partial)
(b) Let B(A) have the Taylor-series expansion
Vn
Then A.26g) appears as
Here we have used the expansion A.26d).
1.6. Show that [q, p] is a canonical invariant.
Answer
dq dp dp dq
dq dp dq dp
In the primed frame, we have
dq dp dp dq ( pq
= 1
dq' dp' dq' dp' \q'p'
which proves the statement.
1.7. Consider a collection of N mass points each of mass m. Let radii vectors to
these points be written r,. Then the radius to the center of mass is given by
mi M
The summation runs over all TV particles. Let
r; = Ti,- R

Problems
63
t
r,
FIGURE 1.23. Coordinates relative to the center of mass.
designate the particle radius vector relative to the center of mass. See Fig. 1.23.
Show that the kinetic energy of the system may be written
2T =
1.8. A rigid weightless wheel of radius a carries a dumbbell of radius 2a. The
dumbbell rotates with its end points constrained to move on the rim of the
wheel. The arm of the dumbbell is weightless, but its end points each have
mass m. The system moves in three-space.
(a) How many degrees of freedom does this system have?
(b) Choose generalized coordinates for this system that include the coordi-
coordinates of the center of mass, X, Y, Z.
(c) The system moves free of gravity. What is the Lagrangian of this system
in the coordinates you have chosen? Hint: See Problem 1.7.
(d) Which momenta are conserved for this system in the coordinate frame
you have chosen?
1.9. (a) With the notation of Problem 1.7, show that the total angular momentum
of a system of N particles may be written
N
(P9.1)
1=1
In this expression
= MR
is the total linear momentum of the system, and
P; = mir;
is the linear momentum of the /th particle relative to the center of mass.
(b) For a finite rigid body with number density «(r), show that (P9.1) may
be written

64 1. The Liouville Equation
where L = RxP and
S = J drfn(rf)r' x p'(r')
1.10. Show that the Poisson bracket relation
is invariant under the transformation generated by
Answer
We find
= sinp', q =
cosp'
Thus
sin p\
cosp_
= — sin p — I I I I —(sin p ) = —
dq' dp' \cos p' / 3^' ycosp'/ 3p'
1.11. Is the transformation generated by
,pf) = A\n[f(q)h(pf)]
canonical? The parameter A is constant and / and h are arbitrary but
continuous functions.
1.12. Show that the transformation
q' = In I — sin p 1 , p = q cot p
is canonical.
1.13. Use the generating function
n ma}q2 , ,
G j = cot q
to obtain the orbits of the Hamiltonian
p2 kq2
H = — + —
1m 1
1.14. (a) Show that the equality
pxdx + py,dy + pzdz = prdr + ped6 + p^dcj)
follows from the transformation equations from Cartesian to spherical
coordinates.
(b) Argue that this transformation is canonical.

Problems 65
1.15. Show that if A is a Hermitian operator, then:
^
(a) Eigenvalues of A are real.
(b) Eigenfunctions of A are orthogonal.
(c) What additional assumption comes into play in your answer to (b)?
1.16. Show that if
A =iL
is Hermitian then:
(a) Eigenvalues of L are purely imaginary.
(b) Eigenstates of L are orthogonal.
(c) A and L have common eigenstates.
(d) What type of operator is L?
1.17. Assuming that ^(k) as given by E.12) satisfies periodic boundary conditions,
show that wave vectors have the values
K
L
where the components of ns are integers and L3 is the volume of cubical
confinement.
1.18. Let a surface be generated by rotating a curve that passes through the points
(jci, y\)\ (jc2, y2) about the y axis. Show that the curve that makes the surface
area a minimum is y = a cosh~](x.a) + b.
1.19. If u and v are two dynamical functions that are both constants of the motion
for a given system, show that [w, v] is also a constant of the motion.
1.20. Show that the Poisson brackets of two Cartesian components of the angular
momentum L of a particle about a specified origin obey the relation
where (/, j, k) are the variables (x, y, z) in cyclic order.
1.21. Constants of the motion for a free particle moving in one dimension are
8\ = P
pt
m
Show that
= 0
dt
1.22. This problem addresses generalized potentials, which come into play for cer-
certain nonconservative systems. Consider that the force Fs in the direction of
the generalized coordinate qs may be written
du d du
tt — I
dqs dt dqs

66 1. The Liouville Equation
where U is the generalized potential. If this condition is satisfied, then the
dynamics of the system is still given in terms of Lagrange's or Hamilton's
equations with
L = T -U
dL
Ps =
H =
Consider a particle of mass m and charge e that moves in an electromagnetic
field. The force on the particle is given by (cgs)
1
5+-(vxB)
c
> ■
where c is the speed of light. Working with the potentials A and 3> given by
C dt
B = V x A
V • A H = 0 (Lorentz gauge)
c dt
(a) Show that the generalized potential for the charged particle is
U = e I O v • A
V c
(b) Show that the canonical momentum of the particle is
p = m\ + -A
c
(c) Show that the Hamiltonian for the particle is given byA.16).
(d) Show that the preceding defining equations for A and 3> remain invariant
under the transformation 3> —> 3>' = 3> — dxj//cdt\ A^A' + Vt/?.
(e) What equations of motion for A and 3>, respectively, do Maxwell's
equations give?
1.23. Three identical particles move in one dimension and are confined to the interval
@, L). At a given instant of time, the joint-probability distribution for this
system is known to be
/3A, 2, 3) = [exV-a(p] + p\ + p])]
x [sin bx\ sin bx2 sin bx3 cos c(xxx2 + x2x3 + X\X3)]
where a, b, and c are constants.
(a) Is this distribution symmetric under exchange of particles?
(b) What is the value of the constant b
(c) What is the pair distribution function F2(x, x') for this system?
(d) What is the conditional distribution function F2(x, x') for this system?

Problems 67
In your answer to (c) and (d), integral expressions will suffice.
1.24. A collection of four identical particles moving in one dimension are known
to be in the following state at a given time t > 0. One particle is moving with
velocity v°} and another with u°. Both these particles were at the origin x = 0
at t = 0. The remaining two particles are stationary at x® and xQA, respectively.
(a) Write down a determinantal joint-probability distribution that describes
this state.
(b) Obtain an expression for /i(jc, v) from your answer to (a).
1.25. A student argues the following: It has been stated that the relation
£ Pdq = £ P dq'
establishes the transformation (q, p) —> (q', p') to be canonical. He argues
that this is in violation of the criterion B.11). Punch a hole in the student's
argument.
Answer
Assuming that the given transformation stems from G\(q, q') leads to incon-
inconsistencies. For with q, q' as independent variables, the transformation gives
p = p' = 0 On the other hand, G2(q, p') may be shown to be consistent with
the given transformation. Consider, for example (for one degree of freedom),
The given relation becomes
pdq - p dq' = p'—dq - p'df = 0
dq
which is seen to be consistent.
1.26. Show that the probability distribution (8.20), relevant to the random walk,
satisfies G.23a).
1.27. Show that the difference equation G.23) gives the diffusion equation22
(P27.1)
dt dx2 ^ ' '
where time t and displacement x are related to the integers n and / through
the interval constants 8} and 8X as
t x
n = T» / = T
o\ ox
and
D = lim
_28t_
22
Suggested by G. K. Schenter.

68 1. The Liouville Equation
is assumed to be a finite parameter.
Answer
Rewrite G.23) as
P(/, n) - P(/, n - 1) = -[/>(/ + 1, n - 1) - 2P(/, n - 1) + P(l - 1, n - 1)]
(P27.2)
Now note that
A[AP(/) = A[P(/
Thus, with An = A/ = 1, (P27.2) becomes
AP _ 1 A(AP/A/)
^7 ~ 2 (A/J
or, equivalently,
AP 82X A2P
At 2
which, in the said limit, gives the diffusion equation (P27.1). Note in particular
that, with correspondence between the difference equation G.23) and the
diffusion equation (P27.1) established, we may conclude that the diffusion
equation describes a Markov process.
1.28. (a) What is the equation of motion in F space for the ensemble density func-
function D that corresponds to the fact that system points are neither created
nor destroyed?
(b) How does this equation differ from the Liouville equation?
1.29. This problem concerns the integral invariants ofPoincare.
(a) Consider a dynamical system with N degrees of freedom. Prove Liou-
ville's theorem for the subsystem comprised of s < N particles. That is
show that
()=■
where (q, p) —> (q\ p') is a canonical transformation for the s-particle
subsystem.
(b) What is the geometrical significance of this statement in F-space?
(c) If we attempt a derivation of Liouville'e equation for the subsystem fol-
following the first derivation given in the text, where does the argument
collapse?
1.30. Consider an N-body system whose Hamiltonian is given by
N
H =

Problems 69
The function Ho is given by D.24) and X and a are constants.
(a) What are Hamilton's equations for this system?
(b) Is the Liouville operator D.20a) Hermitian for this Hamiltonian?
1.31. (a) Construct a generating function G2(qN, p'N) that gives the transformation
1
P\ = "Pi. Pn = Pn
where 2 < n < N.
(b) Show explicitly that the Jacobian of this transformation is unity.
1.32. Show that the Hamiltonian of an N-particle isolated system with conservative
forces may be identified with the energy of the system. Hint: To establish this
result, recall Euler's theorem which states that, if F(x{, x2, ..., xN) is a linear
combination of products of jc, variables with the sum of exponents of each
product equal to n, then
1=1
and employ the Hamiltonian-Lagrangian relation.
1.33. Show that the Gaussian distribution in three dimensions
/t(r, t) =
has the following properties:
(a) fn(r,t)dr = No
(b)
(c) /i(r, 0) = N0S(r)
1.34. If p(n) is a discrete probability density for the rv £, and <j>(a) is its characteristic
function, then show that
(a) 0@) = 1
\)a=0 \ da
(c) \4>(a)\ < 1
1*35. Consider the Poisson distribution for the random variable r,
X'e-'K
P(r) = ——
r\
(a) Construct the characteristic function for this distribution as a sum over r.
(b) Use your answer to obtain S(r) and D(r). (Compare your findings with
values given in the text.)

70 1. The Liouville Equation
Answer (partial)
(a) In0(a) = -A.(l -eia)
1.36. The faces of a six-sided cubical die are numbered 1 to 6, and the faces of
a four-sided tetrahedral die are numbered 1 to 4. Consider the numbers that
appear on the bottom faces of the dice in a random throw.
(a) What are the respective characteristic functions (j)'(a) and 4>"(a) corre-
corresponding to throws of the individual dice?
(b) What is the probability that the sum of numbers on the bottom faces adds
to 8 in a throw of the dice?
1.37. Do the canonical transformations on a system comprise a group? Explain you
answer.
1.38. Show that the discrete random-walk probability P(l,n) given by (8.20) goes
over to the asymptotic form (8.30) for large n.
1.39. This problem addresses classical field dynamics and the Sine-Gordon equa-
equation. Consider a field 0(jc, t) with time and space derivatives denoted by 0,
and 0*, respectively, which has the Lagrangian
L =
1 9
-@, -*!#)- *2 COS 0
dx
2
The field coordinate and its conjugate momentum are @, n), where
8L
Variation on the right denotes functional differentiation.
(a) What is the form of n for the given Lagrangian?
(b) What is the Hamiltonian for this field?
(c) Write down Hamilton's equations for a field (in functional form).
(d) From the relations found in part (c) obtain an equation of motion for the
field 0(jc,
Answer
8L \ l[{4>< + &<t>'J ~ B]dx> ~ \ f[{<t>' ~ B]dx'
(a) n(x, t)= ^^
dip, dip,
where B represents constant terms in the variation. There results
,0= / hi*')
= Jw:
dx'
')8(x -x')dx = 0r(jc,r)
J
Thus
Tl(x, t) = (pt(x, t)

Problems 71
(b) H
= / U(j)tdx -
L
dx
(c)
SH
~sn
SH
(d) We must evaluate
1
2
■/
cos@
dx
{k\4>
xx
sm
There results
SH
+&2sin0)
Combining this finding with preceding results gives
4>tt ~k\4>xx -k2sin(j) = 0
which is the celebrated Sine-Gordon (nonlinear partial differential) equa-
equation. Note:23 The classical field 0(jc, t) may be viewed as appropriate to
a system with an infinite number of degrees of freedom. Thus, for exam-
example, the Lagrangian given in the statement of this problem is related to
an infinite set of freely rotating pendula coupled to each other by springs
with force constant k\. The function 0(jc, t) is related to the angle of a
pendulum at x, t.
1.40. Chebyshev's inequality states that, for an arbitrary probability distribution
with £(x) = x and D(x) = a2,
P(x —x\> Xo) < —
Show that this inequality is valid for a Gaussian distribution.
1*41. A system with N degrees of freedom has coordinates {jc,} and momenta
{Pi}, i = 1,..., N. Consider that {x{} is written as a column vector x. A
For further discussion, see P. G. Drazin, Solitons, Cambridge University Press,
York A983). The phase "Sine-Gordon" was coined as a pun on the Klein-
on equation of relativistic quantum mechanics.

72 1. The Liouville Equation
transformation to new coordinates x is given by
x = Mx
where the elements of the N x N matrix, M, are constants,
(a) If we write
what is the relation between Q and M that ensures that the transformation
is canonical?
(b) Under what conditions will the transformation be canonical if Q = Ml
Answer
(a) For a canonical transformation, in matrix notation, we write
xfdp = xf dp
= t =
where, in general, A is written for the Hermitian adjoint of A. [Recall,
-1
(A \j = A*r] Inserting the given transformations into the preceding
equation gives
(Mx)f d(Qp) = x+ dp
which implies that
= t =
or, equivalently,
(b) If Q = M, the preceding condition becomes
M M = 1
That is, M must be unitary.
1.42. If /v(l, 2,..., N) is translationally invariant,
fN(xi +a, pr,x2 + a, p2;...;xN +a, pN) = /v(xi, pr,x2, p2;.. • ;xN, p/V)
and rotationally invariant,
/v(x1,p1;x2,p2;...) = /v(xj +£ x x,,p,;x2 +£ x x2, p2,..
where e is infinitesimal, then show that
(a) /i(x,,p,) = g(pi)
and

Problems 73
(b) /2(xi,pr,x2,p2) = /z(|x, -x2|,pi,p2)
The functions g and h are arbitrary.
Answer
Consider an arbitrary transformation from variables (x, p) to variables (x, p)
such that (as in this problem)
/v(xi, ..., xN\ Pi, ..., P/v) — /v(Xi, ..., x#; pi, ..., P/
Suppose further that the Jacobian of the transformation is unity, as in this
problem (show this). Then we have the following (for s < N)\
-/
■/
-/
Thus the translational and rotational invariance of fN is obeyed by reduced
distributions as well, and we may write
which implies that
Thus
which is property (a).
To establish (b), first we write
/2(x,, x2, pj, p2) = h(x] + x2, x2 - x2, pt, p2)
Translational invariance of f2 then implies, with y = Xi + x2, that
h(y, x, - x2, pj, p2) = h(y + 2a, x, - x2, pt, p2)
Thus
and the preceding becomes, with w = X] — x2,
/2(x,, x2, p,, p2) = (w, p,, p2)
Let R denote the infinitesimal rotation

74 1. The Liouville Equation
Then rotational invariance of f2 gives
Thus h is independent of the direction of w, whence it can only depend on
|w|, and we conclude
,, x2, p,, p2) = fcflx, - x2|p,, p2)
which is property (b).
1.43. Show that for nonvelocity-dependent potentials, canonical momenta are de-
dependent only on the kinetic energy of the system at hand. Thus, in general,
canonical momentum may be considered a kinematic variable. (An exam-
example where this property does not hold is the case of a charged particle in an
electromagnetic environment, for which the related "generalized potential" is
velocity dependent. See Problem 1.22).
1.44. A particle of mass m moves on a two-dimensional plane surface.
(a) Transforming from Cartesian to polar coordinates, obtain expressions
for new momenta (pr, pe) employing the generating function G2 given
beneath B.15). State explicitly for pr and pe directly from the Lagrangian,
L, written in polar coordinates. State explicitly what the arguments of G2
are for this problem.
(b) Obtain expressions for pr and pe directly from the Lagrangian, L, written
in polar coordinates. State explicitly what the arguments of L are.
(c) Do your answers to the preceding change if the particle moves in a
potential field? Explain.
1.45. A pendulum consists of a weightless rod of length a and a mass m at one end.
The other end is fixed and the pendulum moves in a fixed plane whose normal
is normal to the gravity field.
(a) Write down the Hamiltonian of the pendulum in polar coordinates @, p9)
(with 0 = 0 corresponding to the direction of gravity).
(b) Solve for the motion 0(t) in quadrature.
(c) Sketch the orbits of the pendulum in (#, pe) phase space. Show that orbits
divide into sets: closed (vibration) curves and open (rotation) curves. The
orbit that separates these classes of curves is called the separatrix.
(d) Show that the separatrix [in 0 = (—n, +n)] includes one fixed point
(pe = 0). Such orbits are called homoclynic as opposed to heteroclynic
orbits, which include more than one fixed point.
(e) Show that for motion on the separatrix it takes infinite time to reach the
fixed point.
1.46. Given that dL(q, q, t)/dt = 0, show that the null variation of
/t2 pt2
Ldt and / f{L)dt

Problems 75
give the same Lagrange's equations, where f(L) is any continuous function
ofL.
A. A.
1.47. (a) Suppose M is an arbitrary square matrix. If there exists a matrix C such
that
A = C"'MC
is diagonal, describe the matrices C and A.
(b) If A, B are two square matrices and A is nonsingular, then show that
A. A. A. , A. A. A.
(c) Consider the N-dimensional dynamical equation
v = Mv
where M is a time-dependent N x N matrix, a dot represents time differ-
differentiation, and v is an N-dimensional column vector. Write the solution
of this equation in terms of (exp At) and v@).
(d) If M is Hermitian, how is your answer to (c) simplified?
Answers {in part)
A*. ^
(a) The matrix C is comprised of columns that are eigenvectors of M, whereas
A. A.
A is comprised of the eigenvalues of M.
(c) v@ = etli*\@)
From (a), M = CAC so that
= [expr(CAC)]v(O)
= [exp(Ct AC-])]\@)
with (b) we may then write
A.
(d) The preceding form involves C , which may not be trivial to calculate.
However, if M is Hermitian, then A is real and C is unitary. In this event,
if C is known, so is C = Cf.
1.48. The initial value of an ensemble density is given on the spherical surface in
T-space, E P? + E <l] = a2>t0 be
= ^ q] exp[- ^ Pi2]
System points of the ensemble are comprised of N independent harmonic
oscillators with potential- V = q2/2 and unit mass. What is the solution
£*(<?> ?•> t) to the Liouville equation?
1*49. A canonical set of coordinates and momenta obey the fundamental Pois-
son bracket relations A.12). Expressions for the Poisson brackets of the
components of angular momentum are given in Problem 1.20.
(a) Are the components of angular momentum valid canonical momenta?

76 1. The Liouville Equation
(b) The energy of a rigid molecule, free to rotate about its center of mass, is
given by
L] + L\ L2
E = — - + —
27, 2/3
where moments of inertia, (/], 72, 73) are evaluated in principal axis with
11 + 72 and the origin at the center of mass. Is the preceding expression a
valid Hamiltonian form?
(c) Write down the proper Hamiltonian for this molecule. Hint: Evaluate
[L2, L2].
Answers (partial)
(b) As Lx, Ly, L, are not canonical momenta, the given form is not a valid
Hamiltonian.
(c) The appropriate Hamiltonian is given by
L2 - L2 L2
H = - + —
27, 27,

CHAPTER 2
Analyses of the
Liouville Equation
Introduction
This chapter begins with a derivation of the sequence of equations called the
BBKGY equations (also called the hierarchy). These are equations for the
reduced distributions encountered in Section 1.6.3 and play a key role in the
kinetic theory of gases and fluids. The first two equations of this sequence are
employed in the derivation of the conservation of energy of a gas of interacting
particles. This description serves as a good example of basic techniques used in
kinetic-theory problems. A nondimensionalization procedure is introduced in
Section 2.2, which leads to the notions of strongly and weakly coupled fluids.
Correlation functions are defined through the Mayer-Mayer expansion, and
the Vlasov kinetic equation is found to result in the limit of weak coupling and
long-range interaction. This equation leads to the notion of a self-consistent
solution. The section concludes with a brief account of spatial correlation
functions important to the theory of the equilibrium structure of fluids. These
are the radial distribution and total correlations functions.
In the following two sections, techniques are described that attempt to de-
develop solutions to the hierarchy through perturbation-expansion techniques.
A diagrammatic representation of integrals occurs in the Prigogine technique.
Summing diagrams corresponding to the rare-gas limit gives the Boltzmann
equation. Diagrams relevant to a plasma are also discussed.
Temporal intervals are introduced in the Bogoliubov analysis, relevant to
equilibrium of a fluid. Expanding in powers of inverse specific volume and
keeping leading terms again give rise to the Boltzmann equation.
The chapter continues with a description of the Klimontovich picture of
kinetic theory. This formalism involves phase densities that are delta-function
expressions of the state of the system. Moments of these phase densities are

78 2. Analyses of the Liouville Equation
found to be related to multiparticle distribution functions. The discussion
concludes with rederivation of the first equation in the hierarchy.
In the concluding section of the chapter, Grad's derivation of the Boltzmann
equation from the Liouville equation is presented. It is concluded that all three
techniques are given by Prigogine, Bogoliubov, and Grad contain procedures
for obtaining higher-order corrections to the Boltzmann equation appropriate
to denser fluids.
In Chapter 3 the very important Boltzmann equation is rederived more
directly and studied in detail.
2.1 BBKGY Hierarchy
The reduced joint-probability distributions fs(l, ..., s), s < N, were intro-
introduced in Section 1.6.3. It was noted that f\ and fc determine the kinetic and
potential energy of an aggregate of particles. In Chapter 3 the extreme rele-
relevance of these first two distributions to all fluid dynamics will be discussed.
Thus it is important to obtain equations of motion for these distributions. With
the definition of fs, as given by A.6.10), we see that an equation of motion
for fs is obtained by integrating the Liouville equation over the phase volume
d(s + 1)... dN. Prior to so proceeding, we introduce a few key operators
important to obtaining the desired equations.
2.1.1 Liouville, Kinetic Energy, and Remainder Operators
First we rewrite the Liouville equation A.6.6)
[/*,//]=() A.1)
ot
in terms of a slightly modified Liouville operator
^LNfN 0 A.2)
ot
Comparison of LN so defined with A given by A.4.23) indicates that iLN =
A. With Problem 1.16, we find that LN has purely imaginary eigenvalues
corresponding to oscillatory solutions of A.2). [Recall A.5.7))].
With A.1) and A.2), we find
A.3)
which for an aggregate of N particles gives
9 dH d
) (L4)

2.1 BBKGY Hierarchy 79
Again, the Hamiltonian, //, is given by A.4.27):
N 2 W
Consider the operator
f dp,
Terms in the sum are nonzero only when I — i or I = j. Thus
a d d ^ d
dpi
With the equality
we obtain
d d
3pi
Here we have written
d
1=1
where
*,, =e*(|x,-xy|) A.5a)
is the two-particle interaction potential. We obtain
N ~
P/
A.7)
i (l-7a)
for the force on the /th particle due to the jth particle. With A.7), A.6) may
be written
N
equivalently,
^^ ^^d A.9)

Where Kh the kinetic energy operator, is as implied. As we are interested in
equation for fs, we partition LN as follows:
LN = LS + L,v,,+1 A.10)

80
2. Analyses of the Liouville Equation
s s+1
FIGURE 2.1. Enumeration of terms in the remainder operator
where, the ^-particle Liouville operator, Ls, is given by
'=1
The remainder operator, LN^s+\, is given by
(lid
N
LN — — y^ Ki + R\f A12)
l=s+\
To delineate the On terms in Rn,s+\, we refer to Fig. 2.1. Thus we find
E
A.13)
2.1.2 Reduction of the Liouville Equation
With these operators at hand, we are prepared to carry out the integration of
A.2) as described above. Specifically with A.10) and A.12) substituted into
the Liouville equation A.2), integration gives
d
dt
- L
= f
d(s + 1).. .
N
A.14)
l=s+\

2.1 BBKGY Hierarchy 81
Inserting A.13) gives
RHSA.14)=
a
n
=5+1
N
•+ V
i=S + \
N
7=5+1
The first and third terms in this expression contribute only surface terms and
vanish. Thus we obtain
/
J
A.15)
Inserting the representation A.7) for On gives
s N
RHSA.15) = - / d(s + l)...dNy. Y, Gij-i — - — ijh
As the Pj derivatives run over j > s + 1, they too are surface terms and vanish.
We are left with
fs = ~y d(s + l)...dN y GufN A.16)
J S / . t\__ I V ' / / . IJJ1V \ /
This is a key equation in the present development for the following reasons.
First, we note that up to this point the mass of particles in our aggregate, written
m, many have been written with an index, such as m/ in the expression for the
Hamiltonian A.5). That is, in obtaining A.16) it was not necessary to assume
that particles are of equal mass. Second, it is apparent from A.16) that the
dynamics of the fluid partitions so that the interaction of the subgroup of s
particles, described by fs(l,..., s), with the remaining particles in the fluid is
given by the right side of A.16).
2.1.3 Further Symmetry Reductions
To further reduce A.16), it is necessary to assume that the constituent particles
in the fluid are identical and that fs A, 2, ..., s) is symmetric under interchange
of particle phase numbers. Thus, for example, in fs(\, 3, 2), particle 2 is in state
3 and particle 3 is in state 2. With the preceding assumptions, we may write
/3A,2, 3) = /3A,3, 2) A.17)
To examine the equivalence of the integrals under the j sum in A.16), we must
show, for example, that
d2d3... G12/,v(l, 2, 3,...) = f dl d3 ... G13/aK1, 2, 3, ...)

82 2. Analyses of the Liouville Equation
or, equivalently, that
f rf2rf3GI2/3(l, 2, 3) - f rf2rf3Gi3/3(l, 2, 3) A.18)
Integrating the left side over d3 and the right side over dl gives
f
rf3Gi3/2(l,3)
Changing the dummy variable from 3 to 2 in the right integral establishes
the equality. This result may also be argued geometrically, as demonstrated in
Fig. 2.2. Thus, in A.16) each term in the (N — \)j summation gives identical
terms and we obtain
frf 3p,-
= 0 A.19)
Here we have set j in G; j equal to s +1 to facilitate further reduction. Thus, for
example, with this choice, integration of A.19) over the volume d(s+2)... dN
leaves fs+\ in the integrand and we obtain
A.20)
1 <s < N
The coupled TV equations given by A.20) are called the BBKGY equations.
This abbreviation is written for N. N. Bogoliubov, M. Born, G. Kirkwood, H.
S. Green, and J. Yvon. These equations are also called the hierarchy. We will
use the shorthand notation BY5 to denote the sth equation in the hierarchy.
We first note three basic properties of these equations:
1. They are a coupled sequence of TV equations, the Nth of which is the
Liouville equation for fN.
2. Writing
— = — -Lsfs A.21)
Dt dt J
relevant to a subgroup ofs<N particles, with A.20) we note that
Dt
This indicates that fs(l, 2, ..., s) is not constant along the subsystem
trajectory in f-space. The term DfjDt does not vanish due to interaction
of the s subgroup of particles with the remaining aggregate. As noted
previously, this interaction is represented by the sum of integrals in A.20).
3. The third property is of special significance to the present work. It concerns
the first equation in the sequence A.20), BYp This equation is the generic
form for all kinetic equations. A kinetic equation is a closed equation of

2.1 BBKGY Hierarchy 83
(a)G,2f3A,2,3)
FIGURE 2.2. Graphical representation of force terms in A.16). Arrows represent
momenta; dashed arrows, interaction; and circles, position. Integrating over the phase
variables of 2 and 3 gives equal results.
motion for /j(x, p, t). The manner in which this occurs is as follows. Let
us rewrite BYi as
Pi 9
7i=-Ai/2(l,2)
dt
m
where A\ is the operator implied by A.20) with 5
results if we are able to effect the transformation
A.22)
1. A kinetic equation
A.23)
where J, typically called collision integral, maps functions onto functions.
The simplest way to effect this mapping is merely to write /2A, 2) as
some functional of /i(l). Thus, for example in the Vlasov approximation
to be considered below, we set /2A, 2) = /i(l)/iB). In the Bogoliubov
ansatz (Section 2.4), we assume that /2A, 2) may be written in the general
functional form /9A, 2) = /2[1, 2, /]]. Both these assumptions serve to
close B Yi and produce kinetic equations. The first assumption leads to the
Vlasov equation and the second leads to the Boltzmann equation.
2.1 A Conservation of Energy from BY\ and BY2
We turn now to the explicit form of BYj and BY2:
BY "Jl Pl
BY
A.24)
= 0A.25)

84 2. Analyses of the Liouville Equation
We wish to apply these equations to establish conservation of energy for an
isolated collection of TV interacting particles.
Equations A.6.16) relate kinetic energy density EK and potential energy
density E<&, respectively,to the single and pair distributions F\ and F2 [see
A.6.14)]. In the present discussion it is more convenient to work with velocities
in place of momenta. Thus we write (for a fluid whose mean macroscopic
velocity is zero)
dt dx
dF2 ( d , a \ d d
EK(x, t) = - I mvlFx d\ A.26)
'dx! A.27)
To obtain BYi for Fu we multiply A.24) by TV and remove particle labels.
To obtain BY2 for F2, we multiply A.25) by N(N — 1) and again remove
labels. There results
d i d r (d \ , ,
/ I —* I F2 dx d\ = 0 A.28)
m d\ J \dx I
Y = 0 (I 29)
dt \ dx dx'J d\ o\'
where X and Y are implied interaction terms (which contain F3). To find the
equation for EK, we we operate A.28)with / d\(mv2/2). There results
d
Now note that
2 d\ av \2
Inserting this relation in the preceding equation and dropping the surface terms
gives
. f-mi;2vF1Jv+ f d\d\' dx!\ . ( — | F2 = 0 A.30)
Operating on BY2 as giving by A.29) with
- / dxfd\d\f&(x, x)
and dropping the X and Y surface terms, we find
^) A.31)
^ + \ [dvdvdx&(x,x)(v^
dt 2 J \ dx
Now note that
F2 = — (#F2) - F2 — * A.32a)
3x 3x
F2 = (#F2) F2
3x 3x 3x

2.1 BBKGY Hierarchy 85
^2 = ^*F2) " F2 —* A.32b)
3x' 3x' 3x'
When the first of these relations is substituted into A.31), the d(&F2)/dx term
remains, as x is not integrated. However, the corresponding term in A.32b)
yields a surface term, and we obtain
r) 1 r)
E H • / d\ d\f
2d J
dt 2dx
C
/
J
- - I </v</v'</x'F2(v. — * + v' — *) =0 A.33)
Equations A.31) and A.33) have the form
+ V . QK + / dx'd\f d\F2\<&F2\ • —* = 0 A.34a)
j 3x
3x
— + V-Q*-- f dx/dVd\F2(\ — * + vr. — * ) -0 A.34b)
dt 2J \ dx dx J y }
where the flow vectors Qk and Q$ are as implied. We label the total energies
EK(t)= IEK(x,t)dx
J A.35)
, t)dx
/
Thus, operating on A.34a,b) with / dx and adding the resulting equations
gives
d ~ ~ 1 [ * * , * * , ( a* / a*\
(EK + E&) = - I dxdx d\d\ F2\\ h v •
v 7 A.36)
/* 3*
— I dxdx!d\d\fF2\ •
Owing to the symmetry relations F2(z, zr) = F2(z\ z), where z = (x, y), and
*(x, x') = *(|x — x'|) = *(xr, x), we find that all three integrals on the right
side of A.36) have the same value. Thus we obtain the desired result:
(EK + E*) = 0
_ _ A.37)
+ E& = constant
A more precise statement of conservation of energy in classical kinetic
theory may be obtained directly from the dynamic equation A.1.25). Iden-
Identifying the Hamiltonian H(xN, p^) with the energy of the system, we
find
dH dH
— = [H,H] + —=0 A.38)
at dt

86 2. Analyses of the Liouville Equation
Here we have observed that H is not an explicit function of the time. The finding
A.38) is a more exact result than A.37) in that A.38) identifies the total precise
energy A.5) as being constant as opposed to A.37), which identifies merely
the average of total energy as being constant.
2.2 Correlation Expansions: The Vlasov Limit
2.2.1 Nondimensionalization
In this section we introduce the correlation function representation of dis-
distribution functions. To motivate this representation, it proves valuable to
nondimensionalize BY^ as given by A.20), which, with TV ^> s, we first
rewrite as
where
/- • f
dp/ J
-N T /- • f d(s + l)Gf-,,+I B.2)
ffdp J
With A.11), B.1) is more explicitly written
ds s
EEv) B.3)
/-I I
where O[-} is the momentum operator A.7).
The fluid being considered is comprised of TV particles, which we now
assume occupies a volume V. This permits us to introduce a characteristic
number density
no = - B.4a)
We further assume that we may assign a mean thermal speed, C, and
corresponding temperature, 7\ to the fluid, which are related as
mC2 = kBT B.4b)
where kB is Boltzmann's constant.
The strength of potential 3>o and characteristic scale length ro are defined
by
Gij = —Gij B.5c)
where Gij is nondimensional.

2.2 Correlation Expansions: The Vlasov Limit 87
Finally, we renormalize fs so that1
Fs = Vs fs B.6)
For a spatially homogeneous fluid,
1
/i P V
In general, number density, n(x, t), is given by
n(x, t) = N I fidp = n0 I FY dp B.6a)
To convert B.3) to an equation for Fs, we multiply both sides by Vs to obtain
u x.—\ j^_ x—\ x—\ i. \ _ 1
~ V
The nondimensionalization scheme we introduce is given by the following
relations (barred variables are nondimensional):
x — rox, p — mCp
Note that a number of choices for characteristic length enter this analysis. We
may set ro equal to the range of interaction or we may set
-1/3
or we may set
These choices are noted by the following scheme:
range of interaction
ro ^—V1/3 B.9)
no
-1/3
o
Substituting the relations B.8) into B.7) gives
EE *>) *■ = CW) (^) hFs+l B.10)
The factor r% stems from nondimensionalization of dx, whereas the V factor
in n0 stems from the definition of Fs. It is evident that this equation suggest
!This Fs distribution should not be confused with the s-tuple distribution
introduced in Chapter 1, which was also labeled Fs.

2. Analyses of the Liouville Equation
we identify the parameters
y-x=norl B.11)
BT
Dropping bars over dimensionless variables, B.10) becomes
Q S S \ rs,
TKTY0)F IF B.12)
dt
Consider the coefficient
0 B.13)
Y
With the first choice for ro in B.9), we obtain
A/*o «^
'o =
which is the number of particles in a range sphere. Inserting this result into
B.13) gives
B.14)
The numerator of this expression represents pair-interaction energy per range
volume, and the denominator represents thermal energy per range volume. We
take this ratio to be a measure of the ratio of average potential to kinetic energy
in the fluid. Note that if No is small than few particles in the fluid interact, and
(E&) must likewise be small.
With B.14), we may term the fluid strongly coupled when a/y > 1, and
weakly coupled when a/y <^C 1. Thus we write
1, strongly coupled
B.15)
1, weakly coupled
A representation that well exploits this partitioning is that of correlation
functions discussed below.
First let us delineate coupling domains of the fluid generated by varying
the magnitudes of a and y in the coupling constant a/y. These domains are
shown in Fig 2.3.2
2The s-ordering diagram appeared previously in R. L. Liboff, Introduction to
the Theory of Kinetic Equations, Wiley, New York, A969). More recently, it was
employed in an analysis of galaxy correlations: M. A. Guillen and R. L. Liboff,
Mon. Not. Roy. Astr. Soc. 231, 957 A988), 234, 1119 A988).

2.2 Correlation Expansions: The Vlasov Limit
89
a=<t>o/kBT
t
Strong coupling _.,
or *
low temperature
Weak coupling
or
high temperature
(e-\e)"
(k, e)
(fc~,11)
A,1)'
(E, D
(E
(E,E
-► Y } =
Short range
or dilute
Long range or
dense
FIGURE 2.3. Coupling domains in a fluid. In the (*) boxes, a/y =0A).
2.2.2 Correlation Functions
If particles in a gas are statistically independent, the conditional probability
(see Section 1.6.4)
h\N\l\2,...,N)
B.16)
This follows since the state of particle 1 is in no way dependent on the state
of the remaining particles in the fluid. For B.16) to hold for all particles in the
fluid, we must have, in general,
l<s<N
B.17)
Another way to say that particles in the fluid are statistically independent is to
say that they are uncorrelated. Let us define the correlation function between
two particles, C2(l, 2), according to the relation
f /i /\\ f (\\ f @\ -\- C* A 9^ A 1 S^\
Thus, when C2 = 0, particles are uncorrelated and the preceding returns B.17).
Continuing in this way gives a transformation of distributions:

90 2. Analyses of the Liouville Equation
These relations are given by [repeating B.18)]
/iO)C2B,3) + C3(l,2,3) B.19)
where P(l, 2, 3) denotes permutations of 1, 2, 3.
We may represent these equations with the aid of diagrams. Thus, the
equation for /5, say has the representation
—00000 + 2_] ° o-o-o-o + 2_] o-o o-o-o + 2_^ o o o-o-o
p p p
+ 2_. °~° ° °~° + /_, °~° ooo + o-o-o-o-o B.20)
p p
where, for example, o-o represents C2 and o o represents f\ f\ and P is written
for P(l, 2, 3,4,5).
2.2.3 The Vlasov Limit
We wish to consider the limit <&o/kBT <^C 1 with long-range interaction so that
^ 1-We may describe this limit by setting of —> sot and y~l —> (l/s)y~],
where £ is a parameter of smallness (see Fig. 2.3). Note that a/y is of order
unity. As kBT ^> O0> we suspect that correlation between particles in the fluid
is small. This condition may be described by attaching an s factor with each
coupling bar in the corresponding diagram equation. In this manner we find
B.21)
Substituting these equations into B.12) and keeping terms of O(s), we obtain
[reverting to Fs distributions defined by B.6)]4
d
Y
d
k2 - eadl2] [^A)^B) + eC2(l, 2)]
- -/2[F1A)F1B)F1C) + £^A)^B, 3)
Y
B.22)
3J. E. Mayer and M. G. Mayer, Statistical Mechanics, Wiley, New York A940).
4Here we have assumed that B.21) is written in nondimensional form, in which
case /, = Fs.

2.2 Correlation Expansions: The Vlasov Limit 91
where
s
/•=1
Keeping lowest-order terms in B.22) gives
d
dt
.9+1
B.23a)
F,A)F2B) = -72^A)^B)^,C) B.23b)
y
) nFi(/)=-^ n
These equations compromise N equations for the single unknown F\. For this
sequence to be logically consistent, it follows that these N equations must be
redundant.5 To establish this redundancy for the sequence B.23), we note the
following two equalities:
s c
ri'
x F,(/ + 1)... F,(s) ( - + K, ) F,(/) B.24)
S + \ S
where we have written
d
-■(
9p/ J
d(s + l)G,.s+1 B.25)
Substituting B.24) into B.23c) gives
d
The problem of redundancy in BBKGY expansions was critically examined by
- L. Liboff and G. E. Perona, J. Math. Phys. 8, 2001 A967).

92 2. Analyses of the Liouville Equation
or, equivalently
3 - \ ot
dt 7 - y J=°
With reference to B.23a), we see that the bracketed term vanishes. Thus, if
Fiii) satisfies B.23a), it also satisfies B.23c), and we may conclude that the
sequence B.23 is consistent.
2.2.4 The Vlasov Equation: Self-Consistent Solution
We have found that the equation that emerges from the BBKGY sequence
in the limit a — 0(£), y = 0A/£) is B.23a), which we now write in fully
dimensional form (deleting the subscript on F\):
) F(x, v, 0 ---—•/ dx'dVG(x, x')F(x, v, r)F(x\ v\ t)
J d J
+ v ) F(x, v, 0
dt dx J m
B.27)
[Here we have recalled the equality F(p) dp = F(\)d\.] Equation B.27) is
the Vlasov equation. Its physical significance is revealed when cast in terms
of the number density [see B.6a)],
n(x', t) = n0 I F(x\ v', t)d\f B.28a)
and the mean force field
G(x, 0 = / n{x\ r)G(x, xr) dx! B.28b)
The integral in B.27) then becomes
• n0 f Fltf,\',t)dv' f G(x,*
- — F,(x, v, t) • / n(xr, r)G(x,
F,(x,v,r)-G(x,0 B.29)
3v
Thus B.27) may be written
+ v +
at dx m d\
, v, 0 - 0 B.30)
where G is given by B.28). The preceding equation indicates the following:
F(x, v, t) develops under a force field that is the instantaneous average of all
two-particle forces in the fluid, as given by B.28). This force field is sometimes
called a "snapshot" field. As this cumulative force changes slowly, we may
except particle trajectories in a Vlasov fluid to likewise change smoothly. See
Fig 2.4.

2.2 Correlation Expansions: The Vlasov Limit 93
Interparticle
force
FIGURE 2.4. Snapshot total force in the Vlasov equation gives a smoothly varying
trajectory.
The Vlasov equation B.30) bears a striking similarity to the one-particle
Liouville equation, which is relevant to a single particle moving in an externally
supported force field. The Hamiltonian for this system is given by
H
EL
2m
where
the externally supported force field. The Liouville equation
dF
dt
+ [F, ff] = 0
B.31)
for this system returns B.30) with G replaced by G. That is,
Yt
d G
i
d
dx m d\
F(x, v, t) = 0
B.30a)

94 2. Analyses of the Liouville Equation
The solution to both B.30) and B.30a) is obtained by solving the
characteristic equations
dt = — = —- B.32)
v G
(where a/b = ax/bx — ay/by — az/bz). These are equivalent to the orbit
equations
dx d\
v = —, G -m—-
dt dt
which give
d2x
B.33)
3x
These equations have the integral
m (dx\
B34)
Thus any function of the form
2
F = F
P_
2m
*(x)
B.35)
is a solution to the one-particle Liouville equation B.30a). In this case,
is a given known function.
Returning to the Vlasov equation, we see that for B.35) to be a solution to
this equation O(x) must satisfy B.28):
= -n0 I F(x', v', r)G(x, x') dx! dV B.36)
3x y
So in this case knowledge of O demands knowledge of the distribution function.
An additional requirement for B.35) to be a solution to the Vlasov equation is
that any explicit time dependence of O be negligible. Consider, for example,
the force equation B.33). Multiplying this equation by x gives
mx • x + x • —$ = 0 B.37)
3x
If $ = O(x, r), then
= x-— O + -— B.38)
which does not allow B.37) to be integrated. However, for
d> B.39)
r dx
B.37) returns the integral B.34).

2.2 Correlation Expansions: The Vlasov Limit 95
Guess ct>(x).
Construct
F{
Compare
with <|>{x)
in stepri
3) Construct^:
a
<t> = no(i
FQ (x,x') dx1 dv'
FIGURE 2.5. Flow chart illustrating the notion of a self-consistent solution to the
Vlasov equation.
Another important distinction between the Vlasov and one-particle Liouville
equations is that the former is nonlinear. That is, when writing the Vlasov
equation in the form B.30), we must keep in mind that the force field G is a
functional of the distribution F. Thus we may write
G(x, 0 - G[F(x, v, 0]
which exposes the nonlinearity of B.30).
Let us return to the condition B.36). This is a statement of self-consistency,
which is demonstrated by the flow chart for the construction of a solution to
the Vlasov equation shown in Fig 2.5.
The Vlasov equation is returned to in Chapter 4 where applications of this
equation are made to a plasma.
2.2.5 Debye Distance and the Vlasov Limit
We have found that the Vlasov equation results in the limit
a =
-i
kBT
= 0(8)
ay~l = 0A)
B.40a)
B.40b)
B.40c)
In a plasma comprised of electrons in a neutralizing, fixed, positive background,
the strength of the interaction potential between electrons is the Coulomb form
e2
<D0 = — B.41)
Where we have set the range of interaction r0 equal to the Debye distance kd.
Fory, we write
y-1 =4nn0k3d B.42)
Note that, in the present case, weak coupling corresponds to the limit
4nn0Xd ^> 1. Substituting the latter two relations into B.40c) and solving

96 2. Analyses of the Liouville Equation
for kd, we find
Kd I
B.43)
e"-n0
This characteristic distance comes into play a number of times in the text.6
22.6 Radial Distribution Function
In concluding this section we return to the theory of correlations introduced
in Section 2.2. An important correlation function that often comes into play
in the theory of the equilibrium structure of liquids is the radial distribution
function.7 This is a two-particle spatial correlation function that, for a fluid in
equilibrium, is defined by
B.44)
where g(r) is the radial distribution function.
In B.44), we have labeled
r = |x2 - x, |
and have introduced the Maxwellian distribution8
Mp) =
where V is the volume occupied by the fluid, and the thermal speed C is defined
by B.4b). The Maxwellian B.45 has the normalization
I B.46)
The total correlation function h(r) is defined by
h(r) = g(r) - 1 B.47)
which, with the first to the sequence B.19), gives the relation
B.48)
Thus g(r) is directly related to /2O, 2), whereas h(r) is directly related to
C2O, 2). For a fluid with no spatial correlation between molecules, h(r) — 0
6Thus, for example, the Debye distance is encountered again in Section 3.9 in
application of the diagrammatic approach to a plasma and in Chapter 4 in application
of the KBG equation to a plasma. Numerical values of kd as a function of T/n are
given in D.1.37). Note: With X2d and C2/X2d taken as constant, B.40) et seq. is referred
to as the Rosenbluth-Rostoker limit. (See Phys. Fluids, 3, 1, 1960.)
7For further discussion on this topic, see D. L. Goodstein, States of Matter,
Prentice-Hall, Englewood Cliffs, N.J. A975).
8The derivation of this one-particle equilibrium distribution is given in Section
3.3.

2.3 Diagrams: Prigogine Analysis 97
[r) = 1. As two molecules in a fluid that are separated by a distance large
compared to their interaction length may be assumed not to be correlated, we
write
«(oo) - 1 B.49)
Furthermore, as two molecules cannot occupy the same position in space,
f2(r = 0) = 0, which, with B.44) gives
8(P) = 0 B.50)
These latter two statements are standard boundary conditions in studies of the
radial distribution function.
An important property of these correlation functions is their Fourier
transform, the structure factor:
S(k) = l+n I h(r)eikrdr B.51)
where k is the Fourier transform wave vector and n is the number density of
molecules in the fluid. The function S(k) serves to relate data from scattering
experiments to the correlation functions. For an ideal gas, there is no interaction
between particles, and B.44) gives g(r) = 1, which, with B.51), gives the value
S(k)= 1.
Two key relations involving the radial distribution function, which may be
obtained either from the partition function of equilibrium statistical mechanics9
or BYi (see Problem 3.44), are as follows:
E 3 n f°° A 2 f
— = - H / <&(r)g(rLnr2 dr B.52)
Z LKBL Jq
P n C°°
— / &(r)g(rLnr2dr B.53)
*B* JO
BT
The first of these gives the energy density of the fluid. The second gives the
equation of state of the fluid, where P is scalar pressure.10 Note that the integral
in B.52) gives the interaction potential contribution to fluid energy, whereas
the integral in B.53) gives the interaction potential contribution to pressure.
2.3 Diagrams: Prigogine Analysis
In this section we offer an introduction to a perturbation technique of solution to
theLiouville equation. A diagrammatic representation of interaction integrals
D. A. McQuarrie, Statistical Mechanics, Harper & Row, New York A973). See
T. M. Reed and K. E. Gubbins, Applied Statistical Mechanics. McGraw-Hill,
New York A973). See also Problem 2.12.
. 10The notion of scalar pressure, as well as other dynamic variables, is discussed
Section 3.4.

98 2. Analyses of the Liouville Equation
occurring in the perturbation expansion is given. It is found possible to associate
the topology of these diagrams with the order of interaction between particles
(that is, two particle, three particle, an so on). Application of this formalism is
made in derivation of the Boltzmann equation and the temporal evolution of a
plasma.
2.3.1 Perturbation Liouville Operator
Let us recall the Liouville equation A.2):
ot
= LNf
N
where L# is given by A.3).
LN = [H, ]
The operator LN has the explicit representation [see A.9)]
n n
= e k> + E E
E E
We rewrite this operator in this form
LN = Lo + 8L
where Lo is the free-particle, kinetic-energy operator and
N
a
a
represents the perturbation to Lo.
Eigenfunctions of Lo are given by (see Section 1.5.2)
exp
C.1)
C.2)
C.4)
C.5)
C.6)
where v/ = p//m. Here, as in A.5.10a), we are writing (k) for the sequence
^!,^,...,^) C.7)
; = —ii/ C.7a)
where, with A.5.14), the components of n/ are integers.
We wish to employ the basis functions C.6) to solve the Liouville equation
C.1), with LN given by the general form C.3). Following A.15.16) relevant

2.3 Diagrams: Prigogine Analysis 99
the free-particle case, we write
. . . , iV J I U,(y^\\yM , I J Us (fo\\^A. Jt*- I J.O I
(k)
Terms in this series may be regrouped so that coefficients in this new series
Ibear special physical significance.
Generalized Fourier Series
The regrouped series is given by
yv
jN ~
> 0
V 7=1 k,
x
C.9)
The coefficients in this series have the property that an(kn, pN, t), for exam-
example, contains only n nonzero k vectors. Thus the coefficient <22 contains only
two nonvanishing k vectors. The Kronecker symbol
, k = 0
= 0, k ^ 0
Thus the bracketed terms in the second sum in C.9) contains no contributions
with k, + kz = 0. The volume V = V/Bn)\
Terms in the second summation of C.9) with ky + k/ = 0 are relevant to the
homogeneous limit. This may be seen as follows. If the system is homogeneous,
then fN is invariant under translation of coordinates, (x/) —> (xzr) = (x/ + b).
Thus, for a typical term in C.9), we have
^ e
(k) (k)
This equality is satisfied for all k/ and X/, providing ]Tk/ = 0 for all (k)
Sequences.
Interpretation ofsti Coefficients
integration of C.9) overall x* gives
fNdxN=J
C.10)

100 2. Analyses of the Liouville Equation
The meaning of the double sum in this equation is that for each value of j the
vector ky is summed over all values given by C.7a).
In the limit of large volume,
dk
C.11a)
Furthermore, in this same limit
— [ eikx
dx = 5(k)
C.11b)
so that the second term in C.10) gains a 5(k) factor upon spatial integration.
Thus the only terms that enter this sum are those for which ky = 0. But
@, p^, t) — 0 by definition. It follows that
«o(P , t) —
f
I
N
ao(p",t)dpn = 1
represents the TV -particle momentum distribution.
Let us introduce the momentum integrals of af.
dp'
,t)e
-ico.t
V
I
C.12a)
C.12b)
C.13)
The number density n\(x) is given by
«i(x) =
= N
dpN dx\ dx2 • • • dxi-i dxi+\ • • • dx
whereas the pair distribution is given by
n2(x,x!) = n2(xs,xn)
N(N - 1)
N
C.14)
C.15)
dpN dx\... dxs-X dxs+i ... dxn-X dxn+x ...
We turn first to evaluation of fti(x). Integrating C.9) in accord with C.14)
gives
N
/
C.16)

2.3 Diagrams: Prigogine Analysis 101
In the limit of large volume with C.11) and C.12), we see that all terms in
the y-sum except for the j = s term give a\@, pN, t), which vanishes by
definition.
Each term in the sum over 020*/, k/, p^, 0 contains either a 5(ky) or 5(k/)
factor, which stem respectively from / dSj or / dki integrations. It follows
that all terms in this sum vanish since
again by definition. The same argument applies to all higher-order terms. There
results
N
V
I
ikx
1+ / 5i(k,ry dk
C.17)
In the limit of spatial homogeneity, a\ — 0 and
N
V
(homogeneous case)
Next we turn to evaluation of n2(x, x'). Integrating C.9) over
N(N - 1)
dXj
gives
N(N - 1) | 1
~~ V
2
1
V2
V
a2(ks,kn,t)eilk'x'+k"x"]dksdkn
+
C.18)
The prime on the first integral stipulates that the value ky + k/ has been deleted.
Returning to C.18) and passing to the homogeneous limit gives
- 1)
w2(x, xr) =
2V2
1 + f 52(k, -k, ryk'(M/)
dk
C.19)
The integral term represents the deviation from the state of no two-particle
correlation and is seen to be a function of |x — x! |. In the state of no two-particle
correlation, a2 = 0 and
-1)
2V2
C.20)
Which has the normalization
n2(x, x')dx dxr =
C.21)
This is the total number of pairs of particles in the system.

102 2. Analyses of the Liouville Equation
Having obtained the key relations [C.17), and following] connecting a\ and
<?2 to the fluid variables ri\ and ri2, we return to formulation of equations of
motion for the general a^ coefficients.
23 A Equations of Motion for a(k) Coefficients
Substituting the series C.9) into the Liouville equation C.1) gives
() J2 ^)' C.22)
(k')
Differentiating the left side and then operating from the left with ((k)| and
noting the orthogonality of these eigenstates and the eigenvalue relation C.6),
we obtain
= £ eia)(k"<(k)|5L|(k')>(e'^'''a(k0) C.23)
The operator 8L is given by C.5). Fourier expanding the interparticle potential
we obtain
K(X/~x") C.24)
^ K
so that
K
C.25)
9 a
Inserting this form into the equation of motion C.23) gives
9#(k) 1 ^-^\ ^-^\ ^-^\ ^-^\ . . K(x —x ) f
^^ V (k') K l<n
C.26)
Let us evaluate the matrix element
^ f
C.27)
For sufficiently large but finite volume, we introduce the representation
f dxeikx = V8(k) C.28a)
Jo
Note that
lim V8(k) = B7iK8(k) C.28b)

2.3 Diagrams: Prigogine Analysis 103
We partition terms in the exponential of C.27) as
E
Xj ■ (k'j - kj)
+ x, ■ (k;' - k, + K) + xn ■ (k'n - kn - K)
With the representation C.28a), we obtain
N
: - kn - K) Y[ S(k'j -
= AB/K C.29)
2.3.5 Selection Rules for Matrix Elements
The preceding finding implies a very important result: nonvanishing matrix
elements occur only for (k), (k') sequences that satisfy the relations
= k, - K
= kn + K C.30)
We may conclude that in any nonvanishing matrix element the sequence (kr)
is the same as the sequence (k) except for two kr vectors. Furthermore, these
two vectors satisfy the conservation equation
Equation C.26) now has the explicit form
3<Z(k)
9 V inK
The summation Y^inK ls written for
* AlnKa{k)) C.32)
InK l<n K (k')
For the case of weak interactions, we set
where s < 1. Equation C.32) then assumes the form
- sGa{k) = e J^((k)\8LlnK\(kf))taikf)(t) C.33)
InK
where we have written
">'A/nKtf(k')) C.34)

104 2. Analyses of the Liouville Equation
Note that this matrix element now carries an explicit time dependence
Substituting the perturbation expansion
P2/7B)
into C.33) and equating coefficients of equal power in s gives the following
series of coupled equations.
@) rj A) rj B)
n dfl(k) A @) 0>a(k) A (l)
a* (k) a*
These equations are simply integrated to obtain
C.37)
Setting t — 0 in these expressions and in B8) indicates that <2((jf} is the initial
value of <2(k>; that is, <2(k)@). The symbol ® reminds us that, since G is a matrix,
intermediate summation must be made in effecting the product G(t)G(t").
Thus, the explicit form for a^ is given by
C.38)
We may represent the sum
C.39)
InK
by a diagram as described in the following section.
2.3.6 Properties of Diagrams
As noted above, we may represent the matrix element C.39) by a diagram.
Due to the section rules C.30), such diagrams have one of the six forms shown
in Fig. 2.6. These diagrams have the following four basic properties:
1. Each dot or vertex of a diagram represents an interaction in which two
particles with given k values interact with each other and emerge with two
different but conserved k values.
2. Two lines must always enter and emerge from any vertex.
3. If only three lines are connected to a dot as in the vertex ^~, then one
of the "emerging" k vectors is zero.

2.3 Diagrams: Prigogine Analysis 105
n
n
,' = kn=0
tb)
n -
@
(d)
k,=
^-—/'
(e)
k;=o
(f)
k/, kn, k/', kn — 0
FIGURE 2.6. Six diagrammatic representations of the matrix element C.39).
4. Since diagrams connect the initial a^ state to the present a^f) state, we
may associate the right ket side of any given diagram with an earlier time
than the left bra side. Every vertex is a function of k/, kn, K and times.
As an example of the use of these diagrams, consider that at t = 0 the only
a(k) coefficient that is different from zero is the ax coefficient. Then in equation
C.38), for the second-order coefficient a^, the initial Fourier coefficient a^
becomes a] and its adjacent matrix element has only two possible diagrams,
shown in Fig. 2.7. Due to the matrix product form |(k")) ((k")| in C.38), these
two diagrams can only couple to the five terms shown in Fig. 2.8. We thus
°btain five possible contributions to aB)@- If we ask for af\t), then only two

106 2. Analyses of the Liouville Equation
and
o
FIGURE 2.7. Diagrams coupling aS to
a
FIGURE 2.8. Coupling diagrams of those in Figure 2.7.
diagrams contribute.
ff dt' dt"
a?\t)
C.40)
Summation over (In), (I, n) is tacitly assumed. Each diagram in C.40)
represents a second-order interaction. For example, consider the diagram
k,-K
k,+K
k/,0
SL
k,-K k; + K\ /k;-K k;
SL
k/,0
t"
Consider the example of a homogeneous gas for which /yv(l • • • N) is
independent of x^. We may write
which is seen to be consistent with C.12) [setting fu(xN, pN = V~N
The equation of motion for a0 or equivalently <z0 is given by C.33). Let us
assume that the only <z(k) coefficients that are not zero initially are the ao
coefficients. In the expressions C.37), matrix elements serve to couple the
state @| to the state |0). Some fourth-order diagrams that come into play
in this coupling are shown in Fig 2.9. These terms contribute to the a0 (t)
approximation. The meaning of the first diagram is shown in Fig. 2.10.
The meanings of the four segments of Fig. 2.10 are as follows:
D : @,0,0
SL
K + k
,0

B
A
2.3 Diagrams: Prigogine Analysis 107
a
FIGURE 2.9. Fourth-order diagrams.
\
\
\
\
\
FIGURE 2.10. Interpretation of a diagram.
K + k
2
K + k
K + k
,-K,
2
K-k
,0
2 ) ' ' V 2
<K,-K,0|«L|0,0,0),4
SL
8L
K + k
-K
K-k
h
K,-K,0
The time integration of this term is
f*t f*t\ ptj /*?3
I dt\ I dt2 / dt3 / dt4
Jo Jo Jo Jo
This term is of order s4. Note that each strip (A, B, C, D) involves the interplay
of three particles. Thus, of the four fourth-order interaction diagrams shown
in Fig. 2.9, only the term OCO represents a fourth-order interaction that
includes exclusively two-particle interactions.
2.3.7 Long-Time Diagrams: The Boltzmann Equation
If we restrict our discussion to rare gases, we expect that only two-body in-
interactions are pertinent. Thus, in the homogenous, rare-gas limit, we consider
diagrams like

108 2. Analyses of the Liouville Equation
Let us consider the second-order approximation to a0. In the stated approxi-
approximation, C.38) becomes
ao(O)
dt" J](OO|SL| -K,K)r(-K,K|«L|OO)rflo(O)
-hi
dt" J](OO|SL| -K,K)r@0|SL| - K, K)*,ao(O) C.41)
//iK
Here we have written a complex conjugate equality for matrix elements, which,
we recall, are operators. Thus, for example,
- K, K}r, =
e
@0\8LlnK\ - K, K)r = -— *_,iK • 9ln C.42)
where, since only two k vectors enter these matrix elements,
a)(k) = K • v, - K • vn = K • g,n
Substituting these values into C.41) gives
i-2 ff dt'
dt" J2 I*J2(K • O^e^^'-'^K ■ dlna0@) C.43)
2.3.8 Reduction of Time Integrals
The time integrals in the preceding expressions may be reduced in the following
manner. First note that the time integral in C.43) is given by
7@ = [ dt' [ dt"eia{t'-n C.44)
Jo Jo
The triangular domain of integration is shown in Fig. 2.11. To evaluate the
integral, we change variables to (r, T).
= t'- t"
This transformation carries the Jacobian
t' t"
T, T

2.3 Diagrams: Prigogine Analysis 109
FIGURE 2.11. Domain of integration for C.44).
T = 0, the t- axis
t = 0 the T axis
r
FIGURE 2.12. New domain of integration for C.44).
The new domain of integration is shown in Fig 2.12. There results
lit)=-
lit) = (t
1 /•' f2'-T
- / dx \
2 Jo A
f
Jo
dTe'va = / dxe'ra(t - x)
8
da
Jo
C.45)
dx
In the limit t -> oo,
7@ - 7T
= 7ttS(a)
C.46)
As volume grows large, we may set the sum over K equal to an integral over
K in accordance with C.11a). Thus C.43 becomes
K)K • 6lna0@)) C.47)
Having obtained this explicit form, we conjecture on the form of ao(t) to all
orders in e. For weak concentrations, we expect the relaxation of the system
go as exp —(t/tr), where tr ^ n~\ the inverse of the number density. So tr

110 2. Analyses of the Liouville Equation
decreases as n increases. Thus
ao(t) - e~tn = ^
Z!
This form suggests the following. In the sum C.35), keep all terms that contain
(tnI amplitudes, irrespective of order in s, but discard terms like nm(ntI. From
the expression C.47) for a^\t), we find the nt dependence (note that this is
not a dimensional statement):
tN'
In
Note in particular that the integral term cannot contribute any other V, t or N
dependence. In like manner, we find that the density time dependence of the
third-order term
also goes as Ntn. In this interaction, two particles start with momenta k/ =
kn = 0, then gain wave numbers K and — K, respectively, then through a
second interaction gain wave numbers K', —K, and through a third interaction
reassume the values k/ = kn = 0. The diagram is therefore called a two-body
diagram. On the other hand, the three-body diagram
has the density time dependence Ntn2. The fourth-order product two-body
diagram
ooo - o o
has time dependence N2(tnJ. Continuing in this way, we find that only two-
body diagrams must be retained in the low-density limit. It follows that in the
expansion C.35) of <zo@> terms giving density time dependence which goes
as (tnI, may be grouped in the following manner (with summations over K,
Z, n understood). Note also that explicit time dependence has been extracted
from diagrams.
Long-time limit
O+OO+OQO+ • •
y [ O+OO+ '' •][ 0+00+ • • ->o(O)
... C.48)

2.3 Diagrams: Prigogine Analysis 111
Let us call
s ^0+00+000+ • • • C.49)
Then the above terms may be collected and rewritten
ao(t) = etSao(O) C.50)
Differentiation gives
3flo@
dt
Keeping only the 0(£)
da0
C.51)
dt
dao(t)
-Oo C.52)
f
/
J
x S[K ■ (v, - vB)]K • f ^- - -^-) ao@ C.53)
Note that these terms have the correct dimensions. With V~x = dK, the
dimensions of the right side of C.53) may be written
2 ( \ 1 1 \2 1
To obtain an equation for the one-particle distribution, we operate on C.53)
with
d\\ ... d\r-\ d\
r+\
The contributions from the Z, n summations may be listed in a triangle as shown
in Fig. 2.13. All terms except those on the hatched lines are surface terms and
give no contribution. The remaining terms give the summations
a. 1 - _
a \ /a a
where the three dots represent between the explicit 6 operators. In the Z sum,
the lead d/d\t derivatives give surface terms, as do the lead d/d\n derivatives in
the n sum. (Only the lead derivatives in these sums correspond to divergence.)
So we are left with
o \ /a a

112 2. Analyses of the Liouville Equation
n
1,2, . . . r - 1, /;/-+ 1,
ten
Diagonal
terms
absent
N
FIGURE 2.13. Triangle for enumerating terms in C.53).
Changing the dummy variable n to I in the n sum gives
r-\
a
a
3v,
d
N
d\,
d
l=r+\
N
d
i=\
d\,
d\r 3v/
In this manner we obtain
9/l(Vr)
7T
x K
3v/
(v, - vB))
C.54)
where f\ and /2 are now normalized with respect to velocity. The presence of
/2 in the integrand stems from the fact that vr and vi have not been integrated
out. The above equation may be rewritten
)
C.55)
With molecular chaos, /2(v, vr) = /j (v)/i (vr), we obtain the Boltzmann equa-
equation. Corrections to this equation may be obtained incorporation third-order
The Boltzmann equation is discussed at length in Chapter 3.

2.3 Diagrams: Prigogine Analysis 113
Balescu - Lenard
Vlasov (bare potential) Landau (shielding)
1 Time
Fokker-Planck t = td = ■—
iUp A
FIGURE 2.14. Time intervals and equations for a plasma.
2.3.9 Application of Diagrams to Plasmas
A plasma is a collection of ions and electrons maintained in separation by
sufficiently high temperature. There are two fundamental times for a plasma:
The plasma response time, tp and the dielectric time, td.
If an extraneous electron is introduced into the plasma, the response time of
the plasma to this disturbance is co , where (in cgs)
n Anne2
2 C.56)
m
The parameter cop is called the plasma frequency. So we write
tp = — C.57)
(Dp
A shielding occurs about the foreign particle in what may be termed the
dielectric time,
= - = C.58)
A COnA
Here A is the plasma parameter:
A = —l—r C.59)
where kd is the Debye distance.11 We note the relation
WP C.60)
y m
At high temperatures and low densities, A « 1, and the plasma is said to
be weakly coupled. At low temperatures and high densities, A > 1, and the
plasma is said to be strongly coupled.
The relevant kinetic equations that come into play in these intervals are listed
Fig. 2.14, appropriate to the weakly coupled domain.
So for early times, t < tp,we look for decay behavior like
l\
'The Debye distance was encountered previously in Section 2.5.

114 2. Analyses of the Liouville Equation
<L
FIGURE 2.15. Diagrams contributing to long-range interactions.
The terms correspond to diagrams with (enntI dependence.
For problems concerning the relaxation of an inhomogeneous plasma to a
homogeneous plasma, the initial a\(t) coefficient that comes into play is a\@),
I y£ 0. Due to the long-range nature of the unshielded Coulomb potential, many
particles interact and relevant diagrams are of the form shown in Fig 2.15. Sum-
Summing such diagrams with proper e2nt dependence gives the Vlasov equation
encountered in Section 2.2.
For longer-time behavior, we look for decay like
/!
(On <*>n Mn (Anne1
p nC5 \ m J nC5
C.62)
s
Thus, dependence of relevant terms in this interval is (e4ntI. The kinetic equa-
equation for a homogeneous plasma in this time domain stems from diagrams that
couple the initial interaction jm to a final interaction %C . Predominance of
"grazing" collisions indicates that k = 0 terms dominatethe development of
the plasma. Separation of terms with common (e4ntI dependence gives the
series:
C.63)
This is called the ring approximation and leads to the Landau equation.
Equations describing more general long-time behavior stem from summa-
summation of diagrams that carry dependence like (e2n)p(e2t)q. Summation of these
terms leads to the Balescu-Lenard equation. This equation includes a shielding
kernel in its interaction integral.12
12For more extensive discussion of the diagrammatic technique and its applica-
applications, see I. Prigogine, Non Equilibrium Statistical Mechanics, Wiley-Interscience,
New York A963); and R. Balescu, Statistical Mechanics of Charged Particles,
Wiley-Interscience, New York A963).

2.4 Bogoliubov Hypothesis 115
All these kinetic equations relevant to a plasma are discussed more fully
in Chapter 4. The Landau equation is derived in Section 4.2.4. The Balescu-
Lenard equation is derived in Section 4.2.5. The plasma parameter and the
notions of strongly versus weakly coupled plasmas are more fully discussed
in Chapter 4. Recall that the concepts of weakly and strongly coupled fluids
were discussed earlier in the present chapter (Section 2.1).
2.4 Bogoliubov Hypothesis
2.4. 1 Time and Length Intervals
In this section we turn to the Bogoliubov hypothesis for the equilibration of a
nonequilibrium gas.1314 The hypothesis addresses an enclosed gas and defines
three time intervals, which we label ru r2, and r^. In the interval T\, two
molecules are in each others interaction domain. The interval r2 is the mean-
free-collision times, which is the mean times between collisions. The time T3 is
the average time taken for a molecule to traverse the container in which the gas
is confined. For a mole sample of a gas confined to a macroscopic container,
we may write
r2 <£ r3
We may relate three respective displacements to these three time intervals,
which we label X{, X2, and ^3. The displacement X\ is the range of interaction,15
X2 is the mean free path,16 and ^3 is the edge length of the confining container.
Typical values of these parameters are listed in Table 2.1.
2.4.2 The Three Temporal Stages
The Bogoliubov hypothesis addresses the functional dependence of the N-
particle distribution function, /w(l, • • •, N) relevant to three stages in the
temporal development of the gas toward equilibrium. These stages are defined
in Table 2.2.
The hypothesis relevant to the initial stage in well described by the following
example. Consider that initially one of the TV molecules in the gas is moving
1 o
N. N. Bogoliubov in Studies in Statistical Mechanics, vol. 1, J. de Boer and G. E.
Uhlenbeck, (eds.), Wiley, New York A962), See also E. D. G. Cohen in Fundamental
Problems in Statistical Mechanics, E. D. C. Cohen, (ed.), Wiley, New York A962).
An extension of the Bogoliubov hypothesis to dense fluids was made by R. L.
Uboff, Phys. Rev. A 31, 1883 A985); A32, 1909 A985).
15The range of interaction was introduced previously in Section 2.1, where it was
labeled r0.
16The mean free path is discussed at length in the description of transport
coefficients in Chapter 3, where it is labeled /.

116 2. Analyses of the Liouville Equation
TABLE 2.1. Bogoliubov length and time intervals for a gas with mean molecular
speed of 300 m/s at standard conditions (container edge length is X3 — 3 cm)
cm
sec
3
x 10~8
Ti
102
3
X2
x 10
lO"9
3
r3
10
TABLE 2.2. Epochs in the Bogoliubov Hypothesis
0 < t < r,
T] < t < T2
T2 < t
Initial stage
Kinetic stage
Hydrodynamic stage
in free flight (between collisions) at a speed greatly in excess of the mean
speed of the remaining molecules. The TV-body distribution function for this
nonequilibrium configuration is given by A.6.7).17 In the initial interval there
is no collisional exchange between molecules, and the initial nonequilibrium
state experiences no equilibrating force. Thus it is hypothesized that in the
initial stage no less than the full TV-body distribution function is required to
describe the state of the gas.
In the kinetic stage, molecules experience collisions and there is a ten-
tendency toward equilibration. It is hypothesized that in this interval all s-particle
distributions are functional of /i(l). That is
fs = /,(!,..., s\fi)
D.1)
with explicit time dependence of fs contained entirely in f\. An example of
the functional dependence of D.1) is given by the case where molecules are
statistically independent. For this case, we write
D.2)
In the hydrodynamic stage, it is hypothesized that all distributions depend
only on the hydrodynamic variables n, u, and 7\ where n is number density,
u is macroscopic fluid velocity, and T is temperature.18
Thus we have the following description. As the system passes from the initial
nonequilibrium state to the finial equilibrium state, there is a diminishment in
17Note, however, that in A.6.7) remaining molecules in the gas are at rest. A more
correct distribution would include finite but small speeds of remaining molecules.
18Fluid dynamic variables are defined and more fully discussed in Chapter 3.

2.4 Bogoliubov Hypothesis 117
the level of description appropriate to the state of the fluid. In the initial stage,
the full TV-body distribution is required to describe the state. In equilibrium,
the far less informative macroscopic variables n, u, and T suffice.
2A3 Bogoliubov Distributions
Again we turn to the Fs distribution defined by B.6):
Fs = Vs fs D.3)
Note that, for a homogeneous fluid, Fs is dependent only on momentum.
Rewriting BYS A.20) in terms of these distributions gives
n -s * r ^
LS\FS- —— J2 J 0/.5+i Fs+i d(s+ l) = 0 D.4)
3' 7 V ,-=,
where 0/y is given by [see A.7)]
d d
Ou = -Gy •( — - — ) D.5)
and we have recalled the steps A.15) and A.16).
In the thermodynamic limit,
lim = lim — = - D.6)
^~\ V ) \VJ v
where v, the specific volume (mean volume occupied per particle) is a finite
parameter. In this limit, D.4) becomes
Fs+id(s + l) = 0 D.7)
-I S /»
vf-(J
2.4.4 Density Expansion
In the kinetic stage, D.1) is relevant, and we write, equivalently,
Fs = Fs(l,...,s;F1) D.8)
Since the explicit time dependence of Fs is contained entirely in F\, we write
D.9)
The bracket notation denotes an inner product over all F\ functions. As a simple
example, again consider the case of statistical independence of molecules. In
this event we write
(=1

118 2. Analyses of the Liouville Equation
Substituting into D.9), we obtain
dF1_
dt
dt
D.10b)
As noted previously, the Bogoliubov analysis is relevant to a rare gas. With
this motivation, we introduce the expansion
F (\ s' FA — F° + -
°
H FB) +
B)
D.11)
Substituting D.11) into BYj [obtained from D.7)], gives
dFx
~dt
Pi dFY , 1
12
1
V
12
1
V
dlb
D.12)
12
which serves to identify the AE) coefficients. Substituting the preceding two
expansions into the inner product D.9), gives
dt \SFi
v
D.13)
This expansion may be written in the alternative operational form
1
dt
V
D.14)
where the DE) operators are as implied.
Now we return to D.7) and, inserting the expansion D.11), obtain
dt
v
@)
1
v
D.15)
with this equation at hand, we note that, together with D.14) [which stems
from D.19) and BYj], it comprises two equations for dFs/dt. Setting
D.16)
■ D.14)= —D.15)
dt dt }
and equating terms of equal power in l/v gives the sequence
d(s
D.17a)
D.17b)

n
2.4 Bogoliubov Hypothesis 119
-17c)
Equations D.17) are self-contained functional equations that, in principle,
determine the sequence {F^n)} in the expansion D.11). A key element of this
analysis, as will be found below, concerns the operator Ls. We recall that this
Operator, as given by A.11), pertains to a collection of s particles that interact
with each other but are otherwise independent of the remaining aggregate.
We will not apply the sequence D.17) to find a closed kinetic equation for
F\ in the low-density limit valid to order v~l.
2.4.5 Construction of F^
Our program for obtaining the said kinetic equation is as follows. To O(^), the
relation D.12) becomes
EL dJ± I /rf26ff D.18)
at m ax\ v
Thus, if we obtain F2@)(Fi), a closed kinetic equation for F\ is established. To
these ends, we recall D.17a) and with s = 2 write
£>@)F2@) = L2F?] D.19)
We must solve this equation for Ff\F\) subject to an appropriate boundary
condition. We take this to be the condition that, when carried sufficiently far
back in time, particles in a given aggregate are uncorrelated.
The Liouville operator, LN, was introduced in C.1). For an isolated system
of s we may write
3 F
= [Hs, Fs] s LSFS C.2a)
ot
Whose solution is given by
Fs(t) - etL'FM = AA]FM D.20)
We have found previously in Section 1.5.3 that exptLs propagates particle
phase variables back in time19 through the interval t to values determined by
Hamiltonian for the s-particle system. So we label the operator A(_!,}. Here
some properties of this operator:
Ao = 1 D.21a)
19With reference to Section 1.5.3, we note that As = iLs. Recall also that L,
operates on the phase variables of Fs.

120 2. Analyses of the Liouville Equation
A_,,A_f2 = A_(f|+,2) D.21b)
- = LA_, = k_tL D.21c)
Let us employ the A, operator to express the boundary condition on Fs
described above:
lim A^Fsil, ...,s;Fi)= lim &1) ][Fi(jfc) D.22)
Since this boundary condition holds for all F\(k), it also holds for F[ =
limr^oo ArA)Fi(/:). Substituting this value into D.22) gives
s
lim A^F, f 1,..., s; A^Fi) = lim A(l] ]~f A.A)F,(/t) D.23)
-►00 V / »->00 f 4-
k=\
As ArA) relates to a one-particle system, it operates on free particle phase
variables. Thus
RHS D.23) - lim A(_!,} ^ ^ ' ■ F
= lim
D-24)
*=i m
m
where x^} and pj^} are phase values of the &th particle traced back in time to
t = —oo, under the interaction of s particles from respective starting values
*k + (Vk/m)t andp*.
With these expressions at hand, we return to D.19) and with the identification
of D.13), 4.14 write
A,...,,; Fl) = (^, A^F,)) = (^, (L!^)) D.25)
\ SF I \ SF V //
A tractable representation of the functional derivative in the preceding equation
is obtained as follows. Again we note that since D.25) is true for all F\ it is
a. /1 \
also true for A; F\. This permits us to write
[l,...,,; A<J>F,1 = (-^-y [ti^F,]) D.26)
L J \«[a(!{fJ l j/

2.4 Bogoliubov Hypothesis 121
With the property D.21c), the preceding may be rewritten
/5/r@)
tyhich gives the desired representation
s
-r *i
Substituting this relation into D.17a) gives
/r(°) i c- A( }Fi
With D.20) it is evident that the solution to D.29) is given by
@)
F.
Equivalently, we may write
-tr\
—i s
F
7 * * * 9 7
As this solution is valid for all t intervals, the limiting form
Fs@) [1, ..., j;
lim A_! F
@}
l, ..., s\ A{t
{l)
D.27)
D.28)
D.29)
D.30)
D.31)
is likewise a solution. Incorporating our previously stated boundary-condition
relations D.23) and D.24) gives the final desired solution:
k=\
E) E)
D.32)
Thus, the lowest-order solution F5@) in the Bogoliubov analysis is given by a
product of one-particle distributions with phase variables (xE), pE)) displaced
back in time and coupled through the subspace ^-particle Hamiltonian.
2.4.6 Derivation of the Boltzmann Equation
The final step in our program for obtaining a closed kinetic equation is to obtain
a consistent representation for the L2 operator in D.19).
To these ends, with A.11) we write
D.33)
where, we recall,
Pi 3 . P2 9
— . _ 1 . —
m ox\ m 9x2

122 2. Analyses of the Liouville Equation
and 0i2 is given by D.5). With D.33) we write
f F{0)
L2r2
D.34)
We now find separate expressions for the two terms on the right of D.34).
With these expressions at hand, we return to D.18) to obtain the desired kinetic
@)
equation. We start with L2F2 .
Returning to D.19) and employing D.13) gives
@)
SFY '
D.35)
where with D.12) we write
A@) - -
, p)
m 3x
D.36)
'@)
Employing the solution for F2 given by D.32) in the inner product in D.35)
gives
F
m
B) DB)
2
3
m
B)
n
' Pi
dx\
d
B)
B)
3x
B)Fl
n
l ' Pi
B) DB)
D.37)
We wish now to find an alternative expression for the operator £2. Introducing
the transformation of variables (xi, x2, pi, p2) —> (r, Xi, g, p0, where
r = x2 - X!
P2-P1
m
D.38)
permits k2 to be written
3 Pi
1
3
3r
m
= K B + K
D.39)
The subscript B on the operator KB — g • 3/3r denotes Boltzmann.
Inserting this expression into D.34) and substituting the resulting form into
D.18) gives
_L + El. ^I± - I /
dt m dx\ v J
<;0)
F2<0) dl D.40)
0)
If F20) is homogeneous (that is, independent of Xj and x2) over the interaction
domain, the second integral on the right side of D.40) vanishes and the collision
integral reduces to
jd2KBFf =
(p?) F,
D.41)

2.4 Bogoliubov Hypothesis 123
g before
Before collision:
z =
<0
After collision:
g after
FIGURE 2.16. Asymptotic z values in the Bogoliubov derivation.
where g is the relative velocity defined in D.38). The spatial integral in D.41)
fiaay be evaluated as follows. We work in cylindrical coordinates with the
cylinder axis parallel to g. For the projection of r onto this axis, we have
z =
g
8
D.42a)
The cylinder radius b and azimuthal angle 0 complete the description, and the
spatial volume in D.41) becomes
dx2 = dr = b d(j) db dz
D.42b)
g
9r
8
Integrating the right side of D.41) over z gives
/»OO
J — oo
dz
dz
D-43)
The value z — +oo corresponds to the domain after interaction (see Fig 2.16).
Kpj and p2 are traced back to the interval prior to collision, they assume values
fliat give Pj and p2 after collision. Let us label these values of momenta p\
and p2. The value z = — oo corresponds to the domain prior to collision. What
Values do pj and p2 attain if traced back to the infinite past from this domain?
As particles are assumed not to be in each other's interaction domain prior to
collision they move as free particles and the answer is pi, p2. Thus we write
(p22))
D.44)
Substituting these values into D.41) and then into D.39) (with the L2 - K'2
^tl neglected and Jp2 = m dg) gives the standard form of the Boltzmann

t/*/
124 2. Analyses of the Liouville Equation
equation:
dt m dx\
D.45)
This equation is in a more canonical form than the previously obtained Fourier-
structured equation C.55). Again it is noted that a more direct derivation of the
Boltzmann equation together with a full discussion of collision terms is given
in Chapter 3.
As with the Prigogine technique, the Bogoliubov analysis also provides a
recipe for generating corrections to the Boltzmann equation to higher density
configurations.20
Bogoliubov's hypothesis relevant to the hydrodynamic stage finds strong
corroboration in discussion of the Boltzmann W-theorem in Section 2.5. Here
it is found that the one-particle distribution of a fluid in local equilibrium is
a functional of n, u, and T, as stipulated by the Bogoliubov hypothesis. This
important distribution is called the local Maxwellian and is discussed in detail
in Section 3.3.10.
2.5 Klimontovich Picture
In this section we turn to an alternative description of kinetic theory formulated
by Klimontovich.21 In this formalism, phase densities are introduced, moments
of which are found to be related to multiparticle distribution functions intro-
introduced earlier. Moments of deviations from the mean of these phase densities
are similarly found to be related to previously encountered correlation func-
functions. Applying these algorithms to an equation of motion for the first moment
of the phase densities is found the first equation of the hierarchy (BYi).
2.5.1 Phase Densities
The Klimontovich phase density for a system comprised of TV particles is given
by the following delta-function expression:
n
E.1)
where
E.2a)
20See, for example, E. D. G. Cohen, Acta Physica Austriaca, Suppl. X, 157 A973);
J. R. Dorfman, Physica 106A , 77 A981).
21Yu L. Klimontovich, Statistical Theory of Non-Equilibrium Processes in a
Plasma, MIT Press, Cambridge, Mass. A967).

2.5 Klimontovich Picture 125
^presents the phase point of the /th particle and
5(z - z,) = 8(x - x,)S(p - pd E.2b)
The phase variable z = (x, p) is a point in six-dimensional (for point particles)
xt-space.
The normalization of A/*(z, {z,}) (dropping the {z,} notation) is given by
N r
~ z<)dz = N
r N r
/ Af(z)dz = J2
Integrating over an infinitesimally small domain, Az, of/x-space and recalling
the mean-value theorem gives
A/"(z)Az - AN E.4)
where AN is the number of particles in the infinitesimal phase volume Az.
2.5.2 Phase Density Averages
The average of Af(z) is given by the rule A.6.8), and we write22
/
dzNfN(zu ...,zyv)^5(z-z/) E.5)
= / dz2dz3 • • • fN(z, z2, ...) + / dzx dz3" fN(zuz, z3, ...) H
Collecting terms gives
~~ E.6)
To obtain a parallel relation for the second moment,23 A/r(z)A/r(z/), we first
effect a division of double-sum delta functions as follows:
; - z/M(z/ - zj) + &ij 2^ 5(z - z/M(z/ - z) E.7)
7 i
Again, with A.6.8) we write
dzNfN(zx
X
- z/M(z/ -
22Time dependence of distributions is tacitly assumed.
23That is, moments with respect to the TV-particle distribution.

126 2. Analyses of the Liouville Equation
= N(N - 1) / dzt dzjfite, zj)8(z - z,)S(z' - zj)
+ N8(z - z') / dZiMzi)8(z - z,) E.8)
In the last equality, z, and zy represent two arbitrary terms in the summations.
Integrating gives the desired result:
') - N(N - l)/2(z, z!) + NS(z - z')/i(z) E.9)
Continuing in this manner, for the third moment of A/*(z) we find
J\f(z)J\f(z')J\f(z")
= 3l(^\f3(z, z, z!') + 2!^W - z/)/2(z, z") E.10)
8{z-z")fr(z, z') + 5(z/-z//)/2(z, z!')]+N8(z - z!)8(z! -
Equations E.6), E.9), and E.10) may be concisely written in terms of the
s-tuple distributions A.6.14). There results
E.6a)
0 = F2(z, z') + 5(z - z')Fi(z) E.9a)
NzAf{z')M{zrr) - F2(z, z, z") + 5(z - zr)F2(z, z") + 5(z - z/r)F2(z, z)
8(z' - z")F2(z, z") + &(z - zrM(zr - z/r)F!(z) E.10a)
Continuing in this way, we obtain a mapping between the moments of N(z)
and the ^-particle distribution functions.
2.5.3 Relation to Correlation Functions
Let us introduce the deviation of the phase density N(z) from its mean, which
we label
= M{z) - N{z)
With SAT(z) = 0, we obtain
M{z)M{zf) = N{z)N{z') + 8J\f(z)8J\f(z') E.11)
In the limit N » 1, E.6) and E.9) give
M{z)M{z') - N2f2(z, z!) + N8(z - z')Mz) E.12)
Recalling the first equation in the sequence B.19), which defines the corre-
correlation function C2(l, 2) (written in terms of the z, z! variables), and comparing
with E.11) and E.12), we obtain
8Af(z)8Af(z') = N2C2(z, z') + N8(z - zr)/i(z) E.13)

2.6 Grad's Analysis 127
li similar manner, equations may be obtained connecting the higher-order
orrelation functions with higher-order mean products of deviation from the
pean of the phase functions.
&5.4 Equation of Motion
finally, we turn to an equation of motion for the phase density E.1). With
arguments leading to the Liouville equation A.4.7) taken to be appropriate to
in /x-space, we obtain
dAf(z) dMiz) . dMiz)
h v • hp • = 0
dt dx F 3p
E.14)
A/V)G(x, x') dz!
he two-particle force G(x, x') is assumed to be of the symmetric form
G(x,x') = G(|x-x'|) E.14a)
Combining the latter two equations, we obtain
+ v ^ + f ■ f G(x, OMMVXU = 0 E.15)
dt dx dp J
taking the average of this equation gives
d
I
+ v — + — / G(x, x!)N(z)N(zf) dii - 0 E.16)
dt dx dp
With E.6) and E.9), E.16) gives (in the limit N » 1)
9
/ G(x, x)/2(z, z)dz =0 E.17)
J
+v +
dt dx dp
Here we have dropped the delta-function term in the integral owing to assumed
ymmetry E.14a) of the interaction force (see Problem 2.8. Comparing E.17)
the first of the hierarchy A.24), B Yi, reveals them to be the same.
Grad's Analysis24
%6.1 Liouville Equation Revisited
Ib this point we have encountered two derivations of the Boltzmann equation
Stemming from the BBKGY hierarchy. In the present section, Grad's derivation
. Grad, Hand. d. Physik, vol. 12, Springer, Berlin A958).

128 2. Analyses of the Liouville Equation
FIGURE 2.17. Configuration domain in Grad's derivation of the Boltzmann equation:
(a) domain D; (b) domain D2\ (c) domain Dr.
is described. The starting point is the Liouville equation written in the form
n
dt
-
m
N
a/-
N
0
F.1)
The following formulas are needed in the derivation.
d
|y-x|>A
d
L
A(x,y)dy=
L
i
A(x, y) dy
|y-x|>A
r-x|>A ^x
A dS
|y-x|=A
Ady- (ft) A dS
F.2a)
F.2b)
ly-x|=A
2.6.2 Truncated Distributions
The derivation is further based on the introduction of the truncated distribution
/jA(zi), where i\ = (xj, Vi). This function is the probability of finding no
molecules within a distance A of particle 1, with particle 1 in the state z. To
obtain the equation f}A satisfies, we integrate the Liouville equation over the
domain D. This domain contains all states of particles B, ..., N) in which no
particle is closer than A to particle 1 (see Fig. 2.17). There results
d2...dN <
N
dt
N
m
= 0 F.3)
/.
= / fNd2...dN
Jd
D = {|xi - x2| > A; |xi - x3| > A, ..., |xi -
If Dr denotes the domain |xj — xr| > A, then we may write
F.3a)
> A}
D =
x
x • • x D
N
F.4)

2.6 Grad's Analysis 129
from these equations we obtain
n
> / i o 10 i
F.5)
let 5, be the sphere |x, — Xi | = A. Now consider the first term in the sum
N d
F.6)
1 = 1 °Xi
integral gives
F.7)
!ere we have employed F.2b), identifying the integral on the left of F.7 with
first term on the right of F.2b). The remaining terms in the sum F.6 may
similarly integrated to obtain [with F.2a)]
N
1=2 S
In the force term in F.5), all save the 3/3vi term reduce to surface integrals
%at vanish. The relevant factor in the remaining term is integrated to obtain
f N f
/ Gi/tfd2... dN = V" / Gu(x\,Xi)fNd2...dN
JD i=2 Jd
f G12(Xl,x2)/2(l,2)J2 F.9)
e domain D2 contains all states of particles 1 and 2 in which they are not
ser than the distance A (that is, |x2 — Xi | > A). The two-particle truncated
stribution /2A is defined through the integral
A2)= f fN(l,...,N)d3...dN
JD'
D' = D3 x D4x • • • x DN F.10)
le domain D' contains all states of molecules C, ..., N) in which no particle
closer than A to particle 1.
6.3 Grad's First and Second Equations
-ombining F.5) and F.9) gives Grad's first equation:
s2

130 2. Analyses of the Liouville Equation
g dS<o
(before collison)
[s ds c/cp ]
Every such element
of area has a
projection on
S2 and S2
g • dS>0
(after collison)
FIGURE 2.18. The S2 sphere in Grad's derivation of the Boltzmann equation.
N-l d
-7
l Jd2
Gl2f2Ad2
0 F.11)
In obtaining this form, the surface integrals on the right side of F.7) and
F.8) were recognized to be TV — 1 identical terms. This identity is made for
much the same reasons as were employed in obtaining the N—s identical terms
in BBKGY sequence A.20). The domain D2 in F.11) includes |x2 —Xj | > A,
while the surface integral in this same equation is over the surface |x2 — X\ =
A. Suppose that G12 = 0 for |x2 — Xj| > A (such as for rigid spheres of
diameter A). We then obtain Grad's second equation (see Problem 3.51).
3x
-(N - 1)
l§
f2AdS-(\l-v2)dv2
F.12)
s2
The left side is the total time derivative of the function f*. The surface integral
indicates that the rate at which a set of particles in a given phase element
changes is due only to the loss and gain of a pair of particles as they enter and
leave each other's sphere of influence. Here we are interpreting A to be the
range of interaction, previously labeled r0 in Section 2.2.
2.6.4 The Boltzmann Equation
Grad's second equation is a precursor to the Boltzmann equation. To establish
this property, we consider the sphere S2 shown in Fig. 2.18. The points on £>
are mapped onto the meridian disk by simple projection. Points on the disk
have the radial coordinate (impact parameter) s and azimuth cp. To make the
mapping one-to-one, the two hemispheres are distinguished according to
5+ : g • dS > 0
S2 : g • dS < 0
(after collision)
(before collision)

2.6 Grad's Analysis 131
relative velocity g is given by
F.13)
|or fixed g,
S2 —
l
file point on S^~ that projects onto 0?, <p) is called xj (s, <p), and the point on S2
%at projects onto this point is called x^(s, (p). There results (see Problem 2.16)
— v2) • dS = —
— v2) • dS = +gs as dcp
on
on
F.14)
denotes (v2, xj) and 2~ denotes (v2, x^), then, in the previously described
rdinates, F.12) appears as
F.15)
- /2AA,2 )]gsdsd(pd\2
At this point the following constraints are imposed:
1. Replace the arguments in /2AA, 2+t) by the values they must be at the
start of a binary collision (say 1, 2, F) in order that they have the values
A,2+) at the timer.
2. Set
= (W- 1)
which is the assumption of molecular chaos.
In the four functions /A(l, t\ 7^B", 0, /A(l, 0,and /AB, F),replace
x2, xl5 x^ by Xi and F by t.
lserting these changes into F.15) and neglecting one compared to N gives
Boltzmann equation (recall F\ — Nf\)\
= /
J
F.16)
s completes the Grad derivation of the Boltzmann equation. As noted
fpeviously, the Boltzmann equation is discussed in greater detail in Chapter 3.
Note in particular that as F.12) is a precursor to the Boltzmann equation, the
glected force integral in Grad's first equation F.11) is a measure of the error
sociated with the Boltzmann equation. However, we may easily envision
perturbative technique of solution to Grad's first equation F.11) in which
ad's second equation F.12) [and resulting Boltzmann equation F.16I is
e lead term. Thus, all three techniques as given by Prigogine, Bogoliubov,

132 2. Analyses of the Liouville Equation
and Grad offer a means of obtaining higher-order corrections to the Boltzmann
equation appropriate to denser fluids.
Problems
2.1. Establish the following antinormalization properties for the correlation
functions Cs given by B.21).25
/ C5(l, ..., s)d\ = / C5(l, ..., s)d2 = ■ ■ - = / C5(l,..., s)ds = 0
2.2. (a) Write down BY3. Define terms.
(b) Integrate BY3 to obtain BY2.
2.3. Write down the explicit form of the Liouville equation for a two-component
gas that contains N/2 molecules of mass m and N/2 molecules of mass M.
The interparticle conservative forces are labeled Gm, GM, and G7 relevant to
m — m, M — M and m — M interactions, respectively.
2.4. A dumbbell molecule comprised of atoms with respective masses m\ and m2
that is in a uniform gravity field has the Lagrangian
u k M
B62 + a262 sin2 0) + £2 + (X2 + Y2 + Z2
u k M
L = -(a262 + a262 sin2 0) + -£2 + —(X2 + Y2 + Z2) - MgZ
2 2 2
The center of mass of the molecule is X, Y, Z, and £ is the displacement from
equilibrium separation of the two atoms.
(a) Obtain expressions for the canonical momenta p9, p<p, p^, px, Py, and
pz in terms of generalized velocities.
(b) What is the Hamiltonian of the molecule?
(c) A gas of N noninteracting identical molecules of this type occupy a
cubical volume V = L3. What is the one-particle Liouville equation for
/i, the one-particle probability density of the gas?
(d) Let the molecules interact through the potential V (/?,-_,-), where RtJ is the
intermolecular center-of-mass separation. What is the explicit form of the
Liouville equation for the gas? Leave triple sums intact.
2.5. Consider a system that contains N particles in equilibrium whose Hamiltonian
is given by
Kinetic and potential energy terms have been written as K and <I>, respectively-
Show that if the distribution function is of the product form
25For further discussion, see R. L. Liboff, Phys. Fluids 8, 1236 A965); 9, 419
A966). See also Problem 2.23.

Problems 133
FIGURE 2.19. Prigogine diagram from Problem 2.6.
where
N
i = \
then the Liouville equation implies the canonical distribution
\pN) = Aexp[/3H(xN
fc(x\pN) = Aexp[-/3H(xN ,p
where A and f$ are constants.26
2.6. Consider the diagram in the Prigogine analysis of the Liouville equation shown
in Fig. 2.19.
(a) What is the order of this diagram?
(b) How many particles come into play in this diagram?
(c) Calling the perturbation Liouville operator 8L, write down the product
matrix element (in Dirac notation) that this diagram represents.
(d) What is the integration in time that accompanies this diagram (list only
differentials and limits of integration)?
2.7. Consider a system whose ensemble fluid occupies a simply connected domain
R of F-space. A domain R is simply connected if any two points in R may
be connected by a path contained entirely in R. Show that in the motion of
the system R remains simply connected. That is, R does not separate into
disconnected segments.
2.8. In the Klimontovich formulation, show that BY] as given by E.17) follows
from E.16). Justify all steps.
2.9. A space is said to be a metric space if for any two points (x, y) in the space a
"distance" function d(x, y) exists that maps pairs of points in the space onto
numbers and has the following properties:
A) d(x, y) > 0; d(x, x) = 0; d(x, y) > 0 if x ^ y (positivity)
B) d(x, y) = d(y, x) (symmetry)
C) d(x, z) < d(x, y) + d(y, z) (triangle inequality)
Introduce a distance function for 27V-dimensional F-space. Show that your
choice satisfies the preceding properties, thereby establishing that F-space is
a metric space. {Hint: Recall that F-space is a Euclidean space.)
26This theorem was previously discussed by R. L. Liboff and D. M. Heffernan,
Phys. Letts. 79A, 29 A980).

134 2. Analyses of the Liouville Equation
2.10. (a) What is the Bogoliubov boundary condition on the state of particles at
t = —oo? In what sense is this condition included in the Boltzmann
equation?
(b) Establish the properties D.21) relevant to the time-displacement operator
2.11. What is the structure factor, S(k), for an ideal gas?
2.12. Show that the equation of state of a fluid comprised of rigid sphere of radius
a is given by
P 2nna3
= 1 + —z—g(a+)
nkBT 3
where g(a+) is the value of g(r) at r = a + £, £ —> 0. (Hint: Recall that the
derivative of a step function is a delta function.)
2.13. For low-density fluids, the following form is found to be a reasonably good
approximation for the radial distribution function.
Consider that the interaction potential <I>(r) has a rigid repulsive core and
a long-range "attractive tail." What are the corresponding values g@) and
2.14. Show that the total correlation function h(r) has at least one zero on the open
r interval @+, oo).
Answer
It follows from Problem 2.1 that
C2(\,2)d\d2 = 0
Inserting the defining relation B.48) into this equation gives
/
Introducing the transformation
r = x2 - x,
2R = x2 + X]
(which carries a unit Jacobian), we find
dx\ dx2 = drx dR
and the preceding integral gives
.2
r
I 4nrzdrh(r) =
Jo
Here we have assumed a sufficiently large volume permitting the upper limit
of the integral to be set equal to infinity. As h@) = —1, the above integral
implies that h(r) is zero at least once on the said open r interval.

Problems 135
2.15. What is the radial distribution function g(r) corresponding to the structure
factor
S(k) = e~ak
where a is a constant length.
2.16. Establish the validity of F.14).
Answer
The relation
g . dS = gA2 d cos 0 d(j) cos 6
together with
s2 = A2 sin2 0
gives the desired result. Note that the angle 0 is inscribed between A and the
polar axis in Fig. 2.18.
2.17. A Lie derivative is defined as the form
d
C = A(x) • —
dx
where A(x) is a vector function of x. Show that Lie derivatives have the
property
where
—^— = A[X(x, 0]
ot
X(x, 0) = x
2.18. (a) Show that the s-particle (s < N) distribution function satisfies the
equation
^ + V5.u5/5 = 0
where V5 represents the gradient in s-dimensional F-space. In your
derivation of the given kinetic equation, what does u5 represent?
(b) What is the physical interpretation of the given kinetic equation?
(c) Is the ensemble fluid for this subsystem incompressible? That is, does
V5 • u5 = 0? If not, what is an appropriate expression for this term?
2.19. In the Bogoliubov analysis, we encounter the functional inner product D.9).
What is the explicit representation of the form
18F\(\)F\B) p 3 \
( » — • — ^i (x, p))
\ SFi m dx I
2.20. (a) Is symmetry of the TV-body distribution function in its arguments assumed
in the derivation of the Liouville equation? If so, where?

136 2. Analyses of the Liouville Equation
(b) Is this property assumed in the derivation of the BBKGY sequence? If
so, where?
Answer
(a) This assumption was not made. Recall that it was found that any arbitrary*
function of the constants of motion of the system is a solution to the
Liouville equation.
(b) This assumption was made. Thus, for example, if /3A, 2, 3) is a symmet-
symmetric distribution, then / /3A, 2, 3)dl d3 and / /3A, 2, 3)dl dl give the
same functional dependence on the remaining variable.
2.21. In BY, A.20), discuss the difference between the force terms in the sum on
Otj and the force terms in the integral.
2.22. Obtain BY2 within the Klimontovich formalism. Hint: Write the equation
of motion for M(z)M(z') in twelve-dimensional (z, z') phase space, take the
ensemble average, assume that (z ^ z') and introduce appropriate forms for
terms containing p and p'.
2.23. The correlation normalizations of Problem 2.1 as well as those implied by
Problem 2.14 and equation B.47) are valid for systems with vanishingly small
fluctuations. More generally,27 for the radial distribution function, one writes
[g(r)-\]dr+-=kBTKT
n
where
is isothermal compressibility.
(a) What relation for kt is implied by the preceding integral equation for an
ideal gas?
(b) In what limit is the normalization on the radial distribution implied by
Problem 2.14 valid?
Answers
(a) For an ideal gas, g(r) = 1 and we obtain, nkBTicT = 1.
(b) For a dense medium in which number density, n, is very large for which
KT = 0.
27D. L. Goodstein, State of Matter, Prentice Hall, Englewood Cliffs, NJ A975);
F. Mohling, Statistical Mechanics, Wiley, New York A982).

CHAPTER 3
The Boltzmann Equation, Fluid
Dynamics, and Irreversibility
Introduction
This chapter begins with a review of scattering concepts important to deriva-
derivation of the Boltzmann equation. Collisional invariants are then introduced that
Hid in obtaining fluid dynamic conservation equations and the H theorem.
This theorem in turn leads to the well-known Maxwellian distribution. The
central-limit theorem encountered previously in Chapter 1 is employed in the
evaluation of the distribution of center-of-mass velocities of a finite stationary
jgas of molecules. A derivation of Poincare's recurrence theorem stemming
from the Liouville equation is given and is contrasted with the irreversibility
of the H theorem of the Boltzmann equation.
Transport coefficients are introduced in the context of response to gradient
perturbations of fluid variables. Maxwell's elementary mean-free-path esti-
estimate of transport coefficients is presented, as well as a brief account of their
relation to autocorrelation functions. These notions serve as an introduction to
the Chapman-Enskog expansion of the Boltzmann equation. Values obtained
for transport coefficients through this method are shown to be in good agree-
agreement with experiment. The theory of the linear Boltzmann collision operator
is discussed, and spectral properties of hard and soft potentials are noted. The
chapter next turns to application of the Boltzmann equation to an electron gas
immersed in an ionic fluid in the presence of a dc electric field. The result-
resulting Druyvesteyn distribution is then employed in the derivation of an integral
expression for electrical conductivity.
In the last section of the chapter the notion of irreversibility, previously
discussed in reference to the H theorem, is reintroduced, and more recent
contributions to this subject are noted. Ergodic and mixing flows are described
and Birkhoff's contribution to the theory is reviewed. Action-angle variables

138 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
are introduced together with Bruns's theorem on a limited number of constants
of motion and the closely related concept of an integrable system. Numerical
results relevant to a two-dimensional anharmonic oscillator lead to the concepts
of resonant domains, chaotic motion, and the closely allied, often quoted, KAM
theorem.
3.1 Scattering Concepts
3.1.1 Separation of the Hamiltonian
Basic notions of scattering theory are prerequisite to derivation of the Boltz-
Boltzmann equation. The theory of two-particle collisions is best formulated in the
center-of-mass frame.
First we note that the Hamiltonian of two particles interacting under a central
potential V(r), in the "lab frame," is written
2
+ -^ + V(r) A.1)
2m i 2/7i2
Particle masses are m\ and ra2, respectively, and r is the relative displacement
r = |r2 — r2|
Transformation to the center of mass is effected through the mapping
raip2-ra2pi
P
r = r2 - ri
A.2)
= Pi + P2
mlrl+m2r2
MX. —
m2
Substituting A.2) into A.1) gives
//(R, P, r, p) - £- + f- + V(r) A.3)
2M 2
where [as in A.12)]
P = MR = (wi +ra2)R
is the momentum of the center mass,
m\m2
;
m\ + m2
is the reduced mass, and
//cm = £ A.4)

3.1 Scattering Concepts 139
is the kinetic energy of the center of mass. The remaining part of the
Hamiltonian,
P2
l f
represents the Hamiltonian of a particle of mass /x in the potential field V(r).
JJK\ is the Hamiltonian of the two-particle system relative to a frame moving
with the center of mass. With this notation, A.3) becomes
A-5)
Since H is cyclic in R, we may conclude that P is conserved, whence
HCm — constant
This constant may be absorbed in V(r) without affecting resulting equations
of motion. Thus we may write
P2
% A.6)
2/z
So we find that the original two-body problem is reduced to a one-body
problem of a particle of mass /x moving in the field of a central potential, V(r).
Since the force on this particle is radial, angular momentum is conserved, and
we may conclude that the particle moves in a plane of constant orientation.
See Fig.3.1. Thus the /x particle has two degrees of freedom, which we label
r and 0. The angle 0 is the inclination of r to an arbitrary axis in the plane. In
terms of these coordinates, the Hamiltonian becomes
P2 pl
H = ^ + -^- + V(r) A.7)
2/x 2z
Since 0 is a cyclic variable, p^ is constant, and we set
p<p = L
Furthermore, H is constant and may be identified with the energy of the system
measured in the center-of-mass frame. Thus we write
A.8)
2/x 2/xr2
With A.7), Hamilton's equations A.1.12) gives
= —= J-(E-V)-
A9)
dcj) L Ldt
2' d^
dt iir1 fir
2
1

140 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Trajectory of
center of mass
FIGURE 3.1. The radius vector r moves in a plane of constant orientation in the
center-of-mass frame.
Substituting d(j) for dt in the first expression gives
Ldr/r2
I
- V(r)] -
constant
A.10)
Note in particular that we are addressing scattering or unbound trajectories.
With the convention that V(oo) = 0, an unbound orbit is characterized by
E > 0, where E is the energy as given by A.8). A bound state corresponds to
E < 0 (see Problem 3.1).
3.1.2 Scattering Angle
We wish to apply A.9) to obtain the angle of scatter that two particles suffer in
collision. A scattering event is divided into three epochs: before, during, and

3.1 Scattering Concepts 141
Interaction
domain
FIGURE 3.2. Scattering in the lab frame.
FIGURE 3.3. (a) Relative radius vector r before and after collision in the center-of-
mass frame, (b) Scattering in the center-of-mass frame illustrating relative velocities
of masses mi and ra2.
after the interaction. Both before and after collision, particles are free and do
not interact with one another. A typical scattering in the lab frame is depicted
in Fig. 3.2. The situation in the center-of-mass frame is depicted in Fig. 3.3.
The meaning of before and after collision is given in terms of the potential
of interaction V(r) or the range of interaction r0. Before collision denotes
the interval in which the particles are approaching one another and r > r0,
or, equivalently, V(r) = 0. After collision denotes the interval in which the
particles are receding from one another and, again, V(r) = 0 and r > r0.
It proves convenient in discussing scattering events to introduce the relative
velocity vector before and after collision. Assuming that the interaction occurs

142 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
FIGURE 3.4. Scattering in the frame of the relative vector r. Note that we may write
g' = Sg, where 5 is the scattering matrix 5 = 7— 2aa.
in an interval about t — 0, we write
g = r, t — ±oo A11)
In that V = 0 before and after collision, with A.8) we find
-fig1 =-fig1 A.12)
and conclude that the magnitude of relative velocity is conserved over a
collision.
A scattering event in the frame where the origin of the relative vector r is
fixed (equivalent to the center-of-mass frame) is depicted in Fig. 3.4. The unit
vector a bisects the angle between —g and g'. The impact parameter s is such
that the angular momentum of the system in this frame has magnitude figs.
The impact parameter s as well as the relative velocity vectors g and g' are
properties of the asymptotic collision states. The speed g and impact parameter
s are both preserved in a given collision, and these two scalar quantities may
be termed properties of a collision.
The scattering diagram (Fig. 3.4) is symmetric about the apse, rmin, so that,
at rmin, r = 0. This symmetry is a consequence of the conservation of angular
momentum L = figs = fig's' and the relative speed g — g', so that s = s'.
Inasmuch as r vanishes at the apse, it follows from A.9) that at this point
the denominator of the integral A.10) vanishes. However, the integral remains
bounded.
The angle \/r depicted in Fig. 3.4 is obtained by integrating A.10) from rmin
to oo. There results
2
/ (L/r )dr
, „ „ (I-13)
min J2u(E -V)- (L2/r2)
mm
V) - (L2/r2)
The scattering angle 0 is related to \/f as shown in Fig. 3.4.
0-h2x//=7r A.14)

3.1 Scattering Concepts 143
better representation of the integral A.13) is given in terms of the inverse
adius u — r~x. Furthermore, the angular momentum L = /xgs is related to
he energy E = \[ig2 through
L2
Substituting these relations in A.13) gives
'" sdu
r
Jo
e\1/n
A.18) gives
N
A.16)
o — o V ^^ )
1 - s u = 0
E
For potentials of the form
V(r) = K(r)~N = KuN A.17)
§1.16) becomes
s du
k J\-s2u2-(KuN/E)
Substituting the nondimensional inverse radius
fi=su A.19)
Kid the nondimensional impact parameter
A.20)
f(b)= f d/i = A.21)
Jo JlpQJ/b"
tI -0 A.21a)
SThis general result is employed below.
Cross Section
differential scattering cross section is defined in the following manner. Imagine
uniform beam of particles of energy E and intensity / number/s-cm2), which
incident on a scatterer located at the origin. The number of particles scattered
Lto the element of solid angle dQ about Q is proportional to the incident
itensity / and the element of solid angle dQ. The proportionality factor is a,

144 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
the differential scattering cross section (see Fig. 3.5). That is,
Ia(Q)dQ = number deflected into dQ about Q per second A22)
This number is the same number that passed through the differential of annulus,
s ds d(f), so that
IadQ = I dcpsds A.23)
The azimuthal angle 0, which locates a section of the incident beam, is the
same angle that appears in a spherical coordinate frame fixed with origin at
the scatterer. That is, dQ = dcos d(p. Inserting this equality into A.23) gives
ds(E,8)
cr(E90) = s(E90) . ' A.24)
sin 8 dO
This is the classical formula for the differential scattering cross section. The
functional form s(E,6) stems from the scattering integral A.16).
Whereas a pertains to the number of particles scattered into a specific
direction, the total scattering cross section,
JAtz
aT = adQ = nrl A.25)
relates to the total number scattered out of the incident beam. The range of the
interaction is r0, so particles with impact parameters s > r0 are not scattered
out of the beam. The total cross section oT represents the obstructional area
that the scatterer presents to the incident beam. For a uniform beam of cross-
sectional area A > crT, the quantity <jt/A is the fraction of particles scattered
(in all directions) out of the beam.
An important factor that enters the Boltzmann equation is gad cos 9. With
A.24), we write
go d cos 8 — gs ds A.26)
Recalling the nondimensional parameter b A.20) permits A.26) to be written
go d cos 8 = ( — ) g{N-4)/Nbdb A.27)
V M /
It is evident that the case N = 4 merits special attention. Molecules that
interact under this potential are called Maxwell molecules.l For such molecules,
the scattering weight go dcos8 is independent of g. This property greatly
facilitates calculations related to the linearized Boltzmann equation. These
topics are fully discussed later in this chapter.
1 For Maxwell's introduction of this potential, see Scientific Papers of James Clark
Maxwell, W. D. Niven (ed.), Dover Publications, New York A952).

3.1 Scattering Concepts 145
Incident beam
FIGURE 3.5. Coordinates for a scattering experiment.
The Coulomb cross section
i£t us calculate o for the Coulomb potential
K
V = —
Integrating A.21), with N = 1, gives
b2 — - tan2 \j/
In terms of 0@ + 2\fr = 7r), this relation appears as
A.28)
1 9 17
= -ctn -
4 2
A.29)
fcserting this result into A.27),
a =
dcosO
A.30)
gives
1
cr =
ad cos 0
4EJ sin4F>/2)
dcosil/K2
A.31)
4 cos3 \/fE2
This is the well-known Rutherford cross section for Coulomb scattering. The
divergence at 0 = 0 stems from the long-range nature of the Coulomb force.
Cross section for rigid spheres
The calculation of the scattering cross section for rigid spheres proceeds
directly from geometry. If the two spheres have diameters aOi and aO2, re-
respectively, then interaction does not occur for r > (aOi -\-<jq2)/2, where r is the
displacement between sphere centers. A sketch of the scattering is depicted

146 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
o12
o0i/2
FIGURE 3.6. Scattering of rigid spheres.
in Fig. 3.6. The cross section is obtained directly from the triangular insert in
Fig. 3.6, according to which
s = G\2 sin ^
s ds — o\2 sin ^ cos \fr d\fr
= -^ sin 6>d<9
A.32)
With A.24), we find
oF)
A.33)
So the scattering is isotropic in the center-of-mass frame. The total cross section
is
2n I ad cos 9 — ikj\2
A.34)
-1
the area of a disk of radius a12.
3.1.4 Kinematics
Two particles with respective momenta pi and p2 collide. After collision
the momenta are p\ and p2 collide. Conversation of momentum and energy
equations for this event are
Pi +P2
2 2
pi pi
1m.\ Im.2
1+P2
«'2
Pi ,
2m i 2m2
A.35)

3.1 Scattering Concepts 147
In the event that m\ — ra2, these conservative equations become
v2 = y\ + v2
Solving for v
\ and v2
gives
2
/
, 2
2
= v,
= v2
—
+
—
/2 ,
i ■
ol(ol •
a(a •
/2
I/O
L
■g)
•g)
A.36)
A.37)
The unit apsidal vector a is shown in Fig. 3.4. Subtracting these equations,
we find
g=g-2a(a.g) A.38)
Which, as noted previously in Fig. 3.4, may be written in the scattering matrix
form
g' - (/ - 2oa)g = Sg A.38a)
Inverse and reverse collision
Two symmetry properties of collisions are important in kinetic theory. In de-
Scribing these symmetries, it proves convenient to speak of a collision in terms
of two-particle momenta before (pi, p2) and after (p'j, p2) collision. Thus we
Characterize a collision by the form
-> (Pi>P2)L
The subscript s indicates that an impact parameter is a property of the collision.
The inverse of a collision [(pi, p2) —> (p\, p2)]5 is a collision containing the
final state (pi, p2). We see that if [(pi, p2) —> (p\, p2)]5 satisfies the conserva-
conservation equations A.35) then [(p^, p2) —> (pi, p2)]5 also satisfies them (read the
Conservation equations from right to left).
Symbolically, we may write
inverse of [(pj, p2) -> (p^, p'2)]s = [(p\, p2) -> (pj, p2)], A.39)
See Fig. 3.7.
The second type of collision that is important is the reverse collision. The
teverse of a collision [(pi, p2) —> (p'j, p2)]5 is a collision containing the final
state (—pi, —p2). Again referring to the conservation equations, we conclude
that (—pj, — p2M are the desired initial momenta (see Fig. 3.8). That is, if
("-Pp p2) interact with the impact parameter s, they will yield (—pi, — p2M.
Symbolically, we have
reverse of [(pi,p2) -* (PpP2)L = [(-PpP2) -* (~Pi,p2)L A.40)
We recognize the reverse collision to be the time-reversed orbits of the original
collision (see Section 1.1.5).

148 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Original collision
Inverse collision
FIGURE 3.7. A collision and its inverse.
Original collision
Reverse collision
Time
Time
FIGURE 3.8. A collision and its reverse.
Due to their kinematic equivalence, we may conclude that the differential
cross sections for the original, reverse, and inverse collisions are the same.
These equalities relevant to the lab and center-of-mass frames are graphically
depicted in Fig. 3.9.
The equality of these cross sections may be related to fundamental opera-
operations. Thus, equality of cross sections for the original and reverse collisions is
due to invariance of physical laws under time reversal Equality of cross sec-
sections for the original and inverse collisions is due to symmetric properties_of
the scattering matrix S A.38a). Thus, for example, S = S~\ so that if g' = Sg
then g = Sg' (see Problem 3.2).

3.2 The Boltzmann Equation 149
Original collision
Apse
-Pi
-P2
Reverse collision
P2 r Pi
Inverse collision
g=v1-v2
0=The
scattering angle
FIGURE 3.9. Scattering cross section is the same for all three collisions.
3.2 The Boltzmann Equation
The Boltzmann equation is an equation of motion for the one-particle distri-
distribution function and is appropriate to a rare gas. Various efforts have attempted
to derive this equation from first principles. Three of these, due respectively
to I. Prigogine, N. N. Bogoliubov, and H. Grad were described in Chapter 2.
jJAn assortment of assumptions come into play in all such derivations, which
^renders the analyses somewhat ad hoc.
In the present discourse, we revert to a more heuristic derivation, which
.follows the spirit of Boltzmann's work, that is physically revealing and equally
ad hoc to more fundamental derivations.
3.2.1 Collisional Parameters and Derivation
In the limit of no interactions, particles are mutually independent and F(x, v, t)
(normalized to the total number of particles, N) satisfies the one-particle Liou-
ville equation. This equation states that the net number of particles that enter
the phase element 8\8x following a particle's trajectory, in the time interval
St, is zero. Let us denote this number by 8R. Then we may write
(dF
8R = 8x8\8t hv
V dt
dF K 8F\
dx m d\ J
0
B.1)
where K is an externally supported force field. Consider now that particles
interact. Specifically, let ro denote the range of interaction. This parameter is
defined so that, for interparticle displacement r > r0, interaction vanishes.
When particles enter the interaction domain (r < r0), they experience a colli-
collision. Let the mean distance between collisions be /. This distance is also called
the mean free path. Our first criterion for the validity of the derivation of the

150 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Boltzmann equation to follow is that
/ » r0 B.2)
This constraint ensures rectilinear trajectories between collisions. If C is
thermal speed, then we may introduce the times
I r0
B3)
An equivalent criterion to B.2) may then be written
r » r0 B.4)
Collision terms
If interactions are turned on then the one-particle Liouville equation changes
by virtue of collisions between molecules in the fluid. A measure of this effect
is given by the interaction term in BYi B.1.24). In the present model, this
phenomenon is represented by the net rate at which collisions increase or
decrease the number of molecules entering the phase volume 8x8\. Thus in
B.1) we write
8R = 8R+-8R. B.5)
The number of particles injected into 8\8x due to collisions in the interval St
is 8R+, while those ejected is <$/?_.
First we consider 8 /?_. The velocity of all particles in the gas may be divided
into two groups, the small band of velocities that fall into the interval 8 v about
v and all other velocities denoted by the variable vi. The number of particles
that are removed from the phase element 8x8\ in the time 8t is simply the total
number of collisions that the v particles have with all other particles (that is,
the vi particles) in the time 8t. It follows that to calculate 8 /?_ we must account
for all collisions between pairs of particles that eject one of them out of the
interval 8\ about v, in other words, in every pair with the following properties:
1. One particle is in the phase element 8\8x about (v, x) and the other is in
the phase element 5vi<5xi about (vi, Xi).
2. The vi particles in Sxi undergo a collision with the v particles in 8x in the
time St.
By definition, the number of such particles is given by
SR- = / F2(z, Zi)S\i 8x{ 8\8x B.6)
To construct Sx\ so that it has property 2, we view a scattering event in the
frame of the v particle (see Fig. 3.10). In B.6) we have reintroduced the phase
variable z = (x, v). Furthermore, recall that F2 is normalized to N(N — 1)'
where TV is the total number of particles.
With the aid of Fig. 3.10 we note that all Vi particles in the cylinder shown
of height g St and base area s ds dcp undergo a collision with the v particle in

3.2 The Boltzmann Equation 151
v, particle has
velocity g in
v particle frame-
s ds
FIGURE 3.10. Scattering in the v-particle frame.
the time St. Here we are writing
g
- v|
It follows that
gSt sds d(j)
B.6) becomes
SR.
F2d\\gsds d(j) I S\8xSt
B.7)
B.8)
To calculate 8R+i we must account for all two-particle collisions that send
one particle into the velocity interval S\ about v in the time St. But this is just
the inverse of the original collision
(v,v,
That is,
It follows that
8R
+
= / F2(z\z\)8\\8x\8\f8xf
B.9)
Here it is understood that primed variables refer to the inverse collision of
unprimed variables in B.8).

152 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
The next point in our derivation is to recall the integral invariants of Poincare.
These include the phase volume 8x8\8x\8\\ relevant to an TV-particle system
That is, we may write
SxS\SxiS\i = 8x'8y'8x\8y\ B.10)
Substituting these results into B.5), B.1) gives
DF dF 8F K 8F
DT dt " dx " m 3v
I
-.l_= I[F2(z\z\)-F2(z,zl)]d\lgsds d(t> B.11)
We now assume that F2 is homogeneous over the dimensions of the collision
domain and write
= F2(v,Vi)
B.12)
Finally, we impose the assumption of molecular chaos, which states that
particles in a rare gas are not correlated. That is,
N - 1
F2(v,
B.13)
Substituting B.13) into B.12) and neglecting 1 compared to TV gives the
Boltzmann equation:
nrr rr
B.14)
Here we have recalled that s ds dcf) — a dQ and have introduced the notation
F, = F(vO, F' = F(V), F[ = F(\\) B.14a)
Note also that, with the conservation solutions A.37),
V, = Vj + <X(OL • g)
1 B.14b)
\r — v — ol(ol • g)
These latter three equations, B.14, 2.14a, 2.14b), comprise the Boltzmann
equation.
Key assumptions brought into play in the derivation of this equation are as
follows:
1. Range of interaction/mean free path <SC 1.
2. Particle trajectories are rectilinear before and after collision, as depicted
in Fig. 3.10. This assumption is called the stosszahlansatz.
3. F(x, v) is homogeneous over the range of interaction.
4. Molecular chaos: F2(l, 2) = FA)FB).

3.2 The Boltzmann Equation 153
$.2.2 Multicomponent Gas
(Generalization of B.14) to N species is accomplished with aid of the following
faotation. First we rewrite B.14) as
DF
Dt
fyrhere
B.15)
J(F \G)= if d\lgadQ(F/G\ - FGX) B.16)
Then for a gas comprised of TV species we have
N
^
| F,) B.17)
Note in particular that
ff , ,
J(Fi | Fj) = // dYjgijGijdQijiFjF. - FtFj) B.18)
JJ
The set of N coupled equations B.17) comprises the generalization of the
Boltzmann equation to N species. These relations are returned to in Section 3.7
In the derivation of the Druy vesteyn distribution relevant to electron transport
in an ionic medium immersed in an electric field.
3.2.3 Representation of Collision Integral for Rigid Spheres
Consider a gas comprised of rigid spheres of diameter aOi. With A.33), we
fecall
B.19)
We wish to work with the angle \/f in place of the scattering angle 0 (see
Hg. 3.4).
0 + 2^ = n
so that
cos 0 = 1 — 2 cos2
and
d cos 0 — —4 cos \j; d cos x//
and
dQ = -d(p dcosO =4 cos x/sd cos \/r d<f> B.20)

154 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
The collision integral is a function of v and angles may be defined with respect
to this vector. Thus, with v as the polar axis, Vi is given the polar angle X and
azimuth k. Combining results, we write
J(F | F) = (Jo2! / gv2dv{ d cos X die / cos f d cos f d(j)[F[Ff - Fx F]
B.21)
Recalling the definition of g,
g = vi - v B.22)
we obtain
g2 = v2 + v] - 2vvl cos X B.23)
Referring to Fig. 3.4, we see that
ol • g = g cos ^ B.24)
This permits B.14b) to be rewritten
y\ — Vi + ol cos \j/
\' — v — otg cos
so that
B.25)
F' —F[\ — ag cos /]
To measure a, we choose the polar axis g. The azimuth 0 is measured with
respect to the plane specified by the vectors v, Vi (see Fig. 3.11). The Cartesian
components of a in this frame are
a = (sin \j/ cos 0, sin i/r sin 0, cos i/r) B.26)
In the spirit of this representation, we integrate over ifr, holding Vj and g fixed
prior to integrating over Vj.
Explicit representation of Fr and F[ as given by B.25) requires knowledge
of a • v and a • Vi. With the aid of Fig. 3.11 and the addition law of cosines,
we find
a • v = t>[cos y\r cos X + sin i/r sin X cos 0]
a • Vj = v\ [cos ^ cos(>. + y) + sin ^ sin(>. + y)y cos 0]
where
v? - u2 + g2
cos y =
This completes our representations of J(F | F) for the scattering of rigid
spheres. Note that, in the more general case, in place of o given by B.19)
we have a = cr@, g), in which case a remains in the integrand of the cos $
integration.

3.3 Fluid Dynamic Equations and the Boltzmann H Theorem 155
Plane containing
v,v.
FIGURE 3.11. The coordinates of the apsidal vector ex.
It should be noted that the preceding derivation addresses smooth spheres,
ihich types of spheres exchange no angular momentum in collision (see Prob-
fbms 3.53 and 3.54). We note also that Grad's second equation B.6.12) is better
suited to the study of interacting rigid spheres than is the Boltzmann equation.2
3.3 Fluid Dynamic Equations and the Boltzmann
Theorem
9.3.1
Collisional Invariants
>Ve wish to derive the fundamental fluid dynamic equations from the Boltz-
<&ann equation B.14). Such equations express the basic conservation principles
As has been previously noted, the assumption of Maxwell molecules greatly
Amplifies the collision integral [see A.27)]. With the additional assumption that
%<r = constant, the Boltzmann equation has been solved exactly for a spatially
Uniform fluid: A. V. Bobylev, Doc. Acad. Nauk. SSSR 225, 1296 A975); M. Krook
tod T. S. Wu, Phys. Fluids 20, 1589 A977). See also R. M. Ziff, Phys. Rev. A 24,
•09A981).

156 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
of physics: mass (or number) of particles, momentum, and energy. Thus a nec-
necessary requirement for a kinetic equation to be a valid physical relation is that
it imply these conservation equations. To show that this requirement is sat-
satisfied by the Boltzmann equation, we introduce the notion of collisional (or
summational) invariants.
Toward these ends, we recall the collisional integral of B.14):
J(F) = ffodQdyxg(F[F' - FXF) C.1)
Let us identify the linear operator:
7[0(v)] = ff J(F)<t>(y)dy
If
III
a dQ d\x d\g(F[Fr - Fx FH(v) C.2)
JJJ
Changing variables,
gives the equality
/@) = /@i) C.3)
Consider next the integral
7@') = fff a dQd\x d\g(j)(\r)(F[Ff - FXF) C.4)
The change of variables
carries a unit Jacobian due to Liouville's theorem for a two-particle system.
Therefore,
\ d\f
d\ = d\\ d\f
The measure go dQ is also invariant under this change in variables and C.4)
becomes
/) = fff G' dQ' d\\ dyfg(pf(Fx F - F[F')
= -7@)' C-5)
Finally, exchanging variables (v',, v') —> (vr, v',) gives
7@') = 7@;) C.6)
Combining C.3), C.5), and C.6) gives
47@) = 7@) + 7@,) - 7@') - 7@;)

3.3 Fluid Dynamic Equations and the Boltzmann H Theorem 157
u
Fixed
lab frame
(a)
Co-moving
frame
(b)
Tube fixed
in lab frame
FIGURE 3.12. (a) The absolute frame is fixed in the lab frame, (b) The relative frame
moves with the local mean velocity of the fluid.
Pue to the linearity of the / operator, this latter result may be rewritten
-0' -0;)
C.7)
k function ijf(\) is a summation (or collisional) invariant if
+ $ = $[ + f
C.8)
ITiat is, VKv) is a property of a molecule that is preserved in a collision, such
Is mass, momentum, and energy.
If VKv) is a collisional invariant, then with C.7) and C.8) we see that
I(^j/) = 0, \jr — collisional invariant C.9)
Hie three fundamental collisional invariants described above correspond to
= 1, \j/ = v, and \j/ = v2, and we write
C.10)
S.3.2 Macroscopic Variables and Conservative Equations
frior to learning the manner in which collisional invariants come into play in
obtaining the conservation equations, we turn to the definition of fluid-dynamic
variables and the equations they satisfy. Fluid dynamic (or macroscopic) vari-
variables are commonly defined in one of two frames: the absolute and relative
(or co-moving) frames. In the absolute frame the fluid moves with respect to a
Coordinate frame fixed in the lab frame. In the relative frame, fluid variables are
Measured with respect to a coordinate frame that moves with the local velocity
of the fluid (see Fig. 3.12).
Conservation equations typically appear in the form
dA
~dt
+ v. rA = o
C.11)

158 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
where FA is a flux vector corresponding to the quantity A. Integrating this
equation over a small volume V and employing Gauss' theorem gives
I Adx= -§TA • dS C.11a)
dt
This equation says that the quantity fv A dx can change only by virtue of the
net flow out of the volume as measured by the flux integral over the surface S.
Consider the number density n(x, t) [encountered previously in A.6.12)]
and macroscopic mean velocity u(x, t). These variables are defined by (with
/i(x,r)= / Fd\ C.12)
mi(x,O= / F\d\ C.13)
and obey the continuity equation
dn
C.14)
dt
This equation is an expression of the conservation of matter. It says that the
mass in a given volume can change only by virtue of a net flux of mass out of
the volume.
Note that with C.13) we may write [recall A.6.8)]
u= J-T—r = (v) C.13a)
JFd\
Thus u is appropriately termed mean macroscopic velocity.
Next we consider the pressure tensor p, heat flow vector q, and kinetic
energy density etc, all measured in the absolute frame. These are given by the
integrals3
P— \ Fmwd\ = n(m\\) C.15)
q= F-mu2\d\ = n[-mv2\) C.16)
J 2 \2 /
/I i
F-mv2d\ = n{-mv2} C.17)
The pressure component pxy, say, represents the transport of x momentum per
second across a y surface (the unit vector y is normal to a y surface). Evidently,
q represents a flux of kinetic energy.
The energy density eK was previously introduced in A.6.16) where it was labeled

3.3 Fluid Dynamic Equations and the Boltzmann H Theorem 159
Conservation of momentum gives
a
pu + V • p = pK
C.18)
[ere we have written raK for an externally supported force field that permeates
fluid, where ra is molecular mass and
p(x, t) — mn(x, t)
i
I mass density.
, The conservation equation for energy is given by
V q = pK u
C.18a)
dt
C.19)
rich, together with C.14) and C.18), constitutes the three fundamental
mservation equations of fluid dynamics (in the absolute frame).
13 Conservation Equations and the Boltzmann Equation
obtain the conservation equations C.14), C.18), and C.19) from the
>ltzmann equation B.14), first we rewrite the Boltzmann equations as
DF
~~Dt
DF
J(F)
8F
dF
dt
|rst note that the left side of C.20) gives
d\
DF
Dt
/
\
1
rav
mv2
2
)
dn
dt
—
dt
right side of C.20) gives
d\J(F)
/
1 >
rav
ra n
\
/(I)
ra/(v)
ra
\
<
0
C.20)
C.20a)
C.21)
C.22)
^uating C.21) to C.22) reproduces the three conservation equations C.14),
3.18), and C.19). We may conclude that the Boltzmann equation satisfies the
Squired property of implying the conservation equations.

160 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Relative macroscopic variables
We turn next to rewriting these conservation equations in terms of more
physically pertinent relative macroscopic variables defined in terms of the
deviation-from-the-mean microscopic velocity c given by
c = v-u = v-(v) C.23)
The velocity c is that of a fluid element measured in a frame moving with
the mean velocity of the fluid, u. The relative pressure tensor P, heat flow
vector Q, and (kinetic) energy density EK are given by the parallel equations
to C.15), C.16), and C.17), with v replaced by c. There results
P = I FFmccdc = p(cc) C.24)
C.25)
C.26)
Substituting c as given by C.23) into these expressions gives
p = P + puu C.27)
2
o pu
q = Q + p . u + u(eK - pu2) = Q + P . u + uEK + u^-— C.28)
C.29)
The relation C.27) says that absolute pressure is greater than relative pressure
by the flow energy pun. Consider water flowing in a pipe with u = (wA, 0, 0).
Thus puu = pu2. If p is measured, the fixed pressure meter measures this
additional momentum flux due to the macroscopic flow of the water in the
pipe. This is not what we mean by pressure. We measure pressure in a frame
moving with the fluid. The meter that measures P does not see the macroscopic
flow velocity v.
Inserting the relation C.27), C.28), and C.29) into the macroscopic equa-
equations C.14), C.18), and C.19) gives the normal form of the conservation
equations:
p (— + u • V J u + V • P = pK C.30)
dt
EK + V • (uEK) + P:Vu+V.Q = 0 C.3D

3.3 Fluid Dynamic Equations and the Boltzmann H Theorem 161
;ee Problem 3.3). The tensor notation of the double inner product P : Vu is
follows:
d
P : Vu =Tr(PVu) = P,
jjie right side of the preceding equations is written in the Einstein convention
which repeated indexes are summed. 4
[Equations C.30) and C.31) will be returned to in Section 3.5 in our
liscussion of transport coefficients.
,3.4 Temperance: Variance of the Velocity Distribution
le temperature T of a fluid comprised of molecules with no internal degrees
freedom is given by
f m2 3
Ek= I F(c)— dc = -nkBT C.32)
there
kB = 1.381 x l(T16erg/K
^presents Boltzmann's constant.
The significance of defining T as given by C.32) in the relative frame is as
)llows. Consider a distribution of molecules that are all at rest. Now let the
dd move as a rigid body with a speed u. With C.32), we see that T oc Ek — 0,
it that ex as given by C.29) is greater than zero. So had we defined T with
aspect to absolute variables we would obtain the erroneous result that the
Imperature of a gas of stationary molecules is greater than zero, by virtue of
le macroscopic motion of the whole body of molecules.
Note that, C.23), C.32) may be rewritten
= <(v - (v»2> C.32a)
m
fhich, recalling A.8.2b), indicates that 3kBT/m is a measure of the variance
the velocity probability density. Thus, in general, T is a measure of the
pread of this density about the mean. Thus a wide distribution (many velocities
Pntribute to F) corresponds to a high temperature. If the distribution of speeds
5 sharply peaked about the mean (v), then temperature is small (see Fig. 3.13).
. Having defined the fundamental fluid-dynamic variables (n, u, T, EK, Q,
P) as velocity moments of F(x, c, t) it is evident that such relations may
generalized to define additional macroscopic variables. Thus consider the
4Tensor notation is reviewed in Appendix A.

162 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
t
F[v)
(a)
(b)
FIGURE 3.13. In (a) most molecules move with mean speed (v) corresponding to low
temperature. In (b) the variance of F{c) is large, as is the temperature. Measurement
of v in this instance finds large fluctuations about the mean, (v).
tensor form
wA/IfI-2f. ,/„ = / dcF(c, x, t)chci2... cin
C.33)
The fluid variable A(x, t) is an nth-rank tensor in three dimensions.56 The
indexes ik run from 1 to 3, corresponding to the Cartesian components of c.
The preeminence of the lower-order variables (n, u, T, EK, Q, and P) is
due to their presence in the three conservation equations.
3.3.5 Irreversibility
Preparatory to our discussion of the H theorem, it is appropriate at this point to
discuss the concept of irreversibility. Consider that 1 mole of gas is comprised
of identical inert molecules save for the fact that half of the sample molecules
are labeled color A and the remaining half color B. The entire gas is confined
to an isolated enclosure with A molecules and B molecules separated by a par-
partition located at the midplane of the enclosure. The entire gas is in equilibrium
at some given temperature. The partition is removed (ideally, without incurring
any other perturbation on the system). After sufficient time, the gases mix and
the A and B molecules become uniformly distributed over the entire enclosure
(see Fig. 3.14).
It is universally observed that the starting state of this process does not
reoccur. Why is this observation peculiar to the laws of nature? In Section 1.1.5*
we found that if a system's Hamiltonian has a trajectory [q(t)p(t)] then it also
has a trajectory [q(—t), —p(—t)]. It is evident that we may view the starting
5Thus we have the quip, "A distribution function is worth a thousand macroscopic
variables."
6Grad's method of moments takes advantage of the tensor quality of C.33). See
Section 5.10.

3.3 Fluid Dynamic Equations and the Boltzmann H Theorem 163
•" X X XX X /
\. X XX XX X X >
• • •
• •
•
• * * *
•
w XXXXXXX
0 © •
o
•
* ° * •
^X X XX X X X '
X
YX XXXXX .
X XX X \\\\\\\\\\\\\\
©
e
o
. o . •
IXXXXXX XXXXXXXXXXN
(a)
(b)
GURE 3.14. (a) Confined gases, (b) Gases expand to fill the whole interior region,
mole concentrations, state (a) is not observed to reoccur.
frtate (a) in Fig. 3.7 as the time-reversed state of (b). Thus, for example, say
lat in the state (b), at a given instant, the momentum of all particles in the
ras is reversed. Then, as depicted in Fig. 1.5, the motion should reverse itself,
lereby restoring the initial coordinate configuration. However, such reversal
loes not occur and we conclude the motion is irreversible.
More generally, an irreversible law is characterized by an equation that
las a solution peculiar to a particular direction of time. This is a quality of
reversibility, because if the motion corresponding to this solution is observed,
re may conclude that time is flowing, say, in the forward direction. A reversible
iw on the other hand implies no such distinction.
13.6 Poincare Recurrence Theorem
le evident irreversibility of the process described in Fig. 3.14 also plays havoc
tith a theorem due to Poincare. This theorem states the following: The system
ajectory of a bounded isolated system of finite energy will, after sufficient
le, return arbitrarily close to its initial location in F-space.
To establish this theorem, we consider that the initial state zo = (qo, Po) *s
mtained in the set of phase points £20. That is, z e Q\o. Due to the preceding
lescription, the system point moves on a surface of finite measure in F-space. It
)llows that after sufficient time the path swept out by the set ft0 must intersect
self. Let us show this. Let f denote an operator that displaces ft0 in unit time.
f;ihen, due to Liouville's theorem
p
have the same measure. If these sets did not intersect, the surface on
hich they move would have to be of infinite measure. This contradicts our
assumption.
\ Therefore, we may write
f^nho-ii/0 C.34)
fcr some value of the integers k and at. The empty set is 0. Due to uniqueness of
trajectories, as previously described in Chapter 1, f is a one-to-one mapping.
This property permits us to write
f(AHB) = f(A)nf(B)
C.35)

164 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
F-space
FIGURE 3.15. Shaded domain represents the intersection Tw£l0 Pi
Operating on C.34) with T~n gives
n
With C.35), the preceding equation becomes
0
0 C.36)
Thus, after k — n units of time, the set £2o has a finite intersection with itself.
Taking the measure of £20 to be arbitrarily small establishes the theorem. The
process described by C.36) is depicted in Fig. 3.15. (The group property of the
T operators is discussed in Problem 3.4). Applying this result to the process
described in Fig. 3.14 indicates that, after sufficient time, the system point
returns arbitrarily close to configuration (a).
Another approach to the question of irreversibility is offered by the
Boltzmann H theorem, which is described in the following section.
3.3.7 Boltzmann and Gibbs Entropies
The Gibbs entropy, H,N, is given by
r = / In In
... dN
C.37)
To discover how this function changes in time, we first recall the Liouville
equation
dt dt
Operating on this equation with
[fN,H]=0
C.38)
dl-..dN(l+\nfN)

3.3 Fluid Dynamic Equations and the Boltzmann H Theorem 165
»eveals that7
thus Hm obeys a reversible equation [that is, if H^it) is a solution to C.39),
\o is Hui—t)] and is constant.
The Gibbs entropy is related to thermodynamic entropy through the formula
S = -kBHN C.40)
(This is the kinetic prescription for the entropy of an isolated TV-body system.
the second law of thermodynamics stipulates that, for any process of an iso-
isolated system, AS > 0, the equality holding for reversible processes. Since the
piouville equation is reversible, the fact that it implies S = constant is seen to
|>e consistent with the second law of thermodynamics.
For vanishing correlations,
n
1=1
pnd
N p N
p
^
C.41)
= j fx\nfxdxdp C.42)
lenotes Boltzmann entropy and with C.40) gives the Boltzmann formula
S = -NkBH C.43)
will find later that, away from equilibrium, Boltzmann's equation implies
bat dH/dt = 7i < 0. This is an irreversible equation because, as noted
ibove, it implies one distinct motion with the positive flow of time, that is,
I decaying solution. On the other hand, a reversible equation that implies a
decaying solution must also imply a growing solution. That the laws of nature
*re reversible indicates then that there is no experiment to discern if time is
lowing in one direction. On the other hand, the law H < 0 does imply a
Preferred direction in time, since with time flowing in the forward direction,
7Note that this same result occurs for any functional r/r(fN) with a bounded
derivative \d\/r/dfN\ < oo. See Problem 3.58

166 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
H(t) decreases. If we were to observe H(t) increasing, we could conclude by
virtue of the statement H < 0 that time was flowing backward.
With this background, we turn to derivation of the H theorem (put forth by
Boltzmann near the turn of the century).
3.3.8 Boltzmann }s 7i Theorem
This proof begins with the Boltzmann equation B.14)
K f
///) C.44)
m 3v
where J(f) has been written for the collision integral on the right side of
B.14). (Note also that we are working in velocity space.) Operating on C.44)
with
/ dxd\(\ + In /)
gives
d f f d K f d
- / dxdyf\nf+ / dxdv— .vln/ + -- / dxdv—f\nf
dt J J dx m J d\
= jdxd\J(f)(l+lnf)
= f dxl(l+lnf) C.45)
where / is defined by C.2). Passing to large volume and dropping surface
terms in C.45) gives
ii /»
H= / dxl(l+lnf) C.46)
m dt J
With the property C.7), we may write
4/A + In /) = 7A + In /) + 7A + In /,) - 7A + In /') - 7A + In /,')
C.47)
fif'J \ fif
Thus we may write
4/A +ln/) = - j dv j dvlg<xdn(f[f - /i/)ln
Setting X = f[f and Y = f\f, this equation may be rewritten
/ rt
41 = - f d\ f d\igadQ(X -Y)\n( —
= - f dv f d\\godQ.L{X, Y) C.48)

3.3 Fluid Dynamic Equations and the Boltzmann H Theorem 167
Here we have defined
X
C.49)
With X and Y positive, we examine X = Y, X > Y, and X < Y. For all three
cases, L > 0. It follows that
* 7(l+ln/)<0 C.50)
since d\d\\go dQ is a positive measure. The equality in C.50) corresponds
to X = Y\ that is, ///' = fxf. Inserting C.50) into C.46) gives
1 d
—H= [dxI(l+lnf)<0
m dt J
or, more simply,
dH
1 -T-<0 C.51)
dt
iphich is Boltzmann's H theorem.8 The relation C.51) states that, for arbitrary
initial /(x, v, t), H decreases until
C-52)
whereafter it remains constant.
i
^Source of irreversibility
|Equation C.51) is evidently an irreversible statement. What is the source of
[this irreversibility? To answer this question, we recall that C.51) stems from
|ftie Boltzmann equation C.44), which is irreversible. To see this, consider
^e transformation, the left side of C.44), which is irreversible. To see this,
•consider the transformation x —> x' = x, v —> \' — — v, and t —> t — —t.
Under the transformation, the left side of C.44) changes sign, whereas the
collision integral on the right side maintains its sign. This is so because the
integration measure of J(f) is positive and / > 0.
3.3.9 Statistical Balance
The statement C.52) is sometimes called statistical balance. At this value,
8R+ = 8R- in B.5). That is, this rate of gain of particles into the volume
element 8x8 v is equal to the rate of loss of particles from this volume element.
Note in particular that with
m dt dt
jdxd\f\nf= I dxd\^-(\ +ln/) C.53)
5An H theorem relevant to quantum systems is derived in Section 5.4.3.

168 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
the condition of equilibrium, df/dt — 0, implies that dH/dt — 0. We have
found above that this latter condition is satisfied only when
/i'/' = /i/ C.54)
Thus the conditions C.54) is necessary for equilibrium. For a force-free
spatially homogeneous fluid, this condition is evidently also sufficient for
equilibrium, and we may conclude that statistical balance is necessary and
sufficient for the equilibrium of such a fluid.
3.3.10 The Maxwellian
To find the equilibrium distribution that is a solution to C.54), we first take the
In of both sides. Labeling the solution /0 gives
In /q'j + In /0' - In f0l + In f0 C.55)
This equation asserts that In /0 is a collisional invariant [see C.8)]. Thus we
may write
In /o(v) = -A(v - v0J + In B C.55a)
or, equivalently,
/o(v) = Be-A(y-y°J C.56)
The constants A and B are determined through the relations C.12), C.13), and
C.26).
f
d\ C.57a)
d\ C.57b)
/
With these equations, we obtain
-nKBT = N I B—e-A{y-v°rdc
F0(v) - Nf0(y) = n—^ exp f-(VU) ) C.58)
where9
m
The distribution C.58) is quite important in kinetic theory. It goes by either
of the names the Maxwellian, the Boltzmann distribution, or the Maxwell-
Boltzmann distribution.
9In kinetic theory, the symbol R often denotes the gas constant kBN0, where No
is Avogadro's number.

3.3 Fluid Dynamic Equations and the Boltzmann H Theorem 169
Although the Maxwellian C.58) is a solution to the statistical balance equa-
equation, C.54), there is another more general and physically relevant solution.
Thus we observe that the distribution C.58) remains is a solution to the sta-
statistical balance equation, C.54), there is another more general and physically
relevant solution. Thus we observe that the distribution C.58) remains a solu-
solution to C.54) with the constant values of n, u, and T replaced by corresponding
functions of x and t. The resulting equilibrium distribution is called the local
Maxwellian and is given by
,,0, „ "(X'O / (v-u(x,
Fu(x, v, t) = r^ n^, ^~n exp
[2ttRT(x, Ol3/2 V 2RT(x,t)
C.59)
With the normalization conditions C.57), we find
°(x, v, t)dy C.60a)
w(x,r)u(x,0= / F°(x,y,t)ydy C.60b)
-/i(x, t)kBT(x, 0 = I F°(x, c, ty^- dc C.60c)
In the Maxwellian given by C.58), the parameters n, u, and T are constant
in x and t. To distinguish this distribution from C.59), it is called the absolute
Maxwellian. To recapitulate,
Fo(v) :absolute Maxwellian, n, u, T = constants
F°(x, v, 0 :local Maxwellian, n, u, T — functions of x, t
We recall that H(F°) — H(F0) — 0. Thus both these distributions may be
termed equilibrium distributions. However, thermodynamic equilibrium for
a force-free fluid implies constant values of all macroscopic variables. This
equilibrium is better described by F0(v). Prior to this state, the fluid is in
a local equilibrium described by F°(x, v, 0- This sequence of states in the
approach to equilibrium of a fluid is well described in a sketch of H(t) (see
Fig. 3.16).
We may speculate on the forms of the relaxation times xL and xA shown in
Fig. 3.16. Let the fluid be confined to the volume V = L3, and let I denote the
mean free path of molecules between collisions. For n ^> 1, it is evident that
L. With the thermal speed given by C.32a),
= 3c2 C.61a)
m
we write
/ L
rL --ta-- C.61b)

170 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
t
Local eguilibrium:
:, v, t)
/
Absolute equilibrium.
F = F0(v)
FIGURE 3.16. The approach to equilibrium of a fluid.
Having found that the local Maxwellian C.59) is an equilibrium solution
to the Boltzmann equation, we momentarily return to the Bogoliubov hy-
hypothesis (Section 2.4). We recall that this hypothesis states that, in the final
hydrodynamic stage of a fluid's approach to equilibrium, distributions become
functional of the fluid variables n, u, and T. It is evident that this conjecture
finds corroboration in the structure of the local Maxwellian given by C.59),
which is seen to be a functional of the said fluid dynamic variables.
Local equilibrium and macroscopic variables
We have found that the local equilibrium C.59) satisfies statistical balance
C.55) and therefore renders the collision integral of the Boltzmann equation
zero. However, when substituted into the Boltzmann equation B.14), it is
evident that, in general, the left side of this equation does not vanish. As space
dependence of F° is contained entirely in the macroscopic variables n, u, and
7\ it suffices to find their dependence on space to obtain a complete equilibrium
solution. For non-force-free fluids, this may accomplished directly from the
Boltzmann equation as demonstrated in the following section.
As we will find in Section 3.5, the local Maxwellian also comes into play
in the Chapman-Enskog expansion of the Boltzmann equation. In this proce-
procedure, F° emerges as the lowest-order solution with n, u, and T occurring as
functions of (x, t). Accompanying lowest-order conservation equations serve
to determine these macroscopic variables, thereby establishing F°(x, v, t) as
the corresponding lowest-order solution to the Boltzmann equation in the
nonequilibrum limit.
3.3.11 The Barometer Formula
Consider that an externally supported conservative force field K permeates a
fluid. We may write
K = -

3.3 Fluid Dynamic Equations and the Boltzmann H Theorem 171
us call the equilibrium distribution for this configuration Fo, so that
lFo/dt — 0. The remaining terms in the left side of the Boltzmann equation
J.I 4) give zero, providing
dF
m\
0
dx
dx 3v
C.62a)
more general solution to the statistical balance equation C.55) is given by
-A(v - v0J + In B - 2A\l/(x)
C.62b)
m
Substituting this form into C.62a) indicates that ^/(x) — O(x), providing
fQ . VO = 0. With this constraint imposed, C.62b) is the desired equilibrium
)lution.
Fitting Fo to the constraints C.57) gives
F0(x,
n0
BnRT0)y2
-[m(v-u0J/2
C.63)
Inhere n0, u0, and To are constants and u0 is normal to VO(x).
The equilibrium number density that follows from C.63) is given by
n(x)- / Fo(\, v)dv = n0 exp
kBT
C.64)
io that n0 is the value of n(x) where O(x) = 0. The equilibrium temperature
Is given by
3n(x)RT
I
F0(x, c)c2 dc = 3n0RT0 exp
k*T
C.65)
that T — To = constant. Thus we may write
F0(x, v) =
n(x)
BttRTK/2
exp
(V ~ Uq):
2RT
C.66)
T = To and w(x) is given by C.64).
For a stationary column of gas in a gravity field, we have
Where g is acceleration due to gravity and z is vertical displacement. Inserting
ffiis potential into C.64) gives
mg(z - Zo)
knT
C.67)
This exponential decay in number density is commonly called the barometer
formula (also, the law of atmospheres).

172 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
3.3.12 Central-Limit Theorem Revisited
We wish to apply the central-limit theorem (Section 1.8 to the problem of find-
finding the probability distribution of the total momentum of a fluid of TV molecules
in equilibrium at temperature T. As P denotes probability, to avoid confu-
confusion in terminology, we will call total momentum K and individual molecular
momentum k,. Thus
n
C.68)
If we assume that individual momenta are uncorrelated, then K as given by
C.68) may be viewed as a sum of statistically independent random variables,
and the central-limit theorem is appropriate.
The characteristic function for the random variable k, is (deleting the
subscript /)
0(a)= fdkeik»P(k) C.69)
where P(k) is single-particle probability density. Expanding 0(a) about a = 0
gives
i
0(a) = 1 + f(k) • a - -(kk) : 55 + C.70)
from which we obtain10
1
-a- -(kk- (k)(k)) :aa + C.71)
If the fluid is at rest, then (k) = 0, whereas if the fluid is isotropic, we may
set
<A:O C.72)
which gives the consistent statement
Tr<kk}=Tr-(£2O= {k2)
Substituting these results into C.71) gives
--(k2)a2 C.73)
6
Comparing this finding with A.8.7) gives
£(k) = 0
1 , C-74)
D(k)=-{k2)
10i
Let 0 = 1 + t- Then, with \jr < 1, ln0 = ln(l + V0 = t -

3.3 Fluid Dynamic Equations and the Boltzmann H Theorem 173
With these results at hand, we turn to the central-limit theorem. The gen-
generalization of A.8.29b) to three dimensions for the present problem is given
by
'N) = {inNm)wexp i
~[K-N£(k)]
2ND(k)
Substitution of our findings C.74) into the preceding gives
2 1
C.75)
P(K, N) = — exp
[Bn /3)N <*)F P
2
iN(k2}
3
K
C.76)
With the equipartition theorem, we write
(k2) = 3mkBT C.77)
With M = Nm representing total mass of the fluid, substitution of C.77) into
C.76) gives the desired result:
1 / K2 \
P(K) = — exp ( I C.78)
V } BnMkBTfl2 y\ 2MkBT) }
This finding may be written in a more concise form in terms of the variable
k2 = MkBT
which permits C.78) to be written
\K2/22 C.78a)
(Ittk
With reference to A.8.29), we see that the latter distribution is Gaussian (in
three dimensions) with normalization
/I /»OO
P(K)dK = ——— / 4nK2dKe-K2/2Kl = 1
B7T/C2K/2 Jo
and variance
a2 = k2 C.79)
(which has dimensions of squared momentum).
Consider that the fluid has a mass of 1 g and is at room temperature C00
K). We find
cr 2^2 x 10~7 g-cm/s
Referring to A.8.38), the preceding result indicates that there is a 70% prob-
probability that the center-of-mass speed (K/M) of the fluid is less than 2 x 10~7
cm/s. Furthermore, with the asymptotic relation A.8.39) we find the probability
that this period is in excess of 1 cm/s is given by
Ye ( 1 \
P(K/M > lcm/s^A/ — 1 - — + ...« 1 C.80)
71 A \ Az

174 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
where X — 0.5 x 107. Thus there is an immeasurably small probability that
the fluid surges in a given direction with a macroscopic speed.
Statistical and dynamical pictures
The preceding example indicates that there is a finite probability that measure-
measurement finds K / 0. This is an evident contradiction of the law of conservation
of momentum as the system was assumed to be at rest. This situation arises
as a result of the probabilistic nature of the analysis. Molecular momenta are
viewed as random variables with given expectations and variances. This is
the statistical approach. In the dynamical picture, we calculate momenta as a
function of time from dynamical equations. In this representation, there is no
uncertainty in the outcome of measurement, and momentum is conserved.
3.4 Transport Coefficients
3.4.1 Response to Gradient Perturbations
Consider that a fluid is in equilibrium (or steady state) at given uniform values of
n, u,and T. Gradients of these variables develop under perturbation. The fluid
responds to these gradients in a manner to restore equilibrium (Le Chatelier's
principle). Thus, with TZ written for "response" we may write
= mi D.1a)
D.1b)
D.1c)
That is, fluid motion will develop in response to gradients in density; compo-
components of the strain tensor will develop in response to gradients in fluid velocity;
heat flow will develop in response to gradients in temperature, and currents
will develop in response to a gradient in electric potential.
The coefficients that relate gradients of the perturbation to response motions
are called transport coefficients, which are defined as follows.
Diffusion coefficient
This coefficient stems from the response relation D.1a). It reads
nn = -DVn D.2)
The specific configuration that this equation describes is as follows. Consider
a two-component gas. Let us call atoms of one component, 1-particles and
atoms of the other component, 2-particles. These components have respective
densities n{ and n2. In equilibrium, species 1 and 2 are homogeneously in-
intersperse. Let the gas suffer a perturbation in density subject to the constraint

3.4 Transport Coefficients 175
nx + ri2 — constant. Then velocities will develop in the gas to restore the
homogeneous state in accord with D.2).
In a one-component gas, one may conceptually label a certain subset of
particles with 1 and the complement, 2. The resulting motion of the con-
conceptually labeled subgroup of particles is called self-diffusion. Diffusion in a
two-component system is called mutual diffusion.
Thermal conductivity
This coefficient stems from D.1c) and we write
Q = -kVT D.3)
In the event that internal energy is proportional to T, D.3) indicates that heat
flow will emerge in response to gradients in internal energy.
Coefficient of viscosity
It is conventional to partition the pressure tensor in the following manner:
~P = 7p - 5 D.4)
In this expression, p denotes scalar pressure and S represents the component
of P that emerges in response to gradients in velocity. The following two
properties are assumed for S:
1. S contains no terms other than Vu terms since S = 0 if Vu = 0.
2. S = 0 if the fluid is in a state of uniform rotation. Rigid-body rotation of
a fluid is characterized by a constant rotational frequency vector, f2, such
that the macroscopic motion of an element of fluid at r is
u = f2 x r
Thus, property 2 states that S = 0ifu:=f2xr. This property is satisfied
by the tensor
where a and b are arbitrary constants. With this form at hand, we write
duk 2 3m,
5
OXic OXi 3 OX[ ) OX[
D.5)
In this expression, rj is the coefficient of shear viscosity and f is the coefficient
of bulk viscosity. Recall that a fluid is incompressible, providing V • u = 0.

176 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
The expression D.5) may be written in terms of the symmetric strain tensor:
1 /3m,
2
A^V-u D.6b)
This permits D.4) to be written in vector form:
T = 7p- 2r](K - |7V • u) - £7V • u D.7)
Note that this form has the property
Tr^P = 3p - 2r?[(Tr A) - V . u] - 3£ V . u = 3p - 3£ V . u
For an incompressible fluid,
The term in D.7) containing the coefficient of shear viscosity rj should, by
arguments surrounding D.1), enter as a response force that tends to diminish
gradients in fluids velocity. Let us consider a simple example that illustrates
this property.
The force on a fluid element stems from the pressure tensor P. For an
incompressible fluid, D.7) reduces to
P = lp-2rjA
Consider further that the fluid is in a state of shear given by the fluid velocity
The force on an infinitesimally thin slab of fluid of area Ax Ay that lies normal
to the z axis at a specific value of z is given by
77 n a a du* a a
Fx = PX7AxAy = — 77 AxAy
We see that the force Fx is opposite to the gradient of ux. It slows down the
slab, thereby diminishing the gradient in ux. A molecular mechanism for this
viscous, frictional force is described in Section 4.2.
Momentum equation
An important consequence of transport coefficients is that their defining rela-
relations serve as additional equations that contribute to closing the conservation
11 Multiplying A by At gives the symmetric strain tensor relevant to solid-state
physics. For further discussion, see R. P. Feynman, R. B. Leighton, and M. Sands,
Feynmann Lectures on Physics, Vol. II, Addison-Wesley, Reading, Mass. A964),
Chapters 38 and 39. In general, strain is the deformation of a medium that occurs in
response to stress.

3.4 Transport Coefficients 177
equations. Thus, for example, with D.7) inserted into the momentum equation
C.30), we find
P
du
(u • V)u
+ Vp - r?V2u - (^ + |) V(V • u) - 0 D.8)
This vector equation represents three equations in five unknowns. The conti-
continuity equation that relates p and u together with a remaining scalar equation
in these variables closes the system. With £ = 0, D.8) is part of a system of
fluid dynamic equations called the Navier-Stokes equations, which are more
fully described later in this chapter.
Electrical conductivity and mobility
We recall Ohm's law,
J - oc£ D.9)
where ac is conductivity. The mobility coefficient /x is defined through the
equation
u=(v) = /z£ D.10)
Drude model
The Drude model permits a simple relation between a and \x to be found. We
adopt the following Langevin equation:
m(\) — e£ — vm(\) D.11)
where the second term on the right side represents a friction term. In
equilibrium, (v) = 0 and D.11) reduces to12
D.12)
e
vm
and
e2n
J = enu= —£ D.13)
vm
e2n
a2 = = en/ji D.14)
vm
Values of these parameters for some typical metals are listed in Table 5.1 in
Chapter 5.
12
This topic is returned to in Section 3.7 of the present chapter and in Chapter 5.

178 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Electrochemical potential
Suppose a gradient in density is present in an aggregate of charge, in addition
to an electric field. In this event, two driving forces acton the charges, one due
to electric force and the other due to diffusion. To describe this situation, the
Langevin equation D.11) is generalized to read
m(\) — ee — vmiy) + (FD)
An expression for the diffusion force (FD) is obtained from D.2). Thus
V/i
(FD) = —vm{x)D = vmD
n
Combining the latter two expressions, together with D.13), in steady state we
find13
J=— e + eDVn D.15)
vm
Introducing the electric potential
g — —
and recalling D.14), we obtain
J = — V(T/fDe<I>) D.15a)
e
where
e2Dr]
Gc
^D
<7
and \frD — e<$> is called the electrochemical potential. Note, in particular, that
if t/>o = e<&+ constant, electric force is balanced by diffusion force and there
is no current flow. The notion of electrochemical potential plays an important
role in the theory of the diffusion of ions through semipermeable membranes.14
3.4.2 Elementary Mean-Free-Path Estimates
Basic notions
This formulation of transport coefficients rests on two key concepts. First,
consider a small volume of a fluid in equilibrium with number density of
I3The relation D.15) is often employed in the study of charge carriers in a semi-
semiconductor where it is termed the drift-diffusion equation. For further discussion, see
S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer Verlag,
New York A984).
I4See A. Katchalsky and P. Curran, Non-Equilibrium Thermodynamics in
Biophysics, Harvard University Press, Cambridge, Mass. A965).

3.4 Transport Coefficients 179
1 cm'
t
in
>c
FIGURE 3.17. A flux of nC/6 particles passes through the upper face per second.
particles, n. We introduce the mean speed C,15
(v2) = C2
For point particles, equipartition of energy gives the relation16
mC2 = 3kBT
If particles have the mean speed C, what is the mean flux of particles, F
(number/cm2-s) that move in any of six Cartesian directions at any instant of
time? To answer this question consider a cylinder of height C and unit cross-
sectional area (see Fig. 3.17). Since particles have the mean speed C, on the
average, one-sixth of the particles in the cylinder will pass through the upper
surface per second. If we call this direction the z direction, we have
\»c
D.16)
The second concept that comes into play in this analysis is the mean free
path, Z. It is assumed that the only means for transport of information in the
fluid is via collisions. Thus, for example, equilibration of a gradient in n will
be made through collisions that carry particles from domains of large n to
domains of small n.
Another relation important to transport phenomena relates total scattering
cross section17 a to particle number density and mean free path. It appears as
nol
D.17)
I5More accurately, C here represents the rms molecular speed. In other parts of
the text, for purposes of writing the Maxwellian in concise form, we set mC2 = kBT
[see C.61a) and Problem 3.57].
I6Each degree of freedom of a fluid in equilibrium has kB T/2 units of energy.
I7Previously labeled aT. See A.25). Recall also that ac denotes electrical
conductivity.

180 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
On the average,
one particle
occupies
this volume.^
Area =o
FIGURE 3.18.
n(z + I)
>l
z- I
>l
FIGURE 3.19. A particle that has a collision at z — I has no further collisions until
it reaches z.
The argument that supports this relation is that, on the average, one particle
will be found in the volume la (see Fig. 3.18).
Self-diffusion
Here we are speaking of the self-diffusion of a conceptually labeled subgroup
of particles of a single species. Let the density of such particles be n, and let
there be a gradient of n of these particles in the z direction (see Fig. 3.19).
The flux of particles in the -\-z direction that crosses the z-plane is equal to
the number that reach the z plane from (z — /), minus those that reach it from
(z + /). That is,

3.4 Transport Coefficients 181
With D.16), we have
1
6
21C
6
nuz 2
l(z —
~n(z
1
3
I) - n(z + /)]
-l)-n(z + l)'
icdn
dz
lay conclude
D +
Uc
3
D.18)
Mutual diffusion
We consider two distinct gases with respective densities nx and n2. The gas
occupies a cylindrical volume with unit cross section. Labeling the midplane
of the cylinder z = 0, the system is initially constrained so that particles 1
occupy the domain z > 0 and particles 2 the domain z < 0. This constraint
is removed, and the two species diffuse across the midplane subject to the
previously mentioned conditions:
nx + n2 = n — constant
D.19)
Thus we require that the net flux of particles 1 across the midplane be balanced
by the flux of particles 2 across the midplane.
nxuu
= n2u2
= —r
D.20)
Signs in this equation denote direction of flow. With this constraint and the
defining equaltities
dri\ dn2 ,A^^
T D1 D.21a)
r, T2 D21T
dz dz
we find that the mutual diffusion coefficients are equal:
Dl2 = D2X
D.21b)
Furthermore, in the limit that the two species become identical, Dx2 must
reduce to the coefficient of self-diffusion D.18).
Let us consider the force exerted on gas 1 due to transfer of momentum
between the two components.18 First, let MX2 denote average z momentum
transferred from gas 1 to gas 2 per unit volume per unit time due to collisions.
I8This derivation follows that of J. C. Maxwell, Phil Trans. Roy. Soc. 157, 49
A867). An account of this approach may also be found in R. D. Present, Kinetic
Theory of Gases, McGraw-Hill, New York A958).

182 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
With the law of partial pressures for ideal gases (that is, px — n\kB T), we may
write this force as
dpi = kBT dnx = -Mndz D.22)
We will calculate MX2 for the case of rigid spheres. As was shown in Section 3.1,
the scattering of rigid spheres in the center-of-mass frame is isotropic. So the
average velocity of a molecule in the lab frame after collision is equal to the
velocity of the center of mass. We may conclude that the average decrement
in momentum suffered by a type 1 molecule per collision is
- uc) — fM(u\ - u2) D.23)
where /x is reduced mass and uc is the speed of the center of mass.19
Let o represent the total scattering cross section for the interaction of these
two types of particles. Then the related collision frequency per unit volume is
given by
vn = mn2crCi2 D.24)
where
9 3kBT
C2n = —— D.25)
is an effective two-particle thermal speed. Thus the average rate of momentum
transfer per unit volume is
M12 = n
where
Recalling the equality D.20) permits D.26) to be rewritten as
D.27)
where n = nx + n2 is total particle density. Inserting D.22) into the latter
equation gives
dn\
kBT—- = -M12 = -wri/fi2
whence
Ir — I si v% si m
D.28)
Dn
dz dz
I9Recall A.2); R = uc = (mlul +m2u2)/(ml + m2).

3.4 Transport Coefficients 183
This gives the desired symmetric expression
= (WWf
The cross section for scattering of rigid spheres is given by A.34):
= ncr?2
= n I J = ncr?2 D.30)
where aOi and aO2 are respective particle diameters. The result D.29) then
becomes
= ' 2 D.31)
which is seen to be independent of the proportions of the mixture. In the limit
that the two gases become identical, a\2 —> ^o* the diameter of a sphere, n,
becomes the particle density of a single species (with n\ — n2 = n/2), and
kBT 2kBT _ 2 2
3/x 3m 9
Thus, with D.18) we may write
in the said limit. The approximate form of the equality reflects the rough
estimates made in obtaining these coefficients.
As will be found later in the text, a more accurate form of the mutual diffusion
coefficient compared to the form D.31) is given by
2D.31a)
8
Setting /x = m/2 and on — g0 gives the related self-diffusion coefficient.
Viscosity
We consider a spatially uniform fluid in a state of shear such that
The fluid velocity that is only in the x direction varies only in the z direction
(see Fig. 3.20). Note first that for this configuration D.7) (for an incompressible
fluid) reduces to
dUx
Pxz - Sxz = -r,-± D.32)
dz
Each particle at z — I that suffers a collision and moves in the +z direction, on
the average, carries an x component of momentum from the region at z — I,
that is, mux(z — I). The flux of particles carrying this momentum, again, is

184 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
i
u(z)
7
7
7
>l
Y ^Pxt acts
' on this surface
z- I
FIGURE 3.20. The state of shear in a fluid.
>l
given by D.16). Thus, the mean x component of momentum flux across the -
plane due to transport of particles in the z direction is the difference between
the gain term
1
r~J~ = -nCmux(z -I)
and the loss term
= -nCmux(z + /)
6
The net force per area on the z plane (in the x direction) is
20
xz
xz
xz
= rt -
= \pC2l
O
2/
3
du
Comparing with D.32) gives Maxwell's classic estimate:
n - \
D.33)
With D.17), we see that our expression for r\ is independent of fluid density.
20This force is akin to the retardation of a moving train due to loading of heavy
parcels from rest onto the train.

3.4 Transport Coefficients 185
Thermal conductivity
Continuing in this manner, and assuming a gradient in energy per particle
e/c(z), we find (with TQ — \nueK)
1 deK
Q7 = —nCl
z 3 dz
With
_
Cv ~
written for the specific heat per particle, the preceding equation becomes
1 dT
Qz = -^Cl
3
^nClcv
3 dz
With D.3), we may then write
k = -nClcv D.34)
Setting p — mn and comparing with D.33) gives the relation
- = - = cv D.35)
rj m
A compilation of the preceding expressions is given in Table 3.1.
Kinetic-theory parameters
The preceding relations suggest that certain ratios of transport coefficients are
constants. Thus we find
^ ~ 1 D.36)
K
A closely related parameter, called the Prandtl number, is given by
K D.37a)
K K Cy
Here y is written for the ratio of specific heats, cp/cv, which for monatonic
ideal gases has the value |.
Another closely allied parameter in kinetic theory is the Schmidt number,
C EE 4t - 1 D'37b)
pD
Values of rj, k, and the ratio k/tjcv for some common gases are listed in
Table 3.2. Values of V and C are listed in Table 3.3. In Table 3.2 we see that
the ratio k/t]cv is closer to 2.5 than 1 for monatomic gases. As we will find
in the following section, a more detailed kinetic analysis gives the value |

186 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
TABLE 3.1. Elementary Mean-Free-Path Expressions of Transport Coefficients
Coefficient
Self-diffusion
Mutual-diffusion
Viscosity
Heat conductivity
Ratio
Electrical conductivity
Mobility
Ratio
D =
D,,.
» =
K =
K
V* =
V*
D ~
Expressions
Uc
3
(kBT/3iiI/2
no
-3pci
-nClcy
3
Cy
= Cy
m
e2n
vm
e
vm
e ...
- ^t in. cinrcai n
TABLE 3.2. Observed Values of Viscosity and Thermal Conductivity
X] X 107 K X 103,
He
H2
CO2
CH4
g/cm-s
1875
840
1377
1027
cal/cm-s-K
0.344
0.416
0.034
0.072
K/T]Cy
2.44
2.06
1.64
1.75
4
2.49
1.92
1.70
1.70
for this ratio.21 The last column of Table 3.2 suggests that a uniformly good
approximation for this ratio is the value (9y — 5)/4.22 In Table 3.3 we see that
V and L lie close to unity for all materials listed. A more detailed kinetic study
gives the values V = f = 0.67 and C = | = 0.83.22
3.4.3 Diffusion and Random Walk
In Section 1.7.5, we found that, relevant to the displacement I in n steps of
a random walk, (Z2) = n. Equivalently, with displacement written x and n
replaced by time t, (x2) = t. We wish now to show the relation between this
result and diffusion.
21 See Table 3.4.
22For further discussion see A. Isihara, Statistical Physics, Academic Press, New
York A971).

3.4 Transport Coefficients 187
TABLE 3.3. Observed Values of the Schmidt and Prandtl Numbers
C V
Ne 0.73 0.66
Ar 0.75 0.67
N2 0.74 0.71
CH4 0.70 0.74
O2 0.74 0.72
CO2 0.71 0.75
H2 0.73 0.71
Combining D.2) with the continuity equation C.14) gives the diffusion
(equation
— = DV2n D.38)
Consider that, at time t — 0,
/i(r, 0) - AWr) D.39)
The solution to D.38) corresponding to this intial value is the Gaussian
distribution [recall A.8.34)]
N,
n(r't} = (An DtW £XP ~'~' D-40)
Note that this distribution maintains the normalization
/»OO
/ dr47tr2n(r,t) = No D.41)
Jo
We may identify
47rr2ft(r, t)dr ,A A^
= P(r,t)dr D.42)
the probability of finding a particle in the shell dr about r at the time t. This
probability density permits calculation of the mean square displacement
2 C°° 7 f°° 4nre
(r2) = / P(r,t)r2dr= / — dr = 6Dt
Jo Jo DnDtf2 D.43)
(r2) = 6Dt
which is relevant to a three-dimensional random walk23 and is seen to maintain
the fundamental relation (r2) a t, with 6D playing the role of a proportionality
constant. This result was found previously A.8.31) employing the central-limit
theorem.
23Random walk in various dimensions is discussed by R. L. Liboff, Phys. Rev.
^7,222A966).

188 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
3.4.4 Autocorrelation Functions, Transport Coefficients, and
Kubo Formula
Self-diffusion
In this section we present a brief description of an alternative approach to the
evaluation of transport coefficients. We begin with the diffusion coefficient.24
Our starting equation is D.38).
With the Fourier transform
n(k,r)= I dreik'rn(r,t) D.44)
we find that the transform of D.38) is given by
2Dh
= -k2Dh D.45)
dt
which has the solution
n(k, t) = exp(-k2Dt) D.46)
We wish to recapture the result D.43). From D.46), we find
d2h
dk2
k=0
= -2Dt D.47)
whereas differentiation of D.44) gives [with n(r, t) isotropic]
d2h
3 Jo
dk2 *=° 3
Assuming that
4nr2n(r, t)dr =
4nrn(r,t)dr D.48)
again permits us to interpret w(r, t) as a probability density. Thus D.48) gives
d2h
Jk2
-\{r\t)) D.49)
k=0
Equating this result to D.47) recaptures D.43):
(r2(t)) = 6Dt
If the particle begins its random walk at r0 instead of at the origin we obtain
(|r@-r0|2) = 6Dt D.50)
We wish to express this result in terms of a time correlation function.
24 An extensive review of these techniques is given by R. Zwanzig, Ann. Rev. Phys-
Chem 16, A965). Our presentation follows that of E. Helfand, Phys. Fluids 4, 681
A961).

3.4 Transport Coefficients 189
Suppose that the velocity \(t) of the particle in question is known. Then we
may write
r@ - r0
and it follows that
/ \(t')dt'
Jo
[r@ - r0]2 = f dt' f dt' f dt"\(t")• v(r')
Jo Jo Jo
Taking the ensemble average of both sides of this equation gives
<[r@ - ro]2> = I dt' f dt"(v(t") - v(t')) D.51)
Jo Jo
Assuming the process to be homogeneous in time, we write
(v(r') • v(r")) - <v(r' - t") • v@)) D.52)
itroducing the variable t = t' — t" and T = t' + f" (see Fig. 2.12) gives the
itegral equality
f dt' f dt"g(t' - t") = 2t f (l--) g(t)dt
Jo Jo Jo \ t/
phis relation permits D.51) to be written as
([r@ - VoY) = 2t 1 - - (v@) • v@) dt
whence, with D.50),
6D = 2 I A - - ) (v@) • y(t))dt
assuming rapid decay of the time correlation (v@) • v(?)> permits the
receeding to be written (with i replaced by t) as
1 f
-
3 Jo
00
<v@) • v@) dt D.53)
Which is the desired result, expressing D in terms of the velocity autocorrelation
^function (see Section 1.8.8 and Problem 3.63 addressing "long-time tails").
f Recalling the Einstein relation in Table 3.1, which relates mobility and
^diffusion coefficients, the preceding equation permits us to write25
L
oo
(v@) • v@) dt D.53a)
25 As noted in Section 1.8.4, fluctuation in the number of particles is termed shot
noise. Fluctuation in velocities, such as included in D.53), is termed Johnson noise
(named for J. B. Johnson).

190 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
As a simple application of D.53), consider that
v@ = v@)e-'/T
where r is a relaxation time. Substituting this result into D.53) gives
5/
j Jo
00
Taking the integral inside the ensemble average gives
D = -(rv2@))
Assuming that the relaxation time, r, is independent of the starting velocity,
we find
in agreement with Maxwell's result D.18). Proceeding in a manner similar
to that in obtaining D.53), time-correlation expressions for other transport
coefficients, such as viscosity and thermal conductivity, may also be so
expressed.26
Linear response theory
The preceding development may be generalized to arbitrary Hamiltonian sys-
systems in the following manner. Let A(q, p) be a dynamical function that is not
explicitly dependent on time. With A.1.25), we write
dA - A _,
— - -[//, A] = -LA D.54)
at
whose solution may be written
A(t) = e~tLA@) D.55)
The Liouville operator, L, was introduced previously in the Liouville equation
B.1.3), whose solution in turn may be written
f(t) = elLf@) D.56)
The expectation of A at the time t is given by
(A)= f A@)f(t)dqdp D.57)
26See, for example, D. A. McQuarrie, Statistical Mechanics, Harper & Row, New
York A973).

3.4 Transport Coefficients 191
pHere we have introduced the notation
A@) = A(q,p) D.57a)
iWith D.56), D.57) may be written
(A) = j A@)(etLf@))dqdp D.58)
Consider that the Hamiltonian, //0, of a given system is perturbed so that
H = Ho - sHF(t) D.59)
fwhere s is a parameter of smallness, H(q, p) is time independent, and F(t) is
independent of q and p. The distribution is likewise perturbed and we write
/ = /@) + eA/ D.59a)
Substituting the latter two relations into the Liouville equation gives the
following 0(£) equation:
^ - L0Af - [f°\ H]F(t) = 0 D.60)
ot
(where Lo is the Liouville operator corresponding to Ho. Operating on D.60)
Swith exp — tL0 gives
3
dt
Integrating, we find
Af = f e{'-'')Lo[fm,H]F(t')dt' D.61)
Jo
The average of a response 5 to the perturbation is then given by
(B) = j BAf(t)dqdp
fwhere we have assumed that E) = 0 in equilibrium and have dropped the
jbookkeeping parameter s. Substituting D.61) into this relation gives [in the
notation of D.57)]
E) = f 5@) f e(t-t/)Lo[f(o\ H]F(t')dtfdqdp D.62)
or, equivalently,
E@) = / <t>(t - t')F(t')dtf D.63)
where the function 0 is as implied. As D.63) is a convolution integral, the
Fourier transform gives the following widely employed result:
(B(co)) = <Kco)F((o) D.64)

192 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
The term F(a>) is the transform of the perturbing force, B(co) is the transform
of the response to the perturbation, and 0(&>) plays the role of the transform of
the related transport coefficient.
A concise form of 0(r — t') = 0@ is obtained as follows. With D.62) and
D.63), we write (with t replaced by t)
0@- I B(O)etL°[f°\H]dqdp
A single parts integration gives, with D.55),
0@- J B(t)[f@\H]dqdp D.65)
Again integrating by parts, we find
0@- j[H,B(t)]f@)dqdp
or, equivalently,
0@ - <[//, B(t)]H D.66)
which is the desired result. The zero subscript reminds us that the average is
calculated with the equilibirum distribution, /@).
The quantum analog of D.66) is given by
TT{p[H,B(t)]} D.67)
in
where p is the density matrix and Tr denotes the trace.27 The preceding relation
is often referred to as Kubo's formula. 28
Electrical mobility
Let us apply D.64) in the calculation of the Fourier transform of electrical
mobility. We identify the electric field, e£, as the perturbing force, and particle
velocity v is the response. We will make our calculation with respect to the
Cartesian coordinate jc/. In the notation of D.59), we then identify
H — Xi, Fi — s£i
and the response B = Vj. With these relations, D.64) gives
(vj(co)) = tijiicotfiico) D.68)
where /zy; is the mobility tensor:
= e<f>ji(a)) D.69)
27These quantum concepts are fully developed in Section 5.2.
28R. Kubo, J. Phys. Soc. Japan 12, 570 A957). For further discussion, see R.
Kubo, Statistical Mechanics, Wiley, New York A965), Chapter 6.

3.4 Transport Coefficients 193
The Fourier transform 0(&>) is obtained from D.66). That is, with our
identifications,
j j D.70)
Again assuming a simple relaxation model, we write
where r is a constant relaxation time. Thus D.70) becomes
[where, in the spirit of D.57a), we have set v@) = v. The preceding equation
[gives
(Pjiit) = ^e-"x D.71)
m
Introducing the Fourier transform
= /""
|and inserting D.71), we obtain
D.72)
m
|which is the desired result. In the DC limit, we set co = 0 to obtain
= Vr
m
{Inserting this result into D.69), we find
Iwhich with D.68) gives
(V):
ex
ex
m
e
D.73)
m
These latter two relations agree with our previous mean-free-path estimates
listed in Table 3.1.
With these elementary notions of transport coefficients behind us, we turn
to one of the more powerful techniques of solving the Boltzmann equation.
As will be demonstrated, an integral component of this method of solution
addresses the calculation of transport coefficients.

194 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
3.5 The Chapman-Enskog Expansion
3.5.1 Collision Frequency
The collision integral of the Boltzmann equation B.14) may be written
J(F \F) = -F if crdQgdylF(yl)+ \\ a dQgd\{FfF[ E.1)
It is evident from Fig. 3.10 that the quantity
v(v)= I! odtlgdvxF(vx) E.2)
represents a collision frequency.29
Let us rewrite the Boltzmann equation as
df rr
— = // GdQgdvx(FfF[ -F- FFX) = IF E.3)
which serves to define the collision-integral operator /. With E.2), we see that
/ has dimensions of frequency s~ . Thus we may write
/ = vol E.4)
where I is nondimensional and t>0 has been written for a constant with
dimensions s~l. In this notation, E.3) appears as
DF ^
IF E-5)
The Chapman-Enskog expansion is relevant to the domain of large collision
frequency. With C taken as thermal speed, we write
C 2^lv E.6a)
or, equivalently, with D.17),
v 2^ ncrC E.6b)
The relation E.6a) indicates that large collision frequency is equivalent to
small mean free path, whereas E.6b) indicates that such extremes are attained
in the limit that the product noC grows large.
29Discussion of this parameter is returned to in Section 4.2.1 in derivation °f
the Krook-Bhatnager-Gross equation and in Section 6.4 relevant to the distinction
between hard soft potentials.

3.5 The Chapman-Enskog Expansion 195
3.5.2 The Expansion
30
The first step in the Chapman-Enskog expansion is to write the Boltzmann
equation as
a i
V)F = -IF E.7)
dt
where the dimensionless parameter s <^C 1. This parameter may be thought of
£s a bookkeeping parameter in that it is eventually set equal to 1. With E.5),
we see that the form E.7) is equivalent to stipulating that the gas is dominated
by large collision frequency. The parameter V in E.7) is given by
d d
V = v • — + K • — E.7a)
d d\
Step 2 in the Chapman-Enskog procedure is to introduce the expansion
.. E.8)
The normalization of F follows A.6.14a). Furthermore, F satisfies the moment
relations C.12), C.13) and C.32):
r r 3 r me2
w= I Fd\, nu= / F\d\, -nkBT = / ~Y~Fd\ E.9)
Step 3 in the Chapman-Erskog expansion stipulates that the variables
(n, u, T) are all 0A) quantities and stem from F@), whereas terms in the series
|5.8) corresponding to F(/), / > 0, contribute to the higher moments Q and P.
us we write
F@) v rfv= nxx E.10a)
F(" v \ d\= 0 , i >0 E.10b)
Q = £VQ<'> = - £V / F(l)cmc2dc E.10c)
Fil)mccdc E.10d)
30This technique was developed in a series of independent papers by S. Chapman
and D. Enskog over the second decade of this century. See references. This method
.of solution is also described in S. Chapman and T. G. Cowling, The Mathematical
Theory of Non-Uniform Gases, 3rd ed., Cambridge University Press, New York
A970).

196 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
* ■*•
Expansion of V and J
Substitution of the series E.8) into VF gives
VF = VF@) + sVF(l) +
For the collision integral, we obtain
E.11)
oo oo
n=0
1=0 n=0
It proves convenient to introduce the ordered operator
)W(F(O),
n I
(n+l=s
The expansion E.12) may then be written
E.12)
e2yB)(F@), FA), FB))
E.14)
Thus, for example,
E.14a)
Expansion of the time derivative
The fourth step in the Chapman-En skog expansion addresses the time deriva-
derivative in E.7). It stipulates that the time dependence of F is solely dependent on
the hydrodynamic variables n, u, I so that
dF dFdn dF 3u dF dT
— = 1 1
dt dn dn du dt dT dt
E.15)
The time derivatice is expanded as
dt dt
dt
dt
The physical meaning of this expansion is that lowest-order terms vary most
rapidly, whereas higher-order terms are more slowly varying. Explicit ex-
expressions for the time derivations in E.15) follow from ^-ordering of the
conservation equations C.14), C.30), and C.31). We obtain [recall C.31) for

3.5 The Chapman-Enskog Expansion 197
notation]
don
~dt
drn
~dt
= —V • nu,
= 0, r > 0
i
= _(u . V)u + K V • P
(u )u +
dt p
E.17)
rU -1 =(r) V J
— = —V- P , r > 0
3
5 Vr
uVr[P : Vu~ + V.Q@)]
dt 3 /
r>0
When written in terms of the symmetric strain tensor A D.6a), the last two
equalities in E.17) appear as
\ — [V-Q(
" B E.17a)
A : p" ], r > 0
Substituting the preceding expansions for F, dF/dt, d/dt, VF, and 7(F | F)
into the Boltzmann equation E.7) gives
+ [7@)(F@)) + ^7A)(F@), FA)) + • • •] E.18)
where, with E.16), we write
df(r) df(r)dn Q fir) gu 9 f(r) Qj
= 1 • 1 E.18a)
dt dn dt 3u dt dT dt
Equating coefficients of equal powers of s gives the desired series of coupled
integral equatins for F(r):
E.19a)
E.19b)

198 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
The first-order solution is obtained by solving E.19a):
J(F@) | F@)) - 0 E.20)
First-order solution
Recalling our discussion of the Boltzmann H theorem (Section 3.4), the
solution to E.20) is the local Maxwellian F° C.58):
F" - F ° =
In this manner we find that the first-order solution in the small mean-free-path
approximation is the local Maxwellian F°. This solution may be used to obtain
a first-order set of hydrodynamical equations for the purpose of evaluating n,
u, and T. To construct these, we first calculate the heat conductivity Q and the
stress P. From previous definitions C.24) and C.25), we write
I
P — m I ccF dc
Substituting F° into these two formulas gives
Q@) = 0 E.22a)
= lnkBT E.22b)
To lowest order, there is no heat flow and the pressure tensor is diagonal.
Substituting these values into the conservation equatins C.14), C.30), and
C.31) gives the first-order approximation to these equations, which are called
the Euler equations:
dn
+ V • nxx =0 E.23a)
u + Vp = pK E.23b)
E.23c)
The adiabatic quality of flow E.23c) in this approximation is seen to be con-
consistent with the property that there is no heat flow E.22a). Equations E.21)
to E.23) constitute the first-order solutions to the Boltzmann equation in
the Chapman-Enskog procedure. Solving the Euler equations E.23) gives
n = n(x, t), u = u(x, t), and T = 7\x, t), which when substituted in E.21)
completely determines F°(x, v, t).
Each successive iterate in the Chapman-Enskog expansion yields a more
detailed set of hydrodynamic equations, better suited to higher-order spatial

3.5 The Chapman-Enskog Expansion 199
fluctuations in the fluid. As noted above, the first iterate gives the Euler equa-
equations E.23). Equations stemming from higher-order approxiamtions are named
as follows. The second iterate gives the Navier-Stokes equations [of which D.8)
is the momentum equation]. The third iterate gives the Burnett equations.
3.5.3 Second-Order Solution
The second-order solution is obtained from E.19b):
dt
We introduce the function
E.24)
E.25 a)
and [with E.14a)] the D operator
DO =
= — \J(F°
F°
o d£lg
The starting equation E.24) may then be written
1 /3o
'0
E.25b)
f° \dt
+ V F° = DO
E.26)
Note in particular that D is a linear operator. Terms on the left of E.26) give
dt
n dt
*
dt
dF°
v • = v
dx
Idn
n
3u
dx
2 7 T dt
3\ idT
2 / t dx
E.27)
where31
2RT C2
E.28)
and ^ • 3u/3x is summed over parallel components of £ and u. Replacing time
derivatives in E.27) by related expressions in E.17) gives (with K = 0)
E.29)
£2 - - £ . V In 7
31 In most of the text, mC2 = kBT (save for Section 4.2, where C was used to
denote rms molecular speed). To avoid confusion here, we set mC2 = 2kBT.

200 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
which is seen to be a linear inhomogeneous integral equation for the
distribution <£>.
If this equation is solved for O, then F is known to second order:
F = F°[1 + *] E.30)
The general solution to E.29) is a linear combination of homogeneous, O/?
and inhomogeneous, 4>/, solutions, where
D**=0 E.31)
and 4>/ is a particular solution of E.29).
From the structure of D, it is evident that <t>h is any linear combination of
the three summational invariants.
1 9
<&h = of + C • me + -yra<r E.32)
where a?, /3, y are arbitrary constants. To arrive at the particular solution to
E.29), we note that the left side is in the form
LHS E.29) = X(§) • BRT)l/2V In T + Y(O : Vu E.33)
Since D is a linear operator and O is a scalar, E.33) suggests that we take the
particular solution to E.29) to have the form32
<&,. = A(O • BRT)l/2V In T + 2B(£) : Vu E.34)
Thus, to find the inhomogeneous solution <$>t, we must obtain the vector func-
function A and the tensor function B. Inserting this form into E.29) and equating
coefficients of the different components of V In T and Vu gives the following
equations for A and B :
DA - £ U2 - -\ E.35a)
E.35b)
Note that the only variables in A are £, n, and T. The only vector that can
be formed from these elements is £ itself. Thus we write
A =' bsA(^2)i E.36)
where A is some scalar function.
The linearity of E.35b) and the form of its inhomogeneous term imply that
B is a symmetric, traceless tensor. Again B depends only on £, n, and T. The
32
^. ■ ■ ...
Note that the dimension of □ is frequency and that of A and B is time.

3.5 The Chapman-Enskog Expansion 201
only symmetric traceless tensor that can be constructed from these variables
B
The scalar functions A and B satisfy the integral equations
E.36a)
CK64) = iIt2- -
E.37a)
E.37b)
Returning to the homogeneous solution E.30), we note that the constants a,
f3, and y contained in <£>h are determined by the constraint conditions E.10b).
Inserting
F^ = F°\<t>i, 4- 4>-l E 38)
into these constraint equations yields the three integral conditions (see
Appendix B, Section B.I):
E.39a)
E.39b)
E.39c)
-me2 I dc = 0
In 7 + mC]mc2 dc = 0
I a H—me y I —me dc = 0
V 2I
Equations E.39a) and E.39c) imply that
Of = y — 0
E.40)
whereas E.39b) implies that /3 is in the direction of Vln7\ so it may be
absorbed into the V In T term in 4>/.
The total solution of the Boltzmann equation to terms of second order then
appears as
F = F°[l + BRT)l/2A • V In T + IB : Vu ]
1 + BRT)x/2A^)i • VIn T +
Vu
E.41)
where .4 and ^5 are particular solutions to integral equations E.37).

202 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
3.5.4 Thermal Conductivity and Stress Tensor
With the form of the second-order solution E.41) at hand, it is possible to obtain
expressions for corresponding nonvanishing second-order contributions to Q
and P. These are obtained by inserting E.41) into the defining equations D.3)
and D.7) (with f = 0).
Q = -mBRTK/2 f $2£F dc = 0 + QA) E.42a)
T = mlRT I ]£F dc = lp + PA) E.42b)
Coefficient of thermal conductivity
Substituting E.41) into E.42), we find
mC4 =
Q - —-VlnT • F^2££A($;2)dc E.43)
(Recall that / F° dc has dimensions of number density.) With reference to
(B.B1.3), the preceding reduces to
q = V%L^T f F°$*AG2)dc E.44)
3 m J
With / F°A(%2)%2 dc = 0 [which corresponds to setting C = 0 in E.39b)],
the preceding equation may be rewritten
E.45)
3 m 7 V 2
With E.35a), we find
2klT
q = ( ihL f FoA . aAdc] Vr E.46)
\3 m J )
or, equivalently,
3 m
E.47)
Note, in particular, that the bracket symbol (|) in E.47) includes the local
Maxwellian and an inner product. The latter expression implies the following
form for the coefficient of thermal conductivity:
2klT
K =
B
3 m
(A I DA) E.48)

3.5 The Chapman-Enskog Expansion 203
Viscosity
To obtain the related expression for viscosity, we recall E.42b) and write
E.49,
With E.41), the latter equation may be written (deleting the superscript 1)
P = 4kBT / Fu#(§z)(££ • Vu)££dc E.50)
Employing the results of Problem 3.6 permits E.50) to be written
= 4k BT
5
4k BT
dc E.51)
Vu / Fu££ : B dc
With E.35b), we write
T = —— Vu" / F°B : UBdc
5 J
= 4kBT = .=
n
/
J E.52)
Note that the inner product (|) now includes the trace operation, which renders
it a scalar. Note that we have written [recall D.6a)]
= 1 =
Vu = A /TrVu
3
The shear stress component of D.7) may then be written
P = -
which with E.52) gives
2 = .=
—kBT(B | DB) E.53)
3.5.5 Sonine Polynomials
In this manner we find that transport coefficients depend on the matrix elements
of the interaction operator D. To evaluate these expressions, we work with

204 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Sonine polynomials,33 defined as follows (see Problems 3.7):
(m+n)\(-x)p
E.54a)
Some leading values are
x2
The orthogonality of these polynomials is given by
(m + n)\
L
00
n = q E.54b)
Expansions of A and B{r) are given by
00
A = J2 "rSt/WK = J2 a^r) E.55a)
r = \
00
~B = J2brS5/~2l)^2)H = J2b'W) E.55b)
r = \
Note that A(r) and B{r) are dimensionless.
First approximation for A
To find the ar coefficients, we multiply E.35a) by F°A(/) and integrate:
(A(/) | DA) - I F° U2 - -\i • A(/) dc = noLX E.56)
Note that a?/, as well as A(/), is dimensionless and n denotes number density.
Inserting E.55a) into E.56) gives
00
ar E.57)
r=\
33N.J. Sonine, Math. Ann. 16, 41 A880). First introduced in the present con-
context by D. Burnett, Proc. London Math. Soc. Ser. 2, 39, 385 A935). Subsequently
shown by C. S. Wang Chang and G. E. Uhlenbeck to be eigenfunctions of the □
operator for Maxwell molecules. Univ. Michigan Engr. Rept. CM-681 A952). This
topic is discussed in the present work in Section 6.5. We note further that, apart
from a multiplicative constant, Sonine and Laguerre polynomials are identical. La-
guerre polynomials come into play in the solution of the Schroedinger equation for
the hydrogen atom. See R. L. Liboff, Introductory Quantum Mechanics, 4th ed.,
Addison-Wesley, San Francisco, CA. B002).

3.5 The Chapman-Enskog Expansion 205
The oti are known parameters E.56) as are the A(/) functions E.55a). Thus
E.57) comprises an infinite-dimensional matrix equation for the coefficients
ar.
Varying orders of solution (within the second-order Chapman-Enskog ap-
approximation) are obtained by cutting off the A[r matrix. Thus, for example,
keeping only the leading terms in E.57) gives
not\
Substituting into E.55a), we find
A =
An
whence
2 2
n at
I \ o n at
A | DA) = a2An = l E.58)
Evaluating o?i from E.56), we find (see Problem 3.8)
A I DA) = (l-f) f E.59)
I y 4 J A
Substituting into E.48) gives the following generic form for the lowest-order
approximation to the coefficient of thermal conductivity within the second
Chapman-Enskog approximation:
25 cvkBTn2
k — E.60)
4 An
Here we have written cv for c^/mass, which for particles with no internal
structure has the value 3kB/2m. The An element contains specifics of the
particle interaction. However, a very explicit form of this term may be written
in terms of integration over scattering parameters. We obtain
An =-4>z2ftB'2) E.61)
where, in general (for study of one-component gases)
Q(Lq) _ / *_ / / e-ry2q+3(l - cos; 0)s ds dy E.62a)
V m Jo Jo
where
q2 mg2
2C2 4k B T
With A.16) and following, we recall that the scattering angle 0 = 6(s, g). Thus
E.62a) may be rewritten
m
/
Jo
e-y>y2<+3 q«) d E62b)

206 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
where <2(/) is the weighted cross section.34
/»OO
<2(/) = / A - cos1 0)sds E.62c)
Jo
Note that the dimensions of Q(l'q) are the same as \/n.
Substituting E.62a) into E.60) gives
25 cvkBT
E.63)
First approximation for B
To find the br coefficients in E.55b), we multiply E.35b) by F°S(/) and
integrate.
§ / j E.64)
D§(l)) = / F°j£ : W}dc = tip,
Introducing E.55b) gives
 E.65)
= J2 Blrbr
Again we obtain a matrix equation of infinite dimension for the coefficients
br. Keeping the leading term in E.65) gives
E.66)
With E.55b), we find
_ _ n
(B I DB) =b2Bn =
Evaluating /??, we find (see Problem 3.9)
(B\UB) = —— E.67)
4 fi
Substituting this expression into E.53) gives the following lowest-order ex-
expression for the coefficient of viscosity, within the second approximation of
34Comparison of notation for Q{1) as given by E.62c) [that is, Q(/) here ] with
that found in Chapman and Cowling (CC) and Hirshfelder, Curtis, and Bird (HCB)
is as follows: £(/) (here) = B7r)-IG(/)(HCB) = g-lQ(l)(CC). Note also that: (I)

3.5 The Chapman-Enskog Expansion 207
the Chapman-Enskog expansion.
5kBTn2
E.68)
2 Bu
Further reduction of Bu gives
Bu = -4n2QP) E.69)
and we find
5 kBT
ri = — E.70)
8 QP-^
Comparison with E.63) gives
k 5
— = ~Cy
rj 2
Note that our earlier elementary mean-free-path calculation (Table 3.2) is in
good agreement with this more detailed finding.
3.5.6 Application to Rigid Spheres
We wish to evaluate the integrals E.62) for the case of a gas of rigid spheres
of diameter a0. Recalling A.32), we write
.2
<7n
sds = — d cos 0
4
Substituting into E.62b) gives
f(l - cos' 6)sds
4 li
ro2
o}
1
2-—(-1)')
= -j-Hd) E.71a)
Note that we set /x = cos#. For Q^Lq\ we obtain
-2 /wJ, t\ V2
1)! E.71b)
4 \ m
Thus for interactions of rigid spheres we find \HB)
= 2 2 ( ^bT\
m I

208 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Substituting into E.63) gives
_ 25 cv '-'' ^xl/2
32o|
75 1
E.74)
64 <7n V run
With E.70) we find
5 1 /mkBT\l/2
71 ~ 16 a02 V n )
3.5.7 Diffusion and Electrical Conductivity
As discussed previously, mutual diffusion is a property of a two-component
medium. In this case, coupled Boltzmann equations come into play as described
in Section 3.2.2. Following a procedure similar to that described above, we find
that the interaction integrals E.62) again emerge with the modification
y2 =
E.75)
g = v2 - v,
where, we recall, /x is reduced mass. The relation E.62b) becomes
1I E.76)
For interacting rigid spheres, E.71b) becomes
2/j,
Specifically,
E.76a)
E.77)
where, we recall,
1
-(^01 +^02)
The lowest-order estimate within the second Chapman-Enskog approximation
for the coefficient of mutual diffusion is then given by
3k BT
777)

3.5 The Chapman-Enskog Expansion 209
For spherical molecules, there results
2kBT\x/1
E.79)
n = n\ -\-ri2
This finding was previously cited in D.31a). In the limit that n\ — n2 = n/2
and aOi = aO2 = &o, E.79) gives the coefficient of self-diffusion:
)
2 (
Electrical conductivity
Again stemming from the study of a two-component gas comprised of ions
of charge Ze and electrons of charge e, we find that conductivity is directly
related to mutual diffusion as
ninen -
gc = (Zem - eMJDie E.80)
p2kBT
where p is mass density and M is ion mass. In the limit that m/M <^C 1, this
relation reduces to
^ E.80a)
BTnt
For singly charged ions, ne — nt. Further setting /x 2^ m in E.77), the preceding
relation reduces to
l-^n-} E.8D
8 £2uu
Comparison with the Drude model result D.15) indicates that fi(M) takes the
place of collision frequency times volume per particle. These results relevant
to rigid spheres are listed in Table 3.4
3.5.8 Expressions of £2(/<?) for Inverse Power Interaction Forces
Consider the interaction potential between molecules to be given by A.17),
V = Kr~N E.82)
corresponding to the radial force of magnitude
F = KNr~iN+l)

210 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
TABLE 3.4. Transport Coefficients in the Chapman-Enskog Expansion for Rig
Sphere Interaction3
Coefficient Value
75 1 /kl
k, thermal I
conductivity 64ao \mn
5 1 (mkBT^12
rj, viscosity — —r I
16 On \ n
D]2, mutual ——-
diffusion [tncrn
D, self-diffusion ( I
YlYl Yl
<jc, electrical (ZeM — eMJDie
conductivity P ^bT
nPne2
ac
vie
aWith cy = 3kB/m, these results give K/ncy = f = 2.5,
V = \ = 0.67 and C = | = 0.83, which are seen to
be in reasonable agreement with measured values given in
Table 3.3.
Then, for Q(l\ with A.20) and following, we find
OO
/ A - cos10)sds
Jo
2KX -dl" ' A- cos' 9)bdb
E.8
g-4/NA,(N)
/»OO
A[(N)= / A - cos1 0)bdb
Jo
Inserting this result into E.76) gives
f
E.8
2\
q+2-
where F(x) is the gamma function. Note in particular the simplifying proper
for Maxwell molecules, N — 4.

3.5 The Chapman-Enskog Expansion 211
Nature of approximation
Having come to this point in the Chapman-Enskog expansion, it is evident that
the technique of solution involves nested approximations within each iterate
E.19) of the expansion. As we have seen, the second approximation yields
the integral equations E.35). To solve these equations, an additional approx-
approximation comes into play: casting these equations in matrix form permits an
approximation corresponding to the order of the truncation of the related in-
infinite matrices. In the results presented above, related approximation within
the second Chapman-Enskog iterate involved the lowest approximation, lxl
matrices.
3.5.9 Interaction Models and Experimental Values
Combining preceding results with various models of interactions between
molecules gives the following list for viscosities to lowest-order results within
the second Chapman-Enskog approximation.
1. Rigid elastic spheres of diameter <t0.
162 \ n J
2. Repulsive potential, V — Kr
16ct0
~N
3. Sutherland model.35 Attractive spheres of diameter a0 and interaction
potential V = -K/rN.
(OA + )
16a02 \ J / \ kT)
In E.87), S is the temperature-independent Sutherland constant
36
S - «™ E.88,
Note that S is proportional to the potential energy of the molecules in contact.
The function I(N) is given by
/ P4~NBP2 - 1)A - 32I/2 /
Jo Jo
35W. Sutherland, Phil. Mag. 36, 507 A893); 77, 320 A909).
36S. Chapman and T. G. Cowling, ibid., Sec. 10.41. In this work, the Sutherland
constant is written as E.88) divided by kB. In the present work, S carries dimensions
of energy.

212 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
TABLE 3.5. Viscosity of CO2 and N2 in the Sutherland Model
CO,
rj x 107 rj x 107
T (°C) (observed) (evaluated)
-20.7
15.0
99.1
182.4
302.0
1294
1457
1861
2221
2681
1284
1462
1857
2216
2686
T) X 107 T) X 107
T (°C) (observed) (evaluated)
-76.3
-37.9
16.1
51.6
100.2
200.0
250.1
1275
1465
1728
1880
2084
2461
2629
1269
1469
1728
1884
2086
2461
2633
Some leading values of I(N) are as follows:
N
2
3
4
6
8
|C1n3
§A0 —
rA9n2
8) =
-2)
7T2) =
-88)
1556
0.2337
= 0.2118
= 0.1956
= 0.1722
In applying E.87) to experimental observation, it proves convenient to re-
remove the dependence on numerator coefficients of this expression by rewriting
it in the form
3/2 c , k T
= rKT>)[^\ ^-^ E.89)
T ) S + kB
With values,
7](T = 273K, CO2) - 1388 x 10
rj(T = 273K, N2) - 1654 x 10
and
S(CO2) = 239.7^, 5(N2) = 104.7**
the expression E.89) gives the values shown in Table 3.5, in very good
agreement with experiment.

3.5 The Chapman-Enskog Expansion 213
3.5.10 The Method of Moments
In this concluding portion of the present section, we present a brief description
of an alternative solution to the Boltzmann equation due to Grad,37 which has
come to be known as the method of moments.
We begin by defining another nondimensional microscopic velocity, c,
closely related to § introduced above: c = c/C = \/2£. The nondimensional
distribution function /(x, c, t) is given by
F(x,c,0 = ^/(x,c,0 E.90)
The Maxwellian E.2) may be written
" /°(c) = ^co(c) E.91a)
c2
°>® s lT^me~'C n E-91b)
Tensor Hermite polynomials
In the method of moments, the distribution function /(x, c, t) is expanded in
a series of tensor Hermite polynomials, H^n\c). The double index of //j(n)(c)
reflects the fact that it is a tensor polynomial of rank n in three-dimensional
space. The subscript i denotes a sequence of n indexes (/j, ..., /„), where
ik may have any of the values A, 2, 3) corresponding to the three Cartesian
directions.
Here are some fundamental properties of these polynomials. They are
defined with respect to co{c) given by E.91b). Thus
1 / 9 9 9 - E.92)
(
These polynomials also follow from the generating function:
exp
J_ ^2 2 1 y* H(n) ,, d
cr = c' — c
The orthogonality of the tensor Hermite polynomials is given by
/ H^inH^\jnco(c)dc = nlA^Vj^l E.94)
J " '' 2'"" "
where
37
H. Grad, Comm. Pure andApp. Math. 2, 331 A949).

214 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
for {ik} a permutation of {jk} and zero otherwise.
The first four polynomials of this set are
//@)(c) - 1 E.95a)
fliA)(c) - ci E.95b)
H?\t) = ciCj - Sij E.95c)
E.95d)
Here are some recurrence relations among these polynomials [deleting the
subscript on///n)(c)].
H{n) = IiH{n-l) E.96a)
CiHin) = Hin+l) + hH{n-X) E.96b)
//(n) //(n) 2I//(n) E.96c)
Here we have written I for the identity matrix. It is such that the expression /, A
represents the sum of all products in which / is attached to I. This operation is
demonstrated in application of E.96b) to the case n — 2:
- Sjk) - 8ijCk - 8ikCj
which agrees with E.95d).
The expansion
The distribution /(x, c, t) is expanded in Hermite polynomials as follows:
00
1
/(x, c, 0 - /°(c) Y, -W°(x> Ofll(ll)(c) E.97)
n
n=0
which in explicit form appears as
2\
Inverting the series E.98) gives
vHm + C
E.97a)
al"\x, 0 = I /(x, c, t)H{n\c)dc E.98)
Since ///n)(c) are polynomials in the velocity c, the coefficients aln)(x, t) are
moments of the distribution /. Each such moment corresponds to a macro-
macroscopic fluid dynamic variable [recall C.33)]. Thus, for example, we may make

3.5 The Chapman-Enskog Expansion 215
the following identifications:
a™ = 0
E.99)
py/RT
(a repeated index gives the trace).
Macroscopic equations
With E.97), we see that construction of the distribution /(x, c, t) depends on
knowledge of the tensor coefficients <2.(n). These coefficients are obtained as
follows. The series E.97) is substituted into the Boltzmann equation B.14).
Following this substitution, we operate on the equation with f dcH^n\c) to
obtain (with, we recall, C = V RT)
Dt
Caf+V)
-— + c
C Dt
n) E.100)
Note that all terms in this equation have dimensions of t~l and the nth-order
tensor polynomials. [The coefficient n in the third term of E.100) is tensor
order, not number density.] Subscripts on tensor terms are tacitly assumed
save for repeated indexes, in which case products are summed. Indexes of the
I term are unequal to remaining indexes of a term. Thus, for example, consider
the term
where T{n) is written for an nth-order tensor.
The convective derivative is D/Dt, while the symbol ^ represents the
symmetric sum of gradients. For instance,

216 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
The collision term 7(n) is written for
/<"> = ^ f ~f(c)~f(c{)Iin\c, cOrfcrfc! E.102)
where
ln)= f gadQ[HM]
E.102a)
Nature of approximation
The technique of solution is to approximate the series E.97) with a finite
amount of terms. Inasmuch as the leading term in the expansion is the local
Maxwellian, the closer the has is to the equilibrium state, the less terms suffice
to describe the state.
The simplest approximation is the "second-order" one, where we set
7 = 7
0
1
B)
2-,j ~,j E-103)
Dropping all terms containing a{3) in E.100) gives
.B)
Ot OXr OXr OXr
S^1^ = J'? E-104)
j
If all terms beyond <2B) are neglected in the calculation of 7B), then this
last equation involves {p, u, 7\ af^} or, equivalently, {p, u, T, Ptj] Inasmuch
as a-y- is a symmetric, traceless tensor, there are only six relevant components
of aB). These, together with the five variables {p, u, T], constitute eleven
scalar quantities. These eleven variables define the state of the system in the
second-order approximation.
Equation E.104) comprises six scalar equations. These equations together
with the continuity equation
dp
dt
the momentum equation
+ V • pu = 0
pf-—+u-Vu
and the condition Tr a{2) = 0 are^eleven scalar equations, which serve to close
the system.

3.6 The Linear Boltzmann Collision Operator 217
3.6 The Linear Boltzmann Collision Operator
We return to the linear collision operator E.25) introduced in the Chapman-
Ernskog solution of the Boltzmann equation.
II
F.1)
As evidenced by the expressions E.53) and E.56), this operator plays a key
role in evaluation of transport coefficients. In this section a number of basic
properties of the D operator are derived.
3.6.1 Symmetry of the Kernel
Let us label
d^ = F°(v)dy F.2)
Then F.1) may be written
DO = - I dii\K(Y, Vi)<D(vi) F.3)
where the kernel K is as implied. We wish to show that K is symmetric; that
is,
F.4)
Let 0 and i/r be any two elements of £2 space. The matrix element of the
operator □ with respect to these two functions is given by
/
F.5)
We first wish to establish that D^^ is a symmetric matrix; that is,
D^ = D^0 F.6)
Repeating arguments described in Section 3.1 on collisional invariants, we
obtain
rt^ F.7a)
This may be rewritten
+ (p - <p[ - 0') = D^ F.7b)

218 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
which establishes the validity of F.6).
With F.7a), we may conclude that the diagonal elements D^^ are negative
except when 0 is a linear combination of summational invariants. That is,
□00 < 0 F.8a)
□00 = 0 only if 0 = a + C • v + yv2 F.8b)
Proceeding with our derivation, we consider the difference
i,<t> w jj
Interchanging v and vi in the second integral permits F.9) to be written
0- //rf/xirf/x[A'(v,v1)-A'(v1,v)]0(vH(v1) F.10)
Here it is understood that K(x, y) operates only on functions of y.
Introducing the operator
F.11)
permits F.10) to be written
, Vi)G(v, vi) = 0
F.12)
Suppose T is not identically zero. This means that a function G exists such
that TG / 0 over a domain of v, Vi phase space. It follows that there is a
subdomain D of this domain in which TG does not change sign. Let G = G
in D and G = 0 elsewhere. Then for this choice of function f TG / 0, which
violates F.12). We may conclude that the only way to ensure that / T\j/(j)
vanishes for all ^ and 0 in L2 is for T to vanish identically. Thus K is a
symmetric operator.
3.6.2 Negative Eigenvalues
^ ^^
Let us suppose that the collision operator □ has a discrete spectrum. Then we
may write
F.13)
Due to the symmetry of D F.6), the eigenfunctions \frn comprise an orthogonal
sequence. Thus we obtain
f _ f
J J
= vm8nm\\irm\\2 F.14)

3.6 The Linear Boltzmann Collision Operator 219
where \\\jfm || represents the norm of \jfm. The diagonal elements of □ obey the
properties F.8), and we may write
0 > Unn = vn I d^l = vn\\\{rn
so that
vn<0 F.15)
The equality in this relation occurs if \ffn is any of the five independent scalar
summational invariants. Thus the □ operator has a fivefold degenerate eigen-
eigenvalue as the origin on the real v line. All remaining eigenvalues are negative.
We conclude that the eigenvalues of □ lie on the negative real axis with a
fivefold degeneracy as the origin.
The significance of the negative quality of these eigenvalues is seen from
the following discussion. Let a spatially homogeneous gas suffer a small
displacement from equilibrium in velocity space. We recall the perturbation
E.30)
F = F(O)A + <D) F.16)
Substituting this form into the Boltzmann equation B.14) and keeping terms
linear in O, we obtain
— = D<D F.17)
dt
With this equation at hand, we may conclude that the nonpositive spectrum of
□ implies that O decays in time and that consequently the system returns to the
Maxwellian state. This conclusion, stemming from the linearized Boltzmann
equation F.17), is seen to be consistent with our previous finding concerning
the decay of Boltzmann H and accompanying reduction to the Maxwellian
state, stemming from the nonlinear Boltzmann equation (Section 3.8).
3.6.3 Comparison of Boltzmann and Liouville Operators
It is interesting at this point to offer a brief comparison of properties of the
Boltzmann collision operator (both linear □ and nonlinear J) with those of
the Liouville operator LN B.1.3). The Liouville operator contains the full
Af-particle Hamiltonian, and the related Liouville equation is reversible. That
is, if FN(xN,pN,t) is a solution, then so is FN(xN, -pN, -t) [recall Sec-
tion 1.1.5]. We also found that eigenvalues of LN are pure imaginary (see
Section 2.3.1), which evidently permits oscillatory behavior of the TV-particle
distribution function. Furthermore, by virtue of the Poincare recurrence theo-
theorem [Section 3.6], we may conclude that for a bounded system, solutions to
the Liouville equation exist that, after sufficient time, return arbitrarily close
to their initial values.

220 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
The Boltzmann collision operator, on the other hand, accounts only for
two-particle interactions. Furthermore, owing to the assumption of molecular
chaos [F2(l,2) = Fi(l)FiB)], we may say that particles in collision lose
memory of this collision after interaction. This property evidently contributes
to irreversibility.
Distinctions between J(f) and LN(fN) were also evident in our preceding
discussion of the H theorem (Sections 3.7 and 3.8). There we found that the
Gibbs entropy, as implied by the Liouville equation, is constant, whereas the
Boltzmann entropy, as implied by the Boltzmann equation, increases with time
(recall that S a -H).
3.6.4 Ha rd and Soft Po ten tia Is
We may distinguish between hard and soft interaction potentials in the
following manner. Consider the set of potentials A.17)
V(r) = Kr~N F.18)
where in general the constant K is dependent on the number N. We have found
previously A.27) that for Maxwell molecules (N = 4) the integration measure
ga d cos 0 is independent of the relative speed g. Let us adopt the convention
that potentials with TV > 4 be labeled hard and those with TV < 4 be labeled
soft.
The separation potential with N = 4 corresponding to Maxwell molecules
permits exact evaluation of the spectrum of the □ operator. Results of this
calculation are prescribed in the following section.
3.6.5 Maxwell Molecule Spectrum
As noted above, for Maxwell molecules the integration measure ga d cos 0 is
independent of relative speed g. For this case, the eigenvalue spectrum of the
D operator is discrete. Eigenvalues and eigenfunctions for this case were first
discovered by S. C. Wang Chang and G. E. Uhlenbeck.38
Working in spherical coordinates (§,0, 0), eigenfunctions of the D collision
operator are given by
, 0, 0) - Nnlms\%/2)(^1YPF, 0) F.19)
38C. S. Wang Chang and G. E. Uhlenbeck A952), ibid. These reports are reprinted
in J. deBoer and G. E. Uhlenbeck, Studies in Statistical Mechanics, North-Holland,
Amsterdam A970). A concise description of these findings may be found in G.
E. Uhlenbeck and G. W. Ford, Lectures in Statistical Mechanics, American Math.
Soc, Providence, R. I. A963). See also L. Waldman, Hand, der Physik XIII, 295
Springer-Verlag, Berlin A958).

3.6 The Linear Boltzmann Collision Operator 221
where YmF,(j)) are the spherical harmonics,39 Nnim are normalization
constants, Sj+^^x) are Sonine polynomials, encountered previously in Sec-
Section 5.5 and defined by C.54). With the integration measure J/x F.2), the
sequence F.19) is an orthogonal set and is a basis of £2 space.
Eigenvalues corresponding to the functions F.19) are given by
f
Jo
= vnt = 2n / d6 sin 6FF)
'o
- p rsm -
2 ' \ 2
6F
p
_ p cos
2 V 2
F.20)
where P/(x) are Legendre polynomials (see Table 3.6) and F{6) is given by
[recall A.27)]
1 bdb
FF) = F.21)
d cos 6
Since the azimuthal m number is absent from vnl, we see that each eigenvalue
is 2/ + 1 degenerate.40
As is evident from A.23), all classical total cross sections corresponding to
potentials with unbounded range are infinite. This singular behavior resides
in the eigenvalues F.20) through their dependence on F@), whose properties
are now discussed.
The function F{6) was first evaluated by Maxwell.41 It is given by
(cos26>I/2
F@) = = ^ ^ =r- F.22)
4 sin 6 sin 26>[cos2 6>£(sin 6) - cos26E(s\n 6)]
where
n-6
-.(cos26)l/2k(sin6) F.22a)
z
and K(x), E(x) are, respectively, complete elliptic integrals of the first and
second kind.42
/7T/2 /»7f/2
A -fc2sin20I/2d0, K(k)= /
Jo
An
A concise list of properties of these functions may be found in R. L. Liboff,
Introductory Quantum Mechanics B002) ibid., Chapter 9.
^This degeneracy corresponds to the rotational symmetry of the D operator with
respect to 0. Such degeneracy is often encountered in quantum mechanics; see R. L.
Uboff, ibid.
41Scientific Papers of James Clerk Maxwell, W. D. Niven, (ed.), Dover, New York
A952). See Chapter XXVIII.
42E. T. Whittaker and G. N. Watson, A course of Modern Analysis, Cambridge
University Pres, New York A952), Chapter 22. See also M. Abramoxitz and I. A.
Stegun, Handbook of Mathematical Functions, Dover, New York A965), Chapter 17.

222 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Evaluation of F@) as given by F.22) indicates that it is a monotonic
decreasing function of 6. For small #, we obtain
F{9) ~ Q^e-*2 (l + £-e + :) F.23)
16 . V 24 1
= 7T,
F(n) = -^ = 0.0727
4K2(n/4)
The result F.23) indicates that F@) diverges at small 0, which, as noted above,
is related to the unbounded range of V(r). With the property F.23), we see that
the eigenvalue F.20) likewise diverges at 0 2^ 0. Resolution of this singular
behavior is discussed at the close of this section.
3.6.6 Further Spectral Properties
We return to the notion of hard and soft potentials introduced above. Our
discussion involves the collision frequency E.2) now written relative to a
Maxwellian state.
//
v(v) = // adQgd\iF°(vi) F.24)
We first evaluate v(v) for a gas of rigid spherical molecules of diameter a.
Recalling A.33), we write
a2
<r@) = — F.25)
where o is the diameter of a molecule. Inserting this value into F.24) and
performing the integration, we find (see Problem 3.12)
= v0
F.26)
where erf(§) represents the error function (see Appendix B):
erf(§) = —— I e~x dx
Jo
with § given by E.28) and
At large §,F.26) gives
^(s) ^^ V^^OS' g !^> 1 (o.z/j
whereas at § 2^ 0
JrV 3

3.6 The Linear Boltzmann Collision Operator 223
V{r)
t
V(r)
Hard
Soft
v@)
v@)
FIGURE 3.21. Sketch illustrating the eigenvalue spectra of — □ for hard and soft
potentials, respectively. For hard potentials, the fivefold degenerate null eigenvalue
is seen to be isolated from the continuum.
which gives
v@) - 2v0
F.28)
We may attribute this persistence of collision frequency for a stationary particle
to the fact that the particle is situated in a gaseous medium at finite temperature.
Note in particular that, as T —> 0, v@) —> 0.
The preceding results demonstrate that for a gas of rigid sphere molecules
v(£) ranges from v@) = 2vo > 0 to infinity.
A similar property for hard potentials, in general, has been established by
Grad.43 These properties address the spectrum of the linear collision operator,
D. They are as follows:
1. For hard potentials the null eigenvalue, v = 0 is isolated from the rest
or the spectrum of D. In addition to v = 0, the spectrum of — □ includes a
continuum, which is the values of the collision frequency v(§) and, consistent
with our finding above, ranges from a minimum v@) to infinity. For soft po-
potentials, the eigenvalue continuum ranges from a maximum v@) to zero. These
properties are sketched in Fig. 3.21.
43H. Grad, Phys. Fluids 6, 147 A963), "Asymptotic Theory of the Boltzmann
Equation, II," in Rarefied Gas Dynamics Symposium, vol. I, J. Laurmann (ed.),
Academic Press, New York A963).

224 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
We have noted previously that the eigenvalue vn contributes a term like
exp(—vnt) to the solution of an initial-value problem. The same is true for a
continuous spectrum, where the exponential term contributes to an integral in
place of a sum. Thus, for hard potentials, where v is bounded away from zero,
the slowest decay of the distribution goes as exp[—v@)t]. For soft potentials
(such as for the Coulomb or Newtonian interaction), v is not bounded away
from zero, and we may expect long-time nonexponential decay for such cases.
2. Write D in the form
D = £-v(§) F.29)
where v(§) is given by F.24). Then
■A
(a) The operator /C is bounded. That is, for any \j/ in £2 a constant M exists
such that \\Kf\\ < A* 11^II-
/V .A.
(b) The operator K is completely continuous. A bounded linear operator K
is completely continuous if, for any bounded sequence {\/rn} of elements
■A
of £2 (that is, ||^n|| < R for some R and all n), the sequence {K\/rn}
contains at least one convergent subsequence. As a consequence of this
property and the spectrum theorem** K has a discrete point spectrum
with the origin as the only accumulation point.
3. In Section 6.5, we found that the unbounded range of the Maxwell
molecule force law gave rise to singular eigenvalues. This singular behavior
is circumvented if the force law is cut off at some finite range. The resulting
eigenfunctions are tensor Hermite polynomials, //j(n), encountered earlier in
.A.
Section 5.10. Eigenvalues of the collision operator □ may then be written
l {{ F.30)
Tensor Hermite polynomials are relevant to expansion of the velocity compo-
component of the distribution function in Cartesian coordinates. Sonine polynomials
[as seen in F.19)] are relevant to expansion in spherical coordinates. Thus,
for example, working with the nondimensional distribution, E.90), these
respective expansions appear as
/(x, c, 0 = £ rr ■ r, nn«mm(x, t)e-'cS\%/2)(c2)clY^9^ 0)F.3la)
n,l,m l Kn I /^))
00 j
in\x t)a)(c)H{n\c
/(x, c, 0 = 2^ ~4' (x'f Mc)#i (c) F.3 lb)
n=0 "•
44This theorem and related topics are discussed by H. H. Stone, Linear Transfor-
Transformations in Hilbert Space and Their Application to Analysis, American Mathematical
Society, New York A932); I. Stakgold, Boundary Value Problems of Mathematical
Physics, vol. 1, Macmillan, New York A967), Section 2.10.

3.7 The Druyvesteyn Distribution 225
In either representation, coefficients of expansion, a, are related to fluid dy-
dynamic variables. Factorial factors and weight functions in F.31) are related to
orthogonality properties [recall E.54b) and E.94)].
3.7 The Druyvesteyn Distribution45
3.7.1 Basic Parameters and Starting Equations
To this point in our discourse, we have considered only the kinetic theory
of one-component fluids. As an application of the kinetic theory of a two-
component fluid, we turn to an important problem related to the theory of
conduction of electrons through a medium consisting of heavier ions or neutral
particles.46
Our two basic assumptions are A) electron-electron collisions may be
neglected compared to electron-ion collisions, and B) the background in
distribution is given by the Maxwellian
Mv
M \
2nkBT)
F(vx)d\x
3/2
—
exp
n
G.1a)
Here we have written M and \\ for ion mass and velocity, respectively, and n
is ion number density.
Our starting equation is the Boltzmann equation B.17) for a two-component
fluid comprised of ions and electrons in the presence of an electric field £.
Neglecting electron-electron collisions, only one collision integral remains in
B.17) and, passing to the equilibrium limit, we obtain
— * if
/if <///ri
JJ
G.2)
where /(v) is the electron distribution function and e/m is electron charge-
to-mass ratio. In this discussion we take f(v), as F(v) [see G.1a)], to be
normalized to n.
45Named for M. J. Druyvesteyn, Physica 10, A930). For further discussion of this
topic, see S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform
Gases, A974) ibid.; L. B. Loeb, Fundamental Processes of Electrical Discharges in
Gases, Wiley, New York A947); W. P. Allis, Hand, d Physik vol. XXI, Springer-
Verlag, Berlin A956); M. J. Druyvesteyn and F. M. Penning, Rev. Mod. Phys. 12, 87
A940).
46We will in general call these particles "ions."

226 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Writing
m
a =
CL\
m + M
M
m + M
so that
a + a,\ = 1
and
rav +
G.4)
m + M
is the velocity of the center of mass of a colliding electron-ion pair of particles.
The kinematic relations connecting these velocities may be written
G.5b)
/
!)(v) G5c)
) G.5d)
Energy conservation gives
S = S' G.6a)
whereas conservation of momentum gives
V = V G.6b)
The Jacobian of these transformations G) are all seen to be unity. Thus, for
example, we may write
d\d\i=dgd\ G.7)
From the defining relations
= v - v] G.8a)
= v' - v', G.8b)
\ = a\a]\] G.8c)
V' = av'+a1v'1 G.8d)
we obtain
g - g' = v' - v; - v - v, G.9a)
v'i G.9b)

3.7 The Druyvesteyn Distribution 227
Multiplying Ga) by a and — a\, respectively, and adding to Gb) gives
v/-v = a1(g'-g) G.10a)
v;-Vl = -a(g'-g) G.10b)
From the relations
V = V + aigf G.11a)
v = V + aig G.11b)
we find
va = v2 + 2a1V.g/+ajy2 G.12a)
u2 = V2 + ^V • g + a2g2 G.12b)
Subtracting these equations and recalling the conservation equation G.6) gives
u/2-u2 = 2a1V.(g'-g) G.13)
Our third assumption concerns the electron-to-ion mass ratio m/M. We set
— ^l-a{=s2 G.14)
where s is a parameter of smallness. Finally, we assume thermal equilibrium
between electrons and set
(^) (^) C7.15)
With the preceding relations we find
v\ — sv G.16)
a = s2
ax = \-e2 G.17)
And with G.10) and G.14) we obtain the order-of-magnitude relations
G.18a)
G.18b)
; =0(s) G.18c)
See Fig. 3.22. Having obtained these kinematic relations, we turn next to the
Legendre expansion of /(v).
3.7.2 Legendre Polynomial Expansion
.A.
Let E and v denote unit vectors. Then we define
v G.19)

228 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
g'-V
g-v
FIGURE 3.22. Approximate vector equalities in the scattering diagram for m/M
1. Note that the scattering element of the solid angle, dQ, lies in the direction of g'.
With \x so defined, toward the end of solving G.2), we introduce the following
expansion of /(v) in Legendre polynomials P/(/z) (see Table 3.6).47'48
00
/(v)
G.20)
/=o
In the present study, S is taken to be a moderate field, in which case we expect
small departure from isotropy of /(v). Accordingly, we keep terms only to
I = 1 in G.20) and obtain
/(v) = /0(u) + A6/i0>) G.21)
Substituting this expansion into the left side of G.2), we find (see Problem 3.13)
eS df eS
m d\ m
For the collision terms, we obtain
dfx
3 V dv
v
G.22)
J(f, F) =
,' - fFx)ogd0.d\x
If
G.23)
47This expansion for an anisotropic distribution function was first introduced by
H. A. Lorentz, The Theory of Electrons, B. G. Teubner, Leipzig A909). See Appendix
D.
48The convergence of such expansions is discussed by W. P. Allis in Electrical
Breakdown and Discharges in Gases, E. E. Kunhardt and L. H. Leussen (eds.),
Plenum, New York A983).

3.7 The Druyvesteyn Distribution 229
TABLE 3.6. Properties of the Legendre Polynomials
Generating formulas:
00
s2rl/2 ='
1=0
2
Legendre's equation:
, d2P,(a) dPi(ix)
(i - n)—rF- - ^—t^- + W + i)fl(M) = o
Recurrence relations:
(/ + l)P/+i(/x) = B1
7 d
A - /x2)-—P/(/x) - //xP/(/x) + /P/_i(/x)
Normalization and orthogonality:
2 - .
= m)
1
= 0 (/ / m)
First few polynomials:
2
p0 = 1 p2 = IC/x2 - 1) P4 = ^C5/x4 - 30/x2 + 3)
^ - 3/x) P5 = ^F3/x5 - 70/x3 + 15/x)
Special Values:
where
A note of caution is in order here: we should keep in mind that f\ in G.23)
denotes the first-order correction to / as given in G.21) so that (as opposed
to Boltzmann notation) f[ = f\(vr). However, in keeping with Boltzmann
notation, F, = F(\x) and Ff = F(\\). In parallel form to G.23), we set
J(f I F) - 7o(/o I F) + >,(/, | F) G.24)
where
Wo I F) = ffifoF'i ~ foF^gdQdy, G.25a)
Jdfi I F) = fffa'flFl - LJLfxF{)ogdyx G.25b)

230 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
FIGURE 3.23. The angles of g' and £.
Combining G.2), G.22), and G.24) gives the expanded form
M~ 1" ~ I — 1" "
v
m
+
dv 3 \ dv
Mfo\ F)
F) G.26)
We turn next to reduction of the J\ (f\ \ F) collision integral. With G.6a)
and G.18a, b), we may write
v + 0(e)| = |v' + 0(e)|
whence
v = v' + 0(e)
and
G.27a)
With G.18c), we write
F[ = F, + 0F)
G.27b)
Thus G.25a) becomes
nfi f(ji'
- fi)agdQ + 0(e)
G.28a)
To further reduce this integral, we go into a representation where g is taken as
the polar axis (see Fig. 3.23). We note [recall G.19)]
•) = cos a + 0(e) G.29a)
0(e) G.29b)

3.7 The Druyvesteyn Distribution 231
■A.
From Fig. 3.23 we note the Cartesian components of £ and g',
.A.
£ — (sin of, 0, cos of)
g' = (sin 6 cos 0, sin# sin0, cos#)
where
g' • e = cos of cos # + sin a sin # cos 0 G.30)
Substituting this value into G.29b) permits the £2 integral of the /z' component
of G.28) to be written
/ ix dQ = /// sin^ J^ d(f)(cosa cosO + sin a? sin # cos 0)
which reduces to
' G.31)
lx'dQ=
Substituting this result into G.28) gives
Mfi I F) = -nfxii j(l-cos9)Ggdn + 0(s) G.32)
This finding may be written in the more serious form:
Jx = -nflfMvQ + 0(e) G.33)
where Q is the weighted cross section,
Q = [(I -cos6)adQ G.34)
[compare with E.83)].
Further introducing the relaxation time r and mean free path I [recall B.2)
and following],
r = , / = — = vr G.35)
nvQ nQ
allows G.33) to be rewritten
G.36)
r
With this result at hand, we return to the starting equation G.26). Equating
terms of equal Legendre polynomial order and keeping terms of 0A), we find
e£ (dfx 2/,\
hf1 + — ) = 4>(/o I F) G.37a)
v )
e£ dfo fi
— -^ = -— G.37b)
m dv t

232 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
3.7.3 Reduction of Jo(fo \ F)
It is evident from the preceding two equations that elimination of f\ would lead
to a second-order integrodifferential equation for /0. However, an alternative
path leads to a simpler relation. We note that G.37a) may be rewritten
e£ 1 d 9
( fi J(f\F G.38)
3m v
We wish to integrate this equation from v = 0 to a finite value, which we label
v. Introducing the notation
f d\= f f dvv2dQ
Jv<v Jo J
there results
eS f 1 d z fff , ,
I = — I — — [v fx(v)]d\= HI (fQFx- foFx)ogd£ld\xd\ G.39)
u<u
Switching primed and unprimed velocties in G.39) gives
Jff flF[crgdndYXdv = jjj foFiCrgdQdYidv G.40)
v<v v'<v
Here we have noted the invariance of the measure agdQ d\\ d\ under the said
transformation. Now note that with
v — v — Av
the condition v' < v becomes
v < v + Av G.40a)
where A v is to be determined. Writing
d\ — v2dvdQv
together with the preceding equation, we find
G.41)
•-fff
pv+Av
Jv
To evaluate At>, we recall G.13)
v'2 - v2
and G.18)
To keep this order of magnitude ir
f0Flagv2dv
-2a,V.(g/-
- V, = 0(8)
dQdQvd\
-g)
i mind, we write
£Y\

3.7 The Druyvesteyn Distribution 233
(and eventually set s — 1). To this same order, we find
v'2 - v2 = 2ai e\\ • (V - v)
= 2A -£2)£\\ •(¥'- V)
i/2 = u2 + 2£V, • (v' - V)
i/ = v + ev, • (v' - v) + 0(£2)
Thus the condition v' < v is equivalent to
+
(vr — v) < i)
or
V < V + £\\ • (V — V'
which, with G.40a), implies that
Av = £\\ • (v — v')
That is, Av is an 0(£) quantity. Recalling our previous rule, we set s
obtain
Av
(v - v')
- 1 to
G.42)
3.7.4 Evaluation of /q and I\ Integrals
Having obtained this expression for Av, we return to G.41) and expand fo(v)
about v = v. As the integral is evaluated over a small interval about v, we
write
Mv) =
(v - v)
0
dv
Substituting this expansion into G.41) gives corresponding 70 and 1\ integrals.
We first examine 70.
h -
0
in \r
F\ogv2 dv
G.43)
where ilf(v), as implied, is independent of electric field S.
Next we turn to evaluation of 1\, given by
r
„ dfo 2
(i; — v)——F\ogvdv
dv
G.44)
where we have set
dv
V
dfo
dv

234 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Note that with G.18)
Furthermore, we write
and expand the u-dependent part of a about v = v.
da
g(v) — a(v) + — (v - v) H
dv
There results
pv+Av pv+Av
I (v — v)v3a(g') dv = a(v) I (v — v)v3 dv
J v J v
da
d+Av
a f _ ,
/ (v — v)v (v — v)dv
dv J~v
However, note that over the interval of integration
v — v < Av
Thus
pv+Av pv+Av
I (v — v)v3(v — v)dv < Av I (v — v)dv
J V J V
And we may write
pv+Av pv+Av
I (v - v)v3cr(g') dv = o(v) / (v - v)v3 dv +
J V J V
V3 0
^ o(v) — (Avf G.45)
providing49
da
ll
< Ik
dv
Substituting the result G.45) into G.44) yields
h = V-~ jjj F,a{i,%)[yx '{v-y')]2dSKmvdv, G.46)
Now note that we may write
• (V — V )] = V\(V — V ) • ViVi • (V — V )
49
Note that this condition fails for the Coulomb cross section A.31) at low energies.

3.7 The Druyvesteyn Distribution 235
With the relation (see Problem 3.15)
J 3 7
G.46) may be rewritten
S3</0 fff
dv JJJ
-\')-(\-\')d^dQvd\i
^~3^o fff Fla^(l-\-\f)dQdQd\l G.47)
Noting
v • v' = g • g' + 0(e) = cos 9 + O(e)
and writing
gives, finally,
f f M , /3 ,
/ </nv / FKw,)—uf dv, = 4^ ( - ) nkBT
h = ~JdAAnnllL I a{l-COs6)dn
dv M
which, with G.34), may be written
M
Recalling G.35), 2 = 1/w/, the preceding relation becomes
kBTv3df0
h = 4tt^—— — -— G.48)
M Z J
With G.39) we write
47T e^ 9 _ kBT v3 dfo
v2Mv) = /o + /i = /o(u)^(v) +4^T-p G.49)
M I dv
m M I dv
Let us find an expression for V^(^)- In that this function is independent of £,
we may evaluate it at £ — 0. At this value
\ 3/2 / ~2
27r"^T ' exp'~
so that
kBTJ°
With this value substituted into G.49), we find
m v4
j G.50)

236 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Inserting this value into G.49) with £ / 0 gives the desired equation (replacing
v with v):
lee
3 ra M I dv
Ml
which is equivalent to the integration of G.37a).
3.7.5 Druyvesteyn Equation
Having reduced G.37a) to G.51), we are now prepared to obtain a single
equation for /0. Substituting G.51) into G.37b) gives
1 I /eS\2 kBT v
3 v \ m ) Ml
■"
m v
dv M
0
G.52)
where we have written, with G.35),
Introducing the speeds
Ci
i —
m
_ I
x
2 3 m
(eIL\
permits G.52) to be rewritten
+ C}v^df°- -3
dv
-v3f
o
fo(v) = K exp -
With the mean free path I given by G.35), we may not, in general, assume that
it is independent of v. Integrating G.53) gives the Druyvesteyn distribution
G.53)
where K is a normalization constant.
The Druyvesteyn equation G.52) may be cast in still simpler form when
written in terms of the nondimensional electric field
M ( e£l\2
s = — I I
6m \kBTJ
and nondimensional energy
G.53a)
x =
2k BT
G.53b)
With these parameters at hand, G.52) becomes
G.54)

3.7 The Druyvesteyn Distribution 237
t
foM
FIGURE 3.24. Sketch of the variation of the Druyvesteyn distribution with varying
electric field (s).
which, assuming I to be independent of velocity, is simply integrated to yield
that which we term the absolute Druyvesteyn distribution:
= K(x+s)se
s —x
G.55)
A sketch of this distribution is given in Fig. 3.24 at various values s, illustrating
the manner in which the spread of the energy distribution increases with electric
field. This behavior evidently stems from acceleration of electrons due to the
electric field.
Review of assumptions
A brief recapitulation of the assumptions that led to the Druyvesteyn
distribution G.53) is in order at this point. These are as follows:
1. Neglect electron-electron collisions.
2. Background ions are in a Maxwellian distribution G.1).
3. m/M < 1 G.14).
4. Moderate electric field. Small anisotropy in /(v) G.21).
5. Introduction of phenomenological coefficients r and I G.35), with v 2^
Z/r.
6. Electrons scatter elastically from ions.
3.7.6 Normalization, Velocity Shift, and Electrical Conductivity
The normalization of /0(jc) is obtained as follows. Returning to G.21) and with
G.37b), we write
eSdfo
fiV) = Mv) -
m dv
G.56)

238 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Thus
The second term vanishes and we obtain
•00
fo(v)v2dv G.57)
'o
/
Jo
Converting to x notation, G.53b), we find
3/2
/ Til \
dx
2 \2kBT
so that
K/2 /»
/
*.^, ,/oWV,«* G.58,
2 \ m
With G.55), we write
f
Jo
n = 2tt ( —— ) K I (x + ^)e"V^ ^ G.59)
m /
which determines the normalization constant K. At s — 0 (no electric field),
G.59) gives
o t2kBT^'2
n — 2n
m
n = ( — I K
m
N
K =
BnkBT/mK/2
which is the correct normalization for the Maxwellian [see G.1)].
Finally, we turn to the shift in electron drift velocity and closely allied
electrical conductivity due to the presence of the electric field. With C.13), we
write
nu= I f(\)\d\ G.60)
Substituting the form for /(v) as given by G.56) into the preceding gives
f e£ f
/ fo(v)\d\ /
J rn J
n\x — / fo(v)\d\ / r/x—— \d\
dv
As /o is rotationally symmetric, the first term in the preceding equation
vanishes, and we obtain
n\x — / t dvv3—- / dcosOcosO I d(p(\x,\v,vz)
m Jo dv J_{ Jo

3.7 The Druyvesteyn Distribution 239
Here we have taken £ to lie in the z direction. [Recall G.19).] All but the z
integral vanish, and we obtain
2ne£ f°° A 3df0
So we find
f
/
Jo
4neS f 3df0
/ raw
4neS f 3df0
nu — / raw — G.61)
3 m J dv
Integrating by parts and dropping surface terms gives
4ne£ f 2
nu = / r/of dv G.62)
Substituting the expression for r given by G.35), we obtain
Ane£l f
nu = / fovdv G.63)
Thus we may conclude that a nonvanishing mean speed of electrons is incurred
due to the presence of the electric field.
In the Drude model of conductivity [see D.11) and following], we write
J — enu — oc£
where gc is electrical conductivity. With G.63), we obtain
e2£ln f°° -
gc = 4n—— / fovdv
where G.64)
n -
/o = — /o, v = Cv
With
C
G.64) becomes
gc - a?ac@) G.65)
where a is the Druyvesteyn correction factor,
/»OO
a==4n fovdv G.65a)
Jo
and
mv
G.65b)
is the Drude conductivity as given previously in D.15).

240 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
dS'
dS
H= E+bE
H=E
FIGURE 3.25. The normal displacement Sn between constant energy surfaces varies
as8H/\VH\.
3.8 Further Remarks on Irreversibility
In this final section of the present chapter we return to the problem of irre-
irreversibility encountered earlier in Sections 3.5 and 3.8. First, a brief review of
ergodic and mixing flows is presented. We then pass to a description of action-
angle variables in preparation for a discussion of more recent developments in
the theory of irreversibility.
3.8.1 Ergodic Flow
Prior to discussion of ergodic theory, it is necessary to introduce the follow-
following geometrical notion. As noted in Chapter 1, volumes in phase space are
invariant under a canonical transformation or the natural motion of the system.
In keeping with this property, an energy shell was introduced in F-space to
describe the motion of an ensemble relevant to an isolated system. However,
the language of ergodic theory addresses an energy surface. Accordingly, a
measure-preserving element of surface must be introduced. Let the energy
shell be defined by the surfaces H — E and H — E + SE. The displacement
between constant energy surfaces varies as 8H/\VH\. Thus, if dS represents
differential surface on the energy shell, invariant differential volume of the en-
energy shell dQ, varies as dEdS/\VH\ (see Fig. 3.25). This description permits
identification of
^ (8.1)
as the invariant measure of the element of surface dS.50 Thus we have two
differential elements in F-space that are invariant under the natural motion of
50A quantum version of this construction is given in Section 5.3.

3.8 Further Remarks on Irreversibility 241
the system:
= dQ
(8.2)
The invariant measure of a set, A, of points on the energy surface is given by
MA)=L m <8-3a)
L
The measure of the set of points E comprising the whole energy surface is
(8.3b)
■A.
Let the natural motion of the set A be governed by the operator T [introduced
.A.
in Section 3.6], which we now write as T(t). Then, if in the interval t,A—> A\
we may write
A -* A! = t(t)A (8.4)
By virtue of the Liouville theorem, it follows that
H(A') - n(A) (8.5)
We are now prepared to discuss the ergodic hypothesis.
Ergodic hypothesis
This hypothesis states that almost all orbits (that is, except for a set of measure
zero) on the energy surface pass through every domain of positive measure and
remain in these regions for an average time equal to the ratio of their measure
to that of E.
An equivalent statement may be made in terms of the average of a dynam-
dynamical variable g(z). It is evident that the ergodic hypothesis implies that all
domains on the energy surface of equal measure are equally probable. The
corresponding distribution function is51
fovzonH = E (8.6)
which has the normalization
r
(8.6a)
=E
With (8.3), the phase average of g(z) is given by
/ (8.7)
H=E
51 In equilibrium statistical mechanics, (8.6) is called the microcanonical
distribution. This distribution is further discussed in Section 5.2.1.

242 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
In equilibrium, we may also define a time average of g(z) as
(g)T= lim - / g[z(t)]dt (8.8)
T^°° r Jt0
The ergodic theorem seeks to establish the equality of {g)p and (g)T. For
this equality to be valid, it is evident that (g)z must be independent of the initial
time to and its phase point z@).
Birkhoff's theorem
Birkhoff's theorem addresses these questions.52 The theorem rests on the as-
.A.
sumption that the displacement transformations T(t) for the system at hand
are metrically transitive (or, equivalently, that the energy surface is metrically
.A.
indecomposable). The transformation T(t) on the energy surface is metrically
.A.
transitive if the only sets left invariant under T(t) are the whole set or sets of
measure zero.53 Alternatively, we may say that the energy surface is metrically
indecomposable if it cannot be divided into two invariant parts each of pos-
positive measure. An invariant part of phase space means that all points in this
part remain in that part during the motion of the system. With this assumption,
Birkhoff's theorem states that the limit (g)T exists everywhere on the energy
surface, except for a set of measure zero, and that (g)r is independent of the
initial time to as well as the initial phase point z0.
Furthermore, the theorem implies that
r)r = (8)r (8.9)
Since (g)r is independent of z0, it is constant on the energy surface:
(g)z — A — constant
Thus
and we obtain
= (8)r (8.9a)
52G. D. Birkhoff, Proc. Nat. Acad. Sci. U.S.A. 17, 656 A931). For a more ex-
extensive discussion of these topics, see I. E. Farquhar, Ergodic Theory in Statistical
Mechanics, Wiley-Interscience, New York A964).
53I. Oxtoby and S. Ulam suggest that almost every group of continuous transfor-
transformations is metrically transitive: Ann. of Maths. 42, 874 A941). M. Kac indicates
that it is virtually impossible to decide if a given Hamiltonian generates metrically
transitive transformations: Probability and Related Topics in Physical Theory, Wiley-
Interscience, New York A959). G. Sinai has argued that this property is satisfied by an
aggregate of colliding hard spheres: Statistical Mechanics, T. Bak, (ed.), Benjamin,
MenloPark, Calif. A967).

3.8 Further Remarks on Irreversibility 243
With (8.9), the latter relation gives the desired result.54
r = (8)r (8.10)
The significance of this relation to irreversibility is as follows. First note that
(8.6) is an equilibrium distribution.55 It follows that in writing (8.10), we assert
that, independent of initial conditions, the time average goes to the equilibrium
average (8.7). This result identifies (8.6) as a preferred state, which is an alien
notion to reversible dynamics.
The following property is important to subsequent discussion. A bounded
system with no integral of the motion other than the energy is ergodic.56 The
reason for this is as follows. Let there be an additional constant of the motion,
B, other than the energy for the given system. Then the system point moves
on the hypersurface which is the intersection of the surfaces B — constant and
H = E. This intersection is a subdomain of the energy surface that the system
point is constrained to move on, and the motion is nonergodic. Ergodic motion
results if this constraint is removed.
We may conclude from these statements that, for an ergodic system, points
move about the energy surface visiting all domains of equal measure with
equal frequency with no preference given to any single domain. Whereas
this description suggests that a given distribution will spread with time to
cover the whole energy surface, counterexamples57 exhibit distributions that,
although ergodic, retain their initial geometrical shape and do not approach an
equilibrium distribution.
3.8.2 Mixing Flow and Coarse Graining
Mixing flow causes any initial distribution to spread throughout the energy
surface. Note in particular that mixing flow is ergodic but ergodic flow is not
always mixing. A system is mixing if for any two functions h(z) and /(z) in
C2, and defined on H — E,
lim / A(z)/[z@]rf£= H=E Jh=e (8.11)
± n(E) JH=E U [(£)]2
An immediate consequence of this definition is as follows. In the above equal-
equality, let /(z) be a nonequilibrium distribution with normalization (8.6a). Then
54BirkhofFs theorem does not imply that the system point passes through every
point in the energy surface. Thus, whereas (g)T involves a single trajectory, it is
known that a curve that cannot intersect itself cannot fill a surface in two or more
dimensions.
55In equilibrium statistical mechanics (8.6) incorporates the principle of equi-a
priori probabilities; that is, all states on the energy surface are equally probable. The
corresponding distribution is called the microcanonical distribution.
56For further discussion, see N. G. Van Kampen, Physica 53, 98 A971).
57See, for example, L. Reichl, A Modern Course in Statistical Physics, Chapter 8,
University of Texas Press, Austin A980).

244 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
with (8.11) we may write
(h) = I h(z)f[z(t)]dY —► —— f h(z)dY (8.12)
Jh=e '^±o° V(E) Jh=e
Thus, for mixing flow, (h) approaches an average with respect to the
equilibrium distribution (8.6): / = [/jl(E)]~1.
However, the following should be borne in mind. For either ergodic or mix-
mixing flow, the ensemble density does not change in the neighborhood of a moving
phase point. This is a consequence of the Liouville equation, which, in addition
to incorporating the incompressibility of the ensemble fluid, further implies
an unchanging Gibbs entropy C.39). The mixing flow of an initial ensemble
distribution in phase space with time is often compared to the mixing of water
and oil. After sufficient stirring,58 the fluids appear to be homogeneous. How-
However, more precise observation reveals that the oil and water have remained
separate. Furthermore, we may envision a careful stirring that reestablishes
the original configuration.
Irreversibility in classical mechanics my be attained under coarse graining.
Coarse graining begins with dividing the energy shell into subdomains [En]
such that the nth subdomain has volume Qn. If D(z, t) is the ensemble density
at the time t, then the coarse-grained density is given by
n(z, 0 = — / D(z',t)dzf (8.13)
We note that n(z, t) is constant over each cell and has the same normalization
as D(z, t).
The Gibbs's coarse-grained entropy is defined by
ri(t) = Y tonnn\nYln (8.14)
n
which may alternatively be written
r){t) — I n(z, t)\nU(z, t)dz
r
= / D(z,t)lnTl(z,t)dz (8.14a)
Consider the special case that D(z, t) is initially constant over each cell.59
Then D(z, t) — n(z, 0) and we may write
/ Y\(t)\nY\(t)dz- I
rj(t)-rj(O)= / n(t)\nTl(t)dz- / D@)\nD@)dz
58-
5For consistency of this model, the oil must not separate into disconnected
segments at any time during the stirring. See Problem 2.7.
59In this event, D is a simple set function. For further discussion, see R. L. Liboff,
J. Stat. Phys. 11,343 A974).

3.8 Further Remarks on Irreversibility 245
[D(O In n(O dz - D(t) In D(t)] dz (8.15)
where we have employed (8.14a). Adding
/ Ddz- / Tldz = 0
to the right side of (8.15) gives
n
ri(t) -
dz (8.16)
Since 1 + In y > y and D > 0, we may write
•nit)- 7/@) < 0 (8.17)
We may conclude that, with D initially constant over individual cells, if r\
changes, it decreases. Furthermore, it ceases to increase when I~[(z) is uniform
(see Problem 3.16).
It is evident that the irreversible statement (8.17) is a consequence of the
coarse-graining definition (8.12). In going into this representation, the level
of information on the state of the system is reduced. It is further evident that
Il(z, t) is less informative than D(z, t) is about the state of the system. It appears
to be a general rule that irreversibility occurs when working in a representation
that is less informative than the Liouville picture.
3.8.3 Action-Angle Variables
In this section, we consider finite systems that occupy a bounded region of
phase space and that undergo periodic or nearly periodic motion. A system
of coordinates and momenta particularly well suited to periodic motion are
action-angle variables. These variables are closely related to a formalism called
Hamilton-Jacobi theory. In this formalism, we seek to discover a canonical
transformation that renders all new coordinates (qf) cyclic. As all new momenta
are then constants of the motion, Hamilton's equation A.1.12) is simply inte-
integrated to yield the orbits in the new coordinate frame, that is, q' — q\t). The
dynamical problem is then reduced to algebraically transforming the q' — q'(t)
motion to the original q — q(t) frame.
Such a canonical transformation may be obtained from an appropriate gen-
generating function. Consider specifically the function G2(q, p') (introduced in
Section 1.2.1) relevant to a system with TV degrees of freedom. For uniformity
with commonly adopted notation, G2 is relabeled S. Recalling A.2.14), we
write
Pi = \ F (8.18)
dp, (8.19)

246 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
We assume that this generating function renders all momenta constant. Let us
call these constant momenta aN, in which case S = S(qN, aN). Assuming the
motion to be separable, we write
(8.20)
The relation (8.18) gives
Pi = ~ (8-21)
d
or, equivalently,
Si = / Pi dqt (8.22)
The motion is periodic if pi (qi, aN) is a periodic function of g, (rotation) or is a
closed orbit in the (qi, pt) plane (vibration). In either event, the integrals (8.22)
about cyclic orbits are evidently constant and are taken as the new momenta.
These are assigned the symbol 7, and are called action variables.
(b
i = Ji = (b pi dqi (8.23)
With the new Hamiltonian cyclic in new coordinates, we write
H' = H'(JU...,JN) (8.24)
Coordinates conjugate to action variables, /,, are called angle variables,
labeled 0,-. With (8.19), we write
dS(qN, JNr
Pi =
(8.25)
Hamilton's equations for the new coordinates are given by
= v,-Gi,..., JN) (8.26)
which are simply integrated to yield
@t =ViT + ®i@) (8.27)
To discover the physical meaning of 0,-, we evaluate 5,0y, the net change
in ®j due to carrying q{ through a complete cycle.
5,0 =^rfft = (H) ——- d9/
which, assuming separability (8.20), reduces to
8
(8.28)
j

3.8 Further Remarks on Irreversibility 247
So ®j changes by unity if qt goes through a complete cycle, but remains
unchanged otherwise.
Let T/ be the period of oscillation of qt. Then with (8.27) and (8.28) we have
0i-(ti-)-0i-(O) = i;i-ti- = 1 (8.29)
Thus we may identify i>; as the frequency of oscillation of qt. Angular frequency
is given by coi = 2nv{ so that, with (8.27), 9t — 2n®t may be identified with
angular displacement of the periodic motion. When written in terms of these
angular variables, (8.26) through (8.29) become
. . . ,
(8.26a)
(8.27a)
(8.28a)
2n (8.29a)
In (8.26a), H' was written for 2n x W of (8.26).
A note of caution is in order at this point. In the canonical mapping
(q, p) —> (/, 9), the variable 9 is a many-valued function of old variables
so that the transformation (q, p) —> (/, 9) is not one to one and 9 is not a
well-defined dynamical variable. This situation is easily remedied for peri-
periodic motion. Consider that the 9 interval corresponding to (q, p) periodicity
is 2n. The mapping is rendered one to one either by introducing the function
9 mod27r or by working with any trigonometric function of 9. The function
9 mod 2n of, say, 9 = 9\ + n2n, where n is an integer and 9\ < 2n, is 9\.
It is evident that a well-defined canonical transformation must be a one to
one mapping between old and new variables. Thus, for example, if the mapping
(q, p) ++ (9, J) is not one to one, then knowledge of a constant in the @, J)
frame does not determine an image constant in the (q, p) frame.
As described in Section 3.1, the two-body problem is reduceable to the
motion in a plane of an effective particle with reduced mass /x. It might be
thought that action-angle variables are relevant only to interactions that give a
closed orbit in the plane of motion. The fact is that of all attractive interactions
of the form V = Krn only n — 2 (harmonic oscillator) and n — — 1 (Kepler
problem) give closed orbits.60 However, for nonclosed orbits, r(t) and 9(t) may
still be periodic but with incommensurate frequencies, in which case action-
angle variables remain appropriate. (See Problem 3.17.) We now turn briefly
to the technique of obtaining the generating function S, which renders all new
coordinates cyclic.
60V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag,
New York A978).

248 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
3.8.4 Hamilton-Jacobi Equation
For all generating functions previously discussed, old and new Hamiltonians,
H and //', respectively, are related through H' = H + dS/dt. With S not
explicitly dependent on time, we obtain H = H'. We may identify the constant
H' with, say, the first of the ordered constant momenta of the system. Calling
this constant E, we obtain
/ dS dS \
H I q\, ..., <?#, -—,..., -— I = E (8.30)
V d dqN)
This equation is called the Hamilton-Jacobi equation for Hamilton's charac-
characteristic function S. The solution to this first-order partial differential equation
in TV independent variables is a function of TV constants. Thus we may write
S = S(qu ...,qN;au •••,«#) (8.31)
This is the desired structure of S with {a?,} identified as the new constant
momenta and new coordinates given by q\ — dS/dott. As previously assigned,
the new Hamiltonian H' = o?i, which, as sought, is cyclic in all q[. Note in
particular that
./= dH'
Pi
(8.32)
., dHf da,
dpt da
which give
dS
O(X\
(8'32a)
= ft 0"
Harmonic oscillator
Let us apply the Hamilton-Jacobi equation (8.30) to the problem of finding
the orbit q = q(t) of the harmonic oscillator with the Hamiltonian
H = -*— + -^- (8.33)
2m 2
where m is mass and k is the spring constant. The relation (8.30) becomes
1 /dS\2 kq2
— — I + — = E
2m \dqJ 2
which gives
IE
S(q, E) = Vmk I dqJ q2 + A (8.34)

3.8 Further Remarks on Irreversibility 249
where A is a constant of integration. Differentiating, we find, with (8.32a),
8S 1 f dq
1 f
co J
1
- sin q
co V IE
coj y/{2E/kJ - q2
~K
where we have set
9
co2 = -
m
Inverting (8.35) gives the desired solution:
q =
In action-angle formalism,
J =
J =
as
CO
E
-
V
The new Hamiltonian is
H\J) = vJ
IE
which on differentiation returns (8.27) for angle variable 0
(8.35)
(8.35a)
(8.36)
(8.37)
(8.37a)
3.8.5 Conditionally Periodic Motion and Classical Degeneracy
We return to the discussion of Section 8.1 relevant to systems that are separable
(8.20) and whose independent coordinates execute periodic motion.
Consider a dynamical function of the state of this system, F(qN, pN). When
expressed in terms of action-angle variables, this function is periodic in the
angle variables with the period of each 6t variable equal to 2n. Thus the
function F(qN, pN), when expressed in terms of action-angle variables, may
be expanded as a multiple Fourier series:
00
00
(8.38)
n\=—oo
where {w,} are integers. The time dependence of this function is obtained from
(8.26) and (8.27). Again, setting H' = E gives
r\ jr-i
a
(8.39)

250 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
where a, is a constant. Substituting these value into (8.38) and absorbing {at}
into the Fourier coefficients A^j gives the time-dependent form
F(t) = 2^' •' Z^ A"'-"» exP[/f(n>wi + • • • + nNcoN)] (8.40)
where, with (8.26a), we have labeled
dE
cot = — (8.40a)
Thus we see that each term in the sum (8.39) is a periodic function with
frequency
H \-nNcoN (8.41)
Since these frequencies are in general not commensurate, the sum (8.40) is not
in general a periodic function. However, with Poincare's recurrence theorem
(Section 3.6), we may conclude that after a sufficient time the system, in
developing from a given initial state, will return arbitrarily close to the initial
state. Thus the dynamical function F(t) is said to be conditionally periodic.
Degeneracy
In the event that two or more frequencies are commensurate (for example,
h\Q)\ — h2a>2, where h\ andh2 are integers), the system is said to be degenerate.
For complete degeneracy, all frequencies are commensurate, and we may write
conr..nN = co\(ni + n2k2 + n3k3 -\ \- nKkN) = scox (8.42)
where kt and s are rational numbers and a>\ represents the minimum of the
values {o)\}. In this event, F, as given by (8.40) is periodic. Let us calculate the
frequency of this periodic oscillation. Toward these ends, we set s — ha/hb,
where ha and h^ are integers, to obtain
(^) (8.42a)
So, for complete degeneracy, we may associate a frequency with each term in
the sum (8.40), which is equal to either the fundamental, co\/hb, or one of its
harmonics, ha(co\/hb). Thus the angular frequency of the function (8.40) for
complete degeneracy is co\/hb < co\.
Consider the specific example of twofold degeneracy with co\ — ka>2, where
k is a rational number. Again we set k = h2/h\. The frequency sum (8.41)
becomes
co = ^n + n3co3 + • • • + n^co^ (8.43)
n2
where n is an integer that replaces the h\ and h2 integers in the multiple
Fourier series (8.40). Thus degeneracy leads to reduction in the number of
independent frequencies for the system. As we will find immediately below,

3.8 Further Remarks on Irreversibility 251
a similar property pertains to the number of independent constants of motion
on which the energy depends.
Again considering the case that only two frequencies are commensurate [see
(8.40)], we write
dE dE
ri\ = n2
37i dJ2
This relations yields E = E(h2J\ + nxJ2). Thus
E = E(h2J\ +hxJ2, 73, ..., JN) (8.44)
and we may conclude that, when degeneracy occurs, there is a reduction of the
number of independent constants on which the energy depends.61
In addition to the preceding properties concerning degeneracy, we note the
following. With (8.39), we see that the form
dE dE
e2— (8.45)
dJ
dJ2 x
is a constant of motion. With degeneracy, h\co\ = h2co2, the preceding gives
Bn = -=-^ - hx8x - h282 (8.46)
If either 6\ or 92 changes by 2n, BX2 changes by an integer x In. So replacing
the right side of (8.46) with its value mod 2n gives a single-valued integral of
the motion. On the other hand, for nondegeneracy, it is evident that (8.45) does
not yield a single-valued integral of the motion.
An example of an additional constant for degenerate systems occurs for the
Kepler problem. In this case the additional constant is the Laplace-Runge-
Lenz vector62
A - p x L - /zrV(r) (8.47)
where L = r x p is angular momentum, /x is reduced mass, and the potential
V(r) = —K/r. With the Hamiltonian of the system given by
P2
H = f- + V(r)
2/z
61A similar situation occurs in quantum mechanics. The Hamiltonian of a particle
moving in a central potential is invariant under rotation about the origin. This in-
variance is a twofold symmetry. Consequently, eigenenergies are dependent only on
the principal quantum number and are independent of orbital or azimuthal quantum
numbers.
62Properties of this vector are discussed by R. J. Finkelstein, Nonrelativistic
Mechanics, Benjamin, Menlo Park, Calif A973). See also H. Golstein, Classical Me-
Mechanics, 2nd. ed., Addison-Wesley, Reading, Mass. A980). An additional example
of this degeneracy property is given in Problem 3.17.

252 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
FIGURE 3.26. Angular displacements lie on the surface of a torus, which is a two-
dimensional hypersurface in four-dimensional phase space.
the equation of motion A.1.25) returns A = 0. Incorporating relations among
the constants A and L and the energy E indicates that only one scalar property
of A is independent of L and E as, for example,
A2 = [M2K2 + 2mEL2 (8.48)
This observation permits the following conclusion. The constant A2 is an
example of a dynamical function that commutes with the Hamiltonian but
that, nevertheless, is not a function of the Hamiltonian. When coupled with
A.1.26g), the preceding remarks indicate that whereas G = G(H) =>►
[G, H] = 0 the converse is not in general a valid statement.63
Invariant tori
Consider a system with two degrees of freedom with coordinates q\,q2, which
individually execute periodic motion. The related angular displacements 6\,
02 (mod27r) lie on a constant toroidal surface, which may be identified with
the constant actions J\ and J2 (see Fig. 3.26). If the motion is degenerate,
then h\O)\ = h2(i>2. It follows that in the time interval 2nh2/a)\, over which
6\ makes h2 revolutions, 02 makes h\ revolutions, resulting in a closed orbit.
As the @, J) <-> (q, p) mapping is one to one, a closed orbit likewise results
in (q, p) space, with which we may associate a new constant J value. Again,
degeneracy leads to an additional constant of motion.
63The same rule applies in quantum mechanics. For a particle moving in a central
potential, [Lz, H] = 0, which gives d(Lz)/dt = 0. The degeneracy of the eigenstates
of H stems from rotational symmetry, which infers that H is invariant under operation
by Lz. This is the reason for the null commutation property, and not that Lz = LZ(H).

3.8 Further Remarks on Irreversibility 253
3.8.6 Bruns's Theorem
The three-body problem refers to the following configuration: three bodies
of given masses interacting under attractive potentials and executing bounded
motion. This system has 9 degrees of freedom and, in general, trajectories are
given in terms of 18 constants of motion. Ten such constants are given by
Pt
P, — -R, L, E (8.49)
M
where R, P, and M are center-of-mass displacement, momentum, and mass,
respectively, L is total angular momentum, and E is energy.
In 1887, Heinrich Bruns showed that the only algebraic integrals for the
three-body problem are the ten known integrals (8.49).64 We may conclude
that, in general, the three-body problem is nonintegrable so that the TV-body
problem is nonintegrable for N > 3. [See Problem 3.61.]
Integrable motion may be defined as follows. If a canonical transformation
to a representation exists in which all coordinates are cyclic, then the motion
of the related system is said to be integrable. For bounded integrable systems,
we may identify new momenta with action variables so that the transformed
Hamiltonian appears as
H = H(J\, ..., 7/v)
Therefore, in general, a bounded system is integrable providing a canonical
transformation exists in which the new Hamiltonian is given as above. As
discussed in the previous section, when such constants are expressed in terms
of angle variables, they are referred to an invariant tori (see Fig. 3.26).
3.8.7 Anharmonic Oscillator
We may gain deeper insight into the preceding discussion through study of
certain anharmonic oscillator systems. Thus, for example, consider a system
with two degrees of freedom with the Hamiltonian65
p\ + q\ + q\) + qxq\ ~ -q\ (8.50)
WH. Bruns, Berichte der Kgl. Sachs. Ges. der Wiss. A887). See also H. Poincare,
Les Method Nouvelles de la Mechanique Celeste, Gauthier-Villars et fils, Paris A892,
1893, 1899). Reprinted by Dover Publications, New York A957), Vol. 1, Chapter
V; E. T. Whitaker, A Treatise on the Analytical Dynamics of Particles and Rigid
Bodies, 4th ed., Dover, New York A944). Note: Special solutions to the three-body
problem were shown to exist for restricted motion on a plane by Lagrange A772).
This configuration was cast in a symmetric more revealing form by R. Broucke and
H. Lass, Celestial Mech. 8, 5 A973). For further discussion, see D. Hestenes, New
Foundations for Classical Mechanics, D. Reidel, New York A986).
65M. Henon and C. Heiles, Astron. J. 69, 73 A964). A closely allied problem was
examined by E. N. Lorentz, J. Atmos. Science 20, 130 A963).

254 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
0.4
0.2
0
-0.2
■0.4
-0.4 -0.2 0
-0.4 -I
0
0.2 0.4 0.6
Pi
0.4
0.2
0
0.2
0.4
//.#••.'•••£■
-
• #
•* •* .* •. *
• • ••*• • •
\. *** • **•
■ \k..-.'
i I*—■—-1
w's.'-'-V •.'•••' O\
* /•.; / . • . * •••. /
'•;; ':->^E = 0.16667
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
FIGURE 3.27. Development of chaotic motion corresponding to the Henon-Heiles
Hamiltonian (8.50). From M. Henon and C. Heiles, Astron. J. 69, 73 A964). Used
by permission of Carl E. Heiles.
Trajectories lie on the surface H — E — constant in four-dimensional f-
space. We may obtain a two-dimensional image (that is, a Poincare map) of
the trajectory in the following manner. Observe the trajectory each time it
passes through the qi — 0 surface with p2 > 0 in the q\, p\ plane. (We will
refer to this plane as the q\p\ plane.) If the energy is the only constant of
motion, then the projection of the system point is free to roam about the q\P\
plane corresponding to the surface H = E. However, if an additional integral
of motion exists, then the system trajectory lies on the three-dimensional hy-
persurface which is the intersection of this constant and the energy surface.
The intersection of this hypersurface and the q\p\ plane is a smooth curve.
So if plots of trajectory points in the q\p\ plane show no systematic order,
we conclude that the system has only one constant of motion, H — E. If,
however, the trajectory points follow a smooth curve, we may conclude that
an additional constant of the motion exists.
Three numerical readouts due to Henon and Heiles are shown in Fig. 3.27. At
lowest energy, E = 0.0833, the existence of a smooth curve in the ~q^p[ plane

3.8 Further Remarks on Irreversibility 255
implies, to within computer accuracy, an additional constant of the motion.
Each closed loop in these figures corresponds to one trajectory (that is, one set
of initial data). As energy is increased, at E — 0.125, there is disintegration of
the smooth curves. The random dots correspond to one trajectory. At slightly
higher energy, E = 0.167, nearly all motion is chaotic, and the additional
constant of motion is lost. Note in particular that these numerical data indicate
a fairly abrupt transition to chaos.
3.8.8 Resonant Domains and the KAM Theorem
The loss of a constant of motion with the change in a parameter in the Hamil-
tonian, as in the preceding example, is well described as follows. With aid of
the transformation
Pi —
(8.51)
The Hamiltonian of a general two-dimensional anharmonic oscillator may be
cast in the form
H(JU J2,6U62) = HO(JU J2) + kV(Ju J2,0u02) (8.52)
where k is a coupling constant. The potential V is periodic in 0\ and in 62,
whereas Ho has a polynomial dependence on J\ and J2. At k = 0, in accord
with (8.39), we find66 (for / = 1, 2)
61 = (Oi(J\, J2)t + a,i
dH0 (8-53)
Before proceeding further, we introduce a more definite perturbing potential
in the form of a multiply periodic series [see (8.38)]:
, 72)cos(wi0i +n2e2) (8.54)
/il n2
The sum runs over all positive and negative integers n\,n2. Note that this form
of potential maintains periodicity in 6\ and in 62.
Let us attempt the construction of a generating function SG/, 72', 0i, 02),
which renders 7( and J2' constants of the motion for sufficiently small values
of the interaction parameter k. We choose the form
S = 7@1 + J202 + 2_^£^ B"\m sin(ni6>j + n282) (8.55)
n\ n2
^Note that cot entering the present discussion is not relevant to the entire
Hamiltonian (8.52) but only to the unperturbed term, Ho.

256 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
and determine the coefficients Bnin2 to give the desired cyclic property. With
the preceding expression, we obtain
= J! + J2J2niB"' cos(n,^ + n292) (8.56)
ds
0 = —f (8.56a)
where 7/, 9- are new canonical variables.
Substituting the potential (8.54) into the Hamiltonian (8.52) and first
expanding the resulting form about Jt = 7/, we obtain
H = H0(J[, 72') + —rAJx + —^A/2 + • • • + kV
J\ OJ2
With A Ji inferred from (8.56) and keeping terms to 0(k), we find
(8.57)
Thus, to lowest order in A., 7( and 7^ are constants, providing we set
+
in which case
Expansions for the new action variables are obtained by substituting the
coefficients (8.58) into (8.56). There results
£
£ E , n',":, cos(«i^i + «202) + 0(A2) (8.59)
n2
It follows that the choice of S as given by (8.55) with coefficients (8.58)
has the desired property of rendering J[ and J'2 constants for sufficiently small
X. However, note that with (8.53) coj = &>/Gi, 72), and it is possible that
n\(O\ + n2(i>2 may vanish. In general, the series (8.59) diverges if
|wi<»iGi, J2) + n2co2(Ju h)\ < ^niVn,n2 (8.60)
The region of phase space where (8.59) is obeyed is called the resonance zone,
and n\co\ + ^2^2 = 0 is the resonance condition.61 In the resonance zone, 7/
as given by (8.59) ceases to be a well-defined constant and chaotic behavior
ensues.
Suppose, in the expansion of J' given by (8.59), we exclude small regions
in phase space about each resonance given by (8.60). In this event, we obtain
67This resonant condition is evidently equivalent to degeneracy of Ho previously
described in Section 8.3.

3.8 Further Remarks on Irreversibility 257
a well-behaved expansion for J[. For smooth potentials, the amplitudes Vnxn2
decreases rapidly with increasing n\ and n2, and we may expect that, with
larger n\, n2, smaller regions of phase space are excluded, thereby extending
the domain of convergence of the series (8.59).
In three independent works, A. N. Kolmogorov,68 V. I. Arnold,69 and
J. Moser70 (KAM) established the following: In the limit X —> 0, and pro-
providing Ho has a nonzero Hessian (or, equivalently, that &>/ has a nonvanishing
Jacobian)
det
= det
a2//,
o
37/37/
/0 (8.61)
then the measure of excluded phase space approaches zero, or, equivalently,
we may say that the invariant tori of Ho are conserved.71 In this event, we may
expect orbits to remain well behaved in the limit X —> 0 over most of phase
space. Note that this theorem is consistent with (8.60). In the opposite case
that X grows away from zero, the theorem states that not all invariant tori of
Ho are destroyed. The ensuing motion of the system is a consequence of the
remaining and lost invariant tori.
Two significant properties of resonance zones are as follows: Consider two
points in phase space with arbitrarily small initial relative displacement. For
chaotic motion, the subsequent displacement of trajectories stemming from
these initial points grows exponentially with time.72 This property is relevant
to points within a resonant zone.73 As this property maintains for arbitrarily
small initial displacement, we may conclude that there is no finite-difference
integration accuracy that suffices to retrace a trajectory back to its initial point
in a resonant zone.
Second, it has been ascertained74 that, as the number of resonances grow
dense in phase space, integrals of the motion become nonanalytical or patho-
pathological, as do their intersection. As this intersection represents the system
68A. N. Kolmogorov in Foundations of Mechanics, R. Abraham (ed.), Benjamin,
Menlo Park, Calif. A967). See Appendix D.
69V. I. Arnold, Russian Math. Surv. 18, 9 A963); 18, 85 A963).
70J. Moser, Akad. Wiss. Gottingen II, Math. Physik Kl. 1 A962).
71 For further discussion of this theorem, see G. M. Zaslavsky, Chaos in Dynamical
Systems, Harwood Academic Publishers, New York A985); and H. G. Schuster,
Deterministic Chaos, Physik-Verlag, Weinheim A984).
72The log of this exponential displacement is called the Liapunov exponent. For
further discussion see: S. Chandra (ed.), Chaos in Nonlinear Dynamical Systems,
SIAM, Philadelphia A984).
73This behavior was demonstrated for the Henon-Heiles system by C. H. Walker
and J. Ford, J. Math. Phys. 13, 700 A974).
74I. Prigogine, From Being to Becoming, W. H. Freeman, San Francisco A980).
This work includes a descriptive overview of irreversibility and a comprehensive
reference list.

258 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
trajectory, it is evident that neither the trajectory nor its time reversal are well
defined for this situation.
Problems
3.1. For the Hamiltonian given by A.7) and potential of interaction given by A.17),
with K = —\K\, offer a geometrical argument that illustrates that orbits are
bound for E < 0 and (conditionally) unbound for E > 0. Hint: Introduce the
effective potential
Vcffir) = V(r) + -^
2/xr2
and make a sketch of Vef{(r) versus r.
3.2. (a) Show that the scattering matrix
S = / - 25*f
has the following properties:
f = 5, 5 -5 = 7
(b) What consequence do these properties have on the collision statement
3.3. Show that the external force K drops out in the derivation of C.31).
3.4. This problem addresses the group property of the time-displacement operator
T(t) introduced in Section 3.6 (where they were labeled Tn) and in Section 8.1.
Show that these operators satisfy the group property T(t\)T{t2) = T{t\ + t2)-
Answer
Consider the form T(At2)T(Ati)Q0, where Ar, and At2 are small time inter-
intervals and £20 represents a volume in T-space. Let (q0, p0) e Qo. Integrating
Hamilton's equations A.1.20) over At\ gives
qx = q{Atx) = q0 + Hp(q0, po)Atl = q0 p
with a similar equation for p\. We have written Hp fordH/dp. For the second
displacement, we find
q2 = q} + Hp(qu p\)At2
Substituting for (q], px) in //p gives
q2 = qx+ Hp[q0 + Hp@)Atu Po - H
q2 = qi + Hp@)At2 + OiAtiAh)
Dropping the last term and substituting for q\ gives
+ At2)

Problems 259
with a similar relation for p2. These are the same forms that result for t(At] +
At2)Q0.
3.5. The value of viscosity for liquid D2 at 20K is rj ~ 370 x 10~6 g/cm-s.75
How does this value compare with that obtained from Maxwell's formula
D.33)? The diameter of a D2 molecule may be taken to be ~ 3.5 X. Offer an
explanation for this (good/bad) agreement.
3.6. Show that the following five integrals represent identical tensors. In these
expressions, G is a tensor independent of c, and F = F(c2).
A) / Fcc(cc : G)dc B) / Fcc(cc : G)dc
( Fw(k:G)dc B) f
o o /» Pan
C) — Gs / Fc4dc D) / Fcc(cc:G)dc
15
E) lf}s l(cc:cc)Fdc
Iz5z f
-Gs I
We have written
which is traceless and
A) - ^
which is symmetric and traceless as well. The transpose of A is written A
—
Hint: See Appendix B, Section B.I. Note that if A is symmetric then A = A,
3.7. (a) Show that the Sonine polynomials
(m +rc)!(-jty
occur as coefficients of expansion of the generating function
where s is a positive number less than unity. That is, show that
00
S(x) = J2 S^ix)s"
n=0
3.8. Show that the coefficient a in E.58) has the value -15/4.
Answer
First note that
n jn
75
H. M. Roder et al., NBS Technical Note 641 A973).

260 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
With E.56), we then obtain
With the values of S%\x) given beneath E.54a), we note
_ 5
" 2 ~
Thus
OO
Let x = £2 so that 2£2 dt- = *Jx dx, whence
2
2 E/2)! 2 5 3 1
1! V^2 2 2
15
3.9. Show that the coefficient fix in the equation preceding E.67) has the value |
From E.64),
§4§4
Again, with jc = f2, we obtain
-
VV/ ^' 8
_ 5
~ 2
3.10. Establish E.61). That is, show that Au = -4n2^B2).
3.11. A vial of radioactive O19 breaks. Employing mean-free-path relations, estimate
the velocity (cm/s) with which the O19 diffuses through air.

Problems 261
3.12. Show that the collision frequency v(£) for interacting rigid spheres is given
by F.26).
3.13. Show that substitution of the Legendre expansion G.21)
into the left side of the equilibrium Boltzmann equation G.22) yields the right
side [keeping terms to P
eE
m
df0 1 (dU ^ 2/
dv 3 V dv v
3.14. (a) Obtain an integral expression for the normalization constant K in the
absolute Druyvesteyn distribution
fo(x) = K(x
mv2
.—x
kBT
in terms of n, T, and m. Hint: Recall the normalization
f(\)d\ = n
J
and the connecting formula
zeedfo
f\(v)= —
m dv
(b) What does your expression for K reduce to in the limit of no electric
field?
3.15. Establish the equality
/ g(v2)\\d\= - / g(v2)d\
where v is a unit vector, / is the identity operator, and g(v2) is any scalar
function. (A generalization of this result is given in Appendix B, Section B.I).
3.16. Show that the coarse-grained entropy rj(t) as given by (8.14) is minimum
when n(z) is uniform. Hint: Let n(z) be uniform. Introduce a variation n -»
FT = n exp/c, where k is an arbitrary function of z. With n and W obeying
the same normalization, obtain
= f
Srj= (KeK + 1 -eK)U(z)dz
Then show that this form is greater than or equal to zero.
3.17. Show that for two-particle potential motion, periodic in r, the angle variable
0@ satisfies the relation
0(t + T) = 0@ + A
where T is the period of r and A is a constant.

262 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Answer
With Lz = /zr2#, we obtain
=— I
M Jo
~\
0@=— / r~\t)dt
'o
It follows that
L ft+T
0(t + T) - 0@ = — / r~\t)dt = A
Note in particular that
p9{t+T) pO(t+T)
Je= ped0 = L2 dO = LzA
J9(t) J9{t)
which is constant.
3.18. A two-dimensional harmonic oscillator has the Hamiltonian
z*
Another constant of the motion, independent of //, is given by
A = H}- H2
(a) What is the Poisson bracket of A and //(I, 2)?
(b) Demonstrate a dynamical form that for degeneracy h\co\ = h2co2 gives
a third constant independent of A and //, but that in the event of
nondegeneracy, is not constant.
Answer (partial)
(b) Introduce the complex phase variable
Zj(t) = pj(t) + icojqjit), j = 1,2
The dynamical trajectory may be written
Consider the form
Bz
In the degenerate limit, we
Note that H and A may be
z2
- — ■
find that
written
/ =
1
Z2(
B
is constant
+ z2zV)
A = -(ziz[ - z2z*2)

Problems 263
With these relations, it is evident that the modulus of B is a function of
H and A, but that the phase angle of B is independent of H and A.
3.19. Show that the differential measure of the cross-collision integral of the
Boltzmann equation for a two-component fluid may be written
/»OO
agdQd\\ = I a8(AE)dqd\\
Jo
Here q is the momentum increment,
v',) = m2(v' -v) = q = Mu = /x(g' - g)
= ra1+ra2, /i =
M
and AE is the energy increment,
The integration is over the scalar component of q.
Recall
h(y)
I
h(y)8[f(y)-a]dy =
\df/dy\ j
y=yo
f(yo)=a
Answer
First note that
(See Fig. 3.4) and recall
& I S-^-5- ) = -S(8' ~ 8)
It follows that
/»OO
agdft = / cr8(gf - g)gf dg
Jo
Now
M
so that
a8
= f

264 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Note that
and we may write
agdQdY] = / aS(AE)d\] dq
which was to be shown. Inserting this finding into the cross-collision integral
gives the form [see B.16)]
J(f I F) = f(f'F[ - fFl)a8(AE)dvl dq
3.20. A point particle of mass m with speed v moves inside a spherical cavity of
radius a with perfectly reflecting walls. Initial normal displacement of the
particle trajectory from the origin (that is, impact parameter) is s.
(a) What are the constants of the motion for this system? Is angular
momentum of the particle constant? Why?
(b) Consider the coordinate components of the energy surface in V -space for
this system. Is the particle motion on this surface ergodic? Why? Describe
the exact motion of the particle.
(c) How do your answers to the above change if the cavity is a rectangular
parallelpiped?
Answers (partial)
(a) In addition to the kinetic energy of the particle, angular momentum, L,
is also a constant. The reason for this is that the force due to the wall is
radial.
(b) Since there is a constant other than energy, the motion is nonergodic. The
motion sweeps out an annulus that lies in a plane of constant orientation
and has inner radius s.
3.21. (a) Employ C.40) to calculate the change in entropy corresponding to the
expansion process depicted in Fig. 3.14. Assume that the process occurs
with both gases in Maxwellian states at fixed temperature T. Let the
number of A and B molecules be NA and NB, respectively, with NA = NB,
and call the total volume V.
(b) How does your answer change if labels in the molecules are removed
(Gibb's paradox)?
3.22. Establish the following properties of the symmetric strain tensor D.6a).
(a)
p : Vu = ^P : A
(b) If T = 7/7, then ~P : A = pW . u, where

Problems 265
(c) Evaluate the Tr A.
3.23. Relevant to the Chapman-Enskog expansion [see E.44)], show that
3.24. Show that the functional operation C.2)
iD>) = jj(fL>(y)dv
has the following properties:
(a) 7@)=i/@' + 0;-0-
(b)
3.25. Show that the stress tensor
/=
S = 2rjl A- -/V-u
vanishes if u = ft x v, where ft is a constant angular velocity vector and A
is the symmetric strain tensor D.6a).
3.26. (a) The equipartition theorem assigns kBT/2 units of energy on the average
to each degree of freedom for a system in equilibrium. Using this rule,
obtain the value of cv, the specific heat per molecule, for a gas of rigid
diatomic molecules.
(b) For gaseous N2 at room temperature, viscosity rj = 1.78 x 10~4 g/cm-s.
Employing elementary mean-free-path estimates, obtain the value of
thermal conductivity k for a gas of N2 at these conditions.
3.27. Consider a collection of N noninteracting particles confined to cubical box
of edge length a, with perfectly reflecting walls. The speeds of the particles
have values {%•}, 1 < / < N. Express the motion {x{(t)} of this aggregate of
particles in action-angle variables.
3.28. (a) Construct a simple kinetic model to show that the xx component of stress
interior to a gas at rest is
pxx = 2m / d\± / dvxvxF{\)
where v± = (vy, vz).
(b) The working definition of pxx is that it is the force exerted on a unit area
within the gas whose normal is parallel to the x axis. State a property of
F{\) at the surface of the test area that permits the preceding integral to
be written
pxx =m d\± dvxv2xF(\) = mn(v2x)
J J—00
3.29. Employing the continuity equation, together with the energy equation C.31),
derive the adiabatic law E.23c).

266 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Answer
Our starting equations are
9 \
h u • V Ek + Ek V • u + pV • u =0
dt )
d
u- V I n + nV -u =0
Substituting
p = nkBT = -Ek
into the first of these relations gives
whereas the continuity equation gives
D
V • u = In n
Dt
Eliminating V • u from these latter two equations gives the desired result:
D ( P
3.30. Under what conditions is a dynamical system with N degrees of freedom
integrable?
3.31. The orbits of a dynamical system with two degrees of freedom are given by
4,() cos
q2 = g2@)cos
(a) If co2 = 2(O], what is the nature of the motion in the qx — q2 plane?
(b) What is the minimum frequency associated with this motion?
3.32. A fluid is in a state of shear. A narrow slab of the fluid moves in the x direction
with the velocity profile
where a and r are characteristic length and time intervals, respectively, and
w = y/a. The slab is centered at w = 0. The related xy component of stress
in the fluid has the value
2
-2* (I)
Sxy = 2.6p ( - ) e~w2/2
where p is the mass density of the fluid. If a = 2.4 X, and the thermal speed of
molecules C = l.5a/r, estimate the value of the mean free path of molecules,
/, in the fluid. What are the assumptions of your estimate?

Problems 267
3.33. Employ the Hamilton-Jacobi equation (8.30) to solve for the orbit of a particle
of mass m in the one-dimensional potential well
where a and k are constants.
3.34. Repeating the development leading to the Fourier transform of electrical con-
conductivity D.72), derive a parallel expression for viscosity for a fluid in a state
of shear. Identify all terms in your derivation. Show that your finding returns
Maxwell's expression D.33) in the mean-free-path estimate.
3.35. For attractive potentials A.21a), the nondimensional inverse minimum
displacement appears as
N
Show that this equation has no positive solutions for N > 2, b < 1, and
E > 0.
3.36. Consider the scattering of two particles with respective masses m\ and m2.
The angle of scatter in the lab frame of particle 1 is the angle between p',
and pi. Calling this angle 0\ and the corresponding angle of scatter in the
center-of-mass frame 0, show that
m2 sin0
tan0, =
m\ + m2 COS0
3.37. Calling the cross section in the lab frame aL and that in the center-of-mass
frame <jc, obtain the equality
oL(9x)dcos6\ = ac@)dcos0
3.38. What is ac@) for:
(a) Scattering of rigid spheres with respective diameters aOi and
(b) Scattering of Maxwell molecules of mass m?
(c) Coulomb scattering of particles with charge q and mass ml
3.39. Show that BY] [the first equation in the sequence B.1.20)] implies the three
conservation equations C.14) C.18), and C.19). Note that the pressure tensor
~p and heat flow vector q must be redefined to obtain these equations. See, for
example, B.1.34).
3.40. Show that the Q(Lq) cross section integrals E.62) satisfy the equation
dT
3.41. (a) Equilibrium values of a fluid at rest are h = n0 and f = To. Working with
the Euler equations and assuming one-dimensional motion, obtain linear
equations of motion for the perturbation variables n, u, and 7\ where ,
after a small disturbance, h = n0 + n, u — 0 + w, and f = To + T.

268 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
(b) From your equations, show that the sound speed a is given by the
isentropic derivative
2
a =
dp
where p is mass density.
3.42. A bomb containing 1 mole of argon atoms at pressure p = 3 atm and temper-
temperature T = 300K has a release valve of circular area 10~47rcm2. The valve is
open for 10~4s. The gas cools and then returns to the ambient temperature of
300 k.
Data: Atomic mass of argon = 40 amu. 1 amu = 1.66 x 10~24 g. 1 atm = 106
dyn/cm2.
(a) Let the gas be ideal prior to the valve being open. What is the volume of
the bomb in cubic centimeters?
(b) Write down the Maxwellian distribution function / for the gas prior to
the valve opening. [The units of / are cm~3(cm/s)~3].
(c) Consider the flux /+, of particles that pass through the valve opening. We
may write
The superscript denotes the fact that we are looking for the average veloc-
velocity of particles in the vx > 0 direction. Use this expression to obtain the
mole fraction of particles that escape the bomb while the valve is open.
Sketch the function f(vx) immediately after the valve is shut,
(d) Approximate the immediate drop in temperature, 8T, of the bomb. Hint:
Assume (v2) is constant and guess the value of {vJ from your answer to
part (c). Introduce the speed C2 = 2kBT/m.
3.43. Let X denote an eigenvalue of the JC operator defined by F.29). Imagine circles
of varying radii drawn about the origin in complex A-space. What may be said
about the number of eigenvalues of JC in each domain described by these
circles?
3.44. In Section 2.2 an equation of state was given B.2.53) for a liquid in
equilibrium, which we now rewrite as
PK =nkBT
n2
nl f ,
6 J
Derive the latter expression for P$ from BY! B.1.24). That is, take the appro-
appropriate moment of BY! to obtain the momentum equation and put the resulting
collision integral in gradient form to identify /V

Problems 269
Answer
Operating on BY] with f done and performing a parts integration ont he
collision integral, we obtain
f dc]dc2dx2Gl2f2(\,2) = 0
k-N2 f
+
We may write this equation in the form
where we have set
V.P0 = -N2 £dc]dc2dx2Gnf2(\,2)
(Note that V is written for V]). For a nonequilibrium fluid, we assume
anisotropy in the fluid and write the following generalization of B.2.44):
, 2) = /o(c,)/o(c2)g(x,, x2)
Substituting this relation into the preceding equation and integrating out the
velocities gives
= n2
VP
n2 r
0 = — /
dx2Gl2(r)g(xl9x2)
Recalling the transformation introduced in Problem 2.14 from Xj, x2 -» r, R,
we write g(X],x2) —> g(r, R) and, with r = x2 — xu set <ix2 = dr. In
equilibrium, the radial distribution function goes to the isotropic form g(r),
which causes the preceding integral to vanish. To obtain a finite result, we
return to the form g(r, R) and expand about R = X]. There results
g(r, R) = g(r, *,)+-• V,g(r, x,) + • • •
(Note that r and X] are new independent variables.) Further setting G]2(r) =
rd<$>(r)/dr and substituting these relations into the preceding equation gives
= N2 f N2 f rr
V, • Po = -— J £/r<D'(r)?g(r, x,) - — V, • J dr&(r)-g(r, x,)
As the fluid becomes isotropic, g -» g(r) and the first integral vanishes. There
remains
N2
Taking the trace and passing to the thermodynamic limit gives the desired
result:
1 = n2 C°°
Po = - Tr Po = -— / dr47zr3<S>(r)g(r)
J o Jo

270 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
3.45. (a) Write down an expression for the force F on a small volume of fluid in
terms of a closed surface integral of the pressure tensor P.
(b) What form does your expression for F reduce to if the fluid is in
equilibrium and at rest.
Answer
(a) ¥ =
(b) ¥ = §pl-dS = 0
3.46. (a) An ideal gas of N molecules is in equilibrium at temperature T for which
(E) = \kBT and ((E—)E)J) = kBT(E), where E is single-particle
energy. Using the central-limit theorem, obtain the probability density
P(E, N) for this system, where E represents total energy.
(b) What are the assumptions that make this theorem relevant to this system?
(c) What is the variance a2 of the distribution you have found? In what
manner does cr/(E) make (E) a good thermodynamic variable in the
limit N -» oo?
Answers
(a) From the central-limit theorem, we obtain
1 [E - NE(E)]2
P(E, N) = = exp{-- \r-^
[2nND(E)y/2 2ND(E)
Inserting the given information, we find
1 [E-\NkBTf
3N(kBTJ
(b) Assumptions that permit use of the central-limit theorem are:
(i) E =
(ii) (Ej) are all equal.
(iii) (£>(£,)) are all equal.
(c) From our answer to part (a), we write
(E) = ^
so that
a 1
« 1
(e)
Thus the Gaussian is sharply peaked about E = (E) so that E is effec-
effectively constant and therefore may be taken to be a good thermodynamic
variable. Note: For the system at hand, E must be positive. However, from
part (a) there is a finite probability that measurement find E < 0. The

Problems 271
central-limit theorem addresses the case N ^> 1, in which case (E) grows
large and P(E < 0) ~ 0.
3.47. (a) Obtain the expression for viscosity that the repulsive-core model E.86)
gives for N = 4 (Maxwell molecules). Set K = V0a4, where Vo and
a are constant energy and length, respectively. [Hint: You should find
rj(T) = constant.]
(b) What value of viscosity does your formula give when applied to a gas of
argon atoms at room temperature? Take a = 1 X and Vo = 10 eV. Give
your answer in cgs units. [Hint: You should find rj(T) = AT~]/3.]
3.48. Employing fluid dynamical equations derive the following:
(b) m— + Vh - TVa - -V • 5 = mK
Dt n
where raK is an applied force field and a, h, and n~l are entropy, enthalpy,
and volume per particle. The heat flow vector is Q C.25).
(c) For a = constant and K conservative and S symmetric, show that
[where D/Dt is the conservative derviative given by C.20a)]. From this
result, show that
<b
JC@
ud\ = 0
where C(t) denotes a closed loop moving with the fluid (Kelvin's
theorem).76
Hint: Recall the energy equation C.31), the defining relation for the stress
tensor D.4), and the first law of thermodynamics
dEK = T da — pdn~l
Enthalpy per particle, /z, is given by
h = EK + £
n
and EK denotes kinetic energy per particle.
3.49. Using the moment expansion E.103), show that for the special case of
Maxwell interaction [see A.27)] the related collision integral E.102) reduces
to
rB)
76For further discussion, see L. D. Landau and E. M. Lifshitz, Fluid Mechanics,
Section 8, Addison-Wesley, Reading, Mass. A959).

272 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
where
b\ — n / go sin2 0 cos2 0 dO
Jo
3.50. Determine if the following equations are reversible. That is, show that, if y(t)
is a solution, then so is y(—t). (Primes denote time derivatives.)
/a pt
y(x)dx (c)/= / y(x)dx
Jo
(b)/ = / xy(x)dx (d)/ = / y(x-t)dx
Jo Jo
Answer
In part (a) we question whether y(—t) is also a solution. Substituting y(—t)
for y(t), we obtain
dy(-t)
dt
or, equivalently,
x)dx
which is not Eq. (a), so y(—t) is not a solution and Eq. (a) is irreversible.
Alternatively, we might have shown that the given equation is not invariant
under the time-reversal transformation, t -» t' = — t. In a manner similar to
the first, we find that (b) is irreversible, (c) is reversible, and (d) is irreversible.
Note that differentiation of (a) gives a reversible equation.
3.51. Is the Boltzmann equation B.14) reversible? That is, if / = /(x, v, t) is a
solution, is /(x, —v, —t) also a solution?
3.52. Is Grad's second equation B.6.12) reversible. That is, if /i(vi, t)\ f2(\\, v2, 0
is a solution, is /i(-Vj, -t)\ /2(-V], v2, -t) also a solution? If your answer
is yes, explain this result in light of the fact that Grad's second equation is
approximate.
3.53. Derive an expression for the Hamiltonian of a freely moving rigid sphere of
radius a and mass M in terms of the kinetic energy of its center of mass and
the spin angular momentum S. State explicitly what the variable S2 is written
for.
Answer
From Problem 1.7 we know that the Hamiltonian of the sphere may be written
P
2
where the subscript rel denotes motion relative to the center of mass. This
relative motion is purely rotational, and for a rigid sphere we may write
H
nl

Problems 273
where / is the moment of inertia, which for a sphere is ^Ma2. Angles (#, 0)
specify the orientation of the rotational frequency vector of the sphere, u?, in
terms of which T = IaJ/2. Taking derivatives of the preceding gives
30
which gives the Hamiltonian
T 2
= /(sin 0H =
0
2/ 2/sin20 2/
3.54. This problem concerns "rough" and "smooth" hard spheres, where the quoted
adjectives refer to the nature of surfaces of respective spheres. As noted in
Section 2.3, smooth spheres do not exchange angular momentum in collision.
Let all spheres have volume r = 47ra3/3, mass M, and moment of inertia
/. Consider a gas of N smooth spheres confined to a volume V ^> Nz in
equilibrium at temperature T.
(a) Write down the Hamiltonian for this fluid.
(b) Write down an integral expression for the pair distribution function for
this fluid, /2(x1,x2;S1,S2).
(c) If spheres in the fluid described above are rough, is your answer to part
(a) maintained? If not, why not?
Answers
(a) With reference to Problem 3.53, we write
1 = 1
where
and
0(jc) = oo, for x < 0
0(jc) = 0, for x > 0
(b) The equilibrium N-body distribution function (see Problem 2.5) for this
fluid is given by
fN(l, ..., N) = A
where AN is a normalization constant

274 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
Thus, with r — r12, we obtain
f
/2(P,,P2,S,,S2,r)=
J
Integration over P] and P2 gives the desired result. See Appendix B,
Section B.3.
(c) For rough spheres there is interaction between angular momentum of
spheres in collision, and the above expression for HN is not appropriate.
3.55. (a) Which dynamical systems corresponding to the following Hamiltonians
satisfy the criterion for the KAM theorem?
(b) What is the consequence of the failure of a dynamical system to satisfy
this theorem?
(c) Are any of the orbits corresponding to these systems homoclynic (see
Problem 1.45)?
(i) Harmonic oscillator.
p2 kx:
H = — + —
1m 2
(ii) Pendulum'.
.2
H = -^- - mga cos 0
2ma2
where 0 = 0 corresponds to the gravity direction.
(iii) And Henon-Heiles Hamiltonian:
4\ . 2
T+<M2
where we have written
2 7-2 2 r\
Hi — ^iXi > Pi —
with x and p denoting physical displacement and momentum,
respectively. Hint: In part (iii), introduce the change of variables
Q\ = q\ +qi, P\ = P\ + Pi
Qi = q\ — qi-> Pi — P\ — Pi
to obtain
= Hl(QuP])+H2(Q2,P2)
1 1
3
I
3.56. Show that the Sonine polynomials S{£\x) satisfy the equation (deleting
subscripts and superscripts)
xS" + {m + 1 -x)S' + nS = 0

Problems 275
3.57. With /o denoting the Maxwellian,
fo(v,x)= —
e~/2cl
V B;rK/2C3
where mC2 = kBT and V is the volume of the fluid, show that:
(a) (v) = 3C2
(b) (vJ = £ C2
3.58. Show that the generalized Gibbs entropy
is constant in time, providing the functional \//(f) has a bounded derivative.
Answer
First note that
= A — lni/)
Dt YJ df Dt
Operating on the Liouville equation with
dx//
I
d\--dN{\
df
establishes the result.
3.59. (a) What is equilibrium distribution fx for a gas of N identical molecules of
mass m and temperature T confined to a volume VI Assume that f\ is
normalized to unity.
(b) The gas described in part (a) adiabatically expands to a volume 2V and
comes to equilibrium at the same temperature T. What is the new distri-
distribution, 7i ? Employing the Boltzmann H function, calculate the entropy
change of the gas, AS, due to expansion.
Answers (partial)
(a) With R = kB/m,
___}___ e-v2/2RT
VBnRTf/2
(b) Calculating
AS= -kRN
I
fx\nfxdxdy - I fx\nfxdxd\
gives the result
AS = kBN\n2
3.60. Three particles move on a plane and interact through conservative forces.
In general, are there sufficient constants of motion to render the system
integrable? Justify your answer.

276 3. The Boltzmann Equation, Fluid Dynamics, and Irreversibility
3.61. The Liouville theorem77 (concerning integrability of system) states that for a
system with N degrees of freedom, if N independent constants, including the
Hamiltonian, exist, each of which has zero Poisson brackets with each other,
then the motion of the system may be reduced to quadratures. (See top p. 244.)
(a) Consider Problems 1.1 and 1.8. In each case verify the preceding theorem
and reduce the second to quadrature.
(b) Do the 10 constants for the three-body problem (8.49) satisfy criteria
for the Liouville theorem? How many of these constants do satisfy the
criteria?
Hint: In Problem 1.1 note than px(r, 0) is constant. In Problem 1.8 consider
motion relative to the center of mass.
Note: Previously we have noted that 2N constants are required to specify the
trajectory in f space. The Liouville theorem states that N constants satis-
satisfying certain conditions, are sufficient to reduce the problem to quadratures.
A quadrature is an integral which, for each degree of freedom, implies an
added constant bringing the total to 2N. In action-angle formalism, these 2N
constants are [/,; 0,@)]. See (8.24) et seq.
3.62. (a) Consider a force-free spatially homogenous fluid at rest. Employing the
equipartition theorem, the conservation equation C.19) and the definition
of the coefficient of thermal conductivity, derive the heat equation,
deK 0
— = aV2eK
dt
(b) What is your expression fora?
(c) What property of the structure of molecules in the fluid does your
expression for a assume?
Answer (partial)
2k
(b) a =
3nk
B
3.63. (a) The diffusion coefficient is written in terms of the autocorrelation function
in D.53). This relation is more formally written
1 C°°
D= -lim / dt e~8t (v@)
3 s-oo Jo
Write down the corresponding autocorrelation function definition of the
coefficient of viscosity, rj [see D.7) et seq.], related to shear force in the
x direction due to ^-momentum transported by x velocity components.
In this construction introduce the momentum transfer component Txy =
77For further discussion see, V. I. Arnold, Mathematical Methods of Classical
Mechanics, Chapter 10, ibid A978).

Problems 277
(b) Write down a component force equation in which this expression for r\
enters. What are the dimensions of the expression for r] you have written
down. [Check your answer with Maxwell's result D.33)]
Answer (partial)
= lim -L /"
0 ' V/ /C DM / /\
oo
.St
dte-dt(rxy@)rxy(t))
where the average is taken with respect to an equilibrium distribution.
Note: Numerical simulation calculation78 of the velocity autocorrelation func-
function has observed short-time exponential decay (over a few collision times)
and a slower t~3/2 long-time delay. This secular long-time decay of the
autocorrelation function is labeled "long-time tails.9
78B. J. Adler, and T. E. Wainwright, Phys. Rev. Lett. 18, 988 A967).
79M. H. Ernst, E. H. Hauge and J. M. J. van Leeuwen, Phys. Rev. A4, 2055 A971).

CHAPTER 4
Assorted Kinetic Equations
with Applications to
Plasmas and Neutral Fluids
Introduction
The notion of a plasma was considered previously in Section 2.3 in describing
fundamental plasma intervals and kinetic equations relevant to these intervals.
The present chapter begins with applications of the Vlasov equation to an
equilibrium charge-neutral plasma.
The plasma frequency and Debye length emerge in construction of the
Fourier-transformed dielectric constant. The Debye parameter appears as a
shielding length in the construction of the perturbed Coulomb potential due
to the presence of an extraneous charge in the plasma. The dielectric constant
further permits analysis of unstable and damped modes. Landau damping of
an electric wave in a plasma is described, and the section concludes with the
Nyquist analysis of unstable modes.
Various kinetic equations relevant to the description of a plasma are de-
derived. Interrelations between these equations are obtained and illustrated in a
flow-chart diagram (Fig. 4.9). The Fokker-Planck (FP) equation is obtained
in expansion of the Boltzmann equation about grazing collisions. Equivalence
between the FP and Landau equations is demonstrated. The Balescu-Lenard
equation is obtained from an expansion of the hierarchy equations incorpo-
incorporating construction of the E-field auto correlation function. Divergence of
plasma kinetic equations is discussed, and techniques for removing related
singularities are reviewed.
The Krook-Bhatnager-Gross equation is applied to shock waves in a neutral
fluid, and the chapter continues with a rederivation of the FP equation from the
Chapman-Kolmogorov equation. This derivation illustrates the application of
the FP equation to systems other than plasmas.

4.1 Application of the Vlasov Equation to a Plasma 279
Due to its present-day widespread application, the final section of the chapter
addresses the Monte-Carlo technique in kinetic theory. The discussion con-
concludes with a flow chart for determination of the distribution function of a
fluid immersed in an external force field.
4.1 Application of the Vlasov Equation to a Plasma
In this first section we employ the previously derived Vlasov equation in ap-
application to various wave and particle problems in plasma physics. Thus we
consider a charge-neutral plasma comprised of electrons and ions in equilib-
equilibrium at a given temperature. The plasma suffers an infinitesimal perturbation,
and the resulting electric field and electron distribution are examined, which
gives the very important plasma response dielectric function. When applied
to perturbation in the form of an extraneous charge, this function yields the
Debye potential, which reveals that the Coulomb field in a plasma is exponen-
exponentially attenuated with a scale length equal to the Debye distance. The dielectric
function is next applied to waves in a plasma, and the dispersion relation for
long-wavelength longitudinal waves is found, which reveals that dispersion of
waves grows with temperature. It is then shown that single-peaked electron
distributions (in velocity space) are stable to infinitesimal perturbations.
The Vlasov equation is returned to and, working with a Laplace-Fourier
transform, it is concluded that waves in a plasma experience (Landau) damping.
An expression for the damping rate is obtained, and a physical description is
included, which ascribes the damping of the electric-field wave in a plasma to
increased kinetic energy of electrons.
The section concludes with a derivation of the Nyquist criterion, which,
stemming from a theorem of complex analysis, offers a geometrical construc-
construction for the number of unstable modes in a plasma. Application of this result
is made to multipeaked electron distributions.
4.1.1 Debye Potential and Dielectric Constant
We consider a charge-neutral plasma of electrons and ions in equilibrium at
some temperature T'. Ion (+) and electron (—) distribution functions are given
by (assuming singly ionized atoms)
+
r v2
L 2R+
' V2
2R-
T.
T
R± = kB/m±, / F^d\ = n0

280 4. Applications to Plasmas and Neutral Fluids
We assume that distribution functions satisfy the Vlasov equations [see
B.2.31)]:
DF+ e „ dF+
- +—£ -=0 A.2a)
Dt m+ d\
DF_ e dF.
£ = 0 A.2b)
Dt m- d\
where e is written for \e\ and we have set
d d d
Dt dt dx
Gauss's law links the distribution functions to £.
V £ = Ane[n+ — n_] = 4ne
d\
A.3)
The equilibrium value of the electric field is zero, and n± represent number
densities.
We assume that the plasma suffers a small perturbation away from
equilibrium. Variables assume the form1
£ = 0 E
Due to their large mass, the ions are assumed to remain in their unperturbed
state. Substituting the forms A.4) into A.2) gives the 0(£) equations (deleting
the superscript on Fo~):
3 e dF(\
+v-—g E ^=0 A.5a)
dt dx m d\
V-E = -Ane f gd\ A.5b)
Substituting the transforms2
TV4 f dk f da)^^ k' vy(kx-wf) A.6a)
n p j j
—. f dkdQ)E((o9k)ei0iX-0>t) A.6b)
In) J
into A.5), we find
E =
B:
(a> - k ■ y)g = i-E ■ d-p- A.7)
m \
JThe present discourse is restricted to linearized theory. For further discussion,
see R. C. Davidson, Methods in Nonlinear Plasma Theory, Academic Press, New
York A972).
2Here we are assuming a response that persists indefinitely.

4.1 Application of the Vlasov Equation to a Plasma 281
We may decompose the E vector as
E) (kxE)xk
k2
Assuming that only longitudinal waves emerge in the perturbation, A.7)
gives
- = i(g/m[(kE)/t]kCF0/3v)
to — k • v
With
^o = nofo(v)
we find
df0
^0
d\ d\
and A.9) assumes the form
ien0 (k E)k df0
g = •
m k2(co — k • v) d\
We assume that the perturbation is due to an extraneous electron. The
corresponding charge density may be written
P = ps+Pp A.11)
where
ps = -e8(x-\ot) A.12)
represents the electron source and pp denotes the charge density of the
remaining plasma:
Pp = -e gd\
The transform of the delta function A.12) is
ps(k, co) = -e f S(x - yoOe'^-^dxdt A.13)
Integrating first over x, we obtain
ps(k,co) = -e f e-^0-"* dt
A.14)
ps(k, go) = —2ne8(oo — k
The generalization of Gauss' law to account for the extraneous test charge
A.12) appears as
+ pp) = -4Tte8(x-y0t)-47Te I gdv A.15)

282 4. Applications to Plasmas and Neutral Fluids
With A.16) and A.14), the transform of this equation becomes
ik E = -eSn28((o-k'\0)-4ne gd\ A.16)
With A.10), we obtain
.colk-E f k-dfo/d\d\
ik E = -eSnz8(oo - k • v0) - t " 0 I " ~J"' A.17)
k1 J co — k • v
where cop represents the plasma frequency7* (deleting the subscript on n0):
Anne1
A.18)
Solving A.17) for k • E gives
,k = 8^kv0)
i , *£ r k-dfp/dvdv
1 "•" k2 J w-k-v
The response of the plasma to the test charge may be given in terms of the
dielectric response function defined by
e(k, co) = ps(k, co)/p(k, co) A.20)
The transform of A.15) then gives
ik ■ E = 4"Mk' M) A.21)
s(k, co)
Rewriting A.19) in the form [see A.14)]
, co)
i , ^ r k-dfo/d\d\
' k2 J (D-k-Y
A.22)
permits the identification
e(k, co) = 1 +
col f k-dfo/d\d\
I
k2 I co — k • v
A.23)
which is our explicit expression for the plasma response dielectric function.
The response electric field
With the decomposition A.18), the transform integrals of the electric field
assume the form
E(x, t) = JL_ JJ dkdaie"**-**)^* A.24)
3Encountered previously in Section 2.3.9.

4.1 Application of the Vlasov Equation to a Plasma 283
Substituting the expression A.21) into A.24) gives
E(x, 0 = // i A.25)
BttL JJ k2e(k,co)
Integrating first over co gives
lie f keik<x~yQt) dk
E(x, t) = —- / —-— A.26)
(InJ J k2e(k,k-\0)
With the plasma dielectric constant A.23), and mC2 = kBT, and
cxp(-v2/2C2) 3/o v -
we find
(v - v0)
If the extraneous charge is stationary, Vq = 0, then
C2k2
Thus A.26) becomes
2e fikeikxdk
2e fi
Solving for $>(x) gives the response potential
BttK
-2e feikxdk
a i ^ 1 . p f k • v/ody n
e(k, k • v0) = 1 + t^tt / -— A.
C2k2 J k • ( )
A.29)
A.30)
i
Here we have introduced the Debye wave number
2
A.31)
C2
Setting
E(x) = -V* = -V—l— I i>(k)eikxdk
BnK J
= -tAt [iki>(k)eikxdk A.32)
BnK J
and comparing with A.30) gives the transform
A.33)
A.34)

284 4. Applications to Plasmas and Neutral Fluids
t
V(r)
Coulomb potential
Debye shielded potential
FIGURE 4.1. Bare Coulomb and Debye shielded potentials.
In spherical coordinates with x taken as the polar axis, we find
,/k x
dk
f
= 2n I d cos 0
In r°°2ksmkx
L
00 dkk2eikxcose
f
Jo
—
'o *- + kld
Substituting into A.34) gives the desires result (writing r in place of jc):
\-kdr)
A.35)
V(r) = (-
= - exp(-kdr)
where V(r) denotes potential energy (see Fig. 4.1). This is the Debye potential
that surrounds a charge in a plasma. The distance
is called the Debye distance and represents a shielding length of the Coulomb
potential. [Recall B.2.43).] With A.18) and A.31), we find
C
kBT
kBT
mcoi Anne2
A.36)

4.1 Application of the Vlasov Equation to a Plasma 285
This is an important parameter in plasma physics. We note, in particular,
A.^ = 6.9 ( — j cm (T in K)
(T\x/2
^ = 7401-1 cm (rineV) A.37)
4.1.2 Waves, Instabilities, and Damping
Waves in a warm plasma
We next consider the response of a plasma to an arbitrary perturbation. Thus,
omitting the point-charge delta function in A.16) and A.17), the result A.21)
may be rewritten
e(k,(o)k-E = 0 A.38)
The plasma dielectric constant e(k, co) is given by A.23).
Equation A.38) indicates that a finite longitudinal electric field will be
present in the plasma, providing
e(k, a>)=l + -£ ^ = 0 A.39)
k1 J co — k • v
With A.27), we obtain
/
k v/o d\
°
-0 A-40)
C2k2 J k-Y-oo
To reduce the integral, we choose k to lie along the Cartesian z axis. We may
then integrate fo(v) over vx and vy. There results
POO /»<
J — 00 J —
'oo poo e-/i2/2
dvx dvyfo(v) =
-oo ./-oo V 2jtC
Where4
A.41)
Which permits the dispersion relation A.40) to be written
col 1 f°° dinxe
^=/ =0 A.42)
C2k2
Here we have set
Ca:
4Recalling the nondimensional velocity f C.5.28), we note f2 = /x2/2.

286 4. Applications to Plasmas and Neutral Fluids
We consider the limit /3 ^$> 1 corresponding to co/k/ ^> C. Rewriting A.42)
as
,2
1=
C2k2
-f
^ J
A.43)
then permits the Taylor series expansion
1 =
A.43a)
Only the even integrands contribute to the integration, and we obtain to terms
i =
This equation may be rewritten
CO'
C2k2
1 3
IP2 f>4\
A.44)
co2 = co2p +
A.45)
Again expanding about small k gives
= (jolp + 3Czk
A.46)
This is the dispersion relation that governs the propagation of longitudinal
waves in a plasma at finite temperature and in the said co — k domain.
A theorem for unstable plasma modes
The transform equation A.6) indicates that the growth of waves in time (insta-
(instability) corresponds to Im(eo) > 0, or equivalently, Im/3 > 0. Thus, to examine
unstable modes, we set
= a + /t,
r >0
A.47)
Our theorem states that an arbitrary single-peaked, one-dimensional equilib-
equilibrium distribution /i(/x) gives no unstable modes. Retracing steps leading
to A.42), we find that the generalized dispersion relation appears as [the
one-dimensional velocity /x is defined by A.41)]
1 =
'I
C2k2
'00
-00
/. =
/i
A.48)
where a prime denotes differentiation.5 In a Maxwellian state, f\ —
exp(-^2/2). Substituting the form of $ given by A.47) into A.48) and
'Note that ~f\ is the one-dimensional analog of f0 as given in A.27).

4.1 Application of the Vlasov Equation to a Plasma 287
-\L> 0
FIGURE 4.2. In either case, (/xm - /x)f! > 0 for a single-peaked distribution.
rationalizing the integrand gives
1 =
d\i(\i — a + it)//
2n J-oo (n-ot- /t)(/x - a + it)
1 =
P
2
P
V 27T J-oo
I
L
oo
Here we have set
2 /
C)t
Equation A.49) gives the two equations
2
'OO
-00
0 =
d\x~f[(\i - a)
- aJ + r2
dn~f[
A.49)
A.50)
A.51a)
A.51b)
) be the maximum value of ~f\. Multiply A.51 b) by r ' (a — \xm) and
add the result to A.51a). There results
1 = -
2
P
'In J-
I.
00
- n)~f{
oo
A.52)
0^
K /i(/x) has only a single maximum, then
> 0
A.53)
^ the integrand of A.52) is always positive, whence it has no solution. We
play conclude that an infinitesimal perturbation of a single-peaked distribution
Insults in no unstable modes (see Fig 4.2).

288 4. Applications to Plasmas and Neutral Fluids
4.13 Landau Damping**
If we assume that the perturbation g(t) initiates at t = 0, theng(f < 0) = Oand
E(r < 0) = 0. In this case it is appropriate to work with the Laplace-fourier
transform, and we write
'00
Jcot
g(co, k)= I dxe~ikx f dte'°"g{x, v, f)
v t/ (J / t C A\
A-54)
E(co, k) = / dxe~lKX / dtel(OtE(x, t)
J Jo
The inverse of the time component of g(co) is given by7
= — / g(to)e-ia" dco A.55)
i(O]—oo
with a similar expression for E(f). The line co = ico\ in A.55) lies above the
singularities of g(co) in the co plane. This condition ensures8 that g(t < 0) = 0
(see Fig. 4.3).
Multiplying A.5a) by the exponential operator in A.54) gives
_ 1E .?£ | = 0 A.56,
3
dt dx m
Integrating the first two terms by parts and neglecting surface terms, we find
*f E • ^ + ig@)
g(co,k) = A.57)
co — k • v
where the initial distribution g@) remains a function of v. Inserting this
expression into the divergence equation
/k • E = — Ane I gd\
and recalling A.8) gives
_k.E= J M-ky A.58)
e(k, co)
with s(k, co) given by A.23).
If this form is substituted into the companion equation to A.55) for the
transform of the longitudinal component of E, again we find that its time
development is determined by the zeros of s(k, co) (that is, the poles of k • E).
6L. Landau, J. Phys. (USSR) 10, 25 A946).
7The Laplace transform was encountered previously in the discussion concerning
the resolvent operator (Section 1.5.4).
8Consider the pole, Imco > co\.Fort < 0, exp(—/&>r) gives convergence in A.55)
for Imco > 0. Thus, completion of the contour in A.55) in the upper half-plane would
give a finite contribution to g(t < 0).

4.1 Application of the Vlasov Equation to a Plasma 289
Im
ingularities
of g M
Re co
FIGURE 4.3. The line co = icox lies above the singularities of g(co).
Im
t
\i plane
Re
FIGURE 4.4. The analytic continuation of the integral in A.42) for Im ^ < 0.
Here we envision that the contour of A.55) shown in Fig. 4.3 is closed in a
semicircle in the lower-half co plane.
So once more we are led to the dispersion relation A.42). With the Laplace
l transform A.54), we see that convergence now demands that Imco > 0. Thus
We realize that the integral in A.42) is defined with Im£ > 0.
, However, with A.55) we note that decaying modes correspond to Im/? < 0.
iThe integral in A.42) for this case is evaluated by analytic continuation. This
fis effected through distortion of the contour of A.42) to that shown in Fig. 4.4.

290 4. Applications to Plasmas and Neutral Fluids
There results
1 =
>l
m
/•OO
J—oo
dji jie
+ In if [(ft)
A.59)
Where we have set /\ = exp(—/x2/2). For weakly damped modes, ft hovers
just beneath the Re /x axis, and the pole in the integration is well represented
by one-half the residue at that point. For Im ft <$C Re ft, we obtain9
1 =
"In
L
oo
oo
dji jxe
-H2/2
A.60)
In this same limit, the expansion A.43a) applies and A.60) may be written
1
A.61)
Assuming f[(ft) <$C 1, in the first approximation we neglect the imaginary
contribution, which again yields the solution A.46), which we now write as
ft2 = ft2p + 3 A.62)
This is the first-order solution to A.61). The second-order solution is con-
constructed by reinserting this solution into the imaginary term in A.61). We
obtain
n ~. -
A.63)
or, equivalently,
~ -|l/2
Expanding the radical about small /,' gives
A.64)
To within this approximation, we find
Imj8 =
A.65)
9This equation is exact for Im ft = 0. We assume that it is also appropriate to
the said limit with finite Im ft. Note also that the integrals in A.59) and A.60) are
principle values.

4.1 Application of the Vlasov Equation to a Plasma 291
+ gW)
FIGURE 4.5. Variation in fx due to Landau damping in linear theory.
which, with the assumed Maxwellian form for fu gives the desired result:
\mco =
= -r,
A.66)
The exponential form of the solution as given by A.55) indicates that, at a given
frequency, perturbation fields decay as exp — Ff, where F, the decay constant,
is defined by A.66).
The form of the solution as given by A.65) suggests that damping will occur
for f[ < 0. Thus, with f[ negative for /x > 0, we expect the perturbation to
be damped for v = C/x = v^ = co/k > 0. Here v^ denotes phase velocity.
At these values, the E wave is damped and electric energy is lost to particle
motion. This situation is depicted in Fig. 4.5, which illustrates the manner
in which the perturbation alters the equilibrium distribution function. At the
phase speed i^, the electric field loses energy, and electrons are driven to higher
speeds, thereby depleting the region v < v<p and increasing the region v > v^.
For Vq = co/k < 0, Pp < 0. Since // > 0 for v < 0, again A.65) indicates
wave damping.
Unstable modes
A brief recapitulation is in order: We have solved A.60) for /3 through an
approximation scheme in which its is assumed that Im /3 ^C Re p. The first-
order iterate in this solution A.62) gives the real part of p. This leads to Im ft
as given by A.65).
From this result, we many conclude that damping results if p3pf{(J3) < 0. If
the reverse of this inequality holds, then we may expect the mode at /3 = ft to
be unstable. (Unstable modes are discussed in the following section.)
For a single-peaked distribution, we find fi3pf{(P) < 0 for all ft, thereby
recapturing our previously stated result that single-peaked equilibrium distri-

292 4. Applications to Plasmas and Neutral Fluids
bution functions are stable. A double-peaked distribution may or may not be
unstable.
4.1 A Nyquist Criterion
This criterion is a scheme for determining the number of unstable modes as-
associated with a given unperturbed distribution function. The criterion stems
from the observation that, for an arbitrary analytic function yfr{z) with isolated
poles
A.67)
where Nz denotes the number of zeros of i//(z) within the contour C and Np
denotes the number of singularities within C.
From A.39), we infer that the zeros of the dielectric constant e(k, oo) are
solutions to the plasma dispersion relation. With A.48), we may write
9 A.68)
e(P) = I +/32pF(P)
where the function F(/3) is as implied.
It s(f$) has no singularities in the upper-half /3 plane,10 then, by A.67),
1 f
/
2*i Jc
c
represents the number of zeros s(/3) has in the upper-half /3 plane. But by
previous argument, each such value of /3, with Im /3 > 0, corresponds to an
unstable mode. The contour C includes the entire real /3 axis and an infinite
semicircle in the upper-half K plane (see Fig. 4.6).
Convergence on the semicircular contour A demands11 that fie''/e —> 0 with
increasing /3. With A.68) we find that this is the case and we may proceed with
the evaluation of A.69).
Now
j[W L[d!L A70)
s 2ni
In the 8 plane, the function s has a simple pole at the origin. So the right
equality in A.70) indicates that Nz is equal to the number of times the contour
C encircles the origin in the s plane.
10It is conventional in the Nyquist analysis of a plasma to label e(fi) as H(fi). The
present description stays with e(/3).
11 See, for example, I. Sokolnikoff and R. Redheffer, Mathematics of Physics and
Modern Engineering, 2nd ed., McGraw-Hill, New York A968).

4.1 Application of the Vlasov Equation to a Plasma 293
Re C
FIGURE 4.6. The contour C in A.67) has a circular contribution A and a line
contribution B.
We may use A.68) to define a mapping from the /3 to the s plane. Specifically,
we wish to discern what curve C the contour C becomes in the e plane. Let
us perform this exercise for the single-peaked distribution described following
A.59).
As discussed above, the contour C for the /3 integration in A.69) and A.70)
has two components. An infinite semicircle, which we have labeled A, and the
real axis, labeled B (see Fig. 4.6).
With the single-peaked distribution inserted in F(/3), as defined in A.68),
we see that F(P) -> 0 on A.
On B, with /3 real, we recall A.60), which may be rewritten as
A.71)
where V: denotes "principal part."
Thus the dielectric constant A.68) along B becomes
= [\
A.72)
so that
_«W2/2
A.73)
which is an odd function of p. Furthermore, £_@) > 1 and real. In this manner
we obtain a sketch of the mapping of C into C as shown in Fig. 4.7.
So we find that the curve C in the e plane does not encircle the origin for
single-peaked distributions, and we again conclude that such equilibrium states
are stable.

294 4. Applications to Plasmas and Neutral Fluids
lmC
The contour C C plane
t
Im e
E=1
The contour C
e plane
Re C
Im e
= 0 Re e
FIGURE 4.7. Inferred sketch of C for a single-peaked distribution
A Nyquist analysis for a double-peaked distribution was performed by J. D.
Jackson12 and was further described by T. H. Stix.13 The results of this study
are summarized in Fig 4.8. From these sketches, we see that the inner loop
widens in the e plane and drifts toward the origin as the peaks of ~f\ separate.
We may conclude that a sufficient separation of the two peaks is necessary for
instability to occur. The related phenomena is called the two-stream instability.
4.2 Further Kinetic Equations of Plasmas and Neutral
Fluids
In the discussion to follow, we find that certain limiting forms of the Boltzmann
equation lead directly to kinetic equations relevant to neutral fluids and plas-
plasmas. The first equation derived in this scheme is the Krook-Bhatnager-Gross
equation, appropriate to near-equilibrium states.
For gases dominated by long-range interactions or, equivalently, grazing
collisions, we obtain the Fokker-Planck equation. With the aid of some tensor
properties of the kernel of the Fokker-Planck equation, we obtain the Landau
equation.
Returning to the BBKGY sequence, we finally obtain the Balescu-Lenard
equation appropriate to the "long-time" limit of a plasma. This equation is
found to include a shielding factor in its interaction integral representative of
the Debye shielding discussed previously. In the limit of large number, the
Balescu-Lenard equation is found to reduce to the Landau equation.
12J. D. Jackson, J. Nuclear Energy C, 7, 171 A960).
13T. H. Stix, The Theory of Plasma Waves, McGraw-Hill, New York A962).

4.2 Further Kinetic Equations of Plasmas and Neutral Fluids 295
Im e
Im e
FIGURE 4.8. The loop C encircles the origin in the e plane for sufficient separation
of two maxima of f\. Reprinted with permission of J. Nuclear Energy Cl, J. D.
Jackson, "Longitudinal Plasma Oscillations," copyright 1960, Pergamon Press, pic.
Logarithmic singularities of terms in plasma kinetic equations are discussed,
and a review of various techniques of removing these singularities is included.
Connecting routes between the various kinetic equations derived in this
section are graphically described in Fig 4.9.
4.2.1 Krook-Bhatnager-Gross Equation14
We recall the form of the Boltzmann collision integral C.3.1):
H = ff'f{dlJLl-JffldlJLl
= a dQg d\\
Near equilibrium, the fluid is close to a local Maxwellian state. The primed
component of J(/), relevant to the after-collision interval, describes a situation
closer to equilibrium than the unprimed component of J(f). Thus we set
/
B.1)
14
P. L. Bhatnager, E. F. Gross, and M. Krook, Phys. Rev. 94, 511 A954).

296 4. Applications to Plasmas and Neutral Fluids
BBKGY15
Kirkwood expansion
Bogoliubov expansion
Grad expansion
Prigogine expansion
Grazing
collisions
Diagram
expansion,
short-time
limit
No correlations
BBKGY
Correlation
expansion
Key
BE Boltzmann equation
BL. Balescu-Lenard equation
CK: Chapman-Kolmogorov equation
FP: Fokker-Planck equation
KBG. Krook-Bhatnager-Gross equation
LE: Landau equation
VE: Vlasov equation
Diagram expansion,
long-time limit
FIGURE 4.9. Relations among kinetic equations.
Since
/
= o
we may write B.1) as
B.2)
15J. G. Kirkwood, J. Chem. Phys. 15, 72 A947). N. N. Bogoliubov, Problems
of a Dynamical Theory in Statistical Physics, E. Gora (trans.), Providence College,
Providence College, Providence, R.I. A959). See also Section 2.4. H. Grad, Hand,
der Physik, vol. XII, Springer Verlag, Berlin A958). See also Section 2.5. I. Pri-
Prigogine, Non-Equilibrium Statistical Mechanics, Wiley, New York A962). See also
Section 2.3.

4.2 Further Kinetic Equations of Plasmas and Neutral Fluids 297
Turning to the unprimed component of /(/), we note, in general,
= /
/
B.3)
Next we recall that f° and / have the same first five moments. That is,
0
\
/
= f
d\
B.4)
This common property motivates the final step of our derivation: in B.3) we
set
/
/
B.5)
With the relation B.2) and substituting B.51) into B.3), we find
B.6)
which gives the KBG equation:
Dt
= v(v) [f(v) - f(v)]
B.7)
In this expression, we have written v(v) for the collision frequency:
v{v) =
B.8)
Conservation equations
Owing to the moment equalities between f° and / as given by B.4) and the
defining relation B.8), it may be shown that the KBG equation returns the
conservation equations C.3.14), C.3.30), and C.3.31). Specifically, consider
the case that v is velocity independent.
AT / d\
1 \
m\
mv
V
Dt
/ dn \
+ V nxx
dt
dpu
~dt~
dneK
-p
+ V q
d\
( 1
OTV
mv
(f ~f) =
< 0
0
B.9)

298 4. Applications to Plasmas and Neutral Fluids
It should be noted that, although the KBG equation appears simpler in form
than the Boltzmann equation, it still maintains a strong nonlinear quality. That
is, we note that the local Maxwellian, /°, is a nonlinear function of the first
five moments of /.
Time behavior of /
For a spatially homogeneous gas, the KBG equation becomes
B.10)
Multiplying through by the integrating factor evt gives
dfevt
dt
Integrating, we find
/(/) = f@)e-vt + f A - e~vt) B.11)
Thus, the KBG equation parallels the conclusion of the H theorem and indicates
that the fluid approaches a Maxwellian equilibrium state.
4.2.2 KBG Analysis of Shock Waves
We found previously (Problem 3.41) that relatively small disturbances in a
fluid propagate at the sound speed. This process is isentropic. For disturbances
of larger magnitude, the process becomes nonisentropic and the velocity of
propagation exceeds the sound speed. Such phenomena are called shock waves.
A parameter important to the description of shock waves is the Mach number,
M. This dimensionless parameter is the ration of the local fluid velocity u to
the sonic velocity a\
u
M = -
a
If M < 1, the flow is called subsonic. If M — 1, the flow is transonic. If
M > 1, the flow is supersonic, and if M ^> 1, the flow is hypersonic.
Whereas fluid dynamic equations provide a good description of changes in
macroscopic variables across a shock front,16 a kinetic theory must be called on
to examine corresponding changes in the distribution function. In the following
analysis, we offer a brief description of application of the KBG equation to a
one-dimensional shock wave. The fluid on either side of constant values of the
macroscopic variables: p, u, and T'.
We imagine a very long straight pipe filled with fluid that is at rest with a
piston at one end. The piston undergoes a momentary acceleration carrying
16H. W. Liepman and A. Roshko, Elements of Gas Dynamics, Wiley, New York
A957).

4.2 Further Kinetic Equations of Plasmas and Neutral Fluids 299
Downstream Upstream
us = Ma
M> 1
Shock front
(a) Lab frame
Upstream
up=us
Downstream
ud=us - U
... .,^te
x < 0 Shock front x > 0
at rest
(b) Shock-wave Frame
FIGURE 4.10. Steady-state shock-wave configuration. Analysis is performed in
frame (b) where the shock front is stationary.
it from zero to finite speed. The information that the piston has suffered this
acceleration moves ahead of the piston in the form of a shock wave. In the
subsequent steady-state configuration, the piston moves with constant velocity,
pushing fluid ahead of it with the same speed, up to the shock front. Ahead
of the shock front, information that the piston is moving has not yet reached
the fluid and the fluid ahead of the shock remains at rest (see Fig. 4.10). It is
conventional to view this steady-state configuration in the frame of the shock
front. We call this speed up ("upstream").17 The speed on the downstream side
of the front we label ud. Thus we have the boundary conditions
at x = +oo, u = ud = constant
B.12)
at x = —oo, u = up = constant
Since a steady-state condition maintains in this frame, the KBG equation
reduces to
^ B.13)
Assuming a Maxwellian distribution for velocities normal to the macroscopic
flow, we set
17These labels refer to flow in the frame of the shock wave. Labels are reversed
in the lab frame (see Fig. 4.10).

300 4. Applications to Plasmas and Neutral Fluids
Thus
/ d\±f(x,\) = g(x,vx) B.15a)
and
Substituting B.14) into B.13), with B.15a) we find
dg V V n
+ 8 = S
B.16)
v v
Here it is important to recall that g° is a local Maxwellian that contains actual
values of macroscopic variables. Such variables change in value across the
shock front.
Before examining B.16), we more carefully consider the collision frequency
v. With Maxwell's expression for viscosity C.4.33),
rj = -pCl
and / — C/v, mC2 = 3kBT, we find
nkBT
v =
1
Thus v = v(n, T), and we may conclude that in the present application v varies
with x. This property must be kept in mind when integrating B.16).
Multiplying B.16) by the integrating factor
fiX
exp
/ -dx'
Ja V
where a is an arbitrary length, gives
d
dx
gexp J -dx'\ = (exp J -dx'\ -g° B.16a)
In the microscopic velocity domain: v > 0, we integrate B.16a) over the
interval (—oo, x) to obtain
r d ( r v a r ( r v \ v
/ dx—- I ^exp / -dx I = / dx I exp / -dx -g
J-oo dx V Ja v ) J-oo \ Ja v J v
/x V r-oo px / C* V \ V
-dx' — g(—oo)exp / -dx' = I dx ( exp / -dx' I -g
n Ja V ./.oo V Ja V J V
For v > 0, exp f~°°(v/v) dx' —> 0, and we obtain
g(x, v) = ( exp — I - dx' ) I dx I exp / - dx') -g°
V Ja V J J-oo \ Ja V ) V

4.2 Further Kinetic Equations of Plasmas and Neutral Fluids 301
which gives, for v > 0
g+(x, v) f dx (exp - I - dx') V B.17a)
J-oc \ Jx V ) V
For v < 0, we integrate B.16a) over (oo, x). There results for v < 0,
g-(x,v)\ dxlexp- I -dx')-g° B.17b)
Joe \ Jx V J V
Number density and fluid velocity are contained in the equation
,00 ,0
n(x)u(x) = I dvvg+(x, v) = / dvvg_(x, v) B.18)
Jo ./-oo
Boundary conditions ar given by the relations
/•OO
= I g+(v, +oo)v dv
Jo
ripUp = /
^0
00
' dv
Since ^° contains n, u, and r, which in turn are moments of g, B.17)—B.19)
are integral equations for the distribution function. This formalism was applied
by Leipmann et al.18 to an argon gas. Equations were solved through numerical
iteration, giving good agreement with results of the Navier-Stokes analysis at
moderate Mach number (M = 1.5). At higher Mach number (M = 5.0),
deviation from the Navier-Stokes equation was found to occur in the high-
density side of the shock wave.19
4.2.3 The Fokker-Planck Equation20
Owing to the long-range nature of the Coulomb potential, we find that grazing
collisions dominate in determining the kinetic properties of a plasma. The
kinetic equation that best incorporates this collisional property is the Fokker-
Planck equation. This equation may be obtained through expansion of the
Boltzmann equation about small-angle, grazing collisions.
The cross section for the Coulomb interparticle force,
e2
B.20)
18H. W. Liepmann, R. Narasimha, and M. T. Chahine, Phys. Fluids 5, 1313 A962).
19For further discussion, see W. G. Vincenti and C. H. Kruger, Jr., Introduction to
Physical Gas Dynamics, Wiley, New York, A965).
20A. D. Fokker, Ann. Physik43,912A914); M. Planck, Sitzungsber. Preuss. Akad.
Wiss., p. 324 A917). See also S. Chandrasekhar, Revs. Mod. Phys. 75, 1 A943). The
present derivation is due to H. Grad (unpublished).

302 4. Applications to Plasmas and Neutral Fluids
is given by [see C.1.31)]
k2 dcos
B.21)
2e2
k = — B.21a)
m
The interparticle force given by B.20) describes repulsion between two par-
particles of charge e and reduce mass m/2. This description is relevant to a
one-component plasma (OCP), which is a collection of like charges in a neutral-
neutralizing background.21 Substituting B.21) into the Boltzmann collision integral
gives
.2
(g COS iff)-
B-22)
where, we recall, ^ and 0 are the polar angles of the apsidal vector ex.
With the kinematic relations
Avi = \[ - Vi = -ol{ol • g)
Av = v' — v = ol{ol • g)
we conclude that for grazing collisions, ot{ot • g) may be taken as parameter
of smallness. Consider first
/' = /(V) = /[v + cx(cx • g)]
Taylor series expanding this form about £ = ot{ot • g) = 0 gives
/ = exp(f • V)/ = / + (£. V)/ + l-{£- VJ/ + • • •
= / + (a • g)(a • V)/ + -(a • gJ(a • VJ/ + • • •
= / + g cos V(a • V)/ + -g2 cos2 V^(a • VJ/ + • • • B.24)
For //, we find
/' = exp(-f • V,)/ = /, - gcos^(a • V,)/, B.25)
-g2 cos2 Vr(a-V02 /,+
In these expressions we have written
d\
21 An extensive review of the theory of one-component plasmas is given by S.
Ichimaru, Revs, Mod. Phys. 54, 1017 A982).

4.2 Further Kinetic Equations of Plasmas and Neutral Fluids 303
g
FIGURE 4.11. For grazing collisions, \j/ ~ f- — 8, at which value cos \js ~ e.
The Cartesian components of the unit vector a are
a = (sin i/f, cos 0, sin \f/ sin 0, cos \/r)
with g taken as the z axis [recall C.2.26)].
Substituting B.16) and B.25) into the collision integral B.22), keeping terms
of 0(£2), and integrating over 0 gives
[ ' d<p(f[f - fif)
Jo
l
= it cos2 f\ 2g • (V - V,) + ^[sin2 f(lg2 - gg) + 2cos
[V,V, -2VV, +VV] //,
B.26)
Here we have set
d(f)got • D = 2ng cos
= 2 cos
D
where D is written for V or Vj. Integrating over cos^, we encounter the
forms
I
cos
(
2 ^ cos2 i/f sin2 i/r cos4
cos2 ^, cos2 i/f sin2 i/r, cos
B.27)
At grazing collisions, ^ = 7r/2 — ^, where e is an infinitesimal angle (see Fig.
4.11). This value of ^ corresponds to the lower limit in B.27) and gives rise
to the singularity:
d cos
= — In cos ( —
\2
= — In sin s
= Ins
B.28)

304 4. Applications to Plasmas and Neutral Fluids
For weakly coupled plasmas, we choose £ to be given by
1 n1'2
a — B.28a)
T3
The plasma parameter A was defined previously in B.3.59). Having completed
the angular integrations in B.22), a final integration over Vi completes our
derivation. Integration by parts reveals the form [keeping only the singular
terms in B.26)]
/(/) = -_ . a/ + --— : (bf) B.29)
d\ 2 d\d\ V /
In this expression, a represents the friction coefficient and b the diffusion
coefficient. They are given by
a = 2ttk2 \n£~l / 4/i dv\ B.30a)
J 83
b = 7ik2 In £~l I I ^ j /i dvx B.30b)
In the absence of an external force field, the Fokker-Planck equation then has
the form
B.31)
To discover the physical significance of the a and b coefficients, we consider
them separately and choose the simplest forms. Among the simpler forms we
may choose for the friction coefficients a is that corresponding to Rayleigh
dissipation:
a = -v\ B.32)
With b = 0 and assuming spatial homogeneity, the Fokker-Planck equation
B.31) then reads
— -v— -v/ = 0 B.33)
dt d\ J
Choosing a local velocity frame, v = (u, 0, 0), and setting h = vf, B,23)
becomes (with v and t as independent variables)
^vh B.33a)
dt * dv
The general solution for this equation is an arbitrary function of the solution
to the characteristic equations
v dt dv
~T~ ~ v~

4.2 Further Kinetic Equations of Plasmas and Neutral Fluids 305
t
no)
fit > 0)
FIGURE 4.12. The friction coefficient causes particles to cluster near the origin in
velocity space.
Thus h is constant on the curves vo = v exp vt, and we write
h(v, t) = h(vo, 0) = h(v exp vt, 0)
We conclude that the friction coefficient B.32) causes particles to slow down.
Thus an initially square distribution in velocity space would tend to peak toward
the origin (see Fig. 4.12).
The simplest form we may choose for the diffusion tensor b is that of a
diagonal matrix with equal elements. That is,
b = lb
With a = 0, the Fokker-Planck equation becomes
'1 = I»vV
dt 2 J
B.34)
where V2 represents the Laplacian in velocity space. For the initial distribution
given by
/(v,0) = V-lS(y) B.35)
the solution to the diffusion equation B.34) is given by the Gaussian
distribution22
V-l
B.36)
In these expressions, V represents the volume of the system. So the diffusion
coefficient causes a distribution function initially peaked at the origin to flatten
out (see Fig. 4.13).
Combining both the a and b coefficients gives (with a = — dv and again
working in a local velocity frame)
a/
at
3
2
B.37)
22Encountered previously in A.8.7) and C.4.40).

306 4. Applications to Plasmas and Neutral Fluids
FIGURE 4.13. The diffusion coefficient causes a flattening of f{v) in velocity space.
At equilibrium, this equation becomes
d
~dv
2 av
Thus
dvf + -b— = A
2 av
where A is independent of v. At v = oo, we have
vf =
B.38)
av
It follows that the constant A = 0. Multiplying the resulting form of B.38)
through by the integrating factor
exp
/
2a v
reduces the equation to
3 (eav2/hf) = 0
dv
which gives the Maxwellian distribution
= Be~av2/h
Thus incorporation of the friction coefficient that tends to peak f(v) about the
origin and the diffusion coefficient that tends to normalize f(v) leads naturally
to the Maxwellian distribution.

4.2 Further Kinetic Equations of Plasmas and Neutral Fluids 307
22*
4.2.4 The Landau Equation
The Fokker-Planck equation B.31) may be cast in a more symmetric form
known as the Landau equation. To demonstrate this transformation, we must
first establish some tensor properties of the relative velocity
g = vi - v
and the tensor
2l B.39)
which enters the Fokker-Planck diffusion coefficient b given by B.30b).
These are as follows:
^ B.40)
8 " B.41)
d
d
if fig) is
g o Sfi 3
°v(i
a 9 (*2)V2 l(a2)~1/2
V V
any scalar function, then
3f(g)
a 2gv o
3V(i 2g 3
3/(«)
Next we turn to the calculation of V • T.
d d
"* (JV
.2
°v ~ gv~ gfi
= -2gv + gv
Combining results, we have (see Problem 4.10
23L. Landau, Z. Eksp. i Teoret. Fiz. 7, 203 A937)
B.42)
B.43)
V • f =2g B.44)
g f = 0 B.45)
V • f(g)f = f(g)V • f1 B.46)

308 4. Applications to Plasmas and Neutral Fluids
An illustrative exercise involving these properties concerns the Fokker-Planck
ol and b coefficients B.30). Let us establish the relation
V -b =
B.47)
or, equivalently,
dv
3 C T
£ / —/(Vi)rfVi =
g
B.47a)
= 7tK2
K = 7tK2\ns
With B.44) and B.46), we find
f
g3
l
g
g3
B.48)
which establishes B.47).
Finally, we turn to an explicit representation of t. Specifically, let us discover
the matrix representation of f in a form in which g is parallel to the z axis;
that is g = @, 0, g).
In general, we have
T =
gx gxgy
gygx g ~ gy
\
gyg:
gzgy g gz )
In the said coordinate frame, T is seen to reduce to the simple form
V o ox
T =
\
0 g2 0
0 0 0
B.49)
This form will be returned to in our discussion of the Balescu-Lenard equation.
Symmetrization of the collision terms
We rewrite the Fokker-Planck equation in the form
K Dt
= -2
dvv
/ 1 O O
-kf\dv+- — T—f
g
2 dvv dv.
/
g
d
V
-2/
^
/l^V]
2 31;,
/
g-
f\d\]
B.50)
The second term within the brackets may be expanded to give
u
g
/
g
= ^ / <*v,/,
d
g
2fg
V

4.2 Further Kinetic Equations of Plasmas and Neutral Fluids 309
The second term in this expression combines with the first term within the
brackets in B.50), and we are left with
1 DF 1 d ( f Tuv d
K Dt
The second term may be transformed as follows:
Dropping the first surface term and collecting remaining terms gives the
Landau eauation:
Landau equation:
B.51)
Thus we find an alternate symmetric form of the Fokker-Planck equation whose
nonlinear collision integral more closely resembles that of the Boltzmann
equation.
The question arises as to whether a Max wellian distribution is an equilibrium
solution of the Landau equation. Setting / = A exp(—v2/2C2), we find that,
after differentiation with respect to uM and V\^, the integrand contains the factor
t - g, which by B.45) vanishes. Thus we find that the Landau equation shares
this additional property with the Boltzmann equation.
4.2.5 The Balescu-Lenard Equation24
In Section 2.3.9, we noted that the Balescu-Lenard equation is relevant to the
time interval where shielding comes into play (see Fig. 2.14). This equation
is now derived in a correlation expansion of the hierarchy relevant to a high-
temperature rare-density plasma.
We return to the nondimensionalized BBKGY equation B.2.12) and the
BY, write
-f- ks — avy, I r, = h^s+i
dt J y
24A. Lenard, Ann. Phys. 10, 390 A960); R. Balescu, Phys. Fluids 3, 52 A960).
The spectrum of the linearized Balescu-Lenard equation was examined by A. H.
Merchant and R. L. Liboff, J. Math. Phys. 14,119 A973).

310 4. Applications to Plasmas and Neutral Fluids
where ks is given by B.2.22a), and we recall that the interaction term ISFS+X
has the form
L =
h^ J
d(s
and Gtj represents the two-particle interaction force for particle j on particle /
We examine a one-component rare plasma with
1
- = nor* = 0(8)
y
and weak coupling (or high temperature)
a =
kBT
= 0(e)
where ro is the range of interaction [recall B.2.9) and s is a parameter of
smallness. As particle coupling is small, the correlation expressions B.2.21)
for F2 and F3 are written
F3(l, 2, 3) = Fi(l)FiB)FiC) + e ]T Fx(i)C2(j, k) + s2C3(l, 2, 3) B.52)
P(ijk)
The variables P(ijk), we recall, denote permutations of A,2,3).
The equations BYi and BY2 then appear as
dt J y
KxB) -
cC2(l,2)]
B.53a)
a
y
+
sC2(l, 2)]
F!(i)C2a, k) + ^2C3A, 2, 3)
B.53b)
To terms of O(s), we find (returning to dimensional variables)
d
C2(l,2)
(l) = 0
d
B.54a)
a
m
3v
B.54b)
The technique of solving these equations is as follows: A) Solve B.54b) with
the aid of B.54a) to obtain C2 = C2(t). B) Insert this solution into B.53a) to
obtain a kinetic equation that is valid to O(s3).

4.2 Further Kinetic Equations of Plasmas and Neutral Fluids 311
The solution to B.54a) is the streming solution, which contains only free-
particle trajectories with \{ = \2 = 0. Thus, in this instance,
d ( d
— C2(l,2)= —
dt \dt
d
d d t
+ v2. — )C2A,2)
dt
It follows that B.54b) may be rewritten
dC2 1 9<pA
dt m dx\
= A(l,2,t)
Integrating this equation, we find
C2(l,2)
B.55)
d
d\:
B.56)
C2(l, 2, 0 = I A(t'W + C2(l, 2, -oo)
t/—00
B.57)
If we assume that particles are uncorrelated at t -\—oo, then C2(—oo), then
Ci{—oo) = 0. Changing variables to
r = t - t,
dx = -dt'
gives
'OO
C2(l,2, 0= / A(t-x)dx
B.58)
Proceeding with our solution prescription, we substitute B.58) into B.53a).
There results
d
~dt
a
y
-/, /
K Jo
OO
- x)dx
B.59)
The left side of this equation is identical to that which appears on the left side
of the Vlasov equation B.2.30). Thus B.59) may be rewritten (setting fx = f)
at
edf
m
1 9 f A f
— -—/ d\2 I
m 3vi J J
\L
00
A(t — x)dx
B.60)
where e£ is the Vlasov self-consistent force field. It the two-particle separation
is x]2 at r, then in the straight-line approximation, at t — x, its value is Xi2 — gr.
Thus, with the identification B.56), and writing Df/Dt\VL for the left side of
B.60), this equation becomes
Dt
VL
LA
m 3\
d
f
-eE(xl2) / -
Jo m
- eE(xn - gr)
3v2
f(l,t-r)fB,t-r)dr
B.61)

312 4. Applications to Plasmas and Neutral Fluids
where eE here represents the two-particle force field.
Assuming that /(l)/B) is uniform over the x2, r integration domain, we
find
Dt
= (-) t—/
\m/ 3vi J
dv2(EE)
VL
where
'00
3v
B.62)
(EE) = dx2 dtE(x]2)E(x]2 - gr) B.63)
represents the E field autocorrelation function in the straight-line approxima-
approximation. We may rewrite B.62) in a form that strongly resembles the Landau
equation B.50):
Df =-IQ
VL 9V J V9V - B.64)
- Xi)E*(x - x, - gr)
p \ 2 r r
\ m m
— I I dX\ I
fn/ J Jo
where we have changes notation: X\,\\ —> x, v; x2, v2 —> xi, \\. Note that the
complex conjugate has no effect on the real E field.
Inserting the Fourier transform [see A.32)]
eE(x) = -V*(x) = I dkikeikx$(k) B.65)
BttK J
into the autocorrelation function Q gives
~Q =
B.66)
With
f
eix> (k'-k) = BjrK8(k' -k)
integration of B.66) over k' gives
1 C = C
^7T^ / ^kk|*(/:)|2 /
m2Bny J J=
which gives, finally,
Recalling A.21), we reintroduce the dielectric constant s(k,co), obtained
from linearized perturbation theory of the Vlasov equation, and set
B-68)

4.2 Further Kinetic Equations of Plasmas and Neutral Fluids 313
where s(k, co) is given by A.23):
e(k, co)=l
col r k-dfo/d\d\
k2
co — k • v
Substitution of the form of &(k), so augmented, into B.67) gives the Balescu-
Lenard Q tensor.
*--ibl
dkkk\$(k)\z8(k-g)
j _ <Jp_ r kjdfo/d\)d\
k2 J k \—co
B.69)
The Fourier transform &(k) is that of the bar Coulomb potential energy
[compare A.33) with kd = 0]:
*(*) =
4ne:
I2'
B.70)
With QBL substituted into B.64), we obtain the Balescu-Lenard equation:
B.71)
Note that this equation includes the effects of shielding as contained in the
QBL coefficient.
Reduction of QBL to QL
An instructive example at this point concerns the reduction of QBL as given by
B.69) to the corresponding Landau form B.51). First note that this equation
may be written
Dt
Ql =
Q
L '
d
3v
d
K T
B.72a)
B.72b)
At k » X~\e(k, co)^ 1. In this limit, B.68) and B.69) give
_?4 r kk
Qbl ^ G'bl = =^ / dk—8(k • g)
m2J "k4
We obtain the Cartesian components of this matrix through evaluation of the
integral in spherical coordinates. With g taken as the polar axis, there results
2e4 f°° dkk2 fl f2n =
kbl = —r / —rr / dcosO dc/)kk8(kg cos 0)
mz Jo k4 J_x Jo

314 4. Applications to Plasmas and Neutral Fluids
2eA f dk
r dk r r
h hi
dcosO
8
kk<5(cos<9)
Here k represents a unit k vector. The kk matrix is given by
' O O O
sin 6 cos 0 sin 6 sin 0 cos 0 sin # cos 9 cos 0
O 0 0
sin 9 cos </> sin <p sin 0 sin </> sin 6 cos 0 sin </>
\ cos 0 sin 6 cos </> cos 0 sin 9 sin </> cos2 6
Integration over cos 9 leaves
cos2</> sin</>cos</> 0 ^
cos </> sin <p sin2 <p 0
0 0 0
Completing the 0 integration gives
G'bl =
dfc 1
/
\
g2 0 0 N
o ^2 o
0 0 0
With the identification B.49), the latter equation may be written
27te4f(g)
BL
rfyt
B.73)
Here we have introduced the cutoff parameters
2
e2n
B.73a)
corresponding, respectively, to the wave number of closest approach and the
Debye wave number, introduced previously in A.31). It follows that
B.73b)
3H./2 =4^«^ = X =
eJn
The plasma parameter A was introduced previously in B.3.59). We recall that,
for a weakly coupled plasma, such as is presently considered, A is a small
parameter. We write
Jkd
ko dk
UK
— = In s
and <2'bl reduces to the form
2'BL = —r
K T
8'
— \£L
B.74)

4.2 Further Kinetic Equations of Plasmas and Neutral Fluids 315
We recognize this to be the Landau kernel given by B.72b). This reduction was
obtained in the limit k ^> A.J1, s —> 1, relevlant to a plasma with no shielding
effects, such as described by the Landau or, equivalenty, the Fokker-Planck
equations.
This finding brings our analysis around full circle. As illustrated in
Fig 4.9, the Landau equation follows both from the Boltzmann equation (Sec-
(Section 2.3) and, as shown immediately above, from the Balescu-Lenard equation.
This connecting loop indicates that, as previously described, the Boltzmann
equation too, rests under the aegis of the BBKGY hierarchy.
4.2.6 Convergent Kinetic Equation
It is evident that the kinetic equations we have discussed relevant to plasma
dynamics all suffer divergence. In deriving the Fokker-Planck equation, we
found that the Boltzmann collision integral exhibits divergence at small-angle
collisions or, equivalently, at large impact parameter collisions [see, for exam-
example, B.28)]. When written in terms of impact parameter, this singularity has
the form/ ds/s.
In the limit of large k, the Balescu-Lenard collision operator B.69) exhibits
a ~ / dk/k divergence. Since s(k, co) = 1 in this limit, the Landau equation
exhibits a like logarithmic divergence. At small k, QBL is nonsingular, whereas
the Landau equation remains logarithmically singular.
The regular behavior of QBL at small k is due to the inclusion of dielectric
shielding in its kernel, thereby lessening the effects of long-range collisions.
Its remaining singularity at lark k reflects the fact that the equation does not
incorporate effects of wide-angle collisions. This conclusion follows from
our derivation of the Landau equation from the Boltzmann equation at small-
angle collisions and the demonstration of the equivalence of the Landau and
Balescu-Lenard equations at e(k, co) = 1.
Resoultion of these divergent forms was considered by Hubbard,25 Frieman
and Book,26 Gould and DeWitt,27 and Aono,28 among others.
We first write the convergent kinetic equation in the form
Df ~ ~
J I I + II B.75)
where the collision / terms denote the following:
/0 = Boltzmann collision integral, s < s0
I = Balescu-Lenard integral, k < l/s0
25J. Hubbard, Proc. Roy. Soc. A261, 371 A961).
26E. A. Friedman and D. C. Book, Phys. Fluids 6, 1700 A963).
27H. A. Gould and H. E. DeWitt, Phys. Rev. 155, 68 A967).
28O. Aono, Phys. Fluids 77, 341 A968).

316 4. Applications to Plasmas and Neutral Fluids
70 = renormalization term cancels singularities in /0 and 7
The inequalities on the right denote domains of convergence, with
representing an intermediary value of the impact parameter.
We may write /q as an integral over impact parameter:
/»OO
Io= B(s)ds B.75a)
Jo
For s ^> so, small-angle collisions are predominant and the Fokker-Planck
expansion is relevant, which in turn yields the Landau equation B.51). In
developing this equation, the related singularity was absorbed in the coefficient
K B.47a). Rewriting this coefficient as an integral over the impact parameter,
. . ds
K = 7TK'
indicates that
A
B(s) - -
s
relevant to the large s, grazing-collision domain. The integral A is written for
the Landau form:
Similarly, we write
7 = / G(Jfc)Jfc B.75b)
where G(k) is the implied Balescu-Lenard form. For large k, s(k, to) —► 1 and,
as shown above, the Balescu-Lenard equation reduces to the Landau equation.
Thus, in this domain, 7 suffers the same logarithmic singularity as the Landau
collision form. That is , we may write
G(k) - 4
For 70 to cancel the s singularity in /0 and the k singularity in 7, it should have
a dual form. This property may be satisfied with the aid of the Bessel function
J\ (jc) by virture of its following property.
ds f°° ds
Jl(ks)d(ks)= —
f°° ds f
oo ds / Jx{ks)dk = — /
/ A(x)dx = l <^ ° S ° dk
Jo -"- / Jx(ks)ds = — / J](ks)d(ks) = —
Jo k Jo k
s
dk
dk
Accordingly, we write
/OO /»OO
/ J](ks)dsdk
Jo

4.2 Further Kinetic Equations of Plasmas and Neutral Fluids 317
so that
= A
/So /»OO /»00 /»So
ds dkJi(ks)-\- dk dsJx{ks)
Jo Jo Jo
/ =
L
B{s)ds +
L
/»OO
JSo
oo
Jo
/»OO
Jo
B(s)- A / dkJ\(ks)
ds
G(k)- A / dsJi(ks)
dk
B.76)
The integral over k in the first bracket leaves A/s, which cancels the B(s)
singularity for large s. In the singular domain of G(k), that is, ks0 ^> 1, the
integral over s in the second bracket cancels the A/k singularity of G(k). So
we find that the combination integrals B.76) gives a reasonable model for a
convergent plasma kinetic equation.
4.2.7 Fokker-Planck Equation Revisited
Of the various kinetic equations described in Fig 4.9, the Boltzmann, KBG, and
Fokker-Planck equations are relevant to neutral fluids as well as to plasmas. To
exhibit this property for the Fokker-Planck equation, we revert to the Chapman-
Komogorov equation A.7.7) relevant to scattering processes homogeneous in
time.
Let Y\(\, Av) denote the probability that in the interval At, v —► v + Av.
The Chapman-Kolmogorov equation A.7.7) may then be written
/(v, t)= j f(\ - Av, t - Av) [~](v - Av, Av) dAv B.77)
If we assume that only small velocity changes Av contribute to this integration
then Y\(y — A v, A v) will be sharply varying in its second variable but smoothly
varying in its first variable. This permits expansion of the integrand in B.77)
about v — Av = v and t — At = t. There results
/(v, 0 = / d(Av)
i
3/(v, 0
]\(y, Av) B.78)
3/(v, t) n(v, Av) 1==
- Av • ^-^ h -AvAv :
3v 2
d\dv
Dividing this equation through by Af and then integrating over Av, with
I ]~[(v, Av) dAv = 1
we obtain
= __a_ //Av\ \ i
dt d\'\\At]J) 2
, //AvAv\
2d\d\\\ At
B.79)

318 4. Applications to Plasmas and Neutral Fluids
Here we have set
B.80)
Identifying (Av/Af) with the friction coefficient a and (AvAv/Af) with the
diffusion coefficient b, permits B.79) to be recognized as the Fokker-Planck
equation B.31). However, the present derivation indicates that the Fokker-
Planck equation has wider use than mere application to plasma physics.
A more tractable form of the average quantities B.80) may be obtained as
follows. With B.23), we may write
Av = ot{ot • g)
for velocity change due to collision. For the remaining factor in B.80), we
write
[~](v, Av) rf(Av) = At8v B.81)
Here Sv represents the increment in collision frequency in which a Vi particle
collides with a v particle, causing the change v —> v + A v. This factor is given
by
8v =adQgd\xf(\i) B.82)
Completing the integral in B.80) gives
a = /^\ =[[*<&dyxgct{cx • g)/(v,) B.83a)
Repeating this contraction for the second integral in B.80), we obtain
U ££1 =//„
da dY'g((X'8)W/(Vl) B83b)
Let us employ these expressions to show that, for a plasma, a is given by
B.30a). With B.21),
K2d cos xl/dcp
go dQ = — — B.84)
(g • aK
Then B.83a) becomes
fK2d cos i/dcp -.
— -—a/(vi)rfvi B.85)
(^J
Again, choosing g as the polar axis for the }//, 0 integration gives the Cartesian
components
a = (sin y\r cos 0, sin \js sin 0, cos \jr)

4.3 Monte Carlo Analysis in Kinetic Theory 319
Thus only the z component of a survives integration, and we may write
o CC d cos \l/ g
a = 2jtk2 // 2L±f(yl)dyl B.86)
JJ COS \jf g3
which with B.28) returns B.30a). The same identification follows for the dif-
diffusion coefficient b. Thus we find that the generalized Fokker-Planck equation
B.79) reduces to the plasma Fokker-Planck equation when the Coulomb cross
section B.84) is employed.
4.3 Monte Carlo Analysis in Kinetic Theory
In this concluding section of the present chapter we turn to a computational
method in kinetic theory that has experienced wide use in the recent past. It is
known as the Monte Carlo technique.
4.3.1 Master Equation
Primary use of this analysis addresses a spatially homogeneous fluid in an
externally supported constant force field K. The distribution function for this
fluid, /(v, r), obeys the masterlike equation29
-f + a • -f = / dv'[tu(v', v)/(v', 0 - tu(v, vr)/(v, t)] C.1)
dt d\ J
where w(\\ v) represents the probability rate for scattering30 from the velocity
v' to v and a = K/m.
An important parameter that enters this analysis is the total scattering rate
from the velocity v to all other velocities:
Mv)= f w(v,v')dv' C.2)
To simplify C.1), we introduce
w(\, v') = w(\, \'
C.3)
tu(v', v) = w(\\ v) + [r - A.(v')]<5(v' - v)
29The master equation was encountered in Chapter 1 A.7.20). It is used extensively
in Chapter 5 for problems in quantum kinetic theory; see E.2.28), E.2.86), and
E.3.58).
30For inelastic scattering, we cannot assume w(\, v') = w(\', v). In an inelastic
scattering, initial and final scattering constituents are not the same. Such is the case,
for example, in ionizing collisions or collisions where an atom undergoes transition
to an excited state.

320 4. Applications to Plasmas and Neutral Fluids
where F is an arbitrary constant. The delta function component in these ex-
expressions permits no change in velocity in scattering and is referred to as a
self-scattering mechanism.31
Note that the integration corresponding to C.2) gives
X(\) = I w(\, \')d\' = F C.3a)
Substituting C.3) into C.1) gives
+ a • -f = / dVw(V, v)/(v', 0 - F/(v, 0 C.4)
3v J
In order for the probability scattering rate w to be positive, as the constant F
is arbitrary, we choose
F = max[A.(v)] C.4a)
V
Equation C.4) may be rewritten in the purely integral equation form
/(v, 0 = ff d\'d\" f dt'f{\\ t)w(\\ \")8[\' - v" - (t - t'JL\e-V{t-t!)
C.5)
It is the interpretation of terms in C.5) that forms the basis for the Monte Carlo
analysis in kinetic theory
4.3.2 Equivalence of Master and Integral Equations
Prior to offering an interpretation of terms in the integral equation C.5), let
us demonstrate the equivalence of this equation to its precursor, the master
equation C.4). Towards these ends, we first set
x = t-tf
and integrate C.5) over \f. There results
/(v, t)= I dV f drf(\\ t - r)w(\\ v - ra)e~rT C.6)
Now note that
9 ~ , a
a • —w(\ , v — ra) = w(\ , v — ra)
d\ dr
Operating on C.6) with a • C/3v) then gives
" 3
drf(\\ t - r)
3r
', v — ra)
31H. D. Rees, J. Phys. Chem. Solids 30, 643 A969). For further discussion, see C.
Jacoboni and L. Reggiani, Revs. Mod. Phys. 55, 645 A983).

4.3 Monte Carlo Analysis in Kinetic Theory 321
Integrating the right side of the latter equation by parts (over r), we obtain
a • — /(v, f) = - / dVf(V, 0)w(Y, v - ta)e
d\
i
/ d\'f(\', t)w(y\ v) - / d\' /
', v - ra)
x
Ve-lTf(V,t-T)-e
-Ft
d
dr
f(y',t-x)
= I + II + HI + IV
C.7)
where the Roman numerals stand for corresponding respective integrals. Now
recall that for any integrable function g(t),
dt
I
h(t-T)g(T)dT=h(O)g(t)+ f g{T)^h(t-T)dt
f
Jo
d
C.8)
We are now prepared to reconstruct C.4). First we note that from C.6)
III = -r/(v,r)
Next we employ C.8) with h = /(v', t) and g = w(\\ v — t) and g =
w(\', v — t2L)e~Tx and obtain
d
i + iv = -
-If
dt JJo
/(v\ t - t)u»(v', v -
.rr
ot
O)
the last equality following from C.6). Collecting terms (with II) remaining
intact) returns C.4). Thus we find that C.5) is equivalent to C.4)
4.3.3 Interpretation of Terms
We wish to interpret the terms in C.5). First we note that the probability of a
particles of velocity v undergoing a collision in the time At is
P(At) = i(\)At
It follows that probability, P, that the particles has no collision in the interval
Aris
P(At) = 1 - X(\)At
Over a finite interval t = TV At,
P(t) =
- A.(v2)Ar
C.9)
where
\n = \(nAt) = v@) + an At
and n = 1, ..., N. With C.3a), the preceding becomes
C.9a)
N

322 4. Applications to Plasmas and Neutral Fluids
t
Time
t -
I
v'
FIGURE 4.14. A particle undergoes a collision at t' and changes velocity from v' to
v"(—). In the interval {t — tf), the particle undergoes free-flight acceleration from v"
to v (—).
t
FIGURE 4.15. Three scattering free-flight events.
which in the limit TV —> oo gives
P = e
-Ft
C.10)
Thus, the exponential factor in C.5) represents the probability that the particle
drifts with no collision for the time (t — tf). The delta function tells us that
during this collisionless flight the particle undergoes an acceleration due to the
external field from velocity v" to v. The factor f(V, t)w(\', v") describes a
particle with velocity v' being scattered to the velocity v". This series of events
may be described in a v — t diagram as shown in Fig. 4.14. The continuation
of this collision-free flight process to three such events is shown in Fig. 4.15.
It is evident that the integration C.5) carries this process over all v', v" and
0 < t' < t. In the Monte Carlo analysis the distribution product /(v)Av is

4.3 Monte Carlo Analysis in Kinetic Theory 323
constructed by measuring the time a typical trajectory spends in the interval
Av about the value v, and we write
/(v)Avoc ]T Ar C.11)
v—bin
The summation is over all time intervals Ar that the trajectory spends in the
velocity interval, v, v + Av, which is termed a velocity bin (v-bin).
43.4 Application of Random Numbers
As noted above, in this Monte Carlo process the computer follows the trajectory
of a single particle. The intervals of free flight in this trajectory are related to a
sequence of random numbers in the following manner. Consider that we have a
distribution of random numbers, x, in the interval @, 1) with a uniform proba-
probability P(x) = 1. With C.10), we note that the probability that the particle drifts
for a time t and then suffers collision in the interval dt is equal to the product
{e~rt)Tdt
With t also considered a random variable, we write
P(x) dx = Te~Vt dt = d{\ - e~Tt)
Thus [recalling P(x) = 1] we find
n() C.12)
which is the desired mapping from the random variables jc to the free flight
intervals t.
Another step where random numbers come into play concerns the scattering
rate w(\, v') d\' (dimensions, s~]). Consider now that there is at our disposal
a distribution of three sets of random numbers [all in the interval @,1)]. Let
us call a triplet of such values the vector y. Viewing the final velocity v' as a
random variable permits us to write
w(\,y')d\' = P(y)dy
where again P(y) is taken to be uniform. The preceding equation implies the
relation
v' = v'(v,y) C.13)
which effects a mapping from the random variables (v, y) to the random
variables v'. [As we are constructing /(v), the velocity v is a known vector]32
32
It is this use of random numbers that motivates the gambling label ofthe analysis.

324 4. Applications to Plasmas and Neutral Fluids
Prepare velocity
mesh {Av}
Prepare bins
over velocity mesh
for storage of At in
corresponding Av
C.11)
Initialize
time:
0— T
Pick x; get drift time t from C.12)
[use Pix)]
i
T+t—T
f StopY*-
Print out velocity
histogram f(v) from
storage in bins
in 8
8
Sample velocities Av
and times At in
interval t and store in
bins prepared in
3
v + at
..
Scatter from v' tc
obtain v' from 7 «
with rv y find v" B
[use P(y)]
10
V"—♦ v
iv";
and
1.13)
9
FIGURE 4.16. Flow chart for Monte Carlo determination of distribution function.
Starting given values are constant acceleration a; starting velocity v0; random vari-
variable distributions P(x), P(y); final time Ts. Elapsed time of trajectory in any given
loop is t.
4.3.5 Program for Evaluation of Distribution Function
Having interpreted the terms in C.5) and described the manner in which random
numbers enter the analysis, we are now prepared to construct a flow chart for
the determination of the distribution function. Such a flow chart is shown in
Fig. 4.16. The algorithm command x -» y (also written x = y in FORTRAN

Problems
325
Total time
in each
velocity
bin
Av 2Av
10 Av
v-bin
10.5
9
8.5
9.5
7.6
6.0
7.5
5.5
3.5
2.0
v bin
FIGURE 4.17. Typical bin storage histogram for one-dimensional Cartesian dis-
distribution component, f(v), for force-free fluid. Velocity mesh is 10Av units long.
Smooth solid curve represents the distribution exp — (v/10J.
and BASIC or v := x in PASCAL) means "replace y by jc." Recall also that
the probability distributions P(x) and P(y) are uniform.
Note that the procedure displayed in Fig. 4.16 is directed at the construction
of an equilibrium distribution. Furthermore, owing to the stochastic nature of
these processes, it is assumed that /(v) is independent of the starting velocity
v0 for sufficiently large Ts. If this is so, then the distributions /(v; Vo, Ts) and
/(v; v0 + v, Ts) are equal for arbitrary velocity v. To verify the equilibrium
property of the distribution, we compare /(v;v0, Ts) and /(v;v0, 7^ + r),
which for equilibrium are equal for arbitrary time interval r.
The manner in which /(v) is constructed from time-interval storage in
velocity bins is illustrated in Fig. 4.17.
Problems
4.1. (a) With F(P) in A.71) given by
>oo
d/x
-H2/2
\[2jx Jo
II

326 4. Applications to Plasmas and Neutral Fluids
show that
sinh fry
V : F@) = 1 - !_ / dye~}2/2
V2tt J-oc L 0?
///wf: Introduce the variable
(b) With the preceding result, what is the explicit form of e(fi) for a
Maxwellian plasma?
(c) Establish the following rules concerning principle value:
: I fe(x)dx = Him / ft(x)dx
J —a J£
V. I fo(x)dx=0
—a
where fe(x) and fo(x) are even and odd functions, respectively, and are
regular in the open interval about the orgin.
4.2. What is the explicit form of the response dielectric function s(k, k • v0), as
given by A.28), for v0 ^ 0?
4.3. What does the static Debye potential 3>(r), given by A.35), become for
extraneous charge velocity v0 ^ 0?
4.4. Show that the derivation of the Landau decay constant F, preceding A.66),
remains valid providing F « ^.
4.5. Can the contour in A.69) be closed in the lower-half /3 plane? Explain your
answer.
4.6. (a) Show that
" f]nfdv =
(b) With the preceding property, show that the KBG equation implies an H
theorem.
Hint: Multiply both sides of the KBG equation by / d\ In / and note that,
with part (a),
°-f)\nfdv= [(f°-f)\n(fo)dv
4.7. A longitudinal electric wave with phase velocity v<p = 0.&cop/k and wave-
length A. = 1/zra propagates through a plasma at T = 20,000 K and electron
density n = 4.2 x 1018 cm.
(a) What is the plasma frequency, cop, of this plasma?
(b) What is the Debye distance, Xd, for this plasma?
(c) In how many seconds will the amplitude of the electric wave decay by
the factor e~x due to Landau damping?

Problems 327
4.8. Consider the following one-dimensional plasma distribution:
where /x is the nondimensional velocity given by A.41). Employ the Nyquist
criterion to determine the number of unstable modes corresponding to this
distribution.
4.9. Determine whether or not the Fokker-Planck equation B.79) implies the
conservation equations.
4.10. Establish the validity of B.46),
V • f(g)T = f(g)W • f
where f(g) has continuous first derivatives, the V operator represents single-
particle velocity differentiation, and f is given by B.39).
4.11. Show that the Fokker-Planck coefficients a and b B.30) satisfy the relation
V -b = a
4.12. Show that the Maxwellian distribution reduces the Landau collision integral
[see B.51)] to zero.
4.13. Show that the matrix f lies in a plane normal to g. That is, show that
g-7"=0
4.14. Write down an explicit expression for the integrand G(k) in B.75b).
4.15. What is the physical significance of e(co, k) -> 1 for a plasma, in the limit of
large k. That is, what is the behavior of a wave in a plasma at such kl
4.16. Show that the electric-field auto correlation tensor in the straight-line
approximation bay be written
(EE) = 2ne'
Ins
8
M 0 0 >
0 1 0
0 O
j
or, equivalently
(EE) = 2ne'
Ins (g2l -gg
g
g
Hint: Recall that
f
(EE> = j dx2f JrE(x12)E(x12 - gr)
Use cylindrical as shown in Fig 4.18. Introduce cutoff parameters in the s
integration at the distance of closest approach, k^\ and the Debye distance,

328 4. Applications to Plasmas and Neutral Fluids
t
zaxis
t
g
Straight-line
trajectory of
particle 2
FIGURE 4.18. Configuration for Problem 4.16.
4.17. Relevant to the Monte-Carlo numerical procedure, write a flow chart cor-
corresponding to the output shown in Fig. 4.17. Assume three-dimensional
scattering.
4.18. Determine if the KBG equation [B.7),B.8)] satisfies the conservation
equations. Discuss your answer.

CHAPTER 5
Elements of Quantum
Kinetic Theory
Introduction
This chapter begin with a review of some basic principles of quantum mechan-
mechanics important to the formulation of quantum kinetic theory. In the following
section the density operator is introduced and applied to problems involv-
involving beams of spinning particles. The Wigner distribution is described, and an
equation of motion is derived for this function, which is found to reduce to the
Liouville equation in the classical limit. The Wigner-Moyal equation and the
notion of Weyl correspondence are described as well.
In the third section of the chapter, different manifestations of the KBG
equation are applied respectively to the kinetic theory of photons interacting
with a gas and electron transport in metals. In the former case, the canoni-
canonical criterion for lasing is obtained from the relevant photon kinetic equation.
In application to electron transport in metals, expressions are obtained for
electrical and thermal conductivities and their ratio, the Lorentz number. The
phenomenon of Thomas-Fermi screening is described, and forms for the re-
related screening length and static potential are obtained. Dispersion relations
for waves in quantum plasmas are discussed, and the close similarity between
these equations and their classical counterparts is noted. A quantum modifica-
modification of the Boltzmann equation due to Uehling and Uhlenbeck is introduced.
This quasi-classical equation is found to give the correct quantum equilibrium
distributions. The chapter continues with an overview of classical and quan-
quantum hierarchies, which includes a description of second quantization. With this
formalism at hand, a quantum analysis of electrical conductivity is discussed,
stemming from the Kubo formula derived earlier in Chapter 3. The chapter
concludes with a brief introduction to the Green's function formalism relevant
to quantum properties of many-body systems. Included in this description are

330 5. Elements of Quantum Kinetic Theory
diagrammatic representations of equations of motion for Green's functions.
The discussion concludes with a derivation of the lifetime and energy of a
particle interacting with its surroundings.
5.1 Basic Principles1
5.L1 The Wave Function and Its Properties
In quantum mechanics, information concerning a given system is obtained
from its wave function. Such information is derived from expectation values.
Let
be the wave function of a system comprised of TV particles. Then, if A(xN)
is some dynamical property of the system, the expectation of A in the state
i//(xN, t) is given by
(A) = I \l/*(xN,t)A\l/(xN,t)dxN A.1)
Here we have written A for the operator corresponding to the observable A.
The quantity (A) is interpreted as an ensemble average.
In Dirac notation, A.1) appears as
(A) = (x/f\Axlf) A.2)
Another important manner in which information is derived from the wave
function is through the Born postulate. This postulate indicates that the
configurational probability of the system is given by
A.3)
This probability density is defined in the same manner as that obtained from
the classical TV particle distribution function /n(xn, pN, t):
.N
/
P{x",t) = J fN(x",p",t)dp"
The wave function evolves in time according to the time-dependent
Schrodinger equation,
= Hf A.4)
dt Y
where H is the Hamiltonian operator of the system.
1 For further discussion, see P. A. M. Dirac, The Principles of Quantum Mechanics,
4th ed., Oxford, New York A958); E. Merzbacher, Quantum Mechanics, 2nd ed.,
Wiley, New York A970); R. L. Liboff, Introductory Quantum Mechanics, 4th ed.,
B002), ibid.

5.1 Basic Principles 331
Dynamic reversibility
Reversibility in quantum mechanics is given by the following description.
With H{t) = H{—r), we see that setting t —> t' — —t and taking the complex
conjugate of A.4) gives
3V*(x,-r)
dt
= H1r*(x, -t)
This is the same equation as A.4). We may conclude that if *I>(x, t) is solution
to the Schrodinger equation then vl>*(x, —t) is also a solution.
Time development of expectations
With A.1) and A.4), we obtain
= ([A,
A.5)
Here we have introduced the commutator
[A, //] = A// - //A A.6)
The list of properties A.1.26) obeyed by the Poisson brackets are also obeyed
by the commutator.
Suppose A is not an explicit function of time. Then A.5) indicates that if
A commutes with H then the expectation of A is constant in time. This is the
quantum mechanical analog of the classical theorem discussed in Chapter 1
which states that if a dynamical variable has a zero Poisson bracket with the
Hamiltonian then it is a constant of the motion.
Measurement and operators
A fundamental postulate of quantum mechanics concerns measurement. Thus,
At
if A is the operator corresponding to the observable A, then measured values
of A, which we will call a, are eigenvalues of A. The eigenvalue equation for
A is given by
Acpa = acpa A.7)
where cpa is the eigenfunction corresponding to the eigenvalue a.
The operator corresponding to energy is the Hamiltonian so that energy
values of a given system are obtained from the time-independent Schrodinger
equation:
HcpE = EcpE A.8)
These eigenstates are called stationary states for the following reason.

332 5. Elements of Quantum Kinetic Theory
The solution to the time-dependent Schrodinger equation A.4) correspond-
ing to the initial value \j/(x , 0) is given by (for time-independent H)
xj/(xN, t) = e(-it^)/nxlf(xN, 0) A.9)
Suppose i//(xN, 0) is a solution to A.8). Then
= e(-i'E)/n<pE(xN) A.10)
Forming the expectation {£) as given by A.2), we find
(E) = (irE\HfE) = E A.11)
i
We may conclude that in a stationary state (E) is constant in time.
5.7.2 Commutators and Measurement
An important rule concerning measurement involves the commutator. Thus, if
then A and B share common eigenstates.
As a simple example of this theorem, consider the configuration comprised
of a single free particle whose Hamiltonian is
* 2
P
= —,
2m
It follows that
, p] = 0
Eigenstates of p and H are given by
f2n
corresponding to
p<^k =Hk<pk
h2k2
H(pE = -r—
2m
Thus p and H share common eigenstates. That is, any eigenstate of the form
C exp zk • x is an eigenstate of both p and H.
Note, however, that <Pf(x) as given above is an eigenstate of H but
not of p. This property results from the degeneracy of free-particle energy
eigenfunctions. A Venn diagram well illustrates this situation (see Fig. 5.1).

5.1 Basic Principles 333
Eigenstates of H
\///////> Eigenstates of p
FIGURE 5.1. Venn diagram depicting the eigenstates of H and p for a free particle.
A simple example serves to illustrate the consequence on measurement that
the commutator theorem has. Consider that the free particle is in the state
i/rk(x) = Aelkx
Measurement of the momentum of the particle in this state is certain to find
the value p = hk. Such measurement2 leaves the system in the state ^r(x).
Subsequent measurement of E is certain to find the value E =H2k2/2m.
We conclude that there are states of a free particle in which E and p may be
prescribed simultaneously. There is no uncertainty related to measurement of
these observables in states that are common eigenstates of H and p.
Uncertainty in quantum mechanics
The uncertainty of measurement of an observable A in quantum mechanics is
given by the variance of A [recall A.8.2b)]:
(AAJ = {(A - (A)J) A.12)
For the example of the free particle considered above, in the common eigenstate
Cexp/k • x,
AE = Ap = 0
Clearly, this result is a consequence of the fact that for a free particle [//, p] =
0. More generally, the Robertson-Schrodinger theorem tells us the following:
Suppose two operators have a nonvanishing commutator. That is,
[A,B] = C A.13)
The theorem then states that
AAAB>^\(C)\ A.14)
2Here we mean measurement in the ideal sense, that is, measurement that least
perturbs the system.

334 5. Elements of Quantum Kinetic Theory
The most famous application of this result follows from the fundamental
commutator relation
[x,p] = iH A.15)
Here we are writing x and p for parallel components of x, p. With the preceding
relation, we conclude
H
AxAp>- A.16)
That is, states do not exist in which x and p may be prescribed simultaneously.
Another important illustration of A.14) concerns the Cartesian components
of angular momentum, whose commutator relations are given by
Ux, Jy] = i
[Jy, Jz] = ihJx
[Jz,Jx] = iHJy A.17)
From the first commutator we obtain
H
AJxAJy > -\(JZ
Thus, in general, Jx and Jy cannot be simultaneously designated in any quan-
quantum state. However, from the last of the four relations A.17) we find, for
example, that
AJ2AJZ = 0
So states exist in which J2 and Jz may be simultaneously designated.
5.13 Representations
Consider that the eigenstates of A comprise a discrete sequence. We may then
write
A \an) =an \an) A.18)
where n is an integer. In this notation, \an) represents the eigenstate of A
corresponding to the eigenvalue an. The eigenstates \an) may be taken to be
the basis of a Hilbert space H. That is, {\an)} spans the Hilbert space H. Let
the operator B operate on elements of H. We may represent B by a matrix
with elements
Bnn, = (an\ B \a'n)
These are the matrix elements of B in a representation that diagonalizes A.
This description is due to the following. Consider the matrix elements of A in

5.1 Basic Principles 335
this same basis. With A taken to be Hermitian, its eigenstates are orthogonal,
and we find
Ann' = (an\A \a'n) = a'n(an\an)
A.19)
which is a diagonal matrix.
Suppose the three operators A, B, and C compromise a complete set of
commuting operators. This means that no other independent operator exists that
commutes with A, B, and C. Recalling the commutator theorem stated above,
we may conclude that such commuting operators have common eigenstates,
which may be written \abc). As such states cannot be further resolved, they
are maximally informative and comprise quantum states of the system.
Consider, for example, the angular momentum operators A.17). For a
particle with spin S and orbital angular momentum L, we write
A.20)
s,
J2 = L2 + S2 + 2L ■ S
Commuting operators for this system are
{J\JZ,L\S2}
A.21)
Common eigenstates of these operators may be written \jntjls) with
eigenvalues
J
J
L
S
2
z
2
2
\jmds) =H
rrij/h
1A + 1)
s(s + 1) )
\jmjls)
A.22)
It is quite clear that the four operators A.21) are diagonal in a representation
comprised of the basis {\jnijls)}.
5.1 A Coordinate and Momentum Representations3
It is instructive at this point to include a brief review of the coordinate (x) and
momentum (p) representations and their interplay. Thus let \xf) represent an
eigenket of x. Then with A.18) we write
x\x') = x'\x)
Eigenstates are orthogonal and obey the relation
{x'\x) = 8(x — x)
A.23)
A.24)
3The relations developed here play an important role in Section 5.2.3 on the
Wigner distribution and in Section 5.7 concerning the Green's function formalism.

336 5. Elements of Quantum Kinetic Theory
appropriate to continuous eigenvalues spectra. Matrix elements of x are given
by
(x\x\x') = x{x\x) = xf8(x - x)
The coordinate representation of the ket vector \\//) is given by
A.25)
(x'\\/r)= I (x'\x)(x\i/)dx = I 8(x
A.26)
The matrix of the momentum operator p in the coordinate representation is
7) = —ih—8(x — xf)
dx
A.27)
This relation allows us to calculate an explicit representation of the inner
product (x\p):
p{x\p) = (x\p\p) =
jdx'
(x\p\x')(x'\p)
With A.27), this integral reduces to
p(x\p) =
ldx'
d
—in—8(x — x )
dx
(x'\p)
d
dx
'S(x-x'){x'\p)
(x\p)
Solving the differential equation gives
(x\p) =
2ttH
,ipx/h
A.28)
Let us show that (x\p) is unitary. That is,
/ (p\x)*(p\x') dp = / (x\p)(p\x')dp
= 8(x — x)
which indicates that (x\p) is unitary. We may employ the transfer matrix A.28)
to reestablish A.27).
(x\p\x') = j(x\p\p')(p\x')dp'
= I p'(x\p'){p'\x')dp'= f p'
2nh
Mp'/H)(x-x')
dp
= -m
d
dx \ 2nh J_,
ox

5.1 Basic Principles 337
Here are some simple examples of the preceding formalism. First let us find
the coordinate representation of
'OO
/•oo o
h / dx'-S(x - x')f{x')
^ A.29)
Next consider the matrix elements of [Jc, p] in the coordinate representation.
First we calculate
/•OO /»OO /»OO
(x\xp\x')= / / / dx"dp'dp(x\x\x")(x"\p')(p'\p\p)(p\x')
J — oo J —oo J — oo
x -'x
L
2nh ./-oo
d
dx
Similarly, we find
S(x — xf)
— (x\px\xf) = —ihxf—S(x — x)
dx
Combining results gives
(jc|[jc, p]\x'\ = —ih{x — x')—8(x — x)
x ' dx
= +iH8(x-x) A.30)
which is the continuous coordinate representation of the fundamental
commutator relation A.15).
5.7.5 Superposition Principle
Suppose a system is in an eigenstate \an) of the operator A. If A is measured,
we are certain to find the value an. Suppose the operator B corresponding to
the observable B has a nonvanishing commutator with A. That is,
Then, with A.14), outcome of measurement of B is uncertain. In this case, it
is meaningful to ask of the probability that measurement of B finds the value
bn. The answer to this question is simply obtained by writing the given stage,
an), in a superposition of the eigenstates of B:
A.31)

338 5. Elements of Quantum Kinetic Theory
Clearly, the coefficients (bk\an) represent the projections of the state \an) onto
the state \bn). The probability P (bk) of finding bk in measurement of B is given
by the square of the projection:
Pbk = \(bx\an)\2 A.32)
The expansion A.31) is the essential statement of the superposition principle.
We may describe the state \an) as being partly in the various eigenstates of
B. The degree to which \an) contributes to one of the specific states of B
is a measure of the probability that measurement of B will find the value
corresponding to this specific state. The measure of this probability is given
by A.32).
An informative example of this principle concerns spin 1/2 particles, such
as electrons. Eigenvalue equations for this case are given by
S2az = -?i2otz, Szaz = -a
A.33)
Working in a representation in which S2 and Sz are diagonal, we have the
matrix forms
3 2/l (T
4 V0 17' ~* 2V0 -1
0
- =
Let a beam of electrons be polarized with spins in the +x direction. Measure-
Measurement of Sz is made. What values will be found and with what probabilities will
these values occur?
In the given representation, eigenstates of Sx corresponding to Sx =H/2 are
given by
To answer the said problem, we must expand ax in accord with the
superposition statement A.31). There results
1 1
Thus
p | S7 = +- I = -
It is equally likely that measurement finds either of the two values Sz =

5.1 Basic Principles 339
5.1.6 Statistics and the Pauli Principle
In quantum mechanics, identical particles are indistinguishable. As a con-
consequence, probability densities cannot change under exchange of particles.
Consider, for example, the two-particle state VKxi, X2)- Then we must have
A.35)
Consequently,
, x2) = ±^(x2, xO A.36)
The + sign corresponds to a symmetric wave function, whereas the — sign
corresponds to an antisymmetric wave function.
The Pauli principle state the following. Particles with spin quantum num-
number equation to one-half an odd integer are described by antisymmetric wave
functions, whereas particles with integral spin quantum numbers are described
by symmetric wave functions. Particles in the first class are called fermions,
and particles in the second class are called bosons. Electrons and protons are
examples of fermions. Photons, J~[, and K mesons are examples of bosons.
For a system comprised of TV identical particles, properly symmetrized wave
functions may be put in the form of Slater determinants. Let vk denote the
quantum numbers for the A:th particle in the aggregate. For fermions, we write
1 ^ ■ ■■- - A-37)
The summations run over the permutations of (v\ • • • vN). The symbol |P| is
zero or one depending on whether P is even or odd, respectively, and "
represents X\, " represents x2, and so forth.
Since \j/A is a determinant, it changes sign under exchange of two rows or
two columns. So particle exchange in the wave function A.37) carries a sign
change.
Furthermore, \//A vanishes if two rows or columns are the same. This occurs
it, for example, we set v\ = v-j. Thus, in an aggregate of TV fermions, no two
particles can be in the same quantum state. This property of fermions is called
the Pauli exclusion principle.
The wave function for an aggregate of TV identical bosons may be written
A.38)
This function remains unchanged under exchange of any two particles, so
bosons do not obey the exclusion principle.
5.7.7 Heisenberg Picture
In addition to change in representation through transformation of basis in
Hilbert space, as described in Section 1.3, we also speak of alternative "pic-

340 5. Elements of Quantum Kinetic Theory
tures" in quantum mechanics. In the Schrodinger picture wave functions obey
the Schrodinger equation A.4) and evolve in time through A.9).
Consider the unitary operator
(itH .
— 1 A.39)
with properties
U' =U~X
U^U = 1 A.40)
With U so defined, we may rewrite A.9) as
H A.41)
In the Heisenberg picture, wave functions are related to their Schrodinger
counterpart through the equation
A.42)
With A.41), we obtain the important result
Hit) = U(t)ir(t) = U(t)U-\t)ir@) = V@) = V//@) A.42a)
That is, in the Heisenberg picture wave functions remain fixed at their initial
values.
Operators in the Heisenberg picture.
AH(t) = U(t)AU-\t) A.43)
are seen to vary in time. That is, an operator that is stationary in the Schrodinger
picture varies in time in the Heisenberg picture.
An important result that emerges in the Heisenberg picture concerns the
equation of motion for an operator. In the Schrodinger picture, this equation is
given by A.5). To obtain the corresponding equation in the Heisenberg picture,
first we calculate the time derivative of A.43).
Ar + ^ + A
dt dt dt dt
With A.39), we find
*»=HU
dt
dt

5.2 The Density Matrix 341
FIGURE 5.2. The whole system has the wave function \//(x, y). If \//(x,y)
if\(x)\lr2(y), then system X does not have a wave function. In this event, X is said
to be in a mixed state.
Inserting these relations into the preceding equation gives the desired equation
of motion:
- A.44)
at dt
where we have written4
dA * ,
= U — U~l A.44a)
dt dt
Equation A.44) plays an important role in the interpretation of the correspond-
corresponding equation of motion for the density operator described in Section 5.2.5
5.2 The Density Matrix
Mixed states
Consider a system that is coupled to an external environment, such as a gas of
N particles maintained at a constant temperature through contact with a heat
bath. If x denotes coordinates of the gas and y coordinates of its environment,
then, whereas the closed composite of system plus environment has a self-
contained Hamiltonian and wave function \j/{x, y), this wave function does
not, in general, fall into a product VoOOV^OO- Under such circumstances, we
say that the system does not have a wave function (see Fig. 5.2). A system that
does not have a wave function is said to be in a mixed state. A system that does
have a wave function is said to be in a pure state.
It may also be the case that, owing to certain complexities of the system, less
than complete knowledge of the state of the system is available. This situation
arises for systems with a very large number of degrees of freedom, such as,
for example, a mole of gas. The quantum state of such a system involves
specification of ~ 1023 momenta.
4For further discussion, see R. L. Liboff, Found. Physics 77, 981 A987).
5Additional concepts in basic quantum mechanics appear below. Thus, for
example, second quantization and closely allied Fock space are introduced in
Section 5.5.

342 5. Elements of Quantum Kinetic Theory
5.2.7 The Density Operator
For such cases as described above, in place of the wave function, we introduce
the density operator p. If A is some property of the system, the density operator
determines the expectation of A through the relation
(A)=Tr(pA) B.1)
and
Trp = l B.2)
Let us calculate the matrix elements of p for the case of a system whose
wave function ^ is known. In this case we may write
(A) = {\fr\A\fr)
Let the basis {\n)} span the Hilbert space containing \f/. We may expand \j/ in
this basis to obtain
n))n | V)
n
Substituting this expansion into the preceding equation gives
n
jP"*^" =TrM =TrAp
q n
Have we have made the identification
Pnq = (q I f)*(n | V) = a*an B.3)
The coefficient an represents the projection of the state ^ onto the basis vector
\n). The nth diagonal element of p is
n)\2 = a*an = Pn B.4)
which we recognize to be the probability Pn of finding the system in the state
\n). Thus the diagonal elements ofp are probabilities and must sum to 1.6 This
is the rationale for the property B.2), Tr p = 1.
We note in passing that the representation of p given by B.3) refers to the
basis {\n)}. Suppose these states are eigenstates of the Hamiltonian. As noted
previously, we would say that p as given in B.3) is written in a representation
in which energy is diagonal or, more simply, the energy representation.
Now consider that the system is in a mixed state. Thus we may assume that
the projections an are not determined quantities. In this case we define the
6The off-diagonal elements of p relevant to a given system have been shown to be
related to the long-range order of the system (this phenomenon carries the acronym
ODLRO). For further discussion, see C. N. Yang, Revs. Mod. Phys. 34, 694 A962).

5.2 The Density Matrix 343
elements pnq to be the ensemble averages:
pnq = a*an B.5)
The diagonal element
pnn = a*an B.6)
represents the probability that a system chosen at random from the ensemble
is found in the nth state.
Equation of motion
Suppose again that a system is in a pure state and has the wave function ty.
Again, let TV be a measurable property of the system with eigenstates {\n)}.
Expanding i/r in terms of the projections an gives
n
From the Schrodinger equation for \//, we obtain
Eoan
dt
n n
Operating on this equation from the left with (/| gives
da\ v-—\ da? y—^
dt -^ ar -^ /n n
n n
We may use these relations to obtain an equation of motion for the matrix
elements of p
dpqi dafaq (daf daq\
in—— = in——- = in \-a,—- B.8)
dt dt \dt l dt )
Substituting the expressions B.7) for the time derivatives of the projections an
and setting H*k = Hki, together with forming the ensemble average, gives
- PqkHki) B.9)
or, equivalently,
..3/5
B.10)
ot
As p depends only on time [the projections an in the defining relation B.5)
are spartially independent], the partial derivative in the preceding equation is
understood to be a total time derivative [as is the case for B.12) et seq.].
In the Heisenberg picture, wavefunctions remain fixed [see A.42a)] so that
Ph is constant and we write
^0 B.11)
dt

344 5. Elements of Quantum Kinetic Theory
which, again, is in the form of the Liouville equation A.4.7). Whereas these
resemblances are very strong, we should not lose sight of the fact that in
classical physics /n(xn ,pN,t) gives information of both xN and pN at the time
t. In the quantum domain, simultaneous specification of xN and pN cannot be
made. However, soon after the emergence of quantum mechanics, E. P. Wigner
developed a theory which gives a formal connection between the density matrix
and the classical distribution function. We return to this topic in Section 2.2.
Some elementary examples
Given the form of the equation B.9) some elementary solutions are evident.
Thus, as with the classical Liouville equation, any operator of the form
p = p(H) B.12)
gives
df) r£/ -i n
— = [H,p] = 0
at
so that the form B.10) may be termed an equilibrium distribution. Two
important examples of this distribution are relevant to statistical mechanics:
1. Canonical distribution:
^ B.13)
This distribution is relevant to a system in equilibrium with a temperature
bath at temperature T = \/kBf$.
2. Microcanonical distribution:
p = A8(H-E) B.14)
This distribution is relevant to an isolated system of total energy E.
Let us consider these operators in the energy representation. Energy
eigenstates obey the eigenvalue equation
H\En) = En\En) B.15)
With these eigenstates at hand, diagonal elements of p in the canonical
distribution are given by [recall B.4)]
whereas, in the microcanonical distribution, we find
P(En) = A8(En - E)
All energy states are equally probable in this distribution.
Initially diagonal density matrix
Consider a quantum system whose density is initially diagonal:
pnl@) = Pn8nl B.16)

5.2 The Density Matrix 345
The density matrix evolves according to B.8). For the case at hand, we find
that at t = 0,
ih
dt 0
Summing over k gives
-^ =Hnl(Pl-Pn) B.17)
dt 0
For n = /, we find
dpi
dt 0
= 0
So diagonal elements have zero slope in time initially. For n ^ I, B.17) indi-
indicates that if Pi ^ Pn, off-diagonal elements of p develop and B.16) cannot be
termed an equilibrium distribution. If, however, all states are equally probable,7
then B.16) becomes
which gives dp/dt = 0 so that B.18) is a valid equilibrium distribution.
Random phases
An important case in which p is diagonal enters in the assumption of random
phases. Consider the matrix elements of p B.5):
r\ — n n
Hnm — u-m wn
The indeterminacy of the state of the system may be manifest in a corresponding
indeterminacy of the phases {</>„} of the projections {an}. These phases are
defined through
an — cnel<t>n
where cn and <j>n are real. Consider the matrix element
Pnm = cmcn exp[/@n - 0OT)] = Cmcn[cos((pn - (j)m) + i sin@n - 4>m)]
If phases are random, then in averaging over the ensemble cos@n — <f>m) oc-
occurs with positive value equally often as with negative value and similarly for
sin@n - 0m), so that
cos((pn -4>m) = sin@n - 0OT) = 0
except when n = m. In this case
Pnn = C*Cn COS((pn -</>„)= ~C%C~n
It follows that for the case of random phases p is diagonal
7The condition of equal a priori probabilities.

346 5. Elements of Quantum Kinetic Theory
5.2.2 The Pauli Equation
We wish to examine how an initially diagonal density matrix evolves under
the influence of a perturbation. Matrix elements of p and the Hamiltonian of
the system at t < 0 are given by
Pnl = Pnl@)8nl,
< 0)
B.19)
The vanishing commutator ensures that the system is in equilibrium prior to
t = 0.
A harmonic perturbation is turned on at t = 0 and the Hamiltonian of the
system becomes
H = Ho + H\
t > 0
B.20)
The probability for transition from the nth to the A:th state of the system, in
short interval At after W is turned on, is given by
B.21)
h2
Taylor-series expanding the density matrix about f = 0 gives
p(t) - p@) =
dp_
dt
At +
(At)
0
With B.9), this equation may be rewritten
p(At) = p@) = ^[H, p@)] -
7h
With the commutation property B.19), B.22) may be written
p(At) - p@)
At
1 , Ar
-[H',p@)]-
Ar
M
B.22)
Taking diagonal elements and passing to the limit At —> 0 gives
0
= lim <
Ar
-[//', p@)U-
Tft
nn
B.23)

5.2 The Density Matrix 347
Consider first the leading term:
', p@)]nn = ]T[{w|ff'l<7><<7lp(O)|w> - (n\p@)\q)(q\H'\n)]
= Pnn(O)iKn ~ KJ = 0
Similarly, we find
[Ho, [//', p@)]]nn = 0
With the first two elements on the right side of B.23) vanishing, there remains
dpnn@)
dt Ar^O 2ll
[H'[H'p@)]],
= - lim ^{H\ [H\ p@) - p@)H'] - [H'p@) - p(O)H']H'U
In
= lim ^L[-H'2p@) + H'p@)H' + H'p(O)H' - p@)H>2]nn
In
= lim ^ [2H'p@)H' - H'2p@) - p@)H>2]nn B.24)
In
Expanding products in terms of basis states gives
at in k q
-(n\H'\k)(k\H'\q)(q\p(O)\n)
-(n\p(O)\q)(q\H'\k)(k\H'\n)] B.25)
With B.19), we find
k q
- pnnmnqH'qkH'kn\ B.26)
Summing over q gives
dpnn At t—\
X{ T[2HHi@) - H'nkH'knPnn{Q) - pnn{Q)H'nkH'kn]

- \H'nk\2pnnm
k
With B.21), we obtain
- pnn@)] B.27)

n - PnnWnk)
348 5. Elements of Quantum Kinetic Theory
Since the left side of this equation represents the derivative of pnn at t = 0, we
may generalize this finding to the form
B.28)
Here we have further made the identification w^ = wnk corresponding to the
hermiticity of H'. The latter equation is called the Pauli equation or the Master
equation [compare with A.7.20)]. An H theorem for this equation is derived
in Problem 5.48. Application of B.28) follows.
Density matrix for spinning particles
Spin matrices for a spin ^ particle are given by
0 \ . H ( 0
/5
z~ 2\Q -\ )' bx2\\ 0 )' * 2 ^ 1 0
B.29)
Once again we are working in a repesentation in which S and Sz are diagonal.
These matrices may also be written
S = -<r B.30)
The components of a are called the Pauli spin matrices. Thus
/I 0 \ A / 0 1 \ A / 0 -1 \
&z = l , vx = \ , <Jy = M B.31)
Eigenvectors of 5Z (or, equivalently, az) are given by A.33). We repeat,
1
x 0
where
■■-(£)• A=(i) <2321
az, Szpz pz B.33)
Consider a beam of electrons whose spins are isotropically polarized. The
corresponding density matrix is
'-HJ!)
Since this is written in a representation where Sz is diagonal, diagonal elements
of p give probabilties relevant to the components of Sz. For the matrix B.34),
these probabilities have the values

5.2 The Density Matrix 349
Furthermore, we find
(Sx)=TvpSx =0
(Sy) = (Sz) = 0
These values are relevant to an isotropic beam.
Elementary application of the Pauli equation
Consider an electron beam with the density matrix
a 0
\-a
The beam enters an infinitesimal region of space that contains a uniform B
field that points in the z direction.
B = @, 0, Bo)
Let us find the z-component spin populations when the beam emerges from
the field.
The perturbation Hamiltonian is given by
The electron's magnetic moment is given by
/Ex = -nb&
where jib is the Bohr magneton. Thus we may write
H = fibGzB0 fibBol
The Pauli equation for the case at hand gives
~ Pll) = ™2\(p22 ~ P\\)
dt
* B.35)
' = £_j Wq2(Pqq ~ P22) = ™\2(P\\ ~ P22)
q
With B.21), we have
At . 2
W\2 = W21 = —=-//, 9
n1
Since p is calculated in the basis (a, ft), we find
H[2 = 0
Thus, with B.36), we see that pn and p22 remain constant under the given
perturbation.

350 5. Elements of Quantum Kinetic Theory
More generally, under an arbitrary perturbation, we note that the Pauli
equation B.28) indicates that the Tr p remains constant.
The projection representation
As noted previously, an ensemble description is relevant to a system in a mixed
state. Let the states ^r of the ensemble be distributed with the probability P^.
In this event, we may write the density operator as the following projection
sum over states of the ensemble:
]T B.36)

Consider, for example, that the probability of finding the energy En in
measurement on a given system is Pn. For this case, B.36) gives
n
where \En) are energy eigenstates for the given system. As these states com-
comprise and orthogonal sequence, it follow that p is diagonal with diagonal
elements equal to Pn.
The projection representation of the density matrix given by B.34) is
P = \<*z)\(az
Again we find
Sz = Tr pSz = (az\pSz\az) + {pz\pSz\pz)
If a system is in a pure state, the wave function exists, and we write
P = \f)(f\ B-37)
In a representation with basis {\n)}, matrix elements of p are then given by
= a*an
in agreement with B.3).
The projection representation comes into play again in Section 5.5 in
consideration of generalized hierarchies.

5.2 The Density Matrix 351
5.2.3 The Wigner Distribution
Coordinate representation of p
Another important form of the density matrix is found in the coordinate
representation. The related density matrix appears as8
p(x, x') = (x\if)(if\xf) = V(x)V*(x') B.38)
(x and x' are viewed as the indices of the matrix of p.) The basic property B.2)
is written
Tr p = I S(x- x')p(x, x') dx dx' = 1 B.39)
For the property B.1), we write
{A) = Tr(pA) = Tr(Ap) = J A(x, x')p(x', x)dx! dx B.40)
where A(x, x') is the coordinate representation of A:
A(x,x') = (x|A|xr) B.41)
In a pure state, B.40) becomes [recall B.37)]
Tr(Ap)= f {x\A\x'){x'\if){if\x) dx' dx B.42)
Here we have written (xr|^) for the coordinate representation of x/r, more
commonly written simply as xj/{x') (recall Section 1.3).
The last relation may be rewritten
Tr(Ap)= / (}//\x){x\A\x'){x'\}l/)dx'dx= Ar\A\lr) = (A) B.43)
One further preparatory remark is in order prior to our discussion of the
Wigner distribution. This concerns the momentum representation of a wave
function:
B.44)
The transfer (p|x) is given by A.28):9
(Plx) = t^4w^~/PX/* B'44a)
8This is the form of p as originially put forth by J. von Neumann, Mathematical
Foundations of Quantum Mechanics, R. T. Beyer, (trans.), Princeton University Press,
Princeton, N.J. A955).
9The relation A.28) is relevant to one dimension, whereas B.44a) is written in
three dimensions.

352 5. Elements of Quantum Kinetic Theory
Thus B.44) becomes
This function is the amplitude of the momentum probability density, |<p(p)|2.
The Wigner distribution and its properties
To apply the following formalism to kinetic theory, we extend the analysis to
TV-particle systems. With p written in the coordinate repesentation, the Wigner
distribution is given by10
/ 1 \3N C
F(xN, pN, t) = ( — ) / e2l» y *p(x_, x£, t)dyN B.46)
where
Note that in a pure state B.46) becomes
(LY [
N ,,N m . j. . M. . . M . . \f .^ A'~l\
The inverse of B.46) is given by
F(xN, pN, r)exp F '* dpN B.48)
\ n j
The Wigner distribution serves as a bridge between quantum and classical
kinetic theory. Four key properties that demonstrate this connection are as
follows.11
1. Integration over momenta gives the configurational probability density:
F(xN, pN, t)dpN = \x/r(xN, t)\2 B.49)
2. Integration over spatial coordinates gives the momentum probability
density:
B.50)
10E. P. Wigner, Phys. Rev. 40,749 A932). In this paper we are informed that B.46)
is due to a joint efford of Wigner and L. Szilard. Note in particular that, as F(x, p, t)
is not positive definite, it cannot be view as a probability density.
11 For further discussion, see Studies in Statistical Mechanics, vol I., J. de Boer
and G. E. Uhlenbeck (eds.), North Holland, Amsterdam A962); and B. Kursunoglu,
Modern Quantum Theory, W. H. Freeman, San Francisco A962).

5.2 The Density Matrix 353
3. The expectation of a dynamical function, A(xN, pN), is given by
(A) = I A(xN,pN,t)F(xN,p\t)dxN, dpN
B.51)
The preceding three properties indicate how the Wiegner distribution
bears a striking resemblance to the classical TV-particle joint probability
distribution.
4. The derivation of a dynamical equation for F(xN ,pN ,t) begins with
Schrodinger's equation written for an TV-body system with an interaction
potential V(xN):n
in
dt
k=\
,N
B.52)
Assuming the system to be in pure state, and with the identifications B.52),
B.38), and B.47) in mind, we obtain
dF
(-T f
\jth/ J
N
,N
dy exp
2/p" • y
n
2m V
dx2k
z[V(xN+)-V{xN_)]p(xN_,xN+)
h
B.53)
Next we replace x differentiation with y differentiation once and then
perform a single parts integration over y. Combining this operation with
replacement of p(x^, x^) with its inverse B.48) and noting that
ay/
gives
rn dxk h nh
B.54)
This is an integrodifferential equation for the Wigner distribution function.
[The hierarchy equations for this distribution are given in E.33).]
12
Note that d2/dxj = Vk.Vk.

354 5. Elements of Quantum Kinetic Theory
13
5.2.4 Weyl Correspondence
A note of caution is in order concerning the rule B.51) for obtaining averages
of dynamical variables: For consistency, this expression must agree with B.1).
Such agreement is quaranteed if the dynamical variable A(xN, pN) in B.51) is
appropriately related to the operator A in B.1.)
Without loss in generality, we work in one dimension and introduce the
Wigner counterpart of A, written (A)w, and given by the linear operation
(A)w = 2 f e2ipx/n{x -x\A\x+x)dx B.55)
where (A)w is a function of (x, p). Note that the matrix element of A in B.55)
is written in the coordinate representation. When written in the momentum
representation, B.55) appears as
{A)w = 2 f e2ipx/n(p + p\A\p - p) dp B.55a)
When the dynamical variable A(x, p) in B.51) satisfies the correspondence
= A(x,p) B.56)
Weyl correspondence is obeyed and the expectation of A is given by B.51).
Let us establish this rule. Again consider B.51)
(A) = — J J J dxdpdyA(x,p)e2ipy/np(x.,x+) B.57)
Working in the coordinate representation, we insert B.55) for A in B.57).
There results
{A) = — J J J J dxdxdpdye2ip(y+x)/n(x-x\A\x+x)p(x-y,x + y)
B.58)
Integrating over p and recalling the delta-function normalization B.3.1 lb), we
obtain
2Btt) (h\ r r r
[j / / dx dx dy 8(y-\-x)(x - x\A\x + x)p(x-y, x+y)
r
= 2 1 I dxdx (—xx\A\x + x){x+x\p\x — x)
If
dx-dx+{x.\A\x+){x+\p\x+) =TrAp = TrpA B.59)
13For further discussion, see N. L. Balazs and B. K. Jennings, Phys. Repts, 104,
347 A984); and M. Hillery, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Phys.
Repts. 106, 121 A984)

5.2 The Density Matrix 355
which is seen to agree with B.1). Note that in the last step the change in
variables x, x —> jc_, x+ carries a Jacobian of \. Thus we see that if the rule
B.56) is obeyed the average given by B.51) agrees with that given by B.1)
Correspondence forms
Here we wish to list some relations relevant to Weyl correspondence. First let
us establish that operators of the form f(x) or g(p) obey Weyl correspondence.
That is,
[f(x)]w = fix), [g(p)]w = g(p) B.60)
Consider the first relation for f(x).
[f(x)]w = 2 j e2ipi/n{x -x\f(x)\x + x)dx
= 2 J e2ipi/n f(x - x)(x - x\x + x) dx
= 2 f elip~xlhf(x - x)SBx)dx
= f(x)
A similar argument applies to g(p). An important operator in this discussion
is given by
~> d d d d
dx dp dp dx
where 3/3jc operates to the left, and so forth.
With [A, B]T written for the commutator (—) and anticommutator (+), we
may then write
([A, B].)w = (A)w ( 2/ sinn-d) (B)w B.61a)
([A, B]+)w = (A)w ( 2cos^0 j (B)w B.61b)
Thus, for example,
1 \ B
ifi Jw \h 2
= xOp = 1
whereas
, p]+)w = 2x I cos -0 1 p = 2xp

356 5. Elements of Quantum Kinetic Theory
Thus
(xp)w = xp + —
We may conclude that the operator xp does not obey Weyl correspondence,
whereas the symmetric form [Jc, p]+ does.
Stemming from B.61), we write
(AB)W = (A)w I exp— 1 (B)w B.62)
Here are some additional relations:
f dx dp „
(A) = Tr(pA = / (p)w(^)w B.63a)
,/ 27T/Z
Tr(Af£)= / -(A)UB)W B.63b)
J 2nft
Ef)w = (B)* B.63c)
Let us establish B.63c).
^ = 2 f
^-2//?^(jc+Jc|^|jc-Jc)rfJc
In the last step we employed the change in variables: Jc —> —Jc. Additional
properties of these correspondence froms are left to the problems.
The classical limit
Let us see how B.54) reduces to the Liouville equation in the classical domain
ft = 0. With the change of variables
we may write
X±-X ±j
and the integral term in B.54) becomes
dF i ( i \3N r „ ,M
5 \2nJ J
3' mt
x exp[it=N . (p^ - pN)][V(x») - V(x^)]F(xN, pN, t) B.64)

5.2 The Density Matrix 357
In the limit h —> 0, there results
■ N
—»■ >-
n
and B.64) reduces to
3N
V[xN + mN/2)] - V[xN -
dF
rv. I /•> I <-\ A/ f f r\ A/ >■ '
Of
//
'N
- p'N)]F(x",
dp" J dxN
, t)dp
N
8 //(xw,pA')-/-F(xA',xA',f) B.65)
dp^
Combining B.65) with B.54) gives
dF
= [H, F]c B.66)
ot
where C denotes the classical Poisson bracket. We may conclude that the
Wigner distribution function satisfies the Liouville equation in the classical
domain h = 0.
5.2.5 Wigner-Moyal Equation
The Wigner-Moyal equation in quantum kinetic theory is the analogue of the
particle Liouville equation in kinetic theory. To derive this equation we examine
the Wigner equation B.54) for the case N = I.
Three results
n, A i fci t^y^P'exp -^-f-— [V(x+KV(x_)]F(x,p/,r) = 0
Dt n \7tn J J \ h
B.67)
where
B.67a)
( +
Dt \dt m
We wish to reduce B.67) to a more concise form. Noting the relation
exp ( a • — ) /(x) = /(x + a)
permits us to write
V(x+) - V(x_) = V(x + y) - V(x - y) = [ey(d/dx) - e~y (d/dx)]V(x)
= -2i sin (iy • ^^i V(x)

358 5. Elements of Quantum Kinetic Theory
Thus B.67) becomes
DF 2 / 1 \3 f , , , Biy • (P - P')
f
j
in (i
x sin (iy • ^ j V(x)F(x, p', 0 = 0 B.68)
where d/dx operates on V(x) only. To further reduce B.68), we note the relation
eiyp>iyF(p')dp' = J (j-/yP') Hp')dp'
/
= - I eiyp' — F(p')dp'
J dp' yF F
Thus, if g(y) is a regular function, we obtain
J eiyp'g(iy)F(p')dp' = J jy'g (-jp) F{p')dp
Incorporating this property into B.68) gives
Th J
dy dp exp
Dt n\jiH/j v ft
'h d 3 \
x sin ( - — • — V(x)F(x, p, t) = 0 B.69)
2 dp oxj
Recalling the delta function representation
B
and setting z = 2y/H gives
DF
1— f
■ttK J
dj>'S(j> - P)
Dt
3p/
Performing the final pr integration gives the desired result:
d p d \ 2 /n d d .
+ — • — --sin V(x)
dj n \2d d'
+ sin
dt m dxj n \2dp dx
F(x, p, t) = 0 B.70)
where, again, we recall that 3/3x act only on V(x). The relation B.70) is
sometimes called the Wigner-Moyal equation. It is instructive to note that for
Hamiltonians of the form
P2
H = f- + V(x)
2m

5.2 The Density Matrix 359
B.70) may be rewitten14 (see Problem 5.8) as
dF 2 .
h - sin
dt n
n (d d
d d
_2 \dx dp dp dxj_
FH =0
B.71)
Here it is understood, as in its classical counterpart, that the right operators in
the two products operate on //, whereas the left operators operate on F.
The classical limit of the Wigner-Moyal equation as given by B.71) is par-
particularly simple. Settingh —> 0 and keeping the leading term in the expansion
of sin[ ] returns the classical Liouville equation B.66).
Note that all the above analysis can be generalized to an TV-body problem
with interaction potential V(xN) by setting x -» xN and p -» pN [see E.33)].
Let us return to the quantum one-particle Liouville equation B.70). An
ad hoc kinetic equation may be obtained from this relation by inserting an
appropriate collision form on the right side which, in the Boltzmann formalism,
is written J(F). In the KBG approximation, for example, we write
DF 2 (ft d
sin
n V2
d
V(x)F(x, p, 0 = v[F0(x, p, 0 - F(x, p, t)]
B.72)
where F0(x, p, t) is an appropriate equilibrium distribution and v is
relaxation time.
An additional note is in order concerning the Wigner-Moyal equation B.71).
As x and p cannot be simultaneously specified in quantum mechanics, we
expect the quantum distribution F(x, p, t) to be defined for volumes in F-space,
The Wigner-Moyal equation has found wide application in the analysis of
charge carrier transport in a semiconductor.15
5.2.6 Homogeneous Limit: Pauli Equation Revisited
In this section we employ the equation of motion for the Wigner distribution
function B.70) to derive a kinetic equation for the matrix elements of the
density matrix in the limit of spatial homogeneity. With this motivation and with
reference to B.46), we write the single-particle Wigner distribution function in
momentum representation, which, together with the change in variable 2y -»
k, appears as
p, 0 =
P
B.73)
14A hierarchy of equations for reduced Wigner distributions stemming from the
Wigner-Moyal equation in Af dimension is given in Section 5.5.4; see specifically
E.33).
15 See, for example, The Physics of Submicron Structures, H. L. Grubin, K. Hess,
G. J. Iafrate, and D. K. Ferry (eds.), Plenum, New York A984).

360 5. Elements of Quantum Kinetic Theory
Note that k has dimensions of momentum. Let us call
' k
P+2
With these identifications,taking the Fourier transform of B.70) (and recalling
the convolution theorem) yields
p - |) = p(k, p) B.74)
+ f) ^(k P} " f dk'l sin
J
f dk'l sin (^T ' ir) V(k>(k ~ k> P) =
J n \ 2 dp/
dt m ) J \ p/
B.75)
where V(k) represents the Fourier tansform of the interaction potential V(x).
We wish to obtain an equation for the density matrix relevant to the
homegeneous limit. This is performed through a process of iteration. In the
homogeneous limit, the k = 0 contribution to the integral B.75) dominates,
and we write
d
p(p) = J dk'S(k') V(k')p(-k\ p) B.76)
dt
where
p@, p) = p(p) B.76a)
is written for the diagonal elements of the density matrix and 5(k) represents
the operator
Changing variables k' -^ -k' in B.76) gives [with V(-k) = V*(k)]
', p) B.77)
This equation contains two distributions, one relevant to k = 0 and the other
relevant to k ^ 0. We assume that the k ^ 0 distribution, p(k, p), varies in
time as exp(/&>f), where the interval co~l may be associated with characteristic
times of the fluid. The time dependence of the k = 0 distribution p(p) is
determined from its equation of motion (to be derived). For p(k, p), we return
to B.75) and write16
f
J
dk'V(k')S(k')p(k - k\ p)
16Note that the time dependence of p(p), as well as the co dependence of p(k, p),
is tacitly assumed.

5.2 The Density Matrix 361
Expanding the integrand about k = k' and then keeping the first term with
p(k - k\ p) = S(k - k')p(p), we obtain
P(k, P) = t
1
V(k)S(k)p(p)
Inserting this value into B.77) gives
V(k')S(k')p(p)
B.78)
Now we note
/k a
= sin
dp
k
k
= / sinh —
2
d
a
—
3p
exP x
2 3p
k
2
d
whence
B.79)
Employing this relation in B.78) gives
j dk'\V(k')\2S(Vh
d 2
—P(P) = ~ z-
I
x -
P
k'
~2
k'
~2
B.80a)
Performing the remaining 5 operation gives
[p(p + k') - p(p)] [p(p) - p(p - k')]
-
Changing variables in the second term, k' —> — kr, gives
O 1 /»
-p(p) = ^ J dk'\ V(k')|2[p(p + k') - p(p)]
B.80b)
1
1
B.81)
We may view k' as a momentum transfer in a scattering from the potential
V'(x). The change in particle energy in this scattering is given by
AE =
2m
2m m
B.82)

362 5. Elements of Quantum Kinetic Theory
Thus B.81) may be written
i / _ ~
— n(p) = _ I dk \V(k )|[p(p + k )-p(p)]
dt F7 /i2 7 ^vj
B.83)
As k' has been identified as momentum transfer in a collision, it follows that
h/ AE is a measure of the corresponding collision time. Assuming this interval
to be short compared to characteristic times of the fluid, we write
h 1
AE co
To incorporate this property into the analysis, we set
ico —> s > 0
where e is a parameter of smallness. We may then write
1 1 ±1
Recall the representation (Plemelj's formula)
18
where V denotes principal part. Inserting these relations into B.83) gives the
desired result:
3p(p) In , rfk,^(k,)|2 S(AE)[p(p + k') - p(p)] B.84)
dt n
This completes our derivation of a closed kinetic equation for the diagonal
matrix elements of p relevant to the homogeneous limit.17
The kinetic equation B.84) may be cast in a more physically revealing form
by virtue of the following observation. We note that the factor
w(p, p + k) = -H V(kr)|2 8(AE) B.85)
n
that appears in B.84) is an expression for the rate of scattering from p to p + k'
(or the reverse) (see Fig. 5.3).
With the relation B.85) at hand, B.84) may be identified with the Pauli
equation B.28). In the present case, the kinetic equation B.85) is the momen-
momentum representation of the Pauli equation for which the scattering rate B.85) is
the continuum form of the discrete rate B.21).
17For further discussion, see P. Carruthers and F. Zachariasen, Rev. Mod. Phys.
55, 245 A983); and W. Kohn and J. M. Luttinger, Phys. Rev. 108, 590 A957).

5.3 Application of the KBG Equation to Quantum Systems 363
w(p, p+k')
FIGURE 5.3. Diagrams for scattering by the potential V. For elastic interactions,
the rate for collision and its reverse are equal.
Owing to the symmetric property of the scattering rate w(p, p + k'), B.84)
may be written
= j dk'[w(p + k\ p)p(p + k') - tu(p, p + k')p(p)] B.86)
The first factor in the integrand contributes to scattering into momentum p states
and represents again, whereas the second factor contributes to scattering out of
momentum p states and represents a loss. The latter equation is returned to in
Section 3.6 in the derivation of the relaxation time for charge carrier-phonon
interactions.
5.3 Application of the KBG Equation to Quantum
Systems
5.3.1 Equilibrium Distributions
One of the more significant problems in statistical mechanics concerns the
distribution function for an appregate of particles in equilibrium at a given
temperature T. There cases important in this study relate, respectively, to A)
classical particles, B) bosons, and C) fermions.
For aggregates of noninteracting particles, equilibrium distributions for
these cases, giving mean occupation number per energy level, are given by18
ME) = \ C.1)
P(t-ii) \ a
= (kbTYx
where
a = 0 corresponds to classical particles
a = 1 corresponds to fermions C.2)
a = — 1 corresponds to bosons
18For further discussion, see K. Huang, Statistical Mechanics. Wiley, New York
A967); and D. A. Mcquarrie, Statistical Mechanics, Harper & Row, New York
A976).

364 5. Elements of Quantum Kinetic Theory
and the variable /x is the chemical potential.19
Note that quantum distributions reduce to the classical distribution in the
limit
or, more simply, fo(E) <£ 1. That is, the classical distribution occurs when the
mean occupation number per energy level is small compared to unity.20
The condition C.3) may be shown to be equivalent to the criterion21
^ « ^ C-4)
where
h
^ C.5)
is the thermal de Broglie wavelength, V is volume, and TV is total number of
particles. The condition C.4) states that the classical picture results when the
de Broglie wavelength is small compared to interparticle spacing.
We wish to apply the distributions C.1) to two key problems: A) an
interacting gas of atoms and photons, and B) electrons in a metal.
Planck distribution
Photons are bosons, so the related distribution is given by C.1) with a = — 1.
Furthermore, for photons we may set22 /x = 0. With E =Hco, the appropriate
distribution is given by
^ C-6)
We wish to obtain the number of photons per frequency interval, per volume,
F{co). The distribution C.6) gives the mean number of photons per frequency
mode. Thus we may set
F(a>) = g{a))f{co) C.7)
In this equation, g(co) represents the number of normal modes per frequency
interval, per unit volume. An expression for the density of states, g(co), may be
19The chemical potential enters the first law of thermodynamics for cases involving
varying number of particles: dE = T dS— P dV + ixdN. Here S represents entropy.
20For bosons with /x = 0, the classical limit occurs for small /3. See, for example,
C.68).
21R. K. Pathria, Statistical Mechanics, Pergamon Press. Elmsford, N. Y. A978).
22This is a consequence of thermodynamics. Consider a gedanken process in
which the number of photons in a monoenergetic field at frequency co is doubled as
co -* (jo 12. With dE = dS = dV = 0, and dN > 0, we obtain /x = 0.

5.3 Application of the KBG Equation to Quantum Systems 365
obtained in the following manner. First recall that the momentum of a photon
of angular frequency co is
hco
P = — C.8)
c
It follows that the volume in momentum-coordinate phase space containing
photons with frequencies in the interval co, co+dco is equal to the product of d3x
and the momentum volume corresponding to values in the interval p, p+dp or,
equivalently, 4np2 dp d3x. The number of states in the said interval is obtained
by dividing this latter volume by the minimum of volume per state, h3. Since
photons have two possible polarizations, this minimum volume becomes h3/2.
Thus we may write
, w ,3 4np2dpd3x
g(oj)dojd x = —
co2 dcod3x
TZ2C3
which gives the desired form for the density of states (Rayleigh's formula):
CO2
8(CO) = -5-r C.10)
Combining C.7) and C.10) gives the Planck distribution:
C-n)
Here we have inserted a zero subscript to denote that C.9) is a thermal
equilibrium distribution.
An immediate result that follows from the Planck radiation law C.11) is the
total radiant energy density, U(T), of a radiation field in equilibrium at the
temperature T.
h
U(T)= / Foiooyioodoo C.12)
Introducing the nondimensional parameter
x = f$hto
gives
X (XX
u(T) = r
f
Jo
x2k4
7t2C3H Jn ex —
C.13)
B
\5n c

366 5. Elements of Quantum Kinetic Theory
F . F
E,
E,
(a) (b) (c)
FIGURE 5.4. Three radiative collision processes are included in C.18): (a) resonant
absorption, (b) stimulated decay, (c) spontaneous emission.
[see (B2.23)]. The power radiated per unit area by the radiation field is given
by
Ktafc)
P =aT4
Here c is the speed of light and a if the Stefan-Boltzmann constant, which has
the value
a = 0.567 x 1(T4 erg/s - cm2 - K4 C.15)
5.3.2 Photon Kinetic Equation
We consider a radiation field interacting with a gas of hydrogenic atoms whose
excited states are characterized by the outer atomic electron. The density of
the atoms in the ith excited state is written Nt.
In our kinetic model we discuss photons that are either entirely absorbed by
an atom or give rise to stimulated decay of an atom. Furthermore, phontons
may be generated through spontaneous decay of atoms. Thus the model does
not included, for example, scattering of photons by atoms (see Fig 5.4).
The momentum of a photon of (angular) frequency to is given by C.8),
which we now writes as
where s is the unit vector
P = C.16)
c
s=- C.17)
P
Our kinetic equation is then written23
d
cot
/(x, s,co) + s> V/(x, s, co) = Y" A/n C.18)
23D. H. Sampson, Radiative Contributions to Energy and Momentum Transport
in a Gas, Wiley-Interscience, New York A965). See also D. M. Heffernan and R. L.
Liboff, J. Plasma Physics 27, 473 A982).

5.3 Application of the KBG Equation to Quantum Systems 367
The collision term Ain is defined below.
Let us first concentrate on the distribution function in C.18). It is normalized
with the differential measure C.9). The quantity
2fdpdx
represents the number of photons in the phase volume dpdx about the value
(p, x). In the present description the distribution function / relates to photons
progagating in the direction p with frequency cp/h. As previously described,
the factor 2 in the above expression is included to account for two possible
polarizations at any given frequency.
Returning to C.18), first we note that the double sum on / and n runs over
all atomic states, and we have written
A/n = Nnonl{co)[\ + f(a>)] - Ntcrln(a>)f(a>) C.19)
The first two terms in this expression represent spontaneous and induced decay,
respectively. The third therm represents resonant absorption, and o/n is writ-
written for the absorption cross section at frequency v corresponding to atomic
transition from the /th to the nth state.
Writing gt for the degeneracy of the /th atomic state and recalling the
symmetry relation24
= gn&nl C-20)
permits C.18) to be written in the KBG form:
1 Df(eo)
c Dt
The parameters k and % are written for
C-21)
C.22)
-Y^Nn—a,n C.23)
The distribution f0 appears explicitly as
~r = Hl<n Nn(gl/gn)Vln
Eun^-iglKn/g
Let us examine this distribution for the case of monochromatic radiation, which
is resonant with two atomic states. That is
hco = Eb — Ea C.25)
1 A
H. Bethe and E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms,
Springer, New York A957).

368 5. Elements of Quantum Kinetic Theory
Cross sections in C.24) are peaked about his value of co, and the expression
C.24) reduces to the simpler form
-r Nb(ga/gb)aab
Jo =
a[l- (gaNb/gbNa)]aab
C.26)
(Nagb/Nbga) -
With atomic states taken to be in a Boltzmann distribution,
Na = gae-fiE* C.27)
the relation C.26) is seen to return the Planck equilibrium distribution C.11).
Thus C.23) may be interpreted as a generalized Planck distribution relevant to
the dynamics of an electromagnetic field interacting with an atomic species.
Lasing criterion
Consider a laser beam of radiation interacting with a gas of atoms. Let the
radiation be resonant with two atomic states as described by C.25). In this
case, the absorption coefficient k given by C.22) becomes
C.28)
and /o is given by C.26).
In steady state, our kinetic equation C.18) reduces to
, co) + Kf(s, co) = k~Moj) C.29)
Here we have written s for displacement in the direction of s. The homogeneous
solution to C.29) gives the spatial dependence
f(s, oo) = /@, s)e~KS C.30)
It follows that the number of photons with the resonant frequency co propagat-
propagating in the direction s will grow, providing k < 0. With C.28), this gives the
canoncial lasing criterion25
Na
> — C.31)
gb ga
So, in fact, radiative gain demands something more than simply an excess of
atoms in the higher lasing state than in the lower state.
25 A. Yariv, Quantum Electronics, 2nd ed., Wiley, New York A975).

5.3 Application of the KBG Equation to Quantum Systems 369
.t
a
a
(a)
(b)
(c)
FIGURE 5.5. (a) Free-particle spectrum, (b) Energies corresponding to a periodic
potential. Band energies occur at k = nix I a, where a is the lattice constant and n is
an integer, (c) Related band-gap spectrum of energies.
5.3.3 Electron Transport in Metals
In Section 5.1, the following energy spectrum was obtained for free particles:
n2k2
E(k) =
2m
When plotted as a function of/:, this expression describes a parabola. If particles
are not free, but propagate through a periodic potential, not all energies are
allowed. The resulting E(k) curve exhibits gaps that affect a band structure as
illustrated in Fig 5.5.
The lattice of a metallic crystal presents a periodic potential to conduction
electrons. Consequently, the energy spectrum of these electrons exhibits a bad
structure containing forbidden and allowed bands. In a metal the band of highest
energy is called the conduction band and is only partially filled. An applied
electric field may therefore increase the energy of topmost electrons without
violating the exclusion principle. The band structure of the metal sodium is
shown in Fig. 5.6.
In semiconductors, the conduction band is empty at low temperature. How-
However, due to the relatively small value of the energy gap separating valence
and conduction bands, at higher temperatures, statse in the conduction band
are populated, thereby increasing the conductivity of the sample. Vancancies
left in the valence band due to promotion of electrons to the conduction band
affect the presence of positive charge and are called holes. Such holes may
act as charge carries and contribute to the net current, and we write for the
conductivity
a = \e\{n[xn + p[xp)
where n and p and hole concentrations (cm), respectively, and
are related mobilities.
and /jLp

370 5. Elements of Quantum Kinetic Theory
-10
0)
c
LU
20
-30
Observed internuclear
distance
3.67 5
Internuclear distance, a
FIGURE 5.6. The band structure of sodium as a fuction of internuclear spacing.
Note the overlapping of bands with (extreme) increase in density. Quantum states of
atomic levels are also shown [J. C. Slater, Rev. Mod. Phys. 6, 209 A934)].
The quasi-classical description
For many cases of practical interest, a quasi-classical description of conduction
in solid-state materials proves adequate. Since the presence of band gaps is a
purely quantum mechanical effect, it is important to the self-consistency of a
classical description that electron motion be confinded to a single band.
Let K(k) denote the energy spectrum of a given band. Two parameters play
an important role in the semiclassical model. These are the velocity,
1
v(k) = -
n
and the effective mass tensor
1
m
flV
For motion in one dimension, the effective mass tensor becomes a scalar, which
is labeled m*. For free-particle motion, m* reduces to the classical mass, m.
As with our preceding analysis for photon kinetic theory, the following
quantity is written for the number of electrons in the phase volume dp dx (in

5.3 Application of the KBG Equation to Quantum Systems 371
the conduction band) about the phase point (p, x) (in the quasi-classical limit).
dpdx 2fh3 dkdx dkdx
2/(p, x)^— = -L— = /(k, x)—— C.32)
h5 h5 47T-3
In C.32), the factor 2 stems from the two possible spin projections relevant
to electrons. Since dk dx/4n3 represents a density of states, we may identify
/(k, x) as the mean occupation number of electrons per state.
The Fermi-Dirac equilibrium distribution for an aggregate of electrons at
temperature T is obtained from C.1) and is given by
C.33a)
CT | 1
With the relation
2 x 4np2 dp 4nBmK/2El/2 dE
C.33b)
the preceding two equations give
N _ [ r, X2dp [°° 4nBmK/2El/2dE
This relation serves to relate the chemical potential /x to the number density
of N free electrons confined to the volume V. At 0 K, /x = EF, the Fermi
energy, and the preceding relation gives
4nBmK'2
n =
f
-1 (If "
(If " f^"v"
where we have written n for particle number density. The preceding expression
may be written in terms of the Fermi momentum pF.
Jo
PF 2dp 2 4
■ = —7 X -
C.34a)
Values of EF for some typical metals are listed in Table 5.1.
Working with the distribution function /(k, x, t), the KBG equation assumes
the form26
| f| I C.35)
26In the most solid-state works, this analysis is referred to as the relaxation-time
approximation.

372 5. Elements of Quantum Kinetic Theory
TABLE 5.1. Properties of Characteristic Conducting Metals: Resistivity, Relaxation
Times and Fermi Values
Element Z «A022/cm3)
Li
Na
K
Rb
Cs
Cu
Ag
Au
Be
Mg
Ca
Sr
Ba
Nb
Fe
Zn
Cd
Hg
Al
Ga
In
Ti
Sn
Pb
Bi
Sb
1
1
1
1
1
1
1
1
2
2
2
2
2
1
2
2
2
2
3
3
3
3
4
4
5
5
4.70G8 K)
2.65E K)
1.40E K)
1.15E K)
0.91E K)
8.47
5.86
5.90
24.7
8.61
4.61
3.55
3.15
5.56
17.0
13.2
9.27
8.65
18.1
15.4
11.5
10.5
14.8
13.2
14.1
16.5
<j(lxQ — cm)
273 K
8.55
4.2
6.1
11.0
18.8
1.56
1.51
2.04
2.8
3.9
3.43
23
60
15.2
8.9
5.5
6.8
Melted
2.45
13.6
8.0
15
10.6
19.0
107
39
rA0~14s)
0.88
3.2
4.1
2.8
2.1
2.7
4.0
3.0
0.51
1.1
2.2
0.44
0.19
0.42
0.24
0.49
0.56
0.80
0.17
0.38
0.22
0.23
0.14
0.023
0.055
£/r(eV)
4.74
3.24
2.12
1.85
1.59
7.00
5.49
5.53
14.3
7.08
4.69
3.92
3.64
5.32
11.1
9.47
7.47
7.13
11.7
10.4
8.63
8.15
10.2
9.47
9.90
10.9
kF
(xl08cm-]
1.12
0.92
0.75
0.70
0.65
1.36
1.20
1.21
1.94
1.36
1.11
1.02
0.98
1.18
1.71
1.58
1.40
1.37
1.75
1.66
1.51
1.46
1.64
1.58
1.61
1.70
vF
) (xl08cm/
1.29
1.07
0.86
0.81
0.75
1.57
1.39
1.40
2.25
1.58
1.28
1.18
1.13
1.37
1.98
1.83
1.62
1.58
2.03
1.92
1.74
1.69
1.90
1.83
1.87
1.96
where
/zk = e£
C.35a)
and £ represents electric field strength. The relaxation time r is in general
a function of k [see, for example, C.81)]. Assuming steady state and spatial
homogeneity, C.34) reduces to
C-
h
1 , ,,
= -(/o - /)
r
For short relaxation times, we may set r —► ex, where e is a small bookkeeping
parameter. Repeating the Chapman-Enskog expansion C.5.8),
C.37)

5.3 Application of the KBG Equation to Quantum Systems 373
TABLE 5.2. Thermal Conductivities and Lorenz Numbers
273 K 373 K
Element
Li
Na
K
Rb
Cu
Ag
Au
Be
Mg
Nb
Fe
Zn
Cd
Al
In
Tl
Sn
Pb
Bi
Sb
K
(W-cm/K)
0.71
1.38
1.0
0.6
3.85
4.18
3.1
2.3
1.5
0.52
0.80
1.13
1.0
2.38
0.88
0.5
0.64
0.38
0.09
0.18
k/gT
(W-ft/K2)
2.22 x 10~*
2.12
2.23
2.42
2.20
2.31
2.32
2.36
2.14
2.90
2.61
2.28
2.49
2.14
2.58
2.75
2.48
2.64
3.53
2.57
K
(W-cm/K)
0.73
3.82
4.17
3.1
1.7
1.5
0.54
0.73
1.1
1.0
2.30
0.80
0.45
0.60
0.35
0.08
0.17
k/gT
(W-ft/K2)
2.43 x 10~*
2.29
2.38
2.36
2.42
2.25
2.78
2.88
2.30
2.19
2.60
2.75
2.54
2.53
3.35
2.69
Source: N. V. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart,
and Winston, New York A976); and G. W. C. Kaye and T. H. Laby, Table of
Physical and Chemical Constants, Longmans Green, London A966).
and inserting into C.36) and equating powers of £ gives
/@) =
= -/(
Thus, with C.37), we find, to 0(e),
C-38)
Here we have recognized that the middle expression represents the first two
terms in a Taylor series expansion of the function on the right about xeS/h = 0.
This finding indicates that the Fermi sphere has been displaced by the amount
Te£/H in k-space (see Fig. 5.7). It is evident from this figure that only electrons
at the top of the Fermi sea contribute to conduction. Electrons sufficiently below
the Fermi surface cannot be accelerated to nearby higher-lying levels. Such

374 5. Elements of Quantum Kinetic Theory
FIGURE 5.7. The Fermi sphere in k space suffers a displacement in the presence of
an electric field. Electrons in the darkened area contribute to conduction.
promotion would violate the exclusion principle as these levels are already
occupied.
We may use C.38) to calculate conductivity. Current density is given by
v(k)/(k) dk
4n3
Inserting C.38) into this expression gives
J = -
An*
/
r(k)v(k)£ • -£■ dk
dk
C'39)
Here we have noted that /0 has a vanishing first velocity moment. Let us picture
the Fermi surface, E = Ef, in k space. If dS is an element of area on this
surface, then a differential of volume dk generated by a normal, displacement
8k is |dk| dS.21 The related change in energy is given by dE = \Sk\\ Vk£|.
Thus we may write for the volume element
dSdE C.39a)
dk =
|Vk£|
At 0 K, the Fermi-Dirac distribution C.33) gives
dE
= -S(E - EF)
C.39b)
Thus
dk dE dk
= -\h8{E - EF)
C.39c)
27A similar argument was given previously in Section 3.8.1.

5.3 Application of the KBG Equation to Quantum Systems 375
and C.39) becomes
J = -% / r(k)v(kM • v(k)<5(£ - EF)—— C.39d)
4tt3 J |Vk£|
Integrating over energy gives
r(k)v(k)£ • v(k)
dS C.40)
I v*£|
Writing Ohm's law in tensoral form,
J ix = o[lv£v
and comparing with C.40) implies the following for the conductivity tensor.2*
r(kK(kK(k)
f
JE
C.41)
Here, we recall, dS is an element of area in k space. Only electrons near the
Fermi surface contribute to conductivity. (See Problem 5.52.)
Weidemann-Franz law
If a temperature gradient is present in the solid-state plasma, then the space
gradient term in C.35) is written
df _ df dT
Y"dx~Y' ~df~dx
The analog of C.36) becomes
df dT _ 1
~v * ~r~ — ~
dT ox x
Writing the heat conductivity equation C.4.3) in tensorial form,
dxv
and following the procedure that led to C.41), we find
f r(k)u,(kK(k)^
/ p=r^ dS C.42)
- \2n Je=Ef \VkE\
Providing relaxation times in C.41) and C.42) are equal, we may conclude
k n2 [kB\2 Q o
— = — ( — ) = Lo = 2.445 x 108W-^/K2 C.43)
oT 3 V e I
Electrical conductivity in Chapter 3 was labeled ac. A generalized tensor form
fora is given in Section 5.6. See also Section 7.1.2, 7.1.6, 7.2.3.

376 5. Elements of Quantum Kinetic Theory
where Lo is the Lorenz number (see Tables 5.1, 5.3). The relation C.43) is the
Weidemann-Franz law. An expression for the relaxation time r(k) is obtained
in Section 3.6.
5.3A Thomas-Fermi Screening
In this section we consider the repsonse to an extraneous electron placed in
a charge-neutral degenerate Fermi gas. The analysis parallels the calculation
(Section 4.1.1) relevant to a classical plasma. Converting from velocity to
momentum, together with the transformation relevant to the quantum domain,
permits D.1.23) to be written (in the static limit) as
/•k-a/o
J k
k2 J k p h3
Here we have written /o for the Fermi-Dirac distribution C.33). With
dp mdE
and
dp = 2n{2mf'2Ex/2dE
C.44) becomes
\6n2e2
For sufficiently low temperature, again we write
3/o
dE
and C.45) may be written
= -S(E - EF)
C.46)
where EF is given by C.34) and
6nne2
are the Thomas-Fermi screening wave number and screening distance, respec-
respectively. It follows that the static dielectric constant for a degenerate plasma, in
the quasi-classical limit, is given by
s = 1 + % C.47)

5.3 Application of the KBG Equation to Quantum Systems 377
Repeating the analysis leading to D.1.35) gives the static potential
eexp(—kTFr)
0(r) = -—— C.48)
r
Whereas this form greatly resembles the classical result D.1.35), a more
detailed quantum mechanical description29 indicates that, at relatively large
distance, the interparticle potential oscillates as
cos2kFr C.49)
r3
Here kF is written for the Fermi wave number, EF =h kF/2m.
Nonstatic limit
In the nonstatic, long-wavelength (k2 <£i co2p/vF) limit, we obtain30 the
following dispersion relation for a degenerate plasma:
C.50)
This relation bears a striking similarity to the parallel relation for long-
wavelength waves in a classical plasma D.1.46):
2/,2
C.51)
However, as T —► 0, C -» 0 and C.51) gives nonpropagating standing waves,
whereas in the quantum domain vF persists at T = OK. Progagation of waves
at 0 K in a degenerate plasma have been compared to the phenomenon of
"zero sound" relevant to a fluid of uncharged fermions at 0 K (discussed in
Section 4.2). Furthermore, longitudinal waves in a degenerate plasma C.50)
also suffer Landau damping.31
The similarity in relations of parallel quantities in classical and degenerate32
plasmas is illustrated by the following:
29J. Linhard, Kgl. Danske Videnskab. Selskab. Mat.-Fyd Medd. 28, No. 8 A954).
The behavior C.49) is often referred to as Friedel oscillations.
30L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Physical Kinetics, Pergamon
Press, Elmsford, N. Y. A981).
31 P. M. Platzman and P. A. Wolf, Waves and Interactions in Solid State Plasmas,
Academic Press, New York A973).
32Degenerate here refers to the quantum domain.

378 5. Elements of Quantum Kinetic Theory
Classical
1
knT
col Anne2'
CO
co,
= 1 +
3Czk
2 7,2
CO'
CO'
C2
C.52)
Degenerate
,2 - ATF ~ ^ _2 -
KTF
3 co
co
CO
3v2k
2k2
co\
2
C.53)
5.3.5 Mott Transition33
An instructive application of the Thomas-Fermi potential C.48) comes into
play in the derivation of the criteria for what is commonly called the Mott
transition. Consider a conducting lattice with one free electron per atom, such as
is the case with the alkali metals. The band structure typical to such conductors
is shown in Fig 5.6. Let the crystal undergo an ideal expansion at 0 K. The
question we wish to consider is, at which point in the expansion does the sample
become an insulator? We will assume that the change of state, termed a Mott
transition occurs when a conduction electron of the metal forms a bound state
with an ion in the lattice.
At reasonably good expression for the potential of interaction between a
conduction electron and an ion in a metal for the present model is given by the
Thomas-Fermi potential C.48) (now written as potential energy)
V(r) = -
£2exp(—kTFr)
C.54)
It has been shown34 that an electron and ion interacting under this potential
will permit a bound state providing
kTF <
1
a0
C.55)
where a0 is the Bohr radius. With C.46a), we may rewrite
6nne2
EF
a0
33Named for N. F. Mott.
34J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics, p. 55, Wiley, New
York A953).

5.3 Application of the KBG Equation to Quantum Systems 379
n < y—J n0 C.56)
The criterion C.55) may then be rewritten
2
12
where
3 3
n0 = -na0
is the number density corresponding to one atom per "Bohr volume," 47t<2q/3,
which is relevant to a hydrogen-like lattice. Thus n represents both electron
and ion densities. The relation C.56) indicates that in the present crude model
the Mott transition occurs for ion number density < 0.1n0.35
5.3.6 Relaxation Time for Charge-Carrier Phonon Interaction
For temperatures in excess of — 100A', one of the more important scat-
scattering mechanisms of charge carriers in a semiconductor is that of phonon
scattering.36 Recall the phonons are quantized lattice vibrations. The disper-
dispersion relation for phonons contains two branches corresponding to acoustic and
optical phonons, respectively.37 For relatively small phonon wave number q,
acoustic phonons satisfy the dispersion relation
co^uq C.57a)
where u is constant wave speed, whereas optical phonons in this same domain
satisfy the relation
to ^ coq C.57b)
where co0 is a constant frequency.
For elemental semiconductors (for example, silicon and germanium), charge
carriers interact with phonons through what is commonly termed the defor-
deformation potential interaction or, more simply, the strain interaction. Thus one
speaks of strain-acoustic and strain-optical interactions.
35Whereas experiments are presently underway, the monatomic metallic phase of
hydrogen has not been observed in the laboratory. It has been estimated that solid
H2 undergoes a transition to the metallic phase at approximately 3 Mbar. For further
discussion, see B. I. Min, J. F. Jansen, and A. J. Freeman, Phys. Rev. B. 33, 6383
A986); and D. Ceperely and B. Alder, Phys. Rev. B. 36, 2092 A986).
36 At lower temperature, scattering is dominated by interaction with impurity atoms
in the lattice. For futher discussion, see K. Seeger, Semiconductor Physics 3rd ed.,
Springer-Verlag, New York A985); and R. L. Liboff, J. Phys. Chem. Solids 46,1327
A985).
For further discussion, see J. M. Ziman, Principles of the Theory of Solids,
Cambridge, New York A964).

380 5. Elements of Quantum Kinetic Theory
hk'
Dk'
tik
, k1)
(b) w. (kf k')
FIGURE 5.8. Diagrams corresponding to (a) absorption of a phonon, and (b) emission
of a phonon.
In this section we derive an expression for the relaxation time r(k), in-
introduced in C.35), appropriate to the acoustic strain interaction38 for a
semiconductor in the presence of an applied electric field £.
Neglecting spin, the collision integral for this scattering proces may be
written [compare with B.86)]
/(/) = -JL V f[w±(k\ k)/(k') - w±(K k')/(k)] dk! C.58)
Bny ^ J
where V is the volume of the sample and with C.32) we have written hk
for the momentum of charge carriers. The probability rate or charge carrier
scattering from the k to k' state accompained by the absorption (+) or emission
(-) of a phonon is written w±(k, k'). We may associate diagrams with these
interactions, as illustrated in Fig. 5.8.
The rate coefficient w±(k, k') may be written in terms of matrix elements
of the perturbation Hamiltonian //', as follows [recall B.85)]:
C.59)
Here H^ k is the matrix element corresponding to the transition k -> k'. The
delta function factor in C.59) ensures conservation of energy where, with m
written for effective mass of charge carriers,
AE± =
2m
■2 ;,2
hlk
2m
C.60)
Our plan is to convert the right side of C.58) to that of C.35). To incorporate
the anisotropy of the distribution function to the applied electric field, we recall
38It has been demonstrated that for cubic semiconductors ihe acoustic strain inter-
interaction dominates in the quasi-classical domain. See G. K. Schenter and R. L. Liboff,
J. Appl Phys. 62, 1977 A987).

5.3 Application of the KBG Equation to Quantum Systems 381
C.7.21) and write
C.61)
where, again, /x is the cosine of the angle between £ and k. If we identify fo(k)
as the equilibrium distribution, then by statistical balance (Section 3.3.9) we
may set
tu±(k, k')/o(*) = tu±(k\ k)fo(k') C.62)
Substituting C.61) into C.58) and incorporating C.62), we obtain
p
V
Bn)-
m
P
/ [w±(k\ k)f\{k')ix' — wpm(k, k!)f\(k)ix\dkf
f
Mk)
L/o(*o
wpm(k,k')dk'C.63)
In the high-energy limit, E ^>Hco(q), we may further assume
') ~ fo(k)
') ~ /,(*) C.64)
Substituting these relations into C.63)gives
J(f) = -pr^Y.nMk) f (l - ~) w±(k,k')dk' C.65)
Recalling C.61), we note
Thus C.65) gives the desired expression:
- = t^-j V /" (l - -) ivpm(k, k') rfk' C.66)
Explicit formula for 1/r
To further reduce the integral in C.66), we cite the specific form of the transition
probability rate C.59) for strain-acoustic interactions.39
^q (n, + 1)<5(A£_)] C.67)
E. M. Conwell in High Field Transport in Semiconductors. Solid State Physics,
Vol. 9, SuppL, F. Seitz, D. Tumbull, and H. Ehrenreich (eds.), Academic Press, New
York A967).

382 5. Elements of Quantum Kinetic Theory
where nq is the Bose-Einstein distribution C.6L0 relevant to phonons, and
AE± are given by C.60). The coefficient A is given by
C.67a)
where p is crystal mass density and E\/V is the shift in band edge41 per unit
dilation of the lattice. Note that E\ has dimensions of energy and A that of
energy squared times length.
For sufficiently large crystal temperature, we may write
kBT q
This permits the Bose-Einstein distribution to be expaned42 as
kBT
huq 2
which in turn allows C.67) to be collapsed, and we obtain
nq = ^— -- + ...» 1 C.68)
C.69)
We turn next to the delta functions 8(AE±). Substituting the dispersion
relation C.57a) into AE± given by C.60) permits us to write
AE± = —[(k ± qJ - k2]
2m
H2
2
2m
h kq r q mui
2k Hk 1
m
where we have written
/I = k • q
Taking the delta function of AEpm gives
2m 1
2Jfc/J
Here we have recalled the delta-function relation
m
S(ax)= —8(x) C.71)
\a'
40That is, nq (here) is written for f0[co(q)] in C.6).
41 For a one-dimensional lattice, band edgers occur at k = rut/a, where a is the
lattice constant (see Fig. 5.5).
42Neglecting \ in C.68) gives the phonon equipartition theorem: kBT = nfico =
energy/frequency modes.

5.3 Application of the KBG Equation to Quantum Systems 383
FIGURE 5.9. The angles 0, 0', a, and p for reduction of C.66).
Finally, in our evaluation of the integral in C.66), we seek a relation between
and /a'. To these ends we refer to Fig. 5.9. Vectors are defined with respect
to k taken as the polar axis. With this figure at hand, the cosine addition law
gives
' = \x cos a + sin 0 sin a cos ft
C.72)
Furthermore,
'2
qz = |k-kT = Jr + Jfc - 2JfcJfc'cos or
C.73)
At high energy, k ~ k\ and the preceding relation gives
cos a = 1 —
2k2
C.74)
Substituting this relation into C.72) and neglecting the cos/3 term owing to
the d/3 integration in C.66), we obtain
C.75)
2k1
With the relations C.68) through C.75) at hand, we return to C.66) and
write (with dk' = dq)
1 _ .B
\
fijfc
C.76)

384 5. Elements of Quantum Kinetic Theory
where
2Bn)Wu
With k taken as the polar axis, we write
dp = 2nq dq d/i
and C.76) gives
C.76a)
2nB
- (
(sr
C.77)
Integration over /i leads to bounds on the dq integration, and C.77) reduces
to
/*2(±)
dqq3 C.78)
i(±)
where q(±) are the two solutions at the bounds of the inequalities
mu q
-1 < — =p— < +1
" hk 2k ~ C.78a)
q > 0
There results
2m u
qx{±) = 0, q2(±) = 2k±—- C.786)
n
(see Problem 5.9). Substituting these values into C.78) and integrating yields
■(Hk ± muL
k3H4
In the limit u <C hk/m, the preceding reduces to
- = \6nBk
x
Substituting the value for A C.67a) into B C.76a) gives
This result may be cast in a more revealing form if we identify
/ = fPU\ C.80)
m2kBTE\
as the acoustic phonon mean free path. Thus we may write
C.81)

5.4 Quantum Modifications of the Boltzmann Equation 385
This finding is often employed in the literature.43
Here is a brief recapitulation of assumptions leading to C.81): A) E ^>
ko)(q), k' ~ k, B) kBT ^> hco{q), C)Hk/m ^> u. These inequalities are
satisfied in most practical cases.
5.4 Quantum Modifications of the Boltzmann Equation
5A. 1 Quasi- Classical Boltzmann Equation
Normalizations
In the preceding analysis of photon (boson) and electron (fermion) kinetic
theory a general rule was found describing the transformation of classical to
quantum distribution functions. In the quasi-classical limit, it reads as follows:
dxdp
/CL(x, v)dxrfv-> /QM(x, p)—-=— D.1)
Here we recall that dxdp/ h3 represents the number of available states in the
phase volume dx dp. Note that /qm(x, p) is dimensionless. In the pure quantum
domain, x and p may not be prescribed simultaneously, and the preceding rule
becomes
^ D.2)
(This transformation was used previously in Section 3.4.) In our preceding
examples, the density of states dp/ h3 was increased by a degeneracy factor,
which we now label G. For photons, G = 2, corresponding to two degrees
of polarization, whereas for electrons, G = 2, corresponding to two spin
projections. Thus, for photons, we found C.9)
dp 2 dp aJ dco
h3 h3 n2c3
whereas, for electrons C.33b),
dp _ 2dp 4nBmK/2El/2dE
H3 ~
These notions play an important role in Uehling and Uhlenbeck's quantum
generalization of the Boltzmann equation.44
43See, for example, L. Reggiani (ed.), Hot-Electron Transport in Semiconductors,
Springer-Verlag, New York A985).
"E. A. Uehling and G. E. Uhlenbeck, Phys. Rev. 43, 552 A933).

386 5. Elements of Quantum Kinetic Theory
Collision integral
We concentrate on the collision integral in the Boltzmann equation C.2.14)
(reverting to dimensional form):
J(f) =
/
Here we recall that f f[ relates to a collision in which particles are restored
to velocity volumes d\ about v and d\\ about Vi, respectively. Suppose these
particles are fermions. Then we must ensure that the exclusion principle is
not violated when particles enter respective phase volumes. The number of
available states in the volume dx dp is
dxdi) m
= G
xdi) m
-=- = G—dxd\
h5 h5
D.3)
(Here we are writing dx for the collision-domain volume.) Thus, if / =
Gm3/ h3, the phase volume is filled and there can be no further reentry of
fermions. This property can be incorported in the Boltzmann equation through
the change in collision phase volume relevant to the term f[f:
dx\ d\]
1 -
Gm3
1 -
h3f(Vi)
Gm3
dx\ d\\
D.4)
A similar argument applies to the direct collision product ff\. Inserting these
changes into the derivation C.2.1) et seq. yields the quasi-classical collision
form
Here we have labeled
where
8 = <
h
Urn*
D.6a)
+1 for bosons
0 for Boltzmann statistics
— 1 for fermions
D.6b)
The enhancement value 8 = +1 for bosons stems from a statistical affinity
that identical bosons have to be in a common domain.45
45G. E. Uhlenbeck and L. Groper, Phys. Rev. 41, 79 A932). See also K. Huang,
Statistical Mechanics, ibid.

5.4 Quantum Modifications of the Boltzmann Equation 387
Equilibrium distributions
The condition for statistical balance that follows from D.5) is given by
r \( n \ D7a)
Taking the log of both sides reveals the collisional invariant:
:46
ln(—L-\=l3(v-E)-In \$\ D.7b)
Here f$ = (A^T), E is written for single-particle energy, and /x is a constant
that may be identified with the chemical potential. Inverting the preceding
relation gives
With our definition of £ and taking \S\ = 1 gives the correct quantum
distributions [Bose-Einstein (—), Fermi-Dirac (+)]:
Gdp/h3
fodv= *' D.8)
eP(L-ti) zp 1
In the following section, an equation closely allied to the Uehling-Uhlenbeck
equation is discussed relevant to a Fermi liquid.
5.4.2 Kinetic Theory for Excitations in a Fermi Liquid
Elementary excitations
In quantum mechanics, a weakly excited state of a macroscopic body may be
considered as an aggregate of individual elementary excitations. Such excita-
excitations are a consequence of a quantum mechanical description of the collective
motion of the system and should not be identified with individual atoms.
Excitations behave like quasi-particles with definite momenta p and energy
e(p). We may introduce a nondimensional distribution function for quasi-
particles such that /2 dp/ h3 represents the number of quasi-particles per unit
volume with momentum in the volume dp. In the quasi-classical limit, / may
be written as a function of both x and p. In the present case, the quasi-classical
domain is characterized as follows. Let L be the length over which / varies
appreciably. Writing k ~ 1/L for the equivalent wave number, the condition
may assume that the scattering potential is x dependent. It follows that p
does not commute with the Hamiltonian and cannot be specified together with E in
D.7b).

388 5. Elements of Quantum Kinetic Theory
becomes
hk <£ pF
where pF is the Fermi momentum C.34).
When the liquid suffers a small displacement from equilibrium, we write
/(x, p) = /o(p) + <5/(x, ftp) D.9)
where /o(p) is the equilbrium distribution. In like manner, we write
e(x, p) = eo(p) + &e(x, p) D.10)
where £o(p) corresponds to the equilibrium state. The change in excitation
energy is given by
&e(x, p) = j tf(p, p')V(x, p')^p D.11)
where K(p, p') is the quasi-particle interaction function.
Let the liquid occupy the volume V. For the total energy of the liquid we
write U V so that U is energy density. The variation of U is then given by
f
J
f 2dP
8U = J eSf-^f D.12)
The quantity s(x, p) plays the role of the Hamiltonian of a quasi-particle in
the field of other particles, and its as well as D.12) are employed below to
construct an equation of motion for the distribution /.
Kinetic equation
A kinetic equation for quasi-particles in a Fermi liquid was presented by
Landau in 1957.47 It is given by48
df de df de df
-i- -I . — . — = j(f)
dt dp dx dx dp
47L. D. Landau, J. Exp. Theor. Phys. (USSR) 32, 59 A957). Repringed in, D.
Pines, The Many-Body Problem, W. A. Benjamin, Menlo Park, Calif. A961).
48Note the resemblance between D.13) and the Uehling-Uhlenbeck form D.5).
See also, F. Bloch, Z Physik 58, 555 A929); 59, 208 A930).

5.4 Quantum Modifications of the Boltzmann Equation 389
J(f) = J u>(p,pi;p\p',)[/7i'A -/)d -/i] D.13)
- //id - /')A - fi)]S(s + £,-£'- e\)
<5(p + Pi - p' - p',) dp
|)elta functions indicate that momentum and energy are conserved over a col-
|ison. The function w gives the collision probability and obeys the symmetry
delation
That is, the probability of a collision and its reverse are equal.
As with the Uehling-Uhlenbeck collision term D.5), the Landau form D.13)
likewise ensures that collisions satisfy the exclusion principle relevant to
fermions. Note also that / in D.13) is dimensionless.
The condition of statistical balance obtained from D.13) again yields the
Fermi distribution C.33a).
Conservation equations
Equations of motion for macroscopic variables in a Fermi liquid are obtained
by taking moments of D.14). Integrating over momentum gives the continuity
equation
V() 0 D.14)
dt
where n represents quasi-particle number density,
2 dp
f
and
n(\) = / f\—- D.16)
I wi
is particle flux, where we have set
v=— D.17)
3p
for quasi-particle velocity.
Multiplying D.13) by pi9 and integrating gives
f (BL
J \dXj
The integrand in this expression may be rewritten
g-"(Pi)+l Pi\T!-—-7r-—\-rr=0 D.18)
a / ».\+f*L_±(pi^n D.19)

390 5. Elements of Quantum Kinetic Theory
Integration removes the last term. To transform the middle term, first, with
D.12), we write
dU
f
/
J
It follows that
r ds 2dp d
where we have set
n(e) = J ef-^f D.22)
The relation D.21) permits the middle term in D.11) to be recast and D.18)
becomes
d d y-r
—n{pi)-{ =0 D.23)
(it si \~ •
J u
Here PJ.. represents the momentum flux (or stress) tensor
P| = n(piVj) + Sij[n(e) - U] D.24)
Finally, we multiply D.13) by e and integrate. In like manner to D.20), we
write
dU
There results
dU
+ V q = 0 D.26)
dt
where
q = / efy^r D-27)
represents the energy flux vector.
In this manner, we obtain the classical-like49 conservation equations relevant
to a Fermi liquid:
dt
dt
49The similarity of macroscopic equations in classical and quantum physics is
discussed in J. H. Irving and R. W. Zwanzig, J. Chem. Phvs. 19. 1173 A951).

5.4 Quantum Modifications of the Boltzmann Equation 391
3 U + V .q =
dt
In the quantum equilibrium state, all three flux vectors in D.28) vanish.
Zero sound
wish to apply the kinetic equation D.13) to small perturbations away from
equilibrium of a Fermi liquid at 0 K. If the frequency of the perturbation far
exceeds the collision frequency of quasi-particles, then we may neglect the
collision integral in D.13). With this condition at hand, we obtain
ds0 dSf d8s8 a/o
+ —- • — — = 0 D.29)
dt dp dx dx dp
The quantitities 8f and 8s represent variations away from equilibrium, as in
D.9) and D.10).
At T = 0 K, we have
= -nS(p - pF) = -\8(s - EF) D.30)
dp
Where n is a unit vector in the direction of p. We seek a wave solution in the
form
8f = 8(8 - EF)a(n)ei(k-y-Mt) D.31)
Where n is a unit vector in the direction of p, and a(n), which is to be de-
determined, represents displacement of the Fermi surface in the direction of n.
Combining the last three equations and recalling D.11) for 8s, we find
(co - vFn • k)a(n) = v k f K(p, p')a(ri)8(e -
Where we have set
ds0
\F =
aP
Again transforming to surface area in p-space [see equation following C.39)],
dSDde dSDde p2 dQ.de
dp = —-— = —-— =
|Vp£| V V
permits the preceding equation to be written
(co - vFn • k)or(n) = n • k-^ / K@)a(ri)dQ'
h3 J
Here we have written 0 for the angle between p and pr. With k taken as the
polar axis and setting
CO
s = —

392 5. Elements of Quantum Kinetic Theory
gives
(s - cosO)a@, 0) = cos(9 / «X0)a@', 0') D.32)
J 4
J 47T
where we have set
K = ^f4nK D.32a)
h5
Let us examine the case that K is constant and equal to the value Kq. We may
then assume that the interaction integral in D.32) is likewise independent of
angle. This gives the solution
cos 6
a@) = A D.33)
s — cos 0
where A is an arbitrary constant. The Fermi surface corresponding to this
solution is elongated in the direction of wave propagation.
Substitution of the trial solution D.33) into D.32) gives the condition
n cos(9 In smO dO
f
Jo
s — cos 0 An
Integration gives
= 1
1 5 + 1 1
-s In 1 = -=—
2 s-\ Ko
As s varies from 1 to infinity (corresponding to undamped waves), the left side
of the last equation varies from infinity to zero. It follows that waves exist only
for Ko > 0. In the limit of vanishing interactions, ATo —> 0 and s tends to unity
corresponding to the zero sound speed co/k = vF. For a moderately nonideal
Fermi fluid, the zero sound speed is V
5.43 H Theorem for Quasi-Classical Distribution
In this section we return to the quasi-classical Boltzmann equation D.5) and
present a derivation of the related H theorem. The appropriately generalized
expression for entropy, S = —kBH, is given by51'52
;■)-//In//] D.34)
In this expression, the sum is over all quantum states, gt denotes of the ith state,
and the + sign is relevant to bosons and the — sign is relevant to fermions.
50Yu. L. Klimontovich and V. P. Silin, Zh. Eksper. Teor. Fiz. 23, 151 A952).
51K. Huang, Statistical Mechanics, p. 196, Wiley, New York A967).
52A. Akhiezer and S. Peletminskii, Methods of Statistical Physics, p. 134,
Pergamon, Elmsford, N.Y A981).

5.4 Quantum Modifications of the Boltzmann Equation 393
In the quasi-classical limit, the sum over discrete quantum states reduces to
an integral accordng to the rule
8i^ % /dp {435)
%
Here we are assuming a spatially homogeneous system confined to the volume
V. Employing the preceding relation, D.34) becomes
^ f d^f ± 1) ln(l ± /) - / In /] D.36)
J
B
With this expression at hand, we recall the generalized Boltzmann equation
D.5) and the relation D.1):
DF
= J(f) D.37)
Dt
± /)d ± /i) - //id ± /')d ± /O] D.38)
The distribution functions in these equations are dimensionless.
Employing the relation
^[(a ± f)ln(a ± /)] = [1 + ln(a ± /)]^ D.39)
ay ay
where a is a constant, indicates that to obtain an equation of motion for S, as
given by D.36), we must multiply D.37) from the left by
J
^ J p{( + In /) + [1 + ln(l ±
There results
S/(iZ)*
Recalling the / operator C.3.2),
J
= J J(fL>(p) dp D.41)
[with J(f) given by D.38)] and repeating the symmetry operations C.3.3 et
seq.), again we find
ID>) = /(</>,) = -ID>') = -I(<p[) D.42)
which gives
- lD>') - 1D,')]
D-43)

394 5. Elements of Quantum Kinetic Theory
Substituting this result into D.40) gives
D S V
Dtk
B
2h3
In
f
ln
/i
V
2h3
f
dpJ(f)
D.44)
In
l±/
/i
i±/7Vi±/.7J
Substituting D.38) for /(/) and labeling
dQg
= d/j,
we obtain
D S
V
Dtk
B
2h3
-In
dn[f'f[{\ ±
± /,) -
± /')A ±
l±/\/l±/,
/
ft
i±/'Ai±/,7J
D.45)
Setting
±
±fi) =
permits D.45) to be written as
D S V
Dt
B
2h3
— >0
D.46)
which establishes the H theorem in the quasi-classical domain. Note also that
the implied equilibrium solution, X = Y, returns D.7), which gives the correct
quantum equilibrium distributions D.8).
5.5 Overview of Classical and Quantum Hierarchies53
5.5.1 Second Quantization and Fock Space
Consider the wave function relevant to TV identical particles. In accord with
the Pauli principle, the appropriately symmetrized wave function is given by
= — ^2 (±)P|xi,...,x,v) E-1)
P(\
53 An expanded presentation of these topics is given by G. K. Schenter, Thesis,
Cornell University, Ithaca, N.Y. A988). For further discussion, see L. E. Reichl, A
Modern Course in Statistical Physics, University of Texas Press, Austin A980).

5.5 Overview of Classical and Quantum Hierarchies 395
Hie ket vector |xi, ..., x#) on the right side of E.1) is of arbitrary symme-
symmetry. The sum is over AM permutations of A, ..., N). The + and — signs are
appropriate to bosons and fermions, respectively.
Let the wave function relevant to nx particles at x, n'x particles at x', and
go on, be written |..., nx, .. .)F. The subscript F denotes that this wave func-
function exists in Fock space. This is a space whose independent variables are
occupation numbers. The relation of this wave function to that given in E.1)
is
xN\ N) =
y N E.2)
The only occupation numbers in the right ket vector of E.2) which are nonzero
are those with x-values equal to those in the left ket vector. The wave function
E.2) may be generated by the field creation operator 0f(x). Thus
1 ^ ^ - , E.3)
AM
where \0)F denotes the vacuum state. Note in particular that
..., nx, .. .)F = ^f\±nx\...,A ± nx), .. .
E.4)
0(x)|..., nx, .. .)F = y/n^ ...,
The operator 0f(x) creates a particle at x, whereas 0(x) annihilates a parti-
particle at x. These operators are appropriate to the representation called second
quantization.54 For fermions, nx = Oor 1, whereas for bosons, nx = 0, 1, 2, ...
Commutation relations of field operators are given by
\
[fax)j>\y) T 0f(yH(x)] = [0(x), 0f(y)]T = S(x - y) E.5a)
where the upper sign corresponds to bosons and the lower sign to fermions.
The wave functions of E.4) carry the following normalizations:
xN\x\ '"XfN) = S(xi -x/1)---6(x/v -x^) E.5b)
ri xK • • • E.5c)
5.5,2 Classical and Quantum Distribution Functions
To this point in our discussion, we have encountered: A) the classical
distribution function, fN(xN,pN,t), B) the quantum mechanical density
54An overview of quantum kinetic theory in second quantization is given in T.-J
Lie and R. L. Liboff, Annals of Physics A971).

396 5. Elements of Quantum Kinetic Theory
matrix
pN(xN,yN,t) = (xN\p\yN) E.6)
and C) the quantum mechanical Wigner distribution, FN(xN, pN, t).
With the preceding description of second quantization operators 0f(x) and
0(x), we may introduce a fourth distribution-like form: D) the TV-particle F-
space operator GN(xN, yN, t), where F-space denotes Fock space. It is given
by55
GN(xN, yN, t) = ^(y,) • • • 4?u<3NL>h{*n) • • • <M*i) E.7)
where 0#(x) has been written for
4>H(x, t) = eWl)/*j>(x)e-Wl)/* E.7a)
which is the Heisenberg representation of 0(x) and H is the corresponding
TV-body Hamiltonian in Fock space.
The equation of motion for 0#(x) is given by the Heisenberg equation of
motion A.44):56
u(t>H A ^
^ E.8)
It should be noted that, whereas the distributions fN, pn, FN are defined with
respect to a given number of TV particles, GN is defined for unbounded N.
To obtain a relation between the F-space operator Gs(xs, ys, t) and the
density matrix ps(xs, ys, t), we first note that reduced density operators obey
the relation
ps=Tr^-s)pN E.9)
where Jt(N~s) denotes a trace over the (TV — s) dimensions (s + 1, ..., TV).
Thus, for example in the coordinate representation,
f
ps(xs,ys)= / dxs+{dys+i • • ■ dxN dyN8(xs+l -ys+\) • • • 8(xN-
The presence of the delta functions in this relation indicates that the integral
is a trace operation.
At this point we introduce an s-particle observable equivalent operator. First
we recall that the TV-body potential is written
N
55This operator is closely allied to the TV-body Green's function described in
Section 5.7.
56Note that in E.8), the partial time derivative is used to distinguish between x
and t differentiation. It is not the same as that occurring on the right side of A.44).

5.5 Overview of Classical and Quantum Hierarchies 397
where utj are two-body potentials. Motivated by this form we introduce the
^-particle operator
The time-dependent expectation of A(s) is given by
...s
) j dx?dy'{y'\A1...s\ii?)p,(x',y',t) E.11)
Here we have taken advantage of the symmetry of the system so that A \ s
operates only on the first s particles of the TV-particle sample.
In Fock space, the analogy of E.10) is given by
I f
s\ J
= I f dxsdysGs(xs,ys,0)(ys\Ah.s\*s) E.12)
s\ J
Stemming from the first equality in E.11), we write
^ E.13)
Here we have recalled that Tr[AB] = Tv[B A]. Inserting E.12) into E.13) and
recalling E.7) gives
-hi
sTv(N)[p
= - / dxsdysTr^)[pN(O)Gs(xs,ys,t)](ys\Al..s\xs) E.14)
s\ J
Comparison of this latter result with E.11) indicates that
t) = h Trw[^@)G(x^, /, t)] E.15)
L V •
which is the desired relation between the density matrix and Gs. Note, in
particular, that Gs determines the density matrix ps, but the converse is not
true.
As an elementary example, consider the problem of obtaining pi(x, y, 0) for
a system of TV particles in the pure state |..., nx, ...). Writing pN@) in the
Projection representation B.37), we obtain
j5
Pi(x, y, 0) = 1 TYw[j5a,@)Gi(x, y, 0)]
= -Trw

398 5. Elements of Quantum Kinetic Theory
1
n
A nonzero result occurs only if x = y. Thus we find
...,wx,...>] E.16)
— —-/S
— Ov v
N 'y
This result has the proper normalization
[n* i v-v n
As a second example of this formalism, let us apply E.14) to evaluate the
average total two-body interaction potential
N
Here q, represents the coordinate operator. Thus we must evaluate
(yiy2lw(qi — q2)|xix2) = 5(yi — X\)8(y2 — x2)u(x\ — x2)
Substituting this result into E.14), and with reference to E.7), we find
dx\ dx2 dy\ dy2 Tr(/V)
hlfll
, t)]
- x2) E.17)
Again, as in E.16), let pw@) be relevant to a pure state. Substituting this form
into E.17) gives
1
(V) = -dxldx2{--,nx,...\
, t)u(X\ — X2H//(X2, O0//(xl, 01 • • • » «x, . . .{5.
This integral may be further reduced by setting t = 0. With E.4) there results
{V) = ~ / / dxxdx2nXxnXlu{xx - x2) E.19)
which has a self-evident interpretation. Had we not worked in a pure state,
an additional probability factor would have appeared in the preceding integral
[see B.36)], together with a sum over nXx and nXl.
Let us consider calculation of (//),. With E.13) we write
, = Tr[p(t)H] =
Thus we find that (//> is constant in time.

5.5 Overview of Classical and Quantum Hierarchies 399
§>5.3 Equations of Motion
We recall that the density operator and classical distribution function satisfy
gimilar equations, the Liouville equation A.4.7) and its quantum analog B.11),
free A.44)]
E.20)
dt
E.21)
ot
The classical Hamiltonian for our system is given by
N 2 N
//(I, ..., N) = ]T g^ + J2 E Ml* ~ x;} = kn + Vn E.22)
where KN and Vn are respective kinetic and potential energy terms. Substituting
the quantum analog of E.22) into E.21) and forming matrix elements gives
o .j.,. ..v , — &N\y )J + ITmx )—vN{y )\]Pnkx , y ;
ot in
E.23)
The operators in E.23) are give by
N
i = \
:2
— V\ E.24)
The equation of motion for the Wigner distribution, FN, is given by the
Wigner-Moyal equation B.70):
3Fa/ vN dFv 2 /H d 9 \ »
+ r sin -—-•—- VN(xN)FN E.25)
dt m dxN n \2dpN dxN,
As noted previously, momentum and space derivatives in the sine function
operate, respectively, on FN and VN.
To obtain the representation of the Hamiltonian E.22) relevant to Fock
space, we note the following second quantization rules: With F(x) and L(x, v)
^presenting one- and two-particle functions, respectively, we write
-► J Jx0+(x)F(xHH(x)
E.26)
^ iff
li'Xj)^ 2] J
N

400 5. Elements of Quantum Kinetic Theory
H=
where </>//(x) is the time-dependent operator E.7a). There results
I 4>fH(x)k(x)j>H(x)dx +\f f ftH(x)ftH(y)u(x, y)j>H(yL>H(x)dxdy
E.27)
As an elementary example of the workings of these operators, consider that
the double integral potential-energy operator in E.27), which we label hQ,
operates on the ket vector
There results
-m(x', y') + -«(y', x')
2 J 2
We turn next to construction of equations of motion for 0#(x) and GN. With
the Hamiltonian E.27) at hand, we employ E.8) first to obtain an equation of
motion for 0#(x). With the commutation relations E.5), there results (see
Problem 5.46)
ih 4>H(x) = K(xL>H(x) + f 4>l(y)u(x, yH//(yH//(x) dy E.28)
Employing this equation and the defining relation E.7) gives the following
equation of motion for GN(xN, yN, t) (see Problem 5.47):
- A:w(yW)] + [^v(x") - ^(yw)]}Gw E.29)
ot in
Whereas this equation is equivalent in form to E.23), we should bear in
mind that E.29) is relevant to Fock space, where, as noted previously, TV
is unbounded.
5.5.4 Generalized Hierarchies
In the following, we write z, for the single-particle state (x,, p,). Thus
DN = Dn(z\ , Z2, ..., Z/v) = Z)(x
where DN is written for any of the preceding distributions. Reduced
distributions are given by
N-
Ds(zs) = I dfis+l • • • dfiNDN(z") E.29a)
where the integration measure d/j, is specific to the distribution at hand. In
the following, we assume that DN(zN), as well as the Hamiltonian H(zN), is

5.5 Overview of Classical and Quantum Hierarchies 401
symmetric with respect to z,, z7- interchange. Furthermore, we take
,■ = 0 E.30a)
; = 0 E.30b)
With the preceding relations at hand, hierarchies stemming from the Liou-
ville equation E.20), the density matrix equation E.23), the Wigner-Moyal
equation E.25), and the equation for the Fock space operator E.29) are
obtained. We list first the hierarchy for the classical distributions:
9/,
T\[fs, Kt] + T Ylfs, uu] + (N-s) Y[fs+i,uitS+l]dzs+l = 0
E.31)
[compare with B.1.20)]. This equation poses as a generic form for quantum
heirarchy equations as well. Thus, with Ds representing any of the distribution
functions considered, we write
E.32)
The as, A, Yih an<^ ^A^^+i terms are listed in Table5.3 relevant to the specific
case at hand. Here are some comments concerning the various hierarchies in
Table 5.3.
In the equation for Fs(xs, ps), first note that
d d d d \ ( d d \ d
d dj J
This relation gives the correct correspondence between case (a) and (c) in the
classical limit, h -» 0. As in the example following E.9), the delta functions
in d/jLs+\ stem from the trace operation in E.9).
For a specific example from Table 5.3, we consider the hierarchy equations
for the Wigner distribution. It appears as
t- / t\ c\ r\ c\
n t o o o o
... -+
/
[Fs+huu+l]dxs+ldps+l E.33)
As in B.71), in the double sum in E.33), the 3/3x operators in the sine function
operate only on ul}.
In obtaining the hierarchy for Gs, the equation of motion E.29) was applied
to Gs as given by E.7).

o
to
TABLE 5.3. Elements of Classical and Quantum Hierarchies with Reference to the Generic Equation E.32)
Case D,
a
m
3
m
2 .
- sin
h
n
u,
[ulJ(xl,xJ)- K*(y,,y,)]
dxs+ldps+l
dxx+ldyx+lS(xs+l -
dxs+ldps+l
■dxx+] dyx+] 8(xs+] -
3
3
o
1 r
c
3
3
CD
r-►
o
H
tr
CD
o
(a) Classical distribution; (b) density matrix; (c) Wigner distribution; (d) ^-particle F-space operator

5.6 Kubo Formula Revisited 403
As noted above, the hierarchies for ps and Gs are nearly identical in form.
However, as seen from the generic relation E.32), at s = N a closed equation
jpor Pn results. This is not true for the Fock-space operator Gs. The particle
number TV is unbounded in this case, as are the number of equations in the
hierarchy for this operator.
5.6 Kubo Formula Revisited
Having discussed second quantization in Section 5.1, we return to Kubo's
formula C.4.67) and develop a quantum expression for electrical conductivity.
5.6.1 Charge Density and Current
The Hamiltonian for a particle of charge e and mass m in an electromagnetic
field is given by [recall A.1.16)]
1
H =
- -A(x,r)l +eO(x,r) F.1)
c J
2m
Working in coordinate representation, together with the time-dependent
Schrodinger equation, we obtain
dp
dt
where
+ V • J = 0 F.2)
p = e^if F.3)
J = (V^*V^ - VVVO \rlr*rlr F.4)
2mi me
We may view F.1) as consisting of an unperturbed free-particle component
Ho and perturbed component V so that
H = Ho + V F.5)
where
% F.5a)
V = ^-{p • A + A . p) + -1-A2 + e<t> F.5b)
2mc 2mcl
In second quantized coordinate representation appropriate to Fock space,
F.5a,b) become
/h
^—V2\jrdx F.6a)
2m

404 5. Elements of Quantum Kinetic Theory
—ph „ * p2
j.
— \y\r V • (Ax//) + iff A • Vi/r] H A y y + e&x/f y
2ra*c 2racz
F.6b)
where
Recall that the wave functions for these operators obey the properties E.4).
Stemming from F.4), we write
J = J@) — —A\jff\jf F.7)
me
where
J@) = :[\[rT(V^) - (V^T)^] F.8)
This latter expression permits F.6b) to be written
= / I—J@) • A H -A2\j/f\jf + ^O^ I Jx F.9)
J \ 2 2rac2 /
5.6.2 Identifications for Kubo Formula
With these expressions at hand, we return to the Kubo formula and identify
terms for the purpose of obtaining an expression for electrical conductivity.
Employing C.4.63) and C.4.67) and assuming that {B)o ^ 0, we write the
quantum version of the Kubo formula:
f-
Jo in
(B) - (B)o = f ^-Tr{p@)[H, B(t - t')])F{t')dt' F.10)
J &
where B(t) is given by the quantum analog of C.4.55),
= Qxp(-tS)B F.11)
which serves to identify the superoperator S.51 The exponential commutator
operator in F.11) was defined previously by A.4.20).
The following three relations allow F.10) to be written in a more concise
form. First we note that with [p@), Ho] = 0 we may write
F.12a)
The second relation is given by (see Problem 5.31)
^A^i = Tv{AB} F.12b)
57
Superoperators operate on operators.

5.6 Kubo Formula Revisited 405
und the third by
applying the latter three relations to F.10), we obtain
(B) - (B)o = f ^-Tv{p^[H(a B(t)]}F(tW
Jo &
With the identifications
H(t)F(t)^l-haV(t)
B(f) -> J@
(F.13) becomes
F.12c)
F.13)
F.15)
(J) - (J)o = i / Tr{p@)[j(f), V(tf)]}dtf
iK Jo
Wiere (L)o represents current flow in equilibrium.
5.6.3 Electrical Conductivity
iSubstituting F.7) through F.9) for J and V into the preceding expression, we
find
:,0
(J) - <J>o = t- / dt'Trlp@)
o
J<u'(x, 0 A(x,
me
F.16)
F.17a)
Working in the Coulomb gauge,
V-A = 0,
with
1 O
8= A, B = VxA F.17b)
C dt
and neglecting terms of 0(A2), F.16) reduces to
(J> - (J)o = l~ f dt' j dxf Tr{p@)[j(V, 0, >iO)(x', t')]}Afi(x', t') F.18)
where subscript fi represents Cartesian components and is summed from 1 to
3. We note that the time integral of the a component of F.18) may be rewritten

406 5. Elements of Quantum Kinetic Theory
as
'00
^ J dtf J dxf0(t-t')Tr{---}0(t)Ap(x\t') F.19)
—oo
where
0(t > 0) = 1 and 0(t < 0) = 0
To bring F.18) closer to the form of Ohm's law, we assume that
i
0(t - t') Tr{p(U)[^U)(x, t\ J';\x\ t')]} = Qap(x -x\t- tf) F.20a)
He
and that
0(f)A(x, t) = A(x, t) F.20b)
Inserting these values into F.19) gives
(Ja)(xj)-(Ja)(xj)= f dt' [ dx'Qafi(x-bx\t-t')Ap(x\t') F.21)
J — oo J
We recognize this to be convolution integral, in which case the transform of
F.21) becomes
(Ja)(co, k) = (Ja)o(co, k) + Qap(eo, k)Ap(oj, k) F.21a)
where
C C dk dco
A(x, t) = 4 e A((o, k)
J J B.nr
With F.17b), we may write
£(o), k) = — A(<w, k) F.22)
c
Substituting this transform into F.21a) gives
(Ja)(co9 k) = Ga>0(ttf, k) + -?-QaP(co, k)£p(co, k) F.23)
10)
The first term on the right may be further reduced as follows. Recalling F.7),
we write
(J>o = Tr{p^y} = Tr{po/(O)}- ^
me
= (J(O))o -
= (J(O))
me
It is evident for F.7) that G°H is the equilibrium value of current with fields
turned off. Without loss in generality, we may set this term equal to zero.
Consider the remaining term in F.24), which contains the form
^^ =/i(x,r)

5.6 Kubo Formula Revisited 407
Inhere n(x, t) represents particle number density. The remaining term in F.23)
Wy then be written
n(x, r)A(x;r)
me
e transform of this product is the convolution integral
e1 f f dkdco , , ,
— \ \ TT^n^° - ^, k - k')A(o/, k')
me j j BnL
In the homogeneous limit
n(co -co\k- k!) = BnLn08(co - cof)8(k - k')
Recalling F.22), the preceding expression then reduces to
2 2
e c
—n0A(k, co) — ——no£(co, k) F.25)
me mic
Substituting F.24) in F.23) with (J(O))o = 0 and the remaining term transferred
as given by F.25), we obtain
,2
(Ja)(<O,k) =
C 1 x - e
F.26)
ico mic
This relation is a tensor form of Ohm's law,
(J) = 0afi£ F.27)
[compare with C.41)], and we may write
F.28)
The terms in F.26) may be identified by comparison with F.24), with Qap
given by F.20a).
5.6.4 Reduction to Drude Conductivity
The relation F.26) may be reduced to the Drude formula C.4.14) as follows.
First we neglect the off-diagonal Qap terms in F.26). To incorporate collisions,
we set
ico —> ico —> v F.29)
where, again, v represents collision frequency. In the dc limit, co = 0, and
F.28) reduces to
enr\
a = —- F.30)
vm
in agreement with C.4.14) [as well as with C.7.65)].58
58
In these formulas, conductivity was labeled ac

408 5. Elements of Quantum Kinetic Theory
5.7 Elements of the Green's Function Formalism
The quantum mechanical Green's function comes into play in two classes of
problems:
1. In constructing solutions to the Schrodinger equation for an isolated
collection of few particles.59
2. In calculating average properties of many-body systems.60
As well be found below, for case 2, a significant property of the Green's function
is that it gives few-particle properties of a many-body system. Out discussion
begins with a brief review of topic 1.
5.7.1 Schrodinger Equation Green's Function
This Green's function enters in constructing solutions to equations of the form
G.1)
where R typiclly is a linear differential operator and I(x, t) is a given inho-
mogeneous term. The Green's function corresponding to G.1) satisfies the
equation
R(x, t)G{x, t\x\ tf) = S(x - x)S(t - tf) G.2)
If G is known, then the solution to G.1) is given by
, t)= / / G(x, t\ x\ f)I(bx, f) dx' df G.3)
which is readily seen to be a solution if this expression is substituted into G.1).
Application of the Green's function to quantum theory is chiefly perturva-
tive. Consider the Schrodinger equation for a single particle in a potential field
V(x):
r t\ _ V(VW/YX t) A 4)
at
59See, for example, E. Marzbacher, Quantum Mechanics, 2nd ed., Wiley, New
York A970).
60For further discussion, see P. H. E. Meijer, Quantum Statisical Mechanics, Gor-
Gordon and Breach, New York A966); L. P. Kadanoff and G. Baym, Quantum Statistical
Mechanics, W. A. Benjamin, Menlo Park, Calif., A962); E. Em. Lifshitz an dL. P.
PEtaevskii, Physical Kinetics, Pergamon Press, Elmsford, N.Y A981); and A. L.
Fetter and J. D. Walecka, Quantum Theory of Many Particle Systems, McGraw-hill,
New York A978).

5.7 Elements of the Green's Function Formalism 409
for weak interaction, the lowest-order solution to G.4) is given by the free-
particle equation
~ " :,0 = 0 G.5)
dt 2m
The second-order solution satisfies the equation
n2
m— +
+
at 2m
, 0 = V(xWo(x, 0
G.6)
which has the structure of G.1). Thus we may write
n2 „
+
dt 2m
, r;x\ t') = 8(x - x')8(t - tf)
G.7)
where s is an infinitesimal parameter of smallness that is evertually set equal to
zero and, as we will see, ensures the boundedness of the transform of G(x, t).
We have also written GR, GA for the retarded and advanced Greens functions
and corresponding to +ie and —is in G.7), respectively. This terminology
is motivated belwo. The zero superscript on G^ indicates that it is the free-
A
particle Green's function.
Transformed Green's function
The Fourier transform of GV is given by
dco dk
,co)
2n kltz r a
Taking the transform of G.7) gives (where the limit s —> 0 is understood)
G.8)
h2k2 \
hco ±is) G?(k, oo) = 1
2m ) a
Separating solutions and subtracting, we obtain
1
G.9)
= lim
hoo - (H2k2/2m) + is hoo - (H2k2/2m) - is _
Now recall the relation [preceding B.84)]
G.10)
lim
= V
x
i7t8(x)
G.11)
where V denotes principal part. Employing this relation in G.10) gives
h2k2
2m
This form is generally known as the spectral function as it displays the
dispersion relation of oo and k.
°(k, oo) = i[Gf - G(A0)] = 2tt8 (hoo -
G.12)

410 5. Elements of Quantum Kinetic Theory
5.7.2 The s-Body Green's Function61
The s-body Green's function Gs, relavant to a many-body system is given by
the following form written in second quantization:
isHGs(h ■ • •, s', 1\ ..., s') = (t[4>H(l) • • • 4>H(sL>Us') • • • 01,A')]) G.13)
where
(j) = (Xj,tj) G.13a)
and T is the time-ordering operator:
t
A*
t
4>H(tiWH(t2) h > h
.t
±<pJH(t2)<pH(h) h >
G.14)
which ensures that operators in G.13) are chronologically ordered with the
earliest operator acting first. We recall that (j)H satisfies the equation of motion
E.8). The signs + and — refer to bosons and fermions, repectively. The latter
relation may alternatively be written
^ k ± 0(t2 - h)ftH(t2L>H(h) G.15)
where 6{x) was introduced previously [beneath F.19)]. Note that the
dimensions ofHGs are (volume)"*.
The average in G.13) is written with respect to an TV-body system in the
Heisenberg representation and is given by E.13):
(f) = Tr(N)[pN@)r ] G.16a)
Working in Fock space, we obtain
(f) = ]T (n| ^@) In'^n'l f |n) G.16b)
n,n'
where |n) represents the many-body ket vector on the right side on E.2).
5.7.3 Averages and the Green's Function
In this section we wish to obtain a relation between the average of a s-particle
dynamical variable and the Green's function G.13). Toward these ends, we
note that the Fock-space operator, Gs, defined by E.7), is closely allied to the
61 The many-body Green's function was originally formulated for relativistic quan-
quantum field theory where the use of multiple times is well motivated. See S. S. Schweber,
An Introduction to Relativisitic Quantum Field Theory, Harper & Row, New York
A961). This analysis has found use in nonrelativisitic many-body theory as well.
See, for example, H. Haken, Quantum Field Theory of Solids, North Holland, New
York A976); and C. Kettel, Quantum Theory of Solids, Wiley, New York A963).

5.7 Elements of the Green's Function Formalism
411
/Jreen's function G.13). The average of an s-particle dynamical variable is
Mven by E.14). This expression involves the form
= (Gs(x\f,t))
G.17)
jfcb relate this to the Green's function G.13), first we make the notational change
jfg = xs,ys =x's and let variables include time so that E.7) becomes
G (xs x's) — d)lr(V
G.18)
where again we have set (j) = (xy, tj) with
The form G.18) can be made congruent with G.13) if in the latter we set
t[ > t'2
t's >ts
>tx=t
Let LIM represent the process
lim
\
t's =ts
\ti=t
this limiting process brings the ordering of variables in G.17) in agreement
with those in G.13), and we may write
(isH LIM Gs(x\ x", t)) = isHGs(x\ x", t)
G.19)
Note in particular that averages of one-body observables such as particle mo-
momentum stem from Gi(l, V), whereas averages of two-body observables such
as pair potential stem from G2(l, 2, 1\ 2'), and so on.
Combining the preceding results, we write [recall E.14)]
-f
slj
dxs dxsisHGs(xs, x\ t){xs\A{ • • -s\xs)
G.20)
Note in particular that the s-dimensional summation in E.10) is absorbed in
the integral of G.20).
Let us employ G.20) to calculate A) the total momentum and B) the total
Potential energy of an TV -particle system. For total momentum,
N
and G.20) gives
(P)
= f
G.21)

412 5. Elements of Quantum Kinetic Theory
The coordinate representation of pi is [recall A.27)]
(x'Jpilxi) = -iH—8(x\ -
Integrating G.21) by parts we then obtain
(P) = -(iHJ j dx{ dx[ A-Gi(xi9x'l9t)\ 8(x\ -
/
G.21a)
Our second example addresses total potential energy:
TV
v = Y
Again employing G.20) and dropping the subscripts on utj, we find (with
s=2)
(V) = - dxldx2dx[dx2i2HG2(xuX2,x[,x2J){x\x2\u(xux2)\xlX2)
h f
= — / dx\ dx2dx\ dx'2G2(x\, x2x\, x2, t)u(x\, x2)8(x\ — X\)8(x2 — x2)
G.22)
which reduces to
M
Substituting the defining relation G.19) into the preceding gives
I IT T -■
- ~J h 1 H
which agrees with our previous finding E.18).
The preceding two example illustrate the manner in which the Green's
function determines one- and two-particle properties of a many-body system.
Compilation of relations
One also defines the Green's functions (dropping the subscript onGi)
> ty A.24)
< h>
Note in particular that number density may be written
ij \ f ' / T M A \ 7 / f
= ±tfiG<(l, 1) G.25)

5.7 Elements of the Green's Function Formalism 413
gere is a compliation of relations among the assorted Green's functions
Introduced to this point.
4L G-26a)
G < A, 2) = ± ^ $H BHh A)) G.26b)
, 2) = 0(f, - h)G>(\,2) + 6{t2 - h)G<(\, 2) G.26c)
, 2) + G(l, 2) = 4([0h(D. 0hB)]±> G.26d)
, 2) - G^l, 2) = 4<[0h(D. 0wB)]t> G.26e)
Gs(l, 2) = e(/, - f2)[G>d, 2) - G^l, 2)] G.27a)
GA(l, 2) = 6{t2 - /,)[G<A, 2) - G>A, 2)] G.27b)
G>A, 2) - G<A, 2) = GRA, 2) - GAA, 2) G.28)
5.7.4 The Quasi-Free Particle
In this and the following section we see how the Green's function is related
to distribution functions of momentum. Consider the operators 0(x) and 0 (x)
given in E.4). Fourier analyzing these operators gives
0(x) = J -^elk*a(k) G.29a)
/dk
—-elkxaHk) G.29b)
BttK
where, with E.5a), we write
[fl(x), a\k')]mp = BnK8(k - k') G.30)
The signs (—, +) refer respectively to bosons and fermions.
Note that in this representation the Hamiltonian of a free particle may be
i
written
dk h2k2
BttK 2m
a quasi-free particle, we write
G.31)
f dk
H= / 7^3 £00^A02A0 G.32)
J BnK
Which is a good approximate Hamiltonian for the actual system.
Stemming from the Heisenberg equation of motion E.8), the relation G.32)
gives (see Problem 5.32)
_g_e-(/M)r+ik*a(k) G.33)

414 5. Elements of Quantum Kinetic Theory
We recall that (a*(k)a(k)) represents the number of particles with momentum
k. For a spatially uniform system, we may write
(a\k)a(k')) = BjtK8(k - k')/(k) G.34a)
from which, with G.30), we obtain
\')) = BnK8(k - k')[l ± /(k)] G.34b)
where /(k) represents the one-particle distribution function in k space. These
findings are used below.
5.7.5 One-Body Green ys Function
To obtain the related one-particle Green's function, we recall G.13) with s = 1
and G.15) to obtain
iBG(x, f;x', t') = 0(t - t')D>H(x, 00], (x', t')) ± (tf - t)(ftH(x\ t')J)H(x, t))
G.35)
Substituting G.33) into this equation, together with G.34), we find
iHG(x, r,x\ tf) = f -^- exp (ik • (x - x') - l-E(k)(t - t')
x {0(t - t')[\ ± /(k)] ± 9{tf - 0/(k)} G.36)
Now note the repesentation of 6{t — tf) = 6{x):
/
nco — E ± is
Substituting this representation into G.36), we find
,r;x/,r/)= I [ -^
J J Btt
G(x,r;x/,r/)= I [ -^^exp{ik.(x-xf) = ia>(t-tf)}G(Ka>) G.38)
J J BK 2
G(k, co) = lim
1 ± /(k) /(k)
Hco — E(k) + is Hco — E(k) — is _
G.39)
This equation gives the desired relation between the one-body Green's function
G(k, co) and the one-body distribution function /(k) with normalization.
/dk
/(k)—-3 = N G.39a)
BnK
where N is total number of particles and V is volume.

5.7 Elements of the Green's Function Formalism 415
\7.6 Retarded and Advanced Green's Functions
Retarded and advanced one-body Green's functions are given by
*(l, 1') = 0(tx - tv)U[4>H(l), fa
G.40)
GA(h 1') = 0(tx - tv) (~
Where, again, averages are with respect to an TV-body system and are given by
J7.16). For the quasi-free-particle problem considered above, we find
G; (a>, k) = lim (- ) , ) G.41)
A e^o\Hco — E(k)±ieJ
which, for a free particle, agrees with G.9) relevant to a single-particle system.
Note that the Green's function G.41) obeys the equation
]fico - E(k)]G(k, co) = 1 G.42)
Again, for a free particle, with
n2k2
E(k) =
2m
the Fourier inversion of G.42) returns G.7).
Let us return to the motivation for advanced and retarded terminology intro-
introduced above. With G.37), we see that +ie, —is in the integral representation
of the 0 function corresponds to t > V and t < t', respectively, in the Green's
fonction expression G.35). If we revert back to G.7), it is evident that t' may
be associated with a source. Thus the Green's function in the interval t < t'
describes a response prior to the source being turned on and is called advanced.
For the case t > t'', the Green's function obeys causality and the response is
retarded in time with respect to the source.
7.7 Coupled Green's Function Equations
In this section we construct an equation of motion for the one-body Green's
Junction, which, we will be seen is dependent on the two-body Green's func-
function. The analysis begins with E.28) [dropping the hat and H subscript on
^, r)]; we write
ih
0A) = j dx2ct)\2)u(xux2)<t)B)<t)(\)
G.43)
Where, again, we are using the notation of G.13a).

416 5. Elements of Quantum Kinetic Theory
Now we note the relation (see Problem 5.34) relevant to differentiation of
time-ordered products:
3
= 8A - 1') ± 9(tv - tx)(t>\\')d-^- + 9(tx - rF)^V(l') G.44)
ot ot\
where again the + sign refers to bosons and the — sign to fermions, and we
have set
8A - 1') = 8(x - x)8(t - tf)
Employing G.43) for the time derivative terms in G.44), after some
rearrangement we obtain
/ dx2u(x{,
± / rfx2w(x1,x2)[f0(lHBHtB+Ht(l/)]
G.45)
The time t% of <pf is infinitesimally greater than f2 in order to maintain the
order of operators in G.43). From the defining relations G.13), we write
ihGx{\, l') =
-HG2(l, 2; 1', 2') = (f0(l)^>B)<A+B')</)t(l')} G.46)
Constructing the average of G.45) and substituting the preceding definitions
gives the desired result:
G(h 1') = 8A - 1') ± i f dx2 u(xx, x2)G(l, 2; 1', 2+)
G.47)
Note in particular that when particles become isolated, u(x\, x2) —> 0, and
G.47) returns G.7) relevant to a free particle. Note further that carrying out
the LIM process defined in G.19) returns the first equation in the hierarchy for
Gs, as given in Table 5.2 (see Problem 5.35).
5.7.8 Diagrams and Expansion Techniques
We continue this section with a brief discussion of diagrammatic represen-
representations of equations of motion for Green's functions. Toward these ends,
G.57) must be rewritten in a form that better lends itself to diagrammatic
interpretation.

5.7 Elements of the Green's Function Formalism 417
Consider first that the interaction in G.47) is turned off. The resulting free-
gtuticle equation may then be written
G.48)
Ifhere Go(l, 1') is the solution to the free-particle equation and
G0-'(i, i") = (in
5A-1")
G.49)
us further define
= M(x1,x2M(f1-f2)
G.49a)
tjlfith the preceding identifications, G.47) may be rewritten (see Problem 5.37)
/>d2Go1(l,2)GB, 1') = <5A- \')±i [ d2U(l, 2)GA, 2; 1'2+) G.50)
iDperating on this equation with /d\ GoC, 1) gives
[Jdld2G0C, l)Go'(l,2)GB, 1')
= I d\ GoC, 1MA - 1') ± / I d 1 dl G0C, 1I/A, 2)GA, 2; 1'2+)
fwhich, with G.48), gives the desired relation (exchanging 1 and 3):
. 1' O+-
</2</3G0(l,3)£/B,3)GC,2;l\2+) G.51)
This equation is the starting point for our description of diagrammatic rep-
representation of terms, which is described in Table 5.4. Note that in the box
diagram a dashed line represents the interaction f/(l, 2) and implies integra-
integration. The direction of an arrow is associated with the order of arguments in
the related Green's function. With these rules at hand, G.51) has the following
diagrammatic equivalent:
T = 1
V
ti
G.51a)

418 5. Elements of Quantum Kinetic Theory
TABLE 5.4. Fundamental Green's Function Diagrams
Term
Diagram
1 /-
v
v
fdld2U(l,2)G(l,2;l',2+)
The diagrammatic representation for the equation of motion for G( 1,2; l',2')
is given by
1' 1 .
= 1)
2' 2
The equation corresponding to this representation is given by
G(l, 2; l\ 2') =«G0(l, l')GB, 2') ±«G0(l, 2')GB, I')
±i (/d3d4G0(l,4)£/C,4)GD,2,3;l\2\3+) G.52a)
The explicit equation of motion corresponding to the preceding equation [in
analogy with the structure G.47) is obtained by operating on G.52a) with
fdl Gq 'E, 1). There results
d\ G^E, 1)GA, 2; 1', 2') =H8E - lr)GB, 2') ±/i<$E - 2r)GB, lr)
W
With G.49), and changing 5
equations of motion:
1, the preceding reduces to the desired
ih
d
, 2; 1', 2r) =«5A - lr)GB, 2r
- 2)GB,
±i
, l)G(l,2,3;l/,2/,3+)
Approximations
Let us return to G.51a). For weak interaction it is consistent to expand the
equations in powers of the interaction or, equivalently, expand the correspond-

5.7 Elements of the Green's Function Formalism 419
diagrms in terms of the dashed element. Thus, to lowest order, G.51a)
s
v _ 1
v
G.53a)
fith this simplification, the lowest-order form of G.52) is given by
v
til
2'
V
2'
G.53b)
mploying these diagrams in G.51a) gives the following first-order diagram-
|atic equivalent for the equation of motion for G(l, T)-
r
r
V
V
Iff)
+ m
G.54)
yhich has the topological equivalent
Q,
V
V
- + if) <
V 1
2 V
G.54a)
jp this latter representation, we have joined 2 and 2+ as they represent the same
ce-time point. The equation corresponding to G.54) is
, 1') = G0(l, 1') ± iH I f dld3 G0(l, 3)GOC, l')£/C, 2)G0B, 2+
G.55)
d2d3Go(l,3)GoC,2)t/C,2)GoB, lr
The full equation of motion, in diagrammatic form, for G(l, 2, 3, T, 2\ 3r)
appears as
r
Jt
r
2'
3'
3
2'
2' =/>
3'
3'
2'
+ />
r
G.56)

420 5. Elements of Quantum Kinetic Theory
Furthermore, note that the integration in the final term in G.56) over 4 and
5 leaves a function of A, 2, 3; 1\ 2\ 3') in accord with the left side of this
equation. To lowest order, employing the decomposition G.53b), G.56) gives
(keeping order of T, 2', 3' on the right of each diagram).
2 <«--*■
2'
tfi
r
2'
3'
1 m
2w
/
\
3' —
G.57)
The corresponding functional relation may be written in the following
determinant form:
G0(l,2,3;l',2',3')=fi:
God,
G0B,
GoC,
1')
1')
1')
God,
GoB,
GoC,
2')
2')
2')
God,
G0B,
GoC,
3')
3')
3')
±
G.58)
It may be evident at this point that the equations of motion G.56), G.56),
and G.56) follow a general pattern given by the following rule: In the equation
of motion for the Af-body Green's function, the first N terms are products of
the free one-body Green's function and the (N — l)-body Green's function
with the T variable of Go(l, lr) exchanged in each product with one of the
primed variables of the (Af — l)-body Green's function. In the last term of
the equation, the free one-body Green's function attaches to the (Af + l)-body
Green's function via interaction. This last term contains the factor =b'. The 5th
term (s = 1,..., N) in the sum of products contains the factor +1

5.8 Spectral Function for Electron-Phonon Interactions 421
Following these rules of construction gives the following diagrammatic
Representation of the equation of motion of the TV-body Green's function:62
r
2'
2 t
£':
V
2'
2t
2'
r
N'
/vl J/V' A/1 1/V
/V+1
(A/+1
N'
G.59)
As noted previously, in each column of variables on the right side of the
first N terms of the preceding equation, Y is exchanged with an element of the
right column of a box diagram with the order of the remaining variables main-
maintained. Note also that the space-time coordinate 1 is preferred in the preceding
equation due to the fact that this equation includes implicit differentiation with
respect to xi, t\.
5.8 Spectral Function for Electron-Phonon Interactions
5.8.1 Hamiltonian
We conclude this section with a derivation of an approximate form of the
spectral function [see G.12)] relevant to electrons interacting with phonons in
a lattice (described previously in Section 3.6). The first step in this derivation
is the construction of the related Green's function which in turn is obtained
from its equation of motion. With E.8) and G.13) we see that formulation
of this equation stems from the Hamiltonian of the system. Thus, with bf(q)
representing the creation operator of a phonon with momentum /zq and energy
An alternative development of G.56) follows from Wick's theorem, which ad-
addresses the appropriate expansion of time-ordered product of operators. G. C. Wick,
Phys. Rev. 80, 268 A950). See also A. L. Fetter and J. D. Walecka, Quantum Theory
°fMany Particle Systems, McGraw-Hill, New York A971).

422 5. Elements of Quantum Kinetic Theory
G.32) is extended to read
f dk + C da
/ —- E(k)a\k)a(k) + / —^
+ C*(q)af(k)a(k + q)b\q)] (8.1)
The coefficient C(q) represents the strength of the electron-phonon interaction.
The interpretation of the interaction terms in (8.1) is as follows. Consider, for
example, the third integral on the right side of the equation. In this interaction, a
phonon of momentum/zq is annihilated in scattering with an electron, which in
turn suffers momentum change from/z(k) to /z(k+q). A diagram corresponding
to this process was sketched previously in Fig. 5.7a.
In the Heisenberg picture [recall E.7a)], we write
aH(k, t) = e(i"t)/na(k)e-(i"t)/n (8.2)
This operator has the equation of motion [recall E.8)]
ot
with parallel relations held by d^.bn, and bfH. As <2(k) relates to fermions and
b(q) to bosons, recalling G.30), we write
= B7tKS(k-kf) (8.3a)
[')]- = B7rK<$(q-q') (8.3b)
All other commutators are equal to zero.
Combining the preceding relations and recalling the expansion
/\ /\ /\
gives the following four equations of motion:
j
, t) = E(k)aH(k, t) + j -^[C(q)aH(k - q, t)bH(q, t)
dt v ' ' v - — - ' ■ y Bjr)
+ C*(qJ//(k + q, r)^(q, t)]
8 Af , , _
E(k')a'H(k', t') - I -^i-[C(q)fl;(k' + q, t')bH(q, t')
J B7rr
dt
+ C*(q)aH(k' - q, t')bfH(q, t')] (8.4)
d * * C din +
ih—bH(q,t)=Ha>(q)bH(q,t) + C*(q) / —-,aH(k, t)aH(k + q, t)
ot J (my
, t') = -fi<u(q$,(q, t') - C(q) f -^afH(k + q, t')aH(k, t')

5.8 Spectral Function for Electron-Phonon Interactions 423
0.8.2 Green's Function Equations of Motion
As in 7.13 we introduce [dropping the tilda notation of G.8)]
1 ihG(K t\ k\ t') = (t[aH(K t)al(k\ t')]) (8.5)
the property G.44), we obtain the equation of motion
ih—G(k, t; k', t') = BnKS(k - k')8(t - t') + E(k)G(k, f, k', t')
dt
/I
—5-[C*(q)S'(k + q, t; k', t'; q, t)
+ C(q)S(k-q,f;k',f';q,f)] (8.6)
wher
ihS(k, t;k, t';q,x) = (f[aH(k, t)afH(k', t')bH(q, r)]>
iHZ'(k, t;k\ f';q, r) = (t[aH(k, t)afH(k\ t')bfH(q, r)])
(8.7)
With (8.6) we note that in order to determine G(k, f;k', t') it is necessary
to know S and Sr. Equations of motion for these variables follow from the
defining relations (8.7) and the equations of motion (8.2a). There results
E\K t\k\ r';q, r) = -/^(q)S'(k, t\k\ t'\q, r) (8.7a)
—— C(q)G2(k, r, K, r; k', t\ K + q, r+)
Bny
ih—3(k, t\ k', r';q, t) = «a)(q)S(k, f; k', f';q, r) (8.7b)
OX
—-C*(q)G2(k, r, K+q, r; k\ r', K, r+)
BttK
For differentiation with respect to f, we find
ih—S(k,f;k',f';q, r)
Of
- k')8(t - t')(bH(q, r)> + £(k)S(k, r;k', f';q, r)
f dQ f C*(Q)
J Bny [ in
/dO f C(
B^ I~m{T[a"(k
ih
- Q, t)afH(k', t')bH(Q, t')bH(q,
(8.8a)
The equation of motion for Sr(k, r; kr, rr; q, r) is the same as the preceding
with Z7//(q, r) replaced by bH(q, r). [We will refer to this implicit equation as
(8.8b).] Thus the equations of motion for S and S', (8.7) and (8.8), involve ad-
additional unknowns and, with (8.6), do not comprise a closed system. To remedy
h situation, we set t = x and assume a homogeneous phonon distribution

424 5. Elements of Quantum Kinetic Theory
and write
$ = BnKS(q - q')n(q)
= BnKS(q - q')[n(q)
= 0 (8.9)
= 0
where n(q) is phonon number density. Assuming further that the time-ordering
terms in (8.7) factor into products such as
(t[aH(k + Q, OS*(k\ *'$,«*, t)bH(q, t)])
= {f[aH(k + Q, t)afH(k\ t')]){bfH(Q, t)bH(q, t)) (8.10)
and neglecting the two-body Green's functions in (8.7), we obtain
- S(k - q, r; kr, rr; q, OP^(q) + E(k - q)] + C*(q)n(q)G(k, t\ k\ t')
(8.11a)
, rr; q, r)[^(q)+£(k+q)]+C(q)[rt(q)+l]G(k, r;
(8.11b)
Here we have set
d d d
= — +
Dr dt dr
and, after differentiation, set t = r.
Introducing the time transformation
G(k, r; k\ O = / ^^-'^-')G(k, k\
/"
G(k, k\ tw) = / d(r - r'y!ftK'~'#)G(k, r,
into (8.6) and (8.11), we obtain
Ha)G(k, k\ co) = BnK8(k - k') -I- £(k)G(k, k\ oj)
C(q)S(k-q,k\q,a;)] (8.12a)
HojE(k - q, k\ q, oj) ~ ^w(q) + E(k - q)]S(k - q, k\ q, w)
-I- C*(q)/i(q)G(k, kr, w) (8.12b)

5.8 Spectral Function for Electron-Phonon Interactions 425
HcoZ'(k + q, k', q, co) ~ [-Hco(q) + £(k + q)]S'(k + q, k', q, co)
+ C(q)[n(q) + l]G(k, k\ co) (8.12c)
$.8.3 The Spectral Function
These equations comprise three equations for G,
Dbtain
[hco - E(k) - ]T(k, a>)] G(k, k', cd) =
S, S'. Solving for G, we
(8.13)
where
(8.13a)
x
n(q) + 1
n(q)
hco - £(k + q) +hco(q) hco - £(k - q) -hco(q)_
In the homogeneous limit, we integrate (8.13) over kr to obtain
\hco - E(k) - ]T(k, co)
G(k, co) = 1
(8.14)
In analogy with G.39), we write
1 ± /(k)
/(k)
hco — E(k) — J](k, co + in) ' /ia; — £(k) — J](k, co — in)
(8.15)
and note that in the limit n -> 0, G(k, &>), as given by (8.15), is a solution to
(8.14). Furthermore, in the limit that interaction is turned off, C = 0, whence
= 0and
G(k, co) -> G0(k, co)
the free-particle Green's function.
We now make the identification
lim
, co ± in) = A(k, co) =p iT(k, a/)
(8.16)
where A and F are real parameters and F > 0.
With (8.16), (8.15) may be written
G(k, co) =
1 ± /(k, q)
hco - E(k) - A(k, co + iT(k,
1 ± /(k)
hco - E(k) - A(k, co) - iT(k,
analogy with G.41), we write
1
(8.17)
, w) - A(k, co) ± /F(k,
(8.18)

426 5. Elements of Quantum Kinetic Theory
which, in the limit C —> 0, returns the free particle functions G V as given by
G.41). With (8.18), (8.17) may be written
G(k, co) = [1 ± f(k)]GR(k, co) t /(k)GA(k, co) (8.19)
Explicit time dependence
To revert to the time-dependent Green's function, we write
G(k,f)=
Noting that G#(k, co) has no poles in the upper-half &> plane and that GA(k, co)
has no poles in the lower-half co plane, we write
n
<t
/lh
G(k, 0 = 9(t) <t ^e~iM[l ± /(k)]Gs(k, co)
n
0(-t) <b -^^^[T/(k)]GA(k, co) (8.20)
Here we have written LH to denote a path that includes the real co axis and
a great semicircle in the lower-half co plane, whereas UH includes a great
semicircle in the upper-half co plane.
To recapture the spectral function, with the property
we write
G(k, t)
t
JLH
dco
^-GA
-O(t)(f -
Jlh 2;
f dco
(k,co) = (b —G*(k,
/UH ^7T
e~i(l)t[l dz /Yk)][G/?(l<
n
co)-0
:, w) - GA(k, co)]
r
<b
= 0(-t) <b ^e-i(Ot[±f(k)][GR(k, co) - GA(K co)] (8.21)
With G.27), this expression permits the identification
G"(k, co) = [l± f(k)][GR(k, co) - GA(k, co)]
G<(k, tw) = ±/(k)[G*(k, co) - GA(k, tw)] (8.22)
Furthermore, we note that (8.21) implies that the time dependence of G(k, co)
is contained in the spectral function [recall G.12)]
A(k, co) = i [G*(k, co) - GA(k, co)] (8.23)
5.8.4 Lorentzian Form
We conclude this discussion with a derivation of an explicit expression for the
spectral function (8.23). From (8.18), we write
1
hco - E{k) - A(k, co) + iT(k, co)

5.8 Spectral Function for Electron-Phonon Interactions 427
1
Hco - £(k) - A(k, co) - iT(k, co)_
gives the main result of the preceding analysis:
2r(k, co)
A(k' ^ = in. - £(k) - A(k, op + r'<k, a,) (8-24)
This Lorentzian form is the desired generalization of the free-particle spectral
function G.12). Note that with (8.18a), J](k, co) [or, equivalently, A(k, co)
and F(k, co)] contains the specifics of the phonon-electron interaction for the
problem at hand. Thus, to recapture the spectral function appropriate to no
interaction, we consider the limit
lim
2F(k, co)
_\nco- E(k)-
- E(k)] (8.25)
which agrees with our preceding free-particle expression G.12), with the free-
particle energy H2k2/2m replaced by the quasi-free-particle energy E(k).
To gain deeper physical insight into these results, first we substitute the
general form (8.24) into the Green's function (8.21) to obtain
G(k, 0 = 0@ / ~—.e-l0)t\\ ± /(k)]A(k, co)
+ 0(-t) / -^e~i(Ot[±f(k)A(k, co)] (8.26)
Again consider the limit of no interaction. With (8.25) substituted in (8.26)
and integrating over co, we find
Go(k,0
<8.27)
We note that this form is identical to that obtained from the Fourier inversion
of G.36). Thus (8.27) is a representation of the quasi-free-particle Green's
function.
For further interpretation of the variables A and F in the interaction J] [see
(8.16), let us assume that both these variables are frequency independent. In
this event
2Ffk)
A(k, co) = — ;/ (8.28)
\Ko - £(k) - A(k)]2 + T2(k)
With
2 \hco - E(k) - A(k)]2
Z = ¥

428 5. Elements of Quantum Kinetic Theory
the inverse frequency transform of A(k, co) is written
f° deo . t 1 / (£ + A)A f°° ye~iztdz
A(kj)= / — e-""Afr, to) = — Uxp-iK / Y
A f
/
/ J-
-oc z2 + y2
(8.29)
where y = V/h. Thus
A(k, a>) = — (exp -/ / % (8.30)
Consider the integral
ye~lztdz
I = -
1 /•
n J
+ y2 n J (z- iy)(z
1/1 1
e lz dz
z-iy z -\-iy J
f 1 e~izt f 1 e~izt
(p —: —dz-(p —: —dz
That is, the first integral in the last equality is evaluated on UH [defined below
(8.20) with t < 0 (for convergence), whereas the second integral is defined on
LH with t > 0. Thus we obtain
= - fexp-i(£ +A) ) [0(-t)ert/n+0(t)e-rt/n] (8.31)
h \ h J
Substituting this result into (8.26) and recalling the property 9{t)9{—t) = 0,
we obtain
± 0(-O/OO (exp -/ (£ + AM eri/n (8.32)
V n )
5.8.5 Lifetime and Energy of a Quasi-Free Particle
To discover the meaning of the expression (8.32), we assume that the system
at hand is in the pure state |4>). Consider that at time t = 0 an electron with
momentum #k is added to the system. The initial state of the new system is
then given by
(8.33a)
With the electron now contained in the system, we view it as a quasi-free
particle with energy £(k) + A(k). At t > 0, (8.33a) becomes
(8.33b)

Problems 429
Now we ask the question, what is the probability density for finding the particle
with momentum/zk' at t > 0 if it had momentum/zk at t = 0? This probability
is given by
H (8.34)
With (8.5), we may then write
P(t) = |*7zG(k\r;k,0)|2 (8.35)
. For a system homogeneous in space and time, we write
G(k\ t'\ k, t) = BjrJ6(k/ - k)G(k, t' - t) (8.36)
Substituting G(k, t) as given by (8.32) into this equation, for t > 0, the latter
two equations give
2
- k')[l ± /(k)]exp { -l-[E(k) + A(k)]r -
= ^-2r(k)^p@) (837)
With k = k', P(t) gives the desired expression for the probability density that
the particle remains with momentum/zk at time t. In this sense, the lifetime of
our quasi-particle is given by/z/2F.
With reference to (8.17), we may conclude that the poles of the Green's
function in the complex CD-plane give the lifetime/z/2F and energy E + A of
a particle interacting with its surroundings.
Problems
5.1. (a) Integrate Heisenberg's equation of motion A.44) for a free-particle
Hamiltonian to obtain q(t) and p(t) as a function of <?(()) and p@).
(b) Show that in this case
[q(t),q(O)} = ~-t
m
Answer (partial)
Inserting
P2
H = y
2m
into A.44), we find
dp

430 5. Elements of Quantum Kinetic Theory
in— = [q, H\ = in —
dt m
q(t) = q@) H
m
5.2. A homogeneous beam of electrons has the density matrix
- = {a °
P \0 1-a
in a representation where Sz and 51 are diagonal. If (Sz) = 0.75/i, what is the
value of a?
5.3. (a) If a density matrix is initially diagonal, what are the values of the
derivatives dpnn/dt at t = 0?
(b) Is your answer to part (a) consistent with the Pauli equation B.28)? If
not, explain this discrepancy.
5.4. (a) Show that, in a pure state, p2 = p.
(b) Show that for spinors this property is obeyed by the matrix
cos2 0 ei<tn sin 0 cos 0
P ~ \ e~l<h sin 0 cos 0 sin2 0
providing the phases 0i and 02 are equal.
5.5. In a representation where S2 and Sz are diagonal, the density matrix for a
neutron in a homogeneous beam is
4
-
4
The beam enters a region of space containing a magnetic field B = @, 50,0).
(a) What are the values of (Sx), (Sy), and (Sz) before the beam enters the B
field?
(b) The Hamiltonian for a neutron in a B field may be taken to be of the form
= -fln ■ B
where jln is the magnetic moment of the neutron. Employing the Pauli
equation B.28), determine the probabilities of finding Sz = +(fi/2) and
S: = —(fi/2) after the beam leaves the B-field domain. Data:
where
lxN = 0.505 x 103erg/gauss
and

Problems 431
=A +x
Umax = 1 - A)
FIGURE 5.10. Graphical solution to Problem 5.9. Recall that the relevant domain is
jc>0.
5.6. Show that integration over momentum of the Wigner distribution gives the
spatial probability density. That is, establish the equality B.49).
5.7. Show that the integral of the Wigner distribution function over coordinate
space gives the momentum probability density. That is, establish B.50).
5.8. Show that the Wigner-Moyal equation B.70) goes to the form B.71) for the
canonical Hamiltonian given between these two equations.
5.9. Show that the inequalities C.78a) relevant to the relaxation-time integral
C.78) imply the solutions C.78b). Hint: Employ a graphical technique.
Answer
First label
A =
mu
hk
We must find the minimum and maximum of x (that is, q) corresponding to
-1 <0± < +1
where
0± = A ± x
This function is plotted against x in Fig. 5.10, from which we see that
•^-min —
qmax =2k(\±A)-2k

432 5. Elements of Quantum Kinetic Theory
5.10. Employing the expression for the relaxation time r(k) obtained in Section 3.6
and assuming free-particle motion near the Fermi surface, with C.40) obtain
an expression for electrical conductivity a, in terms of mean-free path, /, and
kF = pF/H, the Fermi wave vector.
Answer
Substituting
together with C.81)
into C.40) gives
dS =
lL —
r(k)
kFdQ
h2k2
2m
Im
~ hk
V
v2
2/ 7,2
e2lk
/
4n3h
Recalling Problem 3.15 (see also Appendix B, Section B.I) we write
1 4n
3 3
which gives the desired result:
e2lk2F
_
5.11. Employing the rule B.51) show that the first three moments of the Wigner-
Moyal equation B.71) return the classical fluid dynamical equations C.3.14),
C.3.18), and C.3.19).
5.12. Show that the s-particle density matrix E.15) may be written as
y\ t) = ^ h Tr(N)[pN(t)Gs(x\ y\ 0)]
5.13. Employing results of Problem 5.12, show that in a pure state, with pN =
(a) Np,(x,y) = Sxynx
(b) Af(N l)P(xXyy) 0$S ±88) Xl X2 X'X2
In part (b), the top term and + sign are relevant to bosons and the bottom
term and — sign to fermions.

Problems 433
(c) Show that (for x, ^ x2)
pi(x,,y2)
P2(xi,x2,y,,y2) =
Pi(x2,y2) Pi(x2,y2)
where (+, —) are relevant to bosons and fermions, respectively. Hint: For,
part (b) note that for fermions we may write
n{\
nB
n2
-n)
-n)
= n
= 0
= n
5.14. Show that the Fock-space Hamiltonian E.27) remains invariant under the
change
0//(x) —> 0(X)
Hint: Note that, in U, H is the Fock-space Hamiltonian in the Schrodinger
representation.
5.15. Obtain a quasi-classical kinetic equation valid to 0(fi3) from the Wigner-Moyal
equation B.71).
5.16. Write down the specific form of the first equation (B Yi) in the hierarchy C.33)
for the Wigner distributions. In what limit does this relation become a closed
equation?
5.17. (a) The thermodynamic properties of a system comprised of N molecules
in equilibrium at temperature T and confined to a fixed volume V is
contained in the partition function63
where the sum on r is over all states of the system, and gr is the degeneracy
of the rth state. Establish the following relations:
d In Z d In Z
(E) = , BP =
V ' dp H dV
S = -kBp2— f-lnZ
(b) The grand partition function, Q, is relevant to an equilibrium fluid under
the same constraints that hold for ZN but whose total number of molecules
may vary. It is given by
N
More generally, ZN = Trexp(—f$H). The expression given for ZN in the
Problem assumes a representation in which H is diagonal.

434 5. Elements of Quantum Kinetic Theory
where
z = exp/?/z
is fugacity and \i is chemical potential. Show that
dz
5.18. Consider a two-dimensional gas of noninteracting spin 1/2 fermions confinec
to an area A at temperature T and chemical potential \i.
(a) What is In Q for this system written as a sum over single-particle moment;
pi (In Q is defined in Problem 5.17.)
(b) Convert the sum in your answer to part (a) to an integral employing thu
following rule:
r, A P
d2p
p
where g is spin degeneracy.
(c) Obtain an expression for the average number of particles (N) as a functioi
of fugacity z and inverse temperature f$ = (kBT)~l.
(d) Obtain an expression for the Fermi energy EF from your answer to par
(c) in terms of n0 = (N)o/A, where the zero subscript denotes zen
temperature. Recall that EF = /x (OK).
Answers
(a)
p
2A C°°
(b) In Q = — / 2npdp ln(l + ze~fip /2m)
h2 Jo
(c)
mA
= —5-
nn ft
mA\n(\
(d) (^V)^o = (N)o = lim
p
mA
E
Thus we find
EF =
m
5.19. Show that in a mixed state
AAAB > -

Problems 435
where A A is the uncertainly
(AAJ = <(A - (A)J)
and A and B are heritian.
Answer
Define
S = A + izB
^ ^ ^
S+ = A - izB
where z is a real variable. It follows that
For a mixed state, with B.36), we write
so that
Choosing the value
we find
+(A2) >0
Now let A -► A - (A) and B
- E). Then
5.20. Show that the uncertainty relation AxAp > h/2 is consistent with the Wigner
distribution function.
Answer
From Problem 5.19, we obtain
AxAp>-\([x,p])\ = -
where
(AxJ = (x2) - (x)
However, as established in B.61a), both x2 and* obey Weyl correspondence so
that averages may be written with respect to the Wigner distribution function
as given by B.51).

436 5. Elements of Quantum Kinetic Theory
5.21. A system is in an eigenstate of its Hamiltonian. Show that the Wigner
distribution is constant in time for this state.
Answer
For an eigenstate, with A.10), we write
Substitution of this form into B.47) indicates that the Wigner distribution is
constant in time for an eigenstate of the Hamiltonian. Note that this conclusion
also follows from B.51):
(A) = / A(x,p)F(x,p,t)dxdp
For a stationary state, (A) is constant, so F(x, p, t) must be stationary.
5.22. Show that the Wigner distribution is real.
Answer
We refer to the defining relation B.46). Deleting vector notation, we write
F(x,p,t)cx f e2ipy/hp{x_,x+,t)dy
Taking the complex conjugate, we obtain
F*(x, p,t)cxj e-2ipy'hp\x_, x+, t)dy
p*(x_,x+,t) = (x_\p\x+y
= (x + y\p\x -y)
Changing variables in the above integral expression for F*(x, p, t) from y -»
—y gives F* = F.
5.23. A student argues that the Wigner distribution function is not consistent with
quantum mechanics for the following reasons. In the absence of interactins,
the equation of motion B.54) reduces to
d p d \
- + -■— F(x,p,f) = 0
at m ox/
As was discussed in Chapter 1, the general solution to this equation is given
by
F = F (x 1 - x0, p0)
\ m /
In particular, we may take F to be a delta function for which F / 0 only on
the system trajectory, in which case x and p may be specified simultaneously
in violation of quantum mechanics. Is the student's argument sound? Why?

Problems 437
Answer
It was established in Problem 5.20 that the Wigner formalism is consistent
with the uncertainty relation between x and p. This finding disavows solutions
B.54) that imply a classical trajectory.
5.24. Show that
fCp)]2)w = [x
Answer
First note that
2L '
Then employ B.61b)
= (A)w (cos^d) (A)w =
5.25. Show that the Wigner distributions given by B.46) is normalized to unity.
Answer
Working in one dimension, we find
f f F{x,p)dxdp = -^ f f f dydxdpe2ipy/n (x_\p\x+)
Integrating over p gives
/ / F(x, p)dx dp= dxdy 8(y) (x. | p \x+)
= I dx (x\ p \x) = Trp = 1
5.26. In quantum mechanics the continuity equation is given by
dp
where p is particle density
and J is particle current
J =
Working in a pure state, show that J and p may be written
m
= dpF(x, p)

438 5. Elements of Quantum Kinetic Theory
where F(x, p) is the Wigner distribution function.
5.27. Show that the z component of angular momentum
Lx = xpy - ypx
obeys Weyl corrrespondence.
5.28. Employing the projection representation B.36), derive the following expres-
expressions for the density matrix p(;c, x') for a canonical distribution [see B.13)]
of (a) free particles of mass m, and (b) particles of mass m confined to a
one-dimensional box of length L.
(~\ nfr y'\ — I p-f*2k2/2m ik{x-x')
(b) p(x,xf) = -
Z n=\
where
2m
2 nnx
= W-sin —
and Z is a normalization factor that ensures that Tr p = 1 (that is, the
partition function).
5.29. Write down the eigenvalue equation that the Fock-space ket vector given on
the right side of E.2) obeys.
5.30. The quantum mechanical s-body Green's function is given by G.13) and
G.14). With s = 1 we obtain
1 ~ .+
G,(x, t, y, t') = -0(t - t'){</>H(x, 003/(y, ?
in
With t' = t + £, where £ is an infinitesimal, show that
,y,r) = ±-(G1(x,y,r)>
where the F-space operator Gx is given by E.7).
5.31. (a) Show that
where 5 is defined by F.11).
(b) With the preceding result, establish F.12b), that is,
^ = Tr{AB]

Problems 439
5.32. Employing the fundamanetal time-development equation E.8), show that
G.33) follows from G.32).
5.33. If the operator R in G.1) is nonlinear, is the Green's function integral G.3)
still a solution of G.1)? Explain your answer.
5.34. Employing the relation G.15) for a time-ordered product, establish the
differentiation rule G.44).
5.35. Show that G.47) coupling the first two Green's functions may be transformed
to the first of the hierarchy for the Gs functions as given in Table 5.3. The
specifics of this tranformation are given in the text.
5.36. This problem addresses composite bosons; that is, bosons comprised of two
coupled fermions. Consider such a two-component composite system with
respective creation operators af and bf with the model Hamiltonian64
h = e0
where Eo and g are constants and a{ and b{ both satisfy fermion anticommu-
tation relations. Let
A = -L Y kbi, ^ = -4= Y b]h] (P36.2)
%
and
Ho = Eo JjS/a, + b%) (P36.3)
where No is the number of free-particle states.
(a) Show that (P36.1) may be written
H = Ho + gNofrA (P36.4)
(b) Show that
[A\A] = I-— (P36.5)
N
where A^ is the operator Ho/Eo.
(c) Given the above findings, what may be said of the bose quality of
composite bosons with strong coupling (g ^> EoO
Answer (partial)
(c) The noncanonical form of (P36.5) indicates that A+ is not a pure creation
operator and in this sense invalidates the interpretation of A1 A in (P36.4)
as representing a quasi-particle number operator. Thus we may conclude
64For further discussion on this topic, see A. K. Kerman, Annals of Physics 12,
300A961).

440 5. Elements of Quantum Kinetic Theory
that to within the consistency of the stated model the composite boson
does not satisfy boson properties.
5.37. Show that G.50) is equivalent to G.47).
5.38. Obtain the explicit time-dependent equation of motion for G(l, 2, 3; T, 2', 3)
[in a form parallel to G.47)] corresponding to G.56).
5.39. Write down the diagrammatic equation of motion for the four-particle Green's
function G(l, 2, 3, 4; 1', 2', 3', 4').
5.40. Write down the lowest-order diagrammatic expansion for the equation of
motion for G(l, 2; T, 2'), as given by G.52), in the topological form given by
G.54a).
Answer
= ti
[:.
" ,-, ]=•[=•
0
[•
The last six diagrams in this expansion stem from the three-body interaction
term in G.52). Consider, for example, reorientation of the term
Maintaining the format of G.52) gives the topologically equivalent form
3

Problems 441
which is the third from the last diagram in the given expansion.
5.41. The Green's function for a quasi-free electron propagating through a medium
is given by (with / denoting a constant length)
W(k - kF)
(a) What is the single-particle distribution for the medium? What is its
significance?
(b) What is the energy of the quasi-free particle?
(c) What is the lifetime of the quasi-free particle? Why is this so?
(d) What is the minimum excitation energy of a quasi-particle described by
the Green's function above?
5.42. (a) Show that
H(u)ftH(t2)} = ±t[4>)l(t2L>H(tl)]
(b) Does validity of the preceding equation depend on the commutation
properties on (j)H and (j)H ?
5.43. (a) Write the correlation function for velocity operator (v(f)v(O)), corre-
corresponding to particles of mass m, in terms of an integral over Heisenberg
operators ^(k, r), aH(k, t).
(b) Show that this correlation function may be written in terms of a two-
particle Green's function.
Answer
, , , ff dk,dk2 h
(a) <v(f)v@)>= // 2 —
J J BnKBnK m2
x k1k2(Sj/(k1, t)aH(ki, t)afH(k2, 0)aH(k2, 0))
(b) First recall
i^G2(i, 2; r, 2') = (f [a//(i)a//B)a;B/)a;(r)]>
with the results of Problem 5.42, the right side of the preceding equation
may be rewritten as
RHS = (f [aH(\yaiH{V)aH{2)a'H{2')])
■* /v + /v /v + /v
Now choose 1' = (k,, f+), 1 = (k,, r), 2' = (k2,0+), 2 = (k2,0) to to
obtain
(v(r)v@)> = // ' 2,
J J B7rKB7T.
x k,k2/^G2[(k,, r), (k2, 0); (k,, r+), (k2,
which is the desired relation.

442 5. Elements of Quantum Kinetic Theory
5.44. We have seen that in both the Chapman-En skog expansion [see C.5.37)] and
in the study of electron-phonon interaction [see C.58)] integral equations of
the following form typically emerge (for elastic interactions):
*(v) = A/(v) - /(v')Mv, vW
R(v) = Kf(\) (P44.1)
These relations define the integral operator K. With R(\) taken to be a known
function, the preceding is a linear inhomogeneous integral equation. Let us
work in Dirac notation with the inner product given by
(We assume that all functions are real.) Given that w is a symmetric kernel:
(a) Show that K is self-adjoint. That is,
(<j>\kf) = (k<p \f) = (f\ k<p)
(b) Show that K is a nonnegative operator. That is,
@ | K4>) > 0
(c) Consider the equation
@ | K4>) = D>\R) (P44.2)
Not all solutions to this equation are solutions to (P44.1). (Two distinct
vectors can have the same projection onto a third vector.) Show that of all
solutions, 0(v), to (P44.2), the solution that satisfies (P44.1) maximizes
@ | K4>).
(d) Show that solutions to (P44.1) render the functional
(P44.3)
minimum.
Answer
(a) We may easily establish
@ | v> \
from which the self-adjoint property of K follows. Setting \jr = <t> into
the preceding representation [with w(\, v') positive and multiplicative]
establishes part (b).
(c) Let 0(v) be a solution to (P44.1) and let ^r(v) be a solution to (P44.2) but
notto(P44.1). Then
o < (@ - i/f) | k(ct> -

Problems 443
k<p) + {f\kf) - <01 kp - & i k<t>)
Thus
| k<p) >
which was to be shown,
(d) To show this, we consider the functional variation
2@
[(8<j> \ K<t>){<t> \ R) - (<p \ K<P)(8<P \ R)]
Thus, for / [0] to be stationary, we must have
£ | R) =
Viewing this equation in function space, with |60) an arbitrary inflnitesial
element of the space, the preceding equality implies that
(P44.4)
We wish to show that this equation implies that
\k(f))=c\R) (P44.5)
where c is a constant. First note that substituting (P44.5) into (P44.4)
gives an equality. Now assume that (P44.4) does not imply (P44.5). Then
we may write
k\4>)=a\R) + \&) (P44.6)
where (8 \ R) = 0. Substituting this form into (P44.4) gives
\8) ((/> \ R) = \R) ((/> \ 8)
which implies that (8 \ R) ^ 0, whence the assumption (P44.6) is incon-
inconsistent, and we may conclude that (P44.4) implies (P44.5). Inserting the
solution (P44.5) into (P44.3) indicates that /[0] is insensitive to the con-
stant c, and we may conclude that /[0] is stationary when K |0) = \R),
in which case
/[0] = ;— (P44.7)
@ I K4>)
As we have shown, @ | Kcj>) is maximum for K |0) = |/?), so we
conclude that (P44.7) is minimum, which was to be shown. Note: The

444 5. Elements of Quantum Kinetic Theory
fact that (P44.3) is minimum affords a means of obtaining an approximate
solution to (P44.1): Choose a trial function comprised of known functions
and a number of arbitrary parameters. Insert this trial solution into (P44.3)
and vary the parameters until a minimum of this form is obtained. If the
trial solution was a good guess, then the solution so obtained is a good
approximate solution to (P44.1).65
5.45. Show that
p(t) = U'
where U is given by A.39).
Answer
Working in a pure state, we write
P(t) = \t)(t\
Recalling A.41), it follows that
p(t) = U* |0) @| U =
5.46. Employing the Fock-space Hamiltonian E.27), derive the equation of motion
E.28).
Answer
With E.8), we write
dt
Consider the kinetic energy term in H given in E.27). Forming the
commutator gives
4»
H(x)f dx'ftH{x')k(x'L>H(x') - f dx'fa(x'
= / dx'8(x - x')K(x')(f)H(x') = K(x)(pH(x)
which is the first term in the right side of E.28). A similar construction gives
the potential integral of E.28).
5.47. Obtain the equation of motion for GN given by E.29) and expressions for
^" and VN(xN).
65This technique has been employed in the calculation of electrical and thermal
conductivity in metals. For further discussion, see A. Haug, Theoretical Solid State
Physics, vol. 2, Pergamon, Elmsford, N.Y. A972).

Problems 445
Answer
Consider the case N = 2. Dropping the subscript on <pH, we write
iH—G2 = ^t^t^^
Ot
With E.28), we obtain
iH-G2 = [-A"(y,) - fi*(y,) - K*(y2) - Q*(y2)
Ot
where Q(x) is written for the integral operator in E.28). Generalizing this
finding indicates that
N
N
i = \
4.48. Show that the master equation B.28) implies the H theorem.
Answer
We label pnn = Pn and define
H = Y, ^ In Pn
n
Operating on B.28) with
gives
dU _
With
^A +\nPn
n k n k
the preceding reduces to
dH
n - Pnwnk)
k n

446 5. Elements of Quantum Kinetic Theory
To obtain the 7Y-theorem, first exchange indices in the second term on the
right. This gives
dH ^^_ . (Pn
dt .
k n
Now we exchange indices in the first term on the right to obtain
dH
k n
Adding the latter two equations and setting wnk = wkn gives
dH 1
dt 2 ,
k n
Noting that [recall C.3.49)]
(Pk-PH)ln[j)<0
gives the desired result
dU
<0
dt
5.49. (a) Show that the Master equation B.28) implies that Tr p is constant in time.
(b) What is the physical significance of this result?
(c) What property of the perturbing Hamiltonian does you proof of part (a)
depend on?
5.50. (a) A system comprises N identical, interacting particles. Write down relation
E.1) and E.2) in the energy representation, in which nE represents the
number of particles with energy E, and Et denotes the energy of the /th
particle.
(b) What is the coordinate representation of the (unsymmetrized) wavefunc-
tions for this system if the particles are non-interacting? Give your answer
in terms of the single-particle energy states, <pEi(Xi).
(c) What is your answer to part (b) if it is known that all particles of the
system are in the ground state?
(d) Under the conditions of part (b), what is the coordinate representation of
the Fock-space vector, |..., nE,.. .)F1
Answer (partial)
(a) \EU ■ ■ ■, En\ N) = ^- J^(±)p \EU • ■ ■, EN)
TV

Problems 447
(b)
Again consider a system of N interacting, identical particles. Wavefunctions
for this system exist in a Hilbert space spanned by products of 1-body func-
functions, {(pn.(Xi)}. In this representation, field operators, 0(x), 0f(x) may be
expanded in the form
0(x) =
(In the preceding and following, hats over operators are omitted.)
(a) What is the representation of the number operator N in this representa-
representation?
(b) What are the companion commutator relations to G.30)?
(c) Employing this representation, write down the form of the Af-body
Hamiltonian E.22) in second quantization.
(d) For which basis functions (pn(x), does the kinetic energy term in your
answer to the preceding question, reduce to a diagonal sum?
Answer
(a) N =
(b) [an,an> ]T = Snn>
(c) H = J2J2 / dxa>
/
, „ 2m
n n' k X
(d) Plane waves.
5.52. Show that if the collision time, r(k) is constant, then the expression for
conductivity C.41) reduces to
where / = rvF is mean fee path on the Fermi surface, and SF = 4nkF2.
Answer
We return to C.40) and assume that the electric field is in the ^-direction. It
follows that
Note further that
= Vf
m
There results
e2rvF n e2l

448 5. Elements of Quantum Kinetic Theory
which was to be shown.
5.53. Consider a dense fluid with zero isothermal compressibility (recall Problem
2.23).
(a) What is the normalization condition of the radial distribution function
under these constraints?
(b) Establish a relation between the radial distribution function, g(r), and the
quantum mechanical two-body wavefunction, t/^Xi, x2).
(c) What is the physical meaning of g(r)? What does your statement reduce
to for an isotropic fluid?
(d) Write down an expression for the number of pairs of particles in the
volume element Ai> <C V, in terms of g(r), where V is the volume of the
whole system.
Answer (partial)
(b) We note that the function |V^(Xi, x2)|2 *s a Jomt probability density. It
follows that the normalization of this wavefunction is given by
With the unit Jacobian change in variables (see Problem 2.14)
r = x, - x2
2R = X! + x2
the preceding normalization becomes
I
The radial distribution function is then given by
g(r) = V
which gives the correct normalization (under the said conditions)
g(r)dr= V
I
(c) The factor g(r) dv/ V gives the probability of finding a pair of particles
with one particle at the origin and the other at r + dr, where dx represents
a volume element. For an isotropic fluid, the preceding probability factor
is given by g(rLnr2dr/ V which represents the probability of finding a
pair of particles with one of the particles at the origin and the other in the
sperical differential shell of volume 4nr2dr.
(d) The number of pairs of particles, AN2, in A V is given by
where r' is a radial measure of A V.

CHAPTER 6
Relativistic Kinetic Theory
Introduction
is concluding chapter addresses elements of relativistic kinetic theory.1 This
mponent of kinetic theory plays a significant role in areas of astrophysics,
-electron lasing, and certain approaches to controlled thermonuclear fu-
:0n. The chapter begins with elementary notions of relativity including
stulates, Lorentz transformation, covariance, and invariance. Hamilton's
uations are written in covariant form. With these preliminaries at hand, the
scussion turns to a covariant description of relativistic kinetic theory. This
alism is employed to obtain a covariant one-particle Liouville equation,
hich when integrated yields the relativistic Vlasov equation for a plasma
an electromagnetic field. A covariant Drude formulation of Ohm's law is
ven in relation to a relativistic monoenergetic beam. The following section
Addresses Lorentz invariants in kinetic theory and a compilation of these forms
p presented in a table. The chapter continues with a derivation of the relativis-
pc Maxwellian. Relativity in non-Cartesian coordinates is discussed in the
Concluding section. Here the reader is introduced to the metric tensor and the
Motion of contra and covariant tensors. Lagrange's equations are derived, from
which a one-particle Liouville equation is obtained relevant to non-Cartesian
ipace-time.
For further discussion, See S. R. de Groot, W. A. van Leewen, and Ch. G. van
ert, Relativistic Kinetic Theory: Principles and Applications, North-Holland, New
fork A980); and P. G. Bergman, Introduction to the Theory of Relativity, Prentice-
~\ Englewood Cliffs, NJ. A955).

450 6. Relativistic Kinetic Theory
t
ct
FIGURE 6.1. The world line of a particle moving in one dimension.
6.1 Preliminaries
6.7.7 Postulates
There are two postulates of special relativity. These are as follows:
1. The laws of physics are invariant under inertial transformations.
2. The speed of light is independent of the motion of the source.
The first postulate states that the result of an experiment in a given inertial
frame of reference is independent of the constant translation motion of the
system as a whole. An inertial frame is one in which a mass at rest experiences
no force. Thus there is no absolute frame in the universe with respect to which
motion of an arbitrary inertial frame is uniquely defined. Only relative motion
between frames is relevant.2
Concerning the second postulate, consider a light source fixed in a frame
S. The frame moves relative to the observer in a frame S' with speed v. The
observer measures the speed of light c, independent of the speed v. This situ-
situation is evidently equivalent to one in which S' moves relative to S with speed
v. Thus the speed of light is independent of the motion of the receiver, as well
as that of the source. This conclusion is alien to our intuitive picture of either
wave or particle motion.
6.7.2 Events, World Lines, and the Light Cone
Einstein defined an event as a point in space-time coordinates. The locus of
events of a particle is called the world line of the particle. For one-dimensional
motion, the world line of a particle is a curve in (jc, ct) space, where c is the
speed of light (see Fig. 6.1).
Of particular interest in the study of relativity is the concept of the light
cone. This is the world line of the leading edge of a light wave stemming from
2Unless otherwise stated, frames referred to in this description are inertial.

6.1 Preliminaries 451
t
ct
0 Elsewhere
\ *
BGURE 6.2. Light pulse is initiated at 0. Points in the domain marked "elsewhere"
Unnot be reached from 0 at speed less than c.
f source switched on at a given instant at a given location. The notion of past
pud future may be defined with respect to the light cone (see Fig. 6.2).
$.1.3 Four-Vectors
An event at a given location x and time t may be described by the four-
Bmensional vector3
x = (x, ict)
•rtiich is called & four-vector. The momentum four-vector is given by
iE
A.1)
P=
A.2)
Here we have written
E = ymc2 = mc2 + T A.3)
for the total energy of a particle (in the absence of potential) of rest mass m,
ith kinetic energy T. The parameter y is written for [recall A.1.19a)]
l
-
C
A.4)
\-P2 ^ c
Note that y increases monotonically from 1 to /3 = 0 to oo at /3 = 1. The
Ifclativistic momentum three-vector is given by
p = ymv,
v ==
dx
It
A.5)
An alternative formalism in terms of the metric tensor avoids use of imaginary
. See Section 6.4.

452 6. Relativistic Kinetic Theory
t
t.
S'
FIGURE 6.3. The frame S' moves at velocity v with respect to the frame S.
where t is time measured in the lab frame.
Let us write /?M for the components of the four-vector p. Thus
iE
= E=
me
A.6)
In this notation we may write
Pu = mut
A.7)
where wM is the velocity four-vector,
y/i dX/i
-i*. i -M A-8)
m ax
and x denotes proper time. This is the time measured on a clock attached to
the moving particle (described in Section 1.5).
Three other important four-vectors are4
" = (J,icp) A.9a)
A.9b)
A.9c)
In these expressions, J is current density, p is charge density, A is vector
potential, O is scalar potential, k is a wave vector, and to is frequency.
6.1.4 Lorentz Transformation
Consider that a frame S' moves at constant speed v in the z direction with
respect to a frame S, as shown in Fig. 6.3. It is readily shown from the two
postulates above that if AM is a four-vector in S an observer in S' observes the
components
4The manner in which electric and magnetic fields are related to vector and scalar
potentials is described in Problem 1.22. See also Section 2.3.

6.1 Preliminaries 453
where
/I 0 0 0^
0 10 0
0 0 y
L =
A.11)
\0 0 -iyfi y
Note that L is orthogonal; that is
L = L~l A.12)
where L is the transpose of L. Note further that
detL=l A.13)
(see Problem 6.1). Thus L effects a rotation in complex four-dimensional
space. Consequently, the "length" of a four-vector is preserved under a Lorentz
transformation. That is
Let us show this formally:
For example, consider the length of the momentum four-vector,
PuPn = Y2m2v2 - y2m2c2 = -m2y2c2(l - fi2) = -m2c2 A.15a)
This returns the useful relation
E2 = c2p2 + mV A.15b)
For x, we find
jcmjcm = x2 - c2t2 A.16)
Such entities, which remain invariant under a Lorentz transformation, are called
Lorentz invariants. Note in particular that the Lorentz invariant jcmjcm as given
by A.16) is a reiteration of the second postulate stated above that the speed of
light is c in all frames.
The Lorentz matrices A.11) have the following group property.
r / o \T(Q \ T (P. \ (\ \H\
where
a i /Q.
A.18)
lP2
(see Problem 6.2). Note that A.18) precludes the speed of any object from
exceeding c. Consider that one of the frames moves with relative speed c, so
that, for example, fi\ — 1. Then A.18) returns the value ^n— 1. Furthermore,
in that L@) = /, the identity operator, and L = L, we see that the Lorentz
transformations comprise a group, called the Lorentz group.

454 6. Relativistic Kinetic Theory
6.7.5 Length Contraction, Time Dilation, and Proper Time
With relative interframe motion again confined to the z direction, Lorentz
transformation of the even four-vector gives
*)(*') A19)
Jet' I ' \ -id 1 / V ict
We find
A.20a)
ict' = iy(-Pz + ct) A.20b)
Let a rod of given length Azf lie fixed in a frame that we label S'. The frame S'
moves with speed fie relative to the frame S. At a given instant, the length of
the rod is measured in the S frame. This means that the locations of the ends
of the rods are measured simultaneously in S. Calling the length so measured
Az = Zb - za, with tb — ta A.20a), gives
AZ = -Az< Az A.21)
Y
The rod moving past the frame S is measured to be shortened by an observer
in S. This is the phenomenon of length contraction.
Consider next a clock that is at a fixed location in the moving S' frame
(z'b — z'a). To find the manner in which intervals x' on this clock are observed
in S, we write the inverse of A.20b):
There results
>T/ A.22)
Thus an observer in S concludes that intervals on his clock, r, are longer than
those on the S' clock or that the Sf clock is "running slow."
An important parameter in relativity is that of proper time. The proper time
of a particle is the time measured on a clock that moves with the particle. Thus,
if t denotes time in the lab frame and r proper time, we write
dt = ydr A.23)
Comparison with A.22) reveals that we have identified proper time with the
single clock fixed in Sf.
6.1.6 Covariance, Hamiltonian, and Hamilton's Equations
It is important to the first postulate that laws of physics be written in a manner
that guarantees that invariance under Lorentz transformation. Relations so writ-

6.1 Preliminaries 455
ten are said to be covariant.5 For example, the covariant forms of Hamilton's
equations are given by6
dxk dH dpx dH
dx apx
For a free particle, the covariant Hamiltonian is given by
H = ^^ A.25)
2m
For a charged particle in an electromagnetic field,
2m
A.26)
where AM is the four-vector potential A.9b).
• In three-vector form, the relativistic Hamiltonian is given by A.1.19). When
working with this form, Hamilton's equations must be written with respect to
the lab time t, as in A.1.12O (see Problem 6.3).
6.1.7 Criterion for Relativistic Analysis
JPrior to our discussion of relativistic kinetic theory, let us ask the question,
when is it necessary to use this formalism? The answer to this question is
given in terms of a simple rule of thumb. Compare the rest-mass energy of
particles whose kinetic theory is being studied to their kinetic energy. Thus,
with A.3), the criterion for nonrelativistic theory is written
T = (y - \)mc2 « me2 A.27a)
or, equivalently,
l<y«2 A.27b)
or, more simply, u«c. The left inequality in A.27b) stems from the definition
ofy.
5Not to be confused with covariant and contravariant vectors discussed in
Section 6.4.
6For further discussion, see H. Goldstein, Classical Mechanics, 2nd ed., Addison-
Wesley, Reading, Mass. A981).
7See, for example, A. O. Barut, Electrodynamics and Classical Theory of Fields
and Particles, Dover Publications, New York A980).

456 6. Relativistic Kinetic Theory
6.2 Co variant Kinetic Formulation
6.2.1 Distribution Function*
We consider a distribution Tix^ /?M) with the following properties:
^>0 B.1a)
T -► 0 as /?M -► oo B.1b)
It has the normalization
p — constant B.2)
where Ex is a hypersurface in four-dimensional x-space, with differential
element do^.
The definition of J-ix^p^) is as follows. The product
d4p B.3a)
represents the probability that the world line of a particle intersects the
hypersurface element da^ at jcm about the point /?M. Furthermore, the product
•H*m> Pn)Pn d°n d*x B-3b)
represents the probability that the world line of a particle intersects the hy-
hypersurface element do^ at /?M about the point jcm, where do^ represents an
element of hypersurface in four-dimensional p-space. As in A.8), the dot in
B.3b) represents differentiation with respect to proper time.
In B.3a), the inner product
'M
dx4) + • • • + u4(dx\ dx2 dx?) B.4)
where, for example, dx2 dx^ dx4 is the element of the hypersurface
parallel to u\.
The relation of J-'ix^, p^) to the classical distribution /(x, p, t) is given by
/•oo
= / J-dp4 B.5)
t/—00
With the normalization condition B.2), we write9
da^ dp dp4 — constant
8B. Kursunoglu, Nuclear Fusion, 1 A961). This formulation takes the four com-
components of p^ to be independent. An alternative description that directly incorporates
the constraint A.15a) was given by Y. Klimontovich, JETP, 37, 524 A960).
9Choosing EA to be a plane normal to the time axis B.6) gives / yf d3x d3p =
constant, which in the nonrelativistic limit returns the classical normalization
/ / d?x d3p = constant.

6.2 Covariant Kinetic Formulation 457
t
t
x-space
p - space
FIGURE 6.4. A three-dimensional depiction of four-dimensional x and p spaces and
closed X)* and Ep hypersurfaces. The net number of world lines entering the closed
fcypersurfaces Hx and £p is zero.
Un don d3 p = constant
B.6)
6.2.2 One-Particle Liouville Equation
Consider that the hypersurface Ex in the normalization B.2) is a closed hyper-
hypersurface. In this case the net number of world lines entering Ex is zero. With
the definition B.3a), we find
i
— 0
See Fig. 6.4.
Gauss's theorem in four dimensions permits B.7) to be written
dx,
A similar argument stemming from the definition B.3b) gives
dAx
I
= I dAx
= 0
Since T =
, /7M), we may write
dT .
dz
ox,
ith Hamilton's equations A.24), the preceding equation becomes
fl r A r X n h n
dx dXa dp n
B.7)
B.8)
B.9)
B.10)
B.11)

458 6. Relativistic Kinetic Theory
Integration of this equation gives
f dJL d<x d<p = f ^Ji d4x d4p + f
J dx J dx^ J
With B.8) and B.9), the right side of B.12) vanishes, and we obtain
dT
I
dx
Passing to zero volume gives
dT
d*x d*p = 0
dx
or, equivalently, with B.10)
= 0 B.13)
=0 B.14)
This equation represents a covariant one-particle Liouville equation. The term
following /?M will be related to electrodynamic field variables, which in turn
will be related to the distribution through the current. With these substitutions,
the preceding equation is more properly termed the Vlasov equation.
6.2.3 Covariant Electrodynamics
To apply the above finding to a plasma, we must first write Maxwell's equations
in covariant form. To these ends, we introduce the electromagnetic field tensor
Fvk. For /, j running from 1 to 3,
Flj=sljkBk B.15a)
where Bk represents the A:th component of magnetic field and s^ is the Levi-
Civita symbol [see beneath (A. 10')]. With £, written for the y'th component of
the electric field, we write
F4j = -Fj4 = i£j B.15b)
These terms correspond to the matrix
B.16)
0 B3 -B2 -i
-B3 0 B\ -i£2
B2 -Bx 0 -i
\ IC\ IC2 IC3 U
The covariant form of B.15) is given
dAv _
' (L OXV
F,v = ^1 - lllH B.17)

6.2 Covanant Kinetic Formulation 459
fcvhere AM is the four-vector potential A.9b). In these variables, Maxwell's
Equations assume the form (cgs)
dFuv An
^L J B.18)
Where Jv is the four-current A.9a). The preceding covariant relation gives the
|wo Maxwell equations:
1 dS 4tt
y-S = 4np, VxB = —J B.18a)
C dt C
The covariant structure of the remaining Maxwell equations is
dF,,v d F\u d Fv\
+ —— + —- = 0 B.19)
dxv d
which gives
V-B = 0, Vx5+ =0 B.19a)
c dt
The covariant form of the Lorentz force law is
e
me
(see Problem 6.4).
B.20)
6.2.4 Vlasov Equation
Combining the Lorentz force law B.20) with the equation of motion B.14)
gives
oj- e
u + F p
3jcm me
or equivalently,
dT e
dx^ c
This scalar equation has the explicit form
dT dT / e \ dT
dt dx ' \ ' c ^ X / dp
oj-
dT
dPn
e
■ + -
c
-0
(v-£)f -0
dp4
B.21a)
Integrating over p4 and recalling the connecting relation B.5) gives the desired
form (with components of p taken to be independent):
f+v|/ + (^+%xB)^ 0 B.22)
f+v|/ + (+
dt ox V c

460 6. Relativistic Kinetic Theory
(see Problem 6.5). Note that the sole difference between the form of B.22)
and the classical Vlasov equation B.2.30) is contained in the presence of p in
B.22), which is the relativistic momentum A.5).
One further note is relevant to B.2). For this equation to be properly termed
the Vlasov equation, the fields E and B must be self-consistently defined.
That this is so in the present case follows from the fact that B.22) includes
the force equation B.20), with field variables coupled to the current density
through B.18). The picture is made complete if current and charge densities
are expressible as functionals over the distribution function. These relations
are given by
J(x, t) = eN f -?- /(x, p)dp B.23a)
J my
icp(x, t) = eN — /(x, p)dp B.23b)
J my
where TV is the total number of particles in the system. Recall that p/my = v,
the velocity in the lab frame. It will be shown that /(jcm, p) and dp/y are both
relativistic invariants.
6.2.5 Covariant Drude Formulation of Ohm's Law
A monoenergetic beam of electrons of number density n and velocity v with
respect to the lab frame has the current density
J = en\ B.24a)
If n0 is density measured in a frame moving with the beam, then
n = yn0 B.24b)
due to length contraction along the direction of motion as given by A.21). With
A.7), we may write
u = y\
and the current B.24a) may be rewritten
J = enou B.25)
With A.7), the preceding expression for current may be written
J = p B.26)
m
which is the vector component of the four-vector [see A.9a)]:
en°
m
This form gives the invariant
JTU = -e2nlc2 B.27a)

6.2 Covariant Kinetic Formulation 461
We may write down a covariant form of Rayleigh dissipation D.2.32) in
terms of proper frequency v$. This is frequency measured on a clock moving
with the beam. The equation appears as
B-28)
Combining this relation with Lorentz force law B.20) gives
dx me
The fourth component of this equation reads
dE
— =eu-£-v0E B.30)
dx
That is, in a frame moving with the beam, energy increases at a rate at which
the electric field does work on the particles and decreases due to collisions at
a rate proportional to the energy.
To obtain the covariant form of Ohm's law, first we assume steady state and
setdpn/dx = 0. Combining B.29) and B.27) then gives the desired relation:
ua
Jn = &Fa{i— B.31)
c
where we have written
a = ^ B.31a)
The vector component of B.31) returns the standard form of Ohm's law,
B.32)
which, we see, is a relativistically valid equation. Note in particular that J,
£, and B in this equation are lab-frame values and that a contains rest-frame
parameters.
The fourth component of B.31), with 74 given by B.27), returns the energy
balance equation [right side of B.30)]
vqE = eu • £ B.33)
6.2.6 Lorentz Invariants in Kinetic Theory
First we note
F(p)d3p = d3p f F(x, p)d3x B.34)
Jv
represents the number of particles in the spatial volume V and in the momentum
volume d3 p about the point p. [Note that F(x, p) is normalized to total number

462 6. Relativistic Kinetic Theory
of particles.] As this is a pure number, it is the same in all reference frames
and thus may be termed a Lorentz invariant.
The same argument leads us to conclude that
F(x)d3x = d3x I F(x, p)d3p B.35)
is likewise a Lorentz invariant.
Consider next how the volume element d3 p transforms under a Lorentz
transformation. To answer this question, we note a general rule of invariance:10
if CM and DM are two four-vectors, then the ratio of two parallel components,
/Di is a Lorentz invariant.
With A.15a), we see that d3p is an element of surface on the hypersurface
/E\2
2 I \ 2 2
[) = m c
2 I \ 2
= P ~ [-) = ~m c
Thus we may write
dpx dpy dpz = da4 B.36)
where da4 is a differential of hypersurface in four-dimensional p-space. With
the above-stated rule of invariance, we conclude that
da4 d3 p d3 p
— = -J- = —£- B.37)
p4 p4 iE/c
is a Lorentz invariant. That is, under a Lorentz transformation we find
d2p d3p'
iE/c iE'/c
or, equivalently,
B.38a)
d3p' = —d3p B.38b)
y
The invariant d3 p/E may be cast in four-dimensional form as follows. First
we write (see Problem 6.8)
/" B.39a)
4f
i E/c P4
which is evidently a Lorentz invariant. In this expression we are treating the
four components of /?M as independent. As in B.22), the delta function in
B.39a) maintains the constraint A.15a). As the integration in B.39a) is only
10L. D. Landau and E. M. Lifshitz, Classical Theory of Fields, 4th ed., Pergamon
Press, Elmsford, N.Y. A975).

6.2 Covariant Kinetic Formulation 463
S'
(x,p)
Lab
S°
Rest
FIGURE 6.5. Particles described by B.41) are at rest in S°.
over Pa, we obtain the alternative form
d3p
iE/c
I
m c )dp4
With the invariants B.34) and B.37) at hand, we write
d2
F(p)d3p =
This equality indicates that EF(p) is a Lorentz invariant. That is, under a
Lorentz transformation
B.40)
In addition to the distribution invariants B.34) and B.35), it is evident that
F(x,p)d3xd3p
B.41)
is likewise a Lorentz invariant. To discover the manner in which F(x, p) trans-
transforms, we must examine the manner in which d3x transforms in kinetic theory.
To these ends, we note that in relativistic kinetic theory it is necessary to dis-
discern between three frames of reference. The independent variables of F(x, p)
are defined with respect to the lab frame, S. An inertial frame, S', moves rel-
relative to the lab frame, where measured variables have values xr and pr. An
additional frame, 5°, is introduced such that particles with momenta defined
by B.41) are at rest (see Fig. 6.5). The proper volume d3x° lies in this frame.
Recalling the description leading to A.21) indicates that the primed frame of
that expression is the frame of the proper volume in the present discussion.
Since both S and S' move relative to S°, where the differential of particles

464 6. Relativistic Kinetic Theory
B.41) are at rest, our previous result A.21) gives the relations
d3x = - d3x° B.42a)
y
d3x' = — d3x° B.42b)
y'
Thus we find
yd3x = y'd3x B.43)
so that yd3x is a Lorentz invariant. With B.41) and B.43), we obtain
F(x, p)d3x d3p = F'(x', p')rfV d3p = F'(x\ pf)d3x d3p B.44)
We may conclude that both the distribution
*Xx, P)
as well as the phase volume
d3xd3p
are Lorentz invariants. A compilation of these Lorentz invariants is given in
Table 6.1. The time dependence of these distributions is tacitly assumed. Recall
also that F is normalized to total number of particles in the system at hand.
The volume elements d4x and d4p are invariant due to the orthogonal property
of the Lorentz transformation, det L = 1 [see A.13)].
6.2.7 Relativistic Electron Gas and Darwin Lagrangian
Let us consider a collection of TV charges interacting with each other through
the Coulomb interaction. Charge and current densities due to the y'th particle
are given by
B.45)
, 0 = qjVj(t)8[x - xj(t)] B.46)
where Xj(t) and \j(t) are written for the coordinate and velocity of the y'th
particle of charge qj.
We work in the Coulomb gauge1 J defined by the condition
V • A0) = 0
where A(y) is the vector potential due to the jth particle. In this gauge, potentials
are given by
[P°)(X'J) B.47)
[dx'
J
The Lorentz gauge was defined in Problem 1.22.

6.2 Covariant Kinetic Formulation 465
TABLE 6.1. Lorentz Invariants in Kinetic Theory
A. Distribution Functions and Volume Elements
F(x)d3x\ F(p)d3p; F(x, p)d3x d3p
F(p)£; F(x, p)
dAx\ dAp\ d3xd3p\ —; Ed3x
E
d3p
iE/c
J p4
B. Invariants for Af-body Systems
N N
1 = ]
or, equivalently/
N
N
With
N
Ptotal — / J Pi
N
E ,-
■'-'total —
i=\
we may write
N
Zp>
?
N
E
= p
'total
total
= -M2c2
where M is called the invariant mass.
C. Collisional Invariants
N
N
1=1
With N = 2, we find5
= -t
= —u
a Contra and covariant notation is described in Section 6.4.
b The terms s, t and u are called Mandelstam variables. Note
in particular that only two of these variables are independent
since s + t + u = 2c2(m\ + m\) (see Problem 6.24).

466 6. Relativistic Kinetic Theory
A0)(x,r)= / dx!dt'
\x — x'\
l d
Jnc'dt'
^ t')
B.48)
With these parameters at hand, the relativistic Lagrangian is written
N
L(xyv,vyv,r) =
\
N
-mc
'-3-E
X
B.49)
We wish to obtain the form B.49) to 0A/c2). Toward this end, as is evident
from B.49), it is necessary only to obtain the vector potential to 0A /c), which
in turn is obtained from B.48). With x written for x,, we write
A0)(x, t)= I dx! dt'
S(t - tf)
c\x — x'\
ar
Performing the t' integration, we find
= t dx'
J
t
J
c|x — xr|
Let us evaluate 0(x, r). Inserting B.45) into B.47) gives
0O)(X, 0 =
+ 0
|X -
B.50)
B.51)
Substituting this expression together with B.46) for J(j) into B.50), we obtain
c\x — Xj(t)\ Ancdt
x — Xr
V-
1
|X'-X;(OI/ \C
B.52)
Now note that
l
[x' - xj
dt \|x' - xj(t)\) |x'-x7(r)|3
Substituting this result into the integral term of B.52) (labeled I) gives
I
~L fax'
Anc J
\x-x'\
Changing variables,
y = x/-xy@

6.3 The Relativistic Maxwellian 467
and integrating by parts, we obtain
Aizc J y3 \rj -y|
Here we have labeled
r
Tj =X-Xj(t)
Integrating the preceding expression gives
B.52a)
which when differentiated gives
1 = -
2c
\j(t)
B.53)
Substituting this result into B.52), we obtain
AU)(x, t) =
1L
2c
V;@
B.54)
Combining this result with B.51) and substituting into the Lagrangian B.49)
gives
N
.ao = E
—me
vt
B.55)
2c2
(vf-
where vf- = V/ (t) and rtj is given by B.52a) with x rewritten xf- [as in B.49)]. The
preceding expression is the desired structure of the Lagrangian of an TV-charged
particle gas valid to O(\/c2). It was first obtained by C. G. Darwin.12
6.3 The Relativistic Maxwellian13
.5.1 Normalization
With A.1.19), we write
H = cy/m2c2 + p2
12C. G. Darwin, Phil Mag. 39, 537 A920).
13J. L. Synge, The Relativistic Gas, North Holland, Amsterdam A957).

468 6. Relativistic Kinetic Theory
2 /
= me J 1
V
mlcl
2 I
= me V 1 +
where
1 -
With C.1), the Maxwellian may be written
= Ae-H(tl)'kBT
me
= Aexp
The normalization of /0 may be written
f fod3fi=l
or, equivalently,
4ttA
^0
OO
exp
mr
= 1
We wish to evaluate the normalization parameter A. Let us define
s2 = 1 + /x2
s ds = \id\i
so that C.4b) becomes
47T.4
/"
Noting that
permits C.5) to be integrated by parts. There results
47rArac2
We recall the integral expression for the Bessel function ^(zI4
/ /^XVT^/I /O\ /*OO
kaz) = K:i , \ L: J e-»{t2 - ly-Mdt
1\ n'
Re|v + -J>0, |argz| < —
C.1)
C.2)
C.3)
C.4a)
C.4b)
C.5)
C.6)
C.7)
I4I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products,
Academic Press, New York A965), Eq. C), section 8.432.

6.3 The Relativistic Maxwellian 469
15
The asymptotic value of Kv(z) for large z is given by
Kv(z)
Employing C.7) permits the normalization C.6) to be written
A =
me
1
T l47tK2(mc2/kBT)_
Our complete relativistic Maxwellian then appears as
fo(u) =
mc2/kBT
4nK2(mc2/kBT)
exp
me2
kBT
6.3.2 The Nonrelativistic Domain
With the definition C.2), we see that
1 + /x2 = 1 +
which, in the nonrelativistic domain, gives
1 + /x2 ~ 1 +
1
Setting
me
z =
permits C.10) to be rewritten
/o(AO =
C.8)
C.9)
C.10)
C.11)
4nK2(z)
Inserting the asymptotic formulas C.8) and C.11) [corresponding to kBT
me2] into the preceding result gives
C.12)
,-mv2/2kBT
C.13)
15
Ibid., Eq. F), section 8.451.

470 6. Relativistic Kinetic Theory
In this same limit,
3 d2v
d
and J3/x/o(/x) —> d3vf0(v), where /o(u) is the classical Maxwellian C.3.58).
The relativistic Maxwellian C.10) may be employed with velocity differen-
differential through the transformation
(mcfd3 [i = d3p = m3y5d3v
(see Problem 6.10).
6.4 Non-Cartesian Coordinates
We close this chapter with a brief introduction to relativity in non-Cartesian
coordinates.
6.4.1 Covariant and Contravariant Vectors
Vectors that transform as the gradient are called covariant vectors and are writ-
written with a subscript, for example, Ba. Vectors that transform as displacement
are called contravariant vectors and are written with a superscript.16 Thus, for
example, coordinate components are written xa.
Consider the transformations of coordinates
x'11 = x^(xa) D.1)
For differential of displacement, we write
3x/fl
dxa D.2)
dxa
Thus contravariant vectors transform as
Aa D.3)
For the gradient of a scalar function <p, we obtain
D.4)
dxa
Thus covariant vectors transform as
16In the following section, we find that a contravariant vector may be transformed
to a covariant vector, and vice versa. A more basic definition of these vectors is given
in terms of the transformation equations D.3) and D.5).

6.4 Non-Cartesian Coordinates 471
A contravariant second-rank tensor Aa^ (with 16 components) transforms
as
D.6)
dxv
A contravariant second-rank tensor transforms according to
dxfv
D-7)
whereas the mixed second-rank tensor Kp transforms as
'a dxv
dx dx
We take the inner product of two vectors to be defined as the product of their
contravariant and covariant forms
B A = B^ D.9)
Note in particular that the
B
inner
A' =
—
—
product
dxa dx/fl
dx'v dxv
dxa
R Av
Dar\
dxv
BVAV = B
BaAv
A
D.10)
is a Lorentz invariant.
6.4.2 Metric Tensor
In A.16), we wrote the invariant internal
(dsJ = (dxJ - c2(dtJ D.11)
This expression is a special case of the more general relation
(dsJ = gapdxa dx? D.12)
Here we have introduced the metric tensor gap = gpa. In Cartesian space, such
as employed in the earlier components of this chapter, gap is diagonal with (no
summation)
gap = gaaSap D.13a)
where
£11 =£22 = £33 = 1, £44 = -1 D.13b)
The contravariant four-vector event is then given by (jc, ct), whereas the
covariant event four-vector is given by (x, —ct). Thus
rMr _ v2 _ s*2t2

472 6. Relativistic Kinetic Theory
An additional property of the metric tensor for flat space is given by
8°" = 8afi
More generally,
vpP D.14)
where 8% is the four-dimensional Kronecker delta symbol.
The procedure for changing an index on a tensor from contravariant to
covariant, or vice versa, is obtained by contraction with the metric tensor.
Thus, for example,
Aa =
T7(X ^CKV A / A 1 C\
tn = Q /!,,« D 1 l)
ctB av By A
p = g gPY Avy
6A3 Lag range's Equations
With the preceding preliminaries at hand, we return to the relativistic action
integral [recall A.1.4)]. In arbitrary geometry, it is given by
S= -mCiJ-gap{x)xaxPd\ D.16)
where X parameterizes the curve
xfl=xfl(X) D.16a)
and
dx11
x* = D.16b)
d\
The effective covariant form implied by D.16) is given by (see Problem 6.23)
L = -mCyJ-gapxaxP D.17)
As noted previously, any function of L (satisfying conditions described in
Problem 1.46), when substituted into the action integral returns Lagrange's
equations. Thus, in place of D.17), we work with
X? D.18)
Elements of Lagrange's equations are then given by
dL '" D.19a)
= — ( gBu) *^M D.19b
2 \dx<* 6^J

6.4 Non-Cartesian Coordinates 473
d I dL \ ,,R dgag „ o
— WtRaBX i 7W XX D.1VC)
Inserting these into Lagrange's equations gives
This is the equation of motion for a free particle in a geometry described by
the metric tensor gap. In flat space, components of gap are given by D.13), and
D.20) reduces to
mgapxp = mgaaxa = 0
(no sum over a). We obtain
m'x\ = mx2 = mJc3 = 0, — mx4 = 0
|f in the last equation we choose the constant of integration to be me, then we
find t = k, and the first three equations return conservation of momentum.
6.4.4 The Christoffel Symbol
The equation of motion D.20) may be cost in a more symmetric form. First
pote that it may be rewritten
Multiplying this equation by gav gives
3x" 3x^ 3jc^
There results
where T^u is the Christoffel symbol:
The relation D.21) is the equation for a "free" particle in a geometry described
by the Christoffel symbol. We may also view it as an equation for the geodesic in
given geometrical environment. A geodesic is the shortest internal between
points in the space-time continuum. This may be seen from the fact that
D.21) follows from rendering the action D.16) stationary.

474 6. Relativistic Kinetic Theory
6.4.5 Liouville Equation
With the effective Lagrangian D.18), we may construct the corresponding
Hamiltonian. First note that canonical momenta are given by
T ruga?* D.22)
dqa
where, we recall, a dot represents differentiation with respect to the parameter
X. Thus we obtain
H = pax" -L = E^Il D.23)
2m
[Note that the preceding reduces to A.25) in flat space.]
The one-particle Liouville equation is then written
ok
dH df df dH
=
dxa dpa dxa dpa
Inserting the Hamiltonian D.23) into this expression gives
9/
L
D.24)
+ =0 D.25)
dk m dxa 2m dx? dpy
which is the one-particle Liouville equation appropriate to non-Cartesian
space. As noted in Section 1.4.5, the most general solution of the preced-
preceding equation is an arbitrary function of solution to its characteristic equations.
These are given by
xa = ££, = SH D
m dpa
j, P-E!L*Jl = _™ D.25b)
y 2m dxy dxy
As the inserts on the right of these equations indicate, they are Hamilton's
equations derived from the Hamiltonian D.23).
The formalism introduced in this section finds application in general
relativity, where space-time curvature derives from large energy-mass
concentration.17
Problems
6.1. Show that the determinant of the matrix L of a Lorentz transformation is unity.
17For further discussion, see C. W. Misner, K. S. Thorne, and J. A. Wheeler,
Gravitation, W. H. Freeman and Co., San Francisco A973).

Problems 475
6.2. If L(Pi) and L(/}2) represent two Lorentz transformations with respect to a
common reference frame, then show that
Pl2 l+
Note that this example establishes the group property of Lorentz transforma-
transformations.
6.3. With the relativistic Hamiltonian for a charged particle in an electromagnetic
field given by A.26), show that Hamilton's equations give
dp „ e
= e£+-\xB
e£+
dt c
dx 1 / e
= V
dt ym V c
6.4. Show that the vector components of the covariant form B.20)
dx me M r
return the Lorentz force law.
6.5. (a) Show that B.21) and B.2la) are equivalent.
(b) Integrate B.21a) over p4 to obtain the relativistic Vlasov equation B.22).
6.6. Show that J4 as given by B.31) is the same as that given by A.9a).
6.7. What is the relation between the conductivity parameter with proper values,
_ e2n0
vom
and a with lab-frame values, a = e2n/vml
6.8. Show that
d3p d? p { ~ ~ -
-f=i-r S(pl+m2c
E c J M
Answer
First write
E\2
=[-\ +p2A
With the expression
~ -a2)=— [8(y - a) + 8(y + a)]
the preceding integral over p4 gives
2

476 6. Relativistic Kinetic Theory
6.9. Obtain an expression for the relativistic Maxwellian for a gas of noninteract-
ing particles of rest mass m confined to a volume V that is immersed in an
externally supported potential field OQc).
6.10. Show that
d3 p = m3y5d3v
Answer1*
With
Pi =
we obtain
dpt
l- = ym[
*u
= ym\8ij +{y2 - 1)-
Introduce the matrix
C2
vv
Q = -
c2
which has the properties
We may then write
/V A. A.
R = yn[I + (y2 - \)Q]
Thus
det R = y3m3 det[/ + (y2 - \)Q]
Now define
K = \n[I+(y2-l)Q]
so that
^ ,(K l)
00
Note that
det£ = y3m3 det e
18
Due to C. Litwin.

Problems 477
Now use the identity
det/ =eTxk
But
Tr K = In y2Tv Q = \ny2
We conclude that
detfl = y3m3e]ny2 = m3y5
. An electron plasma exits at temperature of \06K. Is it necessary to use
; relativistic kinetic theory for this plasma?
6.12. A student argues that relativistic kinetic theory for a many-particle system
is inconsistent because proper time can only be defined with respect to one
particle. Thus the notion of a relativistic distribution function is not well
defined. Is the student's argument valid? If not, why not?
Answer
This problem is obviated if we work with distribution functions with variables
defined with respect to the lab frame. The time that enters this distribution is
the single lab time t. The distributions / and F = Nf encountered in this
chapter were so defined.
6.13. For the monoenergetic beam described in Section 2.5, show that J^J^ =
2 2 2
—e HqC .
6.14. If AM is a four-vector, then show that d4A is a Lorentz invariant.
6.15. (a) What is the value of the relativistic Maxwellian /o(m) at i; = c.
(b) Why is /o(m) physically unrealistic at v > c?
6.16. (a) Argue that an electromagnetic wave that is a plane wave in a given frame
is a plane wave in all inertial frames.
(b) With property (a), argue the existence of the four-wave vector A.9c).
(c) Let the frame S' move with velocity v relative to S. If a stationary oscillator
in S emits a wave of frequency to, then show that the observed frequency
in S' is
CO =
Y
where cos# = k • y/kv. The preceding formula for to' is the relativistic
expression of the Doppler effect. At 6 = n/2, this phenomenon is referred
to as the transverse Doppler shift.
&17. Show that the Lorentz transformation is orthogonal. That is, establish A.12).
6.18. Derive the form that the relativistic Maxwellian C.10) assumes in the
nonrelativistic limit, ft <C 1.
6.19. The relativistic Lagrangian for a neutral particle of rest mass m in a potential
field, V(r) is given by
L = -y~lmc2 - V(r)

478 6. Relativistic Kinetic Theory
(a) Show that in the nonrelativistic limit this form reduces to the classical
expression A.1.3).
(b) Employing Lagrange's equations, show that the given relativistic La-
Lagrangian leads to
— (mu) = -VV
dt
6.20. Show that Lagrange's equations applied to the N-body Lagrangian B.49)
return the Lorentz force law.
6.21. In the vicinity of a black hole, the differential interval in spherical coordinates
is given by
(dsJ = a~](drJ + r2[(d0J + (sin 0 dcj)J] - a(c dtJ
2m G
a =
c2r
where M is the mass of the black hole, G is the gravitational constant, and c is
the speed of light. From this expression, write down the Schwarzchild metric
6.22. What equations do the fourth components of the left equalities in D.25)
correspond to for the case that the parameter A. = r, proper time?
6.23. In flat space, the single free-particle Lagrangian is given by (see Problem 6.19)
L = —y~ me
(a) Does the effective Lagrangian L given by D.17) reduce to the preceding
form in the flat-space limit? Explain your answer.
(b) How can your finding to part (a) be corrected?
Answer
(a) In the flat space limit we find
which is not the desired form.
(b) The canonical action integral is written S = f L dt. So, to bring L to
canonical form, the integral D.16) must be multiplied by dt/dt, resulting
in the flat-space canonical Lagrangian
dX 7 _,
L = L — = —mcy
dt
6.24. Establish the relation
—s — t — u = —2m\c2 — 2m2c2
where s, t and u are Mandelstam variables discussed in Table 6.1

Answer
With given definitions, we write
-s-t-u =
with
(Pi,J =
and
the result follows.
m2c2
™2c2
Problems 479

CHAPTER 7
Kinetic Properties of Metals
and Amorphous Media
Introduction
Solid state matter falls into the categories: metals, semi-metals, semiconduc-
semiconductors and insulators. Structures of solids may be separated into the groups:
single crystal, polycrystal, and amorphous. A sample of single-crystal ma-
material comprises a periodic array of molecules (simple cubic, body-centered
cubic, hexagonal, etc.). A sample of polycrystalline material is composed of
single-crystal "grains" of material separated by "grain boundaries." An amor-
amorphous ("without form") material does not have a regular array of molecules.
The single-crystal phase of alumina, AI2O3, with Fe impurities, is the gem
sapphire, and with Cr impurities, it is the gem ruby. In polycrystalline form,
this material is employed as a high-temperature resistant ceramic. Window
glass is amorphous SiC>2 (fused silica). A common crystalline form of silica is
quartz (the dominant component of sand).
A graphical means of distinguishing between these solid-state phases is
given in terms of the radial distribution function defined in Section 2.2.C. The
radial distribution function for a single-crystal sample is sketched in Fig. 7. la,
that of a polycrystalline sample in Fig. 7.1b, and that of an amorphous material
in Fig. 7.1c.
A homogeneous mixture of two or more metals is called an alloy. The
symbol for a "binary" alloy comprising metals A and B, is written Ai_xBx
which indicates that in the alloy, x mole fraction of B atoms replace x mole
fraction A atoms. Alloys are grouped into two classes: ordered and disordered.
For example, the alloy brass, Cu—Zn, has an ordered phase, "/? brass," which
has a simple cubic lattice with a cesium chloride crystal structure. The alloy
bronze is a multicomponent alloy composed of copper and a number of other
elements, including, tin, aluminum, silicon, and nickel. Other alloys, such

7. Kinetic Properties of Metals and Amorphous Media 481
g(r)
J V
J V
J V
0
ar
(a)
g(r)
t
^/
J v
(b)
@
FIGURE 7.1. Sketches of the radial distribution for: (a) single-crystal, (long-range
Order) (b) polycrystal (partial order) and (c) amorphous material (short-range or-
order). Characteristic intermolecular displacement a, is defined with respect to the
displacement, r. The value g(r) = 1 is relevant to a perfect gas. Note that case (b) is
dependent on the choice of origin-site. As g(r) is an average entity, one may expect
distortion of this curve, beyond a few intermolecular displacement.
as Cu-Pb and Zn-Cd are heterogeneous mixtures of grains of nearly pure
single-metal components.
In the present chapter we are concerned with kinetic properties of met-
metals and amorphous media. In the first of these descriptions, expressions for
temperature-dependent electrical resistivity and thermal conductivity are de-
derived for a class of metals with spherical-like Fermi surfaces. In the second
part of the chapter, concepts are discussed relevant to amorphous media. Such
notions include hopping, percolation, localization, and the related concept of
the mobility edge. The chapter begins with a general discussion of the phe-

482 7. Kinetic Properties of Metals and Amorphous Media
nomenon of thermopower, which relates to the electric field developed in a
metal due to a temperature gradient across the material.
7.1 Metallic Electrical and Thermal Conduction
7.7.7 Background
A number of processes contribute to metallic resistivity: electron scattering
from impurities and lattice imperfections, electron-electron collisions, and
electron-phonon collisions. This latter interaction has both umklapp and nor-
normal contributions.1 In this section of the present chapter, metallic resistivity
of a pure metal will be described, for which electron-phonon collisions are
dominant. Umklapp processes are negligible at low temperature and are ne-
neglected in the discourse. Expressions for electrical and thermal conductivity in
a metal were described briefly in Section 5.3.3. In the present discussion we
are concerned with temperature dependence of these parameters.
Domains of thermal and electrical properties of a metal divide according
to the Debye temperature of the metal, @D. The Debye temperature partitions
classical from quantum behavior of a given material. Classical properties occur
in the domain T ^> @D, and quantum properties in the T < QD interval.
As was first derived by Bloch,2 low-temperature electrical resistivity, p, of
a pure metal (with a spherical, or nearly spherical Fermi surface) at T <C
0£>, varies as T5. A rough derivation of this dependence may be obtained
employing mean-free-path estimates. At low temperature, electron dynamics
is restricted to the Fermi surface. Let lk denote mean-free-path displacement
in A:-space on the Fermi surface and let v denote the electron-phonon collision
frequency so that one may define an effective speed, u a v/lk. One may then
write, kBT ex hulk. Recalling C.4.18) for the diffusion coefficient, we write
Dk a lku a l\v relevant to diffusion in A:-space on the Fermi surface. At
lower temperature it may be shown that3 electron-phonon collision frequency,
v a T3, which gives Dk a T5. For electrical conductivity we recall C.4.14),
a = e2n/vm a e2nvFx/ pF where r is the time for an electron to diffuse
a distance pF on the Fermi surface. In diffusion motion, the mean square
displacement is proportional to time A.8.36) and we may write, pF a D^r.
There results a a ne2p2F/mDk, so that low-temperature metallic resistivity,
p = I/a a T5.
In the high-temperature domain, T ^> 0D, it is generally observed that
p varies as T. This increase in resistivity follows from the behavior of
the Bose-Einstein distribution for phonons at high temperature. As note in
1W. A. Harrison, Solid State Theory, Dover, New York A979).
2F. Bloch, Z Phys. 59, 208 A930).
3E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, Pergamon, New York
A981).

7.1 Metallic Electrical and Thermal Conduction 483
E.3.68) one finds that the number of phonons at high temperature is given by,
nq oc kBT/Hco(q). Since this number of scatterers increases with T so does the
Resistivity.
Here is a summary of these findings.
T « 0D pcxT5
r»0D pcxT A.1)
Whereas electrical conduction in a metal is due only to electron transport,
both electrons and phonons contribute to thermal conduction. However, a sim-
simplifying property comes into effect. Heat transfer in a metal is due primarily
to electron transport, owing to relativity high electron speeds (Fermi speed)
compared to phonon speeds. Likewise, at low temperature, specific heat due to
electrons exceeds that due to phonons. For pure metals, thermal and electrical
conductivity are related according to the Weidemann-Franz law E.3.43)
A.2a)
oT 3
where k denotes coefficient of thermal conductivity. This relation is valid over a
wide temperature domain,4 whereas in the low-temperature domain, T <£ @D,
the Lorenz equation5
^ (rJ (L2b)
dJ
is appropriate, where A is a constant. With A.1) and the preceding relations
we note the following characteristic values of k.
T «0D k ex l/T2
T ^> QD k = constant A.2c)
Measurements of k indicate that at low temperature, k behaves linearly in T
and reduces to zero at OK. This behavior is attributed to impurity scattering.
The main thrust of this component of the present chapter is to derive an ex-
expression for electrical conductivity in a metal relevant to the entire temperature
range. Before doing so, we discuss a closely related phenomenon in metals for
the situation when a gradient of temperature is present.
For future reference we note the following properties of the Fermi-Dirac
distribution E.3.1). We define
n = - f°°(E - »)" (-^) dE A.3a)
n\ Jo V dE I
4N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and
Winston, New York A976).
5J. M. Ziman, Principles of the Theory of Solids, Cambridge, New York, A964),
Section 7.8.

484 7. Kinetic Properties of Metals and Amorphous Media
Then
n poo
Gn =
(kBT)n r
zndz
A.3b)
A.3c)
_ Bbn(kBT)n, neven\
y 0, n odd J
Values of the constant bn are readily calculated, e.g., 2Z?0 = 1, 2b2 =
[See Problem 7.1 and recall E.3.39b).] The following integrals are also noted.
A.3d)
Bn =
/
where Bn is a tensor and f dS represents an integral over the Fermi surface.
With A.3) we obtain,
f
Employing A.3e) we find
1 1
dS
47T3 H
- / rvv
H J
v
EF
7T

= '— (kBTJB0
In the preceding we have recalled the relation E.3.39a)
dSdE dSdE
dk = =
|V*E| Rv
where the right equality is relevant to a spherical energy surface and, we recall,
dS is a surface element on the Fermi surface in k-space.
7.7.2 Thermopower
Thermopower refers to the electric field developed in a metal due to a tem-
temperature gradient present in the material. (A closely allied phenomenon,
electrochemical potential, was discussed previously in Section 3.4. We recall
that in this phenomenon current flow occurs in a medium in which a density
gradient exists in addition to an electric field.) Consider a long, thin metal bar
along which there is a temperature gradient. Due to resulting charge flow, an
electric field is generated in the direction opposite to the temperature gradient.
At equilibrium there is an electric field with no charge flow. This electric field
is called the thermoelectric field. One writes
£=QVT

7.1 Metallic Electrical and Thermal Conduction 485
where Q is labeled the thermopower. In typical metals, observed values of Q
are of order /jlV/K.
Relaxation-time approximation
We wish to study thermopower in the relaxation-time approximation E.3.35).
The equilibrium distribution for conduction electrons in a metal in which there
is a temperature gradient and an electric field is given by the (k, independent
Fermi-Dirac distribution
fo[E(k), T(r), /x(r)] = — \ A.4)
exp[(£ - n)/kBT] + 1
In this model one is concerned with the change in the distribution due to
diffusion effects for which one writes, C/o/3f)Diff = — v • V/, etc.5 The
corresponding steady-state version of E.3.35) is then given by
/ eS dt\ 1
_ (V.V/ + —.— ) = -[/<,-/] A.5)
\ n 3k/ r
where S denotes electric field. We introduce the perturbation distribution
/(r, k) = /o + g(r, k) A.6)
and the relations [see equation above E.3.32)]
1 3 E 3/0 3/0
v(k) = , —=H\— A.7a)
y J n dk dk dE
where v is group velocity. We note the derivative
A.7b)
3/x
Substituting the latter three equations in A.5), and keeping 0A) terms, one
obtains
/3/o 3/o \ 3/0 1
v • — VT H Vu + ec • v — = g(k, r) A.8)
\dT 3/x / dE (£M
With
dE
we find
' E — /jl
dT \ T ) dE
3/o 3/o
A.9b)
5J. M. Ziman, Principles of the Theory of Solids, Cambridge, New York, A964),
Section 7.8.

486 7. Kinetic Properties of Metals and Amorphous Media
Substituting these findings into A.8) gives
T / \ e
For future reference this equation is rewritten as
1
g(k,r) A.10a)
, r) = a\
1
e
= r
dfi
0
dE
A.10b)
A.10c)
To calculate thermal conductivity in a solid one imagines a small region
over which temperature is effectively constant. Recalling the thermodynamic
relation between a differential of heat, dQ and a differential of entropy, dS
dQ = TdS
permits one to write the corresponding relation between thermal and entropy
current densities, Jq and Js
A.11a)
was previously labeled Q). With the entropy relation
TdS =
one writes
TJs = Je — mJ/v A.11b)
where JN is particle current density. With A.1 la) it follows that
J<2 ~ Je — I^Jn (Lllc)
Local energy and particle current-density are given by [with r-dependence of
/ tacitly implied]
A.12)
With A.11 c) we may then write
dk
47T3
r(£)v(k)/(k)
A.13)
As /o A.4) is an even function ofk, only the perturbation distribution g enters
in the preceding integral. With A.10b) we obtain
dk
47T3
Ta(£)v(k)v(k)
1
1
A.14)
Electric current density is given by [recall equation preceding E.3.39)]
dk
J =
4?r3
v(k)/(k)
A.15)

7.1 Metallic Electrical and Thermal Conduction 487
Again with reference to A.10b) we find
dk ^^= fl / 1 \"
—- a(E)\(k)\(k) • — F(—VT) + e \ £ Vu A.16)
4tt3 \_T \ e ) J
It may be assumed that in measurement of electric field in a metal, gradients in
chemical potential are subsumed in the field. Consequently this term is ignored
in the following relations. With A.14) and A.16) we write
A.17a)
A.17b)
where the L/7- coefficients are as implied and are, in general, tensor forms. In the
absence of a gradient of temperature, Ln is electrical conductivity, whereas in
the absence of an electric current, L22 is the coefficient of thermal conductivity.
Here one assumes, |L22| ^> IL^L^/L^I. The off-diagonal elements relate to
the interplay of these two effects when both fields are present. Note that in
general, L2J = 7X12.
For Ln we find
4n3h J J ~"~ dE
which with E.3.39b) and A.3i), may be written
Uf
A.18a)
H J J
J J v
so that
a = e2B0 A.18c)
which is in accord with our previous finding E.3.41).
For L22 we find [with A.3d) and A.3i)]
B Tn2kBB0
L22 = k = j B2 = Tn2k2BB0 A.18d)
which, with A.18c) returns the Wiedemann-Franz law, A.2a).
7.1.3 Electron-Phonon Scattering Matrix Elements
The analysis of the following sections addresses temperature-dependent elec-
electrical resistivity in monovalent metals. These include a subset of the alkali
metals: Na, K, Rb, Cs, which each have bcc structures, and the noble metals:
Cu, Ag, Au, which have have fee structures. Alkali metal atoms include a sin-
single electron outside a closed shell noble-atom electron configuration. Thus,
for example, potassium has the electronic configuration, [Ar] 4s1. Noble metal
atoms include a single electron exterior to closed shell composed of a corre-
corresponding noble atom electronic configuration and a closed d10 shell. Thus, for
example, copper has the electronic configuration, [Ar] 3d104s1. The metallic

488 7. Kinetic Properties of Metals and Amorphous Media
FIGURE 7.2. Fermi surface for a noble metal showing necks at contact with Bragg
planes of the first Brillouin zone (truncated octahedron). Also shown are de Hass-van
Alphen electron orbits due to an imposed magnetic field, //, in the [111] direction.
(Reproduced with permission of N. W. Ashcroft and N. D. Mermin.)
state of both classes of elements is characterized by tightly bounded bands that
lie significantly below conduction electron levels of the metal.
The Fermi surfaces for the alkali metals are spheres and lie, respectively,
within the first Brillouin zone (regular rhombic dodecahedron). The Fermi
surfaces for the noble metals are spherical except for "necks," which extend to
the Bragg planes of the first Brillouin zone (truncated octahedron) in the [111]
directions (Fig. 7.2). As noted earlier, temperature dependence of phenomena
in solid materials separates according to weather temperature in less than or
greater than the Debye temperature. Our analysis will rediscover this property
as well as a residual resistivity, which comes into play near OK.6'7 Characteristic
low-temperature alkali-metal resistance is shown in Fig. 7.3.
Interaction Hamiltonian
Our study begins with the derivation of an expression for the interaction Hamil-
Hamiltonian between electrons and ions in the lattice, in which ion displacement from
equilibrium is incorporated in the analysis. Working in second quantization,
Components of this analysis appeared previously in R. L. Liboff and
G. K. Schenter, Phys. Rev. B54, 16591 A996).
7For a review of these topics see: J. Bass, W. P. Pratt, Jr. and P. A. Schroeder,
Revs. Mod. Phys. 62, 645 A990).

7.1 Metallic Electrical and Thermal Conduction 489
5 ""' ~~10~ 5
Temperature (K)
JL__
20
FIGURE 7.3. Characteristic relative resistance as a function of temperature for an
alkali metal in the low-temperature domain at relatively small impurity concentration.
For further discussion see D. K. C. MacDonald and K. Mendelssohn, Proc. Roy. Soc.
(London) 202, 103 A950); J. S. Dugdale, Electrical Properties of Metals and Alloys,
Edward Arnold, London, 1977 Chap. 10; J. L. Olsen, Electron Transport in Metals,
Interscience, New York A962), Chap. 2.
the potential Hamiltonian component for the system may be written
H= /dx^+(x)^(x)]T<I>[x-y-z(y)] A.19)
** y
where O is electron-ion interaction potential, \//f is electron creation operator,
x denotes electron position, y denotes ion equilibrium position and z(y), ion
displacement from equilibrium. For small ion displacements we write
Hint = H - H(z = 0) =
I dx \j/f
(x)^(x)]Tz(y).^cD(x-y) A.20)
y
Fourier expanding electron operators
iAT(x) =
dk-
f dk
= / —
J Bn
and quantizing ion displacements
A.21a)
n
2MNco
+ ^(-

490 7. Kinetic Properties of Metals and Amorphous Media
gives
n
(=F''qexp/[±q-y-k2-x
x])O(x-y) A.22)
where eM is a phonon polarization unit vector, q denotes phonon wave vector,
a) = uq, N represents total number of free electrons and M is ion mass. The
column vector notation in A.22) is such that the upper <3M(q) term corresponds
to the (+) phonon absorption mode and the lower 2* (q) term corresponds to
the (—) phonon emission mode. In these expressions,
Pm = MnM,
A.22a)
_ n _ N
— * w —
Z V
where yOM is ion mass density, nM is ion number density, n is electron number
density, Z is atomic valence, /x is polarization index and co is phonon frequency.
Now we note that, with u = x — y,
- y) = -e±iqx
u
= - e±iqxi>(±q) A.23)
where V is total crystal volume and
£
A.24)
is the Fourier transform of the potential. The preceding combines with the
exponentials A.23) to give
Btt)
n
± q
IMNgo
A.25)
*(±q)
With this equation we see that only longitudinal modes contribute so we may
set
eM(±q) = =R
Thus, rewriting A.25) in the form
BttN ±
gives the coefficient
- k
A.26)
n
IMNgo ^ V
iq—co, <P(±q)
A.27)

7.1 Metallic Electrical and Thermal Conduction 491
For the shielded Coulomb potential one writes
AnZe2
qz +
where gTF is the Thomas-Fermi shielding wavenumber
A-28)
) d.28a)
and ao is the Bohr radius. With co = uq we obtain
2Z2pMu (q2
where Q is the ion plasma frequency
A.29)
0 4nnM(ZeJ
Q2 = ——- A.29a)
M
As we will see below, the expression A.29) comes into play in electron-phonon
scattering matrix elements.
Scattering-rate matrix elements
Electron-phonon scattering-rate matrix elements are written s££ (with
dimensions of inverse time) and are given by
$ = |(k', n'|//,NT|k, n>|2-^ S(AE) A.30a)
n
8(AE) = 8(E' - E - ahco) A.30b)
where E is electron energy, //INT is given by A.26). [S^, was previously labeled
w(k, k) in E.3.59).] The equality
A.30c)
corresponds to symmetry of the electron-phonon interaction under time re-
reversal and a = (+, —) corresponds to phonon (absorption, emission) in an
electron-phonon scattering event. Momentum conservation in a collision is
given by
A.31)
where /zq is phonon momentum.
In the relation A.30a), |n) denotes the many-phonon state
= \nq, nq<,...) A.32)
where
nq = j. .. T A.33)

492 7. Kinetic Properties of Metals and Amorphous Media
is the Bose-Einstein distribution. For the dispersion relation for phonons we
write
a) = uq A.34)
An estimate of the phonon speed u is given by the Bohm-Staver relation
0 2 ZEp
" - 3 IT AJ4a)
Fast relaxation of phonons to the distribution A.33) is assumed in the analysis
(the so-called "Bloch condition").
For metals, the matrix elements A.30a) have the value8
where with A.29)
\Cq
S(AE) A.35)
2 2 / n
\Cq\2 = RG(q) A.36a)
(q2
H(MQ2J
^ ^
A.36c)
Note that R has dimensions of (energyJ, G, is dimensionless and Cq has
the units of energy. The quantity h Q/2 may be identified with ion zero-point
energy.
With A.33) we see that phonon occupation numbers, nq = 0, at T —
OK. Nevertheless, from A.35) we note that the phonon emission (a = — 1)
matrix element persists at this temperature. Thus, inelastic electron-phonon
scattering maintains at T = OK. The fact that phonon absorption scattering
matrix elements vanish in this limit is consistent with the OK limit. That is, as
the ground phonon state at T = OK is the state of lowest energy, no energy
may be extracted from it in any interaction. The phonon emission at T = OK
is due to zero-point fluctuations which induce an electron to decay, consistent
with A.30b). As will be noted below, this phenomenon gives rise to a residual
resistivity in a pure metal at T = OK.
Assumptions
Here is a brief review of assumptions included in following analysis. In the
absence of an electric field, it assumed that conduction electrons in the metal
are in the Fermi-Dirac distribution
o(k) = A.37)
0V ; l+exp[(E-EF)/kBT]
8F. Sietz, The Modern Theory of Solids, Dover, New York A987); J. M. Ziman,
Electrons and Phonons, Oxford, New York A960).

7.1 Metallic Electrical and Thermal Conduction 493
with normalization
dk
A.38)
BttK
where EF is the Fermi energy.
For metals with a spherical or nearly spherical energy surface we may write
dk = 4nk2dk = —^ A.39)
and A.39) may be rewritten
V = nBnJ(H2/2mK/2 A.40)
With co written for phonon frequency, the following relations are assumed
Ha) <HcoD <£ EF ~ E A.41)
where cop is the Debye frequency. Furthermore, as electron wave vectors lie
predominantly on the Fermi surface, we also conclude that: q <£ kF and
electron scattering is predominantly small angle, or, equivalently, k • q <£ kq.
Scattering matrix elements A.35), A.36), are employed in construction of
the quantum Boltzmann equation presented in the following section, relevant
to derivation of the distribution function for conduction electrons in a metal.
7.1.4 Quantum Boltzmann Equation
Our starting equation is the quantum Boltzmann equation
eS df
T'J£ = JW) (L42)
n dk
/dk'
^ [/'(I - f)S& - /(I - /')$&!] A.43)
which is noted to have the generic form of E.4.5). As noted above, the sum over
ot = +1,-1, corresponds to emission and absorption of a phonon, respectively,
in an electron-phonon scattering event. The electric field is £ and
' = f(k',t)9 etc. A.44)
represents the electron distribution function where k' corresponds to "after"
collision. We recall that the term, A - /) in the "birth" term on the right side
of A.43) is a quantum inhibiting factor relevant to fermions (Section 5.4).
Substituting A.35), A.36) into A.43) gives
/'(I " /) (nq + \ + a I) - /(I - /') (nq + \ - a l-
A.45)
J(f) = T la

494 7. Kinetic Properties of Metals and Amorphous Media
where
iw
A.45a)
):
In this expression, with A.31), we have set dk' = dq and
/'(k) = Z(k') = /(k + aq) A.45b)
so that /'(k) is a-dependent.
Lorentz-Legendre expansion9
To account for anisotropy of the distribution function due to the imposed £-
field, we employ the Lorentz expansion
A.46a)
A.46b)
A.46c)
A.46d)
where hatted variables are unit vectors.
Keeping terms to O(/x) in A.46) and substituting the resulting form into the
collision integral A.45) gives
= k • £ = cos
A.47)
where
(/o-
E
1
2
1
2
JO) uJ0
o
A.47a)
a
2\(i
■« i - /i'/o
A.47b)
Note that
/'(£) = /(£') = / [E + a
A.47c)
Substituting these expressions into A.43) and passing to the steady-state limit,
with the orthogonality of Legendre polynomials, we obtain the two equations
(see Appendix D)
A.48a)
9H. A. Lorentz, The Theory of Electrons, B. G. Teubner, Leipzig A909).

7.1 Metallic Electrical and Thermal Conduction 495
nk dE
f0 = Mfo, fi) A.48b)
It is assumed that fo(E) is the Fermi-Dirac distribution A.38). In that 7o(/o)
vanishes for this choice of /o, A.48a) corroborates the fact that the Fermi-Dirac
distribution is relevant to the zero-field situation, £ = 0. The relation A.48b)
suffices to determine the correction /\.
Reduction to integrals
To reduce the integral A.47b) occurring in the right side of A.48b), we first
note that
/pqD p2n pi
dq= dqq2 dp dcosy A.50a)
Jo Jo J-\
where
q • k = cos y A.50b)
Furthermore, we note that
k
8(Ef - E - ahco) = 8
2E
q t k
2qE
where we have set
cos y — I — a -—\- 8
2k 2qJ_
A.51)
A.51a)
The delta function in Ia restricts the domain of integration. With A.51) we
write
q k
cos y — —a h s — A.52)
2k 2q
Noting that
— 1 < cosy < +1
and that £ is infinitesimally small, we obtain the upper (U) and lower (L) limits
on the q integration:
8k
2
A.53)
qL = 0 + O(s2)
Thus, A.45a) reduces to
Vk f ,
Iai<Pa) = 4^HE I ^tflQlV* d-54)
where qo represents the Debye wavenumber,
= [6n2nM]l/3 A.55)
= — = [6n2
i/t'

496 7. Kinetic Properties of Metals and Amorphous Media
Inserting the preceding expression into A.47b) and summing over a, we
obtain
{( [yfo+)
Wo, /.) =
- /o) + /o] -
'/,(") [e"fo + A - /o] - Hfi
fo
yd ~
eyd
+ /o
where ()± correspond to a = ±1 and
hco hug
y =
kBT
kBT
Q
A.56)
A.56a)
d.56b)
and, with A.47c)
A.56c)
Note further that Q has the dimensions of wavenumber.
In A.56) we introduced
Vk
/ =
With A.36), the preceding is written
A.57)
°
dqqG{q)
A.58)
Since most of electron scattering occurs on the Fermi surface, we may write
k' ~ k. With A.31) we then obtain
* y A.59)
1 -
2k2
Substituting this expression into A.56) permits the starting equation A.48b)
to be written
dE AnEH
Jo
e> - I
2k2
where
L, = f[+\e\\ - /o) + /„] - hW(\ - ft}) + o
+ /^[^/o + A - /o)] - /ikv/0(+) + A - /0(+))] A-61)
L2 = /,(+)kv(l - /o) + /o] + //"'kVo + A - /o)] d-62)
7.1.5 Perturbation Distribution
The relations A.60), A.62) comprise a self-contained integro-difference
equation for the perturbation distribution, f\{E). When written in terms of

7.1 Metallic Electrical and Thermal Conduction 497
nondimensional variables (x, y) the definition A.56c) is given by
/(±)(x) = f(x ± y) A.63)
x = E/kBT, y=ha)/kBT
With A.41) we note
x^>y A.63b)
Thus, in this same limit, A.56c) becomes
A.64)
(both for /o and f\).
Substituting A.64) and A.61), A.61), reduces to Li to zero. For L2 there
results
A.65)
Defining
and
F(E) = - A.66)
dfo/dE
B = —2 A-67)
4nh2
A.60) becomes
e£ QW
F(E) A.68)
where W is the dimensionless integral
>yD ( ey + 1
W(T) = Q'V I dyy3G(y) ( -7—j- ) A.69)
or, equivalently,
Jo
(,69a)
= ^~ A.69b)
kB@D =huqD A.69c)
It follows that

498 7. Kinetic Properties of Metals and Amorphous Media
where the temperature-dependent term
K(T) = QW(T) = QTW(T) A.70b)
Q = kB/hu A.70c)
Note that K(T) has dimensions of wavenumber.10 With A.38) and A.46), the
expression A.70a) gives the corrected electron distribution to the given order
in /x.
7.7.6 Electrical Resistivity
Current density is given by [recall A.15)]
/dk ehk
7T^ /(R) <L71)
m
Substituting A.69) into the preceding we obtain
J =
J Bn
y m
eh - C
£ / dkkkf{(k)k A.72)
J
*
3 m
where 7 is the unit matrix. A double-barred variable represents a dyad. [See
(B.I.I) et seq.] There results
1 eh f f
With A.39) we write
kdk = 4nk3dk = 2n —r) EdE
It follows that
/»
dEEfi(E) A.74)
f
J
In estimating f{ it is further assumed that
A.74a)
dE
Since kBT <£ £>, f0 is sharply peaked in the temperature range of interest
(OK < T < 300K) and A.74a) remains a good approximation. Substituting
10The relation A.70a) contradicts Bloch's principal assumption, f\(E)
adfo(E)/dE, where a is a constant. See: F. Block, Ref. 2.

7.1 Metallic Electrical and Thermal Conduction 499
the preceding relations in A.70) gives the desired solution for the perturba-
perturbations distribution f\{E). When substituted into A.74) this solution gives the
conductivity (a); resistivity (p) expression
16 e2mE\ ( e2mE\\ ( \6hu \
= — = =-£ A.75)
3RK(T) \37tH3R) \kBTW(T)J J
p 3nH3RK(T) \37tH3R) \kBTW(T)
whose temperature dependence is contained entirely in TW(T) [see A.70b)].
As will be shown below, the preceding expression for p gives both Bloch's T5
dependence at (T/@d) <^C 1 as well as canonical T dependence at (T/@D) ^>
1 in addition to a residual resistivity at T = OK.
Properties of W(T) and Si (A)
The function W(T) is singular at T = OK. To expose this singularity first we
note the relation
-1 2
- 1 e> -1
so that [recall A.69)]
>y» dyy4
W(T) =
I
(y2 + jIfJ V ey - l
A.76)
The W2(T) contribution corresponds to the exponential term and is finite at
T = OK. The singularity of W(T) lies in W\(T). To obtain the T-dependence
of this singularity we introduce the variable
There results (relabeling zd = ©d, etc.)
tj, tf + ew r A77)
where Si is the implied nondimensional temperature-independent integral. The
relation A.77) indicates that W(T) has a simple pole at T = OK. Evaluating
the integral Si gives
1 3 ,
S{(X) = 1 + tan A A.77a)
V } 2A+A2) 2A
The parameter ^tf is given by A.28a), qD by A.68c) and we have set tan"' @) =
0.
The function Si (A.) is a positive monotonic function with properties
Si@) = 5J(O) = 0
l i l A.77c)

500 7. Kinetic Properties of Metals and Amorphous Media
For A. <C 1, one obtains
OF A.77d)
Values of S\(k) pertinent to the problem at hand are obtained as follows: First
we note
3tt5\1/3
\16) (*) (L78a)
or, equivalently (with Z = 1)
k = 1.43 x 1(TV/6 A.78b)
where n is electron density in cm.
Among the alkali and noble metals, n is maximum for Cu, for which we
obtain A.Cu = 0.96. In this group n is minimum for Cs, for which we obtain
A.Cs = 0.66. We may conclude that the expansion A.77d) is appropriate to
the metals addressed in this analysis. A more accurate description of Si (A.) is
obtained by curve fitting this function to a parabola in the A.-domain of interest.
There results
Sx(k) = -0.042A. + 0.1U2, 0.60 < k < 1.00 A.78c)
Thus
^ 0.020
A.78A)
^ 0.061
Combining A.77b) and A.77c) we obtain
104Si(A.) = -0.06n1/6 + 0.16 x 10~V/3 A.78e)
values from which are seen to agree with A.78d).
We note that W\ as given by A.76), with A.70b) gives
= Q&DSl = 2£^Jisl(k) A.780
which is independent of temperature. The temperature dependence of the
distribution /i, resides entire in W2.
Temperature dependence of the W2 integral
To examine the finite integral W2 we revert to y dependence and write
W2= —: — — A.79a)
With these results at hand, we consider first the high temperature limit.

7.1 Metallic Electrical and Thermal Conduction 501
Case (a) T ^> 0£>: In this limit, expanding the integral A.78a) about yD = 0
we obtain
X2
In the limit of X <C 1,
A. + O(A.) A.79c)
With the preceding, in the said limit, A.75) gives the result
DSi\ nXA ( HR \ (kB
3tt / H3R \/kBT\n4 &DSi\ nXA ( HR \ (kBT
9 16 \e2mE3F) \hu ) \2 + T ) ' 32 \e2mE3FJ\ hu
A.80)
which is noted to have the canonical form p ex T'.
Case (b) T <£@D:In this limit we obtain
where ^2 is the implied non-dimensional, temperature-independent integral
with the value
-S2 = rE)?E) = 24.886 A.81a)
and r and f are Gamma and Zeta functions, respectively (see Appendix B).
General resistivity expressions
Employing results from the preceding sections we now obtain explicit expres-
expressions for the residual and Bloch components of resistivity. With A.69a), A.75)
we write
y(t\ hrw
A.82)
A A
Note the relations
K
j —
G©TF
A -
XT)
A
coD
u
COjF
u
I6e2m
QTW
A
~q°
— #TF
E3
zF
A.83)
A A.84)
3tiH3R
The parameter A has dimensions of wavenumber, so that K/A has the cor-
correct resistivity dimensions (cgs): time. The parameter R is defined in A.36c).

502 7. Kinetic Properties of Metals and Amorphous Media
Collecting results we write
K(T) = Q
K{T) =
KB(T)
A.85a)
A.85b)
where Ko is independent of T and KB(T) heads to the Bloch result. Inserting
this finding into A.53) gives
P =
Q
'TF
'TF
= Po + Pb(T)
A.86a)
A.86b)
where
Po =
A.86c)
is the component of resistivity due to electron-phonon scattering that survives
at OK and pB is the Bloch contribution. We note that pB may be written in the
more canonical form
Pb =
5 / -7-. \ 5
D \ / i
uA
A.86d)
>TF
where, with A.28a), A.55) one notes that
TF
x6 _
) —
A.86e)
The relations A.57) indicate that po dominates over pB for temperatures
TF
250
A.87a)
or, equivalently
250A.4
1/5
= r(X)
A.87b)
For Cs we find t = 0.25. For Cu we find t
come into play at
= 0.20. Thus, one expects po t0
A.87c)
for the class of metals considered.

7.1 Metallic Electrical and Thermal Conduction 503
7.7.7 Scale Parameters of po
We wish to obtain the manner in which p0 scales with basic metallic parameters.
To these ends we write
A.88)
First, consider the po factor. With A.86c) we write
3n2kBeDH(HnJ
Po =
8 mu
To find the manner in which p0 scales with metallic parameters, in A.89a) we
set all parameters that are constant with respect to change of metallic samples
(e, m, kB, etc.) equal to one. There results
3nZ
— A.89b)
F
To further reduce this relation recall A.22a) and note that
EF a n2/3
@D a n2'3Zl'6/M1'2
It follows that
p0cxZl/6/nMl/2 A.89c)
Large-mass consistency limit
As ion mass grows large, Q —► 0, and electrons do not interact with the lattice
[see A.29)]. It follows that in this limit one should find that po —► 0. To
explore this situation we examine the limit, M —► oo, at otherwise fixed ion
parameters: nM and Z. With these constraints we note that
A.90a)
u2 u
2^3
We recall that u a y/Z/M, q2D a n2^3, Q2 a nMZ2/M, and that
^ A90b)
TF ^
which together with Si (A.) are constant under the said constraints. There results
constant
Po a ~WrT ~* ° (L9Oc)
This property agrees with the preceding observation that in the given limit,
electrons do not interact with the lattice so that p0 -> 0. In this same limit,
Pb -> 0, as T -> 0, providing T/M < 1 [as follows from A.86)].

504 7. Kinetic Properties of Metals and Amorphous Media
7.1.8 Electron Distribution Function
Returning to the expansion A.46) and inserting the solution A.70) gives the
electron distribution
£^ A.91a)
K yl )
?>2e£
D = A.91b)
where fo(E) is the Fermi-Dirac distribution A.37) and R is given by A.36c).
Consider the function K(T) as given by A.85). Let us suppose that there
is no residual term and set Ko = 0. Then as T —> OK, K(T) -» 0 and
the perturbation term in A.91a) grows singular at all E thereby violating the
Lorentz expansion A.46). At T = OK, dfo/dE is zero except at E = EF.
However, with Ko = 0, this zero is divided by KB@) = 0 and the distribution
A.85) is indeterminate. For the case Ko > 0, as found in the present analysis,
this pathological behavior of f(E) is circumvented and, save for the singular
point E = EF at T = OK, a well-defined distribution results for all E.
Reduced resistivity
We note that the reciprocal of the right side of A.75) may be written
p = H(GD, QTF) w A.92a)
where the coefficient H(&D, 0TF) is an implied and dimensionless "reduced"
resistivity is given by
.92b)
and we have recalled that
@TF/ Tf = y2D/X
The term W\ includes the residual resistivity result whereas W2 includes both
Bloch's T5 result as well as the linear T high-temperature dependence. When
p is plotted vs temperature (l/y), the preceding result A.92) exhibits the
characteristic behavior depicted in Fig. 7.3.
It should be borne in mind that the resistivity results presented above are
relevant to ideal monovalent metals free of defects, dislocations and impurities.
7.7.9 Thermal Conductivity
It has been noted that the thermionic properties of metals are complex and not
well explained by standard analyses.8 With the preceding results at hand we
return to the problem of metallic thermal conductivity and describe a model

7.2 Amorphous Media 505
which gives agreement with characteristics of the coefficient of thermal con-
conductivity for a metal over three basic ranges of temperature. We recall that
the Weidemann-Franz law for k/gT as given by A.2a) is valid over a wide
temperature range. Having found a residual resistivity at T = OK, we revert to
this law, which when taken with the preceding results A.86c) returns the ex-
experimentally correct low-temperature thermal conductivity linear-temperature
dependence10
* = — (— T A.93)
Application of this relation to specific metals is derived from the A.-dependence
contained in p0. At higher temperatures but still beneath the Debye temperature,
where W2 [see A.92b)] comes into play, one reverts to the augmented Lorenz
relation A.2b) which returns the correct T~2 thermal conductivity behavior.
Above the Debye temperature, we return the Weidemann-Franz relation and
write
() a94)
P(T) V )
where p(T) again includes the W2(T) component but is now relevant to the
limit, T ^> @d- At these conditions A.94) gives the correct behavior A.2c),
k = constant.
The preceding results imply a thermal conductivity vs. temperature curve
which rises linearly from the origin and then decays as T~2 from a maximum
value and levels off to a constant at T < 0£>, in agreement with characteristics
of measured values (Fig. 7.4).
To assist the reader in this analysis, a list of key parameter-relations and
definitions is presented in Table 7.1.
7.2 Amorphous Media
7.2.7 Background
As noted in the introduction to this chapter, the amorphous structure is a com-
common form of material. We recall that window glass is a fused alloy of Na2O and
SiO2. Other types of glass that are of interest because of their semiconducting
properties are compounds of S, Se, and Te with elements such as As and Ge.
Such compounds are called chalcogenide glasses (compounds of any of the
group 6 elements of the periodic chart, excluding oxides). Common liquids
are amorphous. Liquid crystals are composed of rodlike molecules (length ^>
diameter). In the smectic phase, liquid crystals have long-range order along the
molecular axis, as well as molecular orientational order, but are disordered in
the plane normal to the molecular axis. At higher temperature, in the nematic
phase, the liquid crystal remains only with molecular orientational order.

506 7. Kinetic Properties of Metals and Amorphous Media
K
FIGURE 7.4. Sketch of observed temperature dependence of the coefficient of ther-
thermal conductivity of a metal. At low temperature k ~ T whereas at T ^> 0D,
k ~ constant. At intermediate temperature k ~ T~2. The central peak of the graph
is observed to rise with increased purity of the metal. Fur further discussion, see
J. M. Ziman, Electrons and Phonons, ibid., Chapter 9; H. M. Rosenberg, The Solid
State, 3rd ed., Oxford, New York A992), Chapter 8; G. K. White, Proc. Phys. Soc.
(London) A66, 559 A953).
As shown in Fig. 7.1, the radial distribution function of ions in an amorphous
material has short range order, and is void of long-range periodicity. In what
manner may one expect to find properties of crystal structures in an amorphous
sample? As noted in Fig. 5.5, the energy spectrum of crystalline material has
a band structure. The fact that window glass is transparent to visible light is
indicative of a quasi band spectrum with an effective band gap of the order of
several eV. Liquid mercury, as well as other molten metals, have high electrical
conductivity, again indicative of a quasi band spectrum with a partially filled
conduction band.
An assortment of phenomena are relevant to amorphous materials. These
include the nature of the band structure and related concepts of: localization,
mobility edge, hopping, percolation, and the metal-insulator transition.
Extended states
The notions of extended and localized states play an important role in the study
of electrical properties of amorphous material. Here is a brief review of these
concepts. Consider the Schrodinger equation for a particle of mass m moving
in a periodic potential, V(x) = V(x + a), where a is the lattice constant of the
array.
<p"(x) - Wcp(x) + Kz(p(x) = 0
B.1)

7.2 Amorphous Media 507
TABLE 7.1. Key Parameter-Relations and Definitions for Section 13 et seq.
n N
pM = MnM, nM = —, n = — A.22a)
a0/ \n
2 Z£>
qD =
x =
B-
k =
coD
u
Hco
kBT "
E/!cBT,
Anti1
©TF
©TF
[6tz2
Huq
kBT
CO d
6t)yF
«m]'/3
q
Q
y =hco/kBT
>u - T
kB&fi =Huqp
qTF yTF
u
_
'TF —
d/@tf)
/ \ 6
= f — ) =
2 = mv—)_
M
A.34a)
A.36b)
A.36c)
A.55)
A.56a)
A.67)
zi sew \
A.69b)
A.69c)
(l.77b)
A.83)
A.84)
^V, E
H2 2m
Substituting the periodic function cp(x) = cp{x+a), into B.1) returns the orig-
original equation, indicating that any periodic function with period a is a solution

508 7. Kinetic Properties of Metals and Amorphous Media
to this equation. To gain further information on the eigenvalue problem we set
(p(x) = eikxu(x) B.2a)
The following equation results for u(x).
u + liku - (k2 + W - K2)u = 0 B.2b)
It follows that u(x) is also periodic and dependent on k.
The solutions B.2a) are called Block waves. Each such wavefunction is
present throughout the periodic array and is an example of an extended state.
As depicted in Fig. 5.5, allowed /rvalues comprise a band structure. Electron
wavefunctions in valence or conduction bands are Bloch waves. An example
of localized wavefunctions is given by electronic states of unionized ft-type
impurities in a host semiconductor, such as, for example, antimony atoms
in crystalline silicon. After bonding to the silicon host, one electron remains
weakly bound to the antimony atom, and is in a localized state. Energy levels
of this electron are discrete and hydrogen like. On promotion to the conduction
band, this electron goes into an extended state. If disorder is introduced into
the regular crystal, localization of the wavefunction occurs and k is not a good
quantum number.
As first discovered by P. W. Anderson,11 the wavefunction of an electron
in a lattice of sufficient disorder, is localized. The real part of an extended
wavefunction is shown in Fig. 7.5a and that of a localized wavefunction is
shown in Fig. 7.5b. Note that the envelope of the localized wavefunction decays
exponentially. The exponential coefficient a is called the inverse localization
length. As will be denoted below, Anderson localization is relevant to intrinsic
type material.
7.2.2 Localization
Consider the array of quantum wells shown in Fig. 7.6. For the regular array,
(Fig. 7.6a) the band width is labeled B, whereas for the disordered array,
(Fig. 7.6b), the related width of the spread of states is labeled W. We consider
the parameter
T = W/B
In the limit F « 1, extended states prevail. We work in the tight-binding
approximation in which the overlap of atomic wavefunctions is considered
negligible except for nearest neighbors. This approximation is consistent
with Fig. 7.1c, which indicates short-range order in an amorphous material.
Approximate solutions of the time-independent Schrodinger equation, relevant
nP W. Anderson, Phys. Rev. 109, 1492 A958). See also: J. M. Ziman, J. Phys.
C2, 1230 A969); D. C. Herbert and R. Jones, J. Phys. C4, 1145 A971); N. F. Mott
and E. A. Davis, Electronic Properties in Non-Crystalline Materials, Oxford, New
York A979); P. A. Lee and T. .V. Ramakrishnan, Rev. Mod. Phys. 57, 287 A985).

7.2 Amorphous Media 509
(b)
r\
r\
\
\
-aR
\
FIGURE 7.5. (a) Real part of a Bloch extended-state wavefunction of a single-particle
in a periodic potential, (b) Real part of a localized wavefunction of a particle in a
disordered potential array, exhibiting an exponentially decaying envelope. From:
R. Zallen A983). Reprinted by permission of John Wiley and Sons, Inc.12
a B> W
(a)
Anderson
Transition
W
(b)
B< W
FIGURE 7.6. Energy of states of a particle in a one-dimensional quantum-well array,
(a) For the periodic array the band width is labeled B. (b) For the disordered array the
width of the spread of states is labeled W. Localization occurs in the limit W/B > 1.
From: R. Zallen A983). Reprinted by permission of John Wiley and Sons, Inc.
to electrons in a given band, at F < 1, are given by
B.3)
n

510 7. Kinetic Properties of Metals and Amorphous Media
where an are atomic sites and i/f (r) wavefunctions are centered at these points.
The width B is given by
B = 2zl B.4a)
where z is coordination number and / is the transfer integral
1 = J VKIr-aJWar-a^l) B.4b)
where H is the Hamiltonian of the system. For a lattice composed of hydrogen
atoms, for S-like wavefunctions, one obtains
/ = 2[1 + (p/flo)] exp(-p/flo) B.4c)
where / is measured in Rydberg units (Ry = e2/2ao), a$ is the Bohr radius and
p is atomic separation. This expression exhibits a sharp decay of the overlap
integral with increase in separation, which is consistent with the tight-binding
approximation.
To examine the disordered case for which F > 1, as noted above, one
introduces a random distribution of potential depths at each site, with variance
W. In this limit, the wavefunction B.3) loses phase coherence in passing from
one atomic site to the next and assumes the form
- an|) B.5)
n
where </>n are random phases and cn are constants.
The primary finding of Anderson is that as F is increased, at some critical
value, wavefunctions become localized. Let us call this critical value, Fc. For
coordination number 6, Anderson found Fc = 5. Edwards and Thouless13
obtained the value Fc = 2. More recently, Elyutin et al.,14 found the value
Vc = 1.7.
Localized wavefunctions have the form
* = exp(-|r - r01/£) ^ cn exp(J0nW(|r - aj) B.6)
n
Each such wavefunction is localized about a point r0 in space. The parameter
£ is called the localization length and decreases with increasing disorder, W,
thereby increasing the rate of decay of ^ away from atomic sites.
Metal-insulator transition
As was shown by Mott,15 if the value of F is insufficient to give localiza-
localization throughout the band, at band edges, energies of localized and extended
12R. Zallen, The Physics of Amorphous Solids, Wiley, New York A983).
13 J. T. Edwards and D. Thouless, J. Phys. C5, 807 A972).
14P. V. Elyutin, et al., Phys. Status Solidi B124, 279 A984).
15N. F. Mott, Adv. Phys. 16, 49C A967).

7.2 Amorphous Media 511
M-I Transition
Extended
states
Extended
states
v,m
Efsc
■c,m
FIGURE 7.7. Density of states vs. energy depicting the metal-insulator transition
for an amorphous material. Shaded areas represent localized states. Valence and
conduction mobility-edges are labeled Ev and Ec, respectively and Evm, Ecm denote
respective tail-edges of localized states of valence and conduction band-edges. Fermi
energies of respective semiconductor and metal phases are written ESFC, EF . Tran-
Transition of the Fermi energy from E™ to the domain of localized states corresponds
to the metal-insulator (M-I) transition. As noted in the text, widths of tail edges of
localized states are greatly exaggerated in this figure.
states are separated at critical respective energies, Ev and Ec, labeled "mobil-
"mobility edges." For crystalline metals the Fermi energy, EF, lies in the conduction
band. For crystalline semiconductors (in the "nondegenerate limit"), EF lies
in the band gap. These properties are roughly maintained in amorphous ma-
materials, so that again, for an amorphous metal EF lies in the conduction band.
However, in an amorphous metal, the band-edges have tails, composed of lo-
localized states (Fig. 7.7) and properties of the sample are critically dependent
on the location of Ef relative to Ec. The region between Ev and and Ec is
called the "mobility gap." In typical amorphous materials, tail edges extend to
a very small fraction of the mobility gap.
lfEF > Ec, the material acts as a metal. If EF < Ec, electrical conductivity
of the material decreases, with transport of charge carriers limited either to
hopping or excitation of carriers to Ec. In this case the material may be con-
considered an insulator. Consider that the value of EF can be varied, (Fig. 7.7)
and be made to fall from a value in excess of Ec to a value below Ec. In this
event there is a sudden change of the material from a metal to an insulator.
This transition is called the metal-insulator transition. The value of EF may be
made to change in a number of ways. Thus, for example, EF will change value
by altering the allow composition, by applying external stress to the sample,
or by the application of electric or magnetic fields. In Fig. 7.7 we have recalled
that the Fermi energy of an intrinsic semiconductor lies near the midpoint of
the energy gap. As noted previously, this type of localization is relevant to
intrinsic conduction materials.
7.2.3 Conduction Mechanisms and Hopping
A number of experiments indicate that at low temperatures, conductivity of
amorphous alloys does not fall beneath a temperature-independent minimum

512 7. Kinetic Properties of Metals and Amorphous Media
value of 5 x 105 Q~l m~x. With E.6.30) and Problem 5.52, we write
e2l
S
where SF is a measure of the Fermi surface. The preceding relation is relevant
to extended states in a metal sample. In this event, the mean free path, / > a,
where a is atomic spacing, or the distance over which the wavefunction loses
phase memory. As / reduces to a, states become localized. At the Ioffe-Regel
minimum,16 kl = 1 and 1 = a where k denotes wave vector. Substituting these
relations into B.7a) gives the conductivity
e2
B.7b)
37tzan
This value of a is the minimum on the delocalized side of EC.X1
Mechanisms for conductivity in amorphous media separate according to
temperature. Three basic intervals are: high, lower and very low temperature,
relevant to the respective degrees to which thermal energy can excite electrons
to domains within or above regions of localized states. For conductivity due to
motion of electrons within domains of localized states, the primary mechanism
is that of hopping. This mode of conductivity depends to a large degree on
thermal excitation. In the event that thermal fluctuations are insufficient to
supply hopping activation energy, a fourth mechanism for conductivity may
be available which depends on electrons tunneling to states at nearby or distant
sites of similar energy. (We recall that activation energy refers to the energy
required to carry a particle above a given potential barrier.)
High temperature
For temperature sufficiently high, thermal energy excites electrons above Ec
to extended states in the conduction band and standard ohmic current flows in
response to a potential gradient. In this event conductivity is given by
a = aoexp{-(£c - EF)/kBT] B.8)
where gq is a constant. Such conductivity applies to Ge and Si at room
temperature.
Lower temperature
At lower temperatures, thermal energy is sufficient only to excite electrons
if the interval of localized states (ECnfn, Ec) to the conduction-extended states
16A. F. Ioffe and A. R. Regel, Progress in Semiconductors, Heywood, London
A960).
17For further discussion see, N. F. Mott and E. A. Davis, Electronic Properties
in Non-Crystalline Materials, ibid; S. R. Elliott, Physics of Amorphous Materials,
Longman, New York A983).

7.2 Amorphous Media 513
domain E > Ec. For conductivity to occur, electrons must absorb sufficient
energy, A W\, to enable hopping from one localized state to the next. Activation
energy is, Ec>m + A W\, with related conductivity given by
or = or, exp{-(£c,m + AWi - EF)/kBT) B.9)
The constant o\ is approximately one thousand times smaller than gq. This
mode of conductivity is called, thermally assisted hopping.
Very low temperature
In semiconducting materials the Fermi energy typically lies in the band gap.
In an amorphous semiconductor the same is true and it is argued that there
is a high concentration of localized states in the vicinity of the Fermi energy.
Thermally assisted hopping may still occur for transitions to states close to the
Fermi energy with resulting conductivity
a =G2exp(-AW2/kBT) B.10)
where A W2 is approximately half the spread of the distribution of localized
states about EF. The constants o\ and g2 both depend on the hopping frequency
of electrons.
Variable-range hopping
At still lower temperature, for which thermal fluctuations are insufficient to
supply hopping activation energy, (kBT <^C A W2), there is an additional mech-
mechanism which permits electrical conductivity. In this mechanism, electrons in a
given energy state tunnel to nearby, or possibly distant sites, to states of sim-
similar energy. This process is called, variable-range hopping. The temperature
dependence of this mechanism was first derived by N. F. Mott.18
Since electrons are in localized states, their wavefunction varies with dis-
displacement R, as exp(—a R), where a is a constant. It follows that the probability
of hopping to a site at R, is given by exp(—lot R). Again it is assumed that states
are concentrated at EF. With the density of states written g(EF) the number
of states in the interval dE about EF is Vg(EF)dE, where V is volume. To
obtain the energy of hopping, we assume that V contains one state, and write
Vg(EF)dE = l B.11)
in which case dE is the spread between one state and the next. It follows that
the average separation (in energy) between neighboring states, A W3, is
= l/[Vg(EF)] B.12)
18N. F. Mott, Conduction in Non-Crystalline Materials, Oxford, New York,
A987).

514 7. Kinetic Properties of Metals and Amorphous Media
If we concentrate on states in the volume V = 4nR3/3, the latter expression
becomes
AW3 = 3/[4jtR3g(EF)] B.13)
This expression indicates that the farther an electron tunnels, the greater is
the probability that it finds a site with small AW3. The probability of a given
value of A W3 is exp{ — AW3/kBT}. It follows that the probability of hopping
through the displacement R to a nearby site at A W3 is
P(R, AW3) = Qxp{-2aR - (AW3/kBT)} B.13a)
Substituting the value AW3 from B.13) into the preceding relation gives the
probability
P(R) = exp{-[2aR + C/4nR3g(EF)kBT)]} B.13b)
This probability is maximum at the minimum of the exponent. There results
R = (9/S7tag(EF)kBT)l/4 = (l/nag(EF)kBT)l/4 B.14a)
On substituting this value of R into B.13) and then inserting the resulting value
of A W3 into the generic form
a = a3exp(-AW3/kBT) B.14b)
gives
a =a3exp(-B/Tl/4) B.14c)
() B14d)
This T ~l /4 behavior of conductivity has been observed in amorphous Ge film.J 9
With the preceding discussion it is noted that a plot of log a versus T~l
should reveal four distinct regions, three of which should display straight-line
behavior with respective slopes (Ec — EF) [see B.8)], (Ea + A W\ — EF) [see
B.9)], and A W2 [see B.10)]. This behavior has likewise been observed20 for
conductivity in the amorphous alloy, (As2Se3)o.95Tlo.o5 with slight curvature at
large T~l due to variable-range hopping.
7.2.4 Percolation Phenomena
Consider a network comprising a random array of independent resistors. Each
component either has a finite resistance (probability, p) or an open-circuit in-
infinite resistance (probability, 1 — p). A voltage source is placed across the
network. Current flows providing there is a connecting path of finite resis-
resistance elements between terminals, which occurs at some critical value of p>
19M. H. Gilbert and C. J. Adkins, Phil Mag 34, 143 A976).
20M. R Kotakata, et al., Semicond. Sci. Tech. 7, 313 A986).

7.2 Amorphous Media 515
TABLE 7.2. Applications of Percolation Theory
System
Critical Phenomenon
Mobility edge in amorphous
semiconductors
Dilute magnetic allows
Resistive-open circuit array
Liquid flow in porous media
Supercooling of molten glass
Polymer gelatin
Disease spread in a population
Composite superconductor-metal
materials
Variable-range hopping
Hydrofracturing of a rock
Neural networks
Localized to extended
states
Para-ferromagnetic transition
No current-current transition
Static to continuous flow
Glass transition
Liquid-gel transition
Containment-epidemic transition
Normal-superconducting transition
No current-current transition
Generation of a crack
No current-current transition
The event of current flow at the critical value of p is an example of the phe-
phenomenon of percolation. Another example of this phenomenon is that of the
phase transition of a dilute magnetic alloy from a paramagnetic to a ferromag-
ferromagnetic state, with p now denoting the fraction of magnetic atoms in the alloy.
The occurrence of a phase transition to the ferromagnetic state depends on
the existence of a sufficiently large connected cluster of interacting magnetic
atoms. This transition configuration occurs at a critical value of p. The phrase
percolation was first used by J. M. Hemmersly21 in the study of the passage of
a fluid through a network of channels, some of which are blocked, and some
of which are open. Numerous other examples of percolation are described in
the literature.21'22 (See Table 7.2.)
Bond and site percolation; clusters
Percolation divides into two categories: bond and site percolation. In site per-
percolation sites are either occupied or unoccupied. In bond percolation, bonds in
an array are either connected or disconnected.
Occupied sites fall into a cluster if any two sites belonging to that cluster have
a chain of occupied states connecting them. A measure of the size of a cluster is
given by the number of occupied sites in the cluster. Consider a uniform infinite
square array of lattice points with site occupation probability, p. Three finite
rectangular subsections of this array are shown in Fig. 7.8, with respective p-
21 J. M. Hemmersly, Proc. Cambridge Phil. Soc. 54, 642 A957).
22D. Stauffer, Introduction to Percolation Theory, Taylor and Francis, London
A985); J. W. Essam, Percolation and Cluster Size, in Phase Transitions and Crit-
Critical Phenomena, Vol. 2, ed., C. Dumb and M. S. Green, Academic, New York
A972); H. L. Frisch and J. M. Hemmersly, J. Soc. Indust. Appl Math. 11, 894
A963); G. Grimmett, Percolation, Springer, New York A989); M. F. Thorpe and
M. I. Mitkova, eds. Amorphous Insulators and Semiconductors, Kluwer Academic,
Dordrecht A997).

516 7. Kinetic Properties of Metals and Amorphous Media
p = 0.25
s = 3 —
s = l
(a)
p = 0.50
(b)
p-0.75
S=oo -
(c)
FIGURE 7.8. Rectangular subsections of infinite-two-dimension arrays illustrating
site percolation on the square lattice. Cluster size (s) is shown for three clusters, and
occupation probability (p) is for the three lattices. For the square lattice, percolation
(s = oo) occurs at p = 0.59. From: R. Zallen A983). Reprinted by permission of
John Wiley and Sons, Inc.
values: 0.25, 0.50, 0.75. It is evident that the cluster size for case (c) is infinite
and percolation has occurred. One may conclude that percolation for the two-
dimensional square array occurs at /?-value, 0.50 < pc < 0.75. The critical
value if pc = 0.59.21 If p* and psc represent, respectively, critical /7-values
for bond and site percolation, respectively, then it may be shown that p* 5
/?c5.23 Examples of this property are shown for four periodic two-dimensional
configurations and the Bethe lattice in Fig. 7.9. (At every branching of a Bethe
lattice a stem bifurcates into two branches.)
23J. W. Essam, Percolation and Cluster Size (ibid).

7.2 Amorphous Media 517
/\ /
XXX
Triangular
z-6
)nnd = 0 34 73
*ltP - 0 5000
Square
z=4
pbond = 0 5000
pcslte = 0 59
Kagome
z=4
pbond = 0 45
pcsite = 0 6527
Honeycomb
z=3
pbond = 0 6527
pcslte = 0.70
Bethe Lattice
z=3
pbond = 0 5000
pcsite = 0 5000
FIGURE 7.9. Critical site and bond percolation p-values, and coordinate number (z)
for four two-dimensional periodic arrays. Also shown in the Bethe lattice with effec-
effective infinite dimensionality (no closed loop exists for this lattice). From: R. Zallen
A983). Reprinted by permission of John Wiley and Sons, Inc.
We wish to obtain a relation for the mean size of a finite cluster, with site
probability p. (For bond percolation, p represents edge probability.) To this
end we introduce the number of clusters of size s, per lattice site, ns(p). It
follows that the probability that a given site in a cluster of size s is occupied
in sns{p). (In a lattice of 100 sites, which at a given value of p has, say,
three clusters of size s, ns(p) = 0.03. If s = 10, then sns(p) = 0.3.) Let
P(p) represent the fraction of occupied sites for the infinite cluster. Note that
P(l) = 1 and P(p) = 0, for p < pc. The probability that a site is occupied
for the infinite cluster is pP(p). The probability /?, that a site is occupied is
the sum of occupation probabilities overall cluster sizes. That is,
B.15a)
Since sns{p) represents the probability that a given site in a cluster of size s is
occupied, it follows that the mean size of finite clusters is given by
S(p) =
sns{p)
B.15b)
7.2.5 Pair-Connectedness Function and Scaling
Percolation is closely akin to critical phenomena in statistical physics which
is characterized by the behavior of thermodynamic variables as a function of
the temperature-displacement variable, t = (T — Tc)/Tc where Tc is criti-
critical temperature.24 A universality of such behavior is present in which such
thermodynamic variables scale with f, near the critical point, independent of
24K. Huang, Statistical Mechanics, 2nd ed. Wiley, New York A987); M. Plischke
and B. Bergersen, Equilibrium Statistical Physics, Prentice Hall, Englewood Cliffs,
NJ A989); H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena,
Oxford, New York A987).

518 7. Kinetic Properties of Metals and Amorphous Media
material, but dependent on dimension, d. The exponents of these scaling terms
are called critical exponents and are conventionally labeled: (a, K, y, r], v, a).
Of these six critical exponents, only two are independent.
In a similar vein, percolation phenomena likewise exhibits critical behavior
with related scaling relations.24 To demonstrate these relations the following
analogues with variables of statistical physics are noted. The total number of
clusters per site is analogous to the free energy per site.
B-16)
A parameter relevant to a given cluster is the pair connectedness function,
C(p, r), which is the probability that occupied sites a distance r apart are in a
common cluster. This function is analogous to the pair correlation function of a
statistical physics.25'26 The mean size of finite clusters is analogous to the sus-
susceptibility parameter in magnetic systems. It is then implied that these variables
have the following power-law singularities at the percolation transition.26
\2-a B.17a)
P(p)-(p-pcf B.17b)
S(P) ^ (p - pc)-r BAle)
exp[—r/£(/?)]
rd-2+n
where d is dimension and the correlation length £, which, with comparison to
known physical systems, is expected to diverge as
V B.l7e)
Universality of critical exponents for the percolation phenomenon indicates
that the percolation exponents a, /3, etc., should depend only on the dimen-
dimensionality of the system, not the lattice specifies nor whether the percolation is
site or bond type.
Rushbrooke, Fisher, and Josephson scaling laws
The basic premise of percolation scaling theory26 is that for given correlation
length, §, for p near pc, a cluster size s$ exists which is the dominant contri-
contribution to the percolation functions B.17a), B.17b), B.17c). The cluster size
s% diverges as p —► pc and is assumed to have the power-law behavior
sH-\p-pc\-x/o B.18)
which defines the critical exponent a. It is further suggested that
B.19)
25R. L. Liboff, Phys. Rev. A39, 4098 A989).
26D. Stauffer, Phys. Rev. 54, I A979).

7.2 Amorphous Media 519
where the function (p(x) —> 0, as x —> oo, and 0(jc) —> 1, as jc —> 0 but is
otherwise unspecified. Simulation calculations indicate that for large cluster
size, s, the probable number of occupied sites at the critical percolation state
behaves as, ns(pc) ~ s~T, where the parameter r depends on the dimension, d,
of the system. Inserting this behavior of ns(p), together with B.18) into B.19)
gives
B.20)
With B.17a) and B.20) we obtain for the total number of clusters per site,
= \P- Pcf^ j dx x-T0(x) B.21a)
where
x = s/sH B.21b)
and x is as implied and is the analogue of zero-field susceptibility. Assuming
a convergent integral, with B.17a), we obtain
a = 2 - 1—— B.22a)
G
In like manner there results
y = C - r)/a, 0 = (r - 2)/a B.22b)
These equations give Rushbrooke's scaling law27
a + 2fi + y = 2 B.23)
Another relation among these exponents may be obtained as follows. Integra-
Integration of the pair-connectedness function over the volume of the sample gives a
measure of the mean cluster size, S(p), in dimension, d
S(p) ~\p- pe\-y -jdrr^ (^/^f1) ^\P~ PeV*-* B.24)
which gives Fisher's scaling law28
y = vB- n) B.25)
Lastly, assuming a slowly varying integrand B.18) and B.21) give for the
concentration of clusters of dominant size,
^ \P~ P
x/o
C
27G. S. Rushbrook, J. Chem. Phys. 29, 842 A963).
28M. E. Fisher,P/iys. Rev. 180, 594 A969).

520 7. Kinetic Properties of Metals and Amorphous Media
Assuming that this concentration is inversely proportional to the volume, %d(d),
(in dimension d) occupied by these clusters, gives
cV
\P - Pc\T/a = \P~ P
which gives Josephson's hyperscaling law29
vd = 2 - a B.26)
Equations B.22), B.25), and B.26) are coupled relations for the six critical
exponents relevant to percolation scaling theory.
7.2.6 Localization in Second Quantization
Working in second quantization, the Hamiltonian of a one-dimensional
potential array may be written30
H = -
relevant to an electron in a given band. In the preceding, Sj is the energy of
an electron at j sites, with a corresponding Wannier wavefunction centered
at these sites. The fy matrix couples the ith and y'th sites, and c ■ and c} are
operators which create and annihilate an electron at the y'th site, respectively.
\fij matrices are also called: the transfer integral, the overlap energy integral
or the hopping integral, cf., B.4b)] Sums in B.27) are over the sites of a
perfect lattice with disorder introduced by taking either or both of the Sj and
tij coefficients as random variables subject to a given probability distribution.
We recall the following fermion anticommutation relation rules:
Note also that
H B.27a)
[ct,Cj]+ = S
cJ|O) = |i), cI-|i> = |0> B.27b)
where ,0) includes the empty ith site and the ket vector, |/), describes an
electron at the ith site.
In the tight-binding approximation of B.27) one sets t^ = t, for nearest
neighbors and zero otherwise. Energies Sj are set equal to eA with probability
PA, of sB with probability PB independent of the site j. This labeling provides
a one-dimensional model of a disordered binary allow.
29B. D. Josephson, Proc. Phys. Soc 92, 269 A967).
30A review of one-dimensional disordered systems is given by: K. Ishii, Prog.
Theor. Phys. Supplement, 53,11 A973).

7.2 Amorphous Media 521
Extended states for the uniform case
Let us show that for the uniform case, for which either PA = 1 or PB = 1, and
in the tight-binding approximation, one does not obtain localization. In this
event the Hamiltonian B.27) reduces to
= -t
We consider a lattice of length Na, corresponding to a chain of N atoms and
lattice constant a. The Hamiltonian B.26) may then be diagonalized through
the transformation
.t - x •,.,.,«,„ B.29b)
k =
k
2nn
(Na)
n = 0, ±1, ±2,..., ±[(N/2) - 1], ±N/2 B.29c)
Note that the latter sequence corresponds to periodic boundary conditions.
There results
H =
k k
where nk is the occupation number of the A:th site, 0 or 1. Eigenenergies are
given by
e(k) = sA -2tcoska, for PA = 1 B 31a)
(k) = eB — 2t coska, for PB = 1
For either case energies lie in a band of width At. The eigenfunction corre-
corresponding to s(k) is obtained by operating on the state |0) with the creation
operator b^. There results
\k£)=bfk\0) = -^=^b)exp(ikeja)\O) B.31b)
y/N j
The coordinate representation of this eigenstate is given by the Dirac product
B.31c)
m
This relation permits one to identify the probability amplitude of finding the
site m occupied as the product
(m\ke) = —— exp(imk£a) B.3 Id)
Vn

522 7. Kinetic Properties of Metals and Amorphous Media
It follows that the probability of finding an electron on the rath site is the con-
constant \/N, independent of ra or ke. We may therefore identify the wavef unction
(x\ks) as an extended state.
Disordered One-Dimensional Lattice
Transfer-matrix method
As noted above, to describe the disordered situation one sets energies sm =
sA with probability PA, and em = eB with probability PB, for all ra. We
outline the transfer matrix method31 for examining eigenfunctions and related
eigenenergies, E, of the Hamiltonian B.27) in the tight-binding approximation.
Eigenfunctions are written in the form
N N
\kE) = £ An\n) = J2A*c*n\0) B.32)
n=\ n=\
so that {m\kE) = Am. With B.27a) we obtain
(m\H\kE) = EAm = smAm - t(Am+] + Am_,) B.33)
where ra = 1,2, To satisfy periodic boundary conditions we set AN+\ =
A\ and AN = Ao. Introducing the two-component column vector
B.34a)
gives the recursion relation
B.34b)
Tm(em,E)=[ ' "M B.34c)
£m-E
Note that det 7} = 1, so that eigenvalues, /x( }, of T satisfy the relation
/x+/x_ = 1 B.35a)
which gives the forms
= e±w B.35b)
In the transfer-matrix method, solution for eigenenergies E is reduced to
finding solutions to the equation
, E)fN-i(eN-U E)... fi(eu E)]^/N B.36a)
31J. Ziman, Models of Disorder, Cambridge, New York A979), Section 8.2:
M. Plischke and B. Bergersen, Equilibrium Statistical Mechanics, ibid; M. L. Mehta,
Random Matrices, Academic, New York A991).

7.2 Amorphous Media 523
or, equivalently,
N
J B.36b)
m=\
At
where / is the identity matrix.
Band gap for the uniform array
We wish to demonstrate the existence of a gap in the spectrum of energies for
the case of a uniform array. In this event, Sj = sA for all m. With B.34c) and
B.35) it follows that the eigenvalues of T are given by
/x+ = R + iK = ei0 B.37a)
/x_ = R-iK = e~i9
R = 8a~E 9 k = y/\ - R2 B.37b)
Addition of equations B.37a), with B.37b) gives
E = sA -2tcosO B.38)
Periodic boundary conditions (fiN = 1), imply the discrete ^-spectrum, 6m =
2nm/N.
The condition
\eA - E\ > It B.39)
implies that K is purely imaginary, in which case eigen /x-values are real. For
K purely imaginary, K = i\K\ and /x+/x_ = — 1 which violates B.35a). We
may conclude that for real eigen /x-values, eigenenergies do not exist. This
corresponds to the existence of an energy gap for the uniform potential array.
Disordered chain
For the disordered case, in the product B.36b), transfer matrix components are
/v /v /v
either of the form Tm = TA or TB, depending respectively on whether site m is
occupied by atom A or atom B. To examine this situation we recall the Saxon-
Hunter theorem which states that any spectral region that is a spectral gap for
both a pure A-type chain and a pure Z?-type chain, is also a gap for any mixed
lattice of A- and Z?-type atoms.32 Consider the case that an eigenenergy lies
in such a forbidden domain. In this case, with real eigen /x-values, one again
encounters a violation of B.35a). However, let us consider for the moment that
solutions for this situation do exist.
32J. M.Luttinger, Phillips Res. Repts. 6, 303 A951).

524 7. Kinetic Properties of Metals and Amorphous Media
Let W± denote eigenvectors of T corresponding respectively to the eigen-
eigenvalues, /z±. The column wavefunction ^rn may be written, in general, as a
superposition of these eigenvectors and we write
= a<+> W+ + a<-> W- B.40)
where a^ are n-dependent parameters. It follows that
fa+l = tfa = ct?)ii+W++alr)V>-W- B.41a)
Returning to the situation of real eigen /z-values, we examine the case that
|/x+| > |/x_| so that
|/z+| = l/|/z_| > 1 >|/x_| B.41b)
Consider that a wavefunction of the form B.40) starts at the beginning of the
chain and passes through p such forbidden cells. In this case
fa = (Wo = a^\ti+yW++a[-X^yW_ B.41c)
The larger eigenvalue quickly dominates and ^rp grows as (exp yp) where the
growth coefficient y = In |/x+1 This hypothetical analysis hints at the existence
of localized states in forbidden domains of the energy spectrum.
Extensive numerical work of Ishii30 indicates that solutions do in fact exist in
such domains, and that corresponding solutions are localized. This conclusion
is akin to our previous observation that in a disordered material, localized states
occur at the edges of the forbidden band gap (mobility edge). In a closely
allied, fundamental work by Matsuda and Ishii33 it was demonstrated that all
one-dimensional disordered systems are localized.
Problems
7.1. Show the equivalence between the integral relations A.3b) and A.3a). Hint:
Introduce the parameter F = E — jx.
7.2. Derive the Fermi-Dirac integral series representation A.3e).
7.3. Establish the Fermi-Dirac vector coefficients A.3f)-(l .3h).
7.4. Establish validity of the time-reversal scattering-rate symmetry relation
A.30c).
7.5. Working in second quantization, write down the total Hamiltonian for an
electron-phonon system, incorporating the potential Hamiltonian component,
A.19). Hint: See E.6.6b).
7.6. For each of the cases listed in Table 7.1, indicate if the related transition site
or bond percolation.
33
H. Matsuda and K. Ishii, Prog. Theor. Phys. Suppl 45, 56 A970).

Problems 525
7.7. Employ the transformation equations B.29) to obtain the diagonalized
Hamiltonian B.30) and related eigenenergies B.31a).
7.8. Show that in the tight-binding approximation, the Hamiltonian B.27) reduces
to the Hamiltonian B.28).
7.9. Derive the eigenenergy relation B.33) from the wavefunction B.32) and
Hamiltonian B.27) under the said conditions.
7.10. What is the width of the energy gap for the uniform array described in B.37),
et seq.?
7.11. Show that Gruneisen's34 integral for metallic resistivity
5 [°/T x5dx
p = AT5 /
Jo
reduces to Bloch's form at T <^C 0, and has the correct linear T dependence
at T ^> 0 where 0 represents the Debye temperature.
7.12. (a) Describe conditions under which extended states exist in an amorphous
material.
(b) Describe conditions under which an amorphous metal does not conduct.
In both answers, define parameters.
7.13. A student argues that tearing a piece of paper is percolative. Is the student
correct. Explain your answer.
Answer
Tearing a piece of paper is a local and not a global event. For this reason the
process is not percolative.
34
E. Griineisen, Ann. Physik 16, 530 A932).

Location of Key Equations
Name
Balescu-Lenard equation
BBKGY equations
Boltzmann equation
Chapman-Kolmogorov equation
Conservation equations (in absolute variables)
Conservation equations (in relative variables)
Density matrix, Af-body, equation of motion
Euler equations
Fock-space function, equation of motion
Fokker-Planck equation
Grad's second equation
Green's function equation, coupled one- and two-particle
Green's function, Af-body, equation of motion
Hamilton's equations
Hierarchies of classical and quantum distributions
Kinetic equations, connecting relations
Krook-Bhatnager-Gross equation
Kubo formula, classical
Kubo formula, quantum mechanical
Lagrange's equations
Landau equation
Landau equation, Fermi liquid
Liouville equation
Liouville equation, quantum mechanical
Liouville equation, relativistic
Master equation
Location
D.2.71)
B.1.20)
C.2.14)
A.7.4)
C.14,3.18,3.19)
C.14,3.30,3.31)
E.5.23)
C.5.23)
E.5.29)
E.2.31)
B.6.12)
E.7.46)
E.7.59)
A.1.12)
(Table 5.2)
(Figure 4.9)
D.2.7)
C.4.66)
E.6.10)
A.1.7)
D.2.51)
E.4.13)
A.4.7)
E.2.9)
F.4.25)
A.7.20,5.2.28)

528 Location of Key Equations
Navier-Stokes equation C.4.8)
Pauli equation E.2.28)
Photon kinetic equation E.3.18)
Plasma convergent kinetic equation D.2.75)
Relaxation-time model kinetic equation E.3.35)
Uehling-Uhlenbeck equation E.4.5)
Vlasov equation B.2.30)
Vlasov equation, relativistic F.2.22)
Wigner distribution, equation of motion E.2.54)
Wigner distribution, hierarchy E.5.33)
Wigner-Moyal equation E.2.71)

List of Symbols
(See also Table 7.1, page 487)
A, A Vector and scalar functions in Chapman-Enskog
expansion C.5.36a).
A Vector potential.
A(k, to) Spectral density function E.7.12, 5.8.23).
(A)W Wigner counterpart of A E.2.55).
A(s) s-particle operator E.5.10).
a Acceleration. Friction coefficient in Fokker-Planck
equation D.2.30).
a\ a Creation and annihilation operators.
5 #h Operators a and a? in the Heisenberg representation.
Fourier coefficient in Prigogine expansion B.3.8).
B, B Tensor and scalar functions in Chapman-En skog
expansion C.5.36b).
B Magnetic field.
BE Boltzmann equation.
BL Balescu-Lenard equation.
BYj 5th equation in BBKGY hierarchy.
b Nondimensional impact parameter.
b Friction coefficient in Fokker-Planck equation D.2.30).
C mC2 = kBT in most of text. mC2 = 3kBT when C is
thermal speed C.4.2).
C2(l, 2) Two-particle correlation function B.2.18).
Ci2 Effective two-particle speed C.4.25).
C(q) Coefficient of electron-phonon interaction E.8.1).

530 List of Symbols
C C2 = 2RT C.5.28).
CK Chapman-Kolmogorov equation.
c c = v — (v). Deviation from mean microscopic velocity.
c = c/C Nondimensional microscopic velocity C.5.90).
c Speed of light.
cv Specific heat per molecule C.4.34).
cv Specific heat per molecule per mass C.4.35).
D Self-diffusion coefficient C.4.2). Density of points in
phase space (Chapter 1).
Variance of rv%.
Mutual diffusion coefficient C.4.29).
D/Dt Convective time derivative.
V Differential operator in Boltzmann equation C.5.7a).
E Energy.
Ef Fermi energy.
E Perturbative electric field D.1.4). Two-particle electric
field D.2.61).
E Fourier transform of E D.1.6b).
£ Expectation A.8.2a).
Ek Relative kinetic energy density C.3.26).
E(x) Elliptic integral C.6.22).
eK Absolute kinetic energy density.
F(t) Time-dependent component of perturbation Hamiltonian
C.4.59).
F{6) Term in collision cross section C.6.212).
F(co) Number of photons per frequency interval (Chapter 5).
F(/J) Integral in dielectric constant D.1.68).
Fs 5-tuple distribution function A.6.12).
Fs Fs = Vsfs. Bogoliubov s-particle distribution.
F Distribution function normalized to total number of
particles.
T Relativistic distribution F.2.1).
/ Classical one-particle distribution function in most of
text.
/ Nondimensional distribution function C.5.90).
/°, F° Local Maxwellians.
/o, Fo Absolute Maxwellians.
/o Exponential component of Maxwellian D.1.27).
/i Arbitrary single-peaked one-dimensional distribution
D.1.48). One-particle distribution function.
7i Velocity component of f\ D.1.48).
fo(E) Equilibrium occupation number per energy level
(Chapter 5).
fo(eo) Equilibrium distribution for number of photons per
frequency mode (Chapter 5).

List of Symbols 531
fs s -particle distribution function normalized to unity in
most of text.
/nA Truncated distribution in Grad's analysis (Chapter 2).
G\, G2 Generating functions (Chapter 1).
Gij Two-particle force B.1.7a).
G(x, x') Two-particle force B.2.27).
G(x, 0 Mean force field B.2.28b).
GR,A Green's function, retarded and advanced.
G s-particle Fock space operator E.5.7).
gi Constant of motion (Chapter 1).
g(r) Radial distribution function B.2.44).
g(co) Number of normal modes per frequency interval
(Chapter 5).
g^v Metric tensor (Chapter 6).
g Relative two-particle velocity.
H Hamiltonian.
H-"]2 Tensor Hermite polynomial C.5.92).
H Time-independent component of perturbation
Hamiltonian C.4.59).
h(r) Total correlation function B.2.47).
h\N) 5-particle conditional distribution function A.6.10).
h Enthalpy per particle (Problem 3.48).
H Boltzmann H function C.3.42).
Hilbert space [beneath E.1.18)].
Collision term in BY^ B.2.2).
Collision term in BY^ B.2.25).
I Identity operator.
I Integral [above F.2.52a)]. Tensor term C.5.100).
J(f) Collision integral.
J Current density.
K Externally supported force field C.2.1). Acceleration
C.5.7a). Momentum of center of mass C.3.68).
K Constant in Coulomb interaction D.2.7a). Constant in
inverse radial potential C.1.7).
Kinetic energy operator of 5th particle B.1.9).
Collision-integral kernel operator C.6.3).
Quasi-particle interaction function E.4.11).
Quasi-particle interaction function and constant
E.4.32a).
Elliptic integral C.6.22).
Wave number of closest approach D.2.73a).
Debye wave vector E.3.52).
Boltzmann's constant.
Thomas-Fermi wave number E.3.46).
Is
Ks
K(v,
K(p,
k,k
k(x)
kd
kn
Vl)
p)
0

532 List of Symbols
k, Momentum of ith molecule C.3.68).
k Momentum E.2.73, and following).
KBG Krook-Bhatnager-Gross equation.
/C Operator in Boltzmann collision integral C.6.29).
/ Mean free path C.2.2).
C2 Space of square integrable functions.
LIM Limiting process [defined above E.7 to 5.19)].
L(q, q, t) Lagrangian A.1.3).
LN TV-particle Liouville operator B.1.3).
8L Perturbation Liouville operator B.3.4).
M Mach number (Chapter 4).
M\(n), M2(n) Moments of probability distribution A.7.24).
M\2 Rate of momentum transport per unit volume C.4.26).
A/*(z) Phase density B.5.4).
J\f Number of ensemble points in phase space A.4. lb).
N Total number of particles.
Nz Number of zeros D.1.67).
Np Number of poles D.1.67).
n Particle number density.
Oij Operator in Liouville equation B.1.7).
p Pressure tensor in lab frame C.3.15).
p Scalar pressure C.4.4).
p Probability of step to right A.1.8 to 1.1.14). Momentum
C.4.23).
pN 3 N -dimensional momentum vector relevant to TV point
particles E.2.46).
P/r Fermi momentum E.3.34a).
p^ Four-momentum (Chapter 6).
p Momentum four-vector F.1.2). Momentum three-vector,
Problem 1.40.
p Momentum.
P Relative pressure tensor C.3.24).
P Probability.
V Principal part D.1.71).
q Probability of step to left A.8.14).
q Heat flow vector in lab frame C.3.16).
Q Relative heat flow vector.
QK Kinetic energy flow vector B.1.34a).
Qcfy Potential energy flow vector B.1.34b).
QL Landau collision tensor D.2.72).
Q Collision form in derivation of the Balescu-Lenard
equation D.2.64).
<2bl Balescu-Lenard collision term D.2.69).

List of Symbols 533
<2l Landau collision term D.2.72).
r Interparticle displacement vector. Radius vector.
r Total number of events A.8.21).
rv Random variable.
R kB/m.
R Radius to center of mass (Problem 1.7).
5(k) Momentum operator E.2.76b).
S Action integral A.1.4).
S(k) Structure factor B.2.51).
S^ Sonine polynomial C.5.54a).
S Spin operator (Chapter 5).
S Superoperator E.6.11).
S Response component of pressure tensor C.4.4).
s Impact parameter.
T Temperature. Kinetic energy.
T Tensor of products of relative velocity, g (Chapter 4).
T Integral-kernel operator C.6.11). Time ordering operator
(Chapter 5).
U(T) Total radiant energy density E.3.12).
U Generalized potential (Chapter 1).
f/(l, 2) Single-time interaction potential E.7.49a).
U Unitary operator for Schroedinger equation E.1.39).
u(x, t) Macroscopic fluid velocity (Chapter 3).
ud Downstream fluid speed D.2.12).
up Upstream fluid speed D.2.12).
V Renormalized volume B.3.9).
V Vector in F-space A.4.16a).
V/r Fermi velocity [beneath E.4.31)].
v Microscopic velocity (Chapter 3).
w(y\ v) Probability scattering rate D.3.1), E.3.5).
w(\', v) Augmented scattering probability rate D.3.3).
xbij Scattering rate A.7.20).
ibij Transition probability A.7.22).
wnk Transition probability E.2.20).
x^ A:th particle position vector E.2.55).
xN 3N particle position vector relevant to TV point particles.
xM Four-displacement (Chapter 5).
x Displacement four-vector F.1.1). Displacement
three-vector (Problem 1.40).
ZN TV-body partition function (Chapter 3).

534 List of Symbols
Greek and Other Symbols
a, C, y Constants in homogeneous distribution C.5.32).
a Nondimensionalization parameter B.2.11).
an Matrix element in Chapman-Enskog expansion C.5.56).
a Apsidal vector C.1.38).
a(n) Displacement of Fermi surface in direction of n E.4.31).
ax, etc. Components of spin eigenvector E.1.33).
f}x, etc. Components of spin eigenvector E.1.33).
j3 Dimensionless frequency D.1.42).
fin Matrix element in Chapman-Enskog expansion C.5.64).
Dimensionless plasma frequency D.1.50).
Kronecker symbol B.3.9).
j Kronecker-delta symbol.
<5(r) Three-dimensional delta function.
£ Dielectric constant (Chapters 4 and 5). Parameter of
smallness.
s(x, p) Quasiparticle energy E.4.10).
£ Electric field.
,, Xj) Interaction potential A.4.24).
Fourier transform of interaction potential B.3.24).
Field creation operator E.5.3).
Field operator in Heisenberg picture E.5.7a).
(Pe(*n) Stationary energy eigenstate E.1.8).
y Relativistic parameter (Chapter 6).
Nondimensionalization parameter B.2.11).
Flux vector (No./cm2-s) C.3.11).
Maximum total scattering rate D.3.4a).
Christoffel symbol F.4.21).
Coefficient of viscosity Chapter 3.
rj(t) Coarse-grained entropy C.8.12).
k Coefficient of thermal conductivity C.4.3). Constant in
Coulomb cross section D.2.21a). Absorption coefficient
E.3.28).
ks Kinetic energy operator B.2.22a).
K\2 Coefficient in mutual diffusion expression C.4.26).
k^ Coefficient of thermal conductivity tensor E.3.42).
k(\) Total scattering rate D.3.2).
k(\) Maximum scattering rate C.3a).
Xd Thermal de Broglie wavelength E.3.5). Debye distance
D.1.36).
A.TF Thomas-Fermi wavelength E.3.46a).
A Plasma parameter (Chapters 2 and 4).
A Liouville operator A.4.20).
A Symmetric strain tensor C.4.6).

List of Symbols 535
\x Reduced mass. Mobility (Chapter 3). Chemical potential
E.3.1). Relativistic parameter F.3.2).
Ia(A) Measure of set A C.8.3a).
v Collision frequency C.5.2).
i/i2 Collision frequency for mutual diffusion C.4.24).
Q(n-q) Element of collision integral in Chapman-Enskog
expansion C.5.62).
cjop Plasma frequency D.1.18).
co(c) Maxwellian C.5.91)
vl>(x\ 0 Wavefunction E.1.1).
Electrochemical potential C.4.15a).
Momentum flux E.4.23).
, t | zo, to) Two-time distribution A.7.2).
, 0 Coarse-grained entropy C.8.13).
p Mass density C.3.18a).
p Density operator E.2.1).
pnq Density matrix E.2.3).
g Stefan-Boltzmann constant E.3.15).
<To2 Molecular diameters C.4.30).
Sum of molecular radii C.4.30).
a^v Electrical conductivity tensor E.3.41).
cr(E, 6) Scattering cross section C.1.24).
gF) Scattering cross section C.1.31).
Scattering cross section for rigid spheres C.4.30).
Total scattering cross section C.1.34).
Pauli spin operator E.2.39).
Element of hypersurface in x space F.2.2).
Element of hypersurface in p space F.2.3b).
Element of surface area C.8.7).
Relaxation time E.3.35a; 5.3.79). Proper time F.1.23).
Dimensionless velocity (Chapter 3). Quantum parameter
E.4.6a).
Time-ordering operator E.8.7).
□ Collision operator C.6.1).
Symmetric sum of gradients C.5.101).
Collision term in photon kinetic equation E.3.19).

APPENDIX A
Vector Formulas
and Tensor Notation
A.I Definitions
The vector function of position A(r) is a set of three functions AM; \i = 1, 2, 3.
These represent the projection of the vector A onto the Cartesian axis, so that
A\ = Ax, A2 = Ay, A3 = Az. The tensor notation of the vector A is AM.
More generally, the set of 3n functions
in which each of the n indexes {ip} run from 1 to 3, is called an nth-rank tensor
in three dimensions. A scalar carries no indexes and so is a zeroth-rank tensor.
A vector is a first-rank tensor.
In the text the vector notation for a second-rank tensor 0MV is
I
A popular second-rank tensor has as its nine components the products of the
components to two vectors, A and B and is call a dyad.
(A.I)
V = AMBV (A.2)
A second rank tensor j/r^ is symmetric if
^ixv = ^vu. (A.3)
It is antisymmetric (or skew) if
(A-4)

538
A. Vector Formulas and Tensor Notation
a symmetric or antisymmetric tensor can always be constructed from an
arbitrary tensor in the following manner:
(^ ^ (A*5)
The trace of a tensor is the sum of its diagonal components:
3
(A-7)
A tensor of zero trace can be made from
(A.8)
(Repeated indexes are summed.) The second-rank Kronecker delta symbol
is defined as
8UV = 1 for u = v
= 0
A symmetric tensor of zero trace is
otherwise
" 3
In vector notation, equations (A.8) and (A. 10) appear as
) 1
=0 --(Tr0)/
The third rank Levi-Civita symbol £/VM is defined as
slv/1 = +1 if lv\i is an even permutation of 1, 2, 3
= — 1 if Ivfi is an odd permutation of 1, 2, 3
= 0 if any two indexes are equal
The vector or "cross product to two vectors is defined by
AxB =
Ax Ay Az
Bz
(A. 10)
(A. 11)
The triad of unit vectors (i, j, k) lies along the Cartesian axis with i in the x
direction, j in the y direction, and k in the z direction. In tensor notation the
cross product appears as
(AxB)fl=slxlpAlBp

A.2 Vector Formulas and Tensor Equivalents 539
The inner or dot product of two vectors is defined by
A B = AXBX + AyBy + AZBZ (A. 13a)
A B = AM£M (A. 13b)
The del operator V is written for
d d d
V = i— + j— + k— (A. 14a)
9 9 9
V = — (A. 14b)
OX ^
The gradient of a scalar function 0 is defined by
d
grad0 = V0(grad0)M = 0 (A. 15b)
The curl of a vector is defined by
(A. 16a)
d
(curl A)M = £M/V —Av (A. 16b)
The divergence of a vector is defined by
(A. 17a)
A* (A.17b)
A.2 Vector Formulas and Tensor Equivalents
Some useful vector formulas together with their tensor-notation equivalents
are given below:
V • 0A = 0(V • A) + (A • VH (A. 18a)
(A. 18b)
V x 0A = 0V x A - A x V0 (A. 19a)
a a a
0^ tA A0 (A. 19b)
= BVxA-AVxB (A.20a)
Av (A.2(Jb)
V(A B) = (A V)B + (B • V)A + A x (V x B) + B x (V x A) (A.21a)
d A 9 d A
— AUBU = An—Bu + Bu—An (A.21b)
OXi

540 A. Vector Formulas and Tensor Notation
V x
(Use equation A.25 in
(V
d
i
V
V-
d
i
x A) =
d
dxn
x (V0) =
a a
(V x A) =
d
-- V(V • A)
■ 4 p ,,
0
= 0
^^
-a
A.24b to obtain A.24a.)
V-r = 3
d
(V-
a
V)A
a
i
The symmetric sum of gradients operator ^ is defined by
(A.22a)
(A.22b)
(A.23a)
(A.23b)
(A.24a)
(A.24b)
(A.25)
(A.26a)
(A.26b)
V x r = 0 (A.27a)
d
& = 0 (A.27b)
Xp
(A • V)r = A (A.28a)
d
A5 A (A.28b)
V x (A x B) = -BV A - A VB + AV B + B VA (A.29a)
d d d
AB = BaA[ — AdB[
d
SpkhAkBh BaA[
dxn dxa dxd (A.29b)
=-—(BaAi - AaBi)
dxa
Vx(AxB) =V(BA-AB) (A
(A x B) x (C x D) = (A • C)(B D) - (A D)(B C)
s^kiAkBie^pCdDp = AipCipBkDk - AnDnBjCj (A.30b)
(A x B) x (C x D) = [A (B x D)]C - [A (B x C)]D (A.31a)
(A.31b)
dx
(A.32)
v

APPENDIX B
Mathematical Formulas
B.I Tensor Integrals and Unit Vector Products
Let
Kii -iN = / vhvi2 • • • viNf(v2)d\ (Bl.l)
where vik are Cartesian components of the unit vector v so that the index ik
runs from 1 to 3. Then
Aili2...iN = 0, for N odd
whereas, for N even,
l E / (B 1.2)
where the summation is over the P permutations of the integers i\ --In and
(N + 1)!! denotes the skip factorial
(N + 1)!! = (N + l)(N - l)(N - 3) • • •
With these formulas, the case N = 2 gives
f{v2)dv
For TV = 4, we obtain
1
Aiihhu — 77[<5/ll25/3i4 + <5/,/3<5/4/4 + <5/2/4<5/2/3] / f(v )d\

542 B. Mathematical Formulas
B.2 Exponential Integral, F Function, £ Function,
and Error Function1
B. 2.1 Exponential Integral
The definition of this function is
Ex{z)= / —dt (B2.1)
where | argz| < n. E\(z) has the expansion
Ei(z) = -y-\nz-T ^-^— (B2.2)
~ nnl
n = \
where y is Euler's constant, which is given by
/ 1 1 1 \ r°°
y = lim 1H 1 1 1 lnm = - / dte rlnr
m^oo \ 2 3 m / Jo
= 0.5772157
Here are a few other integral properties of E\(z):
e~at
dt = eabEx{ab) (B2.3)
b + t
Jo
Cz 1 — e~f
L —
I
dt = E\
(B2.4)
E\(t)dt = 2 In 2
is a member of the set of integrals
f°° e~zt
En{z) = / —dt, n= 0,1,2,... (B2.5)
i/i •■
with Re z > 0. These functions have the recurrence relation
En+i(z) = ~[e~z - zEn(z)l n = 1, 2,... (B2.6)
n
Furthermore,
dEn(z)
dz
(B2.7)
'For additional properties, see M. Abramowitz and I. A. Stegun, Handbook of
Mathematical Functions, Dover, New York A970).

B.2 Exponential Integral, F Function, f Function, and Error Function 543
and
En@) =
E0(z) =
1
n-l'
n > 1
,-z
(B2.8)
and
I
OO ,_{y
e~atEn(t)dt = ■
a
n
n-\
-l)ka'
k=\
a > — 1
(B2.9)
The asymptotic expansion of En(t) for large z is given by
,-z
En(Z)
n n(n + 1) n(n + l)(n + 2)
(B2.10)
for (| arg z\ < 3n/2). Two functions closely related to the exponential integral
are given by
e< dt
f00 -e~' dt r e<
Ei(x) = -V I = V I
J-X * ^-OO *
for x > 0, where V denotes principal part, and
(B2.ll)
li(x) = V
r dt
Jo Inr
(B2.12)
for x > 1. Expansion of Ei(x) is given by
00
n = \
nnl
(B2.13)
for x > 0. We note also that
Jo t
dt = Ei(x) — \nx — y
r°° eiat dt
Jo * =f ib
= e
±ab
Ex(ab)
— Ei(ab) + in
(B2.14)
(B2.15)
In the latter relation the — sign on the left side corresponds to the + exponential
and top member of the column and a > 0, b > 0.
F -Function
The r is defined by
'OO
Rez>0
(B2.16)

544 B. Mathematical Formulas
For z = n, an integer,
V(n) = (n- 1)!
(B2.17)
so that
(B2.18)
Here are some typical values:
= 1
FI2 =2
Stirling's asymptotic expression for z
given by
(B2.19)
oo in the domain | argz| < n is
(B2.20)
Riemann ^ Function
This function is defined by
oo
k=\
for Re z > 1 • First we note that
n
z -
(B2.21)
(B2.22)
where the product is over all primes p. We may also write
f (z) =
Here are some special values:
— f
1 V^/ t/0
00
-1
dx
Tf
? I 2 I = 2612
f D) =
7T
90
(B2.23)
- = 1.341
945

B.3 Other Useful Integrals 545
Error Function
The error function is defined by
erf z =
' dt
We also define
erfc z = 1 — erf z =
f
J z
oo
e ' dt
The error function has the series expansions
ry OO
erf z =
Asymptotic expansions are given by
2
jc <$C 1 : erf x ~
1 3
x - -x5 +
10
jc —
Some integral relations for this function are as follows:
'oo 1/^2
/•OO
Jo
1 — exp
—a
~b2
2\ 1
'00
1
/ e~at erf Vbi dt = lf
Jo ay a + b
f°° Ib
/ e-aterfcj-dt = -
Jo M t a
(B2.24)
(B2.25)
(B2.26)
(B2.27)
(B2.28)
e~at zrfbtdt = - exp ( -^- ) erfc (—) (B2.29)
a \4b2 J \2bs
(B2.30)
(B2.31)
(B2.32)
B. 3 Other Useful Integrals
'00
e~y dy = y/jt
Jo
— 00
oo
E{n) = / e~ax xndx, n>0,a>0
E(n)= -T t
2 V 2
£@) = ^*"l/2 EQ) = l-a~2
(B3.1)
(B3.2)
(B3.3)

546 B. Mathematical Formulas
£A) = X-a~x E{4) = l^a' (B3.4)
E{2) = - V7ra/2 £E) = a
/ -—r = T(n)X(n) (B3.5)
Jo x + 1
ex +
00
s E
Green's Identities
Green's First Identity
/ @VV + V0 • V^) dV = f
Jv Js
Green's Second Identity
/ @VV - V^V0) dV = I
Jv Js
Other Identities
/ V xAdV = I
Jv Js
V xAdV = dSxA
s
7 4
= ln2, A.D) = —
w 720
7T2 317T6
A.B) = — A. F) = (B3.7)
w 12 w 30,240
(B3.8)
(B3.9)
<pd\= I dS x V0 (B3.10)
Jc Js
B.4 Hermite and Laguerre Polynomials and the
Hypergeometric Function
Hermite polynomials
These polynomials are defined by the relation
Hn(z) = (-iyV2 (j^e~z2) > * = 1, 2, 3, ... (B4.1)

B.4 Hermite and Laguerre Polynomials and the Hypergeometric Function 547
They satisfy the differential equation
and have the generating function
exp
Recurrence relations are given by
d
n\
dz
2z-
d
d~z
Hn =
Hn =
n
Hn-\
2zHn = Hn+{
Here is a list of the first few polynomials
#0 =
H2 =4z2-2
HA = 16z4 - 48z2 + 12
=2z
= 8z3
- 12z
Tensor Hermite polynomials are discussed in Section 3.5.10.
Laguerre Polynomials
These polynomials are defined by the relation
"
dz"
k, n = 0, 1, 2,..
dz" n+k
These functions satisfy the differential equation
d2
dz-
and have the generating function
dz
K=0
e-zt/{\-s)
oo
Orthogonality relations for these polynomials are given by
./o
oo
e zz
n m
n\
nm
(B4.2)
(B4.3)
(B4.4)
(B4.5)
(B4.6)
(B4.7)
(B4.8)
(B4.9)

548
B. Mathematical Formulas
Hypergeometric Function
These functions are defined by the series (for \z\ < 1)
ab z a(a + l)b(b + 1) z2
F(a, b, c\ z) = 1 + — - + —— -
c 1! c(c + 1) 2!
a(a
2)b(b
c(c+ l)(c
and are related to the differential equation
2) z3
3!
(B4.10)
z(l-
dz-
(c-(a
d u
ab
(B4.ll)
where c ^ —n, (n = 1,2,...). The function F(a, b, c\ z) is the solution of
(B4.11)for<p@)= 1.
The general solution of the differential equation (B4.11) is given by
<p = AxF(a9 b, c\z) + A2zl~cF(a + 1 - c, b + 1 - c, 2 - c\z) (B4.12)
for |z| < 1 and c ^ 0, ±1, ±2,
For large z,
r(c)r(&-<i), x_fl . r(c)r(fl-6)
a> b> c; z)
-b
(B4.13)
- a) r(fl)r(c - 6)
The relation of these functions to associated Legendre polynomials is given
by [for | arg(z ± 1)| < tt]
r2 _ 1 yn/2 / \ — Z
~^2
(B4.14)
whereas the relation to the Legendre polynomials is given by
(B4.15)
Confluent Hypergeometric Function
These functions are defined by the series (convergent in the whole complex
plane)
az
a, c,z) = 1 H
ll
l)z2 t a(a + \){a + 2) z3
{ l)( + 2) 3!
c 1! c(c+ 1J! c(c+ l)(c + 2) 3!
and stem from the differential equation
a
<p =
The general solution of (B4.17) is given by (for c noninteger)
cp = Ai<P(a, c,z) + A2xl~c<Z>(a — c + 1, 2 - c\z)
(B4.16)
(B4.17)
(B4.18)

B.4 Hermite and Laguerre Polynomials and the Hypergeometric Function 549
The relation of these functions to the hypergeometric series is given by
(p(a,c;z)= lim F (a,b,c,-) (B4.19)
b-+oo \ b/
For large z
£i, c\ z) - e-*"*™ z~a + ^ezza~c (B4.20)
F(c — a) F(a)

APPENDIX C
Physical Constants
Velocity of light in vacuum
Planck's constant
h
n
Avogadro's number
Boltzmann's constant
Gas constant
Electron rest mass
Proton rest mass
B
R = Nokb
Volume of 1 mole of perfect gas, at
normal temperature and pressure
Electron charge e
m
M,
2.9979
2.9979
6.6261
6.6261
4.1357
1.0546
1.0546
6.5821
6.0221
1.3807
1.3807
8.6174
X
X
X
X
X
X
X
X
X
X
X
X
108m/s
1010 cm/s
104 J s
107 erg sec
105 eV s
104jS
107 erg s
106 eV s
1023 atoms/mol
103 J/K
106 erg/K
10 eV/K
8.3145 J/molK
8.3145
X
107 erg/mol K
1.9870 cal/molK
22.241
1.6022
4.8032
9.1094
9.1094
liters
X
X
X
X
1019C
100 esu
101 kg
lO8 g
0.511 MeV
1.6726
1.6726
1.0073
938.27
X
X
107 kg
lO4 g
amu
MeV

C. Physical Constants 551
Neutron rest mass
Bohr magneton
Ratio of proton mass to electron
mass
Charge-to-mass ratio of electron
Stephan-Boltzmann constant
Rydberg constant
Bohr radius
Triple point of water
Atomic mass unit
Mn
Mp
m
e_
m
G
1 amu
1.675 x 107kg
1.675 x 104g
939.57 MeV
2.273 x 101 erg gauss
1836.1
1.7588 x 1011 C/kg
5.2730 x 1017 esu/g
5.6697 x 10 erg cm
109,737.32 cm
0.52918 X
273.16 K
1.6605 x 104g
931.5 MeV

^s
K
-4
Useful Conversion Constants and Units
Constants, mks units
1 electron volt
1 coulomb
1 weber per square meter
1 eV/molecule
1 entropy unit = 1 eu
1 poise (unit of viscosity)
1 bar (unit of pressure)
Mo
eV
8.8542 x 1012 F/m
4pi x 10~7 H/m
1.6022 x 10~12erg
1.6022 x 109 J
3.829 x 100 cal
11.605 K
0.1 abcoulomb (emu)
2.9979 x 109 statcoulomb (esu)
104 gauss
96.485 kJ/mol
23.06 kcal/mol
NokB = 2 cal/K
1 dyne-s/cm2
106 dynes/cm2 = 0.9869 atm.

APPENDIX D
Lorentz-Legendre Expansion
In this appendix we wish to outline the precursor relations leading to G.1.48).
The expansion G.1.47) is formally written
oo
= £ • k = cos 6
(Dl.la)
(Dl.lb)
where hatted variables are unit vectors and P/(/x) are Legendre polynomials.
The variables (k,6,<f>) in (Dl.l) are spherical coordinates of the k vector
(Fig. D.I). These unit vectors have the following Cartesian components:
k = [cos(p^/l — /x2, sin (p^/l — /x2, /x]
(D1.2a)
FIGURE D. 1. In this diagram, £, k, and 0 lie in a plane and (/> lies in a plane normal
to £.

D. Lorentz-Legendre Expansion 553
Furthermore,
0 = [/xcos0, /xsin0, —y 1 —
= [— sin0, cos0, 0]
'« d « d
k—+0
(D1.2b)
(D1.2c)
d
dk
dk
d
kdO " (ksinO)d(l)_
(D1.3a)
We choose £ to lie in the z direction so that
d Ir
z k
dk
Z • U 1- Z • (f>
k9# (k sin
(D1.3b)
With (D1.2c) the third term in (D1.3b) vanishes and one is left with
dk
= £
d
d
kdo
(D1.3c)
With the given orientation of £ one may take // = fi(E) in the expansion
(Dl.la). With
d IE d
(D1.4a)
dk
d
~d6
k dE
d
(D1.4b)
one obtains
dk
It follows that
d
dk
or equivalently,
d
(/o + M/i) =
dk
(/o + M/i) =
—
k
IS
k
k
-
-
-
d
d
3£
(D1.5a)
d
d
dE
d/X_
(fo + fifi) (D1.5b)
1 -
3£
(D1.5c)
where it was noted that d/xfi(E)/d/x = fx{E) and that dfo/d/x = 0. The first
few Legendre polynomials are given by
1 o 2
2 ' ^ ^ (D1.5d)
3 3
Keeping Legendre polynomials up to order / = 1, we note that /x2 = |
The relation (D1.5c) reduces to
(D1.6)
dk
This relation gives rise to G.1.48).
dE

APPENDIX E
Additional References
E. 1 Early Works
1. Analytical Theory of Dynamics
Hamilton, Sir W. R.
(a) "Problem of Three Bodies by My Characteristic Function" A833).
(b) "On a General Method in Dynamics, by which the Study of the Motions
of all Free Systems of Attracting or Repelling Points Is Reduced to the
Search and Differentiation of One Central Relation, or Characteristic
Function," Phil. Trans. Roy. Soc, Part 11, 247-308 A834).
(c) "Second Essay on a General Method in Dynamics," Phil. Trans. Roy.
Soc, Part I, 95-144A835).
Jacobi, K. G. J.
Vorlesungen UberDynamik, Berlin, G. Reimer, 1866, second edition, 1884.
Lagrange, J. L.
(a) Mecanique Analytique A788), Oeuvres, 6, 335 A773).
(b) Mem. de L'Institut de France A808).
Liouville, J.
"Note sur la theorie de la variation des constantes arbitraries," J. de Math.,
3, 342A838).
Poincare, H.
(a) Oeuvres, Paris, Gauthier-Villars, 1951-1954.
(b) Methodes Nouvelles de la Mechanique Celeste A892). Reprinted by
Dover, New York, 1957.
(c) "Sur le probleme des trois corps et les equations de la dynamique," Acta
Math., 13, 1 A890); Oeuvres, 1, 272.

E.I Early Works 555
Poisson, S. D.
(a) "Sur la chaleur des gaz et des vapeurs," Ann. Chim., 23, 337A823), Ann.
Physik, 76, 269 A824); English translation with notes by J. Herapath,
Phil. Mag., 62, 328 A823).
(b) "Memoire sur la theorie du son," Nouv. Bull. Sci. Soc. Philomat. (Paris)
1, 19 A807); J. Ecole Polyt. (Paris), 7, 14° Cah., 319 A808).
(c) Nouvelle theorie de Vaction capillaire, Paris, Bachelier, 1831.
(d) Theorie mathematique de la chaleur, Paris, Bachelier, 1835.
2. Kinetic Theory of Matter
Boltzmann, L.
(a) Lectures on Gas Theory, translated by S. G. Brush, University of Cali-
California Press, Berkeley, 1964. This text contains an excellent history of
the theory and a very extensive bibliography.
(b) "Further Studies on the Thermal Equilibrium Among Gas-molecules,"
Wien. Ber.f 66, 275A872).
(c) "On the Thermal Equilibrium of Gases Subject to External Forces," Wien.
Ber, 72,427A875).
(d) "On the Formulation and Integration of Equations Which Determine the
Molecular Motion in Gases," Wien. Ber.y 74, 503 A876).
(e) Wien. Ber., 81, 117A880).
(f) Wien. Ber., 84, 40 (mi).
(g) Wien. Ber., 84, 1230 A881).
(h) Wien. Ber., 86, 63A882).
(i) Wien. Ber, 88, 835 A883).
(j) Jahresb, d. D. Math. Verein, 6, 130 A889).
Maxwell, J. C.
(a) The Scientific Papers of James Clerk Maxwell, Cambridge University
Press, 1890; reprinted by Hermann, Paris, 1927, and by Dover, New
York, 1952.
(b) "Illustrations of the Dynamical Theory of Gases. I. On the Motions and
Collisions of Perfectly Elastic Spheres. II. On the Process of Diffusion
of Two or More Kinds of Moving Particles Among One Another. III. On
the Collision of Perfectly Elastic Bodies of Any Form." Phil. Mag. [4],
19, 19; 20,21, 33 A860); Brit. Assoc. Kept. 29 B), 9 A859); Athenaeum,
p. 468 (Oct. 8, 1859); L'Institut 364 A859).
(c) "Viscosity or Internal Friction of Air and Other Gases," Phil. Trans., 156,
249 A866); Proc. Roy. Soc. (London), 15, 14 A867).
(d) "On the Dynamical Theory of Gases," Phil. Trans., 157,49 A867); Proc.
Roy. Soc. (London), 15, 146 A867).
(e) "A Discourse on Molecules," Nature, 8, 437 A873); Phil Mag. [4], 46,
453 A873); Les Mondes, 32, 311, 409 A873); Pharmaceut, J., 4, 404,
492,511 A874).

556 E. Additional References
(f) "On the Dynamical Evidence of the Molecular Constitution of Bodies"
(lecture). Nature, 11, 357, 374 A875); J. Chem. Soc, 13, 493 A875);
Gazz. Chim. ItaL, 5, 190 A875).
(g) "On Stresses in Rarified Gases Arising from Inequalities of Tempera-
Temperature," Phil. Trans., 170, 231 A880); Proc. Roy. Soc. (London), 27, 304
A878).
(h) "On Boltzmann's Theorem on the Average Distribution of Energy in a
System of Material Points," Trans. Cambridge Phil. Soc, 12,547 A879);
Phil. Mag. [5], 14, 299 A882); Ann. Physik Beibl, 5, 403 A881).
3. Theory of Rare Gases and Solution to the Boltzmann Equation
Burnett, D.
(a) "The Distribution of Velocities in a Slightly Non-Uniform Gas, Proc.
Lond. Math. Soc, 39, 385 A935).
(b) "The Distribution of Molecular Velocities and the Mean Motion in a
Non-Uniform Gas," Proc. Lond. Math. Soc, 40, 382 A935).
Chapman, S.
(a) Phil. Trans. Roy. Soc. A, 277, 433 A912).
(b) On the Law of Distribution of Velocities, and on the Theory of Viscosity
and Thermal Conduction, in a Non-Uniform Simple Monatomic Gas,"
Phil. Trans. Roy. Soc A., 216, 279 A916).
(c) Phil. Trans. Roy. Soc A, 217, 115 A917).
(d) Phil. Mag., 34, 146 A917).
(e) Phil. Mag., 34, 182A919).
(f) "On Certain Integrals Occurring in the Kinetic Theory of Gases,"
Manchester Mem., 66, 1 A922).
(g) Manchester Mem., 7, 1 A929).
(h) "On Approximate Theories of Diffusion," Phil. Mag., 5, 630 A928).
(i) "On the Convergence of the Infinite Determinants in the Lorentz Case,"
J. Lond. Math. Soc, 8, 266 A933).
Chapman, S., and Hainsworth, W.
"Some Notes on the Kinetic Theory of Viscosity, Conduction, and
Diffusion," Phil. Mag., 48, 593 A924).
Enskog, D.
(a) Phys. Zerl, 12, 56 and 633 A911).
(b) Ann. der Phys., 38, 731 A912).
(c) The Kinetic Theory of Phenomena in Fairly Rare Gases, Dissertation,
Uppsala, 1917.
(d) "The Numerical Calculation of Phenomena in Fairly Rare Gases," Svensk.
Vet. Akad. (Arkiv. f. Mat., Ast. och. Fys.), 16, 1 A921).
(e) "Kinetic Theory of Thermal Conduction, Viscosity, and Self-Diffusion
in Certain Dense Gases and Liquids," Svensk. Akad. Handl, 63, No. 4
A922).

E.I Early Works 557
(f) Svensk. Akad. (Arkiv. f. Mat., Ast. och. Fys.), 27A, No. 13 A928).
Hasse, H. R.
Phil. Mag., 7, 139 A926).
Hasse, H. R., and Cook, W. R.
(a) Phil. Mag., 3,911 A927).
(b) Proc. Roy. Soc, 196 A929).
(c) Phil. Mag., 72,554A931).
Hilbert, D.
Math. Ann., 72,562A912).
James, C. G. F.
Proc. Camb. Phil Soc, 20, All A921).
Jeans, J.
(a) An Introduction to the Kinetic Theory of Gases, Cambridge University
Press, New York, 1952.
(b) Kinetic Theory of Gases, Cambridge University Press, New York, 1946.
(c) Phil. Trans. Roy. Soc. A, 196, 399 A901).
(d) Quart. J. Math., 25, 224 A904).
Kennard, E. H.
Kinetic Theory of Gases, McGraw-Hill, New York, 1938.
Loeb, L. B.
The Kinetic Theory of Gases, 2d ed., McGraw-Hill, New York, 1934.
Lorentz, H. A.
(a) "On the Equilibrium of Kinetic Energy Among Gas-Molecules," Wien.
Ber.,95, 115A887).
(b) "The Motions of Electrons in Metallic Bodies," Proc. Amsterdam Acad.,
7,438,585,684A905).
(c) The Theory of Electrons, B. G. Teubner, Leipzig, 1909.
Massey, H. S. W., and Mohr, C. B. O.
(a) "On the Rigid Sphere Model," Proc. Roy. Soc. A, 141, 434 A933).
(b) "On the Determination of the Laws of Force between Atoms and
Molecules," Proc. Roy. Soc, 144, 188 A934).
Pidduck, F. B.
(a) "The Kinetic Theory of the Motions of Ions in Gases," Proc. Lond. Math.
Soc, 15, 89 A916).
(b) "The Kinetic Theory of a Special Type of Rigid Molecule," Proc. Roy.
Soc. A., 101, 101 A922).

558 E. Additional References
E.2 Recent Contributions to Kinetic Theory and
Allied Topics
Abrikosov, A. A., Gorkov, L. P., and Dzyaloshinskii, I. E., Methods of Quantum
Field Theory in Statistical Physics, Prentice-Hall, Englewood Cliffs, N. J.
A963). Reprinted by Dover, New York A975).
Akhiezer, A. I., and Peletminskii, S. V., Methods of Statistical Physics,
Pergamon, Elmsford, N.Y A981).
Arnold, V. L, Mathematical Methods of Classical Mechanics, Springer-Verlag,
New York A978).
Allis, W. P., "Motion of Ions and Electrons," Hand. Physik, vol. XXI, Springer-
Verlag, Berlin A956).
Balescu, R., Statistical Mechanics of Charged Particles, Wiley, New York
A967).
Barut, A. O., Electrodynamics and Classical Theory of Fields and Particles,
Dover, New York A980).
Bellman, R., G. Birkhoff and Abu-Shumays, I., eds., Transport Theory,
American Mathematical Society, Providence, R.I. A969).
Bernstein, J., Kinetic Theory in the Expanding Universe, Cambridge, New
York A988).
Brenig, W., Statistical Theory of Heat: Non-Equilibrium Phenomena, Springer,
New York A989).
Burgers, J. M., The Nonlinear Diffusion Equation, D. Reidel, Dordrecht, The
Netherlands A974).
Burshtein, A. I., Introduction to Thermodynamics and Kinetic Theory of
Matter, Wiley, New York A996).
Carruthers, P., and Zachariasen, F., "Quantum Collision Theory with Phase-
Space Distributions," Rev. Mod. Phys. 55, 245 A983).
Cercignani, C, Mathematical Methods in Kinetic Theory, Plenum, New York
A969).
, Theory and Application of the Boltzmann Equation, Elsevier, New York
A975).
Case, K. M., and Zweifel, P. F., Linear Transport Theory, Addison-Wesley,
Reading, Mass. A967).
Chapman, S., and Cowling, T G., The Mathematical Theory of Non-Uniform
Gases, 3rd ed., Cambridge University Press, New York A974).
Cohen, E. D. G., Fundamental Problems in Statistical Mechanics, Wiley, New
York A962).
Cohen, E. G. D., and Thiring, W. (eds.), The Boltzmann Equation: Theory and
Application, Springer-Verlag, New York A973).
Curtiss, C. F, Hirschfelder, J. O., and Bird, R. B., The Molecular Theory of
Gases and Liquids, Wiley, New York A969).
Davidson, R. C, Methods in Nonlinear Plasma Theory, Academic Press, New
York A972).

E.2 Recent Contributions to Kinetic Theory and Allied Topics 559
de Boer, J., and Uhlenbeck, G. E., Studies in Statistical Mechanics, North-
Holland, Amsterdam A970).
de Groot, S. R., and Mazur, P., Non-Equilibrium Thermodynamics, North
Holland, Amsterdam A962).
Doniach, S., and Sondheimer, E. H., Green's Functions for Solid State
Physicists, Benjamin-Cummings, Menlo Park, Calif. A974).
Dresden, M., "Recent Developments in the Quantum Theory of Transport and
Galvonomagnetic Phenomena" Rev. Mod. Phys. 33, 265 A961).
Duderstadt, J. J., and Martin, W. R., Transport Theory, Wiley, New York A979).
Ebeling, W., Transport Properties of Dense Plasma, Birkhauser, Boston,
A984).
Ecker, G., Theory of Fully Ionized Plasmas, Academic Press, New York A972).
Eu, B. C, Kinetic Theory and Irreversible Thermodynamics, Wiley, New York
A992).
Family, R, and Landau, D. P., eds. Kinetics of Aggregation and Gelation, North
Holland, Amsterdam A984).
Farquhar, I. E., Ergodic Theory in Statistical Mechanics, Wiley-Interscience,
New York A964).
Ferziger, J. H., and Kaper, H. G., Mathematical Theory of Transport Processes
in Gases, North Holland, Amsterdam A962).
Fetter, A. L., and Walecka, J. D., Quantum Theory of Many-Particle Systems,
McGraw-Hill, New York A971).
Finkelstein, R. J., Nonrelativistic Mechanics, Benjamin, Menlo Park, Calif.
A973).
Frenkel, J., Kinetic Theory of Liquids, Oxford, New York A946).
Fujita, S., Introduction to Non-Equilibrium Statistical Mechanics, W. B.
Saunders, Philadelphia A966).
Goldstein, H., Classical Mechanics, 2nded., Addison-Wesley, Reading, Mass.
A973).
Gombosi, T. I., Gas Kinetic Theory, Cambridge, Now York A994).
Grad, H., "On the Kinetic Theory of Rarefied Gases," Comm. Pure andAppl.
Math. 2,331 A949).
, "Principles of the Kinetic Theory of Gases," Hand. Physik, vol. XII,
Springer-Verlag, Berlin A958).
, "The Many Faces of Entropy," Comm. PureAppl. Math. 14, 323 A961).
Grandy, W. T., Foundations of Statistical Mechanics, Vol. 2, Nonequilibrium
Phenomena, D. Reidel, Boston A988).
Groot, S. R. de, and Leeuwen, W. A. Van., Relativistic Kinetic Theory:
Principles and Applications, North Holland, New York A980).
Haken, H., Quantum Field Theory of Solids, North-Holland, Amsterdam
A976).
Harris, S., Introduction to the Theory of the Boltzmann Equation, Holt, Rinehart
and Winston, New York A971).
Harrison, L. G., Kinetic Theory of Living Patterns, Cambridge, New York
A993).

560 E. Additional References
Haug, A., Theoretical Solid State Physics, H. S. H. Massey (trans.), Pergamon,
Elmsford, N.Y. A972).
Hillery, M., O'Connell, R. E, Scully, M. O., and Wigner, E. P., "Distribution
Functions in Physics: Fundamentals," Phys. Repts. 106, 122 A984).
Kac, M., Probability and Related Topics in Physical Sciences, Wiley
Interscience, New York A959).
Kadanoff, L. P., and Baym, G., Quantum Statistical Mechanics, Benjamin,
MenloPark, Calif. A962).
Keldysh, L. V., "Diagram Technique for Nonequilibrium Processes," Sov. Phys.
JETP20, 1018A965).
Klimontovich, Y., Statistical Theory of Non-Equilibrium Processes in Plasma,
MIT Press, Cambridge, Mass. A969).
Koga, T, Introduction to Kinetic Theory. Stochastic Processes in Gaseous
Systems, Pergamon Press, Elmsford, N.Y. A970).
Kogan, M. N., Rarefied Gas Dynamics, Plenum, New York A969).
Kohn, W., and Luttinger, J. M., "Quantum Theory of Electron Transport
Phenomena," Phys. Rev. 108, 570 A975).
Kubo, R., Statistical Mechanics, Wiley, New York A965).
Liboff, R. L., Introduction to the Theory of Kinetic Equations, Wiley, New
York A969); Mir, Moscow A974), Krieger, Melbourne, Fla. A979).
, Introductory Quantum Mechanics, 4th ed., Addison-Wesley, San
Francisco B002).
, and Rostoker, N. (eds.), Kinetic Equations, Gordon and Breach, New
York A971).
Lie, T-J., and Liboff, R. L., "Consideration of Particle Exchange in Quantum
Kinetic Theory," Ann. Phys. 67, 349 A971).
Lifshitz, E. M., and Pitaevskii, L. P., Physical Kinetics, Pergamon, Elmsford,
N.Y. A981).
, and , Statistical Physics, Part Two, Pergamon, Elmsford, N.Y
A980).
Mahan, G. D., "Quantum Transport Equation for Electric and Magnetic Fields,"
Phys. Repts. 145, 251 A987).
, Many-Particle Physics, Plenum, New York A981).
Mitchner, M., and Kruger, C. H., Jr., Partially Ionized Gases, Wiley, New York
A973).
McQuarrie, D. A., Statistical Mechanics, Harper & Row, New York A976).
Montgomery, D. C, and Tidman, D. A., Plasma Kinetic Theory, McGraw-Hill,
New York A964).
Negele, J. W., and Orland, H., Quantum Many-Particle Systems, Addison-
Wesley, Reading, Mass. A988).
Nicholson, D. R., Introduction to Plasma Theory, Wiley, New York A983).
O'Raifeartaigh, L., General Relativity. Papers in Honor of J. L. Synge, Oxford
University Press, New York A972).
Pathria, R. K., Statistical Mechanics, Pergamon, Elmsford, N. Y. A972).

E.2 Recent Contributions to Kinetic Theory and Allied Topics 561
Peletminskii, S., and Vatsenko, A., "Contribution to the Quantum Theory of
Kinetic and Relaxation Processes," Sov. Phys. JETP, 26, 773 A968).
Pines, D., The Many-Body Problem, Benjamin-Cummings, Menlo Park, Calif.,
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Pomraning, G. C, The Equations of Radiation Hydrodynamics, Pergamon,
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Pozhar, L.A., Transport of Inhomogeneous Fluids, World Scientific, Singapore
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Prigogine, I., Non-Equilibrium Statistical Mechanics, Wiley, New York A962).
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Rammer, J., and Smith, H., "Quantum Field-Theory Methods in Transport
Theory of Metals," Rev. Mod. Phys., 58, 323 A986).
Reed, T. M., and Gubbins, K. E., Applied Statistical Mechanics, McGraw-Hill,
New York A973).
Reichl, L., A Modem Course in Statistical Physics, University of Texas Press,
Austin A986).
Resibois, P., and de Leener, M., Classical Kinetic Theory of Fluids, Wiley,
New York A977).
Rickayzen, G., Green's Functions and Condensed Matter, Academic Press,
New York A980).
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Roos, B. W., Analytic Functions and Distributions in Physics and Engineering,
Wiley, New York A969).
Sampson, D. H., Radiative Contributions to Energy and Momentum Transport
in a Gas, Wiley-Interscience, New York A965).
Schenter, K. G., A Unified View of Classical and Quantum Kinetic Theory with
Application to Charge-Carrier Transport in Semiconductors, Thesis,
Cornell A988).
Sinai, Ya. G., Introduction to Ergodic Theory, Princeton University Press,
Princeton, N.J. A976).
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New York A971).
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New York A961).
Uehling, E. A., and Uhlenbeck, G. E., "Transport Phenomena in Einstein-Bose
and Fermi-Dirac Gases. I," Phys. Rev., 43, 552 A933).

562 E. Additional References
Uhlenbeck, G. E., and Ford, G. W., Lectures in Statistical Mechanics, American
Mathematical Society, Providence, R.I. A963).
van Leewen, W. A., van Weert, Ch. G., and de Groot, S. R., Relativistic Kinetic
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Vincenti, W. G., and Kruger, C. H., Introduction to Physical Gas Dynamics,
Wiley, New York A965).
Wing G. M, An Introduction to Transport Theory, Wiley, New York A962).
Wolfram, S., "Cellular Automaton Fluids 1: Basic Theory," J. Stat Phys., 4,
471 A986).
Wu, T. Y, Kinetic Equations of Gases and Plasmas, Addison-Wesley Reading,
Mass. A966).
Zaslavsky, G. M., Chaos in Dynamical Systems, Harwood Academic, New
York A984).
Ziff, R. M., New Class of Solvable-Model Boltzmann Equations, Phys. Rev.
Letts, 45 306 A980).
Zimon, J. M., Electrons and Phonons, Oxford University Press, New York
A960).
, Elements of Advanced Quantum Theory, Cambridge University Press,
New York A969).
Zubarev, D. N., Nonequilibrium Statistical Thermodynamics, P. J. Shepard
(trans.), Consultants Bureau, New York A974).
, and Kalashnikov, V. P., "Derivation of the Nonequilibrium Statistical
Operator from the Extremum of the Information Entropy," Physica, 46,
550 A970).

APPENDIX F
SCIENCE AND SOCIETY IN THE CLASSICAL GREEK AND ROMAN ERAS
Diophantus
(Algebra. Diophantine
.. equations.)
300
200
Galen
(Explains respiration.
.. Shows damage to one side
of brain affects other
side of body.)
Ptolemy
(Draws first map projecting
curved surface of Earth.)
-100
Pax
Romana
Plutarch
Pliny
(Natural History, 37 vols.)
Ovid
Virgil
Cicero
I Vitruvius
J_(De Architectura)
Julius Caesar
-100
Euclid
Punic Wars
("Carthago
delenda est
Cato.)
(Geometry axioms.
Proves number of
primes is infinite.)
Epicurus -•
Archimedes
(Law of buoyancy.
Estimates ji.)
-200
Eratosthenes
(Estimates
circumference
of Earth. Formulates
sieve for prime
numbers.)
Zeno
T Peloponnesian
lWars
T Golden Age
X (Pericles' leadership.)
Democritus
(Proposes atomic
theory of matter.)
Plato
Aristarchus
(Heliocentric
hypothesis.
Estimates
diameter of sun)
300
Aristotle
-400
Socrates
Thales
(Founder of Greek
Pythagoras
(Right triangle theorem,
Co-worker argues that V2
-i- is irrational.)
Heraclitus
-500
science, mathematics,
and philosophy.)
-600
-700
-750
End of Homeric Age
A100B.CE.-750BC.E.)
Homer

Index
Acoustic phonons, 379
Action-angle variables, 245
Action integral, 2, 478
Activation energy, 512
Adiabatic law, 198,265
Alkali metals, 487
Alloy, 480
binary, 480
ordered and disordered, 480, 520
Alumina, 480
Amorphous media, 506/f
Anderson, P. W., 508
Apsidal vector, 142, 302
Atmospheres, law of (See Barometer
formula)
Autocorrelation function, 59, 189,
312,327
and Green's function, 441
Averages (See also Expectation)
ensemble, 343
phase, 37, 241
phase density, 125
time, 242
Balazs, N. L., 354
Balescu-Lenard equation, 313, 316
Barometer formula, 171
Barut, A. O., 455
BBKGY hierarchy, 78
Bergman, P. G., 449
Bethe lattice, 517
Birkhoff's theorem, 242
Black hole, 478
Bloch
condition, 492
waves, 508
Bogoliubov distribution, 117
Bogoliubov hypothesis, 115
Boltzmann equation, 112, 124, 131,
152
assumptions in derivation, 152
quasi-classical, 385, 493
Bose-Einstein distribution, 382, 387
(See also Planck distribution)
Boson, composite, 439
£-Brass, 480
Brass, 480
Brillouin zone, 488
Bronze, 480
Bruns's theorem, 253
Burnett equations, 199
Canonical distribution, 133, 344
Canonical invariants, 15
Canonical transformation, 11, 247
Carbon dioxide, viscosity, 212
Central limit theorem, 54, 172, 270
Chapman-Enskog expansion, 194,
265

566
Index
Chapman-Kolmogorov equation, 41,
317
Characteristic curves, 25
Characteristic function, 48, 69
Characteristics, method of, 24, 94,
304
Chebyshev's inequality, 71
Christoffel symbol, 473
Coarse graining, 243
Collision:
inverse, 147
reverse, 148
Collisional invariants, 155
Collision frequency, 194, 297
Collision operator, linear, 217
negative eigenvalues, 218
spectrum for Maxwell molecules,
220
symmetry, 217
Completely continuous operator, 224
Conductivity, 177, 209, 238
Conservation equations:
absolute, 159
from BY and BY, 83
relative, 160
Constants of motion, 4, 243, 254
Contravariant vectors, 470
Convergent kinetic equation, 315
Convolution integral, 191
Coordinates, generalized, 1
Correlation expansion, 89, 310
Correlation function, 59, 89
total, 96, 134
Coulomb gauge, 464
Covariance, 59, 452
Covariant vectors, 470
Cross section, scattering:
Coulomb, 1?5, 234
differential, 1*43
rigid sphere, 145
total, 144
Current density
entropy, 486
particle, 486
thermal, 486
Darwin Lagrangian, 467
deBroglie wavelength, 364
Debye
distance, 95, 113,326
frequency, 493
shielding potential, 284
temperature, 482, 498
wave number, 283, 314, 495
Deformation potential, 379
Degeneracy:
classical, 250
quantum, 377
Degenerate plasma, 377
Degrees of freedom, 1
de Groot, S. R., 449
Density expansion, 117
Density matrix, 395, 430
equation of motion, 399
Density operator, 331
Deuterium, liquid, 259
Diagrammatic analysis (See
Prigogine analysis)
Dielectric function, plasma (See
Plasma)
Dielectric time, 113
Diffusion, 175
coefficient, 483
mutual, 181,208
self, 180, 188
Diffusion coefficient, 304, 327
Diffusion equation, 67, 187
Dispersion relation:
degenerate plasma, 377
warm plasma, 286, 377
Distribution functions, classical (See
also Quantum distributions)
conditional, 37
joint probability, 36
reduced, 37
s -tuple, 38
Doppler effect, 477
Drift-diffusion equation, 177
Drude conductivity, 177, 239, 407
covariant, 460
Druyvesteyn distribution, 236
absolute, 237
Edwards, J. T., 510
Electrical conductivity, 487
temperature domains, 512
Electrochemical potential, 178, 484
Electrodynamics, covariant, 458

Index 567
Electron gas, 464
Electron mobility {See Mobility,
electrical)
Electron-phonon scattering rate {See
Scattering rate)
Elliptic integrals, complete, 221
Elyutin,P. V.,510
Energy bands, 369
Energy equation, 97
Energy shell, 35, 240
Ensemble, 20
average, 189,343
Entropy:
Boltzmann, 165
change, 275
Gibbs, 164,275
Equal a priori probabilities, 345
Equation of state, 97, 134
Equipartition of energy, 179, 382
Ergodic hypothesis, 241
Ergodic motion, 240, 264
Error function, 545
Euler equations, 198
Euler's constant, 542
Event, relativistic, 450
Exchange transformation, 13
Expectation:
of a classical function, 37
of a random variable, 47
and wave function, 330
and Wigner distribution, 353
Exponential integral, 542
Exponents, critical, 518
Extended states, 506, 521
Fermi-Dirac distribution, 371, 387,
483, 492
Fermi energy, 371, 4343
Fermi liquid, 387
Fermi momentum, 388
Fermi sphere, 374
Field tensor, electromagnetic, 458
Flow chart, 333
Fluid dynamical variables {See
Macroscopic variables)
Fock space, 395, 433
Hamiltonian, 433
operator, 396, 438
Fokker-Planck equation, 304, 308,
315,317,327
Four-current, 453
Four-vector, 451
covariant and contravariant, 471
Four-vector potential, 453, 459
Four-wave vector, 453
Free energy, 518
Friction coefficient, 304, 327
Friedel oscillations, 377
Gamma function, 543
Gauss' equation, 280, 281
Gaussian distribution, 56, 59, 173
Generalized coordinates, 1
Generalized potential {See Potential)
Generating function, 11
Goldstein, H., 455
Grad's first and second equations,
129,262
Grad's method of moments, 213
Grains, 480
Grazing collisions, 301, 303
Green's function equations, coupled,
415,422
Green's functions:
and averages, 410
diagrammatic representation, 416,
440
retarded and advanced, 415
s-body,410,438
and Schroedinger equation, 408
Group property:
canonical transformations, 15
Lorentz transformations, 475
Hamiltonian, 4, 61, 66, 139, 254,
262, 272, 274, 396, 399
relativistic, 5, 455, 467
Hamilton-Jacobi equation, 248
Hamilton's equations, 4
covariant, 455
Hamilton's principle, 2, 76
Hard potential {See Potential)
Heisenberg picture, 339, 396, 422,
429
Henon-Heiles Hamiltonian, 254
anti, 274
Hermite polynomials, tensor, 213, 546

568
Index
Hermiticity, 27
Hessian, 257
Heteroclinic orbit, 74
Hierarchies, quantum and classical,
table of, 402
Hillery, R. K, 354
Histogram, bin storage, 323
Hole mobility, 369
Homoclinic orbit, 74
Honeycomb lattice, 517
Hopping, 506
thermally assisted, 513
variable range, 513
H-theorem, 166, 392, 446
Hydrofracturing of a rock, 515
Hypergeometric function, 548
confluent, 548
Hypersonic flow, 298
Ideal gas, 30, 270
Impact parameter, 142
Incompressible phase density, 24, 136
Inelastic scattering, 319
Insulators, 480
Integrable motion, 254
Integral invariants of Poincare, 68
Integral of motion (See Constants of
motion)
Inverse scattering, 147
Ioffe-Regel minimum, 512
Irreversibility, 162, 167, 240
Jacobian, 17
Jacobi's identity, 11
Jennings, B. K., 354
Joint probability distribution (See
Distribution functions)
Kagome lattice, 517
KAM theorem, 257, 274
KBG equation (See Krook-
Bhatnager-Gross
equation)
Kinetic equation:
convergent, 315
definition, 82
photon, 366
table of, 296
Klimontovich picture, 124, 136
Kronicker-delta symbol, four
dimensional, 472
Krook-Bhatnager-Gross equation,
96, 194, 295, 328
Kubo formula:
classical, 192
quantum, 404
Kursunoglu, B., 456
Lagrange's equations, 3
covariant, 472
Lagrangian, 2, 60, 62, 132
relativistic, 466, 472
Laguerre polynomials, 204, 547
Landau damping, 288, 326, 377
Landau equation, 309, 315
Laplace-Runge-Lenz vector
Large-mass consistency limit, 503
Lasing criterion, 368
Lattice:
Bethe, 517
Honeycomb, 517
Kagome, 517
one-dimensional, 522
Legendre expansion, 227, 261
Legendre polynomials, 229
Legendre transformation, 14
Length contraction, 454
Levi-Civita symbol, 538
Liapunov exponent, 257
Lie derivative, 135
Light cone, 450
Liouville equation, 22, 78, 344, 399
in non-Cartesian coordinates, 474
one-particle, 93
relativistic, 457
Liouville operator, 27, 98
resolvent, 33
Liouville theorem, 17, 276
Localization, 508
length, 510
in second quantization, 520j(f
Localized states, 506
Logarithmic singularity, 295, 303
Long-time limit, 110
Long-time tails, 189, 277
Lorentz
expansion, 494
force, 5, 62

Index
569
gauge, 66, 464
invariants, 453
in kinetic theory, table of, 465
transformation, 452, 475
Lorentzian form {See Spectral
function)
Lorenz
number, 373, 376
relation, 483
Mach number, 298
Macroscopic variables:
absolute, 159
relative, 160
Magnetic alloys, dilute, 515
Mandelstam variables, 465, 478
Markov process, 41
Master equation, 43, 319, 446
Maxwellian, 96, 225, 275, 306, 327
absolute, 169
local, 169
relativistic, 467, 475
Maxwell interaction, 144, 219, 221,
271
Maxwell molecules {See Maxwell
interaction)
Maxwell's equations, 66, 459
Mean-free-path, 179
acoustic phonon, 384
estimates, 178
Measurement:
and commutators, 332
and operators, 331
Measure of a set, 241
Metal-insulator transition, 506, 510
Metals, 480
electron transport in, 369
Method of moments, Grad's, 213
Metrically indecomposable sets, 242
Metrically transitive transformations,
242
Metric space, 133
Metric tensor, 471
Microcanonical distribution, 344
Mixed states, 341, 434
Mixing flow, 243
Mobility
edge, 506, 511
electrical, 177, 192
electron, 369
gap, 511
hole, 369
Monte Carlo analysis, 319, 328
Mott,N. F.,510
Mott transition, 378
Navier-Stokes equations, 199
Neural networks, 515
Neutron, magnetic moment, 430
Nitrogen, viscosity, 212
Noble metals, 487
Non-Cartesian coordinates, 474
Nyquist criterion, 292, 327
Occupation numbers, 395
O'Connell, M. O., 354
Optical phonons, 379
Pair connectedness, 518
Partition function, 432
grand, 433
Pauli principle, 339, 394
Percolation, 506, 5\4ff
bond, 515
cluster, 515
site, 515
Periodic motion, conditional, 249
Phase average {See Averages)
Phase density averages, 125
Phonon polarization, 490
Phonon scattering {See Relaxation
time)
Planck distribution, 364, 379
Plasma:
dielectric function, 282
frequency, 113,282,326
parameter, 113,304,314
Plasma waves, 285
stable modes, 286
unstable modes, 291
Plemelj's formula, 362
Poincare map, 254
Poincare recurrence theorem, 163
Poisson brackets, 10, 64
Poisson distribution, 52, 58, 69
Polycrystalline material, 480
Polymer gelatin, 515
Potential:

570 Index
Potential: (continued)
generalized, 66
hard, 220
soft, 220
Prandtl number, 185
Pratt, W. P., 488
Pressure tensor, 175
Prigogine analysis, 97
Principal part, 326
Projection representation, 350, 438
Proper frequency, 461
Proper time, 454
Proper volume, 463
Pure state, 341, 397, 398, 432
Quantum distributions (See
Bose-Einstein distribution;
Fermi-Dirac distribution;
Planck distribution)
Quantum-modified Boltzmann
equation (See Boltzmann
equation)
Quantum statistics, 386
Quartz, 480
Quasi-classical kinetic equation,
433 (See also Boltzmann
equation)
Quasi-classical limit, 386, 392
Quasi-free particle, 413
energy, 428
lifetime, 428
Radial distribution function, 96, 135,
448, 480
Radiation field, 366
Random numbers, 323
Random phases, 345, 510
Random variables:
definition, 47
sums of, 49
Random walk, 44, 50, 55, 187
Range of interaction, 86
Rayleigh dissipation, 304
Rayleigh's formula, 365
Reichl, L. E., 394
Relativistic Maxwellian (See
Maxwellian)
Relativistic Vlasov equation (See
Vlasov equation)
Relativity, postulates, 450
Relaxation time approximation, 485
phonon scattering, 379
Representations:
coordinate, 335
momentum, 335
Resistivity, metallic, 481
Bloch component of, 501
electrical, 498, 501
low-temperature, 483
reduced, 504
residual, 488, 501
Resolvent of Liouville operator (See
Liouville operator)
Riemann zeta function, 544
Rigid spheres:
collision integral, 153
transport coefficients, 210
Rosenbluth-Rostoker limit, 96
Rough spheres, 273
Sample space, 47
Saxon-Hunter theorem, 523
Scattering:
angle, 140
inelastic, 319
matrix, 142, 258, 488, 493
rate, 362
Scattering cross section (See Cross
section)
Schenter, G. K., 394, 488
Schmidt number, 185
Schroeder, P. A., 488
Schwartzchild metric, 478
Scully, M. O., 354
Second quantization, 395
Self-consistent solution, 92
Self-scattering mechanism, 320
Semiconductor, 369, 480
Semimetals, 480
Shock front, 298
Shock waves, 298
Sine-Gordon equation, 70, 71
Single-crystal material, 480
Small shot noise, 53
Smooth spheres, 273
Sodium, band structure, 370
Soft potential (See Potential)
Solids: categories of, 480

Index
571
Sonine polynomials, 204, 221, 224,
259, 266
Sound speed, 268
Spectral function, 409, 425
Lorentzian form, 426
Spectral theorem, 224
Spherical harmonics, 220
Spin, 338
and density matrix, 348, 430
and density of states, 371
Spinor, 430
Spontaneous decay, 366
Stable modes {See Plasma)
Statistical balance, 167, 381, 387
Stefan-Boltzmann law, 366
Stimulated decay, 366
Stirling's approximation, 57, 544
Stosszahlansatz, 152
Strain, 176
Strain-acoustic interaction, 379
Strain-optical interaction, 379
Strain tensor, symmetric, 176
Stress tensor, 175, 203, 265, 266
Structure factor, 97, 135
Subsonic flow, 298
Superposition principle, 41, 337
Sutherland model, 211
Symmetry properties of distribution
functions, 39, 66, 73, 75
Synge, J. L., 467
Temperature, 161
Tensor:
covariant and contravariant, 472
equivalents, 539
integrals, 471
metric, 471
Thermal conductivity, 175, 185, 202
coefficient of, 483, 487, 505
Thermal speed, 169, 179, 275
Thermopower, 484
Thomas-Fermi potential, 377, 378
Thomas-Fermi screening:
distance, 376, 377
wave number, 376, 377
Thouless, D., 510
Three-body problem, 253
Tight-binding approximation, 508,
520
Time dilation, 454
Tori, invariant, 252, 257
Transfer integral, 510
Transfer matrix method, 522
Transonic flow, 298
Transport coefficients, 174
Chapman-Enskog estimates, 210
mean-free-path estimates, 186
Uehling-Uhlenbeck equation, 388
Uhlenbeck, G. E., 204, 220
Umklapp process, 482
Uncertainty, in quantum mechanics,
333, 334
Unstable modes {See Plasma waves)
Van Leewen, W. A., 449
van Weert, Ch. G., 449
Variance, 48, 161
Variational technique, 442
Vector potential, 66, 452, 466, 475
Velocity autocorrelation function,
441
Viscosity coefficient, 175, 184, 204,
211
observed values, 212
Vlasov equation, 92
for a plasma, 279
relativistic, 459, 475
Vlasov fluid, 92
Walker, C. H., 257
Wang Chang, C. S., 204, 220
Wave number of closest approach,
314
Weidemann-Franz Law, 483
Weyl correspondence, 354, 435, 438
Wigner distribution, 351, 396, 435,
436, 437, 438
equation of motion, 353, 396
Wigner-Moyal equation, 358, 399,
431
World line, 450
Zallen, R.,510
Zero-point energy, ion, 492
Zero sound, 377, 391
Ziman, J. M, 508
Zwanzig, R., 188

Graduate Texts in Contemporary Physics
B.M. Smirnov: Physics of Atoms and Ions
M. Stone: The Physics of Quantum Fields
F.T. Vasko and A.V. Kuznetsov: Electronic States and Optical
Transitions in Semiconductor Heterostructures
A.M. Zagoskin: Quantum Theory of Many-Body Systems: Techniques and
Applications