Classical Kinetic Theory of Fluids

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Classical Kinetic

T heory of Fluids

New York « London « Sydney + Toronto

QC 175.3 7 1977 Copyright © 1977 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine language without the written permission of the publisher. Library of Congress Cataloging in Publication Data Résibois, Pierre M

V

Classical kinetic theory of fluids.

“A Wiley-Interscience publication.” Bibliography: p. Includes index. 1. Liquids, Kinetic theory of. 2. Gases, Kinetic theory of. I. De Leener, M., 1937joint author. Il. Title. QC175.3.R47 532'.05 76-58852 ISBN 0-471-71694-4 Printed in the United States of America 10987654321

Preface

The aim of nonequilibrium statistical mechanics is to determine how the macroscopic properties of matter evolve with time, in terms of the laws of mechanics

that govern

the motion

of its constituents

(atoms, molecules,

etc.). Historically, this ambitious program has taken two directions. (i)

The kinetic theory of dilute gases was started by Kronig, Clausius,

Maxwell,

and

Boltzmann.

From

a few (controversial) assumptions,

they

arrived at a complete description of the approach to equilibrium of these simple systems. The central result of the theory is the celebrated Boltzmann equation that among various consequences, allows us to calculate the transport properties in terms of molecular parameters. (ii) The Brownian motion theory has dealt with the behavior of a heavy particle immersed in a dense liquid. In the traditional approach initiated by Einstein, Smoluchowski, and Langevin, the complicated laws of mechanics

describing the fluid are replaced by simple probabilistic assumptions about the erratic motion of the Brownian particle. These two theories have played a crucial role in the history of nonequilib-

tium

statistical

mechanics,

and

their

influence

can

still be

felt today:

for example, the “generalized kinetic equations” attempt to extend the v

vi

Preface

Boltzmann description to denser systems, and the so-called “autocorrelation-function formalism” is the statistical mechanical analog of the stochastic Brownian motion theory. Statistical mechanics has considerably widened its field of interest in the course of time; problems as different as the physics of superfluids, plasmas, solids, critical phenomena, .. . have been studied with great success. Yet the microscopic description of time-dependent phenomena in classical gases and liquids (i.e., fluids governed by the laws of classical mechanics) remains a prototype of statistical mechanical theories, both for pedagogical reasons and as a subject of great current interest.

This book presents an introduction to the nonequilibrium theory of classical fluids. We limit ourselves to “simple” fluids, by which we mean gases and liquids made of molecules that can be considered as point particles, with no internal degrees of freedom. In most experimental situations, this model applies to the rare gases (argon, xenon, etc). This text has arisen from courses given by the authors at various univer-

sities (Brussels (1972-1973, 1973-1974); Toulouse (fall 1972); Harvard (spring 1973); Leiden (1973-1974)). We started writing when we realized that among the many available texts on statistical mechanics, few if

any give the reader a coherent and self-contained various methods that have made

introduction

to the

nonequilibrium statistical mechanics so

successful. We have already stressed that in the development of this difficult subject, old ideas have served

as extremely

useful guides to new

methods;

con-

versely, the new theories have helped to clarify the earlier points of view. We hope that the book will convince the reader of this deep interconnection. To reflect the different aspects of the theory, the book is divided into four parts. After introducing in Chapter I the tools of probability theory that are required to describe many-particle systems, Part A presents in Chapter II the classical theory of Brownian motion and, in Chapter III, related topics of the theory of stochastic processes. Brownian motion theory furnishes an elegant solution to a simple kinetic problem, but it has an unsatisfactory phenomenological character. Indeed, the approach to equilibrium is introduced as an ad hoc assumption, and furthermore its kinetics involves parameters to which the theory is unable to give any precise molecular interpretation. These two weaknesses are absent from Boltzmann’s analysis of kinetic processes in dilute gases, and his theory, presented in Part B, is the first to fulfill the goals of nonequilibrium statistical mechanics. Even though the derivation of the Boltzmann equation, given in Chapter IV, is based on stochastic assumptions that are hard to justify, its inherent irreversibility, as well as the fact that it permits calculation of the transport coefficients of

vii

Preface

dilute gases (as discussed in Chapter V), make it an outstanding achievement

of theoretical physics. Chapter VI briefly describes how Enskog applied the ideas of Boltzmann to denser systems; this theory is in (surprisingly) good agreement with the computer studies of hard sphere systems.

Part C begins the strictly microscopic study of nonequilibrium processes in

fluids. Chapter VII introduces the main formal tools (distribution functions, the Liouville equation, etc.), and these are used in Chapter VIII to derive the

generalized kinetic equation that governs the time evolution of the distribution of velocities in an arbitrarily dense fluid. This equation is exact but formal and, to give it a physical content, further approximations must be made. In particular, Chapter IX demonstrates that in suitable limits, it reproduces the results of classical Brownian motion theory and of the Boltzmann equation. Chapter X deals with the microscopic generalization of the Boltzmann equation when the density is increased: we show that serious difficulties occur, leading in particular to the nonanalytic character of the density expansion of transport coefficients. Part D is devoted to the autocorrelation-function formalism, which provides us with a unified microscopic (but formal!) description of the response of a macroscopic system to weak external fields or probes. This approach, introduced in Chapter XJ, has found numerous applications in the analysis of experimental data in dense fluids; typical examples are given in Chapter XII. Some mathematical complements have been relegated to the appendices. This book is intended for students at the end of their undergraduate studies (end of “2nd cycle” in Belgium) or at the beginning of their graduate school

(“3rd

cycle”).

It

is assumed

that

the

reader

is familiar

with

elementary thermodynamics, rational mechanics, and quantum mechanics. The mathematical level is that of a first course in quantum mechanics: this includes some acquaintance with complex integral calculus, vector-space formalism, Fourier and Laplace transforms, and the Dirac delta function.

Otherwise the text is self-contained, though some knowledge of elementary equilibrium statistical physics may be helpful. Covering a subject that has evolved over more than a century, we found it inadequate to follow the historical development, and we avoided citing the voluminous original literature. References generally send the reader back to books, monographs, or review papers, where he can find a more complete bibliography; only for some technical results, not yet reviewed, do we quote the original sources.

Many friends and colleagues have helped us in the present enterprise in one way or another, by their lectures, by their writings, by fruitful discussions. Rather than displaying a long list of names, we prefer to thank them collectively, making a single exception to acknowledge the special role of

viii

Preface

our teacher. Prof. I. Prigogine who, with science and creative enthusiasm,

introduced us into this fascinating chapter of theoretical physics. We also thank L. Février and P. Kinet for their technical help.

P. Résibois M. De Leener Brussels, Belgium September 1976

Contents

A

PROCESSES

Introduction to the Theory of Stochastic Processes 1

Statistical Mechanics and Probability Theory, 3 Elements of Probability Theory, 7

2.1 2.2

The Axioms of Probability, 7 Probability Law and Experiment, 7

Random Variables, 11 3.1 3.2 3.3. 3.4

3.5 anAuns

Chapter I

STOCHASTIC

Definitions, 11 Average Value and Variance, 12 Random Variables in Several Dimensions, Independent Random Variables, 14

The Continuous Limit, 15

Stochastic Processes, 18 Markov Processes, 19 The Random Walk, 23

13

x

Contents

Chapter II

ny

The Classical Theory of Brownian Motion The Langevin Equation, 30 The Fokker-Planck Equation, 32 Applications of the Fokker-Planck Equation: Approximate Methods, 38 3.1

3.2 Chapter HI

30

Electrical Conductivity, 39 Diffusion, 41

Gaussian Random Processes

Fundamental

49

Solution of the Homogeneous

Fokker—Planck Equation, 49

Stationary Random Processes, 52 2.1

2.2 2.3.

Definition of Stationarity, 52

Ergodicity, 53 The Power Spectrum and the Wiener-Khinchine Theorem, 54

Gaussian Stationary Processes, 57

3.1 3.2 3.3.

Definition, 57 Joint Distributions, 59 Gaussian Markov Process:

Doob’s theorem, 62

Further Remarks, 64 4.1

Nonstationary Gaussian Processes, 64

THE

BOLTZMANN

4.2

Chapter IV

Back to the Langevin Equation, 66 EQUATION

The Nonlinear Boltzmann Equation 1 3

Introduction, 73 The One-Particle Distribution Function, 75 Two-Body Collision Theory, 77 3.1

3.2

The Mechanical Problem, 77

The Scattering Cross Section, 80

The Collision Term, 85

73

xi

Contents

5

General Properties of the Nonlinear Boltzmann Equation, 90 5.1

5.2 5.3.

Positivity of f1, 90

Collision Invariants, 91 Solution of J(f,, f;)=0 and the H-Theorem, 93

Conservation Equations and Hydrodynamics, 98 6.1 6.2

Microscopic Conservation Equations, 98 Classical Hydrodynamics, 101

Normal Solutions of the Nonlinear Equation, 105

7.1

7.2.

Chapter V

Boltzmann

The Hilbert Principle and Normal Solutions, 106

Sketch of the Chapman-~Enskog Method,

112

The Linearized Boltzmann Equation 1 2 3

118

The Linearized Collision Operator, 118 Macroscopic Definition of Hydrodynamic Modes, 124 Microscopic Expression for Hydrodynamic Modes and Transport Coefficients, 130

Explicit Calculation of Transport Coefficients, 137

4.1

4.2 4.3. 4.4

Introduction, 137

A Variational Principle, 138 The Variational Principle at Work, 141 Transport Coefficients for Special Models, 143

Further Remarks, 146 5.1 The Boltzmann-Lorentz Equation and Self5.2

Kinetic Models, 149

The Enskog Theory of the Dense Hard-Sphere Fluid WMRRWNPR

Chapter VI

Diffusion, 146

Introduction, 156

The Enskog Equation, 157

Conservation Equations, 160 Transport Coefficients, 164 Further Remarks, 168

156

Contents

C Chapter VII

GENERALIZED

2 3

Ensembles, the N-Particle Distribution and the Liouville Equation, 173

Definitions, 186

Reduced Distribution Functions and Average Values, 190

The BBGKY Hierarchy, 192 Conservation Equations, 194

Generalized Kinetic Equations 1 2 3

Introduction, 202 The Formal Master Equation, 205 The Master Equation and the Thermodynamic

4

The Generalized Kinetic Equation, 211

5 6

3 4

Generalized Molecular Chaos, 216 The Markovian Approximation, 218

The Boltzmann Equation, 228 The Fokker-Planck Equation, 233

Hard-Spheres Dynamics and Density Expansion

3

221

Introduction, 221 The Landau Equation, 222

of the Collision Operator

1 2.

202

Limit, 208

Simple Applications of the General Theory 1 2

Chapter X

Function,

Its Link with Thermodynamics, 181 Reduced Distribution Functions, 186

3.2 4 5

173

The Canonical Equilibrium Distribution and

3.1

Chapter IX

EQUATIONS

Distribution Functions in Statistical Mechanics

1

Chapter VIII

KINETIC

Introduction, 240 The Pseudo-Liouville Equation for

Hard Spheres, 241 The Generalized Kinetic Equation for Hard Spheres and Applications, 248 3.1

Formal Results and the Boltzmann

Limit, 248

240

Contents

3.2

4 5

The Physical Origin of the Nonanalyticity in the Density Expansion of Transport Coefficients, 256 Introduction to the Mathematical Analysis of the Divergence and Its Removal, 263

5.1 §.2 5.3.

D Chapter XI

The Binary-Collision Expansion and the Choh-Uhlenbeck Operator, 249

Further Analysis of the Choh-Uhlenbeck Operator, 263 The Ring Operator, 266 The Ring Operator and Transport

Coefficients, 271

TIME-DEPENDENT

CORRELATION

FUNCTIONS

The Correlation-Function Formalism 1

The Case of Brownian Motion, 277 1.1

1.2.

Derivation “a la Einstein”’ of the Green-

Kubo Formula for Self-Diffusion, 277 Correlation Functions and Brownian Motion of an Oscillator Driven by an External Force, 279

2

Linear-Response Functions and Their General Properties, 287 2.1 2.2 2.3.

3 4 5

Linear Response to a Spatially Homogeneous External Field, 287 Linear Response to Space-Dependent Forces, 292 General Properties of the Response Functions, 295

Correlation Functions and Hydrodynamics, 297 Correlation Functions and Inelastic Neutron Scattering, 305 Final Remarks, 312 5.1 5.2

5.3.

Correlation Functions and Other Probes, 312 Correlation Functions and the Thermodynamic Limit, 313

Summary, 314

277

a

Contents

Chapter XII

Calculation of Time-Dependent Correlation Functions 1

Connection Between Correlation Functions and Kinetic Theory, 316 1.1 1.2. 1.3.

2

The Velocity Autocorrelation Function in Dense Fluids, 324 2.1

Moment (or Short-Time) Sum Rules, 325

2.3. 2.4

The Two-parameter Lorentzian, 327 The Two-parameter Gaussian, 328

2.2

3

4

Remarks on Computer Experiments, 330

4.2

Simple Limiting Properties and the Gaussian Approximation, 336 Generalized Hydrodynamics Illustrated, 338

The Van Hove Total Correlation Function, 342

$.1 5.2 6

Zero-Frequency Sum Rules, 327

The Van Hove Self-Correlation Function, 336 4.1

5

Introduction, 316 Correlation Functions and Generalized Kinetic Equations, 317 An Alternative Kinetic Theory, 321

The Hydrodynamic Limit of S(q; w):

the Landau-Placzeck Method, 344 Nonhydrodynamic Regime: De Gennes Narrowing, 347

The Long Time Tails, 350 6.1 6.2 6.3.

The Phenomenological Approach, 350 Kinetic Approach to the Tails, 357 Further Questions, Answers, and Conjectures, 360

wlen:-i 2

APPENDICES Eigenfunctions of the Fokker-Planck Operator, 365

Calculation of the Collision Integral b;,, 372

The Sutherland Formula, 374 Eigenmodes of the Enskog Equation, 376

316

xy

Contents

=

ma

E F

J

K

Some Angular Integrals for Hard-Sphere Collisions, 379 The Decay of the Non-Markovian Kernel G(v,; 7|p,(t—7)) and of the Correlation Term D(v,; t):

a Weak-Coupling Model, 383

Microscopic Expression for the Friction Coefficient {y, 387 Equivalence Between Two Forms of the Hard-Sphere Collision Operator, 389 Divergence of the Choh—-Uhlenbeck Operator in Two Dimensions, 391 Short-Time Behavior of %.q(t), 398

Calculation of zx, 400

BIBLIOGRAPHY INDEX

403 407

Classical Kinetic Theory of Fluids

A Stochastic Processes

I Introduction to the Theory of Stochastic Processes

1 STATISTICAL MECHANICS AND PROBABILITY THEORY*

The aim of nonequilibrium statistical mechanics is to predict the temporal behavior of observable properties of macroscopic systems from an analysis of the microscopic dynamics of the particles constituting these systems. Observable properties are characterized by their dependence on a large number of particles. Consider, for example, the pressure of a fluid enclosed in a vessel. It is defined as the average force exerted by the particles of the fluid on a unit surface. Thus the measure of this pressure does not tell us anything about the detailed motion of the molecules that strike the surface, but only provides us with a piece of global information. A similar remark applies to any experimentally observable property of macroscopic systems,

such as the heat flow or the diffusion current in a mixture. On the other hand, if classical mechanics provides a valid microscopic

description of molecular dynamics,+

the position r,(t) of particle a (a=

* A detailed discussion of the connection between statistical mechanics and probability theory can be found in Penrose (1970). +The general considerations presented here also hold if the system is described quantum mechanically, However all applications in this book deal exclusively with classical nonrelativistic systems.

4

Introduction to the Theory of Stochastic Processes

]- ++ N) at time ¢ is governed by Newton’s equation d7r,(t)_

1

ames

«(1

N)

(1.1)

where m is the mass of the particle and F, denotes the total force exerted on it, which depends of course on the position of all the particles in the system: if we neglect the effect of the walls of the enclosing vessel, we have sony where

oe

(1.2)

V(r,,) is the potential energy between the pair of particles a and b;

for structureless particles (this book is limited to this case), it depends only on the relative distance r, = |tz—r,|. Our program is then, in principle, to calculate the observable properties of a fluid from the solution of (I-1,2).

However we are immediately confronted with two serious difficulties.

(i)

(ii)

A typical macroscopic system involves of the order of 10”? particles, and the mathematical problem of solving 107° coupled differential equations is extraordinarily difficult, even when it is recognized that only very special pieces of information (the value of macroscopic observables) must be extracted from this solution. Except for rather unrealistic models, little hope exists of ever getting an exact solution in explicit form. We should also keep in mind that the equations of motion (I-1) are deterministic: we get different solutions for different initial conditions. Therefore, before even thinking about getting such a solution, we should know the precise position and velocity of each particle at t= 0. No experimental technique can answer such a question: stated in this way, the problem that confronts us seems impossible to solve.

Sometimes however, modern technology almost permits the impossible, and present-day computers do indeed allow the solution of Newton’s equations (I-1) with good accuracy (for given initial conditions) for systems involving as much

as

N= 1000

(“‘almost”

107°!) particles. If we assume

that such

systems already reflect the behavior of macroscopically large systems, these numerical calculations—often called computer experiments—throw considerable light on the proposals made by Maxwell, Boltzmann, Gibbs, and other founders of statistical mechanics, to meet the difficulties just enumerated.

We analyze later some of the detailed results obtained by this method; here, a qualitative summary permits us to avoid a long historical introduction and to suggest the basic postulates of statistical mechanics.

5

Statistical Mechanics and Probability Theory

First, computer experiments clearly indicate that knowledge of the individual trajectories of the particles is essentially irrelevant for our understanding of the macroscopic behavior of the system. Indeed, these trajectories look extraordinarily erratic; their complication defies imagination. In addition, they turn out to be extremely unstable with respect to minor changes of the initial conditions; no, or very little, scientific information can be gained from their analysis. However, if instead of looking at the individual motion of each particle, we examine more global properties, depending on a large number of particles, we find on the contrary that these are generally quite stable and largely independent of the precise initial

conditions. For example, from the solution (I.1) for a sufficiently long time f, we can compute the number, denoted n(v, Av), of molecules having a

velocity |v(t)| comprised between v and v + Av. It is observed that whatever

the

initial

condition,

ratio

the

n(v, Av)/N

becomes,

after

some

time,

approximately time independent and very close to the normalized Gaussian

integral

otdo

{

oo

{

oO

dv v” exp (-av’) (1.3)

dv v? exp (— av’)

provided n(v, Av) > 1. In (1.3) the parameter a turns out to be equal to a

3mN

= 4E*

(1.4)

where E* =¥%_, (mv2(t)/2) is the kinetic energy of the system, which also appears to become nearly constant. Let us stress that the precise value taken by n(v,Av)/N in a given computer experiment depends on the initial conditions chosen and can be computed exactly only after the equations of motion have been solved; yet except for small deviations that cannot be predicted a priori, (1.3) generally gives the correct answer whatever the initial conditions. Moreover, the deviations of n{v, Av)/N from this value

tend to become smaller when the number N of particles in the system is increased. Information like (1.3) is typically statistical ; it does not pretend to give the exact answer for any specific experiment, but it is a “fair” approximation for most of them! These computer results suggest how we can develop a theoretical frame for treating macroscopic systems that bypasses the difficulties mentioned

6

Introduction to the Theory of Stochastic Processes

earlier. We assume, indeed, with support from experimental evidence, that

the precise value of any macroscopic observable quantity changes very little when the initial conditions are slightly modified (though the individual trajectories are drastically changed). Thus we postulate that the precise initial conditions in a given experiment (which are impossible to measure) are irrelevant in the calculation of macroscopic observables ; they may be replaced by statistical assumptions. In doing this, we of course give up the deterministic description in favor of a statistical one; this should give a fair answer for most specific experiments, however.

In the modern approach to nonequilibrium statistical mechanics, this is the only basic postulate added to the dynamic equations (I.1). It is very important and makes statistical mechanics quite different from rational mechanics; yet though it allows us to bypass the impossibility of determining precisely the initial conditions in a given experiment, it does not eliminate the difficult problem of solving the equations of motion themselves. Therefore this modern approach remains quite involved and is dealt with only in Parts C and D. A much simpler point of view results if we add one more basic assumption: the motion of the particles is so erratic and apparently “random” that it can be described itself in statistical terms. This second, very drastic, assumption is exploited in Parts A and B. Until now, we have deliberately remained rather vague in the formulation of the basic assumptions of statistical mechanics; we have used words like “generally,” “fair,” and “random,” with their common-sense meaning. Clearly, if we want to handle these ideas theoretically, we first need

an adequate mathematical framework: this is provided by probability theory. In Section 2 we recall briefly the basic aspects of probability theory and its relation to experiment. To present these aspects in full generality requires mathematical tools (like measure theory) far beyond the level of this text;

therefore we limit ourselves to considering probability theory for denumerable sets of events. We later deal with the continuous case in the most naive (and mathematically unsatisfactory) way, but the results thus obtained are quite sufficient for all later purposes.

Section 3 introduces the concept of a random variable, and Section 4

defines the so-called stochastic processes (or random functions), which are

the basic tools in working with random motion; the simplest nontrivial stochastic process, the so-called Markov process, is discussed in Section 5. Finally, to convince the reader that these formal tools are very useful in the study of well-defined physical problems, Section 6 treats a very elementary application: the problem of random walk and its connection with diffusion, a familiar and important subject of macroscopic physics.

Elements of Probability Theory 2

7

ELEMENTS

2.1

OF

PROBABILITY

THEORY

The Axioms of Probability*

Consider an abstract denumerable set of elements €), €2,..., €,..., called elementary events. This set, denoted by 9, is the sure (or certain) event or the

sample space. Taking an arbitrary subset of these elementary events, we

form an event %; of Y. By definition, # itself is considered as one of its own subsets and, when no elementary event is taken at all, we form the impossible event @.

To each &,, we associate a number Pr (%;) called the probability of €, which is supposed to satisfy the following properties: (i)

Pr(é)20

(1.5)

Gi)

Pr(O)=0

(6)

(ii)

Pr(P)=1

(1.7)

Gv) P(E w= 5 Pri)

if the events %,,

(18)

..., 8, involve no common element ¢, thatis, if 81, ..., &,

are mutually incompatible; in (1.8), n may eventually become infinite. The specification of the set # and of the probabilities Pr (8,) defines a probability law [Y, Pr (&,)]. Probability theory exploits the logical consequences of these very simple axioms. 2.2

Probability Law and Experiment

The axioms of Section 2.1 are very abstract and do not tell us how a probability law can be used in a given physical problem. From this point of view, they are very similar to the axioms of geometry, in which objects like “points,” “lines,” and “distances” remain undefined and, in principle at least, are not related to our intuitive understanding of physical ‘“‘points,” “lines,” or “distances.” Contact with the physical world is made when one

postulates some correspondence between these mathematical entities and some physically observable property. In general, this correspondence is not unique (e.g., the concept of a line may be associated with the trajectory of a mass particle in the absence of forces or with a ray of light in optical geometry), and it is therefore preferable to separate it clearly from the abstract logical content of the mathematical theory. The validity of the * Probability theory, largely relieved from its sophisticated mathematical context, is brilliantly exposed in Feller (1950).

8

Introduction to the Theory of Stochastic Processes

assumed correspondence can be checked only by the correctness of the predictions of the theory. We are concerned here with the application of probability theory to the physics of macroscopic systems, and we are going to become gradually acquainted with the way in which such a correspondence is established. Yet for the sake of orientation we present a few preliminary remarks and illustrate them with the traditional example of the game of dice—admittedly extremely simple compared to the N-particle problems we have in mind. Let us first clarify the idea of elementary events, abstractly introduced in the axiomatic formulation of Section 2.1. Physically, we identify them with the possible outcomes of a given experiment. The definition of “‘outcome,” however, is not unique and must be carefully specified. In the example of the game of dice, an outcome is usually taken as being one of the six faces (1, 2, 3, 4, 5, 6) pointing up. However, it should be stressed that this corresponds to an idealized situation, and more (or fewer) outcomes could be

validly considered as the elementary events. For instance, we could retain the possibility that a die falls on one of its edges, or we could ask a more precise question, such as: what is the place on the table where the die stops? On the contrary, we might be interested only in whether the upward face is “odd” or “even”; we could then legitimately claim that there are only two possible outcomes, “odd’’ and “even.”

Once the elementary events are defined, the events themselves appear as families of outcomes characterized by some common properties; for example, with

the elementary

events

defined

by

1,2,...,6,

the event

“the

upward face is a prime number” will occur whenever the die shows one of the faces 1, 2, 3, 5. The sure event corresponds to “any face up” and the impossible event is ‘no face up.” After the events have been specified, a more delicate task is to assign them probabilities. To discuss this point, let us consider what happens when the given experiment is reproduced many times. In a deterministic situation, which is illustrated by classical mechanics [see (I.1)], it is supposed that the conditions of the experiment can be controlled with an accuracy such that, for given conditions, the same outcome is always obtained. For example, a skillful billiard player knows how to hit the two other balls with his own by looking at the precise position of the balls on the table; he finds no need for probabilistic arguments in his play. Yet in many cases the conditions of the experiment are not well enough under control, and even with the best available control, the same conditions lead to different outcomes; this is the

place for probability. In the case of billiard game, there is of course some element of subjectivity regarding the best available control; it might indeed be better for a bad player to make use of probability ideas before taking up his cue! However the most important applications of this theory deal with

Elements of Probability Theory

9

situations involving more basic and more objective difficulties. We have already encountered an example in discussing the impossibility of knowing the initial conditions in a many-body system. As another example, let us return to the game of dice: because of the enormous difficulty of calculating the dynamics of a cubic object thrown in the air, and the extreme sensitivity of this dynamics to slight changes in the initial conditions, nobody knows how to play a die to ensure throwing a “‘six.” All he can get is a statistical estimate of how many times the six will come out of a series of successive experiments. Two methods are useful to the physicist in obtaining such a statistical estimate: (i)

He can argue that because of the symmetry of the dice, no reason exists to favor one face rather than the others; he then a priori assigns a probability 1/6 to each elementary event; the probability of the other (compound)

(ii)

events

is then

determined

with

the help of (1.8). For

example, the event “the upward face is a prime number” gets probability 4/6. This point of view corresponds to the classical (or a priori) definition of probability. Especially if he is an experimentalist, however, he may be suspicious about such a priori statements. For example, he may argue that despite his symmetrical aspect, the die is perhaps “unfair,” since its center of gravity may not lie at its geometrical center. He will then perform a large number M of experiments and define

Pr (&)) where M, denotes the number

_ M, vid 77

(1.9)

of experiments with the outcome

¢,

(i=1,2,...,6). Clearly this frequency interpretation satisfies the axioms of probability theory, and it may seem preferable to the classical a priori viewpoint because it is more operational. Yet difficulties remain with (1.9); besides the mathematical question of the existence of the limit in (1.9), no infinite sequence of experiments can ever

be performed, and the strict application of (1.9) is impossible. The usual approach is to replace (1.9) by the approximate expression

Pr@=T2

(M>1)

(1.10)

As a matter of fact, in many cases, probabilities are assigned by combining the classical and the frequency viewpoints. Suppose, for example, that 1200

successive throws are performed with a die and that the respective M; are 198, 205, 189, 209, 214, 185, for the six faces 1, ... , 6. Applying (1.10), we

10

Introduction to the Theory of Stochastic Processes

could take Pr (e;)=.165,

.171, .157, .174, .178, .154, respectively; yet a

reasonable mind would not consider these figures significant and would take instead

Pr (e;)=1/6=.1666...

(f=1,..., 6); doing

appeal to the classical point of view.

this,

we

of course

Incidentally, the definition (1.9) clearly shows that it is possible to have

events with zero probability that are different from the impossible event @:

if M, =o(M),* the corresponding Pr (¢,) is zero, although the event €, may occur (even an infinite number of times for M > 00!); we say that e; is almost

impossible. Similarly, we may have Pr (e;)= 1 although ¢, is not the sure

event F; it is almost sure.

Suppose now that we have decided what probability law is adequate for a given experimental situation. Of course the usefulness of this formalization lies in the predictions that can be made from the logical consequences of the axioms of probability. But to check the validity of these consequences, we again confront the same problem: a correspondence must be established between the theoretical predictions and the experiments. Here, of course, the a priori viewpoint is meaningless (otherwise there is nothing to predict!), and we are bound to use the frequency point of view (1.9), or rather its approximate version (1.10). For example, anticipating later results, let us see what happens when we throw two dice independently.} It is easy to show that the probability of finding two “sixes” pointing up is then 1/36. This is verified experimentally by performing this double throw a large number of times M, and checking that indeed M,/M = 1/36, where now M, denotes the number of double throws where the two ‘“‘sixes”’ are face up. Let us make two final remarks concerning (1.9).

(i)

In writing this formula, we have supposed that a large number M of experiments are performed in succession. However another way of reaching the same goal is to take a large number M of replicas of our physical system and perform one experiment involving them all at the same time. In our previous example, we could very well imagine that 1200 dice are thrown simultaneously, whereupon we determine the frequencies from this single throw. The procedure is very artificial in this particular case, but it plays an extremely important role in statistical mechanical applications. Indeed, the N molecules in the system generally play an equivalent role and are subjected to the same

* We use the standard notations: f(x) =o(g(x)) if f(x)/@(x) > 0 for x +00 and f(x) = O(g(x)) if f(x)/e(x)> constant for x > 00. + Here again, the correspondence between mathematical independence, as defined below in (1.34), and the physical independence of successive throws—which was tacitly assumed in (1.9)}—should be made precise. However we take it for granted that the first concept is the correct axiomatic translation of the second.

il

Random Variables

Gi)

probability law: hence the frequency interpretation is often applicable to one single experiment. This point is illustrated later on. A famous theorem of probability theory, which we do not prove here, clearly indicates the evens and odds of formula (1.9). Suppose that we perform independently L times the same experiment corresponding to elementary events e, with probability Pr (e;). This succession of experiments, taken as a whole, can be considered as one single compound experiment, and is itself subjected to a probability law. In particular we can define the event “the outcome e; appears L; times in the compound experiment” and, from this, the event characterized by the property L; |E-Prie) L

2.

The Classical Theory of Brownian Motion

36

Thus in the limit At— 0, the equation of evolution (II.14) becomes

3,0, +¥

am,

——s

a

—

(&

)

mre}ts

es | —

1a—«

+-—

a (& Se

|

wee

)

(11.30) .

The parameter &q can be related to the friction coefficient Jy and to the

absolute temperature Tof the surrounding fluid, by assuming that, whatever the initial state of the system, the probability density ®,(r, v, ¢) tends, for sufficiently long times, to the equilibrium Maxwell-Boltzmann distribution

2G. vlaseca) PPA ”(-aia7)

:

OMG, v= ==5 QakaT!) uot Br, v, NS=o" Here 2

a

II. (11.31)

is the total volume of the system, and kg is Boltzmann’s constant.

Although we expect the reader to be somewhat familiar with the Maxwell— Boltzmann distribution, it is perhaps worthwhile to derive it here as an illustration of the concept of independence, rather formally introduced in Chapter I, Section 3.4. This derivation follows in fact the original calculation of Maxwell, and it played an important role in the history of the kinetic theory of gases. Let us denote by ¢,(v,) the probability density that the x-component of the velocity of a given particle is v,. Because there is no preferred direction in space, the same function y, describes the probability density of v, and v,, Let us also introduce the probability density F, that the velocity vector takes the given value v. Taking again the isotropy of space into account, this function can only depend on the length of v

F, =F, (|v) =Ve2 +03 +0)

{II.32)

Maxwell assumes that the probability densities for the velocity components ,, vy, 0, are independent; he writes

F, = 9,(0,) 91(2,)¢1(2,)

(11.33)

We now show that (II.33) implies (II.31). Indeed, taking first the logarithmic derivative of (I1.33) with respect to v,, we get (for v, #0)

1 0F, le 1 _ —ot lv dv] F,

1 ag(v,) x,

82,

1 —-

glo,)

(11.34

Differentiating again this equation with respect to v,, we obtain (for v, # 0) @o1aF, ened 1 le 5)

(11.35)

1 oF iF 1 soe Mal

(11.36)

aly| |v alvl Fi

Hence

The Fokker-Pianck Equation

37

where —2a is the (unknown) more, we arrive at

constant of integration of (I1.35). Integrating once

F,= (2) TT:

3/2

exp (—av”)

(11.37)

where the factor in front of the exponential has been determined by the normalization condition

(11.38)

fav F,(v)=1 To fix a, we calculate the mean kinetic energy of one particle EX

nie

M |

A

3M

dv v°F,(\v/)= 7

(11.39)

and we relate E* itself to the thermodynamic temperature by identifying the perfect-gas law p = nkgT for the pressure (n = N/Qis the density), with the result of the classical virial theorem* for the equilibrium pressure pQ = 2E*/3. Hence we obtain

a

M 2kpT

(II.40)

If we furthermore take into account that, at equilibrium, in the absence of external

forces, all positions in space are equiprobable, we immediately obtain the expression

written in (11.31) for f(r, vy).

The necessity of the assumption (II.31) is of course very unsatisfactory: the approach to equilibrium of the 8-particle with the fluid should come as a consequence of any good theory of nonequilibrium; it should not enter as an ad hoc input. Unfortunately, we must await a discussion of Boltzmann’s ideas (Chapter IV) before we can even think about properly formulating the basic problem of irreversibility, including the validity of (II.31). The best we can do here is to exploit the consequences of this assumption and to show that it leads to nontrivial results that can be verified experimentally. If (11.31) is a stationary solution of (11.30), we must have

0

-4,(% 1a,0(é aa (Sra; 3) +35 aa ee )

(1.41)

or fa IMk,,T! be )2. (2 av vb eq

* See, for example, Ter Haar (1954).

=0

(IL.42)

38

The Classical Theory of Brownian Motion

For this equality to be satisfied identically for any v, we must have fa =2kpTly

(11.43)

Equation (II.30) becomes then the Fokker—Planck equation

a, +Vv

2Firs _ by 2, [ v+ kT ey2]

®,

(IL.44)

Its proof can be generalized without difficulty to the case where an external force F(r) acts on the 8-particle; we get then

aPitve

a®, Fie) 90) _ fy 0 [ ‘eT 2) ay Mav LYM vl!

(11.45)

The Markovian character of the classical Brownian motion theory is not immediately apparent in the present calculation; in particular, our starting point (II.5) is independent of this assumption. A closer examination reveals that this property is deeply rooted in the statistical properties assumed for the Langevin

equation

(thus for the transition probability y,,); however

we postpone the discussion of this point to Chapter III and examine here some consequences of the Fokker—Plank equation.

3 APPLICATIONS OF THE FOKKER-PLANCK EQUATION: APPROXIMATE METHODS The Fokker—Planck equation (II.45) is linear; therefore its solution can be

studied by the well-developed eigenfunction-expansion method

methods of linear analysis (e.g., the and the Green’s function method). In

particular, in the absence of external forces, this solution can be obtained in closed form.

However, rather than starting with an exact calculation, and deducing from it the physically interesting consequences, we prefer not to hide the physical problems behind a rather heavy mathematical apparatus and to attack directly these physical questions by approximate methods, which have a much wider range of applicability than the Fokker—Planck equation. To check the validity of these approximate methods, we tackle the same questions from a more satisfactory mathematical point of view in Appendix A and in Section 1 of Chapter III. Before starting these calculations, it is convenient to consider a system containing Ny %-particles, instead of a single one. The so-called oneparticle distribution function (d.f.)

At, v; 2) = Ng.

v, 2)

(11.46)

Applications of the Fokker-Planck Equation: Approximate Methods

39

is then such that f,(r, v; t) Ar Av gives the probable number of 8-particles lying in the volume element Ar Av around (r, v). The local density of 8-particles, na(r; f), is obtained by integrating f, over

all possible velocities

nalts t) =| dv fi(r, v3)

(11.47)

and an additional integration over r leads to | dr na(t; t) = Ng

(11.48)

since ®, is normalized to 1. If the 8-particles are sufficiently dilute (ie., Na« N,, where N, denotes the number of fluid molecules), the Fokker—Planck equation for ®, remains valid and we have

F

Of, +

itv

Mav

M

where the Fokker-Planck operator O is defined by

a2. (® (ketst’ 8 ) O=>

(11.50)

The Fokker-—Planck equation (11.49) has the form of a kinetic equation, governing the time evolution of the one-particle df. Let us now examine various consequences of this equation. 3.1.

Electrical Conductivity

Suppose that the %6-particles are charged—the charge per particle being eZ—and

that we submit them

to a constant electric field E,, along the

x-axis. We expect that after some transient time, a stationary state will be reached in which the particles move with a constant velocity; we want to find the corresponding electrical conductivity. We soon see that this problem can be tackled with the help of the Fokker-Planck equation, but there is no need for such a sophisticated description. Indeed, let us consider the Langevin equation (II.3), suitably generalized to take the presence of the external field into account

do, __Met tw, ,eZEx, fxlte) a MM M

(11.51)

Taking the average of this equation, we get div.) __ Sx

a

eZE,

Mota

(11.52)

40

The Classical Theory of Brownian Motion

where (II.24) has been used. A stationary solution (d(v,)/ dt = 0) is

(v,)=

ZE,

(11.53)

ie)

The corresponding average electric current per unit volume is

See)

_

ZB, to

qa hweZ(o,) = ne

(11.54)

where My is the (constant) density of 8-particles; the electrical conductivity og, is thus

oe "OF"! Ueda n f= Ge

(11.55)

Of course it is unrealistic to consider a single charged species; we should instead investigate a mixture of species a=1---/ characterized by the

parameters n,, Z,, £4; this is the only way to satisfy the electroneutrality

condition

i

L Zn

a=l

=O

(11.56)

The generalization of (II.55) to this case is trivial; one finds

a

1

272

Ie a

aml

(11.57)

ar

This result is so simple because in the calculation of the average velocity of a %8-particle, the fluctuating Langevin force disappears. Nevertheless, to gain some familiarity with the Fokker—Planck equation, it is worthwhile to rederive this simple result on the basis of (11.49). If we

again neglect transient effects and look directly for a stationary state, we get in a spatially homogeneous system eZE, ofi _ ox M

av,

M

Of;

(11.58)

We multiply this equation by v, and integrate it over v; simple integrations by parts lead to

kpT

@

~eZE, | dvf,=- fe/ av(“7 a+o)h

(11.59)

In the right-hand side, only the second term in the parentheses leads to a nonvanishing contribution, which is —¢(v,)ny/M, where the average

Applications of the Fokker-Planck Equation: Approximate Methods

41

velocity (v,) is defined by

w=4

ny

| dv vf;

(11.60)

Inserting (11.47, 60) into (11.59), we indeed recover (IJ.53); here also, the

generalization to a set of different species is trivial. Incidentally, the friction coefficients are parameters of the theory, and for charged particles conductivity measurements often can be used to determine them experimentally. Another procedure often followed in practice is to use the result of the macroscopic theory*

Sg = Oana

(11.61)

where 7 is the viscosity of the fluid and a the radius of the 6-particle. This formula usually gives satisfactory results. However, from a theoretical point

of view, there is no evidence that this formula should hold beyond the limits

of macroscopic hydrodynamics, on which it is based. Except when the %8-particles have a macroscopic size, (IJ.61) should not give more than an order of magnitude. 3.2.

Diffusion

We have already seen how a diffusion equation for the density of the %-particles emerges from the random-walk model. Here we derive the same

result from the Fokker—Planck equation. Let us start from (II.49) in the absence of external forces and define the

Fourier transform of f;

f,(v3t) =| dre

'*"f,(r, v; t)

To simplify, we take the vector q along the x-axis: readily Ofgl¥; t) = (-igo, +20)

(II.62)

q= q1,. We obtain then

0; t)

(11.63)

For the sake of compact notation, let us introduce an abstract vector space (Hilbert space) |f) to represent any function f(v) of the velocity v. By definition we have

fw=lf)

(II.64)

Moreover, we define the scalar product of two vectors lf) and lg ) by

(flg)= | dv oi'(v) 'f*()g(v) * See, for example, Landau and Lifschitz (1963).

(11.65)

The Classical Theory of Brownian Motion

42

where ¢%4(v) is the Maxwell-Boltzmann equilibrium velocity distribution function [see (II.31)]. €4(4)

eile)

=

( ({a/M) small enough, the solution (II.82) will be given by

(11.90) ', and provided q is

|fa(0) = cB exp (Az Iv) +0 (exp (£2) q>0,

(2ey)

’

d.91)

Taking the time derivative of (II.91), we get

alfa = Ab f(t)

490

(11.92)

Since the Fourier transform ng,q(£) of the B-particle density is [see (II.47)]

nyqlt) =| dv favs ) = (eT fat)

(11.93)

The Classical Theory of Brownian Motion

46

we get also

(11.94)

970

dry q(t) =Agnaalt)

ion equation (1.106), Comparing this equation to the macroscopic diffus which reads in Fourier space

(11.95)

Ati q(t) = Dany q(t)

tion of this diffusion we see that the eigenfunction method leads to a deriva we get an explicit er, equation from the Fokker-Planck equation; moreov More precisely, ’). expression for the diffusion coefficient D as (—Aé/q because q was taken very small, we have

D=lim 4

q

(II.96)

nt from zero. Before We see presently that this limit exists and is differe fication that has been demonstrating this, let us stress the considerable simpli ¢ becomes large and q brought into the problem: by taking the limit where problem to the small, we have reduced the solution of the Fokker—Planck in the limit where calculation of one single eigenvalue of the problem (11.67) q becomes small.

evaluate Aj by a The fact that q is small immediately suggests that we perturbation calculus. We write

tq? Ag + GA S+ AS=A

+

(11.97)

m, and the correcwhere A®= 0 is the eigenvalue of the unperturbed proble or (—iqv,) asa operat ian ermit non-H tions are calculated by considering the small quantity.

and the closure Just as we have shown that the orthogonality property ed we use provid ors, operat ian ermit relation can be generalized to non-H the stanthat ately immedi proved be right- and left-eigenfunctions, it can from elementary quandard formulas of perturbation calculus, well known

tum mechanics,* remain valid with a non-Hermitian merely quote the final resultt

g A$? = (we|- iqvslwo)

a? AG = 4? n#0E, (Wboslvn) x01 n Wn lesa) * See, for example, Messiah (1963).

+ Note that (| = (#%] since O is Hermitian.

perturbation. We

(11.98)

(11.99)

Methods Applications of the Fokker-Planck Equation: Approximate

47

We first notice that The evaluation of these expressions is straightforward.

symmetry imposes

(1.100)

AY’ =0

restriction on the Second, using this property (11.100) to eliminate the

summation in (II.99), we write

GAP= 4? lim Y WAedbzo—0(wAle|wd 0+ altn

A,~

&

—U-101)

« #0. because the term n = 0 added in this equation gives zero for With the help of (II.84) and of the closure relation

(11.102)

LleoXwrl = 1 we can formally rewrite (II.101) as

AS? = 4? lim Co) ve *

ue —é

vlv)

(11.103)

Equation (I1.96) thus becomes

D= lim (wole:|xe) Ene

= lim | dv vu,X.(¥)

(11.104)

1 HO)° =a pore ke)"=D

I1.105 (11.105)

e704

where |y.) is defined by

Indeed, let us The matrix element in (II.104) can be calculated exactly.

multiply (11.105) by — (fg/M)O +e

with (olv,; we obtain

(v3

o,f _

and take the scalar product of the result

O+ 2)

x.) = Wolve hbo)

(11.106)

form of the Using the definitions (11.65, 86) together with the explicit simplified be can operator O (see (II.50)], the matrix elements of (11.106) considerably. We obtain

(+)$94.6) || dv

oxe(¥) =Pz

(11.107)

The Classical Theory of Brownian Motion

48

Inserting this formula into (11.104) and taking the limit ¢ >0,, we finally arrive at

D

akeT Sx

(II. 108)

This is the well-known Einstein relation for the diffusion coefficient. It is usually derived by very different, often simpler, arguments (see Chapter XI). However the merit of the present method is that it can be generalized with no effort to much more complicated transport phenomena (e.g., transport processes in the dilute gas).

Ill Gaussian Random Processes*

1

FUNDAMENTAL SOLUTION OF THE HOMOGENEOUS FOKKER-PLANCK EQUATIONt

Besides the eigenfunction method discussed in Chapter II and in Appendix A, other techniques have been developed to solve the Fokker—Planck equation exactly. This section obtains this solution in closed form when the initial condition is independent of r and is a delta function in velocity space, that is, when

®,(r, v, 0) = 2 5(¥—¥6)

(III.1)

Let us first make the following simplifications, which lead to a more convenient notation without essential loss of generality. (i)

With the initial condition (III. 1), it is clear that the probability density ®,(r, v, t) remains independent of r for all ¢. It is then simpler to

consider the velocity probability density ¢,(v; f) (also called velocity

* This chapter can be omitted on first reading. + For this point and for many other aspects treated in this chapter, see the excellent book by De Groot and Mazur (1962). 49

Gaussian Random Processes

50

distribution function: see Chapter VII), which is defined by gly; o=|

dr (rr, v, t)

(111.2)

2

Gi)

Since the initial condition (III.1) becomes now

(II1.3)

gilv; 0) = 8(¥— vo)

the solution ¢,(v; t) of the Fokker-Planck equation obviously gives

the velocity-probability density at time 4, when this velocity was equal to vo for sure at ¢ = 0: it may be interpreted as a conditional probability density. To keep this in mind, we write this particular solution as (11.4)

(¥, tivo, 0) ¢1(¥; t)= Gin

This type of solution is known in mathematics as the fundamental solution (or Green’s function).

(iii)

Moreover we limit ourselves to one dimension, writing v for 2,.

We look thus for the solution of

kaT 2)o,(0,t100,0) apesil0, too, = 28 2(y442F

(ILS)

11(8, O]vo, 0) = &(v — v9)

(III.6)

such that

Let us define the variable u by

tn!) y

u=exp = ( M

(11.7) 7

and the function

t

xu, tlv9, 0) = eu(exp( —!2f), tlvo, 0)

(III.8)

An elementary calculation then transforms (III.5) into

a,x(u fv, 0)=S2{ 1 +exp

2a!)

Mi)

eT

M

a

au x(u #00, 0) FA)

(III.9)

an equation that is readily solved by Fourier transforming x with respect to u. If we put Xw(t|vo, 0) =|

+20

co

du exp (—iwu)x(u, t|vo, 0)

(III.10)

51

Fundamental Solution of the Homogeneous Fokker-Planck Equation

“Efron

(III.9) leads immediately to

(III.11)

lXw

Wo

M)M

1-exp

M

aXw

The solution of this equation, subject to the initial condition [see (III.3)]

(111.12)

Xw(O|vd9, 0) = exp—iwvo is

kaTw’

.

Xw(tlvo, 0)=exp | —iwo, +4220"

,

{'

Loot!

[ dt’ exp (742")))

(11.13)

This result can be verified by differentiating it with respect to 4. With the abbreviation y(t)

=

és) _

=exp ( M

(III.14)

1

the function y is thus given by the inverse Fourier transform

{

x(u, t|d9, 0) cei kgT

+0

dw exp [iw(u— v9)

2

(111.15)

om VOW]

a Gaussian integral which can be performed by completing the square in the

exponential; we get

_

M

X(u, t|vo, 0) = (=n)

ir}

Sal

exp ( fat) exp [

_Mu- Do)"

Tk. Ty(). |

(HIL.16)

Turning back to the velocity variable v, we obtain the required solution

I(t)” ue exp [ _ 20Moo -F())kaT ee]

M _ Pip(®, tlvo; 9) -(=q =e)

(111.17)

where the function I(t) is simply

-

T(t) = exp (

fs!) M

(III.18)

The Gaussian character of (IIJ.17) is an important property, which we

investigate more carefully in coming sections. Random processes of this type are classified under the general heading of Gaussian random processes.

Since, in addition, the Fokker-Planck equation describes a Markov process

(see Section 4.2), we speak here of a Gaussian Markov process. A very

52

Gaussian Random Processes

simple meaning can be given to ['(¢) in terms of the conditional expectation of v(t, €) defined by

(0) vo = [ We have indeed

dv ve i(2, tlvo; 0)

(111.19) (111.20)

{v(t))u9= Pol(4)

which shows that I'(t) describes the decay of the average value of the velocity

v(t, €) when its initial value is given. Also, we observe that, for > 00, giji tends to the Maxwellian distribution; this is of course no surprise because we

have constructed the theory of Brownian motion in such a way that this property is automatically satisfied [see (II.31)]. 2

STATIONARY

RANDOM

PROCESSES

This section introduces a few more definitions and properties of random processes.* Though these topics might have found a suitable place in Chapter I, we prefer to put them here just before their application in Section 3, which deals with stationary Gaussian processes. 2.1

Definition of Stationarity

Arandom process Y(t) is stationary (in the strict sense) if its joint probability distributions satisfy the condition

Daas

ATS Yas batTS

65 Yar be FT) = DalVas £15 Vos t25 ++ + 5 Yow In)

(1.21)

(for all n and r). Thus the statistical properties of the process are independent of the absolute value of the time: they only depend on the time differences t; — tp, to— ts, .... In particular, ®,(y,, f) = ,(y,) and the expectation of ¥(t)

(yoy

fay yPi(y, =e

(II1.22)

is a constant c. Similarly, we have

(y(¢+7)y()) = | dy, dy2 yiy2P2(y1, £473 Yas f) =R(r)=R(-7)

(IIL.23)

* Anextensive but elementary discussion of the topics treated in this section can be found in the book of Papoulis (1965).

Stationary Random

Processes

53

(we used property (i) p. 19). The function R(7), called the autocorrelation

function

(or covariance) of the process Y(t) only depends on the absolute

value of the time difference 7. It satisfies the very important inequality

(11.24)

|R(z)|=R(0)

This can be seen by taking the random variable ¥(r) + Y(0) for fixed 7. We have

0t, >)

(III.92)

It is well-known that the solution of this functional equation is an exponential [see (11.33) for a similar example] and we have

T(t)=exp(-eltl)

(c= 0)

(111.93)

The condition c = Q is required by (III.26), and the dependence on \¢l comes from the symmetry ['(t) =T'(—2).

64

Gaussian Random Processes

4

4.1

FURTHER

REMARKS

Nonstationary Gaussian Processes

We have limited ourselves thus far to statiqnary situations, but nonstationary Gaussian processes, which can be defined as well, are of some interest for

acquiring a better understanding of the Langevin equation (see Section 4.2). Taking for simplicity a one-dimensional process Y(t) with zero

expectation—otherwise, we simply replace Y(t) by (¥(¢)—(y(t)))—we say

:

facta”

the process is Gaussian if it has the following joint distribution: nA (V1, (V15 £1515 Yas Vos b23te

+

5 JYn fe)) =| Say (27)"|Rl

x exp [-3 x ye(Ru] 2 kimi

where R''

(n=1,2,...)

(111.94)

is the inverse of the matrix R with elements (111,95)

Ry = R(t th)

depending only on the times ¢, and 4. These matrix elements (which can always be taken as symmetric with no loss of generality) are arbitrary except for the positivity condition

LD ye Rey,=O kimi

(yt ++ y, arbitrary)

(III.96)

which ensures that (III.94) vanishes at infinity. First of all, we should convince ourselves that (III.94) defines indeed a probability distribution; that is, that properties (i) to (iv) of Chapter I, Section 4, are satisfied. Among them, (i) and (ii) are obviously true, whereas

(iv) is a consequence of the equality between (III.74) and (III.75). Property (iii), namely fa.

DaCyast15 665 Yate baa 13 Yrw bee) = Bn

as fs 0

3 Yaa

Pn)

(11.97) and a few other features of Gaussian processes are readily established when

the so-called characteristic function of ®,, which examined. It is defined as the Fourier transform

we

denote

dy, exp (-i x YaP) Ba Vy f05 =

Yu i)

by x,,

is

Xn(Pisfis - +3 Pm ba) = [ayy

kel

(III.98)

Further Remarks

65

and is thus such that

Diyas 5 «+6 3 Ynw tn)

=a | dos dp, exp (iS vape)xalDre thi. --5 Puta) (11.99) 1

n

(27)

kel

The calculation leading from (III.68) to (11.64) can be immediately applied to compute y,, with the very simple result

Xn (Pas ths «+3 Pas t)=exp(-5

1

(III.100)

y a) i wl

Three important properties follow. (i)

The autocorrelation function of the process ¥(t) is the function R(t, t)

(y (4) ¥(t)) = R (tes &)

(III.101)

Indeed we readily see from (III.98) that a8

(y(t) = -[2 S— Xn(Pisfis ~~~ 5 Pas 1) OP

OP

pie

pana

(IIT.102)

which, supplemented by (III.95, 100), leads to (III.101). Incidentally, we can now understand why (III.94) was, strangely enough, defined in terms of

an inverse matrix R™': this inverse is a rather complicated function of the times ¢,,..., ¢,; on the contrary, the direct matrix R has a simple physical

meaning and naturally appears in the characteristic function x,.

(ii) The correlation functions of order n > 2 vanish for n odd and are products of pair correlations for n even. The general proof is tedious and we limit ourselves here to n = 4; from (III.98), we find

(yay ye )y(&))

‘Lae ou

OPk OP OP, APs”

I pi= paw

-[ pee (ERG wrnan)| =

* e Lana R(t,(te, t)Xn tn + top ap. ap,

Pie

R(t

pnw O

(ty ta) bu) B, Pu 2 YR (t,t)(bes be) PoX,PoXn

]

Pim =pa=0 (111.103)

66

Gaussian Random Processes

In the first term, we again differentiate y,, with respect to p, and p,; in the second term, these two derivatives have to act on the factors p, and p, in front of y,,; otherwise we get zero when we put p;= +--+ =p, =0. We find then

(y(t y dy (6) y (t6)) = R (tes OR (bb) +R (te HR =

(f+

R (te ts).(te fy)

2, I (yy) al pairs Gi)

(111.104)

This is of course the generalization of (III.59) to nonstationary processes. (iii) Equation (111.97) is satisfied. as follows Bi

3.

Yna

Indeed, let us define the function ®;,_,

fe) = { a, (yi,

f15 +. 5 Ym

ba)

(111.105)

With the help of (III.99, 100) and the representation (III.66) for the Dirac delta function, we immediately obtain the required result. By

(Yi 85-65 Yn

fr)

1

= (Qn)"

:| dp: +

Pat

werp(-i3' wa) exp(-E"S miken) n~l

1

kel

= B11, 15 6

4.2

nal

kel

Yaa

be)

(III.106)

Back to the Langevin Equation

Let us reconsider the Langevin equation (11.3) which, for simplicity, is taken here in one dimension

dv__fw

aM’

fee) M

(t>0)

(IIT.107)

The solution of this equation is v(t, €) =exp ( a Exo)

+2 j dt' exp (2

)ie: 2]

(IIT.108)

which depends of course on the velocity vo of the 8-particle at f=0; this velocity may itself be a random variable, as is made explicit in the notation. We here carefully discuss the properties of the stochastic force f(¢, e) and

justify from these the various assumptions made in deriving the Fokker Planck equation in Chapter IJ, Section 2; by the same token, we rederive ina

Further Remarks

67

much simpler way the expression (III.17) for the fundamental solution of this Fokker—Planck equation. We assume that (i)

f(t, €) has zero average

(111.109)

{F(z))=0 di)

For ¢>0, f(¢ €) is uncorrelated to vo(e); in particular

(III.110)

{f(2)09) = 0 iii)

f(t, €) describes a stationary Gaussian process with a spectrum inde-

pendent of w (a white spectrum) and equal to S(w) = 2ky Thy

(111.111)

Let us examine the role of these assumptions. Condition (i) was discussed in Chapter II, Section 1: it appears because the whole systematic effect exerted by the fluid on the 8-particle is included in the friction force (—£qv). Notice that for a fixed initial velocity vo, this sole

condition is sufficient to prove the known result [see (II1.18, 20)].

(0(0)a4= vo exp (=— 48tel)

(1>0)

(111.112)

from the solution of the Langevin equation: we merely have to take the expectation of (III.108).

Condition (ii) says that the force exerted by the fluid is independent of the initial velocity of the 8-particle. Though reasonable, this has the unpleasant feature that the initial time ¢ = 0 thus plays a special role. Condition (iii) appears very strong indeed. From (III.59) and (III.35), we

know that it is equivalent to the two following statements: 0

Hed

n odd

TGM= 5 y TEgadG)) aye

({(0)flé2)) = 2Laky 78 (6, — 6)

neven

To get a better understanding of these statements,

—_—(IHL.113) (111.114)

let us first notice

that (III.113, 114) imposes that f(t, €),..., f(t, €) are independent when

t;# +++ #£,3a process of this type is known as a differential process (we can of course imagine differential processes which are not Gaussian). The assumption that f(¢, €) is differential is very attractive because it guarantees,

independently of its Gaussian nature, two important properties.

68

(i)

Gaussian Random Processes

The privileged role of t=0, explicit in (III.110), is now destroyed: f(t, €) is independent of v(to) for f >to, with to arbitrary; for example, we have {f()v{to)) =0 {f> fo) (111.115) This is readily seen by using (III.108) for ¢ = f, multiplying it by f(¢, €)

and taking the average; the differential character of the process ensures that

>

GOEM=0

(111.116)

y>F'/)

and this, together with (IJI.110), leads to (III.115). (ii)

We can now also prove that v(¢, €) is a Markov process. Indeed, from (III.108) again, we have for any time t) 3]

(1.117) ;

and from {III.115), we know that the statistics of f(t’, e) (t>t'> to) is independent of v(t,, €) for t; 0) [see (11.31, 43)] and Section 1 of the present chapter showed, starting from this Fokker—Planck equation, that the velocity v(t), €) is a Gaussian process.

Reading the Langevin equation as

dv “ €) iG) _ Se v(t, €)+— = MM”

*See Wax (1954).

(111.119)

69

Further Remarks

we conclude that the Langevin force is the sum of Gaussian processes and,

by the lemma of Section 3.2, is therefore itself Gaussian. Incidentally, this theorem allows us to discover that the hypothesis (II.27), made in Chapter II on {(¢, €), is incorrect: the higher moments of f(t, «) do

not vanish but must satisfy (III.113). Yet this slight inconsistency is irrelevant because the property (II.27) was only used to show that jim and

this property

remains

valid

(Av" dar

~G

when

=0 f(¢,¢)

(111.120) is a Gaussian

differential

process. Having discussed with some detail the random force in the Langevin equation, we now show that the direct solution of this equation leads to the results derived in Section 1 on the basis of the Fokker—Planck equation. By the same token, we can illustrate the difference between a nonstationary and

a stationary process.

First, let us suppose that the velocity vp in (III. 108) is fixed; we then have

[ov e)~enp (=!)6] = 2 exp (=)

x [ dt’ exp (£)te, e)

(111.121)

The right-hand side of this equation is a weighted sum of Gaussian variables; thus the left-hand side also represents a Gaussian process. This process is entirely determined by its expectation and its correlation function, which are readily calculated with the help of (111.109, 110, 114); we find

(v()-vo exp (=)

-0

(111.122)

and

([210-v0esn (9) [oer eoew (=), = MFe exp[ 4] M

i dt, { dt’ exp [see] (fenced)

= Mw exp [=a]

{ dt, exp (2x

= 122 exp [2]

1- exp (= tet")

2hykaT

(e>¢')

(111.123)

Gaussian Random Processes

70

In the left-hand side of these equations, we have added a subscript vo to the bracket, to indicate that the average is taken at fixed initial velocity. From (III.94) taken for n = 1, we can also calculate the probability density

for the variable [v(t)—exp (—Zt/ M)vo], which is of course the conditional

probability density ¢11:(v, t\vo, 0); we then recover (III.17).

However we can adopt a different point of view and consider that vp itself

is a random variable, independent of the random force f(t, e) for > 0. If we want the process to be stationary, the condition ¢(v(t)) = ¢1(vo) forces us to

adopt the probability density (III.118) for @,(v); indeed, we have imposed this form when r> 00, Equation (III.108) expresses thus v(t, €) as a sum of Gaussian variables, from which it is again Gaussian. Moreover, the process is stationary and has zero average; this is easily seen by averaging (III.122, 123) over v9, which yields

{vo(t))= J dv (0(t))uP1(Bo) = 9

(111.124)

and

Rt,

= (vot)

= exp (=a)

| dug v9 1(¥o)

tao =BHL= Oe =224]} fare kyT

—fy(t—t'

_keT “M ex

[-f9(t-1) | M

=R(t-¢)

(t>?)

—2Let'

(111.125)

In agreement with Doob’s theorem, we now find an exponentially decaying

covariance [see (III.84)].

B The Boltzmann Equation

IV The Nonlinear Boltzmann

Equation

1

INTRODUCTION

The stochastic description of the preceding chapters does not fulfill the program of nonequilibrium statistical mechanics, which is to make the link between Newton’s equations (I.1) describing the motion of the individual molecules and the time evolution of the macroscopic properties of the system. Indeed, we have merely invoked the “complication” of the deterministic molecular motion—in some imprecise sense—to replace its description by a probabilistic picture. In doing this, we have lost track of all the parameters characterizing the molecular properties of the system (e.g., the law of interaction).

A satisfactory theory of nonequilibrium phenomena has to go far beyond this simple-minded viewpoint; it should tell us how the observed irreversibility of macroscopic systems emerges from the microscopic dynamics. Such an ambitious program is very hard to realize, even in the simplest case of the dilute gas, and we postpone its discussion until the Part C. However the possibility of developing such a program is very strongly suggested by the kinetic theory of the dilute gas proposed by Boltzmann, a little more than a century ago.

73

74

The Nonlinear Boltzmann Equation

Boltzmann’s theory lies at a level somewhat intermediate between the purely stochastic analysis discussed in the preceding chapters and the fully microscopic treatment: it faces the dynamical problem in its simplest aspects (i.e., the two-body problem) but avoids its most delicate features (i-e., the apparent randomness of molecular motion) by introducing extramechanical,

probabilistic assumptions. These assumptions concern the distribution functions describing the statistical properties of the gas, not the individual motion of each molecule. Countless criticisms have been raised against the theory of Boltzmann; yet the successful predictions of this theory leave little doubt of the validity of the Boltzmann equation to describe the statistical behavior of dilute gases. This equation still provides today the key to our understanding of the dynamics of many-particle systems, even for problems that are more sophisticated than those involving a low-density gas. Hence despite its rather

limited range of applicability, the Boltzmann equation deserves a careful and detailed study, which is presented in this and the following

chapters. Section 2 introduces Boltzmann’s definition of the one-particle distribution function; though equivalent to that given in Chapter II [see (11.46)], this definition is interesting because it clearly shows the necessity of a statistical description of N-body systems. Section 3 gives a brief account of the two-body problem in classical mechanics, with special emphasis on scattering theory; this background is used in Section 4 to establish the Boltzmann

equation, which governs the time evolution of the one-particle distribution

function in the dilute gas. The main general properties of this nonlinear integrodifferential equation are studied in Section 5 and are used in Section 6, to derive the conservation equations for the particle, momentum, and energy densities. The connection of these conservation equations with the well-known equations of hydrodynamics is also discussed in Section 6. Section 7 demonstrates that in general the calculation of the transport coefficients appearing in these hydrodynamical equations requires the solution of the Boltzmann equation. This is a rather formidable mathematical problem, and even in the simplest but most important case where the macroscopic properties of the system are slowly varying in space, the Hilbert and Chapman-Enskog methods— devised to obtain this solution—are extremely involved. We limit ourselves here to a very sketchy description of these techniques because fortunately most experiments of interest to the statistical physicist involve systems that are only slightly perturbed away from equilibrium. In such

cases it is legitimate to use the linearized form of the Boltzmann equation, which is mathematically much simpler; its study is the content of Chapter V.

The One-Particle Distribution Function

75

2 THE ONE-PARTICLE DISTRIBUTION FUNCTION

As stressed at the beginning of Chapter I, the macroscopic observables (e.g., density, pressure, average velocity, temperature) depend on statistical properties of the system, not on the detailed motion of the individual molecules.

We learn later that in the dilute gas the calculation of these properties requires only the knowledge of the one-particle distribution function (d.f.)

f(t, v; ¢) defined, according to Boltzmann, in such a way that file,v; 2) drdy

(IV.1}

is the number of molecules that lie at time ¢ within a volume element dr around point r and have a velocity in the element dv around v. This simple definition requires some comments because it is useless taken too literally: because of intermolecular repulsion at short distances, there can be either

zero or one molecule with its center inside a mathematically infinitesimal volume element dr, and this number is expected to vary extremely rapidly with time; moreover it depends on the shape of the element. We should consider instead a ‘“‘physical” element Ar Av that is “macroscopically infinitesimal” but “microscopically infinite”, that is, small enough for all macroscopic properties not to vary appreciably over its extension, but big enough to contain a large number of molecules. Even so, the number of molecules within a physical element remains a strange object: albeit large, it changes discontinuously with time whenever a molecule enters or leaves the element; it still depends on the shape of this element. However we intuitively expect that the relative fluctuation of this number remains small during a time interval At short at the macroscopic scale but large compared to the average time spent by each molecule inside the element: then the one-particle d.f. f, would more properly be defined in such a way that f; Ar Av represents the most probable number of molecules in the physical element Ar Ay during such a time interval Az. It is now safe to assume that this statistically defined distribution function f, is independent of the shape of the element and is sufficiently regular to allow the usual manipulations of calculus—in particular differentiation and integration with respect tor, v, and £. To illustrate this definition, consider a gas under normal conditions of

temperature and pressure (JT = 273°K, p= 1 atm); its density is = 3.10" mol./cc. A cubic volume of edge Ar ~ 10°? cm is certainly smaller than any volume that is studied experimentally; yet it contains of the order of 3. 10°° molecules and, even if we look only at molecules that have a velocity within

Av around a given v, where Av is, for example, 107° times the average velocity (Av/(v) = Av/VkgT/ m= 10 5) we still have the enormous number

76

The Nonlinear Boltzmann Equation

f; Ar Av~3.10°.

Moreover,

just by virtue of their free motion,

these

molecules cross this volume in a time At + 107’ sec, much smaller than the

duration of any observation. By averaging over such a time, we may expect that the fluctuations in the number of molecules will be completely washed out; thus the most probable d.f. f; is a relevant concept.* In many experimental situations, we have no way to measure directly the one-particle d.f.; we have control only over macroscopic quantities, defined

as average values taken over this d.f. For example, the local density n(r; t),

defined in such a way that n(x; 1)Ar is the number of particles in the element Ar, is given by nt; o={

The total number N

dv f(r,v; t)

(IV.2)

of particles in the system is such that [ar nr; o=|

drdvfit,v;J=N

{IV.3)

Similarly, the local velocity u(r; £) will be defined by

je

u(r; f)= ned

|

.

dy vf, (r, v; 0)

(IV.4)

More generally, we see later that in a dilute gas any local macroscopic quantity (A),,, appears as the average

_JavAW)fir, (An

v5)

(IVv.5)

of a suitably chosen function A (v). Even though most observations only involve such averages, the oneparticle d.f. remains the central object of the theory. Indeed, it provides a unified description of the various macroscopic quantities, and its time

evolution can be written in closed form, which is not the case for these average values.

The Boltzmann equation governs this time evolution of f,. To derive it, we work in a six-dimensional 42-phase space, where each molecule is rep-

resented by a point x = (r, v). We consider first the fictitious case in which the

molecules

evolve

without

interacting:

point x has then the “velocity”

* It is true that with the help of microscopic probes (e.g., photons or neutrons), we can study a system over distances much smaller than 107° cm and for characteristic times much shorter than 10°’ sec. However as we show in detail in Part D, what is really measured then is an average over a large number of such “microscopic observations,” and the present concept of a distribution function, therefore, remains meaningful (see also the ensemble theory formulation of Chapter VII).

Two-Body

Collision Theory

77

x = (=v, ¥=F/ m), where F denotes the external force. The corresponding equation for f, takes the form of a continuity equation in 4.-space: the rate of change of the number of particles inside a volume w in y-space is the integral of the flow J,, of molecules across the surface S of w a{

wo

ax fit, Vv; )= -|

s

dS-J,

(IV.6)

or, by Green’s theorem

a

we

ax f,= -{

wo

axV,°5,

(IV.7)

where the flow J,, is given by L,=%f1

(IV.8)

and the six-dimensional divergence is

=(2 4) v.-(22

(IVv.9)

In the limit where w becomes small, (IV.7) yields

(IV.10)

afitV, + §,=0

or

4, f1 = Of

ow = ay

fs)

GE

F

9

ah

(IV.11)

Of course in reality, molecules do interact with one another, and (IV.11) should be replaced by afi = (8 f now + (rf

eon

(IV.12)

where the second term represents the effect of these collisions. 3

TWO-BODY

COLLISION

THEORY

We expect that the essential effect of the intermolecular interactions in a dilute gas will be to produce binary collisions, well separated in space and time. This fact makes it necessary to discuss here some aspects of classical scattering theory.

3.1

The Mechanical Problem

Consider two particles 1 and 2, with identical mass m. Their positions and

velocities are denoted by rj, v; and rp, v2; they interact through the central

potential V(|r,—r|), which is supposed to have a strong repulsive core.

78

The Nonlinear Boltzmann Equation

Owing

to Galilean

invariance,

the study of the motion

of these

particles can be reduced to two decoupled one-body problems. (i)

two

The first (trivial) one describes the linear and uniform motion of the center of mass with the velocity

V.

co

_Vt+¥2

2

(IV.13)

which is a constant of the motion.

(ii)

The second problem corresponds to the motion of a fictitious particle with the reduced mass

wed

(IV.14)

This particle follows the relative trajectory t=F,—8,

(IV.15)

in the fixed potential V({r|); it has the velocity g=Vi-Vv2

{IV.16)

We suppose that at the initial time this particle moves with a velocity gp toward the field of force but is still in a region of space where the potential vanishes (formally, we put the particle at infinite distance from the origin). It is then convenient to represent the trajectory by polar coordinates (r, g, ) with the polar axis lying parallel to go. (See Fig. IV.1.) Because of the central

Figure IV.1_

Defiection of a particle in a central field.

Two-Body

Collision Theory

79

character of the force, this trajectory lies in a plane passing through the origin and we may take ®=0 without loss of generality. Of course, the energy of the particle is conserved* E

“$L(3) +7) ]+ v= Al

and so is the angular momentum

Vir)

a

r

a

=constant

({IV.17)

/, since the potential is central

d i ur(2) = constant

(IV.18)

The value of the constants in (IV.17, 18) are readily determined from the initial condition

B= 28 2

2

(IV.19)

and

1 = peob

(IV.20)

where b denotes the impact parameter defined as the perpendicular distance between the center of force and the initial velocity. From (IV.17, 18) we get

a) *GGLE-YO-a}”

(¢

e+

he E-V(r)

Dart

(IV.21)

The differential form of the trajectory g = g(r) can be written do=

ldr

? ~ ur(drl db)

(IV.22)

We now integrate (IV.22), with the help of (IV.21), between the initial condition (r = 00, » =0) and the point of closest approach (r =Tmin, @ = 0),

which is the point on the trajectory where r is minimum. From (IV.21), we see that the distance of closest approach rj, is the largest root of 2

E~ V(rmin) ~—ez-=0 2urmin

(IV.23)

or, with the help of (IV.19, 20)

1- (+) Tin

2

—2V Cain) =0 2-4)

(IV.24)

* More details on the two-body problem can be found, for example, in the book by Goldstein (1950).

80

The Nonlinear Boltzmann Equation

Clearly (dr/ dt)Prax

(IV.28)

this function is well defined for b Tnax

(IV.29)

The Scattering Cross Section

In many experiments involving atoms and molecules, we are not interested in the trajectory of a given particle but rather in a more global information: given an incident homogeneous beam of particles sent on the fixed potential, all with the same initial velocity gg, we ask for the number dN of particles that are deflected during the interval dt in a solid angle around N=(y, ®); thus © represents the polar angles of the final velocity go. This number dN is obviously proportional to dt, to dQ defined by dQ=sin y dyd®

(IV.30)

and to the incident flow per unit surface

I= ngo

(IV.31)

Two-Body Collision Theory

81

where n is the number of particles per unit volume. We define then the

scattering cross section o(Q); go) by the relation

aN = Io (Q; go) dQ at

(IV.32)

It has a simple geometrical interpretation: o(Q; go) dQ is the surface in a plane perpendicular to the incident flow such that the molecules that cross this surface end up with a velocity go within the solid angle dQ.

Figure FV.2

Schematic description of a scattering experiment.

The relationship between o(Q; go) and the deflection angle x is readily established once we observe that all particles with an impact parameter in

the range (6, b + db) are deflected in the angle (x, x + dy), independently of ® (hence: o(Q; go)=a(y; go); see Fig. IV.2]. We have thus

dN = b(x) db(x)I dtd®

(IV.33)

and, identifying this with (IV.32), we get

oe [ex O(X3 80) = inte d(x) a

(IV.34)

where we have taken the absolute value of db/ dy because the scattering cross section is obviously positive.* * We suppose here that b(y) is a uniform function of y, as is the case for purely repulsive potentials; otherwise we would have to sum over those values of 6 that lead to the same x

The Nonlinear Boltzmann Equation

82

The analytical calculation of o(y; go) can only be done for a few potential laws. Even with the simple repulsive law

V(r)=ar"

(IV.35)

the complete solution can be obtained only in the cases n = 1, 2, which are very unrealistic—except of course for charged particles where n= 1— because the repulsion is too weak at short distances and decreases too slowly at large distances. Nevertheless, for arbitrary n, partial information can be obtained; we use

the variables

_ 5.

xan

y

=(

Lied )""

2d

2in

bgo

(IV.36)

in (IV.25b, 24); we get

a=ay)=] with

xo

wy]

dx[1-x?-2(2) |

0

ny

1- x3—1(*2)" =

t/2

(IV.37)

(IV.38)

ny

We sce that 6 only appears through the combination bg@’" and, even though (IV.37) cannot be evaluated in closed form, we have

db| _ (2x)

dx =

ere

80

2

"y(x) dy

(IV.39)

That is, the go-dependence of the cross section is exactly known. Let us also notice that for n =4, the go-dependence entirely disappears from (IV.33)

[see (IV.31)]; this case, which corresponds to so-called Maxwell molecules,

has played an important role in kinetic theory because of the ensuing mathematical simplicity; it has little physical significance however. Another case of interest is the hard-sphere potential. Here we have =

co

r 0, as is geometrically obvious. As a

final remark,

let us briefly discuss

the connection

between

the

deflection go> gi and the inverse process gy => go. As Fig. IV.3 indicates, one

The Collision Term

85

Figure IV.3

Direct and inverse collisions.

process is derived from the other by making an inversion of the trajectory with respect to the origin O and by reversing the velocities. We have seen

already that [go|=|go|; moreover

the deflection

angle y is obviously

unchanged in these transformations; the cross sections of the two processes

are thus the same.

7 (80 > 80) = 7 (Bo > Bo)

(IV.60)

Notice however that the unit vector along the apse line changes sign;

nevertheless, consistently with (IV.60), we have

s(€'; Bo) = S(€; Bo)

(IV.61)

with €’ = —e as can be seen from (IV.45, 53, 57).

4

THE

COLLISION

TERM

Our discussion of Section 2 indicates the need to evaluate (0, f;).on defined in

such a way that

(8: frdeon Ar Av At

(IV.62)

is the change due to collisions, in the time interval (¢, ¢+ Ad), of the number of molecules lying in the element ArAv, around the point (r, v), in p2-space.

The Nonlinear Boltzmann Equation

86

In the low-density limit (IV.63)

nr«1

where ro is a parameter characterizing the range of the intermolecular forces (e.g., “9 =a for hard spheres), it is very natural to assume that only binary collisions have to be retained; indeed, the probability of having more than two molecules in their mutual range should be much smaller, by a factor proportional to nra. Moreover, such a binary collision takes place over a distance of the order of ro, which is very small compared to the length Ar of the physical element introduced to define the distribution function f,; ro/ (v), is also very similarly, the duration of this collision, of the order of 7,

small compared to the “smoothing” time At used in defining f;. Therefore, we formulate

Assumption I. In the low-density limit, we can limit ourselves to binary collisions and consider them as instantaneous and local in space. This allows us to write

(IV.64)

(A: freon = C"- C"

where the following definitions obtain. (i)

(ii)

C’ dr Av At is the number of binary collisions in the time interval Af, where one molecule lying in the range (r, r+ Ar; v, v+ Av) is deflected

to any other velocity v. C” Ar Av At is the number of binary collisions in the time interval Az, where one molecule lying in the element Ar around r, with an arbitrary initial velocity v’, ends up after the collision with a velocity in the given range (v,

v+ Avy).

A more delicate assumption is needed to get an explicit expression for C’ and C". Indeed, it is clear from assumption

I that we need to know the

number of pairs of molecules, both lying in the element Ar, which are going to collide in the time interval Ar. This number can be evaluated if we make Assumption II (“Molecular-chaos assumption” or “stosszahlansatz”). The number of pairs of molecules in the element Ar with respective velocities in the range (v, v+ Av) and (v1, ¥, is given by

+ Av), which are able to participate in a collision,

fir, v; 0 Ar Av f(t, vi; 2) Ar Av,

(IV.65)

The Collision Term

87

This assumption is very hard to justify because it introduces statistical arguments into a problem that is in principle purely mechanical; it lies at the heart of all the criticisms voiced against the Boltzmann equation. It cannot find any rigorous support within the frame used here. All we can do is to give it an intuitive meaning by realizing that in a low-density gas, a binary collision between two molecules that have already interacted together, either directly or indirectly through a common set of other molecules, is an extremely improbable event. Indeed, colliding molecules come from differ-

ent regions of space and have met, in their past history, different particles; they are therefore entirely uncorrelated. Let us stress that assumption II is \ needed only for molecules that are going to collide; after the collision, the scattered particles are of course strongly correlated. As we see later, however, this is irrelevant for the calculation. Misunderstanding of this point is at the origin of many unjustified criticisms of Boltzmann’s ideas.* With these two assumptions, and our knowledge of scattering theory, the calculation of C’ and C” is elementary. We consider one given molecule lying in Ar and having velocity v and we move along with it: this molecule plays the role of the target in the scattering experiment of Fig. IV.2. The molecules having a velocity in the range (v;, v,; + dv,) that may collide with this target are uniformly and randomly distributed on the scale of rp; they form a homogeneous incident beam with intensity [see (IV.31)]

dI= gfi(r, v4; 1) dv,

(IV.66)

Here we have used the molecular-chaos assumption; moreover we have dropped the subscript in the relative velocity g=v-Vv

(IV.67)

because from now on we are interested only in the velocities before (v, v, and g) and after (v’, v, and g’=v{—v’ the collision and never in intermediate

velocities during the scattering process. By definition of the scattering cross section, the number of molecules deflected by our target in the solid angle dQ is [see (IV.32)]

aN = fi(t, v1; 0) dv, go(x; g)dQ Ar

(IV.68)

in the time interval Az. If we notice that any change in the velocity of an incident molecule implies a corresponding change in the velocity of the target molecule (because v+v,=v'+v}), we see that C’ Ar Av At is obtained by multiplying (IV.68) by *It remains extremely instructive to read Boltzmann’s refutation of the arguments against molecular chaos in his famous Lectures on Gas Theory (1964, English translation); an elementary historical discussion can be found in ter Haar (1954); see also Brush (1965, 1966).

The Nonlinear Boltzmann Equation

88

the number f,(r, v; t) Ar Av of target molecules and by integrating over all deflection angles © and all velocities v,. gelfilt, v15 OFC, v; t) Ar Av Art]

C’ Ar Av ar=| dv, | dQo x;

Equation

(IV.69)

(IV.69) involves the total scattering cross section (IV.43), a

quantity that generally diverges. Though this causes some mathematical difficulties in a rigorous discussion of the Boltzmann equation, we see later that this point is of little physical significance: thus we ignore it by assuming provisionally that the interaction potential is cut off at some finite distance Tmax < Mr, as was done in (IV.43). The right-hand side of (IV.69) is then well

defined.

The calculation of C” runs along the same lines: we consider a molecule

with a given velocity v’ and we look at all binary collisions that lead to the given final velocity v for this particle. Thus we examine the inverse collision depicted in Fig. IV.3. We find

C' dr avar=|

do

( vev(v'

vi)svtAv)

dv'dy, a(x; g')g'

x [fir vis OF, v5 Ar Ar]

(IV.70)

The integral over v’, v; runs over values such that for given Q, the final velocity v is in the given interval (v, v+ Av); itcan be transformed into a more

transparent expression if we reexpress it in terms of the variables v and yj.

To do this, we use (IV.26), (IV.49), and the conservation of the center of

mass velocity [see ([V.13)]. The Jacobian of the transformation is 1 [see (1V.47-49) or (IV.73)]

vi) _ atv’, atv, v1)

(IV.71)

This allows us to write (IV.70) as

C" Ar Av At= { dv, | dQ a(x; gg lf, vis Of (, v's t) Ar Av Az] (IV.72) where of course the velocities v; and v’ must be expressed as functions of v;

and v for given ©, through the law of collision. For example, we can use (IV.47) to write vi=v, —e(g- €) v'=v+e(g-e)

(IV.73)

The Collision Term

Combining equation

89

(IV.11,

12,

ALFof

Ofit fit

ov

69,

72),

we

readily

arrive

at the

Boltzmann

=| av, J dQ o(x; gg x LAG, v5 OA, vi O-AG, v3 Of v5 OD] (IV.74)

This equation plays such a fundamental role in kinetic theory that it is worthwhile to display a few equivalent forms of the collision term, which are all of some use. (i)

If we describe the scattering with the help of the impact parameter, we get from (IV.34)

(8, fidcon =| dy, { bdb i" 49 gfifia-fifial

(IV.75)

where we have used the abbreviation

A=AGyv9,

fi=Alnvso

fir =A, v52),

fir =A,

vist)

(IV.76)

In (IV.75), v’ and v; should now be expressed as functions of v, v,, 5

(ii)

and ®. Alternatively, it is often convenient to use the unit vector € along the apse line to describe the scattering. Using the equality between (IV.33) and (IV.56), we have

(ike [dvi {Pegs fifia-ffial

V7)

where (IV.73) can now be used directly to compute vj and v’.

(iii)

Finally, the physical meaning of the collision term appears rather nicely if we express it in Wi(v, v1; Vv’, vi). We can write

(Adan

terms

of

a

“transition

| av, dvidvi Wovisv WOOf fia fifa

probability”

(V.78)

where [see also (IV.13)]

Wy, v1; V, vi) = WU, vis ¥, vi) = W(v1, ¥; Vi, Vv’) =

g's.

,

)

1

=olarc*—>; of 8g &g] 6(VG-V G G) 4

2 gr § ~8 ) 2

( IV.79 )

The Nonlinear Boltzmann Equation

90

Proving the equivalence of {1V.78, 79) with the previous forms is the matter of some elementary transformations. (We use the variables Vg and g’ instead of v' and v; and perform the integrals over VG and g’ with the help of the Dirac delta functions.) With this notation, the Boltzmann equation is reminiscent of the gainand-loss master equation [see (1.76)] of stochastic theory; yet its nonlinearity gives it remarkable properties that even qualitatively are completely absent from the master-equation description; the analogy is very superficial.

Before turning to a detailed study of these properties, let us point out that although each term of (8, f:)con Separately diverges for potentials with infinite range [see the remark after (IV.69)], the sum of these two terms can be expected to be well defined when r,,.,>°. This is most easily seen on (IV.75); of course, we have

{ pdb = Tm Qo

©

fOr Fax? 00

(1V.80)

but when b > 00, the deflection angle y ~ 0; in this limit the velocities after and before the collision become such that (vj;—v,)>0, (v’—v)>0, and

fi fia-fi fisJ> 0. Ignoring mathematical difficulties, it is thus reasonable to assume that

lim [sant fiacfi fial= finite

Fmax-?0O

(IV.81)

Hence provided we always consider the difference [fi fi.i—fi fii] as a whole, no difficulty is to be expected when rpax > ©.

5 GENERAL PROPERTIES OF THE NONLINEAR BOLTZMANN EQUATION

5.1

Positivity of f,

The one-particle d.f., representing the number of particles in a unit volume in «space, is semipositive definite: f, = 0. If the Boltzmann equation is to make sense at all, it must preserve this property for any initial condition that satisfies it: fit, v,0)20

(whole j.-space)

That this is indeed the case can be shown “ab absurdo.”

(IV.82)

General Properties of the Nonlinear Boltzmann Equation

91

Suppose that there exists a point (ro, vo} where, in the course of time, this condition

is first violated; this occurs at time f

fi (fos Vo3 to) 1)

(IV.210)

* There is of course no contradiction in taking p, of order unity and still depending on 6. For example, the function exp (—V 5) depends nonanalytically on 5 but is of order unity as long as 6 is finite.

The Nontinear Boltzmann Equation

114

Summing these equations over n clearly reproduces the starting point, as in the Hilbert method. However the physical and mathematical contents of the present theory are very different. Consider the order 8 ', known as the first Chapman—Enskog approximation: the solution of ([V.208) is the Maxwellian (IV.106), but the condition (IV.204) entirely determines the parameters n, u, T in terms of the

moments p,, [see (IV.173)]; we have n(r;t

yell. m

u(r; 1) pilts t) pi(rst)

kpT(t; t) = 2m3 ests t) loi; t) Moreover it can operators Dare full moments p,.

(i= 2, 3,4™ 4, y, z)

L

pitts 6)

(IV.211)

i=234

22ers)

be readily checked from (IV.207) that the nonlinear precisely the Euler operators (IV.184), now acting on the

In the next order, corresponding to the second Chapman-Enskog approximation, we have to solve the linear inhomogeneous integral equation (IV.209). However a difficulty seems to appear: whereas the Hilbert method

allowed a choice of the moments p® in such a way that the solubility

conditions of this equation were satisfied, we have no such freedom here because the left-hand side of ([V.209) is completely determined from the

first approximation. Yet as we prove presently, the choice (IV.207) made for the 8-expansion of the conservation equations guarantees the automatic fulfillment of these solubility conditions. Let us demonstrate the slightly more general statement: if all five (m 2!), and one is generally satisfied with obtaining the structure of the macroscopic equations: their solution is quite a separate problem, involving the competence of classical hydrodynamicists. Let us see a little more explicitly how the formalism works for n =1. We first

rewrite (IV.209) as

3 alnf® : ae nft (Dog)

J+:

Pa

22)

—

EH

f)

(IV.216)

where we have used the fact that f° depends on r only through p,. Moreover we have written the equation in such a way that the collision term is Hermitian [see remark (ii) after (IV.179)]. From the definition (IV.106), we have alin fo |

on

alnf _

n(r; t)”

au,

mg

an fi” _

kg T(r; t)

aT

il

eK)

Tri” (IV.217)

where we have used the notation {IV.130) as well as the abbreviation Ke; t OO

(me ag

ere

2

3 2)

2

Ue B IV.21 ES

Equations (IV.211), together with the rules of chain differentiation, then lead to

as

Ame

dp,

p

aln fy” _ 9p;

ain

aps

u +on( 2Hu? -1)|

kaT

3kaT

m (6-224) pkaT

fy” _ 2me*

~ 3pkyT

\”

3

(1Vv.219)

where we have dropped the (r; t) dependence of the hydrodynamic variables. We now insert (IV.219) into the left-hand side of (IV.216); we also use (IV.184)

for the operators D®. The lengthy expression obtained in this way is simplified

Normal Solutions of the Nonlinear Boltzmann Equation

117

considerably after some algebra; the result is*

FIA)

Eee -§ tt) AG)

fo

me?

5\aT

(IV.220)

T ses T 2 pone ril

The linear collision operator on the left-hand side is isotropic when expressed in terms of the variable & (see Chapter V); thus the solution f(” has the same

ro, aloe)

&-symmetry as the right-hand side and can be written as

(IV.221)

=

. aes

The explicit calculation of the scalars A and B, which only depend on é = |g|, requires more detailed analysis; yet even with this formal notation, the orthogonality conditions {IV.205) are seen to be satisfied.

Inserting (IV.221) into (IV.207) (with n= 1), we can use the symmetry of the angular part of the integrands to find

Di=0

pias F

Di=

.

a pean”

a

0

Set

ar

Aa[(itait)-2oy( 5 2) an,

du\

2

ar,

ar)

3°"

YS

a

gece,”

Ou;

[fau,ou\

[ (24+

an

an)

ear

2

) 2 5e(

3

ar.aan

(2,3

au,

|

\ Sand

;

|

%H2)

(IV.222

where 2

n=7 k,T | dt

EFA (EF?

(IV.223)

(independent of the particular components i #/ €x, y, z chosen) and

(independent

of i), respectively,

represent

(v.24)

(E-2")aore

xo] ae the

shear

viscosity

and

the

thermal

conductivity of the dilute gas. Thus we recover exactly the macroscopic results [see {IV.152-154)], as well as an explicit expression for 7 and x, and a vanishing bulk

viscosity. We dwell no further on the calculation of A and B, nor on the symmetry arguments leading to (IV.223, 224) because a quite similar problem is treated in detail in the next chapter. * The solubility conditions (IV.212) for n = 1 are easily verified on this equation.

)

V The Linearized Boltzmann

Equation

1

THE

LINEARIZED

COLLISION

OPERATOR

Consider a situation in which the gas is only slightly perturbed away from

absolute equilibrium. We have

=fii+6f,;

with

Poet

(V.1)

With the help of J(f1", f{°) = 0 and of the symmetry property (IV.90), the

Boltzmann equation (IV.74) yields (in the absence of external forces) a

6, Of; +¥> 2

noes,

(V.2)

to first order in 5f,. Here we have written

2

Cof, = 72

fv

FY)

=| av, fan oty: gelefot +o afi. afets— oP Af] (V.3) 118

The Linearized Collision Operator

119

This relation, written with a notation linearized Boltzmann collision operator

velocity distribution

a eq

similar to (IV.76), C, in terms of the

defines the Maxwellian

fe _ 1 ( mv? ) fl n QakpTim)= exp - kp.

(V.4)

From 9{7@14 = ¢i4¢44 we have the equivalent formula

Cf = fav, [ao a(x; geeies

(A+B

4-9]

ws

This linear operator C is much simpler to handle than the nonlinear

Boltzmann collision term, and a great deal can be said about the solutions of (V.2). Let us first examine a few properties of C.

For two arbitrary functions k and h, we have

favor

"k(v)*Ch(v) = fav dy, fan ox; gee ety

(Ga) +Ge),-Ge)-) (oe) +Ge),-Ge)-(A)] (V.6) as a consequence of (IV.91, 92).

This property suggests that the abstract Hilbert space introduced in Chapter II, Section 3.2, remains useful in the present analysis. With the scalar product

(k|h) = fav giv) 'k*(v)h(y)

(V.7)

(V.6) shows that C is a Hermitian operator

(k|C]h) = (h|C|k)*

(V.8)

Hence the eigenvalues a? of the problem nl?) = All?)

(V.9)

(hiC|h)=0

(V.10)

are real. Moreover, we have

120

The Linearized Boltzmann Equation

because the integrand in (V.6) is semipositive definite for inequality holds for the eigenvalues

k

yo wee PilnClo))—50

*

(g7l67)

=h; the same (v.11)

and, again from (V.6), the equality sign holds only if (b)/ @ °) is a collision invariant.* ‘Thus C has five and only five zero eigenvalues. Denoting them by oael-

5); they are such that the ratios (¢2/g{") are linear combina-

tions of 1, v, v”. With the orthogonality condition

(O7loy) = 8

(V.12)

we readily obtain their explicit form

b4(¥) = 9$(v)

ere

(=2,3,4=x, y, z)

sijm

sae F—

Sov)

(V.13)

We soon discuss the other eigenfunctions and eigenvalues j¢ {a}. Provi-

sionally, let us assume that the spectrum of the Aj’s is discrete and that the ;’s provide a complete basis in our Hilbert space

L SiW)*Of(v) = 6(v—V) piv)

(V.14a)

x |e;)e;|=1

(V.14b)

i

or, formally,

The solution of the linearized Boltzmann equation for a spatially uniform

system then becomes trivial. From

we get with

0,5f = nCof,

(V.15)

dfi(v; )=L cf Pj (w)exp (478)

(V.16)

,

= ($716f:(0))= Jav Gi() BF (W)*5fi(¥50)

(v.17)

See (II.82) for a similar calculation. * This property, as well as (V.6), obviously remains valid if the full Boltzmann collision term is expanded around /ocal equilibrium: this justifies statement (ii) made in Chapter IV, p, 108.

The Linearized Collision Operator

121

In the limit ¢ > 00, we see that all contributionsj¢ {a} vanish exponentially and we are left with

af, {[2?>Oanl = 3 eb)

(V.18)

It is readily checked that (V.18) is merely the first-order correction in the

series expansion of

Fea _

m

r= (n+on)_

in powers of dn, 5T, and

|

ke

__m(v—édu)’

P| eee

(v.19)

u; these parameters are entirely determined by the

initial condition, for example,

én = fav 6f(v; 0)

(V.20)

We find here the linearized version of the phenomenon already discussed in connection with the Hilbert principle: an arbitrary uniform deviation from equilibrium exponentially reaches a new equilibrium state, which is entirely determined by the five conserved moments of the distribution. Although the eigenvalue problem (V.9) plays an important role in many developments, the 6?and A; remain largely unknown. We can however use the fact that Cis an isotropic operator in velocity space (i.e., commutes with the rotation operators defined in this space) to write the eigenfunctions as

7 () = brin(¥) = $10) Yi (Our By)

(V.21)

while the corresponding eigenvalues A% only depend on r and i. Here

Yim (6,, ,) are spherical harmonics, functions of the polar angles (6,, ®,) of v with respect to an arbitrary direction; ¢,(v) depends on / (not on m) and

on the modulus of vy, and it is characterized by the supplementary index r.

For later use, let us remind the reader that the spherical harmonics are

given by*

Yin = Atm P!"'(cos 6,) ef’Pe

(-l 0. In this regime the A@ can be expanded as*

At=a,q + bq? +.0(q°)

(V.53)

Substituting this form into (V.51) and setting the different powers of q equal to zero, we get

04°):

aq (Bias tap +42)=0

4).

oa’:

=

ba

_ ag

tb: +£i))+ar€i9 3a2+Bimitaip

(V.54)

* For consistency, the terms of order ¢ cannot be retained in (V.53) because similar terms have

been neglected in writing the starting hydrodynamic equations; they would be meaningful only at the level of the Burnett equations (see Chapter IV, Section 7.2).

Macroscopic Definition of Hydrodynamic Modes

127

These equations have three solutions ay =

4) = i(Byui tap)”

=p--l

b=

b=

Ale

aoa

—Bimidr_ ae

7]

as=0 = bs

aps) (Bigr +aip)

(V.55)

Replacing the parameters by their values (V.39) and using the thermodynamic formula*

_¢ _ Tleplan?

pop

(V.56)

(ap! ap)r

(CG, and C, are the specific heats per unit mass at constant pressure and

volume, respectively), we arrive at the five eigenvalues

Ma= Ficq—Tq?

Age

(V.57a)

2

(V.57b)

p

Mag? 3 2G

; (V.57c)

where

«(22° “LC,

is the sound velocity and

1 [2

3pl3

=—]—+

v.58

\ap/r( 1

Tete

.

=)

Ge

_-_-

|

(v.59)

is the sound-absorption coefficient. The meaning of these modes is clear: Af describe damped sound wave propagations, A$, correspond to diffusion of the transverse velocity, and AZ describes heat diffusion. This interpretation appears most clearly if we calculate the corresponding eigenfunctions 3 and 7. Given an eigenvalue A{, the systems of five linear homogeneous * This equation is proved, for example, in Landau and Lifshitz (1958).

128

The Linearized Boltzmann Equation

equations (V.44, 45) are readily solved; we find in the small-q limit l/p +

°

te,

lin . ta (52)c,\

tim

7.

Fi

,

a)

eC,

tin t.= (52) Uke

tim $7.2

nl

(2)

\8T/ o

OT! p

0

Him 3 =

| (kp \ "2

(7)

zg

’

x) 2

TL

(V.60a)

Mm

lim @3 =p ling 03 (V.60b)

L r As lim 4

Tr

0

2G,

| o

ogg

(7) Op!

0

2C,

cy

oO 0

sg lim 4

. : =|

(*)

mt ket in

B

0

(V.60¢c)

p

-

7

G

tim = ( CG

G)'

2

Vp

o 0

oO

0

fo

.

52

Emes=

(6

=a)"

(Ae

0

0

7H) (V.60d)

_

L

Macroscopic Definition of Hydrodynamic Modes

129

These results can be used to get an explicit form of the expansion coefficients c2(0) [see (V.48)]; from (V.41, 47, 49), this leads in turn toa complete determination of the

macroscopic variables in terms of their initial value. Though

this calculation is

straightforward, the general expressions arrived at are rather awkward.

Here we

merely quote the specific solutions that emerge for special initial conditions of particular interest, leaving it as an exercise for the reader to verify that the following results hold true.

{i)

Initial condition P4(0) # 0, u,(0) = 9, T,(0)=0.

C,

gt P(t))=|— [Eos ‘cat

C,-¢.

exp ( —T,q7t q°t) +(

G

We get

:) exp (

xq7t

aC,

p,{0)0)

(V.61 (V.61)

This important formula shows that the density evolves in time through the double

mechanism of sound propagation and heat diffusion. {ii)

Initial condition Ugy #0, p4(0) = uy,.(0) = u,,,(0) = T,(0)=0.

g(t) = exp (= ‘ 44,(0)

We find

(V.62)

which illustrates that the modes A%, correspond to the viscous damping of the transverse velocity. (iii)

Initial condition U,,.(0) #0; p,(0) = u,,,(0)

Ug.x(t) = cos (c,qt) exp (iv)

u,.(0)= T,(0)=0.

(-T',q*t)u,..(0)

Initial condition u,(0) = 0; p,(0) 40, T,(0)#0.

defined by [see (IV.158-160)]

*

(2) (2) dp. Pa

(V.63)

If we look for the entropy Sq

Nar, a@

= -4(2) we find

We have

G

=~ Bar) Pe tp «q’t

5,(t) = exp (- ts, (0)

Mi) (V.65)

which justifies the name “heat-diffusion mode” for d%: the entropy (or the heat Ts,(t)) is entirely propagated by this mode. In an arbitrary reference frame (V.62, 63) remain valid provided we make

following substitutions:

the

(ug)

Ug. > Ug = “st

gy Ug, =U ta) q

(v.66)

130

The Linearized Boltzmann Equation

3 MICROSCOPIC EXPRESSION FOR HYDRODYNAMIC MODES AND TRANSPORT COEFFICIENTS We now show how, in the limit of small wave numbers and long times, the

solution of the linearized Boltzmann equation leads to the hydrodynamic description analyzed in macroscopic terms in the previous section. In addition, we establish explicit microscopic expressions for the transport coefficients of dilute gases. In macroscopic theory these transport coefficients were introduced as phenomenological constants. Except for minor technical difficulties, our strategy is the same as the one we followed in Chapter II to establish the diffusion equation from the Fokker-Planck equation. We first write the linearized Boltzmann equation in Fourier space. Defining f,(vs 0) ={ dr ef

,(r, ¥; ft)

(V.67)

(we again take q= q1,), we get fg Now

suppose

that the solution

+ iu,f, = nCf, of (V.68)

(V.68)

has been

obtained.

Then

the

macroscopic variables p,, u, and T, {i.e., the vector s,(r)} are given by the integrals 1

H(t) = m{ dv f,(v;t)|

Vv

?

mov?

3kap

(V.69)

(the equilibrium mass density p is given in this linearized problem). If we can identify these expressions with the macroscopic formulas (V.47-49), we can

gain a well-defined microscopic definition for the modes A%, hence for the

transport coefficients. We should not expect this identification to be valid for all wave numbers and all times, however: macroscopic hydrodynamics is expected to hold only in the double limit q > 0 (i-e., small compared to the inverse of characteristic molecular lengths) and t > 00 (i.e., large compared to molecular relaxation times); only in this regime does the comparison make sense.

The solution of (V.68) can be obtained from the non-Hermitian eigen-

value problem

(nC- iqu;)|69)=A4|6%)

(V.70)

Microscopic Expression for Hydrodynamic Modes and Transport Coefficients

131

As discussed in Chapter II, we should also consider the left-eigenvalue problem

(B9|(nC — iqu,) =Aj64|

(V.71)

Precisely as for the Fokker-Planck equation [see (JI.74, 75)], it can be readily shown that*

M=at, — (v1d2)=(v1b4)* = (1b;

which saves us the trouble of studying (V.71).

(V.72)

If we assume that the eigenfunctions ¢7(v) form a complete basis, the solution of (V.68) is [see (I1.82)]

Ls

D=L cf (oF v) A

= Le4(0) exp (a4) 64 (v)

(v.73)

i

with

¢}(0) = (631 f,(0)) =| dy of'(v)

67 (v) f,(v; 0)

(V.74)

For q=0, we already know that in (V.73) five eigenfunctions have zero eigenvalue while all the others correspond to relaxation with a finite characteristic time. Thus if we assume that the eigenvalues Af can be expanded in powers of q, five of them (A%, a = 1 : - « 5) will tend to zero with q, the others remaining finite. Hence only the corresponding five terms will survive in (V.73) for

g>0, t>00

flv; 6) 100,

q700

z €2(0) exp (ARN) b2(v)

Although (V.75) is an asymptotic formula, we do hydrodynamic limit g > 0, t > 00 with (q72) finite, as was diffusion [see (11.83)]; this is because we expect (V.75) modes, which would oscillate infinitely fast in this limit, vanishes as q for g>0.

(V.75)

not take the strict done in the theory of to contain the sound since their frequency

Inserting (V.75) into (V.69) and comparing with (V.47, 49), we see that

the eigenvalues AZ can be put in correspondence with the macroscopic eigenmodes.

*Tt is readily checked that in the dilute-gas limit where (IV.164) is valid, a similar symmetry

holds for the macroscopic modes (V.60) if we express them in the dimensionless units Pal 2, ug Vig Ti m, Til ( vin).

132

The Linearized Boltzmann Equation

We have thus a well-posed problem: follows:

we expand the AZ and the $2

Ad = igh —a?ap+ 162) =|b2)—iqlo)+---

as

(V.76)

and we solve the eigenvalue problem (V.70) by a perturbation calculus (i.e., an expansion in powers of q). To do this, we suppose that the unperturbed problem (V.9) has previously been solved. A small technical difficulty appears because this perturbation calculus is

degenerate: A? =A$=---=A3=0.

It is well known that the first step in the

perturbation scheme must be to remove this degeneracy by solving exactly the problem (V.70) within the subspace spanned by the degenerate eigenfunctions. We then write

(nC — igv,)\64) = A164)

with

Ibad= 2 Coalbz?

(V.77)

(a= 1-5)

(V.78)

Since C\p°) = 0, (V.77) becomes

0164) =Adlba)

(V.79)

with A, =—igA‘,. Incidentally, note that the non-Hermitian character of the perturbation has been entirely absorbed in the definition of AJ: therefore the eigenfunctions |¢{) will be orthogonal as if the problem were Hermitian. Inserting (V.78) into (V.79) and taking the scalar product with be), we

get with the help of (V.12) 5

z Caa'l(h3'0zlbe) —AL8p «] =0

(V.80)

This system of five linear homogeneous equations has a nontrivial solution only if the following determinant vanishes:

Kbploslbe)—A.8 pial] =0

(v.81)

With (V.13), the calculation of the matrix elements of v, is very simple; they vanish, except for

($5 |v4163) = (b5|v,|6%) = (#7)

v2

d3lvs162) = (Slo.168)= (82)

(v.82)

Microscopic Expression for Hydrodynamic Modes and Transport Coefficients

133

With these results, we find the following solutions to (V.81): T

A=

p= (“et ) 3m

1/2

=

AS=AL=A5=0

(V.83a) (V.83b)

the last equality in (V.83a) following from (V.58) together with the perfect-

gas values (IV. 164). The corresponding eigenfunctions are found by solving (V.80). 1

%

Ibi)= Fe LVaI6%)—162)+ V9 05) 1

I62)= Fe WVlot) +168) + V3105)] 163) =|6)

16) =|6) 165)= v2 16) —V369)]

(V.84)

(b51bi) = bE.

(V.85)

Notice the strong analogy between these formulas and (V.60) taken in the dilute-gas limit. We have of course the orthogonality property Let us also point out that three of these eigenfunctions remain degenerate; but this is now harmless because the perturbation has no matrix elements

between different 64,’s

(balveloa) =ALbne

(V.86)

With the following basis as a starting point

ana slt) 1O)=4)

le?)

6

ela}=(1v...5)) .

(7 {e})

(v.87)

the perturbation calculus proceeds in a straightforward manner. We find

AW = AL to first order, and the second-order eigenvalue is , ne ptl '

AQ= — yee

dng v,16a)

(V.88)

i

It is convenient to get rid of the restriction j¢ {a} with the help of the trick used in (11.101); we write

AP =~ e770 lim ajF (BilexI6) 0 (remember that C seminegative definite). Thus we get

AD =~ tim (6% | oe —(0e-2$)| 64) nC~e

is

(v.91)

270,

Equations (V.76, 83, 91) give us the perturbation expansion of the eigenvalues of the linearized collision operator that tend to zero with q. To identify them, let us show how these expressions can be rewritten in a way that combines the transport coefficients y, ¢, and « and the specific heats

C, = 3kg/ 2m,C, = 5kp/ 2m. For A}, this is; particularly simple: we have from (V.7, 13) 1

AS. = -——

lim { dv v,v, aCe —— 7 Bnet

(V.92)

ae £705

which is identical to (V.57b) if we take

n= pasa

(V.93)

The identification of the mode a2 with (V.57c) leads to =

K=PGAS

(2)_ _>7 Kap

“m dm

of

_

|

vE[-(_ mv?

javyei)

mov? -3)]

nm

kpTee we

Cres

a 3]

a1

2) "noe

fa

(ar 2?) do(% S22),

1

(g-52),

m?!*nC-e\2

2m

J"?!

ea

Taking into account that

1

J av,

(e

sks‘)

3

m

Jee

cae

=|

2(z Sten) ea

5

dv vy,

1

(e

om

JP!

0

we arrive at nim

Ka = TT nr

2

v

tim | av— Ose e — ND

ks?)

Om)

PPI ii

4

V.96 (V.96)

Microscopic Expression for Hydrodynamic Modes and Transport Coefficients

135

A similar calculation shows that

1 [% ( 1 of1 ) | 2 Mo, ten PeeayHay pl 3 + ft GG K

(V.97)

which proves that ¢ = 0, as we had anticipated. To arrive at this result, the following equality is needed

v\

1

v?

4

1

{ dv (02-4) aCue (02-2 )os =3/ AY v,dy A

.

baby" (V.98)

a direct consequence of the rotation invariance of C [see (V.21)].

Thus we have proved in the dilute-gas limit that the linearized Boltzmann equation leads to the (linearized) equations of macroscopic hydrodynamics, with microscopic expressions for the transport coefficients, as well.

The e-limit involved in (V.92, 96) is awkward in explicit calculations. It can be eliminated, however, as we now illustrate in the case of shear

viscosity. Equations (V.92, 93) give

nm.

n= a

|

pe

dv u,v,

yy

(Vv)

{V.99)

where

xe Xe) (v)

—

1

emacs

eq

(v)

(V.100)

Multiplying both sides of this equation by the (nonsingular) operator

(nC— «) we get

(nC-e)yP =0,0,9)°

(V.101)

Since the only solution to (nC— )f = 0 (with ¢ > 0) is zero, theorem (i) page

108, tells us that v2” is uniquely determined by (V.101). Moreover, taking the scalar product of this latter equation by ¢2, we see that

(bax?) = { dv ei (uv) ‘ot(v)y?(v) =0

(V.102)

since both (¢2|C and (¢2|v,0,¢§% vanish, the latter property being easily checked by direct computation. From the way the factor ¢ was introduced [see (V.88, 89)], we know that the limit

x=lim xe

(V.103)

exists and is unique and well behaved. From (IV.102), it satisfies

@2lx")=0

(w@=1--+5)

(V.104)

The Linearized Boltzmann Equation

136

Suppose that we take the limit e > 0. in (V.101) nCxy*” = 0,0, 91"

(V.105)

The theorem p. !08 implies that the most general solution is

WP + YL yahalv)

WET

ael

(V.106)

Among this infinite set of solutions, however, it is easy to choose the unique solution (V.103) because it has to satisfy the subsidiary conditions (V.104): these completely fix the constants y,. Therefore we can write nm?

n =mm

dy u,v," *y (v)

( V.10 7)

where x*” is entirely determined by (V.104, 105).

Similarly the thermal conductivity (V.96) can be written nm 2 =|

«=

v2 x dv > Ux (v)

(V.108)

where y*(v) is the solution of v° 2

x

Skpl\

nO" = »({5- 7 oro) subject to the subsidiary conditions

(palx*)=0

(@=1---5)

(V.109) (V.110)

Before closing this formal discussion on transport coefficients, let us briefly compare the present expressions with the formulas derived in the nonlinear case by the Chapman-Enskog method. Symmetry shows that the solutions y*” and y* can be written

X°7(¥) = 00,4 (0) oF) 2

x)= (527

)a ero)

(V.111a) (V.111b)

Comparing (IV.220) with (V.105, 109), (IV.205) with (V.104, 110), and (IV.221, 223, 224) with (V.107, 108, 111), we readily see that these two sets

of expressions are identical except that in the Chapman-Enskog method, 4 and « are defined in a local-equilibrium state, characterized by the local parameters p(r; t), u(r; ¢) and T(r; t), whereas in the present linearized case

they are defined with reference to absolute equilibrium with p and T constants and u=0. Hence as suggested before, there is no advantage in

Explicit Calculation of Transport Coefficients

137

dealing with the full nonlinear case as far as the calculation of transport

coefficients is concerned; indeed, the present approach is much simpler: it requires no more than the mathematics of elementary perturbation calculus.

4

EXPLICIT CALCULATION OF TRANSPORT COEFFICIENTS 4.1

Introduction

Although the calculations of the preceding section considerably clarify the microscopic interpretation of transport coefficients, the computation of

these coefficients still requires the solution of nontrivial linear integral equations [see (IV.105, 109)].

Of course if we knew the eigenfunctions and eigenvalues of the collision

operator [see (V.9)], this part of the calculation would be very simple. For example, writing the shear viscosity as [see (V.88, 93, 13)]

7

2

-

eT yaa?J koo9? le})|

—__

|

2

(V.112)

we would be left with the task of computing the nonvanishing matrix elements (v, vp" $5).

Unfortunately the solution of (V.9) is not known, except for Maxwell molecules. Despite the unrealism of this model, it is instructive to see what (V.112) gives in this case. Using (V.22—26, 30-34), and choosing the polar axis of v along the z-axis, we have

v,vyp{(v) = v’y{(v) sin’6, sin dy cos ,

=1(82) 2bhaT mv? )s9 ( mo”) 15 2kyTT!” \2kgT. x_

28, bv) — Y2,-2(O, bo) ]

; F b8a2- $02,-2) -at

(V.113)

From the orthogonality of the $%,,, we then obtain the extremely simple result nkyT n= “a7

where AQ. is given by (V.35).

la?

V.114 Wale

138

The Linearized Boltzmann Equation

In general, the radial part of $4, is not known, and we need approximate methods to compute the transport coefficients. Yet since the Sonine polyno-

mials furnish an exact solution for Maxwell molecules, we expect that they

will provide a useful tool in these approximate calculations. 4.2.

A Variational Principle

Suppose we have the abstract operator equation

(V.115)

nClx)=1Y) subject to the subsidiary conditions

(V.116)

(a@=1---5)

(x)=0

for the unknown vector |y) with given C, |@2), and | Y). In addition, suppose that the operator C has the following properties:

(file) = 0

while at the same time the two species become mechanically identical. Thus, we imagine a fluid in which all molecules have the same molecular properties but carry a “label” that allows us to distinguish a few of them from the others. Experimentally, such a label can be provided for example by the decay of a dilute radioactive isotope (neglecting the small mass difference) or by some nuclear-spin property. The macroscopic transport phenomenon that describes the motion of such a dilute species is called self-diffusion. If the species 1 is sufficiently dilute, the interactions between molecules of this type will become negligible (as in Brownian motion). Then the behavior of this species will be reduced to the motion of one particular molecule in a fluid of identical particles. We have already emphasized that we have no way of studying the deterministic motion of such a single molecule in a single experiment. However statistical mechanics does give useful information about the behavior of one molecule in a series of experiments or about an ensemble of (independent) identical particles in one experiment.

Self-diffusion and related phenomena are particularly simple properties of fluids and have attracted much attention, especially with the recent development of computer experiments (Chapter XII). In the case of dilute gases, the proper description is offered by the Boltzmann-Lorentz equation, which we now briefly discuss. We suppose that the tagged particles are described by a one-particle distribution function, here denoted f, ;(r, v; ¢).* To simplify, we assume that, at ¢=0, the molecules of species 2 are at thermal equilibrium; their distribution function, simply written f,, is then [see (IV.122)]

Ail, v; 0) = n2@i(v)

(V.167a)

* In the limit where species 1 becomes infinitely dilute, f, , describes the statistical properties of one given molecule in a fluid of identical particles; the dynamics of such a given particle is often called seif-motion ; hence the subscript s.

Further Remarks

147

We can now reproduce step by step Boltzmann’s arguments for the calculation of the time evolution of the nonequilibrium d.f. f,1- Important simplifications occur for the following reasons. (i) (ii)

The probability of collision between two particles of species 1 is extremely small compared to the probability of collision between Particles of different species (by a factor ,/n3). Similarly, the d.f. of species 2 is only slightly perturbed (again by a factor n,/n2) from its equilibrium form by the very rare collisions of particles 2 with particles 1; hence (V.167a) remains approximately valid at all times:

filt.vs)= nsei%(o) + 0(!)

(V.167b)

2

With these two observations, we readily obtain otfeitr,v; t+ v-

fst

or =n2|

dv,

| dQo(y;g)g

Xia Vs 09M) falts vide MoNI+O(Z) — (v.168) ny

nz

Here a is the collision cross section between molecules of species 1 and 2, which is independent of the species, since they are mechanically identical.

When n, > 0, n2 becomes the total density n, thus we can rewrite (V.168) as

afl, V5 ty

Ss. nC

fe,

(V.169)

where the Boltzmann-Lorentz collision operator

Cp.=f

dv | ado

gdeliat, V3 De M(o) fale, v5 DeH(v,)] (V.170)

is linear.

Similarly, for the Fourier transform of fea

foals p=| dre f(r, v; t)

(taking q=q1,), we have

Ofsgt igvy fig = nw”

fig

(V.171) (V.172)

We leave it as an exercise for the reader to prove that with the scalar

product (V.7), we have

(AICPA 00, the density of tagged particles

Nsg(t) =| dv fiq(v; t) is the only conserved

diffusion equation

mode

of the problem;

(V.175) it will of course

a,n,.q = —q’Dn, q(t) where D

obey

the

(V.176)

is the self-diffusion coefficient.

The formal analogy between (V.172-174) and (11.63, 85, 86) saves us the

task of deriving the expression for D; we can simply use (11.103) to get

D=-lim (a3 e70y

OAD “a

eu

$°)

(V.177)

Following the method used in Section 3 of this chapter, this can be rewritten as = | dv v,x;(v)

(V.178)

aCe? x, = 0,9 1(0)

(V.179)

where x, is the solution of

subject to the subsidiary condition

i=

dv x,(v) =0

(V.180)

The problem posed by (V.179, 180) is of the type studied in Section 4.2 and the variational method proceeds in a straightforward manner. We merely quote the result in the first approximation. Oo

Bm

3keT

QM

(V.181)

Further Remarks

149

For hard spheres, it follows from (V.153) that this reduces to 3

k.

Pos Epa)

2

Other aspects of the self-diffusion problem are discussed later. 5.2

wae

Kinetic Models

The aspects of the Boltzmann theory that we have investigated are the most

conventional ones: both in the nonlinear and in the linear cases, we have

shown how this equation leads to a justification of macroscopic hydrodynamics, with well-defined expressions for the transport coefficients. Despite the importance of these problems, in particular in view of their possible extension to dense systems, they fail to cover a variety of interesting phenomena that at least in principle can also be described by the Boltzmann equation. We have in mind all the processes that vary rapidly in time and/or space:

boundary layers, strong shock waves, and so on. In the nonlinear

theory the Chapman-Enskog procedure seems deceptive for these problems

(the Burnett equations have had little success), and in the linear domain the

perturbation expansion in powers of q is clearly useless in obtaining the eigenvalues of the inhomogeneous Boltzmann operator at high wave numbers. What we really need are alternative methods of solution. An example of these is the attractive Grad’s moments method, which allows the analysis of

flow problems well beyond the Chapman-Enskog gradient expansion.* This approach is not discussed here, however, because such flow problems belong to hydrodynamicists rather than to statistical mechanicians. In any case, it is clear that the difficulty in solving the Boltzmann equation in general, whether in the nonlinear or in the linear regime, is largely due to the intricacy of the collision term. Hence it is very tempting to try guessing model equations with the same basic features as the Boltzmann equation, but simpler to solve. One of these equations (the BGK

model, originally

proposed by Bhatnagar, Gross, and Krook) was introduced in Chapter IV, when discussing the concept of normal solutions in Hilbert’s theory. Indeed, this model amounts to writing [ef. (IV.190)]

afity: wh vrelf to imitate the Boltzmann

collision

frequency

equation. Here

{of the order of ran(v))

*See Grad (1958) as well as Cercignani (1969).

fil

(V.183)

v,.; is a velocity-independent and

the parameters

n(r; t),

150

The Linearized Boltzmann Equation

u(r; 4), T(r; f) that appear in the local Maxwellian f\” are such that f© gives the correct macroscopic moments [see (IV.204)]. This equation reproduces the two basic features of the Boltzmann equation. (i)

The conservation equations are trivially satisfied because by definition

[ avunti=[ artes? where the y,(a@ =1(ii)

(a=1---5)

(V.184)

+ +5) are the collision invariants (IV.95).

It satisfies an H-theorem because of the property Vel | dv (ff)

In f, = Vet | dy (FY-f) In

+ raf av (ffi) ln f? Vn no

we thus write

Hamiltoseparate that they between (VII.37)

PNana( Hy, + Hy, + Vue) © ON ena Hn, + Hy.)

= pn, (An, )on,(An,) * See, for example,

Goldstein

(1950) and

Khinchine

(1949). Notice,

moreover,

(VII.38) that in this

elementary discussion we make no distinction between the so-called global and local invariants [see Farquhar (1964)]. We discard here the trivial invariants—the total momentum and the total angular momentum of the system—which we take to be equal to zero.

The Canonical Equilibrium Distribution and Its Link With Thermodynamics

183

Taking the derivative of the expression with respect to Ay,, we have OP Ni+No op MM SPN SPN P™?eq aHy,+N dy,

and

OP NxN2 = OPN EN Ne

dHy,

VII.39 (VI1.39)

eq OPRK

dHy,

Paty,

2

( VIL.40 )

We thus obtain

den /AHN, _ 8pN2/AHN, _ PR PN,

—B

(VII.41)

which defines the constant parameter 8, independent of Hy. By integration

of (VIIL.41) we get

pN(Ay) =

exp (—BHy)

(VII.42)

Zn

where the constant Zjy is determined by the normalization condition (VIL8), which yields

Zy=|

dr dv exp (— BHy)

(VII.43)

The distribution function (VII.42), known as the canonical distribution, is a

fundamental tool of statistical mechanics: it allows calculation of the value taken at equilibrium by any macroscopic observable, through (VII.18), which becomes

(A) = i dr dvA p3

(VI1.44)

Equation (VII.42) involves the parameter 8, which we must interpret physically; in doing this, we establish the link between equilibrium statistical mechanics

and

thermodynamics.

Let

infinitesimal changes of B and 2.

us

6 In Zy= — 6B( Hy)

take

+60

the

olin Zy ——* 30

variation

of Zn

(VIL45)

‘The average value of Hy appearing in the first term in the right-hand clearly the energy of the system, which is known thermodynamically internal energy E. The second term is more difficult to analyze, but as below, it can be rewritten as B5Q(J,,)°4/O, where the phase function can be interpreted as the microscopic definition of the pressure

B

ain

Zn

1s

5g We =P

for

side is as the shown J,,/Q

(VII.46)

184

Distribution Functions in Statistical Mechanics

To simplify, let us suppose that the system is in a cubic box with sides 0'” and let us use rectangular coordinates r, = (x,, y,, Z,), and so on, with axes directed along the edges of the cube; we have

B 12a Ziy = (3)

fal

_

"

yy, . {

"dey exp(

arv,)|ze (VIL47)

where we have used the abbreviation

ZR

| dv,-+-dvy exp (—BHy)

(VI.48)

With the new vartables x Zz, x! 7 = que Lees N=t 58

(Vi1.49)

we obtain alnZ,; pM zupziy' la | ax; ' --- [azn

x exp [ ~BXA vireo") |} anb

N =—-

Wen

['

Zn

0

og

dx! ---| *

[’ f)

dz

a

x exp [ -BYA vier”) a/ vanish by

The BBGKY Hierarchy

193

integration over r, (we assume that py vanishes at the boundaries), and we are left with

= Sve

piles

i

a#l

a

tM

5D

(VII.106)

In the second term of the right-hand side, we consider three separate

cases. (i)

a,b1. Since all the particles /+1,/+2,...,N guishable, these terms yield a contribution (N-)D

t

¥ | 141

AV 141 Onsen J tjii2° +

a=1

(VII.107) are indistin-

dtndv¥ii2°** AVNOn

i

=(N-1)

ZI

Arp

AV 141 OaresPrziChay

6 +9 Breas Vin

e +) Ween; 8)

(VII.108) (iii)

@,b5>L An integration over y, and y, gives trivially zero (because Pn > 0 when v, > 00).

Combining these results and multiplying by a factor M//(N~2J)!, we obtain

an equation for the generic d.f. f;: OAlti,..8b Vis -- + V5 2)

t =— Vvat

F)

a=l

Alt, -- ste Vi-

Cl

VO

i

+.

LY

+2

t

-a=1

ae@l

Gaefilts,.--. te Vise VO)

J Fi41 AVi41 Oaterfier (ty, «62s trey Vises

+) Vers A) (VII.109)

which, when we take /=1,2,..., forms the BBGKY hierarchy. To determine the dynamic behavior of f,, we need to know f)4,, which in turn

requires fj42:+:. The exact solution of (VII.109) for any / necessitates solving the whole hierarchy of equations for /+1,/+2,...; the only state

194

Distribution Functions in Statistical Mechanics

where this hierarchy is exactly cut is for /= N (because with N particles, there is no fy1!). But then we are back to the original Liouville equation, and no progress has been made. The only way to make (VII.109) useful is to look for some good reason to truncate the hierarchy for / sufficiently low (i.e., to neglect f,,,, or to express itin terms of the f,’s with /’ [=

a=1

2

me

x

2 b#a

Vrae)]8(¢-Fa)

(VII.116)

In (VII.116) we have arbitrarily ascribed half the potential energy V(ra») to each molecule of the pair (ab). This arbitrariness is of little importance *For a phase function A (tr, 0) that depends on both the ['-space coordinates and on the position r of a point in the system, we often write simply At).

196

Distribution Functions in Statisticat Mechanics

because when we calculate the energy AE inside a volume AQ through the integral

AE=|

dré(r)

An

(VI1.117)

it only plays a role for the coupling between the molecules inside AQ and the

molecules outside AQ: for AQ large compared to the range of the forces, this is only a small surface effect. The average value of these microscopic quantities can be expressed in terms of the reduced d.f. From (VII.72, 73) we get immediately

nies =e), = | dvfle,vs0 iG; d= Gon

| dv mvfi(r, v; 1)

(VII.118) (VIL119)

&(r; 1) = (E(n)), = (EX), +E"),

(VIL.120)

aK mv? (é* (1), =| dv She, y; t)

(VIIL.121)

é"(@), = af ar’ dv dv' Vir—r'|fa(r,r', v, v3)

(VII.122)

with

and

corresponding respectively to the kinetic- and potential-energy densities.

Except for the potential term (€"(r)), all these macroscopic conserved quantities are expressed in terms of /,, as for the dilute gas. To establish the time evolution of these conserved quantities, we first

consider the one-particle averages (f,(r)), (w = 1 - - - 5) where [see (IV.95)]

dale) = 3 Yalva)8(e—r,) awl

is such that

Wate=

| dv dafile,v30)

(VIL.123)

(vIL.124)

From the first hierarchy equation (VII.110), we get

adba(t)). +2

J dv viba(v)fi(r, v3 t)

=-[ ar | avav be)

!1 ove aVUle el oe, v,v, 52)

m

(VII.125)

Conservation Equations

197

We now consider separately three cases. (i) %:=1. The right-hand side of (VII.125) identically vanishes; with p(r; t)=mn(r; t) and i, (0; ) =p(r; u(r; 1), we immediately arrive at the continuity equation

a(t; +e. (ii)

Yj, =v,

@=2,3,4=x,y,z).

[e(r; t)u(r; )]=0

(VII.126)

Here we find

alot; thui(r; n]+s 5 J dv mu;vf (8, ¥; t)

= -| ar’ | dvdv' TED i

A, rey,v3t)

(VI1.127)

We now must transform the right-hand side of this equation to make the

conserved character of pu; transparent, that is, to put (VII.127) in the form

Lots ute; )]+ F

fexy.z

oT; )=0 OF

(VIL.128)

where by definition I],, is the momentum -flow tensor. We remark that the right-hand side of (VII.127) represents the average

force exerted on the particle located at r by the other particles in the fluid. If f2 were even in the variable

Ar=r-r

(VH.129)

at fixed r, this force would vanish for symmetry reasons. To use this fact, we rewrite this term, now denoted J,

i 5

av’ f dv dv' MND,

r,v,v5!)

(VIL130)

where

J@r,v,v30)=[fole,e+ dr, v, v5 2) —f(r,—Ar,v,v; 0] (VII.131) The symmetrical character of f, in its primed and unprimed variables allows us to rewrite j as

J=[AG, rt Ar, v, v5) -fr— Ar, x, v', v5 4)

(VII.132)

198

Distribution Functions in Statistical Mechanics

We now add the assumption that f, varies slowly in a simultaneous translation of r and r’; that is, we consider a slightly inhomogeneous fluid.* We have thus

fi(r—Ar, r, v’, v5 £) = f2(r, r+ Ar, v’, v; 1)

r

jexyz

(2

r, vv; 0) 4 felts, Vs ) cs or;

or;

+0(Ar’)

(VII.133)

We introduce (VII.133) into (VII.132) and the result is in turn substituted

into (VII.130). Using the dummy character of the variables v and v’, we

integrate by parts over r’ and find

t= y

I x

oN ae |

ar, 2

avdy evi)Cg —Dhle ty, v5.0) (VII.134)

The combination of (VII.127) and (VII.134) leads indeed to (VIE.128),

with the following definition of I;

Ths(r; 2) = | dv muy;f (tr, v; 1)

(\r-r'l) 3lf J dr’ | dv dv OV a,

F

Ae,

Doe

r,v,v'; 0)

= (fin),

(VII.135)

where

Bai fly(t)= ¥ ( moaites 32X,5 ew)

rass)8(0—Fa)

It is convenient to separate from the momentum-flow

(VIT.136)

tensor the con-

vective term due to the average velocity u(r; t) [see (IV.130-133)]. We then

arrive at the equation of motion in the usual form

a.le(r; Ou(r; n]+e +[ouu+P]=0

(VIL.137)

where the local pressure tensor P is defined by

Py(t; 0) =(Py(r, #)),

(VII.138)

* This assumption is not essential, as is shown by the general treatment found, for example, in

the book by Massignon (1959). However the resulting equations are more complicated.

Conservation Equations

and

a P(t, =

199

N Vv 5 [ (vas ~ ua t))(0a,5 — uj »)-2 x ae) rani] (0 Ta) 2ea Oa; (VII.139)

avi

Let us point out that at equilibrium

these definitions coincide with

(VIL46, 54). Indeed, in this case, u(r; t) = 0, hence

(Bile,

0) = Thy)

(VII.140)

Moreover, the fluid is isotropic and translationally invariant; thus

(Pylr, 0) = T1135 o al dr (fey 6X 1,»

#8

Hi

= po

(VIE.141)

(ii) ws = 07/2. Using the same method as for uv, we can establish the following equation for the kinetic-energy density:

ane

aN). +2 jes N+oele; =O where

iar; 1) =| dy vf, 2

a7) -3| ar’ | dv av(v . avie—¥D) —,'

(VU.143)

xO-—P flr, v, v5) oa(r; t) =5/ dr’ | dv av[v-» Let us stress that, because

5 ~| fiolt,t’,v, V5.0)

of the source

(VII.144) term

oz, (VII.142)

is not a

conservation equation: this is not surprising because in general kinetic energy alone is not conserved. To get the total-energy density, which is

conserved, we need also to consider (é “(r)),. Its time evolution is determined from (VII.122) by the second hierarchy equation (VII.111). After simple

manipulations, we find

aE), +2 Me; )-oeles )=0

(VIL.145)

200

Distribution Functions in Statistical Mechanics

where

i203

= 2{ dr | dv dvvV(r-r') for, rv, v5.0) (VIL.146)

Comparison of (VII.142) with (VII.145) clearly shows that oz represents the exchange between kinetic and potential energy. Combining these two equations, we arrive at the energy equation

d,€(r; +e "hes O=0

(VII.147)

where the energy flow

fs

0 = 105 +505

= Ge),

(VIL148)

is the average of the microscopic energy flow, given by

je(r) i.()

=

i

Evala

mv2

3

1

2,

S45

(ran) )-ve } -¥

Vira

1.

dV (ra5)

2.

He,

+5

a

a = (tas)

|e

Fa |5(r—ta)

(VII.149) Here again it is useful to separate in (VII.147) the convective contributions,

defining the internal energy per unit mass e by zen.

ae; 1) = pln; [Es

et; 0]

(VI1.150)

and the heat flow jo by

jo; ) = (jolt, )),

with

(VII.151)

jolt = 2X five—u(e; y| eM Ss Virg)| -

“py

—(va — u(r; £)) + 5

1

-r)}6(e —r)

(VII.152)

Then we arrive again at (IV.136), which we repeat here for completeness:

ad ote [ED

2

se o]} +2. [ou(4-+e)+P : utio| =0 (VII.153)

Thus

we showed

that the conservation

equations of fluid dynamics

(VII.126, 137, 153) can be givena general microscopic basis. This extends to

all densities the analysis of Chapter IV. A striking difference appears,

Conservation Equations

201

however: in the case of dilute gases, all the relevant macroscopic quantities

are obtained from the one-particle d.f. f,, whereas in general one needs to know f, and f2. We also learned that these d.f. obey the BBGKY hierarchy, which is not closed. We may therefore wonder whether it is possible to derive a closed set of equations for f, and f,. This can indeed be done with

the help of techniques very similar to those discussed in the next chapter for spatially homogeneous systems. However the formal character of these exact equations is such that it is very hard to obtain practical information from them. Moreover, as Part D reveals, simpler methods can be used if we

are only interested in calculating transport coefficients.

VII Generalized Kinetic

Equations

1

INTRODUCTION

With the language introduced in the preceding chapter, we are now ready to consider the fundamental theory of the nonequilibrium behavior of classical fluids. Our aim is to explain this behavior starting from the laws of mechanics that govern the motion of the constituent particles. This problem can be attacked from two pedagogically different viewpoints. (i)

(ii)

By studying simple models first, like the weakly coupled gas or the

dilute gas, where the various assumptions can be checked explicitly, and generalizing afterward to more complicated cases with the help of the intuition gained from these simple models. By developing a formal but very general frame that applies in principle to the most general case, using at various steps plausibility arguments that we hope to be able to verify on simple examples.

The first approach, which follows historical development,* may appear to be more promising for the beginner in the field, but it suffers from two serious drawbacks. * See, for example, Prigogine (1963). 202

Introduction

(i)

(ii)

203

Explicit calculations are very awkward and the amount of technicality

that must be reproduced for each specific model is rather large; the

physical meaning is easily lost in the mathematical manipulations. Worse yet, even the simplest situations in fluids cannot be treated with

mathematical rigor. Such rigor can be attained only for very unrealistic models, which tell us very little about the behavior of a real fluid.*

We therefore prefer to present the second approach: the calculations are formal but easy, and instead of mathematically unclear “proofs,” we introduce a series of well-defined assumptions that are physically plausible. The next chapter shows that this formal frame is at least reasonable since it reproduces the kinetic equations we already know (e.g., the Fokker-Planck equation and the Boltzmann equation). At various places in the remainder of the book we see that this frame is also operational because it allows us to

tackle new problems with reasonable success. Yet it is only honest to warn the reader that this formal frame is not the magic key to the solution of any

problem

in fluid theory.

When

the formalism

is developed,

most of the

theoretician’s work remains to be done: using his physical intuition, he must introduce mathematical approximations that will lead to his final goal (ie., obtaining numbers that can be compared to experiment). Underlying the theory is the basic idea that most of the information in the

N-particle d.f. px is not observable; as shown in Chapter VII, Section 3.3, the important information is contained in the reduced d.f. fis fo,..., from

which all observable quantities can be computed. Hence the whole problem of kinetic theory is that of deducing from the Liouville equation a contracted description for these reduced d.f. To make things as simple as possible, we deal with the one-particle d.f. fi

only, limiting ourselves to spatially homogeneous systems. (Some aspects of

the inhomogeneous case are treated in the frame of linear-response theory, Part D.) Then f, takes the form [see (VII.72,74)] Filer, V13

) = n@ilvi; t) =n{

ae, +++ dtydv2-+++ dvnpn(t,0; 0)

(VIII.1)

It is convenient to make the reduction from py to g; in two steps.

(i)

First derive a formal master equation distribution function

for the N-particle

onto; )= | deen(e, 950

* See Prigogine et al. (1973) and references quoted there.

velocity

(vii)

204

Generalized Kinetic Equations

(ii)

Then integrate g,, over the (N— 1) velocities v2, . > Yn, to arrive at the so-called generalized kinetic equation for ¢,(v;; ft).

As the next paragraph reveals, the derivation of the master equation for oy,

keeping N fixed, is exact and straightforward. However this equation is not

very useful without further assumptions:

it is indeed a formal identity so

general that it describes equally well physical problems as different as the motion of the moon-earth system, or the dynamics of liquid argon! In a macroscopic system irreversibility is expected to emerge from the erratic motion of the many interacting particles. To introduce this essential feature in the theory we must take the thermodynamic limit where the number of particles N and the volume 2) go together to infinity while their ratio, the density n, remains fixed. If we studied the dynamical behavior of a single system with a prescribed initial condition, the thermodynamic limit would also be required to avoid the paradox raised by the Poincaré recurrence theorem. This theorem states that, whatever the

initial condition (corresponding to a point (t, D,) in phase space),* any finite system will ultimately return as close as desired to this point. Clearly, such a recurrence is incompatible with irreversibility. However, model calculationst strongly suggest that the Poincaré recurrence time grows rapidly when the number of particles increases, thus becoming much larger than any physically interesting time scale. Here though, we analyze the time evolution of the distribution function py for an ensemble of systems, and this function is assumed smooth in phase space. Under these circumstances it can be shown¢ that the Poincaré theorem puts little constraint on the (possibly irreversible) behavior of gy. Nevertheless we see later that this limit is

essential for justifying a series of assumptions made in the development of the theory.

Section

3 demonstrates

that when

taken at the level of the master

equation for yy the thermodynamic limit leads to meaningless results. However as mentioned earlier, gy is only an intermediate tool for calculat-

ing 1, the physically important quantity. At the level of g,, a well-defined kinetic equation results from the thermodynamic limit and this equation can be interpreted by comparing it to the now familiar Boltzmann equation for the dilute gas; this is done in Section 4. Applications are given in subsequent

chapters.

* Except for conditions of measure zero.

t See, for example, Frisch (1954). $ See Lebowitz (1971).

The Formal Master Equation

2

205

THE

FORMAL

MASTER

EQUATION

Our aim is to deduce the equation obeyed by ¢,(v¥,; ¢) from the Liouville equation (VII.15), which we rewrite as

id,pn(t, 0; t) = Lypn(t, 9; t)

(VIII.3)

using the Liouville operator [see (VII.15, 1-3)]

Ly =LN+A6Ln = i{Hn, ++}

(VIIL4)

with

LY=-i

N

Y ver =

a=l

Og

N

d LY

aa

(VIILS)

and

=y The operator Ly written in the form

N

a

(VIIL.6)

is introduced here of (VIII.3) is very

equation of quantum mechanics

ha, Vv(ti

because the Liouville equation reminiscent of the Schrédinger

= AYE Ns

(VIII.7)

This analogy is often very useful. For example, the formal solution of the Schrédinger equation is

Wyle; f) = exp ( — iNew lts 0)

(VIIL8)

where the exponential of an operator Aj is defined by the expansion 2

exp (-iAnt)= ¥ 7 tmobt

iAne)!

(VIIL.9)

Similarly, the formal solution of the Liouville equation (VIII.3) is [see also

(VIL.29)]

pn{t, 0; t) = exp (—iLnt)pn(t, 0; 0)

(VIIH.10)

In addition, if we associate a Hilbert space with the phase-space functions

that obey periodic boundary conditions at the limit of the volume 2 (or

vanish at these boundaries) and go to zero when any of the velocities v, goes

206

Generalized Kinetic Equations

to infinity, it is readily checked that for any fixed N, the operator Ly is

Hermitian.*

{ drdv gtr, v)Lyhy(t, 0) =| dv dvhy(t, v)Ligh{r, v)

— (VIII.11)

The choice of periodic boundary conditions is of course rather artificial, but the hope is that when N becomes large, the behavior of the system in the bulk will not depend on these surface effects. The similarity of (VIIL.11) with the well-known hermiticity property of HY in quantum mechanics, strongly suggests that the formal language and the tools developed in the quantum many-body problem may be useful in statistical mechanics. We now perform the first part of the program sketched in the introduction, namely, deriving a formal master equation for the N-particle velocity

d.f. gy. It is convenient to introduce the operator Px, which transforms an

arbitrary phase-space function gy(r, v) into its average over position space Pygn(t, D) =a]

dt gn(t, v)

(VIIL.12)

This integral operator is a projector, since we have

PR=Py

(VIII.13)

Pr=Ph

and the same is true for its complement

Ov™1-Py

Moreover,

Qx=Qy

OQv=ON

(VIII.14)

comparing (VIII.12) with (VIII.2), we see that except for the

constant factor

2%, Py extracts from the N-particle d.f. ax the part we are

interested in (we call it the “relevant” part); Qxpy = (1 — Px)pn is then the

“irrelevant” part.t If we now multiply the Liouville equation (VIII.3) by Py and by Qn, we obtain two coupled equations for these two parts of pyt i0,Pypn(t) = PrLnpn(t)

= PyLyPxpn(t)+ PrlLnOnpn(t) 10,Qnpn(t) = QnLnPyon(t) + QnLnOnen(t)

(VIII.15a) (VIII.15b)

* Notice that the two members of (VIII.11) generally become ill defined in the thermodynamic limit; therefore the reality of the eigenvalues of Ly, which is an immediate consequence of (VIIL11) for N finite, has no immediate bearing on the (possibly irreversible) behavior of the system in this limit. t The definition of the “relevant” and “irrelevant” parts depends on the problem at hand. If we were to study f2,

f3,...,f; (for / finite), some relevant information should also be extracted

from the “irrelevant” part. + We drop the (r, p)-dependence to shorten the equations.

The Formal Master Equation

207

To eliminate the irrelevant part Oxpa(t), we note that (VHI.15b) is an inhomogeneous linear equation for it, with the following formal solution. Qnen(t) = exp (—iQnL nt) Onn (0)

-i{

o

dr exp (—iQnL yt) OnLnPrpn(t—7)

— (VII.16)

This can be verified by differentiating it with respect to t and by checking that it is correct at r=0. The substitution of (VIII.16) into (VIII.15a) yields for the relevant part of Pw aclosed equation that simplifies when the following properties are used:

LRPx =0

(VIII.17)

PyLyPy =0 PyL2=0

(VII.18) (VIII.19)

The first of these relations is a trivial consequence of the definitions (VIIL.5, 12); the second follows when we notice that in (VIII.6) the pair potential

V(r) vanishes when

r goes to infinity; finally, to assert the validity of

(VIII.19), we have to use the assumption that any distribution function py over which the operator PyL}, acts, obeys periodic boundary conditions (or vanishes at these boundaries). We thus arrive at

10,Prpn(t) = PrAdLy exp (—iQnLnt}Onpn(0)— if dz PyASLN O

x exp (—iOnL yr) OnASLNPypn{t — 7)

(VHI.20)

which, through the definitions (VIII.2, 12), can be rewritten

9: Pn (D; 1) =

[ar Guto; Dew(ost

7) + Dy (0; t}Onpn(0))

— (VIHII.21)

Here the operator in velocity space Gy is defined by Gy(v; 7) = PrASLN exp (—iONL NT) OnABLN

(VIII.22)

and Q, is the following function, depending on the initial value of the irrelevant part only:

(05 | Onpn(0)) = FO%PAASL exp (—iQuLnt}Qnpw(0)

—_(VII.23)

Equation (VII.21) is known as the formal master equation ;* it is a formal identity for the velocity d.f. gy in any system of N particles interacting * See Prigogine (1963), Résibois (1967), and Zwanzig (1960)

208

Generalized Kinetic Equations

through arbitrary interactions. Since this equation is so general and has been obtained so easily, it is clear that the complexity of the N-body problem has merely been hidden in the definitions of the objects Gy and Dy. To extract any practical information from the formal master equation, we must still analyze these objects in detail; this is of course a difficult problem, because Gn and By involve the operator exp (~iQnL yt), which is very complicated.

Moreover we already know that although correct in general, this master

equation should be relevant only to describe irreversible phenomena in the

limit where N-> oo. Rather than trying to interpret this equation at the present stage, therefore, we find it preferable to begin by investigating its behavior in the thermodynamic limit.

3 AND

THE MASTER EQUATION THE THERMODYNAMIC LIMIT

To demonstrate the problem posed by the formal master equation in the thermodynamic

limit, and the way out of it, let us examine

the N- and

-dependence of the kernel Gy defined by (VIII.22). Clearly this definition has a meaning for any finite value of N. Thus for a system of two particles (a and 5) we can define

(VHI.24)

G.=G For a system of three particles (a, b,c), we then write

G3=G34+G9+G2+GR

(VIII.25)

which defines G‘j). as the part of the kernel G3 which involves the three particles simultaneously. We can similarly decompose G4, Gs, ...and finally Gy by writing

Gyo;7)=

N

YL

GE vas Vos 7)

beael

+

N

YL

cwbwa=l

Gives Vos Ves TIFT

+ GYD. Vi, Since these definitions ensure that in GY

YN 7)

(VIII.26)

..., n P particles PP appear explicitly Pp: in the interaction operator 5L, = Y3~a-1 6L'*” (see (VIII.22)], it is easy to guess the N- and Q-dependence of the corresponding contribution to (VII.26).

The Master Equation and the Thermodynamic Limit

209

Let us start with the first term

L Gas Y Pras

“

exp [-iQ LYn( +L 9+ ASL )r]Qnash

-_

(VIIL.27)

The sum runs over N(N— 1)/2 ~ N? pairs of particles. On the other hand [see (VIII.12) and (VIII.6)], we have the integral

a

3

| dry+ ++ dtydL@?-- a

(VIII.28)

since the particles a and 6 have to be in the Tange ro of their mutual

interaction for SL” not to be zero. Thus we find 2

= Go~ ‘ ~naN

(VIII.29)

Similarly, we have 3

» since there are ~N°

G68.~3~1N

(VIII.30)

triplets, and they must interact together, yielding a

factor (r8/)’. Things become different in the four-particle terms. We now have two kinds of contributions. (i)

“Genuine”

four-particle

( (ii)

terms,

where

all

four

particles

interact

together, in the same region ~r§ of space; these behave as

z

d=c>b>a

Gea)

N(R)

3) 3

~n°N

(VHL31)

again of order N in the thermodynamic limit.

Pairs of binary events, where two particles (say, a and 6) interact in one

region ro [whence a factor (r3/)], while the remaining particles c andd interact independently, in any other region of space, yielding the same factor (r9/). We thus have the contribution

(= Ga d>c>b>a

) sae

ra\2 ~N*(2)'~n?N?

(VIN.32)

growing now as N*—that is, infinite compared to the preceding contributions in the thermodynamic limit!

The catastrophe continues when we go on through the series (VIIL.26): for

L.so>a GY...,

we would

find that (if | is even)

it contributes

a term

corresponding to //2 independent binary contributions and diverging as

210

Generalized Kinetic Equations

N'(r8/0)'? ~n'?N.

Clearly the series (VIII.26) has a very singular

character in the thermodynamic limit. We are thus in a rather embarrassing situation with the master equation. Either we take it for a finite system, in which case it is well defined, but we do not expect it to be relevant to describe irreversible behavior; or we go to the thermodynamic limit, in which case the equation becomes ill-defined! This difficulty is evidently attributable to the fact that the master equation

describes the time evolution of the velocity d.f. of the N particles in the system, therefore accounting for the interactions involving groups of particles that are arbitrarily far from one another in space: the number of such processes grows without bound as N increases. Yet this difficulty must disappear when we integrate the master equation over (N —/) velocities to obtain an equation for the reduced df. g; (in particular with / = 1). Indeed,

the velocities of a specified group of / particles can be influenced only by the interactions wherein these particles play a part, whatever happens in the rest of the large volume. This argument is easy to put on mathematical grounds; consider, for example, the case of g;. When the master equation is integrated over v2,..., Vx, taking into account that | dv, dv, 6L'”+-+-

=0

(VIII.33)

[as can be checked by using the definition (VIII.6) of 5L‘*””] we see that

J ava+- avy

x

dva- dvs

GB-=[

ome

Since the sum is thus restricted to the (N—

N

yr G@..

ba2

(VIII.34)

1) ~ N particles 4, the argument

[see (VIII.27-29)] developed previously now leads to the result

[aves

dvs

x GQ

~Ran

(VIII.35)

which remains finite in the thermodynamic limit. Similarly, for the threeparticle contribution

involving

G&.

and for the “genuine”

four-particle

term, we now find that they behave as n? and n°, respectively. Finally, the

contribution (VIII.32), proportional to N’, now gives identically zero, since only one of the two pairs can involve the labeled particle 1; for the other pair, the property (VIII.33) applies. This analysis is of course very schematic [a rigorous analysis of fdv2---dvy Gn(o; 7) +++ in the thermodynamic limit has not yet been made], but we hope it will help the reader understand why, although the

formal master equation does not have good mathematical status in the

thermodynamic limit, the equations obtained from it for the reduced d.f. (by

The Generalized Kinetic Equation

211

integrating over a large number of velocities) exhibit well-defined behavior

in the same limit.* The conclusion is that although the objects Gy(v; r) and In(v; t}Onpn(0)) do not exist when N and QO go to infinity, the functions

iim { dvi. °+ + dV¥y Gy(v; T)on(0; t= 1)

(VIII.36)

im { dM41°°° d¥n Dn(v; t]Onpn(0))

(VHI.37)

nevertheless remain well behaved for any finite 1+ Here lim, indicates that

the thermodynamic limit (VII.68) is taken. In the absence of a rigorous proof of these assertions, we henceforth accept them as heuristic assumptions, 4

THE

GENERALIZED

KINETIC

EQUATION

The analysis of the preceding section indicates the way to obtain the

equation governing the time evolution of the one-particle velocity d.f. ¢;:

we integrate the formal master equation (VIII.21) over (N—1) velocities and take the thermodynamic limit afterwards. The result is

8 i(v15 t)= -[ dr BUv,; t, 7) + Dv; 0)

(VIHI.38)

where Biv; 47) ~iim { dv2+++dvxy Gu(v; T)@N(0; t—7)

(VITI.39)

and

Dv; t= tim { dvz+ ++ d¥n Dn(; tlOnpyn(0))

(VIHI.40)

This formal result is not yet very useful because (VIII.38) is not closed: it

relates ¢, to the complete velocity d.f. Suppose however that the following assumption is valid (at least for N large):

en(0i)= TT onlvast)

(vit 4t)

* Notwithstanding these mathematical difficulties, the formal master equation has been extensively studied, keeping constantly in mind that in the thermodynamic limit, it can only be used to

calculate reduced properties. In particular, many remarkable formal properties are discussed in

Prigogine et al. (1973) and the references quoted there. They are not needed for our elementary and pragmatic discussion of the kinetic theory of fluids. + The argument for Gy, can indeed be reproduced word by word for the function Dy.

212

Generalized Kinetic Equations

We then arrive at the so-called generalized kinetic equation ‘

0,¢1(v13 t) = -{

0

ar G(v3 tlei(t—7)) + D(w1; 2)

(VIII.42)

where we have introduced the nonlinear functional of ¢) N

Gv; tles(1) =iim{ dv,- ++ dvyGy(o; 7) T] eilvast) a

ami

— (VIIL43)

Equation (VIII.42) provides the most general (but formal) answer to the

question of the evolution of ¢;. Of course this equation rests on the validity of the very strong assumption (VIII.41), which we investigate in the next section.

Let us provisionally accept the generalized kinetic equation and try to clarify its physical meaning by comparing it to the Boltzmann equation, to which it reduces in the dilute-gas limit (as the next chapter shows). In a homogeneous

(VIH.1)]

system the Boltzmann

equation becomes [see (IV.74) and

aeivsd=n| dvr | doles ge leuvis Nerlvss Del; Deus O) (VITI.44)

We see that the right-hand sides of both (VIII.42) and (VIII.44) involve a

nonlinear dependence on the one-body distribution g). It is thus natural to interpret the operator G as the generalization of the Boltzmann collision operator.*

However

there

is a particularly

striking

difference—namely,

although the same time appears throughout the Boltzmann equation, the collision operator G connects the time evolution of ¢, to its value at earlier

times. This memory

or non-Markovian

effect reflects the fact that the

collisions responsible for the evolution of ¢; last a finite time 7,, which is generally not negligible compared to the relaxation time 7,.; of the d.f. This can be understood on the basis of the following very simple dimensional argument. Starting from the low-density (or Boltzmann) limit, let us estimate 7,., aS in (IV.191) and the duration of a binary collision as

Tee (remember

(VIII.45)

that ro is the range of the interaction and (v) is the average

velocity). We see that

7. co nr3

(VIII.46)

* Since the functional G(v,; 7|p,()) is defined for an arbitrary d.f., we can consider the object

G=G(v¥;; 7|---) as a nonlinear operator in the space of the d.f.

The Generalized Kinetic Equation

goes

to zero

negligible.

when

n—0;

213

that

is, the

duration

of a collision

becomes

When we now increase the density, we expect the relaxation time 7,,; to be modified by a correction proportional to (or at least growing with) the dimensionless parameter nr3. If we keep terms of this order in 7,2, we must

consistently take into account that the collisions are not instantaneous,

For hard spheres, for which we would expect such simple kinetic ideas to apply, the estimate (VIII.45) is clearly wrong because the two-body collision is instantaneous: T. =0!

To

define

a sensible

collision time, we

must

consider more

complicated

events, like the three-body collision represented in Fig. VIII.1, where a particle 1 has

Figure VIII.1

three successive event is roughly The geometrical that the process

A typical three-body collision for hard spheres

collisions with two other particles 2 and 3.* The duration of this of the order of r, = R/(v) where R is the distance between 2 and 3. constraint imposed because 1 has to be scattered back on 2 implies has sizable probability only if R ~a.t We have thus the estimate T.

Pod

(VIIL47)

(v)

and the foregoing argument can be reproduced with rp replaced by a.

If the non-Markovian

character of the generalized

kinetic equation

reflects the finite duration of the collision, we understand why no such effect * As Chapter X reveals, the process by which particle 1 collides with 2 and 3 with no further scattering with 2 is a succession of two independent binary collisions and is already included in the Boltzmann equation. + When this argument is put on more quantitative grounds, it appears much more delicate than is apparent here (see Chapter X, Section 4).

214

Generalized Kinetic Equations

persists in the Boltzmann limit. More generally, we expect the operator G to decay to zero with a characteristic time of the order of 7-.

(Vili.48a)

(7 >t)

Gy; 7lei())>0

This property should at least apply for a velocity v, of the order of (v) and for a distribution function ¢, not too different from the equilibrium Maxwellian. The statement (VIII.48a) plays a crucial role in the development of the theory, and therefore, a neat proof of it would be most welcome. It would

even be desirable to have a somewhat stronger version telling us how G tends to zero; for example, is there some bound of the following type?

(VIIL48b)

(4%)

Gwitle@)0

we expect the following property, {t>7,)

(VIII.52)

Precisely as for the collision operator G, no rigorous proof of (VIII.52) is

available and, again, confidence in this assumption rests on model calcula-

tions: Appendix F verifies that (VIII.52) is satisfied for the weakly coupled

gas with exponential repulsion.

Let us stress that the argument leading to this assumption is crucially based on the existence of smooth correlations extending over molecular distances: this concept is fundamentally of statistical nature and it would become meaningless if we were to study a single mechanical system, Similarly, the model calculations of Appendix F rest upon the smoothness of the N-particle d.f. py at = 0 [see (F.20)]. For a single system (which obeys

the Liouville equation (VIII.15) with a singular initial condition Pn(t, 0; 0)= 6(t—t9)5(p — v9), this Dirac delta function following the trajectory of the

system for ¢ >0) the property (VIII.52)—and its consequences—would not

apply.

To end this section, let us explain how the present approach handles the

famous reversibility paradox

(which, together with the Poincaré paradox,

raised the most severe criticisms of Boltzmann’s ideas and their generalization®). This paradox asserts that irreversibility cannot be a consequence of the laws of mechanics (without any additional ingredient), since these are invariant with respect to time reversal. If we change ¢ to ~¢ and consistently put —v, for v, in Hamilton’s equations (VII.6), these equations do not change. Therefore, if there exists a solution toward some equilibrium, there must also exist the corresponding “antisolution” away from this equilibrium. Various applications of the present approach later show that an irrevers-

ible behavior is always linked to the forgetting of the initial correlations

through the way assumed in (VIII.52). However it is clear that (VIII.52)

* For a historical introduction, see ter Haar (1954), Brush (1965, 1966) and references quoted there; see also Balescu (1975).

216

Generalized Kinetic Equations

cannot be valid for arbitrary initial correlations: in our qualitative argument justifying this equation, we have explicitly used the intuitive idea that correlated particles tend to separate. Suppose now that we adopt an initial

condition such that (VIII.52) is valid; as time goes on, the dynamical evolution of the system builds new correlations, which are very sensitive to

the detailed motion of each molecule and extend over larger and larger distances. To get an ‘‘antisolution’’ of the type considered in the reversibility

paradox, we must, after an arbitrary time ¢, invert exactly the velocities of

every particle in the system, preserving these very special correlations. This new initial state, which obviously is impossible to realize in practice,” is a very singular function of the phase-space coordinates, and it clearly does not fall into the class of initial conditions for which the argument leading to (VIII.52) holds. With such an initial condition the contribution @ does not

vanish after a time 7, and ensures the return of the system to its initial state at time 2t.

Although this qualitative picture is supported by model calculations,t it is

true that we lack a general characterization of the initial conditions for which

(VIII.52) holds true; it is observed empirically that this property is valid for all physical situations of interest. In any case, as stressed in Chapter VI, questions about initial conditions cannot enter into conflict with the dynamical laws. Henceforth we always assume that (VIII.52) holds.

5

GENERALIZED

MOLECULAR

CHAOS

The factorization (VIII.41), which plays a crucial role in the derivation of a

meaningful (i.e., closed) kinetic equation for ¢,, is often called generalized molecular-chaos assumption, by comparison with (IV.65). This terminology is somewhat misleading because Boltzmann’s stosszahlansatz was proposed in the context of his description of the inhomogeneous states of dilute gases and taken as a local property of the system, whereas (VIII.41) is here used

for systems that are arbitrarily dense but spatially homogeneous. It is clear that the statement fat, Be, V1, V25

= fat, Vi5 Of (ta, Vo5 £)

Q?)

(VIII.53)

cannot be correct for all densities, on all space and time scales. But for

homogeneous systems the assumption (VIII.41) can reasonably be expected

to hold in most situations of practical interest. Indeed, gv(v; ¢) is the integral of px(t, 0; f) over the spatial coordinates in the whole volume ©. If correla* Except in computer experiments: see the interesting work of Orban and Bellemans (1967). t See Balescu (1975).

Generalized Molecular Chaos

217

tions between particles extend only over finite distances [as we assumed to

get an estimate of the time scale for the decay of D(v;; t), Section 4], the

weight of these correlated configurations should go to zero in the limit O00,

Moreover, if (VIII.41) is assumed to hold at any one time fg, it can be shown in a weak sense (made precise later) that this property is maintained

at any later time: molecular chaos persists. In fact, the persistence of molecular chaos has been studied on models and this has led to one of the very few rigorous results of nonequilibrium statistical mechanics. Indeed, Kac*

proposed

collision operator

a model

master

Gn(0; 7) =

equation

with

a two-body

, W(¥a, Ve)6(7) bani

Markovian

(VIIT.54)

with a suitable choice of W(v,, v,). Kac showed that for any initial distribu-

tion such that

ft

him evi,

‘v5 t=0)= TT jim eilvast#=0)

— (VIII55)

a similar property is maintained for t>0 by the master equation. This is not the place to give the details of Kac’s proof. A similar result—though obtained by much less rigorous methods—can be achieved within the present formalism.t To sketch this proof, let us go back to the formal expansion (VIHI.26); using (VIII.34) and similar properties for the higher-order terms, we can expand B(v,; ¢, 7) as follows [see (VIII.39)]:

Bivi347)=lim Y | dvz+++dvw GR(v1, ve; r)on(v; t~-7) co bl]

tlim CO

XY

c>bet

| dvo+++ dvy Gini, Vs, Ve; TEN 3 t—T) +++

=limoO b#1Y | dv, GR(v1, v5; 7)¢2(vi, ves t—7) a +lim CO

EY

cbs]

3) | dvedve GiR(V1, Ves Vos T)03(V1,Vin Ves . tT)

P5500

(VIII.56) * Kac (1959, 1973). t Of course Kac’s model ignores the difficulties related to microscopically derived master

equations applied to finite systems.

+The calculation is given in Clavin (1972).

218

Generalized Kinetic Equations

Now it seems reasonable to assume that even in a dense fluid, this series will

converge: we do not expect to have relevant contributions from collision processes involving an “infinite” number of particles. If this crucial assump-

tion is correct, it follows that we do not need (VIII.41) to arrive at the generalized kinetic equation, but only the weaker statement lim ¢(v1,...,¥50= oo

i

TL lim ¢,(v_; 0)

awl

o«

(VIII.57)

valid for any finite /.

But if (VIII.57) is assumed to be correct at ¢ = 0, it must apply at any later time, since its violation can be due only to interactions between the specified

/ particles; these interactions have a vanishing weight in the thermodynamic

limit (/ and ¢ remaining finite) compared to that of the interactions involving some of the remaining (N—/) particles. This argument can be put on mathematical grounds by studying the operators B,(v1,..., v5 t, 7), defined in analogy with (VIII.39) by integrating the kernel Gy over the velocities Vivi. +

3 ¥N(E> 1), 6

THE

MARKOVIAN

APPROXIMATION

The generalized kinetic equation (VIII.42) is believed to be the most general

equation governing the time evolution of the one-particle velocity d.f. But it is still formal, and no explicit expressions exist in general for the quantities G and Q. Therefore little can be obtained from it without further approximations. For example, the question of the existence of an H-theorem (Chapter IV, Section 5.3) remains unanswered. The only result obtained in this direction is that the Maxwellian distribution (V.4) is indeed a stationary solution in the long term. That is, it satisfies the condition t

lim { dr Gv; tTei*) = 0

100 Jo

However, even the proof of this weak statement is far from trivial.*

Nevertheless,

the generalized

kinetic equation

is important

for two

reasons. First, as the next chapter shows, we can derive from it, by taking

appropriate limits, the important equations heuristically introduced earlier for various particular problems (e.g., the Boltzmann and Fokker—Planck equations).

Moreover

the first corrections

to these

equations

(e.g., the

so-called Choh-Uhlenbeck correction, which takes triple collisions into account in a moderately dense gas) often can be found. * See Résibois (1967).

The Markovian Approximation

219

Second, when we deal with dense systems, the generalized kinetic equation provides a formal structure that suggests reasonable approximations that would have been hard to guess from the (Markovian) kinetic equations

studied in Parts A and B. This point of view is developed in Part D, within

the framework of linear response theory. One of the difficulties encountered in these applications of the generalized kinetic equation is related to its non-Markovian character: the time evolution of ¢, is related to the value of this distribution at earlier times. Although

this can lead to important effects (see the model discussed in Chapter XII, Section 2), the disappearance of this memory effect in the Boltzmann limit

indicates that in certain cases it is legitimate to replace (VIII.42) by its Markovian approximation. To see when this happens, let us go back to Section 4, where we introduced the duration of a collision T, and the

relaxation time 7,.,, the latter being the characteristic time over which the

d.f. g; decays, and the former being identified with the time scale of the kernel G [see (VIII.48)]. Let us suppose that these times are very different; that is, that

Te & Tres

(VIII.58)

We are then interested in the kinetic equation (VIII.42) for times ¢ 7,1 (otherwise ¢, has not changed) and [see (VIII.52)] D(v,; #) can be neglected. Moreover, if G decays rapidly to zero for times larger than 7, [as in

(VIII.48b)], we can expand ¢,(v; t—7) around ¢ as follows:

gilv;t—r)=e1(v; )—7

te) sf SEND or

=e1(¥; o[ i+o(2)]

=9,(y; ofa +0( te )]

(r 7,, which would certainly be correct if (VJIJ.48b) held true.

Unfortunately, as already noted, this is seldom the case, and the Markovian approximation should be used with great care to describe relaxation phenomena. Nevertheless Chapter XII demonstrates that the Markovian collision term plays an important role in stationary transport phenomena (independently of the assumption 7,/7,-:« 1); it thus appears as a central object in kinetic theory.

We end this chapter with two remarks.

(i)

It is possible to generalize the “Markovianization” procedure and to

(ii)

expansion.* This chapter has been limited to the study of the kinetic behavior of the

get formal expressions for the corrections of order (r./Trei), (t./Tre)”, .... However it is difficult to assert the convergence of this

one-particle d.f. ¢,. However we know (Chapter VII, Section 3.3) that

in dense systems some observable properties of the system also depend

on the two-particle d.f. f.(t2, V1, V2; £), for which a kinetic equation is

needed. This equation can be obtained with techniques similar to those employed for ¢).

* See Prigogine (1963); Résibois (1967).

IX Simple Applications

of the General Theory

1

INTRODUCTION

The general theory introduced in the preceding chapter furnishes a framework for the examination of particular problems. First, we want to rederive the kinetic equations that were previously set forth for simple situations (essentially the dilute gas and Brownian motion), using a less

fundamental point of view. Second, we would like to consider more difficult

cases (e.g., the dense gas) or, at least, to study the corrections to the simple ones (e.g., modification of the Boltzmann equation at finite density). Questions of the first category can be considered to be answered, and this chapter describes the answers. However the second part of the program is still at a rather early stage, and it involves enormous difficulties, First, there are technical problems, which we do not intend to treat in any detail. But

more fundamental difficulties have appeared: it turns out that the simple limiting cases do not correspond to the leading terms in series expansions. We give an example of this phenomenon in the next chapter, where we show that the transport coefficients cannot be expanded in powers of the density n: these coefficients are not analytic functions of n near n =0; therefore, an order-by-order computation of the corrections to the Boltzmann result is meaningless and must be replaced by more sophisti-

cated, less systematic, methods,

221

222

Simple Applications of the General Theory

This chapter considers only the “simple” problems. Section 2 derives the Landau equation—that is, the kinetic equation of a weakly coupled gas; although this is a very unrealistic model, it has a definite pedagogical interest and plays an important part in the kinetic theory of fluids of classical charges. Section 3 deals with the dilute-gas limit, that is, with the Boltzmann equation. Finally, Section 4 describes the microscopic theory of Brownian motion (the Fokker—Planck equation).

2

THE

LANDAU

EQUATION

Let us consider a classical gas, as described by the Hamiltonian (VI.1). We

want to derive the kinetic equation corresponding to the situation in which the interactions are very weak. In the spirit of perturbation calculus, this amounts to expanding all quantities in powers of A and keeping only the lowest-order terms. Supposing that the potential is finite everywhere and has a finite range ro, we expect this approximation to have meaning when this potential energy is small compared with the kinetic energy

AV(O) « mio

(IX.1)

A crude dimensional argument convinces us that this is also the condition for the particle trajectories to be dominated by the free motion (corresponding to the kinetic Hamiltonian

Hy),

that is, for the deflections in a typical

collision to be small. Indeed, if the velocity change in a binary collision is Av, we have small deflections if Av«(v)

(IX.2)

We can estimate Av from Newton’s law Av= £ At m

(1X.3)

where an order of magnitude of the force F is aa Vir) wA v(0)

F= ano

ro

(IX.4)

while At is of the order of +,, the duration of a collision. Remembering that 7 *fo/{v) and substituting these estimates into (IX.2), we indeed recover

condition (IX.1).

The Landau Equation

223

Of course this dimensional argument makes sense only for smooth and finite potentials and does not apply to a realistic neutral classical fluid, where there exists a strong repulsion at small distances, causing a large deflection in the trajectories. The Landau equation therefore corresponds to a very unrealistic model, and we derive it here mainly for pedagogical and historical reasons. In one situation, however, it plays an important role: it is the case of the plasma, that is a fluid of charged particles interacting through the weak and longrange Coulomb forces. Because of this long range, the collisions are most efficient at large distances, where the potential is weak. But a new difficulty appears: the perturbation expansion fails to converge because many-body effects are important and lead to the screening of the Coulomb potential at large

distances.

Balescu,

Lenard,

and

Gurnsey

have

shown

how

these

collective effects must be incorporated into the frame of the Landau equation, and the resulting kinetic equation is important in plasma physics.* To obtain the Landau equation we start from the Markovian approxima-

tion (VIII.60-62) to the generalized kinetic equation. This Markovian form applies in the weak-coupling limit because even though the duration of a collision

1,ro/(v)

is independent

of A,

the

superscript L stands for Landau) goes to infinity as collision term

(VIII.62)

relaxation

is (at least) proportional

time

Tra

(the

A? when A > 0, since the

to A”, and

74, can be

estimated by writing 4,9, = @ 0.

Let us thus assume that the collision term can be expanded in powers of A

near A = 0; that is, let us write

B(vilpi(e)) = A? EW, |py(1)) + OA?)

(IX.5)

where €°” is obtained by replacing Ly by its “unperturbed” (A =0) expression Ly in the exponential. Remembering that PySLyPy = Py =

LiPy = 0 [see (VIIE. 17-19)] we discover that the projectors appearing in the combination (1 — Py) disappear identically. From this we get the result 6

vi \01(0)) = -[

x ‘0

dt lim i dv, --+ dvy PrdLye 8 6Ly

ry I gil¥a3 2)

(IX.6)

Here and in the following text it is useful to work with a Fourier space language. For any function or operator A(x, ) defined on the N-particle *See Balescu (1964),

224

Simple Applications of the General Theory

phase space, we introduce the following Fourier transform:

WA @It) DIE) ==e on |f

ae,---dewtvexp|—i exp(-i ¥Y ke Ka -1s)ote

ar

A

x Athy sot

Vises

N

¥n) exp (i z Kits) bel

(IX.7)

where we use the abbreviation f=k,, ko, ..., ky.

Assuming as usual periodic boundary conditions in a cubic box of volume

Q, the wave numbers k, (and k;) take the discrete set of values

2 k= 917M.

(IX.8)

where n, is a three-dimensional vector with integer components. In the thermodynamic limit (where

> 00), the discrete set (IX.8) tends toward a

continuous spectrum, and any sum over this set becomes an integral as follows:

2 xL> 8x f dk,

(IX.9)

Ka

The objects (£ A (v)|t') closely resemble the matrix elements of the quantum

mechanical Hilbert space; in particular, they obey the usual rules of matrix multiplication:

HA OBO) =F CAE

BO))

(IX.10)

This Fourier space language has a number of advantages. For example,

from the definitions (VIII.12) and (IX.7), we see at once that

PA (er, ») = (0|A (@)|0)

(IX.11)

that is, projecting with P,, is the same as taking the diagonal matrix element

with the “vacuum”’ vector

[0)=[k, =0,k,=0,..., ky =0)

(IX.12)

Moreover, the matrix elements of Land SL, are easily computed from the definitions (VIII.5-6), and we get

qe=(F ke -v.) fats N

N

a=1

bel

Pa

0x.13)

and

(letwit)=

£

beanwl

(srr)

(IX.14)

The Landau Equation

225

Here

;

2

1/a

5

i

(HBL (ab)] IE)ge = —5 (a — a) View ea (2-2) Stee sien A, a

.

(IX.15) where

V, is the Fourier transform of the interaction

Y= { dre™™ V(r)

(IX.16)

(In fact, since the potential is spherically symmetric, V, depends only on the length of k.) In these expressions, we used the well-known representation of the Kronecker delta function

buy I dre’* “

(IX.17)

oO

Looking at (IX.13-15), we notice that Lis diagonal in Fourier space: its

matrix elements are zero unless k, =k; (for every

a =1--+ N). As for Ly,

it is “nearly” diagonal: only two wave vectors, corresponding to two interacting particles, can change. Note also that as a result of the translation

invariance of the system, all matrix elements contain a Kronecker delta

expressing the conservation of the sum of the wave numbers:

(IX.18)

Lk =Lk,

Since to calculate the collision term (VIII.62), we are interested in matrix elements between |0)-states, and since the nondiagonality of the matrix elements of 5Ly is small, we usually encounter intermediate states [in

products of the type (IX.10)] where only few wave numbers differ from zero. The following convention then makes the notation simpler: for a vector Ik, Ko, ...,ky),

we

only

write

the

wave

vectors

that

differ from

zero,

indicating the particle label as a subscript; for instance, for a state where k,; =k, k24= —k and all other wave numbers are zero, we write simply |k3, —kz4), instead of |0, 0,k,0,..., —k,0,..., 0). Let us now apply these definitions and conventions to (IX.6). First, we

remember that we study a reduced distribution function; the same argument as that which leads to (VIII.34) implies here that

av, +++ dvy Y SL... bra

=| av, ++ dvy

N

Y SL)...

(IX.19)

ba?

That is, only the terms in 5Ly that involve particle 1 remain, and we arrive at

226

Simple Applications of the General Theory

the following expression:

Cv, |e1()) = -{

dr tim { dv,++-d¥x

0 N

x ¥ EY CO6Lo” Ik, —k,) exp [-ik - (v,—vs)r] be2k

x(k, —k,|6L°(0)

N

T] give; 8)

(IX.20)

ael

Now, all particles 6 = 2 - - - N play the same role and their coordinates and

velocities are dummy variables; therefore, the sum over b gives (N—1)=N

times one of the terms, say b = 2. The integrals over v3 - - - vy are then trivial to perform, because of the normalization of ¢,:

(1X.21)

(Va; t)= 1

{ Ava

Finally, we go to the thermodynamic limit, using (IX.9), With the expression (1X.15) of the matrix element of 6L"”’, we arrive at €°

,

|e)

n

= Pl

.

ér| dv,

| dkkV,

a

- ay,

eye aa oxi; destvas 2) eens kV: (2-2 dV; Ve

x

(IX.22) where V)> = V, — V2. In (1X.22) the integral over + cannot

be done

before

the k- and v2-

integrals. However if the function of 7 obtained after integrating over k and

v, is assumed to be integrable—that is, if (IX.22) has any meaning at all—we

are allowed to introduce a convergence factor, by replacing exp (— ik ¢ v,2T) by exp (—ik+v,.7— 7), with « going to 0,. at the end of all calculations. Then the time integral can be performed at first. This mathematical trick can be rigorously justified by the method of Laplace transforms sketched in Appendix F. More intuitively, it should be correct if our guess applies

(Chapter VIII, Section 4), that the time integral of G converges over times of the order of the duration of a collision 7,; for such short times, the

convergence factor e “ cannot have any effect when ¢ goes to zero. The integral lim {

270+

rd Ig

dre

™" =

lim -

er04 IX TE

= 175_(x)

(EX.23)

is known as the “function” 6_(x) (multiplied by 7r). It is a distribution and

The Landau Equation

227

only takes a meaning when it is integrated over x. We have* 78

.

€

-

(x)= lim xe?

i lim wae

(IX.24)

or eof 1 m5_{x) = 75(x)— ia?)

(IX.25)

where 6(x) is the Dirac delta function [see (I-52)] and the principal part P(1/x) is defined as follows: for any smooth function f | "ds A)

Fox = lim [f

er0-Ll,

: ax)

+ [ af)

(IX.26)

a and b being arbitrary positive real numbers.

Substituting (IX.23, 24) into (IX.22), we notice that the principal part

gives no contribution, since it is odd with respect to the change of sign of the

integration variable k. Thus we are left with the (real) Dirac delta contribution, which yields our final result, the Landau kinetic equation

Xm

a O1(¥)3 8) =A]

dv, | dkkV, =

x Pil¥i; f@r (v2; 1)

a

1 78(K+ Via)kV;,

)

(2

a

1 2)V2 (IX.27)

This is one of the simplest kinetic equations of irreversible statistical mechanics. Comparing it with (II.44), we see that the Landau and the Fokker—Planck

collision terms are similar; however the former is a non-

linear functional of the d.f. g, and depends on the velocity in a more complicated way. We do not examine the properties of the Landau equation in any detail;

indeed, besides the important fact that it satisfies an H-theorem, similarly to

the Boltzmann equation (Chapter V, Section 5.3), not much is known about it. Even its linearized version, obtained by writing p,= {+ 69, and retaining only the linear contributions in 5g,, has complicated spectral properties that make it harder to solve than the linearized Boltzmann equation with strong repulsive forces. More important, even the derivation of the H-theorem does not tell us anything new because the Landau equation is nothing but the weakcoupling (or small-deflection) limit of the Boltzmann equation itself. This was in fact the basis of Landau’s original derivation and is formally proved in * This can be found in any modern mathematical textbook; see, for example, Dennery and Krzywicki (1967).

228

Simple Applications of the General Theory

Section 2, when we rederive this Boltzmann equation. This may seem strange at first because we did not assume the density to be low in the present calculation; however the assumptions at the basis of the Boltzmann equation

(Chapter IX, Section 4) also apply here. (i)

(ii)

By restricting ourselves to the A?-contributions, we have only taken binary collisions into account, as is clear from the quadratic dependence on ¢, and from the factor n in front of the collision term. These collisions are considered

Markovian approximation.

to be instantaneous,

leading to the

To make this point clearer, let us remember that the relaxation time in the

Boltzmann limit has been estimated as [see (IV-191)]

Tre = (nrp(v)) |

(IX.28)

and this time is the average time between two successive collisions, because hard collisions bring the system toward thermal equilibrium very efficiently. On the contrary, in the low-coupling limit every collision produces only a small deflection, and there must be many collisions to induce the system to relax. Writing as usual 4,9; ~ — ¢,/7% to obtain a dimensional estimate of

the relaxation time ri, in the Landau approximation and taking in the collision term of (IX.27)

we arrive at

V, =r3V(0)

(IX.29)

T= E Tre

(IX.30)

where the “efficiency” E of a collision is AV(0) i (2)’ ea (ACO. as (——

mioy'/2)

~\io)

IX.31 (3)

As expected, in the weak-coupling limit, the time between two collisions

remains the same as in the Boltzmann equation but the relaxation time rey goes to infinity, as A ~, because E> 0.

3

THE

BOLTZMANN

EQUATION

We now turn to the case of a dilute gas, where the interactions are arbitrary but the density goes to zero: nr}>0. We want to show that the generalized

The Boltzmann Equation

229

kinetic equation then reduces to the Boltzmann equation for a homogene-

ous system, that is, to (VIII.44). Here again the duration of a collision 7, ~rpo/{v) is expected to be much smaller than the relaxation time 7,.) = (r2n(v)) |, since 7./T:2: is precisely nrg. We are therefore allowed to start from the Markovian approximation (VIII.60-62), in which we want to retain the lowest order in n. We examined

this density dependence in Chapter VIII, Section 3. The leading contribu-

tion is obtained from (VIII.26, 35, 43, 61), and we get

4, ey); 1) =n} (v,|9,(t)) + O(n’)

where ¢

_

n€(vi\e())=—-|

“ 0

(IX.32)

4s delim Y | dv.-++ dvy ©

x PyASL

bs2

exp[—i(1—

Py)L O's]

N

x (1= Py)adL TT] oylvq;6) al

The

complete

two-body

Liouville

operator

(1X.33)

L“°”=L00+L®+,6L"”

appears in the exponential. As in the weak-coupling case, we can simplify this expression by noting the following. (i) (ii) (iii)

Since all particles 6 play the same role, we have (N- 1) = N times the same term, say for b = 2. The integrals over v3: -- vy then yield trivially 1 [see (IX.21)]. Both

factors

(1—Py)

can

be

dropped.

In the

Landau

case,

they

disappeared identically; here, however, they become negligible only in the thermodynamic limit. To show this, we first note that only the relative distance between the two particles 1 and 2 appears (through 6L).

It is then natural to use, instead of r, and r, the coordinates rtr,

55 R=02, +

Vo a,

r=r,-r, =r,-

g=V,-¥

(IX.34)

For later use, note that the two-body Liouville operator then becomes [see (VIII.5, 6)]

wy .8 a LY) +10? = ~iNo + sp igs ga iPa2L9VO

or

a

Og

(X.35)

(IX.36)

Simple Applications of the General Theory

230

{the first term in the right-hand side of ([X.35) will never contribute,

since we are dealing with a homogeneous system]. The action of the

operator PxySL"” on any Q-independent function f(r, g, Vc) then gives

PSL"

Fr, g, Va) =a J dR J dx HOV . =f g, Va) 3

(IX.37)

72\__, =o=o( ra) 20

since the integral over R gives ©, while the integral over ris cut off ata distance of the order of the range of the interparticle force. Hence

PySL" can be neglected compared to 6L“” itself. Of course we should not conclude that PyéL°” is always negligible: this would make (1X.33) identically zero! There, however, the OQ”!

factor from

(1X.37) is compensated by the factor N from the sum over particles b. This is a (trivial) example of the dangers of the compact formal methods we are using in this part of the book.

The collision term (IX.33) now becomes in the thermodynamic limit n}(v, |e)

= -n|

0

dr| dv | dr ASL! @ th"

x Livi; Nei; 8)

(IX.38)

The second operator ASL"? has been replaced by the total Liouville

operator L",

since L{? and Lo’ give zero when acting on a space-

independent function

(LY +LD (v5

Nei(va;

=O

(1X.39)

Let us note that the collision operator (IX.33) differs from (IX.6) only in

the replacement of L”) in the exponential by its A =0 value. This was

indicated in the preceding section: the Landau collision operator is the weak-coupling approximation to the Boltzmann one. To show that (IX.38) is identical to the right-hand side of the Boltzmann equation (VIII.44), we have to go through a series of formal transforma-

tions. Discarding mathematical evaluate the time integral

co

0

-iLar “L% drew*

which yields

=[

2

0

subtleties, we assume

4 pe arith" ar

that we can first

= Ti+00 Nim i(e EP- 1)

(IX.40)

(¥45 Depil¥a; 8) |g (t)) =-in | dv, | dr ASL" Jim, eT E? Tig, nv, (IX.41)

The Boltzmann Equation

231

The term (—1) in (1X.40) has disappeared because the integral over r of

6L? is zero.* Let us now

assume

that (IX.41)

can be handled

as if the interaction

potential V(r) had a strictly finite range r,,,,. More precisely, noting that

5L

involves the force —aV(r)/ar, we write

| dr 5L°)---=

lim Frmax

00

1 artilim oN1, | av' | dr, dv'f {exp{-i(1 — Py)L’r\(1— Put Hf” dr’ exp[—-i(1—

Py) L“(7-7')1 — Pdf

x exp [—i(1—Py)L%7"J1- Pyyst'} TiUy eva) (IX.74) and

Xe= |

dr lim ar ow |

dv’ | dr, dt'f

if’,

xtf

ilo

;

dr’ exp[-i(1—

Py) L’(r7’)

a

— Pr)

or,

N

xexp[—i(1— Py)L/r'(1 — Py)8L" TL 91%.) ae2

(IX.75)

The isotropy of the fluid implies that both these tensors are proportional to the unit tensor Xy=LgU Xy= fg (IX.76) If we can show that they are related as follows fg = 2kp Tn

(IX.77)

then the Fokker—Planck equation follows (cf. (I1.44)]: on

Note

< -( kgT @ ) git M oy, Jy, yt B= = Mw,v,

that the present derivation

furnishes

3

+ Oy)

the microscopic

. (IX.78) expression

(IX.75) for the friction coefficient fy.

It is not surprising that (EX.77) is indeed correct: Chapter II indicated that this relation is required to ensure that ¢,(v,; f) tends toward the equilibrium Maxwell—Boltzmann d.f. when t> 00. However the formal proof of (IX.77)

involves rather complicated operator algebra; we sketch it in Appendix G. Asa by-product, we show there that the friction coefficient can be cast in the elegant form 1

f= 3kpT i

dr lim (f(r) + 1(0))7"

(EX.79)

The Fokker-Planck Equation

239

where [see (IX.72) and (VII.31)]

Ka)=e "Ff

(IX.80)

is the total force exerted at time ¢ by the fluid on the 8-particle located at some fixed point r,. The bracket (oc ope

dt’ dv'--- p74

(IX.81)

represents the equilibrium average over the fluid distribution evPHs P= ta

pp Jdt' dv’ ef

(IX.82)

in the presence of the Ssparticle atr,: at the Hamiltonian Hy is

Hy=

N

MDa

an2

2

(IX.83)

braa2

The analogy of these results with those of the stochastic theory is very

striking (cf. (IX.77,

79) with

(I1.43, 25)].

However

the meaning

of the

average is very different; in addition, (1X.77) is now proved, not postulated. The microscopic expression (IX.79) of the friction coefficient is an example of the so-called Green-Kubo formulas for the transport coefficients, which relate these coefficients to the time integral of the autocorrelation (A @A (0))*4 of some quantity A

the central subject of Part D.

(here the force f). These formulas are

X Hard-Sphere Dynamics and Density Expansion of the Collision Operator

1

INTRODUCTION

One of the important questions of kinetic theory concerns the modification of the Boltzmann equation at higher densities. Although this question can be studied formally for arbitrary repulsive forces, explicit calculations have been done mainly in the case of hard spheres, where the purely geometrical nature of the collisions considerably simplifies the problem. However for hard spheres Hamilton’s equations (VII.6)—hence the Liouville equation (VII.15) and the generalized kinetic equation

(VIII.42)—are ill defined because the “potential” of interaction

vo={o

rsa r>a

(X.1)

is too singular. Thus, we must discuss first the special questions raised by the dynamics of hard spheres. Section 2 shows that in this case, the phase-space

density py

satisfies

a pseudo-Liouville

equation

that is singular but well

defined. Since the formal properties of this pseudo-Liouville equation are analogous to those of the Liouville equation for smooth potentials, there is no 240

241

The Pseudo-Liouville Equation for Hard Spheres

difficulty in deriving from it the generalized kinetic equation that describes the time evolution of the one-particle velocity d.f. in a hard-sphere fluid. Section 3 supplies this derivation, and we start examining the low-density limit of the (Markovian)

collision operator:

to lowest order, we recover

immediately the Boltzmann equation. We also show how proper machinery—the so-called binary-collision expansion—allows the analysis of the first correction: we sketch the derivation of the hard-sphere form of the famous Choh-Uhlenbeck triple-collision operator. If we then consider the next-order term in three dimensions (correspond-

ing to quadruple collisions) or if we analyze the Choh-Uhlenbeck operator

in two dimensions, we discover that these formal expressions are actually ill

defined: this is the microscopic manifestation of the nonanalytic character of the density expansion of transport coefficients, mentioned in Chapter IX. These difficulties are profound and the way out of them is not simple. Section 4 tries to pin down the physical idea underlying the problem and Section 5 sketches how it can be attacked by the present formalism. 2 THE PSEUDO-LIOUVILLE EQUATION FOR HARD SPHERES

Because of the pathological form of the potential (X.1), the Liouville equation (VII.15) has no clear meaning for hard spheres. One possible way to bypass this difficulty would be to work with a smooth potential, for example,

vir)= va(2)" and, at the end of the calculations, take a limit (here n-00)

potential however, To this equation

(X.2) where

this

tends to the hard-sphere interaction. This procedure is awkward, and we prefer a more direct approach. end we have to remember that the formal solution of the Liouville is given by [see (VII.26)]

prt, 05 1) = py(t-» -,5 0)

(X.3)

where r, =1,(r, v), 0, =,(r, v) denote the coordinates in phase space at time

t for a point that is at (r, p) at time t=0. Although the trajectories are impossible to calculate in an N-particle system, it is clear that they remain well defined in the hard-sphere limit, provided we confine ourselves to initial conditions such that for all pairs of particles le; —rl>a@

(all i, f€1,2,...,N)

(X.4)

242

Hard-Sphere Dynamics and Density Expansion of the Collision Operator

Calling the region of phase space where (X.4) is satisfied the physical region

(P.R.), we conclude that for any (x, v) in P.R. the corresponding (t_,, v_,) is

well defined and also in P.R. However the variables r, » in the function py(t, »; t) may take arbitrary values in phase space; in the hard-sphere limit, what is then the value of Pn(t, 0; t) in the unphysical region (U.R.) where at least one of the conditions (X.4) fails to hold? The answer is clearly arbitrary, since no relevant physical questions about a hard-sphere system can involve the unphysical

configurations.* Since we nevertheless wish to work with a distribution PN

that is uniquely specified over the whole phase space, it is reasonable to attribute zero probability to any configuration in U.R. for all time; there-

fore, instead of (X.3), in the hard-sphere limit we take oy

J Pn (ta

ent Bs = {

9-43 0)

(t, v)EP.R.

(1, v)E UR.

(X.5)

Even though (X.5) determines the d.f. py, the functions r_, and v , are

quite complicated, and it is preferable to replace these equations by an equivalent differential equation (with specified initial conditions) precisely as, for smooth potentials, we preferred to work with the Liouville equation rather than with its formal solution (X.3). We first remark that hard-core interactions are instantaneous: in a sufficiently small time interval At, we

never have more than one binary encounter. Therefore, to derive a differen-

tial equation from (X.5), it suffices to consider the dynamics of a two-body

problem; we even can take a one-body problem by eliminating the center of mass motion! The generalization to N particles is then completely trivial. Consider therefore the one-particle d.f. f,(r, v; £) fora particle submitted

toa hard-core potential centered at the origin. Equation (X.5) becomes now

yf

fievi)

fiG-.v-650)

{6

(vePR.

(, vEPR.

(X.6)

We want to write down the explicit solutions r_,(r, vy) and v_,(r, v) for ¢=0.

For any given point (r, y) in P.R., we can define the impact parameter b (Fig. X.1; see also Chapter IV, Section 3) b=r- re

(X.7)

rey a v

(X.8)

v

with

* This property led to a variety of choices, all dicussed and compared in a fundamental paper on these questions by Ernst, Dorfman, Hoegy, and Van Leeuwen (1969).

The Pseudo-Liouville Equation for Hard Spheres

243

collision cylinder Figure X.1_

Geometry of a hard-sphere collision.

If b0, the particle has collided with the hard core, and we can write the following definitions:

(i)

The vectorof impactae (where € is a unit vector), determining the point where the trajectory passing through r with velocity v touches the

sphere:

ae=b+y\~

(X.9)

y= Va?—b?

(X.10)

with (ii)

The time t* needed for the particle to reach r starting from the point of

impact ae:

eater

(X.11)

From the laws of hard-sphere collisions [see (VI.8)], we obtain the following

solution r_, and v_,:

(i)

If b0 and t>¢* (these inequalities define the collision cylinder) v_,=v—2e(e-v)=v' r_, =ae—vi(t—t*)

(ii)

(X.12)

In all other cases inside P.R. r,=r-vt

(X.13)

244

Hard-Sphere Dynamics and Density Expansion of the Collision Operator

With these results, the d.f. (X.6} becomes

fi, v5) = filr—ve, v; 0) {1 - O(a — b[O(7,+ y) — OC) vp} t+(fie-vit—1*), v5 0)—filr—ve, v; 0)]@(a — 5) @(t—£*)O(t*) (X.14) where

@(x) is the Heaviside function (IV.59).

Indeed,

this equation

ex-

presses the fact that f,(r, v; t)=f,(r—ve, v; 0) in the whole phase space except for the following cases: (i)

In U.R., which is characterized for given v by the conditions bn>-y

(X.15)

There we have chosen f; =0. (ii)

In the collision cylinder, where (X.12) is obeyed.

Differentiating (X.14) with respect to ¢ and to rp we use the identity a —O(x) Ox

= 5(x)

(X.16)

to obtain the following expressions:

a filt, vs 1) =[a,file—ve, v; OI{1 — O(a — b) [O(y+ y) - OC yp} +[8.f:(e—v'(t—-£*), v'; 0)-a,fi0r—ve, v; 0)] X O(a — b)O(t - 1*)O(t*)

+[file-vi(t—1*), v5; 0)—file—ve, v; 0)] X O(a —b)8(t— *)O(t*) and

(X.17a)

— vt, v;0 oF 1 = [te —e(a-on'004+ 9A w}

—of (r— ve, v; 0)O(a — b)[E(n4j+ yy) — 8 (ry - yy]

afi(e-v'(t—0*), v5.0)

* [-

ary

af. (r—vs, v3 0)

°

ory

|

xO(a—b)O(t— f*)O(t*)

+[file-v(t—#*), v5 0)— fe — ve, v; 0)]O(a — b) x[{-8(t-*)O(t*) + OC - ese

(X.17b)

The Pseudo-Liouville Equation for Hard Spheres

245

Adding these two expressions, we see that most terms cancel. For example, we have [see (X.11)]

Oe

¥5, fe

aN

a

vi(t—2*), v’'; 0)

a

a hile

LON Pe

v(t—#*), v'; 0)

=—a,file—vi(e—t*), v; 0)

(X.18)

since the derivative with respect to r, is taken at € (hence v’) fixed. We use also the fact that for t>0, @(t—¢*) is 1 when multiplied by 5(¢*):

O(t-1*)8(t*) = 6() = o( 0) =vd(y-y)

(> 0)

(X.19)

We are thus left with the following result:

af i(t, ¥; t)+v- SAle, v; 1) = vO(a-b)[f,—v't, v'; 0)5(r—yy) —filr—ve, v; 0)5(r,+ y)]

(¢>0)

(X.20) The right-hand side of this equation is zero everywhere except at the two points where the straight line passing through r and parallel to v crosses the surface of the sphere. When extended at t = 0, this expression is ill defined, since f, is dicontinuous on this surface [see (X.6)]. But (X.20) is an equation

for f, in P.R. (in U.R. it must reduce to 0 = 0); therefore both points on the

surface must be considered as being in P.R. To eliminate the ambiguity, we simply replace the right-hand side of (X.20) with

vO(a—b[fia-VE+n,), V5 DS(%— yp) —fia—v(t+ 74), ¥; 05+ yy]

(X.21)

where the time ¢ has been shifted by a small positive quantity 7, (which we let go to zero at the end of ail calculations). In the first term in (X.21) it is then clear that r—v'(¢+ »,), with 7 = yj,

traces backward the straight-line trajectory before the collision (Fig. X.1), and we have [see (X.6)}

Aie—Vet a), V5 6G)— wWHAiC-v'n4,.V5 D5(y—y)

(K-22)

A similar relation holds for the second term.

Ae—v(t+ 74), v5 DS + yy) =Ale—vas, vs NS(y+ yy)

(K-23)

A more compact form of (X.21) is obtained by relating the functions f, appearing in (X.22, 23) to f\(r,v; ¢) with the help of the displacement

246

Hard-Sphere Dynamics and Density Expansion of the Collision Operator

operators

a

d,,,

=eXp (-n.v o =)

(X.24)

and

b. =ex [-2« ve “ ue .

2] av

(X.25) .

Such operators are defined by their series expansion; we have

exp(a2)fay= ¥ Ha*)"Kxapeta) P\ aa) Hon! ax! 1 a

(X26) ,

and therefore [see (X.12)]*

exp (-mv-2)

a

BAG

hie v0 = fen, v0)

(K.27)

O=AG, v5 6)

(X.28)

The pseudo-Liouville equation (X.20) thus becomes

a fult, V5 DEVS fil V5 = Ky, fll, ¥5 0)

(X.29)

where the operator K,_ is defined by

K,,. f= 2, O(a — b)6(y—

ybe— 5(y+ yp an. fi

(X.30)

Two properties of the operator K,, are important. First we have K,.K,, =0

(n,, 7/, are arbitrary positive numbers)

— (X.31)

This equation expresses the very simple property that a given particle can collide only once with the hard-core potential; its formal proof is based on the remark that once the displacement operator d,,, has acted on the delta function in K,,, [see (X.26)], the conditions imposed by the delta functions in K,,, and in K,,, cannot be simultaneously satisfied. Second, we notice the following convenient representation K, fi = a | d’e(e- v)O(E- v)[S(r— ae)b,— 5(r+ ae))dy4fi (X.32)

where d’e denotes the differential solid angle associated with the unit vector e. A proof of the equivalence of (X.32) and (X.30) is given in Appendix H. * Note that by convention, the operator /év in b, acts only on the function on its right (not on the factor in the exponential itself [see (X.26}]}.

The Pseudo-Liouville Equation ior Hard Spheres

247

The generalization of (X.29) to an N-particle system is obvious: since within a sufficiently short time only one collision can take place, the total effect of the hard-core interactions is simply additive. Therefore we have

3:0 thy2 Va 2,7 PN = =

a

xF

b>a=t

ga Kon

(X.33)

where

K)py =a? { a e(E + Van) OLE * Vas) X[8(tgs — 2€)bS” — 8(r,, + a) Joy (t, 0; 2)

(X.34)

Here the displacement operator b%°? is defined by

b= exp {~te . vase : (2 -2)]}

{X.35)

To simplify the notation, we have neglected a factor Gea exp (—74¥ap * 0/ar,,) at the right of the bracket: indeed, in later calculations, this factor plays a role only in giving a meaning to two successive factors K in expressions of the type K?

exp (—vae

: ~

ab

\e>

{X.36)

which vanishes, even at ¢ = 0 [see (X.31)]. If we keep this property in mind,

the factor d“*” can be forgotten.

The remarkable feature of (X.33) is of course that the formal structure of

the Liouville equation (VII.15, 16) has been preserved. In particular, we can rewrite (X.33) in a form analogous to (VIII.3). idpy = Expn

(X.37)

Ey =L3,t+6Lhy

(X.38)

with

and N

N

dL = poke Les iz am Equations

(X.37-39)

sphere N-particle df.

define the pseudo—Liouville

equation

(X.39) for the hard-

248

Hard-Sphere Dynamics and Density Expansion of the Collision Operator

3 THE GENERALIZED KINETIC EQUATION FOR HARD SPHERES AND APPLICATIONS 3.1

Formal Results and

the Boltzmann Limit

There is no need to derive the generalized kinetic equation for hard spheres; we can simply translate (under the same assumptions of course!) all the results obtained in Chapter IX, by merely replacing ASL” by 6£”. Thus we obtain

aoi=-[

dr G(v\; t7|g,(t—7)) +B;

(X.40)

where

G(s 7] @(t-7)) = lim | dva+++ dvy{i8(r) PybLy

+ Pyby exp[-i(1~Py)EwrK1—Py)dLn} I] olvas) (X.41) and &

has a definition analogous to (VIIL.40, 23). Comparing with (VIII.43,

22), we see that the only difference with the case of smooth forces is the appearance of a supplementary term involving

8(r) PydLy #0

(X.42)

while the corresponding expression for regular potentials vanishes [see (VIIL.18)]. The reason for this difference is easy to understand: whereas a

soft force cannot give rise to a collision process with zero duration, the instantaneous nature of the hard-core process allows for a contribution proportional to 6(7). The Markovian approximation [see (VIII.60)] reads again with

2,01 = (v4 | e1))

(X.43)

€=- { dr Givi; te.)

(X.44)

Asa first, very elementary application of (X.43) leads to the Boltzmann equation for limit. We again expand @ in powers of the VIII, Section 3, and in Chapter IX, Section

this formalism, hard spheres in density m as we 3. To obtain the

let us show how the low-density did in Chapter dominant term,

The Generalized Kinetic Equation for Hard Spheres and Applications

249

we need to retain only the contributions that involve particle 1 and one dummy particle 5 [see (VIII.34, 35)]; we write them as n@

= n@4

ne)

(X.45)

where we have separated the terms

n@Owv,| 90) =—i lim | dv,-++dvy Co

and

n@"v, |p ()) =-|

ea 0

N

_

N

Y PrdL™ TI eilva; 0)

ba2

a=

(X.46)

ae dr lim | dvz-++dyy Y Pdi C)

be?

Xexp [-i(1— Pr)(LYP +L P+ 6L%)7] _

N

x(1— Py)dL°) TT gi(va; 2)

(X.47)

ael

However, expanding the exponential in (X.47),* we see that 6” involves

at least two factors 6£” [or K”, from (X.39)] separated by free motion.

From (X.31), we know that such contributions vanish for all times; that is,

n@"v, | p(t) = 0

(X.48)

Now all that remains is ¢"“”, which is readily evaluated using, in particu-

lar, (X.34, 39); the result is

nO| 91()) = na? | d€(€ + ¥12)O(E + Vi2) xLeilvis Der(v2; )-—e1(¥135 Delve; t)]

= (K.49)

This is indeed the form of the Boltzmann collision term for hard spheres [see (IV.77, 58)]. 3.2. The Binary-Collision Expansion and the Choh-Uhlenbeck Operator The ease with which the Boltzmann equation for hard spheres was recovered

in the preceding section encourages us to look at the first correction the density expansion of the collision operator

Ean +n? OPH... * See (X.53).

6 in

(X.50)

250

Hard-Sphere Dynamics and Density Expansion of the Collision Operator

Chapter VIII, Section 3 indicated that this n?-term consists of contribu-

tions to @ that involve particle 1 together with two and only two dummy particles (hereafter denoted 6 and c). From (X.44, 41), we find* WO},

l0,(D)) = -|

dr lim | dv,-+:dvyn

lo

es

x{ YD PySE° N

exp[—i(1 —Py (LQ + LG? +L?

=|

}

b¥c#l

+8£°"4 SE + BE) 711 — Py SE a

+8L'"

-

N

+30}

TI ¢:(¥,3t)

a=1

(X.51)

This expression simplifies at once for three reasons. (i) (ii) (iti)

All particles 2--- N play the same role. gy (vq; t) is normalized [see (IX.21)]. PyL§ =0 [see (VIII.19)].

We get

WOU", | (0) = =|

F dr lim ~ N?

dv, -++ dvyP,6L"”

xexp {-i(L$+ (1 — Py)(6L"? + dL) + 8) Jr} X(1— Py (SEO) + 8£°) + SE) Xpilvs; Hei (V2; eoi(vs; 1)

(X.52)

where we have used the abbreviation L3= LYV+LP+L®. The evaluation of (X.52) requires the solution of a difficult three-particle

scattering problem. Mathematically, this calls for the proper handling of the exponential operator in (X.52); we accomplish this by expanding it formally in powers of (1— Py)6L((a,

b) € 1, 2, 3) with the help of the identity

exp [—i(A + B)t] = exp (—iAn) +t {

0

dt' exp [—iA (t~t')]B exp (-iAr’)

+ terms involving two B’s, three B’s,...

(X.53)

This is valid even for noncommuting operators A and B. * In contrast with the general cluster decomposition of Chapter VIII, where the three-particle term G&. was introduced through (VIII.25), there is no need to subtract from (X.51) the contributions where only particle 6 (or c) interacts with 1: in view of (X.31), these terms vanish

identically.

The Generalized Kinetic Equation for Hard Spheres and Applications The proof of (X.53), which

following theorem: if

is familiar in quantum

251

mechanics,

is based on the

U,=exp[-i(A +B)t]

(X.54)

then the following identity holds:

U, = exp (~iAt) +t | at' exp [—iA(t—r)]BU,

(X.55)

oO

To check this result, we differentiate (X.55) with respect to ¢ and we get 8,U,=—-i(A+B)U,

{X.56)

while for t= 0 we have Uy = 1. These two conditions are also obviously satisfied by

(X.54) and determine U, uniquely. The expansion (X.53) is then obtained by iterating (X.55) with U\” = exp (—iAn) as

the ‘first approximation.”

In the present case, (X.53) leads to

exp {—i[L$+(1~ Py)(6L0 + 5L°°) + 6 )}7} 1{* : = exp (-iL$7) +5 { dr’ exp [-iL3(r-r')](1 — Py)

«(£0

+8L

0

+ 6E) exp (—iL$7')

+terms involving two 6Ls

or more

(X.57)

Inserting (X.57) into (X.52), we see that the three-particle process is now

described as a set of sequences of two-body collisions between which the three particles move freely: such a description is known as a binary -collision expansion. Such expansions can be established for arbitrary smooth forces, as well, but it is very difficult to make precise statement about their convergence. Fortunately, in the case of hard spheres, the remarkable

theorem has been proved that no term contributes to ¢

that involves more

than four successive SLs; this means that to get the exact 6”, we must

simply keep in the expansion (X.57) the terms involving zero, one, or two 6Ls.

The proof of this theorem

is tricky.* Moreover,

since we do not

intend to perform any explicit calculation with ¢@, we merely display the

formulas as they arise from the first two terms in the expansion (X.57): the physics of the problem is apparent on these examples. Using (X.31) again, we arrive at

GP = GP) 4 Gu)

* See Sandri, Sullivan, and Norem (1964); Cohen (1966).

(X.58)

252

with

Hard-Sphere Dynamics and Density Expansion of the Collision Operator

m4"

Jou) == |" dtm N? f diva» dvnPyE" x exp (-iL$7)(1 — Py (6L" + 6°)

XpilVi3 eilve; Ner(ys; 2)

(X.59)

and

n? 6" (yp, (t)) = -[° dr lim N? | dvz +++ dvyP,6L"” 0

x : { dr’ exp [-iL&(r-1')]

x {(1— Py)6L" exp (-iL3r')(1 — Py )(8E" + 8 £°)

+(1—Py)6L@ exp (-iL$r'\(1— X o1(¥15;

or(V23

PSE"

Neilvs; 1)

+6£)}

(X.60)

Although they are already considerably simpler than our starting point, these equations are still rather difficult to grasp, and a picturesque representation of the physical processes they are describing is very useful. We use graphs in which the free motion of each particle is represented by a straight line moving upward as time increases; whenever a pair of particles (ab)

comes into contact and interacts through a binary collision 6‘, we break

their straight-line trajectory. For example, the term in (X.59) involving the

combination 6£''”)- - - &£° is represented in Fig. X.2a, whereas the term in (X.60) corresponding to the sequence 6L"?’-- - 6£'°?)- - . 8L"” appears

in Fig. X.28,* Figures X.2 immediately reveal a striking difference ¢® and in 6". In the former each binary collision particle that has not collided before and is therefore lated to the others.t Such an uncorrelated sequence already accounted, however, by the iteration of the

between the terms in involves at least one completely uncorreof binary collisions is Boltzmann collision

operator n@” [see (X.49)]. Indeed, the formal solution of

4,91 = nEMW, | 9,(0)

(X.61)

* The operator 5L®” [see (X.34, 39)] involves two terms: one, called the direct-collision term in which the particles come in with the velocities v, and v,, and the second, called the inverse -collision term, where they come in with the velocities vi, and v;. In detailed calculations [see Sengers (1973) and references quoted there], it is convenient to use a separate graphical representation for each of these terms; however for a general discussion the present graphical picture is quite sufficient. + Remember that the initial condition in (X.59, 60) is the factorized distribution I 1 G1%a5 £).

iN IX

The Generalized Kinetic Equation for Hard Spheres and Applications

253

(a)

Figure X.2 Graphical representation of the binarycollision expansion. (a) A term in €); (b) a term in Gerd,

leads to gilttr)= gilt) t+ nO,

[e(0)

+o7

2!

By,

| CGN)

=

(X.62)

and the last term displayed in (X.62) precisely corresponds to two successive uncorrelated collisions. Hence to ensure that these processes are not

counted twice, we must have

EM=0

(X.63)

On the contrary, Fig. X.26 shows a correlated sequence of binary collisions,

and particles

1 and 2, which collide last, have already encountered

each

other in the first collision. This distinction between correlated and uncorrelated processes can still be made (but is slightly more delicate) in more complicated situations, involving 4, 5, .. . particles, and it leads to the very important property that the collision operator © describes all possible correlated sequences of collisions between any number of particles; that is, the contribution of uncorrelated sequences vanishes. In the proof of this property, a crucial role is played by the projector Q, =(1— Py), as we now illustrate by establishing (X.63). We consider the following factor, which appears in (X.59)

= N*PySL

exp (—iL$7)(1— Py(SL°9 +6L°”)

(X64)

From the definition (VIII.12) of Py and the fact that 5£°°” depends only on the relative distance r., =I —Ts, we get 2

= x f dr, dy, drs, 6L°(ry9)

xexp[-i(LQ2 +L) per

3) + 6E (7,3)

a)a aris BL" 3) a)1 | drs; o£

(eh)|

(X.65)

254

Hard-Sphere Dynamics and Density Expansion of the Collision Operator

where the property PyL{? = 0 [see (VIII.19)] has been used to eliminate the

factor exp (iL $s). A superficial look at this expression might suggest that the last two terms in the bracket, involving a factor 2"', become negligible when the volume becomes large. However this is wrong because these terms (in contrast to the

first two terms in the same bracket) are independent of r,3, which allows fora

free integration over this variable, leading to a compensating factor 2. We get thus I'=0

(X.66)

which proves (X.63).

We leave it as an exercise to the reader to show that for the corresponding factor in @” [see (X.60)]

I"=N?Px6L" x {(L—

exp [-iL(7 —7')]

Py)d£"® exp (-iL$r')(1 — Py) (80? + 8E 2)

+(1-Py)6L?” exp (-iL$r')(1 — Py(SE + 8£)}

(X.67)

no such compensation occurs for the operators Py appearing in the combination Qy =(1— Py): in (X.67), we simply set (1— Py)=1 in the thermodynamic limit. With the help of these results, we arrive at the following formula:

nevilec)=i|

lo

dr limN* | dv2+++dvy Py ey

{[" dr! 8£""? exp[-iL(r—-7')] x{6E

exp (-iL$r')(6L" + LE)

+8£° exp (-iL87')(6E 0? +86£)} +terms involving four ais} eil¥1; Delve;

Ne (v3; t)

(X.68)

This expression can be made more explicit by using the Fourier space representation introduced in Chapter IX, Section 2. These calculations involve nothing new, and we can be very brief. (i)

(ii)

We introduce the representation (IX.11) for Py.

We use the matrix-multiplication rule (IX.10), with the help of (IX.13) and of the analog of ([X.15), which is [see (IX.7), (X.39, 34)]

The Generalized Kinetic Equation for Hard Spheres and Applications

(HSL) =F p(ab)jgrn

_

1

(ab))

ia?

= al

255

ale ) gr

dele: Vab OCE * Vay)

x {exp [—ia(k,—k,) > Jb” — exp fia (ki, ka) €]} XOtetenacen,

(iii) (iv)

I] Suche:

(X.69)

c#ab

This equation defines the binary independent of the volume.

collision

operator

t@,

which

is

We go to the limit of a large volume [see (IX.9)]. We perform the time integration with the help of a convergence factor

as discussed before (IX.23).

Thus we arrive (with the notation explained on p. 225) at

n? 2 Ev, | p(t) “Gyn?

-

{oe

5 lim | a

C2 Ik,

{ dv, dv; 1 rvs)

ke)

1 x [deed eee tk, Helo) 12~ He)

E(k,

(ki l¢

|e)

ks)

(ak

+(~k,|¢2>4

I)

(23)

(hee

1

ews —k,)

i)

(Ke, le10)

1 a a5 (ky, “He"?)0)

1 3)

vv,;=ie)

(k,, “ksleI0)]

+ terms involving four srs} x O(¥1; Depry; Deilvs; f)

(X.70)

which was established here for d =3 but remains valid for any dimension-

ality d, provided the ¢”-operators are properly modified [see (X.99)]. This

rather lengthy result can be shown to be equivalent to the famous Choh-

Uhlenbeck

(C.U.) expression, when

collision term is inserted.*

the explicit form of the four-binary-

* See Sengers (1973) and references quoted there.

256

Hard-Sphere Dynamics and Density Expansion of the Collision Operator

We have thus far considered a system in which all the particles play a symmetrical role. Yet Chapter V demonstrated that it is often useful—in particular for the self-diffusion problem—to consider a system in which at the initial time, only particle 1 is out of equilibrium:

s.1(¥15 0) # GI(v,) gil¥as 0)= Gv.)

(a #1)

(X.71) (X.72)

In fact, we are going to illustrate the further developments of the theory with

the example of self-diffusion. The foregoing results are modified in this case as follows. In the Markovian approximation, we get 9Gs.1(V15 ) = C51

(X.73)

where C, is the generalized Lorentz collision operator. This operator, which is linear, has a formal expansion analogous to (X.50):

Coe nCP tn? CPt:

(X.74)

Here nC‘ is the Boltzmann-Lorentz collision operator [see (V.170)] for

hard spheres, and n?C® is the C.U. correction

VCP

n>

(0 = Om

Jim J ax dv dv,

x{-. he. (¥1; NET (vs)ei(v3)

(X.75)

which is obtained from (X.70) by the mere substitutions

AWVSOD> eas)

and

— gi(¥as Neva)

(a =2, 3). To avoid repetition, dots inside the brackets have been used to

represent the expression put inside similar brackets in (X.70). Needless to say, this generalized Lorentz kinetic equation is valid under the same assumptions and restrictions as in the symmetrical case.

4

THE PHYSICAL ORIGIN OF THE NONANALYTICITY IN THE DENSITY EXPANSION OF TRANSPORT COEFFICIENTS

Recalling the difficulties already encountered with the nonlinear Boltzmann equation (Chapter IV), we should not be surprised that almost nothing is known about the nonlinear three-body collision operator (X.70). In particular, no H-theorem is available, and we can show only that the Maxwellian

Density Expansion of Transport Coefficients

257

distribution is a stationary solution:

Ew

|e4) =0

(X.76)

However we have seen that in the dilute-gas limit, transport coefficients can be calculated from the linearized homogeneous Boltzmann operator (Chapter V), which is simpler to handle. Similarly we expect that transport coefficients in the moderately dense gas can be obtained from the linearized Choh-Uhlenbeck operator or, for self-diffusion, from its Lorentz form. For

example, how should (V.177) for the self-diffusion coefficient D be modified

when the density is increased? The natural conjecture is an expansion of the following type:

slo

D2—-(D+n +n?D? D+...)

(X.77)

where the coefficients D, D®.-- are density independent and the term n'D™ describes the dilute-gas limit, denoted simply D in Chapter V. What is then the microscopic interpretation of the corrections D®, D® .--? A priori, we expect two types of effects.*

(i) (ii)

Instead of retaining only two-particle collisions, we should now successively add triple, quadruple, . . . , collisions. We should take into account the spatial correlations between the particles, introduced by their interactions.

In Part D (Chapter XII), we present the formal support to this intuitive

surmise. Here let us accept it without further discussion; moreover, let us

limit ourselves to the collisional aspect (i) which poses the most serious problems and involves the important physical phenomena. How should multiple collisions affect transport coefficients? Again, let us rely on intui-

tion. We remark that in the dilute-gas limit, all transport coefficients involve matrix elements (taken with respect to suitable functions in velocity space) of the inverse linearized Boltzmann (or Boltzmann-Lorentz) operator: for self-diffusion, we can write (V.177) schematically as

Lp n

u

nc?

(X.78)

* For the other transport coefficients (e.g., viscosity) we should also take account of the

potential parts of the dissipative flows (cf., e.g., (VII.138, 139) for the pressure tensor and its dilute-gas limit (IV.133)); there is no such contribution for the self-diffusion flow, which is of

purely kinetic origin.

258

Hard-Sphere Dynamics and Density Expansion of the Collision Operator

The simplest generalization of this result is 1 DxG

. (X.79)

which should be valid at arbitrary density. With the help of (X.74) and (X.77), we recover (X.78) by expanding (X.79) in powers of n; we get

moreover*

D® c=1 Pam1

(X.80)

and similar formal expressions for the higher-order terms D®, .... Equa-

tion (X.80) and the corresponding expressions for the other transport coefficients (which involve the linearized form of the C.U. operator ¢'”) show that the calculation of the first correction in the density expansion

of transport coefficients requires the detailed analysis of C? and ¢'”

(linearized), It is remarkable that despite their complexity, these operators

can be handled with success, leading, for example, to a numerical expression

of D’, Needless to say, such calculations are very hard to perform.t

One of the main difficulties is the large number of integrals; this is why the same problem was investigated for two-dimensional hard disks, where the formal developments are the same but the integrals are simpler. It came as a big surprise that Dx

C2

=00

(d=2)

(X.81)

with a similar (infinite) answer for the other transport coefficients. It was also found that the same divergence occurs in three dimensions for the correction

D”’ (and the four-body operator C®)

D?xC?=0

(d=3)

(X.82)

Of course, at least in three dimensions transport coefficients can be measured experimentally: they are not infinite! Equations (X.81, 82) merely reflect the inadequacy of the analytic density expansion (X.77). Indeed, it was rapidly realized that the expansion (X.77) should be replaced in three dimensions by

D=2(D

nD?

+n? Inn D+:

-)

(d=3)

(X.83)

* Because we ignored the spatial correlations between the particles, there is actually a contribution to D® that is missing in (X.80). Yet this extra term shows none of the pathologies discussed below. t See Sengers (1973) and references quoted there. + For a detailed historical survey, see Brush (1972).

Density Expansion of Transport Coefficients

259

with a logarithmic term that cannot be expanded around n=0. (The coefficient D"® is difficult to calculate and no explicit value yet exists.*) In two dimensions the problem is even more delicate and largely unsettled; in particular, no experimental evidence is available. However it is generally believed that according to the model considered, transport coefficients do or do not exist. For example, for the case of moving hard disks the conjecture is D = 0; on the contrary, for the so-called Lorentz model, where

one molecule moves among fixed scattering centers, there is a convincing (though not rigorous of course!) calculation of the self-diffusion coefficient.

The result ist

D= (1) [aes (6) ‘(any In (a?n)+ o(n’)| (d =2, Lorentz model)

(X.84a)

and its expansion starts like

D= “(D4 ninn D'?+-.-)

— (d =2, Lorentz model) (X.84b)

The corresponding expression in three dimensions is D

—44(kal ) 12 [wa°n 3

\2arm

+c,(a°n)? +0.215 +++ (a°n)? In (a°n)+---J! (d = 3, Lorentz model)

(X.85)

where c, is an unimportant numerical coefficient.

To give a theoretical foundation to these unexpected and important results, it would be hazardous to jump too soon into the mathematical formalism. The problem is difficult, and it is easy to get lost in the calculations. Therefore it is more appropriate to seek first some physical insight into the origin of these divergences, in the remainder of this section. Section 5 sketches a means of casting our qualitative picture into mathematical terms. Since we have already learned that dimensionality plays an essential role in the problem, let us reconsider the physics of the C.U. three-body collision operator for an arbitrary dimensionality. Section 3.2 indicated that this operator gives the frequency of a correlated sequence of binary events between a given particle 1 and two arbitrary particles 2 and 3. Suppose that at time zero, a collision (12) occurs; the average frequency of such a process is [see (IV.191)}

Yrea=a* 'n(v) *See Pomeau and Gervois (1974) and references quoted there. t Van Leeuwen and Weyland (1967, 1968).

(X.86)

260

Hard-Sphere Dynamics and Density Expansion of the Collision Operator

In the later time interval between 7 and ++ dr, the probability of collision between 2 and an arbitrary particle 3 is »,., dr.* However if we impose that

this process (23) be such that 1 and 2 collide again in the future (this is the

condition for having a correlated sequence: Fig. X.26), the velocity of 2 after the process (23) has to point out into the solid angle under which 1 is seen from the location of the collision (23). For large 7 this reduces the probabil-

ity of these collisions by a factor a7 '/((v)r)* ' (the separation between 1

and 2 at time 7 is taken as approximately (v)7), and from this it becomes

(a*"')n dr

(X.87)

Multiplying (X.86) by (X.87), we see that the frequency of a three-body collision where the intermediate binary collision occurs in the time interval (7,

t+dr) is

(at

' Pn? dr

(X.88)

To obtain an estimate of the total three-body collision frequency, we have

to integrate (X.88) over 7, starting from a lower limit of the order of a/(v)

(where our dimensional analysis becomes too rough). This yields Md-, f°

2c.

nes

(v)?

2

{Lvecs a 7

X.89

(X.89)

Comparing with the two-body frequency (X.86), we obtain the ratio 22)

nC;

ac

oo

dz

iT

an j eT

(X.90)

where we have introduced the dimensionless time

i (0) a

(X.91)

The integral in (X.90) is finite in three dimensions, and we recover the property asserted in Chapter VIII: three-body collisions appear as corrections of the order (a’n) to the Boltzmann limit C°. On the contrary, for d=2, we obtain

=a

(d=2)

(X.92)

and the integral multiplying the factor (a7) is infinite! From (X.80), this *In this dimensional analysis, we can ignore the distinction between the probability of a collision in the small time interval dr and the probable number of such collisions.

Density Expansion of Transport Coefficients

261

leads in turn to the announced result D®=0o, Since we are going to compare various diverging integrals, it is convenient to replace the infinite upper limit of integration by a dimensionless time 7, which is arbitrarily large but finite; integrals diverging at infinity are then replaced by large T-dependent quantities that are easier to compare. Thus we rewrite (X.92) as n?

A

2)

Toye

=a *n}

Hann?

(d=2)

(X.93)

Now it is easy to understand why this divergence has no physical meaning. Indeed (X.93) shows that it comes from configurations in which particles 1 and 2 (refer to the specific example of Fig. X.25) travel a long distance (R ~aT > ©) before recolliding. But the three particles 1, 2, and 3 are not isolated; during such long paths, particles 1 and 2 will encounter other particles i (i # 1, 2, 3) which will scatter them away from each other. The

foregoing description is valid only as long as particles 1 and 2 can move freely, and this time is roughly of the order of the time between two collisions, that is, the dimensionless relaxation time

Fei=(a?n)'

— (d =2)

(X.94)

For 7 > 7,1, the other particles in the system will act as a screen, preventing the recollision. Thus it is reasonable to assume that the integral in (X.93) has a natural cutoff at 7~7,.,. We get then, instead of (X.93) 242)

mC

owt

an

{

(azn)

rl

lye

ar —a’n In(a’n) 7

(X.95)

which explains the logarithmic dependence in (X.84). A similar analysis prevails for the four-body collisions (corresponding to

the super-Choh-Uhlenbeck (S.C.U.) operator C®). A process of the type

depicted in Fig. X.3 leads to a contribution of the order of Cc?) rou

which diverges as In T for

ue =(a4n)

j

pr?

(X.96)

d=3 and immediately leads to (X.82). Intro-

ducing the cutoff at 7,.;~(a4n) ', we obtain

CO lemon =_ ac

in agreement with (X.83).

—(a°n) In (a?n)

(d= 3)

{X.97)

sg

262

Hard-Sphere Dynamics and Density Expansion of the Collision Operator

Figure

X.3

A

contribution

to the

super-Choh-Uhlenbeck operator.

Before closing this qualitative discussion, let us make a few more remarks.

(i)

The argument leading to (X.95) seems reasonable but it leaves no room for the possibility indicated earlier that transport coefficients might not exist in two dimensions. It is based on the assumption that the propagation of disturbances in a fluid is cut off at distances of the order of the mean free path A,.1~47,¢;; yet Chapter V demonstrated

the existence of transport modes that are weakly damped and propa-

gate over distances

much

larger than

A,,.). These

modes

generally

invalidate the argument leading to (X.95) for two-dimensional systems: the divergence remains because the screening effect due to the

particles in the fluid is not strong enough. Only for the Lorentz gas

is our argument valid in two dimensions because in this model it can be shown that the transport modes are not important; there the result (X.85)

holds true. For three-dimensional

these transport modes

(ii)

systems it turns out that

are not efficient enough

to invalidate our

qualitative analysis; this is checked in the next section.

Our discussion has neglected higher-order recollision processes; in the

case of C®, for example, these correspond to terms involving four binary-collision

operators

[see

(X.68)].

A

dimensional

argument

shows however that these terms are less diverging than the ones we have considered, and can be ignored. For instance, it is easy to show that the graph of Fig. X.4 leads to a correction of order (a

(iii)

a\() n)

d#

; Pay

(X.98)

which converges even for d = 2. From the estimate (X.96) for d = 2, we see that the divergence of (oS) is worse than that of Cc, therefore we must remove this divergence

Introduction to the Mathematical Analysis of the Divergence and Its Removal

2

1

Figure

X.4

263

3 Example

of

a

four-binary-collision contribution to n?C??,

too, and the same is true for C?, C®, .... Similarly, in three dimensions we find worse and worse diverging terms in the sequence

Cc”, C®, C®,.... The next section reveals that the mathematical

formalism allows us to eliminate these badly diverging contributions altogether. 5

INTRODUCTION TO THE MATHEMATICAL ANALYSIS OF THE DIVERGENCE AND ITS REMOVAL*

5.1

Further Analysis of the

Choh-Uhlenbeck Operator C? Although the C.U. operator is well behaved for d=3, the discussion of Section 4 indicates that it diverges for d= 2. Moreover, even for three dimensions, a detailed analysis of this operator is very suggestive.

Limiting ourselves to the Lorentz problem, we start from (X.75), which is

equally valid for d = 3 or 2 (provided of course we properly interpret dk « : -

as d°k +--+ or d’k - --, respectively); the only difference lies in the explicit form of the matrix elements t. In three dimensions these elements are defined by (X.69), whereas for d =2 they are given by

OP) =a { de (€ + Van) O(€ -Vay) {exp [—ia(ki,~ kg) + €] 2” —exp[ia(ki—k,) + €T} She eats ts

II , oe:

ca,

(X.99)

where € is a unit vector in the plane and de is its differential solid angle. *This section may be omitted on first reading.

264

Hard-Sphere Dynamics and Density Expansion of the Collision Operator

A crucial simplification is suggested by the dimensional analysis of Section 5: if that picture is correct, the divergence comes from collision processes extending over large distances, or, in Fourier language, from small wave

numbers. In examining (X.75), we then limit ourselves to the part of the k-integral with k lo>Aakjtayp*—e)

0, for any dimensionality.* Similarly, for d=3, the hydrodynamic

contribution (X.141) involves the integral

=n

(d =3)

(X.142)

in the small wave number region k « A;.1, and is therefore finite. On the * What we cannot prove easily is that this finite expression behaves as n7~!

Inn.

274

Hard-Sphere Dynamics and Density Expansion of the Collision Operator

other hand, for d = 2, we get the diverging integral*

edhe _ «0

(d=2)

k?/n

(X.143)

As discussed earlier, the long-range propagation of hydrodynamic perturbations prevents the efficient screening of the divergence of the C.U. operator in two dimensions. We are thus in a bad situation with two-

dimensional

systems

because

there

does

not

seem

to be

any

physical

mechanism capable of eliminating this divergence. Therefore it is believed that the (Markovian) collision operator—hence the linear transport coefficients—simply do not exist in two dimensions (see also Chapter XII, Section 5.4).

This discussion indicates that the analysis of the ring operator and the justification of its use are in a rather primitive state; it may even be that this approach is not the proper way of tackling the divergence problem. Yet even so, this operator would remain an important object of kinetic theory because as we see in Part D, it also plays a crucial role in the understanding of the slow decay of the time-dependent correlation functions. To avoid leaving the reader on a deceptive note, let us briefly return to the Lorentz

model,

where

the

situation

seems

rather

clear.

Because

the

scattering centers are fixed, the classification of the diverging contributions to the density expansion of the collision operator is slightly different from the one discussed here: some graphs, which are not of the ring type, also lead to badly diverging terms. Yet the resummation procedure is very similar to the one presented earlier; moreover the resulting expression can be evaluated explicitly and confirms our qualitative arguments, except for one point: even in two-dimensions we find no divergence because the hydrodynamic mode of the Lorentz gas does not contribute in the k = 0 limit [see footnote after (X.143)]. The final expressions were displayed in (X.84, 85). It is hoped that an analogous expression will emerge from the detailed analysis of the ring operator for moving particles in three dimensions.¢ Finally, despite their theoretical interest, these logarithmic terms seem numerically small and hard to detect by the analysis of experimental data and computer simulations. t Equation (X.143) applies only if the factors in (X.141) on the left and on the right of (Ak, +A,"~ 6) | do not vanish when k > 0: Thisis true in general, but in the equivalent Lorentz gas calculation, one can show that these factors go to zero with k, preventing the divergence (X.143). +See Van Leeuwen and Weyland (1967, 1968); Hauge (1975). + Though ail the calculations in this chapter deal with hard spheres, the fact that the divergence arises from long-distance collision processes, for which the detailed nature of the interaction should not be important, makes it clear that the same effects occur with more general interaction laws, though the theory is slightly more involved.

D Time-Dependent Correlation Functions

XI The Correlation-Function Formalism

1

THE

CASE

OF

BROWNIAN

MOTION

Before presenting a microscopic analysis of time-dependent correlation functions, it is very instructive to consider these functions from the point of view of the theory of stochastic processes. This approach, partly covered in Part A, nicely illustrates how these functions naturally enter into many problems of nonequilibrium physics. 1.1 Derivation “a la Einstein” of the Green-Kubo Formula for Self-Diffusion

Let us consider a Brownian particle immersed in a fluid of light molecules. The random-walk theory of Chapter I led to the result that the mean square displacement of the 8-particle is given by [see (I.108)]

(Ax7,)=2Dt

— (t>0)

(XL1)

where the displacement in the time interval ¢ is now denoted

Axi, = x(t, €)— x10, €) [To simplify, we work

in one dimension;

the random

(X1.2) coordinates of the 277

278

%-particle

The Correlation-Function Formatism

are x,(t,€), v),,(t, €)]. The

phenomenological

cient D thus is such that

diffusion coeffi-

2

D= 120 tim S212 = Dt

(XI.3)

We now can rewrite the right-hand side of (XI.3) as the time integral of the velocity autocorrelation function of the 8-particle [see (III.23)]

R(t) = (0,,x(1)01,,0))

(X1.4)

obtaining in this way the so-called Green-Kubo formula for D: p=|

dr R(r)

0

(X1.5)

The proof of (XI.5) is easy. From the definition of the velocity vx, we have

(XL6)

Ax. = { at’ 0). {t', €) °

and therefore

t

t

(Axi) = { dt’ { dt"(v,,.(t'Y0,,.(t")) 0

0

(XL7)

Since the random process is stationary, we can write [see (II].23)]

iC)oreY= REPS R12)

(XL8)

and splitting the time integrals into the domains ¢' £3/2M. The constants c, and c_ are determined by the initial condition (x,(0, €), 1,,(0, €)):

HEU, +

= 01.2(0, €)

ron

oa ai,

€) + 01.0, €)

ron

(X1.22)

The analysis of the process characterized by the statistical properties

(X1.17, 19) is far-reaching, but we are mostly interested here in the average response (x,(t)). Taking the average of the solution (XI.20) with the help of (X1.17, 19), we see that only the term proportional to the driving force

remains (clearly, when this force is absent, (x ,(t)) is zero], and this allows us to write

euoy=[

dt’ Xee(t~

VFL)

(X1.23)

282

The Correlation-Function Formalism

where the retarded response function y,,,, is given by

xonl

-

Wa

(Si (Gi) 2m) @} -22)")4]i}

ear

| fe

-col -(Ze+(ga)-ai)")} 2

1/2’

ona

that is, the sum of two decaying exponentials for wo < ¢g/2M and a damped

oscillation for wo >—/2M.

Notice that y,,,,(¢) is only defined for t>0,

although it is sometimes convenient to extend its definition for ¢ 00 with the (real) oscillatory force

F(t) = Fy cos (wt) = Fa(@m te)

(X1.34)

We get Fo tot P(t) _fo3 (e+e+

_ iwF5 4

iwt

a

d

Xero)

9 (OO Ka) te

Fo

foot ~

ior ~

[Rerns(@)~Xeyey(-@) + Xeres(@) 0—2iwt — eyey(—w)€ Plat) (X1.35)

The last two terms in the right-hand side oscillate in time and average to zero over

one

period.

With

dissipated systematically

the help of (X1.32),

P(t) = WX",

we

then

get for the power

(w)F()?

(X1.36)

where the bar means the average over one period. Notice that the physical requirement P(t) > O imposes

y;’,,,(@) > 0 for w > 0, which is indeed satisfied

[see (XI.31b)]. Incidentally, there is no power dissipated at zero frequency [¥,,,,(0) = 0]: when submitted to a constant external force, the oscillator stabilizes in the

long term at a fixed position:*

Tim (21 (9) = Xe121(0)Fo= Xi1x,0)Fo

Fo = Mur

X1.37 (X1.37)

(iii) Kramers—Kronig relations. We already noticed that ¥,,,,(g) is analytic in the upper half plane S*. This is a general property of Laplace

transforms

and

reflects

here

the

causality

of the

response:

in (XI.23),

* This is no longer true for a free Brownian particle, which is formally obtained by taking w. = 0. There (x,(1)) Ss t

The Case of Brownian Motion

Xex(f—’)

285

appears only for t—f'>0; this led us to limit the integral in

(XI.26) to positive times—that is, to introduce a Laplace transform rather

than a Fourier transform. This analyticity of X,,,,(z) implies remarkable relations between the real and imaginary parts of ¥,,,,(#) known as the Kramers—Kronig relations. To derive them, consider the integra!

nal) 1G) ily” en date

(X1.38)

with g in S*, Closing the contour (—00, +00) by adding a large semicircle in S*, which

to the integral because ¥,,,,(z')>0 when

does not contribute

z'|> 00, we use the residue theorem. The only singularity is the pole at 3’ = g and we have

(X1.39)

Ges")

1g)=Xane)

If we now let g approach the real axis from above and use the identity [see (IX.23)] lim - | - = 9(+) +ins(x) x

(X1.40)

we get .

.

.

,

_

1

+00

Xeiz(@) = lim funlo—is)=5-9 | .

deo Xeyxilto’) 4 Xanlo)

ow

3

(X1.41)

Taking

the real and imaginary

parts of this identity, we arrive at the

Kramers-Kronig relations

ialoy=Zto[” arial co

alot

o-w'

dw Fale”

(X1.42)

wet These relations are important: from the knowledge of ¥,,,,, they allow us to calculate ¥,,x,, and vice versa.

Fluctuation -dissipation theorem. We illustrate next a famous theorem (iv) that connects the dissipation resulting from the action of an external force to

the spontaneous fluctuations at thermodynamic equilibrium.

286

The Correlation-Function Formalism

Consider the correlation function (x,(#)x,(0)) in the absence of external

force (F(t)=0). From (XI.20-22), we have xi i(t,(t,€)= €)

r.x1(0, €-)—01,.(0, €) ent —rx1(0, €)+01,.(0, €) — : a

ore

+[ dt Nex (t OC, €)

(X1.43)

Multiplying this expression by x,(0, €) and taking the average according to the statistical properties (XI.18, 19), we get*

(e(O)=_— or

kpT es! ev Grr (SE)

1

(e(OxO)=KeT[ dl xnnlt)

cx.aa)

—@>0)—(KLAS)

In writing the last equality, we have used r,r_=wo and the expression (X1.24) of the response function. If we now substitute the Fourier expansion (see (XI.25b)]

Kualt) =x

1 |

dae

r

Xy,x(@)

(X1.46)

in (XI.45), we can perform the time integral explicitly. With the factor “(t'>0, © > 0) to ensure convergence at infinity, we get

(x(x (0)= 422 tim | dno) 27 e040 ~~ iw

0X14)

We now remark that the function y,,,,(t') defined by (X1I.46) vanishes for t' 0)

O=kaT | Xan(l') de -t

_keT

27

.

620,

ke T

= Faq

[a

lim

eiricn

a

$x +2

7

|

io i@titie ei"

(w)

sie Mt

dw io’ aieytn—@)

(X1.48)

* The correlation function (X1.44) of the process x,(t, €) is not a simple exponential. There is

however no contradiction with Doob’s theorem (Chapter III, Section 3.3) because the process x,(4, €), taken alone, is not Markovian; only the two-dimensional process x (1, €), 01,.(4, €) is Markovian, but in this case Doob’s theorem does not apply in the simple form presented earlier.

Linear-Response Functions and Their General Properties

287

where we have used the variable w’ = —w in the last equality. Adding (X1.48) to the right-hand side of (XI.47), with the help of (XI.32), we have -_

{x1 (#)x,(0)) =—— lim £04

+00

S00

dos’

e

tlaie)t

> Xe) (w—ie)

(X1.49)

Since y/,,,(@) goes to zero when w > 0 and since ¥,,,,(0) = 0 [see (XI.31b)], the «> 0, limit is smooth and leads to the famous fluctuation-dissipation theorem

ceo = 222 {* de tele)

(X1.50)

This theorem shows that the Fourier transform of the correlation function of x(t, €)—that is, the power spectrum

[see (III.33)}—is proportional to the

power dissipated in the system in the presence of an external force [see (X1.36)). Most of these properties of the susceptibility of the Brownian harmonic oscillator have been established with very little reference to the specific features of this model. As we see later, they are indeed valid for most non

equilibrium systems that are slightly perturbed away from equilibrium. 2

LINEAR-RESPONSE THEIR GENERAL

2.1

FUNCTIONS AND PROPERTIES

Linear Response to a Spatially Homogeneous External Field

We now consider the statistical mechanical problem of the linear response of

aclassical N-body system to a weak external field. The total Hamiltonian is

Ayorailt) = Hy + Ayen(t)

(XL51)

where Hy is the internal Hamiltonian [see (VII.1)] and

Hyex(t) = -AF(0)

(X1.52)

describes the linear coupling of the “force” F(s) to some phase function of the system A=A(ct, v) (as usual x and v standforr,...,ty andv,,..., Vy). For example, if a fluid of charged particles (with charges Zae) is submitted to an external electric field E(t) along the x-axis, we have Hyex(t)= E() Da

eZaXa-

The state of the system is described by the N-particle d.f. p,, which obeys the Liouville equation i0,Pn = En totalPn (X1.53)

288

The Correlation-Function Formalism

with the definitions Ly, total =Ly

+ Ly, ext(f)

Lyen’** = LH yexv ---}

(XI.54)

and Ly given by (VIII.4-6). Suppose that the field has been turned on at time ip, before which the system was at equilibrium. Puan (X1.53) then should be solved with the initial condition py(to) = pv [see (X1.14)]; we have of course i0,.pN= Lypn =0

(X1.55)

If the field is weak for f> fp, the system is assumed never to get very far from its equilibrium state. We therefore write

(X1.56)

pnt) = pnt Apn(t)

and linearize (XI.53) by keeping only terms proportional to F or Apy

© F

itself. Notice that there is a rather subtle point here. Indeed, we want to solve

the Liouville equation over macroscopic times, during which any force considered to be ‘“‘weak” by the experimentalist generally leads to a catastrophic deviation of the trajectory of every molecule from its unperturbed motion.* This individual motion certainly cannot be described by a linear theory. However we are interested only in computing macroscopic properties of the system as averages over the distribution py. We have already remarked that although the individual trajectories are very erratic, the df.

Pn is expected to be smooth; for the same reason, the deviation of py in the

presence of a weak external field should be smooth and small. It is thus reasonable to assume that the solution of the linearized equation 3, Apn(t) = LyApn(t)+ Luexlt)on'

(XL57)

allows us to calculate correctly any observable property of the system, to first

order in an expansion in powers of F. The formal solution of this equation, with the initial condition Apy(to) = 0, is

Apnit)=~| dt! exp[-iLu(t—W)lLnen(t)o

(X18)

to

(This result can be checked by differentiation.) We are interested in the response of the system, which is the average of some dynamical function B= Bit, v) over the nonequilibrium state of

the

system;

in

our

current B al, += y

electrical

example,

B

would

be

the

electrical

, ¢ZqVa,x. We therefore need (B), = fax dv Bt, p)Apy(r, 0; ¢)

(X1.59)

* A model example, illustrating very interesting remarks, is given by Van Kampen (1971).

Linear—Response Functions and Their General Properties

289

(where we have used the convention of subtracting from B

equilibrium average). Expressing Ly.ex: in terms of the force

Lyen

its eventual

= if..., AJ)

(XI.60)

we find that the response per unit volume can be written B

t

Bf

dt’ xpa(t—') F(t’)

(X1.61)

to

where the response function yg4(t) gives the response per unit volume, for t>0, to a unit impulse F(t) = 6(t) and is defined by

Xea(t) =3 | dr dv Be™™™{p%3, A}

(X1.62)

The following identities

fax dv A{B, Ch= fac dv B{C, A}= ac dv C{A, B} fax dvde™B={

dr dvBelwA

(X1.63) (X1.64)

are readily established by integrating by parts the derivatives involved in the Poisson bracket {...,. . .} [see (VII.16)]. These relations allow us to cast the response function into the following equivalent forms:

Xa(t) =4 fax dv {p%3, A(O)}B(t)

(X1.65a)

=4f dx dv {B(s), p%3}A (0)

(X1.65b)

or

1

“

A

xeal=5 fax dv pA (0), BOO}

(X1.66)

where [see also (VII.31, VIII.4)]

Bi)=e"™B = B(r,(r, 0), 0,(t, 0))

(X1.67)

describes the evolution of the observable B in the absence of external field according to

@,B(t) ={B(O, Hy} = iLyB(t)

(X1.68)

290

The Correlation-Function Formalism

With the notation (XI.15), vg (¢)}—as given by (X1I.66)—takes the very

elegant form

xoald = (A), BON

(X69)

Still other forms are obtained if we notice that

to, A) = SE (Hy, A(O}= Boa a

a)

iy

A

A

.

= BeHA(0)

Then (X1.65) leads to

(X1.70)

Xpa(t)= £ (A(0)B(t))*4 = EA (0) B(n))°9 which, incidentally, proves the useful identity

(XL71)

(X1.72)

(A(O)B(n)* = “AO BO)

The response function appears thus as the equilibrium average of dynamic

functions taken at different times, and these functions obey

the laws of

mechanics with no field present. Although the computation of expressions like (XI.69, 71) is a full N-body problem, an important step has been achieved in deriving these linear-response forms. They play a role similar to the partition function in the equilibrium case: it remains to find approximate methods to evaluate them. As for the Brownian oscillator discussed earlier, we can introduce the

generalized susceptibility

Xpa(@)= { ate'ypa(t)

(X1.73)

such that the Laplace transform of the response

(B),, = [° dte™(B), satisfies the linear relation

.

(X1.74)

°

Boe ena w)Fw) For instance, in our electrical example, with A=B=

(X1.75) Sex =P4eZava,x, the

conductivity o.(w), which is the coefficient of proportionality between the macroscopic current per unit volume (J, ,),,/9 and the electric field E(w), is thus vot _[? Bel o,(w) = i

dt

ry

F.(0)F,,.(0))

(X1.76)

Linear—Response Functions and Their General Properties

291

Such a relation is another example of a Green-Kubo formula [see (X1.4, 5)}

expressing a transport coefficient as the time integral of a Green~Kubo integrand (an autocorrelation function), here weighted by the oscillatory

factor e,

Another function often encountered in the literature on linear-response theory is the relaxation function

Raa(t)= |

dt’ Xgalt')

(X1.77)

If the system has been submitted to a constant force F since fo = —00 and the force is suddenly switched off at ¢=0, the response will relax to zero according to [see (XI.61)]

(B).>0 ={ 2

oO

dt’ xo4(t-t)F =Rp4 (OF

(X1.78)

Using the last form in (XI.71), we see however that

Rea (t)= EuA (0)B()*4-(A (0)B())*}= nBRga(t)

(X1.79)

where Rga(t) is the correlation function between A and B [ef. (1I1.23)]*

(X1.80)

Rea(t)= iA (COB) To

write

the

second

equality

in

(XI.79),

we

have

used

the

hypothesis that A and B become independent in the long term

him (AMB()"= tim (AOXBO)

(ergodic)

(X1.81)

and we have remarked that (B(t))4= fax dv pH e"™B = fac dv Be p34

which is zero by definition.

J dr dv Bp}

=(B)4

(X1.82)

* When the quantities A and B involve the sum over the N particles in the system—which is always the case in applications, except for self-diffusion [see (XI.4)}-—it is convenient to introduce the factor N_' in the definition of the correlation function, to have a finite limit for this function when the thermodynamic limit is taken (see Section 5).

The Correlation-Function Formalism

292

We see that up to a factor nf, the relaxation function is identical to the usual correlation function and the distinction between these two functions is really not necessary. Both have been introduced because this simple relationship only holds in classical mechanics, not in the quantum mechanical

case.*

2.2 Linear Response to Space-Dependent Forces

We here generalize the analysis of Section 2.1 to the case of a system coupled to a set of / different space-varying fields. The external Hamiltonian is

Hyew= — jul3, fae AP GHe, PM 1 1

Se

fm ee gq

where

we have

function A®

ag

.

A®Q(, v) FY)

(X1.83)

introduced the spatial Fourier transform of the dynamic

A(t, v) = far e 19°F AMEE, v)

(X1.84)

and a similar definition for the force. For technical reasons it is convenient to work in a (large but) finite volume © and to assume periodic boundary conditions: the wave numbers q then take discrete values as in (IX.8).

We want to compute the Fourier component of an arbitrary response

(AP ).= fax do Adc, v)pu(t, 0; #)

(X1.85)

to linear order in the forces. Because of this linearity, the generalization of the results of the preceding section is nearly obvious. (i)

The response (XI.85) is the sum of the individual response to each

(ii)

Similarly, each term in the sum over the wave number qin (XI1.83) will

(iii)

separate field F.

lead to an independent response. Since the internal Hamiltonian is translationally invariant, the Fourier

component of the force F® will induce a response only at the same wave number:

(A%), = (AP), 685

* See, for example, Martin (1968).

(XI1.86)

Linear—Response Functions and Their General Properties

293

Hence we shall have

(A®),= 5 { dt x, ,0(q; t-1)FO() fel

“to

(X87)

To simplify the notations, in the following we write only the formulas corresponding to the case of a single response

(Badr = fac dv By(t, »)py(t, 0; £)

(X1.88)

of an arbitrary observable B to the external interaction with a single external force 1

Hyen = ~~ A(t, 0)Fa(t) 2

(X1.89)

Except for the additional subscript q and the factor 0 ! this is the same

problem as discussed in Section 2.1. We find [see (X1.69, 71)]

xoales = 2 UA (0), BAO =F (A (OB 0" =

BA 4(0)B,(t))"*

(X1.90)

Here B,(t) is the solution of [see (XI.68)]

0,Bq(t) = iLyB,ft)

(X1.91)

B,(t)=e""B,

(X1.92)

(Bao = { dte“(By,

(XI.93)

that is, With the Laplace transform

Oo

and similar formulas for Fy

and Xga(q; w), (X1.87) becomes of course

(Bao = Xpa(@; ©)Fqe

(X1.94)

Let us illustrate these formal developments on a specific example: the current induced in a system of charged particles by a longitudinal electric field. In a gauge where the vector potential vanishes, the electric field is minus the gradient of the scalar potential U

E..=—VU

(X1.95)

294

The Correlation-Function Formalism

The charges Z,e are coupled to the field according to the Hamiltonian

Hneu(t) = y Z,eU (ta; t) =[adt Z

Z,¢6(t—4r,) U(r; t)

i

Oy Peal U.() where

N

beqg= L

a=!

.

Zee lv"

(X1.96) (X1.97)

is the Fourier component of the charge density. We see that (XI.96) is of the form (XI.83), though the external “force” — U,(t) is of course not a physical force. If we are interested in the Fourier component of the current density N

heqg= X Zaevae 1"

(X1.98)

awl

the corresponding response function is

recs =F G-a( Deal)"

(x1.99)

Since the time derivative of the charge current is related to the electric current by the continuity equation [cf. (VII.126)]

(X1.100)

Peg =—!4 hea

we find, as expected, that the induced current is proportional to the external electric field Eextq= —iqUa:

Geade= [dl xicolas -01-U Ae) =| at 0.45 1-1) + Baal) t

0.

(X1.101)

where the tensor o, has the following components

ol =F kg all)

Cajexyz)

(1.102)

Remark that og, is proportional to the autocorrelation function of the current Je.q and is not a retarded response function ya: this is due to the fact that the microscopic “force” is (—U,) and not the physical force Eex.q-

Linear-Response Functions and Their General Properties

Let

us

also

stress

that

the

Laplace

These

two

fields are

transform

295

of

(XI.102),

which

generalizes (X1.76) to space-dependent fields, generally cannot be identified with the experimental conductivity tensor because the latter quantity is defined by the ratio of the current to the total electric field E,, not to its

external

part Eexiq.

of course

connected

through

Maxwell’s equations (which themselves involve the induced charge and current densities), but because of the long-range character of the Coulomb

forces, this connection is not simple.* 2.3

General Properties of

the Response Functions

If we assume that the Laplace transform defining the generalized susceptibility

Xpa(Q; @) =|

0

dte"'yna(q; 1)

(X1.103)

exists for w real (which implies that the function %45(q; z) defined replacing w by the complex variable z is analytic in $*), most of properties established for the Brownian oscillator are easily shown to valid in general. The only new feature is that there exist symmetry relations between

by the be the

response functions yg, (q; ¢) and y48(q; ¢). To obtain them, let us start with

the response function as given by the last equation in (XI.90)

Xpa(q; 1) = -£ (A_,(0)B,(0))*

(X1.104)

By stationarity, this can be rewritten as

Xpa(q; = -F (BO A_{-0)"= —xXan(-q;—8)

(X1.105)

In the last equality, we used the second form for the response function in

(XI1.90); indeed, though this function was used until now for t>0 only, the various expressions (XI.90) define it for t—#) and

spatial reflection (r=> —r or q=> —q). From the definition of the various A,(¢) and B,(¢), it can be shown that all these quantities are either even or odd in

these transformations. Therefore, (XI.105) leads to

Xpa(Q; 1) = —epaXan(G ft) “See Martin (1968).

(X1.106)

296

The Correlation-Function Formalism

with eg, = +1; these are the famous Onsager relations. Yet the calculation of the signature eg, is tedious,*and in the simple fluids we are studying these general relations are not very useful. We therefore limit ourselves to the case B= A, where no such questions arise. The susceptibility y,4 then has the following properties (cf. Section 1):

(i) Symmetry in frequency and in wave number. the observable Aare, bv) imposes the relation

The reality assumed for

(X1.107)

A(t, 0) = A*,(t, 0) and this immediately leads to [see (XI.32)] Xa

(X1.108)

(Qs ©) = Xa,a(-G; —-w)*

(ii) Work done by the external force. The calculation proceeds as for the Brownian oscillator. If the system is driven by the force

F(r; t) = Fo cos (q-r-wt)

(X1.109)

the power dissipated per unit volume is

P(t) _ =I) dr (4 77 Wy) F(r; . t) © O

(X1.110)

When averaged over one period, this power is proportional in the long term to the imaginary part of the susceptibility:

fim 2O - wee (q; 0) FF

(XL111)

wo

Of course since this result is quadratic in F, we cannot simply add up expressions of the ine oS 111) when there are several fields coupled to different variables A™, ..., A; we do not need the formula corresponding to this more general case. (iii)

Kramers-Kronig

relations.

The

analyticity

of

guarantees that relations analogous to (XI.42) are valid:

Kaalqs @) = -tef T .

]

+00

to +O

XAG w=tof im

* See De Groot and Mazur (1962), Martin (1968).

~ iH

.

dey alee) o-@ ,

~

5

du ca

i

,

Ysa(q;z)

in

S*

(X1.112) (X1.113)

Correlation Functions and Hydrodynamics

(iv) Fluctuation-dissipation function of A, as

297

theorem.

If we

define

the autocorrelation

Real N= ZA (OAK)

(X1.114)

the last equality in (XI.90) can be written Xaa(Q; 1) = —nBd.Raa(Q; t)

(XE.115)

which is already a form of the fluctuation-dissipation theorem: Raa describes the spontaneous fluctuations of A, at equilibrium while y,, is related to the dissipation in the presence of an external force. We can cast this result into a more familiar form by writing (X1.115) as

kaT(* , Raal@ t= be dt! xa(q5#) ;

(X1.116)

t

where we have assumed that R,4(q; ©) = 0 [see (XI.81, 82)]. This equation

is completely analogous to (X1.45) and can be treated in the same way; the result is

Raalqs= ‘et | 3

+00

—0o

a

.

dw enw tdalgic)o)

(XL.117)

CORRELATION FUNCTIONS AND HYDRODYNAMICS

The correlation-function formalism describes the linear phenomena produced by external forces with arbitrary space and time variations. On the

other hand, when these space and time variations are sufficiently slow (i.e.,

when the wave number g and the frequency w go to zero), the conserved variables are known to obey the relatively simple laws of hydrodynamics. Of course hydrodynamics is not limited to the linear regime: for g and small, the amplitude of the phenomena may be large. However linear-response theory and hydrodynamics do have a common domain, where both g and w are small and the amplitudes of the nonequilibrium deviations are also small. By identifying these two descriptions, we gain from the response functions microscopic expressions for the transport coefficients.* Yet a difficulty appears at once: the dissipative behavior described by hydrodynamics is generally not induced by external forces but by the gradient of local thermodynamic quantities. For example, the thermal * See for example Kadanoff and Martin (1963) and Forster (1975).

The Correlation-Function Formalism

298

conductivity is introduced through Fourier’s law . Jo

oT mK

(X1.118)

and the temperature gradient is provoked by imposing a constraint at the boundaries of the system, not by any coupling to an external force!

Nevertheless, the method works if we invent artificial external forces that

would give rise to such thermal flows if they existed in nature. More precisely, we make two assumptions. (i) (ii)

That the macroscopic laws of linearized hydrodynamics are correct for q and w > 0 and close to equilibrium. That these laws are not modified in imaginary situations where nonphysical forces are coupled to the conserved macroscopic variables.

Then the correlation-function formalism provides microscopic expressions for the transport coefficients, which take the form of Green-Kubo formulas, as we suggested (but did not prove) for self-diffusion [see (XI.5)]. That is, they appear as the integral of the time-dependent autocorrelation function of some microscopically defined flow. We work out in detail the case of shear viscosity. To this end, let us consider the linearized Navier-Stokes equation in Fourier language [see (V.38b, 39)]. We can decompose the Fourier component u,(¢) of the velocity

field in its longitudinal and transverse components u,*

pe,

ee

x

x

oe

(X1.119)

This terminology stems from the fact that uj, is parallel to q [its inverse Fourier transform satisfies V xuj(r)=0] while u,, is perpendicular to q [V+ u,(r) =0]. The transverse component obeys the separate equation

Lt atta = al

@2(2) Via

(X1.120)

where we have added to the right-hand side the acceleration term due to the transverse component of the external force

F, ,(¢) = F*@) 92

(X1.121)

We can take the Laplace transform of (XI.120) with the help of [see (X1.25)]

i, g(w) = i. dte™u, o(t)

(XL.122)

Correlation Functions and Hydrodynamics

299

Assuming that, at ¢= 0, u,q(0)=0, we obtain readily

b, {w) -___1_# =

(w)

m(-io+q°n/p)

(X1.123)

[the transform of the force is defined similarly to (XI.122)]. If we could find, or eventually invent, a Hamiltonian Hy... that couples

our system to a transverse force F, q, then the linear-response theory would tell us that [see (XI.94)]

tt, q(@) = Xe (q; &)F,,q(o)

(X1.124)

where A is the dynamic variable that is coupled to F, q [as in (XI.89)] and B is such that (By): =u, = a(t). Identifying (X1.123) and (X1.124), we would then get an expression for 7 in terms of the microscopically definedYg, (q; #). Alternatively, we can take the double limit of (X1.123) when q > 0 first and then w 0 (i.e., g>0, @ > 0 with w » nq?/p); we then have 2

fs g(w) = (So

VB

(o)

(X1.125)

If (IX.124) takes the same form in the same limit, the microscopic definition of 7 will be determined by comparison.

Now, here is the difficulty: there is no physical transverse field that couples to the hydrodynamic variable u,, in a neutral fluid. Indeed, mechanical forces usually derive from a potential: F=—dU/dr, which becomes in Fourier space F,=—iqU,,; this is clearly longitudinal! Suppose, however, that the particles in the system have a small electric charge e; then a transverse electromagnetic field, described by a transverse vector potential f(r; ¢) (such that 4/ar + ef = 0) with a zero scalar potential, would submit the particle to a force which has a transverse component.

and magnetic fields

Eoalts

1

)= — 7 d.cf(e; 0)

Hewlt; t) = Sx ott t) are clearly transverse. An example of such a chromatic and polarized plane wave: Ar;

Indeed, the electric

(X1.126)

field is the following mono-

t) =al, efor)

iw

E. a(t; f) = aac!

e

i(wt~qx)

Haglt 1) = —igel e6e™

(X1.127)

300

The Correlation-Function Formalism

This situation is clearly not physically realizable: all the particles having the same charge and the total momentum being conserved by the internal Hamiltonian,

a current,

once

set up,

would

never

stop

and

the static

electrical conductivity would be infinite. Yet we need not worry about such an unrealistic behavior and we depart even further from “real’’ physics by letting the charge e of each particle become infinitestimal (¢ > 0). This allows

us to identify the external fields E,,, and #,,, to the internal fields that

appear in the macroscopic equation (XI.120): the differences between these fields vanish when e>0. Moreover the Lorentz force acting on a fluid particle with coordinates (ra, Va)

F(a, Va; t) = e[Eexi(Fa; t) + Va X Hox(Fas £)]

(XI.128)

simplifies in this same limit because the average velocity (v,), is proportional

toe, therefore the second term in (XI.128) leads to effects of order e ? and is

negligible: we are then left with a purely transverse force. Similarly, the electrical interaction between the charged particles is also of order e*. We are thus led to calculate the nonequilibrium average fluid velocity [see (VIL.115)]

ue; D=2 G0),

(X1.129)

or rather the Fourier transform of its transverse component, induced by the external force F(t; f) = eE,,,(r; f). To use the response-function formalism, we should now look for the proper interaction Hy corresponding to this force. Such a calculation is rather tedious because account must be taken of the redefinition of the momentum

variable as pz = mv, +(e/c)

in the

presence of the vector potential.* Rather we shall guess the result from the electrical example given at the end of Section 2.2 where we obtained the relation (XI.101). We reason as follows.

(i)

We notice that although this relation was derived for a longitudinal field in a gauge where »f=0, it must be gauge invariant and, in particular, it must be correct in a gauge where U=0 but #0.

(ii)

We assume that the same result also applies if the vector potential is

(iii)

transverse. We note that when all particles have the same charge, the particle current and the electric current are simply related by

(XI1.130) * The full calculation can be found in Luttinger (1964).

Correlation Functions and Hydrodynamics

301

Then from (XI.101, 102) we read off

i, glo) = 6,(q; o)F.q(o)

(XI.131)

where 1? Glas

o)=—

|

dt eo.

= [dre

(43 1)

ECs es Oowai(O™ — (41.132)

Here Ip, 1 is the transverse component of the momentum density A

Jiq=

N

L

awl

mvae

7H,

(X1.123)

Note again [see the remark after (XI.102)] that G, is not a generalized

susceptibility but rather the Laplace transform of the autocorrelation of the

mass current j,,.,q, that is it has the form of a Green-Kubo integrand.*

For an isotropic fluid, a, depends on the scalar q’ only, not on the vector q, and it is convenient to rewrite (XI.131, 132) ina reference system where q is along 1, [see (X1.127)]. Taking the y-component of (XI.131), we have

with

(XI.134)

By q(@) = Foyy(Qs @)Fy,q(@) Fo.yy(Q3 @) = j de

iwt

above Oiosall ee

(XI.135)

As is clear from (XI.123), this quantity is singular when q, w > 0. This

is of course related to the fact that the total momentum in the system is a conserved quantity. Mathematically, this means that the equation of motion [see (XI.91)] (X1.136)

arfo.y.a = iLNioy.a = —{Hn, Tech

becomes, at ¢=0, 4f,y0=9:j,y,0= L_M"Uay has a vanishing Poisson bracket with Hy. We show below that near q = 0, (XI.136) becomes

Afoya = —~iq¥ Lay q(t) where fi, q(t) is the Fourier transform momentum-flow tensor, that is,

of

Tha =e" Tyg * For a different point of view, see Forster (1975).

(X1.137) the

xy-component

of

the

(X1.138)

302

The Correlation-Function Formalism

with

Tyg = | dre'*'TL.y(t)

(X1.139)

where II,,(r) is defined by (VII.136).

With the help of (XI.137), a lengthy but straightforward calculation, which we give below, shows that (XI.135) can indeed be put in the form suggested by (X1.125):

-

7.

yal

ang; e) mp?

Goyy(q3 @) = ima +

(XI.140)

In the limit where q and w go to zero, we find for the shear viscosity n = 1(0;0)= np {

0

dt R,,,.3,,(t)

(X1.141)

where the correlation function Ry, ,,,(t) is

Ie

np

ave

Raga (= Say Og OY with

and

S=e""F,

WG je(xy,z))

(XI.142)

(X1.143)

Jj = | dr nw ant

bea

ary

(X1.144)

Equation (XI.141) is the Green-Kubo formula for the shear viscosity. We present here the main steps leading from (XI.135) to (XI.141). First, we have

to establish (XI.137). The shortest way is offered by the conservation equations

derived in Chapter VII. In our present notation, we rewrite (VII.128) as

.

a

af). + jony2 LM), OF

=0

(X1.145)

But for any phase function A we have [see (VII.18, VHI.10, XI.64)}

(A), =

drdv A(t, v)px(t, 0; t)

= [ax dv A(t, v) e Loto.(r, 0; 0)

a | dt dv(e™*A(x, »))py(t, 0; 0)

(X1.146)

Correlation Functions and Hydrodynamics

303

Therefore (XI.146) is equivalent to

[ acavla a Fo Us aren= ft: Donte»: 0)=0

(X1.147)

with

Salts ) = ef, s(n)

Tales ) =e"

fly (r)

(X1.148)

Since (XI.147) holds for any initial condition p,(t, p; 0) that describes slowly varying phenomena, we need to have C)

(X1.149)

dioalts 1) +E7 OF;— Tyr; 2) =0

whenever such slow variations are considered. Taking the Fourier transform of (X1.149), we arrive at (XI.137), while (XI.148) is equivalent to (X1.136, 138). Second,

to cast &,,,(q;@)

(X1,135) and

in the form

(XI.140),

we use the identity between

5 1 Goyy(Q3 0) = Af a] nye Ee Oona) em]

(X1150)

Integrating this equation twice by parts, we get Gp, yylGs @) = ~O2[ae abe ma”
00, these two terms can be evaluated with the help of the following averages: fay n(O)

N z

=

m0,,0py eit(ra-te)yea

N LD m(v7

"

Coy a

abel

oe

= m?NkaT

(X1.152)

and

Goya since 0, = F,,(t)/m

.

N

OF o.yq(0))° = 2 F 07( 04 Dp, x eT

YET =

(X1.153)

[see (1.1, 2)] is independent of v and (v,,,)°* vanishes.

304

The Correlation-Function Formalism

In the integrand of (XI.151), we use the relations (X1.72) and (X1,137), which yield

Goa-al Vf yal OV = — Goyal Of yal = GT

-q( OE. (8)

(XI.154)

Combining these results, we arrive at (XI.140) with

nigia=| a OS, (Orta) 0

(X1.155)

In the limit q, # > 0, this leads to the Green-Kubo formula (XI.141).

A similar reasoning can be followed to establish the Green-Kubo formulas for the other thermal transport coefficients. For the bulk viscosity {

{or rather for the combination 47/3 + ¢, which appears in the longitudinal

part of the linearized Navier-Stokes equation (V.38b)], we obtain

Sega pn | de Rieaill

(1.156)

Ria (O= Li lorheoy"

(XI.157)

with

In the canonical ensemble considered here,* we have

Sc=5. —pO+ (Hy —(Hy)°9(2)

(XL.158)

where J,, is defined by (X1.144), and p and @ are the equilibrium pressure and energy density, respectively. To obtain the corresponding formula for the thermal conductivity, the only new problem is to invent an external field that provokes a temperature

gradient. Surprisingly enough, such a field does exist in nature: it is the gravitational field. The reason is that an energy density €(r) behaves in relativity as a mass density €(r)/c?, from which it interacts with a gravita-

tional field. These effects are extremely small, but this does not matter; in fact, if such a coupling 4

—UG;t

Anext = -| dr e(r) (—

(X1.159)

* The form of the flow J‘, in (XI.157) depends on the equilibrium ensemble chosen (here we always take it canonical}. This dependence comes from the delicate nature of the q, # +0 limit taken in the formula for 4n(q; w)/3+¢(q; w) analogous to (XI.155).

Correlation Functions and Inelastic Neutron Scattering

305

(where U is the gravitational potential) did not exist, one could as well invent it for the purpose of the proof. Following the same reasoning as for the viscosities, we then arrive at

Re iF i dt Rig,Jo,(t)

(X1.160)

with irs

A

Rodos) = Vox (O)Ja,.())

(X1.161)

Here the quantity Jos is the total heat flow in the system A

a

N

Jox = Fax —h i=lY vax

(X1.162)

which is obtained from the total energy flow [see (VII.149)]

k= { dr je.(0)

=5 eae aml

2

r Virw))—Y Ly Mra)

sea

rar

Tob,x

]

;

X1.163

;

by subtracting the flow of the equilibrium enthalpy per particle

h _&t+p

(X1.164)

n

Finally, the self-diffusion coefficient is obtained by coupling a specified particle to an external field: we then recover the Green-Kubo formula (X1.5) in its statistical mechanical interpretation.

4 CORRELATION FUNCTIONS AND INELASTIC NEUTRON SCATTERING

When

a beam

of particles

(neutrons,

photons,

electrons)

encounters

a

many-particle system, the particles are scattered and transfer energy to the system. The cross section for these inelastic scatterings is directly related to correlation functions. For simple fluids, the best-suited probes are neutrons

and photons (laser beams). Although we are limiting ourselves to classical fluids, we discuss mainly the scattering of neutrons, which is a quantum phenomenon. The corresponding analysis for light is sketched in Section 5.

306

The Correlation-Function Formalism

Eup

Figure XI.2

h(ubew)

Scattering of a monochromatic wave by a macroscopic object.

The diffusion of a particle by a macroscopic object is schematically

represented in Fig. X1.2, where

Vin Cexp [i (kp + r— wot]

our

exp ti[(Ko +g) + (wot @)t}}

(X1.165)

are the monochromatic plane-wave functions of the ingoing [with momentum p,, = #k, and energy E;, = hw] and outgoing [with pou = #(Ko+q) and Eu. = fi(w)+)] particles, respectively; moreover,

the direction of pou: iS

determined by the angles y and ©. The momentum and energy transferred to the particle are thus Ap=hq,

AE=hw

(X1.166)

Energy and momentum are related by some dispersion relation such that

@o= (ko),

wo tw = e(|ko+ql)

(X1.167)

For neutrons, we have

o(k)= a (My

(X1.168)

is the mass of the neutron), and for photons

o(k)=ck

(X1.169)

In an inelastic scattering experiment we measure the energy hw and the momentum fq that the macroscopic system exchanges with the probe particle. To judge a particular probe, we should ask about the range of q and w values at our disposal. The upper limit is of the order of ky and wo. [For q » ko, the w values obtained by varying the angle of q—see (XI.167)—lie within the uninteresting narrow range Aw ~[d¢(q)/dq]kp around the energy ¢(q).] The lower limit is determined by technical problems, that is, how precisely w@ and kg are given (monochromaticity) and how precisely w and q can be detected (resolution). In many cases these two limits are close: the

Correlation Functions and Inelastic Neutron Scattering

307

measurable values of g (and w) are near ky (and wo). Thus for the measurable energy and momentum transfers we have approximately

(X1.170)

g~ko

ow,

and they are then related by the dispersion relation w ~ ¢(q). This relation makes it easy to understand why low-energy (“thermal’’)

neutrons, corresponding to ky~10%cm™’,

thus w)~10

"eV, are very

appropriate to measure fluctuations that extend over molecular distances

(=10 *cm)

at ordinary

temperature

(T~10?°K

or

kgT~10

7eV).

Indeed, this length scale corresponds to wave numbers q of the order of

10° cm”! and, by (XI.168, 170), this means fa ~ 10 ?eV~kgT.

For photons having the same wave numbers (i.e., X-rays), we find from (X1.169) fw) * 10° eV » k,T and any thermal-energy transfer is too small to be detected. Fortunately the development of laser technology allows us to dispose of the restriction (X1.170): with these coherent and highly monochromatic beams, high-resolution spectroscopic techniques make possible the measurement of g and w values ranging from the incident order of magnitudes (q = ky= 10° cm”, hw = hwy = 10 eV) down tog

+ 10 cm” ‘and

hw ~10 '* eV, a very wide range indeed. Letus now show how the neutron-scattering inelastic cross section is related to the density-density correlation function of the macroscopic target. We start from the inelastic cross section d’a/dQ. dw describing the scattering in the solid angle dQ (around y, ®) with an energy in the range [hw, h(w + dw)]). This quantity is related to the transition probability W for the neutron to be scattered from the initial state p,, to a final state pour

through the well-known formula*

do Ode

Pour Wom

MiQiox eer

(X1.171)

where 1,.. is the total volume of the scattering experiment (Oho, > ©, where

0. is the volume of the target that contains N atoms). Assuming that the coupling between the target and the neutron is weak,

we can take for W the result from the second-order perturbation calculus, that is, the famous “golden rule:”

. W=E Pi flVbPow AP 2 CE: Ey— ho)

(XL172)

Here the quantum mechanical states have been denoted | ) to avoid confusion with notation used previously in this book; the indices i and f * See, for example, Messiah (1963); a more advanced and very careful treatment of the present problem can be found in Goldberger and Watson (1964).

308

The Correlation-Function Formalism

characterize the initial and final states of the target, with respective energies

£, and E;; finally, V is the interaction between the neutron and the target.

Equation (XI.172) applies to the case where the initial state i of the system is given; we sum over the final states f, since they are not detected experimentally. However if the macroscopic target is at thermal equilibrium, the initial state / is not well defined and we must take the average of (XI.172) over the

canonical distribution for these states. We then have*

e PF

hw) F— W= zy Kin: lV [Pow f) Pate, —E,-

(XL.173)

As a matter of fact, things are even more complicated because the neutron has a nuclear spin, which is not taken into account in (X1.173).t To simplify, we forget about all spin variables and merely indicate their effect on the final result. It can be shown that the dominant interaction of a neutron with an atom is with its nucleus; because the wavelength of a thermal neutron is much larger than the range of the nuclear forces, the scattering by the atomic nuclei in the system is s-wavelike and can be described in the Born approximation by a Fermi local pseudo-potential, usually written as

=e Y

— ta bablt~te)

‘X1.174 (X1.174)

5,8 (ty

where 6, is the scattering length of the atom a. The normalized plane-wave function for a neutron with momentum p is (Qo) '’? exp (ip + r/h); hence [see (XI.166)]

Pins fl VbPow A= sf drive’ aC roe

/)

a=

(X1.175)

We also use the representation of the Dirac delta function

8(E:-E)-hw)=28(2 =o) +00

_

-4/ dt exp [-i(E5#-w) | 2tth Sco h and we note that, for any operator A we have

efF™ FA le) eH" = (fA eI) = (FIA (Nk)

(X1.176)

(XL177)

* We use here the quantum mechanical generalization of (VII.42): the probability of finding an

equilibrium system with the energy E; is exp (—8E,)/Zy where Z, =Y; exp (-8E;); see for example Landau and Lifschitz (1958). + Of course the target particles also have spin degrees of freedom (both electronic and nuclear), but these could be formally incorporated into the indices i and f.

309

Correlation Functions and Inelastic Neutron Scattering

where A(t) is the operator in the Heisenberg picture, that is, the quantum analog of exp (iLyt)A. We thus see that the scattering cross section can be written

|

Pou N’ do Aida mn, 2a)

-iqt-w)

{~ HP Tage

Test)7/.

(XL.178)

with Fle t) Fie;

th=

4 i N dr Zrye < | | “Pei

dy

%

a=)

nO)

a's]

b,6(e+e —t, of]

- N 2, babar | de (8(r'—1.(0))6("

tr, (4) (XL.179)

To arrive at (XI.179), we used the closure relation Y,||f)fl|= 1 as well as the definition of the quantum mechanical equilibrium average [cf. (VII.42, 44))

(Ay =

TrAe Phe _ ¥ (illAlli) oO

Z

‘No

waa i

(X1.180)

How should these results be modified to incorporate the spin variables? The point is that the scattering length depends on the neutron spin oy and on the nuclear spin a, of the atom a; we have

b, = 6 +b%o,

+ ow

(X1.181)

We must then correct the definition of W by taking the average of (XI.175) over these spin variables (we denote this average by a double bar): both for oy and @,, we have an unweighted average over all spin states for the following reasons: (i)

We limit ourselves to the simple case where the incident neutron beam

is unpolarized and the eventual polarization of the outgoing beam is not measured.

(ii)

The nuclear spin variables do not appear in the Hamiltonian Hy of the target (their contribution is negligible).

The result (XI.178, 179) is then merely modified by the replacement

baba > baba’

(X1.182)

310

The Correlation-Function Formalism

From (X1.181) we get bb.=

bP?

44 Op 2

iO,

+b (oy Ov

)

Ga)

(X1.183)

Since the nuclear spin of the atoms (like that of the neutron) has no preferred

orientation, the second term vanishes. For the same reason, in the third term we have

(G4 * GaGx + Fa) = (Oy * Oa) boa

(X1.184)

In addition, it is easily verified from (XI.181) that

B= beh

B2= 6!" +b?(oy-o,) = b°

from which

(X1.185) (independent of a)

$2

OF

baby= b +(b°—b

eke

baa

(X1.186)

With this supplementary spin average, (XI. 179)i is conveniently written as GF (ir; t)= eb “Gn; j+(be - b \g, (r; ¢) where @ and G are known as Van Hove’s functions, respectively. They are defined by 1

G(r; 1) = af dr

N

Y

abe=t

total

(X1.187) and

self-correlation

(8(r+r'—r,(t))6 (0 —r4(0)))"

-1 f dv (A(etr’s t)A('3 0) = “cate t)A(O; 0)"

(X1.188)

and 1

N

AUS o=4] dr Y (Sete —41,(1))5(e'—14(0)))* awl

- | dr’ (5(e-+e'—1,(0))8(r —",(0)))*4 (X1.189)

= (6(r—r,(2))8(r, (0)

In deriving the second and third equalities in (XI.188), we have used the

definition

of the

particle

density

[see

(VII.113)]

and

the

translation

invariance of the fluid; in (XI.189) we took account of the symmetrical role

of all the particles and of translation invariance again.

Correlation Functions and Inelastic Neutron Scattering

311

A decomposition similar to (X1.187) exists for the cross sections (XI.178); we write ao do ao

dQdw

=

dQdwlcon

+

dNdalincon

(

X1.190

where the coherent cross section

dadeal| con _ Pow N Pin —b 2

GO

"¥(q; w)

(X1.191)

is related to the double Fourier transform of the Van Hove total function:

S(q; 0) =| dr}

+00

dte*

GR, t)

(X1.192)

and the incoherent cross section is

with

d’a |non Pupe HO? NG By AG) Tal

X1.193 (X1.193)

L(g; 0) = { ar |

(X1.194)

;

co

dt eh" °P'E (er; 1)

The adjectives “coherent” and “incoherent,” respectively, indicate that the first cross section arises from the interference of the waves scattered by the different atoms in the target and the second is the sum of individual contributions from each particle. A delicate experimental problem is to separate the coherent part from the incoherent part: in a single 1 measurement, only the sum (X1.190) is determined. Suffice it to say that b and BE vary differently with the isotopic number of the atoms whence, by varying the isotopic composition of the target, it is possible (in principle at least) to get separate information on ¥ and ¥,.* Let us also notice that in the long term we have

Ge; 1) — “(ale 1)°(A(0, 0)" =n

(X1.195)

and that this constant term in the Van Hove function leads to a singular contribution to the cross section L(q; @) = (21r)*nd(w)8(q) + regular terms ing andw This

singularity

is harmless,

however,

since for

particles are mingled with the unscattered beam.

* See for example, Egelstaff (1965); Copley and Lovesey (1975).

q=@ =O

(XI.196)

the scattered

)

312

The Correlation-Function Formalism

Although the scattering by a neutron is a quantum process, the behavior of the fluid itself is in general classical, and this book only deals with this case. The averages (X1.188, 189) defining @(r; 1) and Y,(r; #) then take the same

meaning as for other correlation functions introduced earlier. 5

5.1

FINAL

REMARKS

Correlation Functions

and Other Probes

It is clear from the derivation of Section 4 that the connection between neutron inelastic cross sections and correlation functions of the target depends very little on the specific properties of the neutron. The basic ingredients of the theory are all in the golden rule (XI.172), which correctly describes the action of any probe* on a macroscopic system, provided the interaction between them is weak enough. This coupling involves both the internal and the translational degrees of freedom. In the neutron case, the internal variable (i.e., the nuclear spin) and the position of the atom could be treated as independent, whence the cross section was related to the correla-

tion function of the number density only. In general, however, this is not true and one is forced to study correlation functions also involving these internal degrees of freedom: these questions cannot be treated by the model of simple fluids for which, by definition, internal degrees of freedom are neglected.t With this restriction, the only technique of interest other than neutron scattering, is Rayleigh—Brillouin light scattering, where light is scattered by the density fluctuations in the fluid. This process can be understood in classical terms

if we

remember

that in the

long-wavelength

limit elec-

tromagnetic waves are scattered by the fluctuations of the dielectric constant e€q(r; t): the coupling takes the form

V(r; 1) 0'7/(v) would this ratio go to zero. However we do not know how

to study finite systems in any clean way; that is, the thermodynamic limit is

inevitable. We are then led to understand (XI.202) as

.

i.

D= fim, = lim (Axi.r,)

(X1L.203)

where lim,, denotes, as usual, the limit (VII.68). Indeed, the limit rejects to infinity the time after which (Ax7,) will stop growing. Similarly, (XI.5) should be understood as

D= We assume

lim {

T17®

Ty, Jy

dt lim R(t) ees

(X1.204)

that this is the correct procedure in general: the Laplace

transform of any correlation function R4,(t} must be written

jim { 1200 Jy

Ty,

dte™ lim Rag(t) eS

(X1.205)

to describe the corresponding observable property. Unfortunately this rule is difficult to apply rigorously (in general, we do not even know whether these limits exist). Nevertheless it is important to keep in mind, in particular when applying the method of kinetic equations to correlation functions (Chapter XII, Section 1).

5.3

Summary

All the responses of a macroscopic system, initially at equilibrium, to weakly coupled external probes (either external fields or beams of particles) can be expressed in terms of correlation functions

Raalt) =7(A (0)B(o)

(X1.206)

Table XJ.1 lists the most important of these functions for simple fluids and their connection with experiment. Notice that computer simulations are included among these experiments; they have made a significant contribution to our understanding of correlation functions, as Chapter XII indicates.

“(907

ig

Vv kuy

=a ° (ase ZN =u

N

=D

(4I-3)ONA ="u

Ae

PA 9°7 'z =F

7Aa7

S452au ) ee a] ee ry7 =

(PAC as 1 1

+

YY 2a.)

9 2P

° w+

o

sursu, (7 “4)%% parouap] (+)"™y UONDUN-JJ9s BAO} UBA OY}

(4)""y

‘

[a fuw:ns

m

oO

J gu =(@)?0

je j =(M'b)'f

(2) a LD

‘Te fu = u

UOISNYIP-J[2S,

& YISOISIA IBD!

Ayradorg yeatskyg

syuatoyjaos yiodsues |,

Jre

°

TTX F198.

34st

uonenuns s9yndut0Z

(uinojiiig—ysia[Aey)

vemos)

Zursayeos

quazayor

,1uaTayoouy

Suirayje0s uOMAN,

AVAQONpUOD [eoLNIIe[q

(JaquINUDARM O49Z 12)

Ayanonpuos feusayy

rorro|

j ey Ory ap J "CL 4 (2)

Aysoosia ying rsPy

"yap

ros

7 (2) npr"Hy ep [a =}+¢/ley Ly

ta

wre |

(4)°"

uolssaldxq UOTSUN-UONe[a1I0D,

(2)

we (2) amass2 +P

uoNouny WONe]a109-OINe AYOOJIA ay JO VOHIUYEp a4} Ul PIONPONUI AY A 40192] AY SOUOEN y

LX) UORIUYap oY) UL paoNposuT ,_ AY 10198} DY} [soURD O} [a29y mas]

90,

pu [azoymasyo (4) y parouap] (+) "yy

[

DAZ (uc ee

(Gp) ((*H)—"H) +04 "pa"

4 730

rege TPAC pay = _ !a!Pau 1,

(Satyr)

suoloun aseyd

spmiy ajdung 105 suopouny uOpeIes10D UIE ML

315

XII Calculation of Time-Dependent Correlation Functions

1

CONNECTION BETWEEN CORRELATION FUNCTIONS AND KINETIC THEORY 1.1

Introduction

The previous chapter indicated how time-dependent correlation functions are related to various experimental situations, and established some of the properties of these functions (Chapter XI, Section 2.3). The great generality of these properties indicates that little can be learned from them about the behavior of a given physical system. To get more precise information, we have to devise approximation schemes to calculate the correlation functions for specific models. This chapter illustrates such calculations. Of course the general methods of kinetic theory introduced in Chapters VIII and X can be applied to the correlation functions. This section describes this application on the simple example of self-diffusion. Yet linear-response theory does not correspond to the most general situation, only to small deviations from equilibrium. Consequently specific methods have been developed for them; they are reviewed in this chapter. 316

Connection Between Correlation Functions and Kinetic Theory

317

1.2 Correlation Functions and Generalized Kinetic Equations To

be

specific,

we

consider

autocorrelation function

the

calculation

of the

normalized

(01x). CO)" (v.09)

Tw)=

velocity

(XIL.1)

from which the self-diffusion coefficient can be computed by a time integration [see (XI.5)]: D= “eT [ m

do

drT(r)

(XII.2)

To make the connection with the general analysis of Chapter VIII, we use

the identity (X1.64) to rewrite (XII.1) as

T=

[ax dvv,,e

MO. Tes

(XIL.3)

or

Fa)= J dv, 0; .8951(¥1; 2)

(XII.4)

with the following definitions: ,

89,,:(¥15

= | dv, -++ dvy dr, +++ dry dpn(t, 0; t)

Spy (t, 0; 1) =e

*™ Spy(r, v; 0)

- 9) — Mex

Spn(t, 0; 0)

eq

kT PN

(XII.5)

(XI1.6) (XI1.7)

These formulas readily show that 5¢, ;(v,; £) is related to 6px in the same way as the self-velocity distribution ¢, ; is related to the N-particle distribu-

tion py in general.* Moreover (XII.6) reveals that Sp,(r, 0; ¢) satisfies the

Liouville equation (VIII.3). However Spy is not a probability distribution in phase space: it does not have the proper dimensions, it is not positive definite and its integral over t and v is equal to zero. These differences are of no importance in the formal developments and

the reduction of the Liouville equation for 5p, to a kinetic equation for de, ; * The self-distribution g, , is defined in terms of py exactly as the usual velocity distribution [see (VIL74)]. The only difference is that ¢, is introduced when py is a symmetricat function of all the particles, whereas in the “self’’-problem, particle 1 plays a special role (see also Chapter V, Section 5.1.).

318

Calculation of Time-Dependent Correlation Fanctions

is exactly the same as the one from py to ¢,. Thus the results of Chapter VIII can be used at once. In particular, we obtain from (VIII.38) 9,895,1(¥13

=

[ ar B(y,; t, T) +D(v,; t)

(XII.8)

The only differences with Chapter VIII are that in the definition (VIII.39) of B, we should make the following replacement:

on > Bon = {dr dovle, 050)

(xIL9)

while in the definition (VIII.40) for 2, we must change px(0) into dan(0):

pr (0) => 6pn(0) = To

obtain

from

(XII.8)

“ps2ed

(XII.10)

Dix

kaT

a closed equation

following factorization property of Spy:

for 59,,,

we

must

N

Son (0; t) = 59,1(¥1; £) TT ee.)

use the

(X11.11)

For gy we indicated that the corresponding property (VIII.41) can be shown

to be correct, if it is assumed to hold at t = 0. The same is true here; however

we must not make any assumption on the initial condition (XII.10). Since on = Tr. 195°(vq), (XII.11) is clearly correct at t=0. With (XIL.9, 11), (XII.8) becomes the kinetic equation (cf. (VIII.42)} t

8,805,1(¥15 1) = -| dr G(¥1; T)8G.1(¥15 $7)

°

+B, (V5.0)

(XIL.12)

Here the linear non-Markovian collision operator G, is

G.(v,5 7) = tim | dv3 +++ dvy Gy(0; 7) Here.)

(XIL.13)

and the inhomogeneous term is DAV; 0 = lim | dv2...dVy Duo; t|ndpn(0))

(XII.14)

where Gy and Qy are given by (VIII.22, 23) respectively. The linearity of the collision operator is not a surprise, since this property was

observed

in our previous analysis of self-diffusion (e.g., Chapter V, Section 5.1).

What is more remarkable is that linearity is not a special feature of self-diffusion: we

Connection Between Correlation Functions and Kinetic Theory

319

would also find it if we studied the other Green-Kubo integrands. This point deserves

some comment. Far from equilibrium, the generalized molecular-chaos assumption

is written

on(v; t) = Terres t)

(X11.15)

which has to be understood in the more rigorous form (VIII.57). In a self-diffusion

problem it is argued that all particles except 1 are at equilibrium, and this leads to (XIL.11). Consider instead a problem where all particles play the same role; then (XII.15) immediately leads to the nonlinear generalized kinetic equation (VIII.42).

But in the frame of the linear-response theory, we are not studying an arbitrary nonequilibrium problem: we limit ourselves to small deviations from equilibrium. If

we write

Pilva; £) = P%(v,) + 59, (a5 £)

(XIL.16)

we find that the deviation of gx from the Maxwellian is N

den = ¥ 89,(vq; t) O 93(v5) awl

(XIL.17)

wa

to first order in 69,. This Jinear expression of course leads to a linear collision operator, quite analogous to (XII.13).

It is clear from (XII.2, 4) that to compute I(t) and D we need the solution of the kinetic equation (XII.12), for which we need explicit expressions for

G, and 9,. This program cannot of course be fulfilled exactly, except for very special cases. It is nevertheless instructive to pursue the formal analysis a

little further. We take the Laplace transform of 5¢,,, [see (X1.25)]

86.0133)

| dteSe,.,(v;; t) 0

1 55,1041; =f

2a

IAA dg e”"39, (v3.3)

(cs

(XII.18a) (XIL.18b)

where G is a parallel to the real axis above the singularities of the integrand. From (XII.12) and from the initial condition

$@,1(¥1; we get

. * 1 8G5,1(¥i, 3) RCT)

mv,

t= 0) = kT e1"(v)) gl

[

(XIIL.19)

ue eq a kaT Pio) +B; ly» ¥) ] (XI1.20)

where the Laplace transform G, and &, are defined by equations analogous to (XII.18a).

Calculation of Time-Dependent Correlation Functions

320

From (XI1I.4), we then get the following formal solution for the transform of the normalized velocity auto-correlation function: | keT eq [ 1 =m =. G,(v135)

dv, “WE

TG) =".

wig) oly

DixPi (e+

(XI1.21)

Therefore the self-diffusion coefficient (XII.2) can be rewritten

pak! Mm

jim fie)

¢70+

. kgT » e te1 [oueien+ ® G,(vy;ie)| (x01.22)

=-lim. { dv, e370.

where

C,=— erlimOe G,(v1; ie) =" { dr G,(¥13 7) 0

(XII.23)

is precisely the Markovian generalized Lorentz collision operator (X.74) (here written for arbitrary forces).

These formal results are interesting in many respects.

(i)

(i)

They give the formal support to the intuitive analysis of Chapter X. The transport coefficients (here D) are of the form suggested in (X.79), involving the inverse of the Markovian limit of the generalized collision operator. Let us stress that this property is an exact consequence of the theory and does not result from vague heuristic arguments, such as those we gave in our general discussion of the approach to equilibrium in Chapter VIII, Section 6. If we substitute in (XII.22) the (formal) density expansions

p=1p+D®+-

a

n

C=

nC +n?CP

+- +:

+n G+

--

G, = 1G

(XI1.24)

we find to lowest order inn D=-

. im

1

f av Mie Ga,

«

Peron)

(XII.25)

We leave it as an exercise for the reader to verify, following the method of Chapter IX, Section 3, that C° is the Boltzmann-Lorentz

Connection Between Correlation Functions and Kinetic Theory

321

collision operator (V.170). Thus (XII.25) is identical to (V.177): there

is complete equivalence between the correlation-function formalism and the method of hydrodynamic modes. When Hes densities are considered, we find terms involving

C?, C®,...,

and terms involving 3, Q,....

D® =

lim

dv; rd

0s

2

=a

1

1

a

Cah

s

Ss

For example, e

D111)

(XII.26)

50%,; ie)|

The first term in the integral has the form (X.80), and by working out

the density expansion of (XII.13), we would verify indeed that C® is the Choh-Uhlenbeck operator, which is displayed in (X.75) for hard spheres. The second term, involving 9Bb, was discarded in the analysis of the divergence problem because it remains finite, even in two dimensions. Gii)

Equations

(XII.21-23)

show

that the simplicity of the correlation-

function formalism is largely illusory because when it is worked out explicitly, one immediately encounters the same difficulty as in the general kinetic theory: a need for explicit expressions for G, and S,, which can be obtained only in simple cases. 13°

An Alternative Kinetic Theory

In Section 1.2 we used the same formalism as in the general theory of Chapter VIII: the only difference is that the initial condition is here specified in terms of the equilibrium distribution [see (XII.7)]. In particular, the projector Py, which appears in the definitions of Gy and Dy [see (VIII.22, 23)], makes no reference to the equilibrium average. It is often convenient to take more benefit from the fact that we are dealing with a linear-response problem; this is done by using new projectors. In the problem of sel-diffusion, for example, the projector Pees

osi5

dvy:° -dvw | dr:

(XII.27)

is often used. It obviously satisfies P\y= P.* This definition is chosen such

that, as can be seen from (XII.5) and (XII.7),

d9,1(V1; 0) = oS * Pi,

is also

self-adjoint

(P= Pt)

with

respect

(flan) =J de dolos)”'FRdr, v)en(, v) [ef. (V.7)].

Pio m0) to

the

(XII.28) phase-space

scalar

product

322

Calculation of Time-Dependent Correlation Functions

and

Qnbpn (0) = (1 — Pr) den(0) = 0

so that all the “relevant information”

(XII.29)

is contained in Pydpn(t),

and the

initial condition is entirely contained in the “relevant subspace” spanned by Py. The calculations that led to the formal master equation in Chapter VIII [see (VII.15-21)] can be reproduced step by step with this new projector; with the help of (XII.28), they lead to an alternative kinetic equation for

85.1

' 0,59.1(¥15

with the one-body VIII.22)]

= -{

0

dr Gv;

non-Markovian

1)69,,1(¥1; t- 7)

collision operator

(XIL30)

[cf. (XII.13

Giv,; 7)= lim ar | dvz:°° dvy | dv[8Ly exp (—iQnLyt) OnLy]

and

en giv) (XIL31)

There are two noticeable differences between (XII.30) and our previous result (XII.12).

(i)

There is no equivalent to 9, (v,; 2) in (XII.30): the reason can be traced

(ii)

present approach. Because LXPWis different from zero, the whole operator Ly appears at

back to (XII.29). This is, a posteriori, the best argument in favor of the

the right of the kernel G; in (XII.31), not simply Ly. Moreover, this

straightforward derivation makes no appeal to the molecular-chaos property (XII.11).

To judge the power of the present formulation, let us calculate again the Laplace transform 5¢,1(v1; 7). We get now 1

mv. keT ?'io| )

( XII.32 )

= P1950 a) ~ig 1201" (01)

( XI1.33 )

— 01.0501)

(XI1.34)

66,0013 Gs.1(¥13 5)9) == Givis 9) —ig and from this a

m

r ) =f kT

avy 101, oeGis

1

and D=-lim

e+0,

| dv, Dia

Cyne

Connection Between Correlation Functions and Kinetic Theory

323

with CQ=-

lim Givi; ie)= -|

dr Giv,; 7)

(XI1.35)

The elegance of the present formalism clearly shows in these formulas: at all densities we find for the self-diffusion coefficient the same formal structure as in the dilute gas [see (XII.25)]. Yet both (XII.33, 34) and (XII.21, 22) are exact, therefore equivalent! The point is that this elegance was attained at

the price of introducing the projector Px, which involves the equilibrium distribution px/ and is therefore a much more complicated object to handle than Py.

The choice between the two formulations is largely a matter of taste. It also depends on the type of approximation considered. For example, as we now briefly sketch, a reasonable approximation comes readily out of the present kinetic theory in the case of hard spheres. As we discussed at length in Chapter X, Section 2.3, hard-sphere interactions can be treated as smooth potentials if we replace 5L, by 6Ly, the singular pseudo-Liouville operator (X.39). This leaves the kinetic equation (XII.30) unaffected except that the kernel G{(v,; 7) now involves a supplementary impulsive term [see (X.41)]:

Givy3 1) = im | on

dy,"-: avy | dr[i8(r) hw

+6L£y exp (-iQME yt) ONE y aac 5

(XI1.36)

Assume now that a good approximation to Gi(v,; 7) for all 7 is furnished by its short-time value, which is dominated by the impulsive term (oc 5(r)}. Thus we write

1 _ Gilvy; 7) =6(7) lim aI dy... dv dx i8Ly

PN

er)

=GFv.;7)

(XIL37)

Quite remarkably, (XII.37) is equivalent to the Enskog approximation (Chapter VI). Indeed, it is readily verified from the definitions (X.34, 39) that

G®(v,; 7) = -8(r)CF

(XII.38)

where the Enskog operator CF is defined by (VI.76)! This result throws light on the empirical proposal of Enskog: since the approximation (XII.38) becomes obviously exact at ¢=0, we see that close to equilibrium at least, the Enskog theory is the short-time approximation to the exact kinetic equation (X11.30).

Finally, the similarity of (X1I.33) and (XI1.34) allows us to guess the approximate

time dependence of I(t) in the Enskog theory. With (XII.38), we get indeed

A

Ps)=

DFE

kaT/m

7"|

B

m

zr kat in

av

1

aeee Vi PTV1)

1

| dv, 0), "CEL Ce,"

xP (v1)

(XII.39)

324

Calculation of Time-Dependent Correlation Functions

(the superscript E in the left-hand side refers to the Enskog approximation). Now Chapter VI (VI.77) gave in the first Sonine polynomial approximation E E Diy = 3Tyet keT/m 2

(XII.40)

B

where the Enskog relaxation time is

k

W241

|

T= [4vmarn Y*(n) (=)

(XIL41)

The analogy between D® and T suggests that Pa

1

(XIL.42)

MO@=FZ an?

and, taking the inverse transform 1

ro=z|

c

1

ge" 3TE

we arrive at the approximate formula

TF) =18,(0) = exp (=

(X11.43)

= 2t

a )

(XI1.44)

rel

Detailed calculations confirm this result; it is of use in Section 3.

2

THE VELOCITY AUTOCORRELATION FUNCTION IN DENSE FLUIDS

The formal approach of Section 1 is of no help for dense fluids, and much simpler, mostly phenomenological methods are called for. The most naive assumption about the behavior of the velocity autocorrelation function of a particle in a fluid is that it relaxes exponentially: Fi) =e %"

(XIL45)

where v,. is some relaxation frequency. This corresponds to the Markovian behavior

=

— rel(t)

(XII.46)

For example, (XII.45, 46) are valid for a Brownian particle, with v,.)= 3/M

[see (III.125)]; however in the Brownian motion theory the evolution of the

%8-particle is assumed to be very slow on the scale of the molecular times of the fluid: this is certainly not true for self-diffusion, where particle 1 is mechanically identical to all the others. The general qualitative discussion of Chapter VIII, Section 2, then suggests that the Markovian behavior (XII.46)

The Velocity Autocorrelation Function in Dense Fluids

325

should be replaced by the non-Markovian form aT(t) = -[ drv(r)T(t—7) dt 0

(XII.47)

where v(r) is some memory function. In terms of the Laplace transforms I'(y) and P(g), we have (XII.48)

Having physically motivated (XII.47, 48), we can now forget about this “derivation” and consider these equations as providing the definition of the memory function v(t) [or #(g)]. Since #(g) reduces to the constant {3/M in the Brownian limit, it will presumably remain a fairly simple function of gin general. The strategy is then to make some simple (hopefully reasonable) phenomenological assumption on the analytic form of #(;)

P(g) = Dprenlg |@1, 2, -- ») where @,, a@2,..., are adjustable parameters. Once

parameters

are determined

properties are of two types. 2.1

from some

(XII.49) Donen iS chosen, these

exact properties of I'(g). These

Moment (or Short-Time) Sum Rules

The reality and parity properties ['(¢) =I'(#)* =I'(—1) ensure that the com-

plete Fourier transform

+00

r= |

dte™T(t)

(XI1.50)

T, =2Ref(w)=T_,

(XI1.51)

—20

is real and even in w:

The even moments with respect to I’, [odd moments vanish by (XII.51)] are defined by

(w?")p

_ [ede w"T,,

"edo P,,

= 2/

do w" ReT(w)

(XII.52)

We have used the fact that 1

+00

ro=+ | dwoT,=1 2a Jo

For any given #(g), these moments can in principle be calculated from (XII.48). Yet from elementary properties of Fourier transforms (based on

326

Calculation of Time-Dependent Correlation Functions

the correspondence d/dt =

—iw), they are also given by

(™*)r=(-S)'T0],.0 2

n

(X11.53)

and thus are obtained from the short-time behavior of the system. But as we show

below,

these time derivatives are related, if n is small enough,

to

relatively simple equilibrium averages. For example, the second moment is equal to the following integral involving the equilibrium pair correlation function g$%(r):

w=

(xI1.54)

|" dregs n(nVv 2aV r dr

3m

From such relations the adjustable parameters a, a2, . . ., introduced in the

assumption (XII.49), can be expressed in terms of equilibrium properties. To establish (XI1.54), we start from (XII.53); using (X1I.72), we get 2 (w")+

—

=

a(t) dP

V6

mo

i«

—

Ka Fees

O)21.(0))

mele

Ere”)

eq

(XIL.55)5

2yeq

which, from Newton’s law, is also equal to

1

{opr= mk

(( 2Bs are) \" Pies

XIL.56 (XII.56)

To proceed, we first use the definition of the equilibrium average and the isotropy of the fluid to write

yp =

(or

1

|

3mk,T

dtdo

NaV(r) ra 1,

=

NaV(ry,) e7BHn

a

-_

ry,

Zn

(

XIL.57

and we then perform the following transformations:

NaVrie) la

2

oor= Sif ae aere a,

_ { 3m

esate

a @ PMN

ar,

Ly

FP Virg} € PMN

i=

+ Oia

Zy

(XIL-S8)

From the definition of the equilibrium pair correlation function [see (VII.75, 77)], we

arrive at (XII.54) after going to polar coordinates. A similar calculation leads to (op=

1

{

N

3m? \age2

@ Vira), PV (ris)

Oy ar,

Oty ar

+68, )

which depends on the three-particle correlation function 23".

(XII.59)

)

The Velocity Autocorrelation Function in Dense Fluids

2.2

327

Zero-Frequency Sum Rule

From (XII.2) we have

="sTpPa

ro y= 0)

If all the parameters in a

(XII.60)

@,...) have been determined by the

short-time sum rules, (XII.60) provides a “calculation” of the self-diffusion

coefficient. This procedure is rather paradoxical because a long-time quan-

tity, the transport coefficient D, is thus obtained from the short-time behavior of the system. Consequently the value of D obtained from (XIII.60) can be expected to be rather sensitive to the choice made for ¥phenMoreover, since any reasonable choice for Pye, depends on at least two parameters, the method requires (at least) the knowledge of (w 2), and (w*)p,

and the fourth moment involves the equilibrium triplet distribution, which is poorly known. Yet the results of such a calculation are reasonable, as we show below. Alternatively, if we are interested in the behavior of I'(¢) itself, D can be

considered to be an experimentally given quantity: (XII.60) then gives us an additional sum rule to determine the parameters of Pypren(gla1, @2,...). Let us now illustrate this phenomenological method taking the two most popular assumptions for >(;). 2.3

The Two-Parameter Lorentzian

The relation

Dy)=

~ im, corresponds to an exponential timece

v(t)=" exp (=att) TL

TL

(XII.61)

(XII.62)

Notice that v(t) depends on |t| (to satisfy the required symmetry with respect

to time reversal), and this makes v(t) nonanalytic at t=0 @v/dt\,- o is discontinuous while d?»/det? |,x9 is infinite). As a consequence, the moments

(w?"), calculated from (XII.61) do not exist, except when n = 1; this is a bad defect of the Lorentzian approximation. To see this, there is no need to perform the frequency integrals in (XII.52). Indeed,

from (XI1.53) and (XII.47), we readily obtain oes

aT) dt’

lino

= v(0)

(X11.63)

328

Calculation of Time-Dependent Correlation Functions 4y = {o*)r=

__ &r(t) a

d“T(t)

non

dr‘

dE

=

5 +y(0)?

(XII.64)

which show that (w*), is infinite in the Lorentzian model, whereas the exact result (XII.59) is of course finite for smooth potentials. Notice also that whenever I(t) is

analytic at ¢= 0, the even character of this function imposes dv

dtl-o

aT

df

fee)

=0

(XII.65)

which is of course violated by (XII.62).

Thus from the short-time sum rules, we only gain one condition

(w?)p ==

(XII.66)

TL

To get a second condition, we need to use (XII.60), which yields

_keT eames

(X11.67)

Computing (7); from the available data on the equilibrium pair correlation

functions, and v, from the experimental value of D, we determine the parameters of the Lorentzian (XII.61), hence I'(w). For argon at 91.4°K and

1.428 g/cc (which is close to the triple point),* typical values are

{w*)- = 6.2

107° sec?

tL = 1.7210"

sec

(XIL.68)

The function Re f'(w) is plotted in Fig. XII.1, where it is compared with computer experimental

results discussed in Section 3, and with the two-

parameter Gaussian model. 2.4

The Two-Parameter Gaussian

We take

22

Re #(w) = vg exp ( 278)

(XII.69)

and the imaginary part of 7 is determined from the Kramers-Kronig relation [see (XI.113)). In time variable, this corresponds to _¥o (= 2 ) v(t)

To exp

ra

(XII.70)

The Velocity Autocorrelation Function in Dense Fluids

329

2Ref(w)10"sec)

W(10" sec) 0

1

2

Figure XII.1 Spectral function of the velocity autocorrelation function in dense argon: , computer experiment (cf. Section 3); ------ , Lorentzian memory function; ---,

Gaussian memory function.

Here the derivatives at ¢ = O are all finite, and we obtain from (XII.63, 64)

76TT = (wy), aT v,

=F = (@*)p— (wt

276

(XUL.71) (XII.72)

From (X1I.60), the self-diffusion coefficient is also related to the second and fourth moment, as follows:

D

7 2y2y1/2 _ ket a ) V2(w 4) nos Wn)

(X11.73)

For argon in the same condition as for (XJI.68), a numerical evaluation of

the equilibrium average (XI1.59) gives (w*)r = 9.92 x 10°! sec”*. This leads

to

D=2.43x 107° cm?/sec

(XII.74)

which compares reasonably well with the computer experiment datum,

1.78 x 10° cm?/sec. The corresponding spectral function is plotted in Fig.

XII.1. In view of the crudeness of the model, the results are not unsatisfac-

tory except at low frequency.*

* In the Lorentzian model the correct value of Re (0) is of course imposed by (X1I.67).

330

Calculation of Time-Dependent Correlation Functions

3.

REMARKS

ON

COMPUTER

EXPERIMENTS

Although neutron- and light-scattering experiments give important information on time-dependent correlation functions in liquids, this information

is often difficult to obtain [e.g., neutron scattering necessitates the separation of coherent and incoherent effects, see (XI.190)]. Moreover, they allow

us to measure only the Van integrands

for the transport

Hove functions, whereas the Green-Kubo

coefficients, for example,

theoretical interest but cannot be measured directly.

also are of great

We mentioned in Chapter I, Section 1, that a new kind of data has become

accessible through the development of fast computers. The idea underlying these computer experiments is very simple: to solve Newton’s equations (1.1) for asystem of N particles with given initial phase-space configurations, chosen randomly. From this solution, any equilibrium property can be calculated as the time average of the dynamical function associated with this macroscopic quantity; this procedure is based on the assumption that the system is ergodic (see Chapter III, Section 2.2).* Since time-dependent correlation functions are defined as equilibrium averages of dynamical functions (taken at different times), they can be calculated by this computer technique. In fact, the data obtained in this way are among the best presently available. It would be out of place to give a detailed

account

of these

methods,t

nor

do we

analyze

the

enormous

amount of results. Yet it is worthwhile to make a few remarks about these

computer experiments and to display some data for the velocity autocorre-

lation function we studied in Section 2. Other examples are considered later.

In the first step of a computer experiment we solve Newton’s equations of motion (1.1) by a finite-difference numerical approximation. Hard-sphere interactions are more appropriate than smooth potentials for such calculations because their dynamics consists of straight-line motions, interrupted by sudden deflections obeying simple geometrical laws [see (X.12, 13)]. Numerical integration is much harder for smooth potentials, and particular

care is needed with respect to round -off errors—that is, the accumulation of

smail errors coming from the finite number of figures memorized at each step by the computer.

The second step is the (computer-time-consuming) statistical analysis of these solutions. For example, we calculate the velocity autocorrelation * Equilibrium calculations can also be made by numerical Monte-Carlo methods, which sample the phase-space integrals defining the equilibrium average (VII.44), This radically different approach has found little application for time-dependent phenomena. t See the excellent review by Wood (1975) and the references quoted there.

Remarks on Computer Experiments

331

function from a single solution (x(t), v(t)) as

(01,.(t)0,,.(0))4= where initial One forces

TL

Lit

[3 z vaelt +7967)

(XII.75)

the sequence of times 7,,... , 7, are sampled sufficiently far from the time that the system can be assumed to have reached equilibrium. of the great advantages of the method is that the intermolecular are completely specified (usually as hard-sphere or Lennard-Jones

interactions). Thus we avoid one of the uncertainties of real experiments,

where the interaction law is not precisely known. Another advantage is that in principle at least, any time-dependent correlation function is accessible: for example, we can compute separately both the Van Hove functions, an impossible feat with neutron scattering. Turning now to the drawbacks of the method, it should be kept in mind that in addition to the numerical errors involved when solving Newton’s equation and the statistical errors implied by (XII.75), intrinsic errors arise because the number N of molecules studied is not very large (of the order of 10° at most), and we do not know whether they correctly represent a macroscopic system; in particular, the boundaries of the system play a role. It is natural to think that these spurious effects start to appear when on the average any molecule has felt the presence of the boundary, and this occurs approximately after the time required for a sound wave to cross the system. After a time of this order the computer data cannot be compared to real experiments nor to theories made in the thermodynamic limit. Similarly, when studying space-dependent properties [e.g., the Van Hove

functions F,(q; w) and ¥(q; w)] these size effects prevent us from consider-

ing the hydrodynamic long-time and long-wavelength limit, and we are usually limited to wave numbers such that gry 1 (where rg is the range of the forces), comparable to the range accessible by neutron scattering but far larger than those attainable by light scattering (Chapter XI, Section 4). Let us now briefly illustrate the results obtained for the velocity autocorrelation function I(t). We first consider the case of hard spheres; Fig. XII.2 plots the celebrated data of Alder and co-workers* at a few typical densities. Time and density are expressed in the dimensionless units

Fey, where

resp

+2, is the Enskog relaxation time (XII.41) and

(XII.76) Vy =a3/V2

is the

volume per hard sphere at close packing. Moreover, since we expect the

* Alder, Gass, and Wainwright (1970).

Calculation of Time-Dependent Correlation Functions

332

ae ar (tt) 0.02+-

—— xs5

oor : Ool-

oa 0.03

Tae ~~ T 10

va ey N

\

.

N

_

-

rd

re

7

wo

a7

q T 20

SS AEG Le

Figure XII.2 Deviation of the velocity autocorrelation function for 108 hard spheres, from its Enskog value at various densities. After Alder et al. (1970).

Enskog formula (XII.44) to furnish a good first approximation, it is natural to concentrate on the deviation

(X11.77)

alt= 77 )=TO-THO The following features are noticeable. (i)

(ii)

(iii)

The deviations from the Enskog theory are small, even at the highest densities (x = 1.6 is very close to the crystalization density of hard spheres). This is also apparent in Table XII.1, which gives the values of the self-diffusion coefficient [see (XII.2)] at various densities. When the density increases, the deviations change sign and shift toward larger times. Though small, these deviations do not follow a simple relaxation law and decay unexpectedly slowly. This can be seen even more readily in a two-dimensional calculation (for hard disks)* where

T= Freee = * Alder and Wainwright (1970).

(d = 2, hard disks)

T>1

(XII.78)

Remarks on Computer Experiments

333

Table XII.1 Ratio of the Observed Diffusion Coefficient to Its Enskog Value at Various Densities x

D/D®

100 3

1.02 1,22

2 1.6 15

1.14 0.76 0.55

=_E it F(t) é & é

0.1

4 Aa

4 a

0.01-

0.001

N

1.0

Figure

XII.3

Velocity

we

0.10

0.01

autocorrelation

function

for hard spheres at x=3 on a log-log plot. The straight line is drawn with a slope corresponding to

7 ?/?, After Alder and Wainwright (1970).

334

Calculation of Time-Dependent Correlation Functions

In three dimensions this slow decay is more difficult to observe,* but there is evidence (Fig. XII.3) that 1

Tere) Xap

(d = 3, hard spheres)

(XII.79)

A general explanation of these observations is still to come, but probably all

such

behavior

is rooted

in the laws

(XII.78,

79), which

are rather well

understood and are discussed in Section 6. For the case of smooth potentials, we display in Fig. XII.4 some results of computer experiments with the Lennard-Jones interaction [see (V.163)].+

(a)

{b)

Figure XII.4 Velocity autocorrelation function with Lennard-Jones potentials. (a) n* = 0.65, T* = 1.43 (—), T* = 5.09 (---); (6) n* = 0.85, T* = 0.76 (—), T* = 4.70 (- ~~). After Verlet and Levesque (1970).

Reduced units are now defined with respect to the parameters €p and ro of this potential: a= t _t * 3 T* kaT Jo =—— T (mr2/48e9)2 ae n = (nro) 7 = ¢ (XII.80 )} Of course for this model, we have no good first approximation against which to compare the data. The following qualitative features are apparent however. (i)

At low density and/or at high temperature, the decay is almost exponential; this is quite natural because (a) if the density is suffi-

*See the discussion in Wood (1975). T See

Verlet and Levesque

(1970).

Remarks on Computer Experiments

335

ciently low, Boltzmann’s equation applies and this Markovian description

(ii)

(iii)

leads to an exponential

decay,

and

(5) if the

temperature

is

sufficiently high, the attractive part of the forces is unimportant and we are back to a hard-core system, with its Enskog-like behavior.

At n*=0.85, T*=0.767 (close to the triple point) we observe a negative region, analogous to, but more pronounced than for hard spheres. A marked difference from the hard-sphere fluid is that I(t) behaves

here as 2” near f = 0 (instead of |t|); however this fact has little influence

on the later behavior of F(t).

(vi)

More interesting yet, the decay of the velocity autocorrelation function is also “slow” at high densities. Of course we have no clear-cut

way to define a characteristic relaxation time, but if we estimate this time as r*,~1 (Fig. XII.4), we see that correlations of the velocities

persist for times +* > 7%,. In fact, this is why the Gaussian approxima-

tion (X11.69) fails at low frequency (or for long times), as indicated in

Fig. XII.1. Careful analysist has shown that to get a precise fit (better than 1%) of the data, the following three-parameter memory function had to be used

v(7*19) = (wp exp (

B,

s

#2:

) + Aor** exp (—ar*)

(X11.81)

with ad-hoc empirical formulas for Ag, Bg, and a. There is no point in

giving these formulas here, but it is important to stress that the time scales (VBy)

' and ag‘ are very different: the second term in (XII.81)

describes a much slower decay than the first one. For example, at n* =0.85, T* = 0.76, we find (o 2

(2) 0

5.98

pa, To 1/2

= 0.46,

0.88

Ao=— To

ao'=1.60

This is not unlike the difference we have observed between the Enskog decay of I'®(7r&,) [which vy (7re,) = 28(7)/37=,"] and the much slower decay 5T. This point is supported by a careful simulation,§

(XII.82)

for hard spheres corresponds to of the deviation which has shown

This point corresponds to p = 1.428 g/cc and T = 91.4°K and was discussed in Section 2. + Verlet and Levesque (1970). § Levesque and Ashurst (1974),

Calculation of Time-Dependent Correlation Functions

336

that for Lennard-Jones molecules the behavior (XII.79) also appears

in the long term. The analogy is not yet fully understood. 4

THE

VAN

HOVE

SELF-CORRELATION

FUNCTION

4.1 Simple Limiting Properties and the Gaussian Approximation

As the simplest example of correlation functions of conserved quantities, we

discuss the Van Hove self-correlation function (XI.189). Its Fourier transform

Gglt) =e 18 HO ef Oye

(X11.83)

is known as the intermediate (incoherent) scattering function.

The simplest approximation to (XHI.83) is that of an ideal gas, where Ar, (t)=1,(t)-11(0) = vyt

When we insert this into (XIJ.83), the canonical average formed and leads to

Pk G, 4(t)= exp( 4 Ast)

(XII.84) is readily per-

(XIL.85)

This formula expresses the idea that a free particle moves over a distance

caricon'?= (2)

nes)

or in a time interval ¢, This model should hold either in very dilute systems is 1 particle of for very short times. Indeed, when ¢ is small, the motion interacby dominated by its initial velocity v,(0), since it is not yet affected tions with the other molecules. Notice that for large q this short-time

if q is behavior is the only relevant information: (XII.85) guarantees that play. to start very large, G,. is exponentially small when the interactions q Another extreme case, valid in any interacting fluid, is the limit of small

to and large t. There we expect Y,,,(#) to describe a diffusion process, that is, obey (XI1.87) —Dq?F.q(t) = OGq(t) q70,t70

or

q(t) = exp (—Dq’Id))

(XI1.88)

The Van Hove Self-Correlation Function

337

These formulas correspond to the propagation law

(Arj(0))'? = (6D1)'?

(XIL.89)

Qualitatively, we expect that the time scale over which 4 q(t) decays to

zero becomes shorter when q increases. We then see that (XII.85) should be

valid in one limiting situation [¢ « 7,2, q > ((v) T7e1) "J, and (XII.88) should be valid in the opposite limit. It is remarkable that a simple interpolation formula covering the whole range of q (and ¢) can be obtained from the so-called Gaussian approximation. We write

@,q(t) = exp (—q7w(t))

(XII.90)

and determine the function w(r) in such a way that both limiting behaviors are satisfied.

In fact, w(t) is directly related to the velocity correlation.

Indeed, we can expand (XII.90) around q = 0 as

G4(t)=1—q*w(t)+---

(XI1.91)

whereas from the definition (XII.83), and taking q along the x-axis, we have

Galt) = 1497 Cr (rs (0) —r1 (Ory ,O))+ +++ — (XII.92) Identifying the q?-factor in these two expressions, we get

o(t)=

~En Or.) —11,.O)r (0))*

(XII.93)

From (XI.6-10) we then obtain

o()= [ dr (t~7){0,.(7)0),,(0))* or

(XII.94)

°

w(t)= ‘eT i dr (t-1)T(r)

(XIL.95)

Thus &,,, is entirely determined if I(t) is known either experimentally or theoretically (Section 2). Note that Y,, has the following proper limiting behaviors. (i)

t/te1>0.

We have then I'(t)> 1 and

w(t)> (427) er

(XII.96)

in agreement with (XIIJ.85). (ii)

t/t,:> 00.

Here we use [see (XI.4, 5)] t

i dr (t—7)T(r) ~[

O

Dt

dr T(r)= kaT/m

(X11.97)

338

Calculation of Time-Dependent Correlation Functions

which Jeads to

w(t)= Dt

(XII.98)

in agreement with (XII.88). The

numerical

computation

of ,,(t) from this Gaussian

approximation

compares fairly well with computer simulation and scattering data, in the whole q-f (or g-w) plane; as expected, the agreement is the worst in the intermediate region. To arrive at a qualitative fit there, we must replace (X1I.90) by more complicated interpolation formulas.* Notice also that, in the dilute-gas limit, the solution of the linearized Boltzmann equation allows in principle an exact calculation of Y(t). This solution is obtained in practice with the help of kinetic models, as discussed in Chapter V, Section

5.2;-agreement with experimental data is excellent, provided we take a

sufficient number of terms in the model operator (V.194).7

4.2

Generalized Hydrodynamics Illustrated

Despite its success, the Gaussian approximation is not always the most convenient approach. Indeed, it is expressed in terms of the variables q and #,

and comparison with the experimental data requires a numerical Laplace transformation over time; this tends to hide the physical meaning of the

results. In addition, no similar approximation can be developed for the Van Hove total function, because the existence of propagating sound waves

proportional to exp (+ ic,qt) destroys the Gaussian behavior in the small-q

limit. A memory-function approach has been proposed that circumvents these

difficulties; in the present context, which deals with conserved quantities

treated at arbitrary wave number and time (or frequency), this approach is called generalized hydrodynamics. The most original application of this method is the approximate calculation of the Van Hove total function, but the technical problems are rather tricky because the couplings between the five conserved quantities vary according to the values of q and ¢. For pedagogical reasons, we describe the method as applied to the simpler

problem of the Van Hove self-correlation function.

The idea is straightforward; for arbitrary q and ¢, we write

8,G.q(t) =—q? { dt D(q3 1)Gq(t-7)

(XII.99)

* Verlet and Levesque (1970); Skdid et al. (1972). t Chen, Lefevre, and Yip (1973); let us also point out that the function i,,(w) discussed in Chapter V, Section 5.2, is identical to the Laplace transform of &, ,(t).

The Van Hove Self-Correlation Function

which defines

the memory

339

function D(q; 7). The rationale behind this

procedure is to generalize, in the most natural way, the diffusion equation (XIL87): when the wave number increases and the time gets shorter, nonlocality effects in space and time should start to appear. The hope is that in the form (XII.99), a simple choice for D(q; 7) will give a good approximate description of Gq.

A two-parameter approximation (e.g., an exponential in time) for D(q; £) itself would be too simple. The reason is apparent if we remember that in the

small-g limit

p=KsT/ m

which suggests that

“9

dt r=] kpT

D(O; t)

7

dt D(0; t)

0

rt)

(XII.100) (XIL.101)

Section 2 indicated the need for a memory-function description to obtain a

fair approximation to I(t); similarly, we should write here [cf. (XII.47)] the non-Markovian equation

a,0(q; t)= -|

0

drv(q;r)T(q;t—7)

(XII.102)

for the function

.y_P@s9

Tq; = koT/m

(XII.103)

Hopefully simple approximations to the new memory function »(q; f) will then lead to a reasonable form of Pq; t) [or D(q; 1)], thus of Y,,(¢). Despite

the property

F(0; 1)=[(4) suggested

in (XII.101), the function

except at q = 0, differs from the local velocity autocorrelation function*

Ril) kyT/m

_ (or g(0) ev, (0) ery" k,T/m

I'(q; 1),

(XII1.104)

Indeed, taking the second derivative of (XII.83) with respect to ¢, we use (XI.72) to

get

Galt) = GPR iG)

(XI1.105)

and, in Laplace transform, this leads to

.

1

caren

2

© Baas)

(XII.106)

* The subscript j,,, comes from the definition of the local self-current j,, = v),,5("—1,).

340

Calculation of Time-Dependent Correlation Functions

whereas (XII.99) yields g,

(0)

1

=o

XII.107

io +q°D(q; 0)

CatD

Comparing these two expressions, we obtain

Rissinx (43 ereCRe k_T/m

(@eor,

mw

1

1

(XII.108)

)

T(q;@)

which shows that R deaf /(kaT/m) is identical to hq; w) only when q = 0.

We now proceed with the memory function »(q; t) as we did in Section 2

for the function v(t). For instance, we can choose the Gaussian form v(q3, t=

0

¥(9) ex

Folq)

( —at? )

(XII.109)

? \478(@)

and determine the parameters vg(q) and 7¢(q) from the short-time behavior

of Y(t). This calculation is more tedious than for the velocity autocorrela-

tion function because v(q; #) is only indirectly related, through (XII.99, 102, 103), to ¥q(¢). As is sketched in Appendix J, we find

megs O28

5 ego. [ood —o98-+3w02(3(00%)p-+ 202)

(XI110)

where 2

w2= thst

(XI1.111)

m

and (w”)p and (w*), are the (g-independent)

moments

of the velocity

autocorrelation function [see (XH.54, 59)}. We can readily (XII.110) has the proper limits (XII.71, 72) when q > 0.

check

that

In this way we obtain all the elements required to compute the incoherent scattering function

¥,(q; @) =2 Re %,,(w) _

2w2F'(q; @)

[w? — w5—w0"(q; w)F +[wr'(q; @)

(X11.112)

Indeed, from (XII.109), the real part of #(q; w) is

22

5'(q; @) = v¢(q) exp (-222@)

(XI1.113)

The Van Hove Self-Correlation Function

341

and the imaginary part can be computed from the Kramers~Kronig relation [see (X1.113)]

sq0)-+0| WT

+00 x

wie

deo’

or

Ge) ao~-@

(XIL.114)

Figure XII.5 plots the height ¥,(q; 0) and the half-width at half maximum

®,,1/2(q), a8 a function of g for liquid argon: the quantities are measured in

units of the corresponding diffusion approximation 2

F,(q; 0) ditt = ge

Lae

5, sou(q,0)0a7 2

Lab

13

13h

2b

4 a

IFoo

1

nn ° a 1 0% :

1

oar 08"

Figure XLS

ost

+

n

oS

t

LD 5

+} 5

°

ot

By 5

-1,* aX)

can

({b)

Height ¥,(q; 0) and half-width «, 1/2(q), compared to their diffusion value,

calculated from (XII.112) for liquid argon (T= 85.2°K) (—, neutron scattering data). After Skold et al. (1972).

Moreover,

.

2

ost

(a)

(XI1.115)

W, S2, .(q)/0q2

te

°

>

@s,1/2(4) aise = Dg”

rather than

using the badly known

theory; +, computer data; O,

(w*);, we

have fixed this

fourth moment with the help of (XII.73) and the experimentally determined value of the diffusion coefficient (D = 1.94 107° cm?/sec). Figure XII.5

‘also presents the corresponding data coming from computer experiments*

and from neutron scattering. t

It is seen that the agreement between theory and experiment is not very

good except for the qualitative feature that the deviations from the simple diffusive behavior are rather small. The Gaussian assumption obviously is * Verlet and Levesque (1970). + Skéld et al. (1972),

342

Calculation of Time-Dependent Correlation Functions

not sufficient to get quantitative agreement. In particular, in the limit of high wave numbers, we have seen that the free-motion approximation becomes exact [see (XII.85)]; we have thus f

—ry2

(XI1.116)

L(g;@) q72= 2Wo exp ( 205“) In the same limit (XII.110) yields 7

2m)

¥6(4) =f 3) which,

when

inserted

into

1/2

_

#744)

(XH.113,

114,

(2)

Ng)

112),

1/2

does

1

Gg,

(XII.116). Yet the difference is not very large; for example,

qi lous _ 21 45... SAGs Nleracr qooV3

Improved

approximations

to v(q;t)—of

the

type

(XT) not

reproduce

xuL.t18)

(XII.81)—have

been

proposed, but they are not much more than ad-hoc formulas. To end this section let us point out that our empirical approach to generalized hydrodynamics can be formalized with the help of projectionoperator techniques and continued-fraction methods.* Yet, this formal approach, which in principle gives microscopic definitions for the various memory functions, cannot be pursued very far when looking for numerical predictions. We must resort to approximations that are equivalent to those we discussed from a more pragmatic viewpoint. 5 THE VAN HOVE TOTAL CORRELATION FUNCTION

The spatial Fourier transform of the Van Hove total correlation function (X1.188) is 1/XN

.

N

er.

o\"

G,(t)= a Dy eisai yy eaiarrel ) N\a=1

b=

(X11.119)

At t= 0, this function is simply related to the equilibrium pair correlation

function g5%(r). Indeed, isolating the terms a = b in (XII.119) and using the definitions (VII.73, 75, 77) for the terms a #b, we readily find for the

so-called static structure factor (at q #0)

G,(0) = 1+ngs

(X11.120)

where g$‘is the Fourier transform of g3"(r). *See Copley and Lovesey (1975) and references quoted here; see also Forster (1975).

The Van Hove Tota! Correlation Function

343

It is interesting to notice the Y, (0) can be directly measured by scattering experiments; indeed, if the scattering function S(q; w) is narrow enough in frequency (which turns out to be the case if g is small enough), we have

Pout * Pin in (XI.191) for all the relevant values of w. By integrating (XI.191) over all energies, the so-called elastic cross section is obtained

do

oT)

+00

si.

2

2

Nb |

d‘a con” 40 70 do

Dr

+0

AE)

=2 = Nb @,(0)

(XI1.121)

Neutron scattering [or light scattering for which, from (XI.200), a similar result holds] thus gives interesting information about the equilibrium properties of the sytem. Another remarkable feature of (XII.120) is its connection with thermo-

dynamics in the q = 0 limit: it can be demonstrated that

Palim G,(0) = 1+ ng§?= nk Txr

(XII.122)

where y; is the isothermal compressibility

1/a

xr= -(*)

T

(XIL123)

The proof of this so-called fluctuation theorem is tricky and is not presented here.* Its validity, however, is suggested from the results of Chapter VII concerning moderately dense gases. From the virial equation (VII.94), we get indeed

Xr=1/(nkgT(1+2nBy(T)+ O(n”)]

= (nkgT)'[1-2nB A(T) + O(n?)]

(XH1.124)

and the limiting property (VII.86) implies

lim g7°= | dr {exp[—BV(r)]— 1}+ O(n) = -2B(T)+ O(n)

(XIL.125)

where (VEI.95) has been used; comparison of (XII.124) and (XII. 125) shows that (XII. 122) is indeed satisfied in a moderately dense gas. * See, for example, Landau and Lifshitz (1958).

344

Calculation of Time-Dependent Correlation Functions

5.1

The Hydrodynamic Limit of S(q; ): The Landau-Placzeck Method

In general, ,(¢) {or A(q; w)]is difficult to calculate even approximately, but in the small wave number and long time (or small-frequency) limit, a simple macroscopic argument, due to Landau and Placzeck, considerably simplifies the problem. To see this, let us rewrite the Van Hove function as

G4 (0) = 3 (Bq(008-a(0))"

(XI1.126)

where N

(XII.127)

5

pa(t)=m amlY eft

is the microscopic mass density, which depends on the detailed motion of all the particles

in the system.

Formally,

we

average involved in (XIJ.126) in two steps. (i)

(ii)

can

perform

the equilibrium

We make a partial phase-space average with the constraint that a given mass-density fluctuation p, is imposed.* We then take the properly weighted average over the fluctuation pg.

Using the definitions (X1.15, 92) and (VII.42), we thus writet

G0=

1 5-95 1

a -ityta 5 | dt dopye NG ge OF -BHy

= pita]

Nm where

dba

a

de doy

elie

LNB 68 (pq

A

Bq) @ OF

dp, {o-«P(oa)| { dt dv py e*n'p\io.)]}

3 dv(pg— 5(0,-p) fq) eP™ i = [dr Ca Pal Pn(Pq)

I (XI1.128)

(XI1.129)

* Strictly speaking, we should also impose a velocity u, and a temperature T,. Here we anticipate that these fluctuations do not couple to pg. + From (XII.127) we know that 6,= 64+ i#q is a complex quantity that satisfies the reality conditions y= 6*,. The integrals over pg and the Dirac delta functions in (XII.128-130) thus should be expressed in terms of the variables p, and pg. We ignore this point in our formal calculation, treating pq = p*, a8 a single real variable.

The Van Hove Total Correlation Function

345

and P(p,) _ jdt

_ 6.)

e7Plin

dv 5(pq~ pa) eT ™

ZN

(XII.130)

Clearly, P(p,) can be interpreted as the probability density of a massdensity fluctuation pg. Moreover, (XII.129) reveals that py describes an equilibrium phase-space distribution except for the constraint that the local

density varies as pq e‘*". If q is small enough, we can think of our system as being divided into cells of size R such that q~'« R« rp (r9= molecular

interaction length): in each cell, py describes an absolute-equilibrium state with a constant density and is thus a local-equilibrium distribution (see also Chapter [V, Section 7).

The time evolution of

PMpg; 1) =e ppg)

(X11.131)

is entirely due to the slow spatial variations of the mass density; in the absence of these, p would be a constant in time [see (VII.33)]. But since this density is a conserved quantity (lim,..9d,p4 = 0), it evolves very slowly if q is small; on this slow time scale, it is then legitimate to assume that pM 03 t) adjusts itself continuously and remains the local-equilibrium distribution corresponding to the macroscopic local mass density p,(t), which obeys the linearized hydrodynamic equation (V.61):

Prog: 1)=pn(Oa(é))

(X11.132)

With this crucial assumption, (XII.128) readily leads to

1

G() = Nm? | pg Pal t)P-qP( Pq)

ee

&

GAG

*

(XI1.133) In going from the first line to the second, we have used (XII.128) (at t

and (XII.122) to get*

1 al m

dp gPaP-_P(Pq) = G (0) q70 > nkgTxr

=0)

(XI1.134)

* We could also use macroscopic arguments to evaluate the probability distribution P(pq) for small q [see Landau and Lifshitz (1958)].

Calculation of Time-Dependent Correlation Functions

346

The scattering function Y(q; ) thus takes the form

#(q; 0) =2 Re G,(w) =

=

kare

w-0

C,-C,

pe

CG,

2q°«/ pC, EP

wo +(q?x/pCG)”

ql. + (es2y ts > ]

XII.135 M139)

Cr wicgye@Ty)

This is the famous Landau—Placzeck formula. It shows that in the low wave number limit, the scattering function has a three-peak structure: (i) (ii)

Acentral (or Rayleigh) peak, describing isothermal heat propagation. A (Brillouin) doublet around w = +c.q, describing damped adiabatic sound propagation.

This structure, illustrated in Fig. XII.6, furnishes a wealth of information. S(q;) (q FIXED

T l ' iy 1 ' ! ( 4 \ '

Figure XII.6

(iii) (iv)

2754

2q?K IPC,

! -C,4q

(i) (ii)

AND SMAI

0

+09

w

The function /(q; @) in the hydrodynamic limit

The total area under S(q; w) allows us to estimate the compressibility. The position of the Brillouin peaks gives the sound velocity, and their width gives the sound-absorption coefficient. The width of the Rayleigh peak gives access to the ratio «/C,. The ratio of the area under the Brillouin peaks and the Rayleigh peak is equal to (C,—C,)/C..

The Van Hove Totat Correlation Function

347

These results are very well confirmed by light scattering experiments, where the required small wave numbers are easily accessible.* Neutron scattering is less adequate because the transferred wave number is generally too large.t 5.2.

Nonhydrodynamic Regime: De Gennes Narrowing

When the wave number increases (and for shorter times), the macroscopic Landau-Placzeck argument ceases to be valid; the sound oscillations pres-

ent in (XII.133) progressively disappear, and in the high-wave number limit, where g¢'—0, we recover the free-particle result already obtained for the self-correlation function [see (XII.85)] 242

qt te?) G,(t) = exp(-T% |

(XI1.136)

Figure XII.7 illustrates the smooth changes in Y(q; #) when q ranges from zero to infinity. The curves are for the typical values of q (compared to the Gq

|

p-------}---|----A-X---

o&

(i)

uu

5

(ii) ci

(iV)

(a)

An

Sl; Ww), 2{a)

i

H

{9}

Wes

WI, fo

on

{(Q'p)

uu

w

iii)

Slaw)»

1

turf)

(a)

Her

fA)

(iv)

Figure XII.7 Schematic picture of /(q;) at various wave numbers in a dense fluid. {a) The static structure factor G, (0); (6) (q; w) for the four values of q indicated in (2) @12(4) is the half-width of S(q; w)). * See, for example, Fleury and Boon (1969). f See, however, Bell et al. (1975).

Calculation of Time-Dependent Correlation Functions

348

inverse molecular length ro') where the static structure factor @,(0) of a

dense fluid (obtainable from the known equilibrium pair-correlation function) manifests the most characteristic features.

(i) (ii) (iii) (iv)

qro« 1. The thermodynamic identity (XII.122) holds, and we have the Landau—Placzeck formula (XII.135) for #(q; @). The structure of the fluid starts to play a role; A(q; w) can no qro*1. obtained from macroscopic arguments, though there is still be longer ce of the sound-wave peaks. reminiscen some The function G,(0) shows a strong maximum, manifesting gro~2m. the quasi-lattice structure of the #(q; w) with neighbors at a distance r =r (see Fig. VII.1c); A(q; #) has a single peak. The function Y,(0)~1 (there are nearly no correlations); qrg%1. Y(q; ) is the Fourier transform of (XII.136).

Except in the dilute-gas limit (where successful kinetic models exist), it is

difficult to guess approximations that are free of adjustable parameters and are satisfactory in the whole q-range; the best agreement is obtained by the method of generalized hydrodynamics,* already discussed in relation to the self-correlation function in Section 4.2. The resulting formulas being rather

awkward, we limit ourselves to the much simpler moment analysis (which is

of course an ingredient of generalized hydrodynamics too) and show how it leads to qualitative predictions that are well supported by the more sophisticated methods and in agreement with experiment. The moments of S(q; w) are defined by (w")y,

«

(tS dw w*"S(q; 0) 7 fFS dw S(q; «)

any

_= fp dw w" Re G(w)

IP dw Re G,()

XII.137

(11.137)

and, in analogy with (XII.53), this is equivalent to

(-8)'0h. a\"

(w?" \y,

=$———j G,(0)

(XII.138)

qkpT

(X1I.139)

The calculation of the time derivatives of $,(t) is simple and we merely give the results

(w)9,= Tg © * See, for example, Aifawadi, Rahman, and Zwanzig (1971).

The Van Hove Total Correlation Function

4a

(oy, =

349

g*ksT] 3kaT+n | de g3 in( m™G,(0)

Ee eye *] ax?

(X1I.140) With no better argument than that of simplicity, let us first assume that far from the hydrodynamic regime (i.e., for q large and ¢ small), %,(t) has the Gaussian form

= Hw.) 5 €,(t)lcauss = %(O) exp (

(XI1.141)

or

$(q; ©) Gauss = 8 (0)( Se oar ) 2

o) $4

exp (==)

(XI1.142)

The characteristic half-width w,,2(q) is

winla= on)?=a pao) eo)

(xI1.143)

°

In the high wave number limit, (0) tends to 1 (Fig. XII.7@) and we recover

the free particle behavior (XII.136). On the other hand, the strong peak of @,(0) near qro= 27 indicates that in this region the half-width (XII.143) should show a rather deep minimum: this phenomenon, known as De Gennes narrowing, is indeed observed (Fig. XII.8a).

). genes

units)

ia ny an ot a ou soy moa toy poy 4 i'

t) Ww

2

LANDAU-PLACZECK REGIME.

(a)

(b)

Figure XIL8 Some additional properties of S(q; w). (@) Half-width w ,,2(¢) (—) compared to the static structure factor G,(0) ( ); {6) comparison between a Gaussian ( ) and a Lorentzian truncated at w, © 5w/2(q)} (—).

350

Calculation of Time-Dependent Correlation Functions

Now it is readily verified that with the Gaussian (XII.141), we have the exact relation (e*) y,,.Gauss =o

VF

Gauss

(X11.144)

Hence the validity of the Gaussian assumption can be tested by checking how well (XII.144) is satisfied by the exact moments (XII.139, 140). To geta

qualitative answer to this question, we disregard the second term in the

bracket of (XII.140): it is a smooth function of g that can be shown to be at most of the order of a few times kgT. Thus we have

(co*)y, = 3(w)%, G, (0)

(XII.145)

Comparing with (XII.144), we see that the Gaussian approximation is seriously in error when qrg=27. There the exact fourth moment is much larger than for a Gaussian, which means that the ‘‘wings” of the true spectral function S(q; w) are underestimated by (XII.142): (q; w) should have a more Lorentzian character. In fact, with a pure Lorentzian, the wings would be overestimated (all even moments 2n > 0 diverge), but the point is correct

qualitatively: a transition from Gaussian-like to Lorentzian-like behavior is experimentally observed when gro 27. We have illustrated this in Fig. XII.85, which plots a Gaussian Y(q; w) together with a Lorentzian S(q; w) truncated at w, = 5w1/2(q). The difference in shape of these two functions

[which have the same integral |*% dw S(q; w)=27G,(0) and the same second moment] is striking. Nevertheless the width w, ,2(q) is always qualitatively given by (XII.143). Of course in the hydrodynamic limit q > 0, this width ceases to be very useful because of the appearance of sound-wave peaks. 6

6.1

THE

LONG

TIME

TAILS

The Phenomenological Approach

An unexpected result of the computer experiments has been the discovery of

the slow power-law decay of the Green-Kubo integrands—in particular, the

velocity autocorrelation function—in the long-time limit [see (XII.78, 79)]. This result is crucial both fundamentally and practically. For instance, it

forces us to reconsider our naive ideas about the approach to equilibrium, which we assumed to be governed by processes localized in time (i.e., lasting approximately during a relaxation time) and in space (extending over distances

of the

order

of the

mean

free path).

Furthermore,

this slow

power-law decay seems to be responsible for the deviations of the velocity autocorrelation function (and of the other Green-Kubo

the short-time (or moment-analysis) predictions.

integrands) from

The Long Time Tails

351

The study of these long-time effects—and of their consequences—has concentrated many efforts in nonequilibrium statistical mechanics in the last few years, and active investigation continues. It would be out of place to present here a complete account of present-day research; we sketch the main ideas, on which there is overall agreement, and we suggest various fields in which these ideas are applicable. The reader is referred to the

literature for the details.*

In its crudest version, the theory of the slow power-law decay is very simple: it uses macroscopic arguments, which were suggested to Alder and Wainwright by an analysis of the pattern of the molecular motion observed in their simulated system. Let us concentrate again on the velocity autocorrelation function. Suppose that at

t=0, molecule 1 has some velocity v,,,(0)

(Fig. XH.9a). The other molecules are at thermal equilibrium, but since this

“a Qe

WwW.

Ww

Tel

t=o

t= Tel

t wT at

(a)

(b)

(c)

Figure XH.9

t

Schematization of the slow decay of I'(1).

isotropic distribution does not contribute to any transport process, we may as well suppose that they are at rest. Particle 1 interacts with its neighbors,

and after a short time 7,,, it will have shared its initia! momentum with the molecules lying in a small volume w,,,, around it (Fig. XII.9b); hence it

moves with a velocity

Dia Fes) eM)

(XI1.146)

The further decay of v, ,(¢) can occur only because as ¢ increases, the volume

@,, within which the fluid molecules have been set into motion, grows larger

and larger. Now however complicated the initial expansion of w, may be, we

expect in the long term that this expansion will be governed by hydrodynamics. The velocity field then propagates by two mechanisms: (i)

Longitudinal

sound

waves,

neglected here [see (V.63)].

which

are

fast

processes

and

can

be

* See, for instance, the report by Pomeau and Résibois (1975) and the references quoted there,

352

Calculation of Time-Dependent Correlation Functions

(ii)

Transverse shear modes [see (V.62)], through which the linear dimension R, of w, will grow by velocity diffusion, in a way analogous to

(XI1L.89) (Fig. XII.9c):

R, rex (2)

1/2

© (R,)*

(2)

(X11.147)

Thus in d dimensions we get

@

fa

d/2

(XI1.148)

and

v(t) =

D1,x(0) paren 1x0 —

(XI1.149)

Inserting this result into (XII.1), we obtain on the average

© So«< ———a nGnloy™

Ti)

(XIL.150)

However in this rough argument we have assumed that particle 1 was staying

at the center of w,; we should also take its own diffusive motion into account.

We then find

Tx

1

n(D + n/p)”

in agreement with (XII.78, 79).

(XIL.151)

This qualitative picture can be put into quantitative terms by a

generalization of the Landau-Placzeck

slight

method of Section 5.1 (which can

also be applied to the other Green-Kubo integrands). The calculation starts with the following trick: we write the spatially homogeneous function I'(¢) as the integral of a spatially inhomogeneous quantity

r= paul de de (j,.(0; Diz. (30)"* ———_ (XIL.152) where the self-current density j, is defined by

jolt; ) =v,(t) 6-1, (0)

(XII.153)

We then remark that two conserved quantities are involved in the dynamics of the tagged particle 1: {i)

Its locality density A,(r; t)= 6(r—r,(2))

(XII.154)

The Long Time Tails

(ii)

353

The local momentum density j,(r; t) [see (VII.115)] of the whole fluid

(only the momentum of the whole system, including the tagged parti-

cle, is conserved). Equivalently, as second conserved variable, we can

take the local velocity

ar; N=

1

N

D v_.S(r—r, (2)

(XII.155)

a=]

(in a linear approximation, we neglect the variation of the density).

In the spirit of the Landau-Placzeck method, we perform the canonical average in (XII.152) in two steps. First, we take a partial average, fixing the local density of the tagged particle and the velocity field (or their Fourier transforms 4,, and ii,); then we average over these fluctuations. To avoid irrelevant mathematical complications, we work ina large but finite volume

©. Thus in analogy with (XII.128) we obtain

m

Tw= kak

(n dn.gd u,) (0

P(n,q)P(uy)

x (i ar dr’ { dt dv f,.(r; 0) ej, (r'; 0)

Xpnllteg, ta))}

(XII.156)

where we have limited the constraints to wave numbers q smaller than a

(small) cutoff go to ensure that the local equilibrium states vary slowly in space. The distribution en {rg}, {u,}) is the generalization of (XII.129):

enna}, {Ug}) =

[E40 5(Ms.9 7 Figg) 5 (tg —8,)] e P%™ Jac 40 [I] y 00 (if 1 >4).

We showed that a way to circumvent the difficulty with C, is to resum all

the ring graphs to generate the ring operator 6C“""® [see (X.113)]. By analogy, let us introduce

8G, (v3 TH" =X GPW; 1)” ted

(XII.182)

with the hope that this (partial) ring resummation will eliminate the divergence effects noticed in (XII.181). That is, we hope that the same mechanism that suppresses the divergence in the density expansion of the transport coefficients also eliminates the time pathologies in the density

expansion of G,.

We are then left with a rather straightforward calculation, which we now

sketch. From the first equality in (XII.23) applied to the ring terms, we get

for the Laplace transform of 6G,(v,; 7)"

Jim 8G,(v,; ie)" = -8Cf"?

(XII.183)

where the right-hand side is given by (X.113). Yet by the replacement ie > 3, it is readily seen that a similar formula holds for any complex g in the upper half-plane S*, namely*

86,(¥13 5) = — (2x) x

|

k/?-effects. Thus

the long-time behavior I'(t) is determined by the small-z behavior of the ring

operator (XII.184).

To proceed, we use a decomposition analogous to (X.140)

Here

BG.(v15 3)" = 5G(v45 a NSB + 8G.V45 3B ao (XIL.190)

5G,(V13 g )igek Ss.

=35 a

xz

ak | av,(ole"I0) -k

k

Patvi)bal¥)

X PI(V5)

"PEG

* See, for example, Doetsch (1950).

1

TeFAL

+ iy

a

a

dy; dv, gi"(v1)

*(V3NOlt 10) 50, (v1) @5%(05)

(XIL.191)

360

Calculation of Time-Dependent Correlation Functions

describes those contributions to the ring operator where the intermediate state is a product of hydrodynamic eigenstates of Ct) and C$” [see (X.141)]. The remainder

5G

vdeo involves contributions with at least one fast-

and/or j’# 1-+

+5 in the expansion (X.135).

relaxing mode in the intermediate state: these terms correspond to j # t Inserting (XI. 189, 191) into (XII.188) and performing the -integral by

an elementary residue calculation, we readily see that SG yaro leads to negligible (i.e., exponentially decaying) contributions because the corre-

sponding eigenvalues AK and/or Ay remain finite when k > 0. We are left with mn} a:

ro = ee), |. Al explat rasa

(XI1.192)

where 1

B= | dy d¥2 01 00 has

a Laplace transform with a (jw)'/? singularity near # = 0. Thus

Pw)-FO) &~, Gw)'”

(X11.204)

BQ,a)-D

(XII.205)

and

x, (iw)'?

In fact, the situation is even worse! A coupling between sound-wave modes becomes possible when q is different from zero, and as a result, when q and @

go to zero (with w oC q’), we have in fact

Bqe)-D * See, for example, Doetsch (1950).

«wag?

woq?+0

(X11.206)

The Long Time Tails

363

instead of q, as predicted by (XII.199). Inserting this into (XII.201), we find

(XI1.207)

Asg = G [1D +q'?D" ++] which shows

that the Burnett correction is dominated

by mode-mode

coupling effects.* This theoretical conjecture has not yet been confirmed experimentally.

(iii) The existence of transport coefficients in 2d.t A serious difficulty arises in two dimensions (except for the Lorentz model, see Chapter X, Sections 4 and 5.3) because with I(r)~1t7', we find

= tT m

+o

at T(t)=00

(XI1.208)

Thus the calculation of I'(#), based on the assumption that a diffusive mode exists, leads to the contradictory result that the diffusion coefficient does not exist.

The way out of this paradox is not yet clear. A rough self-consistent argument leads to the proposal that in two dimensions, the transport coefficients should be replaced by time-dependent coefficients growing as vin ¢ (e.g., D would become D(t)~~In t for t +00). Indeed, such an extra

time dependence in the argument of the exponential in (XII.172) yields

ro~f

41}

dQo(x;8B.B, — 7,74]

(B.5a)

or, by (IV.34)

Ly = [

~

le b do

0

a0(,p, — P.P,)

oO

(B.5b)

It is the xy-component of the tensor }

1~|

= oO

bab |

aw oO

a (pp — pp’)

(B.6)

For symmetry reasons, | must be of the form

'= a(p)pp + B@)U

where a and 8 are scalar functions of j. Taking the trace of (B.7), we get

pines

Ba

|” bal”

aap)

(B.7)

(B.8)

The integral on the right-hand side vanishes by energy conservation, and we have

3B =~ap*

Left- and right- scalar multiplication of (B.7) by p yields 1 B ou1f* 20 ae

per

rnat

ana

F bdb ,

49(p*—

(B.9)

(p -p')”)

= [oa fr aea —cos*y)

(B.10)

by definition of the deflection angle y. From (B.9) and (B.10), we find

a=3a{ and, from (B.7, 5a), we arrive at

bdb(1

cos*x)

(B.11)

,

1, = 30p,B, [ b db(1—cos?y)

(B.12)

Inserting this result into (B.4) and performing the integral over the angles of §, we finally get

b,,=29(k eT) [lo app’ e*|[ b db(1~ces"y)| 5 m 0

which is exactly what emerges from (V.146,

calculated along the same line.

147,

(B.13)

148). The other integrals 5, are

APPENDIX

C The Sutherland Formula

From (IV.25b, 27) we get for the deflection angle °

dr

xnmn2 \° Pll) VOke TPT

a

where V(r) is given by (V.164). We distinguish two cases. (i)

&>a)

(C.2)

and the contribution to the integral 6"(p) [see (V.148)] is of order e”, which we neglect. (ii)

&0)

JIO= { d*u e€(e-g)"

(n>0)

K= | d*u ee(€ + v)(€-g)

(E.1) {E.2) (E.3}

where v is an arbitrary vector, independent of e.

E.1_

CALCULATION

OF I”

From (VI.9) we have

r= «| de(g- €)"*'O(g- ee

(E.4)

By symmetry, I° is a vector along g

IP =aM(g)g

(E.5) 379

380

Appendix E

Hence

lw. gu { 92 ay in) (g)= sal “BO deg- n+l €)"*'O(g + €)

(E.6)

In a polar-coordinate system with g as polar axis, this integral is elementary, and we find

a"(g)=

8"

{E.7)

2aa7g" naa &

(E.8)

or I fn)

In particular, we used this result in (D.6) for n = 0.

E.2,

CALCULATION OF J”

From

JM= a? | d’e(g - €)"*'O(g - eee

(E.9)

we conclude that J® is of the form

I= af (gest yi(g)e"U

(E.10)

Taking the trace of this tensor and its contraction with gg, we obtain, respectively,

Tr J = (ar $(g) + 3-y9(g))g? = a|

de(e + g)"*'@(g + €)

2ma” ae

(E.11)

and

gS

g= (apg) + yS(g))e* =? f de(e +g)" =

2na* n +48

a

O(g- €)

4, ¢

E.12

Solving these equations for a$” and y$”, we arrive at 2ga7-1

iny__= wtDinrae” 270° +egt+

g7U]

(E.13)

)

Some Angular Integrals for Hard-Sphere Collisions

E.3

381

CALCULATION

OF K

From K=a? | ae(g + €)*(v + €)O(g : elec (E.14) we deduce, by symmetry and from the linearity in v, that K must have the form

K=[vg+gv]g’a;(g)+ [egys(g) + Ug*5s(g)I(v - g)

The scalars @3, y3, and 6, are calculated from the following contractions:

(E.15)

Tr K= g7(v + g)[2a3+ 3+ 53] - af d’e(v - e)(g - €)’Og - €) =v-I?

(E.16)

v-K-g=[(v-g)?+17g7]g7a5 + (v- g)*g%y3+ 85)

-a°| delg-6)"0- 608-6)

=e dOry

eK: g=g*-g)(2a,+ 73455) =a’

(E.18)

d@e(g- €)*(v- €)O(g- €)

vel?

Combining

(E.17)

(E.19}

these equations with (E.8,

equations determines a3, y3, 83:

13), we find that the following system of

2a; t+ ¥3+383> ma”

28

[w- g)+ v?g7]a; +.

8)"(y3+ 55) a ime

2

2

(3@- g)’+ v2g?}

ma?

2astyst8s= aE

(E.20)

Its solution is a3 = y3= 5; = wa?/12g; thus aa?

K= Tog let evle? +[get+g°U](v - g)} E.4 With the help of

SOME

SPECIAL

(E.21)

CASES

vi =v, —€(€ -g) v=v+el(e-g)

(E.22)

382

Appendix E

which implies in particular

oP = vi 2+ v,)(e-g)+(€-g) v? =v? +2(e-vyle-g)+(e+g)’

(E.23)

I”, J, and K allow us to evaluate all the integrals that appear in the Enskog theory. For example, we have

Jeu (vi, ty) = 2Fy, - J

{E.24)

and from the xy-component of this tensor, we obtain (VI.58). We also have

Jeu ee[(vi—v,)-v]=—-K

{E.25)

and taking for v the unit vector along the y-axis, we get (VI.62). The

following

conductivity:

consequence

of (E.8,

13)

is useful

in calculating

aru etor+o9 =7= {rote Zp2g-werervd+S8} .

2

2

2,

the

thermal

6.26)

APPENDIX

F The Decay of the

Non-Markovian Kernel

G(v,; 7/9,(t—7)) and of

the Correlation Term 9 (v,; ¢): A Weak-Coupling Model

There is no satisfactory proof of the gener al statements (VIII.48a, 52), that is

Gs rle(t-))>0

ra,

Blv,;1)>0

i> 4,

(F.1) (F.2)

where r, is the characteristic durat ion of the collision. These assum ptions can be checked on a few models, however. A simple model corresponds to the weakcoupling approximatio

n (Chapter IX, Section 2) with the (very unrea listic) potential Vir) = V, e777 (F.3)

where ry is the range of the forces. The Fourier transform of this potent ial is

_8aVy

1

(F.4) (k? +452)? The weak-coupling approximation to G(v,; 7]g,), denoted VG y,; rlp,), is obtained by replacing the full Liouville operator Ly by its unperturbed value Lyin

ety

383

384

Appendix F

the exponential of (VHI.22). We have thus VG;

rilei(t—7)) = tim | dv,+ +> dvyPybLy e rsh, "°

N Tl ¢:(va3¢—7)

ant

(F.5) We used (VIII.17, 19) to derive this expression; moreover we found it convenient to

introduce two variables + and 7,, which must be set equal at the end of the calculation, to get the correct G°”. This trick allows us to define the Laplace transform of (F.5) with respect to the variable 7, only. We write thus

Gs

slout—7)) = | dr, e™'G?(v,; rile (t—7))

(F.6)

in such a way that 29

Gy;

Te, (t—7)) =

1

at

J dg e“""G"(y,;

(ft -7))

(F.7)

where G is parallel to the real axis and above the singularities of the integrand. Inserting (F.5) into (F.6), we notice the close similarity between the function Gen

and the Landau collision operator €‘” defined in (IX.6). All the manipulations performed on @°” are also applicable here; obtain

Gin

slot) =_

5 |

0

in particular, in analogy with (IX.22), we

dr,{ dv, | dkky,

f) ; aa +>ov;—exp[—i(k+vi2—s)ri]kV, - (2-2) eu t—r)pi(¥23 1-7) dav, 2

(F.8)

The z-dependence of this expression is entirely contained in the tensor

Poasd=[ dr f dkek Vi expl—itk-vis-a)riIkV, Ul

f dkkV, —1__. “i(k *Vi2—3)

V,

(F.9)

«

,

To evaluate (F.9), we use a cylindrical coordinate system (k,, b, 8) with the z-axis parallel to v,,. To simplify, let us limit ourselves to the component F”,,; with (F.4), we have

A Palen

Sn)'Vo \~ dk. a ki (vi2 Ls Lis) p= ied, {™ do [~ Eb)*(k 7 bdb pe)5) (ro? +24 ,012— (F.10)

The b- and @-integrals are readily performed, yielding =

F’

webs d=

jz=

Ot’ Voois {*

Gia)

u? du

| ohsatu*Faa)

F.11 eM)

A Weak-Coupling Model

385

where u = k,v,2. Since the inverse Laplace transform (F.7) is defined for g above the

singularities of its integrand, we need to calculate the integral (F.11) for s in the upper half-plane. This is an elementary residue calculation; the result is

F0139) =

Sati

it

3

i"

[sy +i@r2/ro))

i(042/1o)

[e+ ilor2/t)P

2(012/ro)*

by ti(012/r0)P

}

(F.12) To evaluate G°”(v,; 7,19,(t 7), we need to know the inverse Laplace transform

of (F.12)

1

ya

Fy(012 Lsm=5- { dg e"""'F1(v12 1,53) tig

(F.13)

and similar expressions for the other tensor components. This integral (F.13) is again calculated from the residue theorem by closing the contour © witha large semicircle in the lower half-plane; we get

4

2

F042 1.5 71) = = Viro{1 ott LA ("2") ] ens To

To

(F.14)

A similar result is obtained for the other components of the tensor F’. Therefore, for v, in the thermal range (v) we have

G's rle(t—7)) < eT

(F.15)

Thus (F.1) is indeed satisfied in this model, where the duration of the collision is of

course 7, =ro/{v). Notice that we even have the stronger result (VIII.48b), but this depends very crucially on the exponential character of the potential (F.3). A similar analysis can be made for the correlation term D(v,; t). Replacing again

Ly by Lin

the exponential of (VIII.23), we have from (VIIL40)

BW, O= lim | dyv,+-- dvyt

ONPyABLy e thay O(A))(1 — Py) py (0)

(F.16)

To get the correct weak coupling limit of 2, we must keep in mind that p,,(0) itself

depends on the interactions. At equilibrium, for example, we have (see (VII.79)]

pe = F520) TL 72.)

(F.17)

with

Pile) = f dvexpexp(—BAVy) -(BAVy) _

1-ABVyt ++

~ ONL AB/O) FdrVy AB

+]

f acvy+--)

(F.18)

386

Appendix F

where we have expanded #% in powers of A (ignoring possible difficulties in the limit Q.> 00). Out of equilibrium, it is of course difficult to make a realistic choice for p,(0), but in many cases we expect the correlations between the particles to be of thermal origin. Therefore a reasonable situation corresponds to N p= zee1 (1-aBv +28A | de Vat) Hetsi0)

19)

where only the velocity part is taken out of equilibrium. In this case we find from the definition (VIII.12) of Py 1

qd — Pr)pw(0) =A os (-ov.+4

N

dtVy+-: ) au ei(¥.;0)

(F.20)

and the weak-coupling limit of @ is of order A?: :

NB;

1

iat

t)= lim J v0 ++ d¥y— PydSLy e*™ (1— Py\-ABVy) °°

ig

TT ei(¥a3 9)

ant

(F.21) Here again, the formal similarity with the Landau collision operator (IX.6) can be used to cast this equation into the following form: Qo?)

so)=

iO

np

im

a F - ay, exp (ik + Vi2t) Vig lv; 0) 9,(v2; 0)

{ dy, | dkkV,

t

(F.22)

To study the time dependence of (F.22), we introduce its Laplace transform By 9) = |

at

@°%v,; p=

0 1

2m

dt e'™" Dv; |

1)

(F.23)

stat

Je

dz ee" D"(v 3 3)

(F.24)

From (F.22, 23) we have

43)

ng

DB’ “y, sz) 3) = — Brim Bis

f

J

dv, | dkkV,

a

1

._— av; ——__ i(k vias)

Vi.

(v1; : O)ei(v2; . 0)

(F.25) the y-dependence of which is governed by the vector ~ F'(v,2; (Vi25 #) = | ak

5 1 viz "ike — vis—9)

( F.26 )

This quantity is very similar to F’ and can be treated by the same method. We leave the detailed calculation to the reader and simply mention the final result, which

closely parallels (F.15). For v, =(v), we obtain

Dy; which indeed satisfies (F.2).

1) Ov

- €)5(r—ae)

+1

= a 1

Qn

d(cos @) {

oO

a® v cos 6 @(v cos 8)6(r%—a cos 6)

x 6(b— av 1—cos? 6 cos &)8(av 1 — cos? 6 sin &)

(H.2) We eliminate the ®-integral with the help of the last Dirac delta function; the only

. acceptable solution is @ = 0, since the other one, ® = 7, prevents satisfaction of the second Dirac delta condition (6 is positive). From the well-known formula gly.) [’ ay a(y)aLfon=5 EO

(4.3) 389

390

Appendix H

where the y,’s are the zeros of f(y) lying in the range (y’, y"), we obtain 1

I

ao

0

dy y8(r,— ay)5(b — av 1— y?)

1

av1-y?

(H.4)

with y = cos @. The second delta function in (H.4) is satisfied for y = y, where a2—h?\1/2

y= +(—)

(H.5)

provided b < a (otherwise there is no solution). Since y lies in the interval (0, 1), only

y. has to be retained. Using (H.3) again, we get

1 = vO(a -6)8(,-Va?—b*) which, in view of (X.10), is the same as the left-hand side of (H.1).

(H.6)

APPENDIX

I Divergence of the Choh-Uhlenbeck Operator in Two Dimensions

Because the linearized Boltzmann operator C4” [resp. the Boltzmann-Lorentz operator C{!] defined in (X.104) [resp. (X.106)] involves four [resp. two] different terms, the complete analysis of 6C’, as defined by (X.109), is quite long.* To be brief, we illustrate here how a clean proof of the divergence (X.1 10) can be obtained,

by discussing one typical term of 6C” only.

Take for example the contribution to (X.109) coming from the last factor in the bracket of (X.104), which defines C$). It reads

ay

(22)4 a

\.

2 J

ak)

av, (lr

azygy to) (k- Vi2— ie)

xat! Jan] d*e(€ + ¥,,)O(€ « ¥23)95%(v5) 1

x ke-vzie) (Ole 10)0..1(v1; Nes%(v5)

(1.1)

Even for d =2, the number of integrals in (1.1) is high and the calculation is not easy. Let us first take v,, as a new integration variable , instead of v3. For e #0, the

‘integrand is regular and the order of the integrations may be changed at will. We then * This calculation in two dimensions can be found in Haines et al. (1966); the corresponding analysis for the S.C.U. operator in three dimensions is given by Pomeau and Gervois (1974),

391

392

2

Appendix I

rewrite (I.1) (for d = 2) as

(1.2)

lime704 | dv, (Olt! 10} %(o.)L.(v1, Vas ko)

Aar*

with L,(¥1, ¥2, k=

| dra

k1/K.

A further expansion of the bracket in (1.15) then leads to

{ | aeacerey(!to2)-1+6(4))

oct

x

(1.16)

394

Appendix! and therefore

i,(u, u', u', Ko, x) -C, Cx «| . i,(u, Ko x)—

kouse

dx

1

€

&_(1_£) NK kon

1.17 (1.17)

Thus in regime (i) i, remains finite when ¢ becomes small.

(ii)

je” -1| 0,.

The 6-integral in (1.28) cannot be performed analytically. However it is readily checked that a singular behavior of (1.28) can come only from the region @« 1/K; thus we add only finite terms to the result, if:

(i) (ii)

We multiply the integrand by (1467). We then replace the limits of integration +1/K by oo,

In this way, we arrive at 10) zs

“2 / 0

>

du’ flu’, 2) |

oD

dga—1_ arctan Kou6 J+eu 0(1+ 6?) e(1t+a)

‘The 6-integral can now be performed exactly* and leads to .

po

2

foo

k

’

42 J du’ fu’, m) In [ee] +0(1) uly e(utu’)

* See, for example, Gradshtein and Ryzhiz (1965).

(1.30)

(1.31)

Appendix I

396

where the definition a = u/u’ has been used. Therefore

I~

ne

(1.32)

and this logarithmic divergence obviously cannot disappear by the further integra tions implied by (1.2).

Since ¢ has the dimension of the inverse of a time, the present calculation confirms our rough analysis leading to (X.93) in the text. Similarly, if we suppose that the convergence factor ¢ in (X.109) introduces a cut-off in (X.110) at k =e/{v), we get

|

ke

ein

es(v)

€

(1.33)

in agreement with (1.32). However this last argument is oversimplified and may lead to wrong results! Indeed, (1.33) should apply as well to all the terms of the binary-collision expansion (X.109); in particular, we could use it for the combination J

1 e?k——_j (k-v.— ie)?

(

1.34

)

where the same denominator (k - v,.~ie)”' appears twice, corresponding to terms where the velocities v, and v, are not modified by the operators Ct!) and CY’. However from (1.8), we see that (I.34) is the same as i, (u, u, 0, ko). A direct residue

calculation on (I.10) leads for this quantity to* a

i,(U, i.(u, u,u, 0, k 0) =ws,

(role

y dx

aa Gtx

(1.35)

independent of e when « > 0,. Thus in this case the dimensional argument leading to (1.33) is wrong.

Physically, the existence of a divergence for x = a only can be understood on the basis of the geometric requirements for a recollision (12) in the process described by (1.1) and illustrated in Fig. AI.1. Indeed, let us work in areference frame moving with

particle 1: after the initial collision (13), particle 3 goes away with velocity y;,; at time 1, it collides with 2, and 2 emerges with velocity v,,. Clearly, 2 will collide with 1

Figure AI.1

by (1).

The process described

* Equation (1.35) also proves the statement made after (1.12).

Divergence of the Choh-Uhlenbeck Operator in Two Dimensions

397

(after some time 7,) only if the condition V31T1 = "Vy 2T2 (1.36) can be satisfied. But this is equivalent to xX =a, when 7, (and r,) become large.

It is difficult to keep track of these geometric requirements when working in

Fourier space, and to obtain correct estimates of the various terms, it is necessary to

perform the detailed calculations carefully.

APPENDIX

J Short-Time Behavior

of & (t)

If we write the short-time expansion of &,, as

Gq

Q= 1

the coefficients @, a4, @,...

Ga

2

ia

+ Os TO

re

Gt B09

(J.1)

can be calculated from the derivatives of (X11.83),

taken at t=0. With the method used in the text for the velocity autocorrelation function [see (XII.55, 59)], we find a,=0%

a,= (305+ (w")p)ws Ag = ((w*)-+ Swiw)p+ 1506)@6

(J.2)

where w2, (w”),, and (w*), are defined by (XII.111, 54, 59), respectively. From (XI1.99, 103) we readily see that

T@;0=4 ‘02 aw,

a0 (4s Ol-0=

-a,taz wo

2

a,—2a,a,+a3

ATs D0 2 398

J.3 (J.3)

Short-Time Behavior of Gal

399

Hence we obtain from (XII. 102) a4 2

»(q3 0)= — aT (9; Dl no=

a¥(4; No=

3

— GT (4; O],-o+ (TG; Oho)”

_ 96 ~ 44/05 + (2ara4~ wo

2

a3) —a2/w?)

Combining (J.2) and (J.4), straightforward algebra leads to (XII.110).

(J.4)

APPENDIX

K Calculation of pi

In the small-k limit, the operator is the Maxwellian

conserved

eigenfunction

of

the

Boltzmann-Lorentz

H21(¥1) = G5%(0,) [see (V.173,

(K.1)

174)], and the conserved eigenfunctions of the linearized Boltzmann

operator have the form [see (V.78)]

BEV) = Y Caabear(¥2)

(K.2)

where the functions #2. are displayed in (V.13).

With the help of (X.69) and (X.104, 106), we have

f dv, (Olt 10) p7(v,) P22)

=a?( dv, | Pele -V,)O(E + vill Pio b2vs) ~P(v,)bo(v2)} = CPGMY)— CPO)

=~ C921) since }2{v,) is an eigenfunction of C“”, with zero eigenvalue. 400

(K.3)

Calculation of x2

401

With these results, when k is small, (XH.193) yields ui=|

1

dv, naan

| dv, (Olt

re

«12:

0p i9(v,)o.*(v2)

=-[av, Oren1 POM) =-n" | dv, 0, ,.05*(v;)

(K.4)

To evaluate the integrals (K.4), we must remember that the eigenfunctions

b,(v;) = lim oxy)

{K.5)

were written in the text for k along the x-axis. To obtain the corresponding expressions for an arbitrary orientation of k, we should make the following substitutions in (V.84) [see also (X1.119)]: u>

and

v,°k k —0,

kk, + d, y(ki+ k=

kVe+k?

0,

kk,

(K.6) which provide a decomposition of v, in three orthogonal components, one taken along k and the two others perpendicular to k and to one another. By elementary integration, we then find first 1 (zk Bo) 7) x2 N27, 272 peal lek on?\ m x

K.7 (K.7)

These coefficients, however, are not interesting since they correspond to sound-wave modes that give an exponentially small contribution to T(s) in the long term.

Similarly, we find

lee =0

(K.8)

which shows that the thermal mode does not contribute to T(t). Our last results

lai?

kaT kk? =— n?m k2+k?’

combine and lead to (XII.194).

et?

kpT k?2 = n?m k?+k?

(K.9)

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Index Apse line, 84, 85, 89 Average value, 12, 13, 17, 34 in ensemble theory, 279

at equilibrium, 183, 190 Autocorrelation function, 53, 54, 65, 297 normalized, 53 and power spectrum, 54 velocity, see Velocity, autocorrelation

function see also Green-Kubo integrand

Bhatnagar, Gross, and Krook model, for Boltzmann equation, 149 for Boltzmann-Lorentz equation, 153 for self-diffusion, 154, 338 and transport coefficients, 151

Binary collision, expansion, 251, 268 Binary collision operator, for hard disks,

263 for hard spheres, 247, 255, 379 382, 389 Bogolubov, Born, Green, Kirkwood, and Yvon hierarchy, 192 Boltzmann equation, linearized, 119 eigenfunctions, of collision Operator, 120, 133 hydrodynamic modes, 130-135, 401 spectrum, of collision operator,

120

nonlinear, 85—89

H- theorem, 94

microscopic derivation, 229-232

normal solutions, 105~117

Positivity of

fi, 90

Boltzmann-Lorentz equation, 147, 320 Boundary conditions, in computer

experiments, 331

and I] - theorem, 97

in phase space, 205

Brownian motion, classical theory, 30-38,

66. See also Diffusion; Fokker-Planck equation; and Langevin equation ensemble description, 279 microscopic theory, 233-239 of oscillator, 279-287 and random walk, 23-29 Burnett equation, 115, 361

Canonical distribution, 183. See also Distribution function

Chapman-Enskog method, 112~117, 136 Chapman-Kolmogoroff equation, 21, 24

Characteristic function, 60, 64

Charge density, 294

Choh-Uhlenbeck collision operator, 255, 259, 321 in two dimensions, 391—397 Closest approach, distance of, 79

Collision, cylinder, 243 direct and inverse, 84, 85, 232

duration of, 212 . integral, 142, 372373 invariant, 92, 108 Collisional transfer, 157

Collision operator, see Binary collision operator; Boltzmann equation;

Choh-Uhlenbeck collision operator;

Enskog equation; Hard disks; Hard

spheres; Kinetic equation; Landau equation; Non-Markovian collision

operator; Ring collision operator;

and Super-Choh-Uhlenbeck operator

Compound

experiment,

11

Computer experiment, 4, 330— 335,341, 356 407

Index

408 Conductivity, see Electrical conductivity, Thermal conductivity

Conservation equation, 194, 302 and Boltzmann equation, 98-101 and Enskog theory, 160-164 Conserved quantities, 196 Constant of motion, 182 Continued fraction, 342 Continuity equation, 103, 197 for charges, 294

Convection, 100 Convergence factor, 226, 396 Correlation, 15 effect on kinetic behavior, 87, 214,

385-386.

See also Molecular chaos

effect on transport coefficients, 257, 321

of sequence of collisions, 253

Correlation function, 291. See also Autocorrelation function; Hydrodynamics;

Green-Kubo integrand; and Pair correlation function Coulomb forces, 223, 295 Covariance, see Autocorrelation function Covolume, 157 Critical dynamics, 364 Current, see Electrical current; Energy; Heat; Mass; Momentum; and Seif-

current

Deflection angle, 80 De Gennes narrowing, 349 Density, expansion, of collision operator,

249, 357 of transport coefficients, 256-274, 320 see also Charge density; Energy; Momentum; and Particle density Diffusion, of heat and entropy, 127, 129 of particles, 28

self-, see Self-diffusion of transverse velocity, 127 Diffusion coefficient in Brownian motion, classical theory of, 41-46

Ejnstein relation, 48 Green-Kubo formula for, 278-279 random

walk theory of, 28

Dirac delta function, 18, 60 Distribution function, absolute

Maxwell-Boltzmann, 36, 42, 97, 189 canonical, 183 generic, 187 local equilibrium, 345, 353 local Maxwell-Boltzmann, 94 N-particle, 175, 203 one-particle, 38, 75, 187 pair, 188 reduced, 187, 192, 203

spatial, 187

specific, 187 two-particle, 187 velocity, 50, 187 Doob’s theorem, 62 Ejnstein formula, 28 Einstein relation, 48, 278

Electrical conductivity, in Brownian motion theory, 39, 371

Green-Kubo formula for, 290, 295 Electrical current, 40, 288 Energy, current, 200

equation, 103, 200 internal, 183, 277 local, 94, 195 Ensemble, 174, 279 Enskog equation, linearized, 164 eigenvalues and eigenfunction, 376

microscopic basis for, 323 and velocity, autocorrelation function,

323, 332

nonlinear, 159

Entropy, at local equilibrium, 102, 129

non-equilibrium, in dilute gases, 95-97

thermodynamics, 95, 185 Equilibrium, absolute, 97, 181-186 local, 94, 101, 111, 345, 353 Erogodicity, 53, 179, 291, 330 Euler equation, 101 Event, 7-10 Expectation, conditional, 52

see also Average value

Flow, see Current Fluctuating force, see Langevin force

Fluctuation-dissipation theorem, 285, 297 Fluctuation theorem, 343 Fokker-Planck equation, 32—38, 238 eigenfunctions and eigenvalues, 369

in external field, 369

Index

409

fundamental solution, 50—51, 67 microscopic derivation, 233-239 Fourier law, 101 Free energy, 186 Friction coefficient, 30, 41, 280 Green-Kubo

formula for, 238, 388

Gaussian process, Markov, 51, 62 nonstationary, 64-66

Stationary, 51, 57-63 Gibbs postulate, 182 Green-Kubo formula, see Diffusion; Electrical conductivity; Friction coef-

ficient; Self-diffusion; Thermal

conductivity; and Viscosity Green-Kubo integrand, 291, 298, 332~334,

350-361.

See also Autocorrelation

function; Velocity,

function

autocorrelation

Hydrodynamics,

101-105

and correlation functions, 297—305

generalized, 338, 348

linearized, 105, 124

Ideal fluid, 101. See also Euler equations Impact parameter, 79, 89, 242

Independence,

15, 35, 36, 67

Kinetic equation, 31 generalized, 204, 211-216 and correlation functions, 317 for hard spheres, 248

for self-diffusion, 256, 317, 319, 322

Markovian, see Markovian generalized kinetic equation

see also Boltzmann equation; Choh-

Hamiltonian, 173 in external field, 287, 292 Hamilton’s equations, 175 Hard disks, binary collision operator, 263 transport coefficients, 258 velocity autocorrelation function, 332 Hard spheres, binary collision operator, 247, 255, 379-382, 389 Boltzmann equation, 249, 264 Boltzmann-Lorentz equation, 265 collision cross section, 82, 84 generalized kinetic equation, 248 law of collision, 243 Liouville equation, 247 Pressure tensor, 163, 191 transport coefficients, in dilute gases, 143

velocity, autocorrelation function, 331 Heat, current, 200, 305 in dilute gases, 99 in Enskog theory, 162 in ideal fluids, 101 diffusion, 127-129 quantity of, 185 Helmhoitz free energy, 186 Hilbert principle, 108—112, 115

Hydrodynamic limit

limit, 130-135, 401 in Enskog theory, 376

, 44, 369

for Van Hove functions, 336, 346

Hydrodynamic modes, macroscopic theory,

124-130 microscopic theory, in Boltzmann

Uhlenbeck operator; Enskog equation;

Fokker-Pianck equation; Landau equation; Ring collision operator;

Super Choh-Uhlenbeck

operator

Kinetic model, 149-155. See also Bhatnagar, Gross, and Krook model Knudsen number, 111

Kramers-Kronig telation, 285, 296, 328,

341 Kubo formula, see Green-Kubo formula Landau equation, 222, 228, 230

Landau-Placzeck formula, 346 Landau-Placzeck method, 344-345 Langevin equation, 31, 66, 68 with external force, 39 for oscillator, 279 Langevin force, 31, 35, 66, 280 moments of, 67—69 Large numbers, law of, 11, 54 Laser, 307

Legendre polynomials,

121122

Lennard-Jones interactions, 145, 334 Light scattering, 312, 346 Liouville equation, 176, 205 for Brownian motion, 234 formal solution of, 180, 205 for hard spheres, 240-247 Liouville operator, see Liouville equation Lorentz model, 259, 274

Index

410 Markovian generalized kinetic equation, 219 for hard spheres, 248 Markov process, 20 and Brownian motion, 38, 68 Mass, current, 195 density, 344 Master equation, 22 and Boltzmann equation, 90 formal, 205~—208

Maxwell-Boltzmann distribution function, at absolute equilibrium, 36, 42, 97, 189 at local equilibrium, 94 Maxwell molecules, 82 and linearized Boltzmann equation, 122 and transport coefficients in dilute gases, 138 Mean square deviation (or fluctuation), see Variance Memory effect, see Non-Markovian effect Memory function, 325

for Van Hove function, 338, 340 for velocity autocorrelation function, 325, 335 Mode-mode coupling, 360 Molecular chaos, 86 generalized, 217, 319 Moment, analysis, 325 first, see Average value

second, see Variance sum rule, 325 Momentum, current, 197, 301 density, 195, 301

Motion, equation of, for continuous fluids, 198 for particles, 4, 175 Navier-Stokes equation, 103

with external force, 298, 304

linearized, 124

Neutron, scattering, 305, 311, 341, 347

thermal, 307 Newton equation, 4 Newton law, for pressure tensor, 103 Non-Markovian collision operator, 204, 318, 357, 383-385 Non-Markovian effect, 212

Onsager relations, 296

Pair correlation function, 188 at equilibrium, 189, 326, 342

Particle density, 76, 195 Partition function, 186 Phase-space, , 76

T, 174 Poincaré recurrence, 204

Poisson brachet, 176 Postulates, of statistical mechanics, 6, 175, 182 Power, dissipated, 284, 296 Power-law decay, see Green-Kubo integrand Power spectrum, 54-57, 59, 287

white, 67 Pressure, 102, 183, 198 at equilibrium, 191 in hard spheres fluids, 162-163, 191 in ideal fluids, 101 in perfect gas, 37, 99, 100

Principal part, 227

Probability, axioms of, 7

classical interpretation, 9 frequency interpretation, 9, 11, 28, 177 conditional, 14, 20 density, 17 distribution, 12, 13, 17, 18.

See also

Distribution function law, 7 Projection operator, 206, 321, 342 Pseudo-potential, 308

Random process, see Stochastic process purely, 19 Random variable, 12, 13, 14 Random walk theory, 23-29 continuous limit, 27 moments, 25, 26 probability density, 27 Rayleigh-Brillouin scattering, see Light scattering Reflection, spatial, 295 Relaxation, function, 291 time, in Brownian motion, 234 in dilute gases, 110 in weakly coupled gases, 228 Repulsive power-law force, and scattering cross section, 82

and transport coefficients in dilute gases,

144 see also Maxwell molecules

Index

411 Tails (long-time), see Green-Kubo integrand Temperature, absolute, 185 local, in dilute gases, 94 in nonequilibrium fluids, 102

Response function (retarded), 289

for Brownian oscillator, 282 symmetry properties, 295 Reversal, of time, 215

Thermal conductivity, 102

of velocity, 85 Reversibility paradox, 215

Ring collision operator, 266—271, 358-360 Sackur-Tetrode formula, 96

314

Sample space, 7 Scattering cross section, 80-85, 87 for light, 312

Three-body collision, see Choh-Uhlenbeck collision operator Transition probability, and Boltzmann

for neutrons, 311

equation, 89 and Markov process, 21 in scattering theory, 307

total, 88

Scattering function, coherent, 346 incoherent, 340 intermediate, 336

per unit time, 22

Transport coefficients, density expansion, 256-274 effect of spatial correlations, 257

Scattering tength, 308

Self-current, 352 Self-diffusion, 146 and BGK model, 154

in two dimensions, 258, 262, 363

and variational principle, in dilute gases, 138 of Van der Waals fluid, 363

and density expansion, 257, 320 in dense fluids, 323

in dilute gases, 148 in Enskog theory, 170 generalized, 362

see also Diffusion; Electrical conductivity;

Friction coefficient; Self-diffusion;

for hard disks, 258 for hard spheres, 258, 322

Thermal conductivity; and Viscosity Two-body problem, 77—80

Solubility conditions, in Chapman-Enskog

method, 114, 117 in Hilbert method, 109

169

propagation, 127-129

velocity, 127 Standard deviation, 13

memory-function approach, 338 total, 310 Gaussian approximation, 349 hydrodynamic limit, 344

Stationarity, 52-57

Stochastic differential equation, 31

Stochastic force, see Langevin force Stochastic process, 18

Variance, 13, 63

Variational principle for transport coef-

joint probability density, 18 stationarity, 52—57

ficients, in dilute gases, 138 Velocity, autocorrelation function, in

Structure factor, 342

325

Super-Choh-Uhlenbeck operator, 261 Susceptibility, generalized, 290

for Brownian oscillator, 282 Sutherland modei, 145, 372—

373

Van der Waals model, equation of state, transport coefficient, 363 Van Hove correlation function, self, 310 Gaussian approximation, 337 and generalized hydrodynamics, 338

Sonine polynomials, 123 Sound, absorption coefficient, 127

Sum rule, short time (or moment), zero frequency, 327

for dilute gases, 117, 136, 142 in Enskog theory, 167 Green-Kubo formula for, 304 Thermodynamic limit, 186, 204, 208, 313

Brownian motion, 278 and computer experiments, 331 in Enskog theory, 323, 332

and generalized kinetic equation, 317 local, 339 power-law decay of, 332-334, 350-360 distribution function, 50, 187