Citation
Fluctuations in systems far from equilibrium

Material Information

Title:
Fluctuations in systems far from equilibrium
Creator:
Marchetti, Maria Cristina, 1955- ( Dissertant )
Dufty, James W. ( Thesis advisor )
Bailey, Thomas L. ( Reviewer )
Broyles, Arthur A. ( Reviewer )
Hooper, Charles F. ( Reviewer )
Chen, Kwan Y. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1982
Language:
English
Physical Description:
vii, 207 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Correlations ( jstor )
Distribution functions ( jstor )
Gas density ( jstor )
Hydrodynamics ( jstor )
Kinetic equations ( jstor )
Mathematical variables ( jstor )
Navier Stokes equation ( jstor )
Particle density ( jstor )
Thermodynamics ( jstor )
Velocity ( jstor )
Dissertations, Academic -- Physics -- UF
Hydrodynamics ( lcsh )
Irreversible processes ( lcsh )
Physics thesis Ph. D
Shear flow ( lcsh )
Statistical mechanics ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
A unified formulation of transport and fluctuations in a non-equilibrium fluid is described at both the kinetic and the hydrodynamic levels . The method is based on the analysis of a generating functional for the fluctuations of the set of observables of interest around their average values . It is shown that the first functional derivatives of such a generating functional obey an inhomogeneous form of the macroscopic regression laws (nonlinear Boltzmann equation or nonlinear Navier-Stokes equations) . From this equation the dynamics of multi-space and -time fluctuations may be obtained by suitable functional differentiation , In particular the equations for the second order correlation functions of fluctuations at different and equal times are obtained to illustrate the method . The dynamics of a non-equilibrium fluid is governed by three sets of coupled equations: (1) nonlinear equations for the macroscopic observables, (2) linear homogeneous equations for the two-time correlation functions, and (3) linear inhomogeneous equations for the equal time correlation functions . The description obtained constitutes a precise statement of the generalization to non-equilibrium states of Onsager's assumption on the regression of fluctuations . The results apply to general non-stationary non-equilibrium states, either stable or unstable. The description of non-equilibrium fluctuations obtained rests on the same theoretical basis as the macroscopic description of the system (Boltzmann kinetic theory for a low density gas and hydrodynamics for a general fluid) .
Thesis:
Thesis (Ph. D.)--University of Florida, 1982.
Bibliography:
Bibliography: leaves 203-206.
Additional Physical Form:
Also available on World Wide Web
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Maria Cristina Marchetti.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
028821388 ( AlephBibNum )
09483361 ( OCLC )
ABW4348 ( NOTIS )

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FLUCTUATIONS IN SYSTEMS FAR
FROM EQUILIBRIUM


BY

MARIA CRISTINA MARCHETTI


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1982


















Digitized by the Internet Archive
in 2009 with funding from
University of Florida, George A. Smathers Libraries



A Teresa e Piero M.


http://www.archive.org/details/fluctuationsinsy00marc









ACKNOWLEDGEMENTS

I would like to thank Professor James W. Dufty for his constant

guidance and support in the course of this work. He suggested this

problem to me. It was exciting and challenging to work with someone who

has such a contagious enthusiasm for physics. I have learned much from

him and he has also been a good friend.

My gratitude also goes to all those who have helped me, both as

colleagues and as friends. In random order they are: Greg, Mark, Bonnie

and Pradeep, Robert, Vijay, John, Annalisa M., Annalisa P., Martin,

Carlos, Simon, Bob, Teresa, Jackie, Attilio R., and many others.

Without the love and support of my parents I may not have had the

courage to venture to the United States. A special thanks goes to them

for having always been near me during these three years.

I would like to acknowledge the support of a Fulbright-Hays Travel

Grant, which allowed me to come to the University of Florida, and an

Educational Award from the Rotary International for the year 1981-82.

Finally, I thank Viva Benton for her quick and accurate typing of

the manuscript.










TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS .....................................................iii

ABSTRACT ................................................................vi

CHAPTER

I INTRODUCTION .....................................................1

II NONEQUILIBRIUM STATISTICAL MECHANICS
OF CLASSICAL SYSTEMS ............................................17

1. The Nonequilibrium Distribution Function:
Average Values and Fluctuations ..............................17
2. Reduced Distribution Function Formalism ......................24
3. Generating Functional for Nonequilibrium
Averages and Fluctuations ....................................28

III HYDRODYNAMICS ...................................................34

1. Generating Functional for Hydrodynamics
and Nonlinear Navier-Stokes Equations ........................34
2. Correlation of Fluctuations ..................................46
3. Equal Time Fluctuations ......................................53
4. Discussion ...................................................57

IV KINETIC THEORY ..................................................60

1. Generating Functional for Phase
Space Fluctuations ...........................................60
2. Low Density Limit and Boltzmann Equation .....................66
3. Correlation of Fluctuations ..................................70
4. Equal Time Fluctuations ......................................75
5. Hydrodynamic Limit ................. .........................81

V TAGGED-PARTICLE FLUCTUATIONS IN SHEAR FLOW .......................91

1. Definition of the Problem ....................................91
2. Transformation to the Rest Frame .............................97
3. Two-time Velocity Autocorrelation Function ..................100
4. Equal-time Velocity Fluctuations ............................103
5. Results and Discussion ..................................... .105

VI DISCUSSION......................................................113





APPENDICES

A DERIVATION OF THE X-DEPENDENT NONLINEAR NAVIER-STOKES
EQUATIONS............................................. .. ....... 118

B EVALUATION OF THE SOURCE TERM FOR
HYDRODYNAMIC FLUCTUATIONS......................................137

C DERIVATION OF THE X-DEPENDENT BOLTZMANN EQUATION ................145

D SCALING METHOD FOR A LOW DENSITY CLOSURE
OF THE HARD SPHERES BBGKY HIERARCHY.............................155

E HYDRODYNAMIC "NOISE" FROM KINETIC THEORY ........................187

F DETERMINATION OF R.. FOR MAXWELL MOLECULES ......................191
ij
REFERENCES..................................... .... ........ ... .......203

BIOGRAPHICAL SKETCH............................... ..................207










Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


FLUCTUATIONS IN SYSTEMS FAR
FROM EQUILIBRIUM

by

Maria Cristina Marchetti

August, 1982

Chairman: James W. Dufty
Major Department: Physics

A unified formulation of transport and fluctuations in a

nonequilibrium fluid is described at both the kinetic and the hydro-

dynamic levels. The method is based on the analysis of a generating

functional for the fluctuations of the set of observables of interest

around their average values. It is shown that the first functional

derivatives of such a generating functional obey an inhomogeneous form

of the macroscopic regression laws (nonlinear Boltzmann equation or

nonlinear Navier-Stokes equations). From this equation the dynamics of

multi-space and -time fluctuations may be obtained by suitable

functional differentiation. In particular the equations for the second

order correlation functions of fluctuations at different and equal times

are obtained to illustrate the method. The dynamics of a nonequilibrium

fluid is governed by three sets of coupled equations: (1) nonlinear

equations for the macroscopic observables, (2) linear homogeneous

equations for the two-time correlation functions, and (3) linear






inhomogeneous equations for the equal time correlation functions. The

description obtained constitutes a precise statement of the

generalization to nonequilibrium states of Onsager's assumption on the

regression of fluctuations. The results apply to general nonstationary

nonequilibrium states, either stable or unstable. The description of

nonequilibrium fluctuations obtained rests on the same theoretical basis

as the macroscopic description of the system (Boltzmann kinetic theory

for a low density gas and hydrodynamics for a general fluid).









CHAPTER I
INTRODUCTION

A large class of macroscopic irreversible processes in many-body

systems is adequately described in terms of relatively few (compared to

the number of degrees of freedom) macroscopic variables, changing contin-

uously in time.1 This description applies when the time scale of

interest is large compared to a microscopic time scale characteristic of

the problem and the system considered. The macroscopic variables can be

identified with the quantities measured in an experiment, and their

values at a time t completely characterize the state of the system at

that time. Examples of such a set of observables are the hydrodynamic

densities in a fluid and the concentrations describing a chemical

reaction. On this macroscopic scale static and dynamic properties of

the system are expected to be governed by a closed set of macroscopic

laws. In particular the time dependent properties should obey a set of

differential equations of first order in time. It is then a

deterministic description in the sense that the equations assign fixed

values to all the macroscopic variables, once the values at t=0 have

been given. The completeness of a set of macroscopic observables for

the description of a particular problem on a particular time scale is

difficult to determine a priori and is established only by the

derivation of the macroscopic equations themselves.



*The complete set of observables often coincides with the set of
conserved variables in the system. However nonlinear coupling between
the hydrodynamic modes in a fluid can lead to the appearance of memory






2

The role of statistical mechanics consists in providing a micro-

scopic basis for the macroscopic description, defining its limit of

applicability, and establishing a precise connection between the

dynamics of the observables and the time evolution of the positions and

moment of all the particles in the system.

A derivation of the macroscopic laws from the microscopic dynamics,

governed by the Liouville equation, can be accomplished by introducing

the idea of several well separated time scales in nonequilibrium fluids.3

A system in an arbitrary nonequilibrium state is considered. On a very

short time scale, of the order of or smaller than the duration of a

collision, the dynamics of a system of N particles is very complex and a

large number of variables (the 6N coordinates and moment of the part-

icles in the system or, alternatively, all the reduced distribution

functions) is necessary to describe it. However, after a microscopic

time, of the order of the duration of a collision, or mean collision

time, Tc, the system relaxes to a kinetic regime, where the dynamics

may be described more simply in terms of the average density in the

single particle phase space.4,5 Therefore, when the time scale of

interest is large compared to T a contraction of the formal descrip-

tion of the dynamics of the N-body problem to a description in terms of

the one-particle distribution function is obtained.




effects in the macroscopic equations of motion: this indicates that the
chosen set of variables is not complete and needs to be enlarged (for a
discussion and references on this point see Ref. 2).

*For intermolecular forces with a finite and nonvanishing range, t is
defined as the ratio of the force range, a, and the mean thermal speed
of the molecules, v ,-i.e. T =o/v .
o c o





3

At larger time scales (for gases larger than the time between

collisions, or mean free time, tf, ) a further contraction of the

description takes place. The system relaxes to a state near local

equilibrium and its dynamics is described in terms of an even smaller

number of variables, identified with the local conserved densities,

e.g., mass density, energy density, flow velocity.6 This is the

hydrodynamic regime, which describes variations on time scales of the

order, for example, of the time required by a sound wave in a fluid to

cross a region of macroscopic interest .

The macroscopic regression laws governing the relaxation of a

system displaced from equilibrium can be derived in principle by averag-

ing the microscopic equations of motion and introducing approximations

suitable for the time scale of interest. For example, in the limit of

times large compared to a collision time (i.e., the kinetic regime), the

nonlinear Boltzmann equation for the one-particle distribution function

is obtained by a low density closure of the BBGKY hierarchy.5,8 The

Chapman-Enskog expansion provides then a systematic method for deriving

the hydrodynamic equations for the average densities in the gas, on the

longer time scale.8'9 Alternatively, the hydrodynamic equations can be

obtained directly by averaging the microscopic conservation laws,

6,10
bypassing consideration of the kinetic stage .

The contraction of the description of the dynamics of a many-

particle system in terms of few average variables naturally introduces

the concept of fluctuations. From a microscopic point of view in fact





*Defined as t = Z/v where Z is the mean free path of the molecules.
For liquids, t can Be of the order of T and the separation of time
scales is no longer useful.







4

the dynamical variables fluctuate instantaneously around their average

values. The physical origin of the fluctuations can be different for

different systems. In general, however, the amplitude of their space

and time correlations controls the adequacy of the macroscopic

description and can be measured in a variety of experiments. Thus the

statistical mechanics of irreversible processes in a many-particle

system should also provide a framework to describe fluctuations in an

arbitrary nonequilibrium state.



The description of fluctuations in equilibrium systems is well

established from both phenomenological considerations,1 and first

principles calculations for some limiting cases (e.g., low density

gases).12 A precise connection between the spontaneous fluctuations in

the equilibrium state and the macroscopic response of the system to a

small external perturbation is established by the fluctuation-

dissipation theorem.13,14 This result is concisely expressed in

Onsager's assumption on the regression of fluctuations,15 which states

that spontaneous fluctuations around the equilibrium value decay in time

according to the same linear laws governing the relaxation of a

macroscopic state close to equilibrium. In other words the linearized

regression laws apply regardless of whether the nonequilibrium condition

is prepared or occurs spontaneously. The adequacy of this assumption is

a consequence of the fact that the size of the equilibrium fluctuations

is limited by their thermal origin. Furthermore the space correlation

of the equal time fluctuations can be calculated directly from the

equilibrium ensemble.





5

The relationship of correlation functions and response functions to

a great number of measurable properties of macroscopic systems in equi-

librium is also well understood and a wide variety of experiments is

available for comparison.16 The interpretation of Rayleigh-Brillouin

light scattering experiments in fluids17 constitutes a well known

example of the success of the equilibrium fluctuations theory. The

detailed evaluation of time dependent correlation functions remains in

many cases a difficult many-body problem, but the theoretical formu-

lation can be stated clearly and is founded on the basic principles of

statistical mechanics.



The corresponding theoretical and experimental study of fluctua-

tions in a nonequilibrium system is much less complete and several new

problems arise, even from a purely theoretical viewpoint, in addition to

the difficult computational aspects.

The systems considered are in general open systems, in contact with

several reservoirs, which are not in equilibrium among themselves.3'6

These reservoirs impose external forces and fluxes on the system,

preventing it from reaching thermodynamic equilibrium and keeping it in

a nonequilibrium state, which is stationary if the external forces are

time independent. In contrast to the equilibrium system, which is

specified by a unique thermodynamic state, a large class of nonequili-

brium states must now be considered. The first problem encountered when

dealing with systems out of equilibrium is therefore the adequate

characterization of the particular nonequilibrium phenomenon of

interest.

Furthermore, the formulation of the macroscopic description of the

system and the derivation of the nonlinear regression laws for a






6

complete set of observables, such as the nonlinear hydrodynamic equa-

tions, presents several difficulties. At the hydrodynamic level the

evaluation of the form of the macroscopic fluxes as functionals of the

thermodynamic forces is required. These constitutive relationships are

highly nonlinear and only a formal expression can be provided in the

general case.10,19 An explicit evaluation has only been performed in

specific model cases (shear flow0',19-24) or under precise limiting

conditions (Burnett coefficientsl820) On the other hand, in near

equilibrium situations the formalism of linear response theory provides

expressions for susceptibilities and transport coefficients in terms of

the correlation functions of equilibrium fluctuations (Green-Kubo

relations).3'6 Nonlinear transport phenomena also can be related to

nonequilibrium fluctuations by introducing the concept of nonlinear

response functions. The understanding of the dynamics of fluctuations

in the nonequilibrium state is therefore intimately related to the

description of energy, momentum, mass and charge transport in systems

far from equilibrium.

Other characteristic nonequilibrium phenomena where fluctuations

play an important role are the appearance of instabilities, bifurcations

and turbulences, and the related onset of nonequilibrium phase tran-

sitions.25 A system macroscopically displaced from equilibrium and

brought outside the region of applicability of the linear response

theory (into a region where the macroscopic regression laws are non-

linear) evolves through a succession (or branch) of nonequilibrium states

as the magnitude of the parameter measuring the strength of the external

perturbation increases. At some critical value of this parameter the

state of the system may become unstable. The system will then make a





7

transition into a new branch of states. The occurrence of such

instabilities may lead to a change of symmetry of the stable state of

the system. In this sense the phenomenon is analogous to a phase

transition in a system in thermodynamic equilibrium. Therefore, as the

mechanism of equilibrium phase transition can be understood by recog-

nizing the importance of the fluctuations near the transition point, the

occurrence of bifurcations (and other nonequilibrium phase transitions)

is intimately related to the fluctuations in the system at the

instabilities.

A large variety of physical systems shows this general behavior.

The most popular hydrodynamic example is the Bdnard instability, occur-

ring in a liquid layer heated from below in the gravitational field.

Other examples are found in nonlinear optics, autocatalytic reactions

and biological systems. The formulation of a theory of nonequilibrium

fluctuations which does not impose restrictions on the size of the

fluctuations and applies near instabilities is an essential starting

point for the understanding of all these nonequilibrium phenomena.

However, even for the simplest case of hydrodynamic fluctuations,

it is not obvious how Onsager's assumption should be generalized for

such nonequilibrium states. Because of the nonlinear character of the

average regression laws, the dynamics of the macroscopic observables is

no longer the same as that of their fluctuations. Furthermore a non-

trivial problem is the determination of the source of the fluctuations,

which may no longer be simply thermal in origin and whose knowledge is

required for the specification of the equal time correlation functions.



There have been several recent studies of fluctuations in nonequi-

librium stationary states that are similar in spirit and objectives to






8

the present discussion. It therefore seems instructive to put the

present work in context by presenting a brief overview of these

approaches. They can be classified in general into two broad categor-

ies: stochastic approaches,26-33 where the microscopic fluctuating

variables are assumed to define a random process, characterized by a set

of probability densities, and microscopic approaches, 46 which attempt

to derive from first principles the dynamics of nonequilibrium

fluctuations.

The stochastic formulations are essentially based on a generaliza-

tion of the Onsager-Machlup regression hypothesis to the nonequilibrium

case. The time-dependent properties of the system are governed by a set

of equations for the probability densities (Fokker Planck equations) or

for the stochastic variables themselves (Langevin equations).

A stochastic theory of hydrodynamic fluctuations has been developed

by Tremblay, Arai and Siggia26 by assuming that the fluctuating

hydrodynamic equations proposed by Landau and Lifschitz4 '8 can be

applied to describe not only fluctuations in the equilibrium state, but

also small fluctuations around a nonequilibrium stationary state. The

time evolution of the fluctuating variables is then governed by a linear

Langevin equation whose coefficients depend on the nonequilibrium state

of the system. The correlation of the random forces is chosen to have

the same form as in equilibrium, except with local thermodynamic

variables for the nonequilibrium state. The characteristic feature of

the Langevin theory is the separation of the time variation into a

slowly varying and a rapidly varying part. The basic assumption is that

the correlations of the fast variables are determined by a local

equilibrium ensemble. This Langevin theory has been successfully






9

applied by several authors2649-51 to the evaluation of the scattering

of light from a fluid with a temperature gradient and/or in shear

flow. However the limitation on the size of the fluctuations clearly

restricts its applicability to regions far from instabilities and

critical points. Furthermore in Ref. 26 the example of a case where the

theory fails is given: electrons scattering off impurities to which a

potential difference is applied. Presumably this is a case when the

local equilibrium assumption for the fast variables does not apply.

A formal theory of nonequilibrium fluctuations and nonlinear

irreversible processes has been developed by Grabert, Graham and

Green.27,28 Their approach generalizes to the nonlinear regime the

functional integral expression for the transition probability between

two macroscopic states introduced by Onsager and Machlup'l15 as an

extension to the time dependent domain of Boltzmann's relationship

between entropy and probability. The basic assumption of the theory is

that the probability of a given fluctuation from one state to another is

measured by the minimum increase in action associated with the change

among the two states considered. This minimum principle provides the

possibility of constructing a path integral expression for the

conditional probability which appears as a natural generalization of the

Onsager-Machlup functional. However this minimum principle is purely an

assumption and has not been justified on any microscopic basis. Only a

posteriori can it be seen that the results obtained agree with those

derived through different formulations. In particular the conditional




*It should be pointed out that the results of Ref. 50 for the Brillouin
scattering in temperature gradient are not in agreement with the results
of the other calculations 6.





10

probability can be shown to satisfy a Fokker-Planck equation. This

equation has also been applied by Grabert28 to evaluate the spectrum for

light scattering from a fluid in a temperature gradient. His results

agree with those derived by other methods.

Within the framework of this generalized Onsager-Machlup theory,

Graham25 has also analyzed in detail problems associated with the

stability and the breaking of symmetry in nonequilibrium stationary

states.

Several authors29-32 have formulated studies of nonequilibrium

fluctuations based on a master equation in an appropriate stochastic

space. With the aim of generalizing Landau-Lifshitz's fluctuating

hydrodynamics to the nonlinear region, Keizer30 has rewritten the

microscopic conservation laws in a fluid in the form of master equations

by describing the fluid in terms of elementary molecular processes. His

results support the Langevin theory proposed in Ref. 26. Coarse grained

master equations for inhomogeneous systems have been assumed as the

starting point to describe phase space fluctuations by Onuki31 and

Ueyama.32 For systems near equilibrium the linearized Boltzmann-

53
Langevin equation5 is recovered. Furthermore by using the Chapman-

Enskog expansion method fluctuating hydrodynamics can be derived.

Finally, van Kampen29 has applied the method of expansion in the size of

the system to the solution of master equations for several problems,

from the Boltzmann equation to the rate equations governing chemical

reactions. The principal limitations of the master equation approaches

are the necessity of assuming (instead of deriving from first



*This is in effect a generalization of an analogous master equation
proposed by Logan and Kac52 for homogeneous systems.






11

principles) the basic equation, and the strong dependence of this

equation on the model considered.

Finally, a generalization to the nonlinear regime of the linearized

Boltzmann-Langevin equation53 has been proposed33 to describe

nonequilibrium phase space fluctuations in low density gases. This

method is similar in spirit to the fluctuating hydrodynamics of Ref. 26,

since the correlation of the random forces is again assumed to have a

local equilibrium form.

The microscopic studies are based on several different techniques.

First of all the methods of the kinetic theory of gases have been

used34-38 to describe phase space fluctuations. To make any progress

with the formal theory the limitation of low density has in general to

be imposed. This restriction, however, allows the introduction of

controlled approximations. Furthermore, the kinetic theory is more

general than the hydrodynamic description since it describes fluctua-

tions on shorter space and time scales and it incorporates the latter.

In particular Ernst and Cohen37 have applied the methods of the nonequi-

librium cluster expansion to derive kinetic equations for the equal and

unequal time two-point correlations of phase space fluctuations for a

gas of hard spheres. The meaning and adequacy of their expansion will

be discussed in Chapter IV and Appendix D, where an alternative method,

leading to equivalent results, is presented.

An alternative microscopic theory of fluctuations in nonequilibrium

steady states has been developed by an M.I.T. group.39-42 For states

not too far from equilibrium (up to second order in the parameters

measuring the deviations from equilibrium) they apply the method of

nonlinear response theory to express nonequilibrium averages and






12

correlation functions in terms of higher order equilibrium correlation

functions.40 These correlation functions are then evaluated (in the

hydrodynamic case) by assuming a separation of time scales in the

system, implying that the correlations involving the dissipative fluxes

decay on a time scale shorter than the macroscopic time scale of inter-

est. Their results are also generalized to nonequilibrium stationary

states arbitrarily far from equilibrium by employing the projection

operator techniques .41 They conclude that in general a simple

generalization of the equilibrium fluctuation-dissipation theorem to a

local equilibrium form does not hold. The formalism used however

obscures many of the physical hypotheses entering the derivation.

Furthermore the separation in fast and slow variables is not well

founded for states arbitrarily far from equilibrium.

Finally Kirkpatrick, Cohen and Dorfman43-46 developed a

hydrodynamic theory of nonequilibrium fluctuations in stationary states

based on the use of projection operator techniques and on the Kadanoff-

Swift mode-coupling theory for the evaluation of the equal time

correlation functions. They also have applied their description of

hydrodynamic fluctuations to the problem of light scattering from a

fluid subject to a temperature gradient, for both the cases of small and

large gradient.

The formulations briefly described here suffer from several limita-

tions. Their applicability is often restricted to nonequilibrium

stationary states and to small fluctuations around the states. The

latter condition implies that the system has to be far from insta-

bilities and critical points. A precise connection between the

stochastic and the microscopic approaches, as is possible for the case






13

of equilibrium fluctuations, has not been established. In the

stochastic methods it is often not clear how the macroscopic information

on the nonequilibrium state, as specified in an experiment, enters the

problem. In general there is a need for developing an exact treatment

of nonequilibrium systems whose validity extends to both the kinetic and

hydrodynamic regimes and that establishes a precise connection between

the theoretical and the experimental description.



The objective of the present work is to provide a unified

formulation of the description of nonequilibrium fluctuations founded on

a microscopic basis and constituting a precise statement of the general-

ization of Onsager's assumption on the regression of fluctuations to

arbitrary nonequilibrium states. The problems mentioned above are

addressed explicitly. An open system in contact with external

reservoirs is considered. The reservoirs prepare or maintain the system

in a general nonequilibrium state. The results derived apply to

nonstationary states, either stable or unstable.

The method rests on the definition of a generating functional whose

first functional derivatives are directly related to the complete set of

macroscopic nonequilibrium observables relevant to the problem consider-

ed. Higher order functional differentiation generates the correlation

functions of the fluctuations of the chosen dynamical variables around

their average values. The first advantage of this approach is that it

makes it possible to discuss the problems of average behavior and

fluctuations in a unified context. Characterizing the nonequilibrium

state amounts to providing a statistical mechanical derivation of the

macroscopic regression laws. This can be accomplished both at the






14

kinetic and hydrodynamic level in such a way that the properties of the

generating functional are preserved.

A set of nonlinear equations for the first functional derivatives

is first obtained: these equations have the same form as the macroscopic

regression laws, the only difference being the appearance of additional

contributions arising from transient effects in the system whose life-

time is short compared to the time scale of interest. By functionally

differentiating these equations, equations for the correlation functions

are obtained.

After defining precisely in Chapter II the class of problems

considered and the general form of the associated generating functional,

the specific cases of hydrodynamic and kinetic regimes are analyzed in

Chapter III and Chapter IV, respectively.

The appropriate macroscopic variables for the hydrodynamic descrip-

tion are the five average conserved densities. For simplicity attention

is limited to a simple fluid whose nonequilibrium state is adequately

described by the nonlinear Navier-Stokes equations. The set of equa-

tions for the first functional derivatives can be derived by using the

nonlinear response theory, as shown in Appendix A. The equations are an

inhomogeneous form of the nonlinear Navier-Stokes equations. Additional

contributions appear as extra terms in the irreversible heat and

momentum fluxes. By functionally differentiating these generalized

Navier-Stokes equations, a set of linear equations for the correlation

functions is obtained. The additional contributions to the irreversible

fluxes survive only in the equations for the equal time correlation

functions, derived by a limiting procedure on the equations for the

multitime fluctuations.





15

A similar procedure is carried through at the kinetic level in

Chapter IV. Here the macroscopic variable of interest is the one-

particle distribution function, interpreted as the average of the

microscopic phase space density. A low density kinetic equation for the

first functional derivative of the associated generating functional is

derived in Appendix C by following closely the nonequilibrium cluster

expansion used to obtain the Boltzmann equation from the BBGKY hier-

archy. Again the equation obtained differs from the nonlinear Boltzmann

equation by the presence of extra short-lived terms that only contribute

to the equations for the equal time correlations. The equations for the

correlation functions are obtained by functional differentiation of this

generalized Boltzmann equation.

At both levels of description, kinetic and hydrodynamic, the same

structure is obtained: nonlinear equations for the averages and linear

equations for the correlation functions. The coefficients in the linear

equations depend on the solution of the nonlinear problem: the dynamics

of the fluctuations is then entirely governed by the nonequilibrium

state. The amplitude of the noise, or source of fluctuations, enters in

the form of an inhomogeneous term in the equations for the equal time

correlation functions and is derived without introducing any assumptions

other than the ones entering in the derivation of the macroscopic

equations.

The formulation presented here starts from the Liouville equation

for the system, but all the equations considered for the macroscopic

description involve only averaged quantities, either observables or

correlation functions. The information on the nonequilibrium state

enters through boundary and initial conditions for such macroscopic

variables: these are the parameters controlled in an experiment.






16

The linear character of the equations for the correlation functions

appears as a general result in the present discussion, not as the

outcome of a linearization around the nonequilibrium state, applicable

only when the fluctuations are small in size. In this sense the results

obtained here agree with the conclusions of the M.I.T. group: the

dynamics of the fluctuations is, as in equilibrium, determined by the

state of the system in a form that naturally generalizes Onsager's

regression hypothesis.

As an application of the general description of nonequilibrium

fluctuations obtained by the generating functional method, the velocity-

velocity autocorrelation function of a tagged particle in a fluid in

shear flow is evaluated in Chapter V, for arbitrarily large shear

rate. Substantial simplifications occur in the problem as the result of

three conditions: (1) low density, (2) uniform shear rate and (3)

Maxwell molecules. In fact it is well known that the irreversible part

of the stress tensor may be evaluated exactly from the nonlinear

Boltzmann equation for a gas of Maxwell molecules in uniform shear

flow.54 Consequently, the hydrodynamic problem can be solved exactly to

all orders in the shear rate.19

Here the velocity-velocity correlation function is evaluated by

solving the appropriate set of coupled kinetic equations. The velocity

correlations do not decay to zero at long times, since the asymptotic

velocity of the particle is equal to the velocity of the fluid at the

position of the particle. Nonvanishing position velocity correlations

are then present in the fluid at large t. Also, due to the presence of

the shear in the fluid, equal time correlation functions which are zero

in equilibrium are now nonvanishing.









CHAPTER II
NONEQUILIBRIUM STATISTICAL MECHANICS OF CLASSICAL SYSTEMS

II.1 The Nonequilibrium Distribution Function:

Average Values and Fluctuations

To prepare or maintain a system in a nonequilibrium state external

forces have to be introduced in general; nonequilibrium statistical

mechanics involves the description of a system in interaction with its

surroundings in addition to specified initial conditions.3'6 The external

forces are assumed to have the character of reservoirs, in the sense

that they can be specified independently of the state of the system. In

the case of a fluid, for example, the most general situation is repre-

sented by interaction with particle, energy and momentum reservoirs.

In classical mechanics a system of N interacting particles enclosed

in a volume V is described in terms of the canonical coordinates,

ql...qN, and the canonical moment, pl, .. N, of the particles. The

state of the system is represented by a point in the 6N-dimensional

space of the canonical coordinates.

In the language of statistical mechanics the state of the system at

the time t is described by the N-particle distribution function,




PN(x ...,xN,t) = PN(xl(-t) ...,x (-t)) (2.1)



where x. = (qi,pi) Here xi(t) are the canonical coordinates of the

i-th particle as evolved from their initial value x.(0) x., according
to the dynamics of the system interacting with its surroundings. The
to the dynamics of the system interacting with its surroundings. The







18

distribution function pN represents the probability that the system lies

in a neighborhood dr = dxl...dxN of the phase space point

F = (xl,...,xN) at time t. The time evolution of the distribution

function is governed by the Liouville equation for a system in

interaction with its surroundings,3,6



S+ }PN PN 0 (2.2)
pn


where here and in the following summation over repeated indices is

understood. The operator C in Eq.(2.2) is



C = L(x ,...,xN) + P -I- (2.3)
bPn


where L is the Liouville operator for the isolated system, defined as



LpN = {PNH} (2.4)



Here (..*} indicates the classical Poisson bracket and H is the Hamil-

tonian of the system. P is an external force representing the effect
n
of the reservoirs on the n-th particle of the system: it depends in

principle on all the degrees of freedom of the system and reservoirs and

on the details of their interaction. The force n is in general
n
nonconservative and time dependent, and can be assumed to be applied

everywhere at the boundary of the system. This assumption does not

constitute a serious restriction since it corresponds to most


experimental situations of interest.







19

Equation (2.2) can be integrated formally when supplemented by an

initial condition at t=0,



pN( ...,xN,0) = PN,0(X1(0) ...,x (0)) (2.5)



In general pN, will be assumed to represent an arbitrary nonequilibrium

state. Also, the distribution function pN is normalized at all times,



I f dx ...d NPN(x, ...XN,t) = 1 (2.6)
N=0


As already mentioned in Chapter I, a macroscopic description of a

many-body system only involves a set of relatively few (compared to the

number of microscopic degrees of freedom) conveniently chosen variables.

The appropriate set of observables depends on the system considered and

on the macroscopic space and time scales of interest: the one-particle

distribution function is the object to consider in the kinetic theory of

gases and the five average conserved densities are the appropriate se:

of variables to describe hydrodynamics in a simple fluid. In any case

the choice is guided by the requirement that the macroscopic description

is closed on some time scale when given in terms of the variables

considered. A discussion of this point can be found for example in

Ref. 1.

Within the formalism of statistical mechanics the macroscopic

measurable quantities are defined as ensemble averages of corresponding

microscopic dynamical variables, {A (x1,... ,xN, s, functions of the

phase space coordinates,








A (t) = (2 .7)



where <...;t> indicates an average over the nonequilibrium distribution

function at time t,



= J dr A (x,..., N)pN(xl ,...,xNt) (2.8)



and f dr = f dxi ...dxN. Here and in the following a caret
N=0
indicates the microscopic variable as opposed to the averaged one.

An alternative expression of Eq.(2.8) is obtained by observing that

the time evolution of the dynamical variables {A } is governed by the

equation of motion,



} A(r,t) = 0, (2 .9)



where the operator has been defined in Eq.(2.3). Equation (2.9) can

be formally integrated with the result,



A (r,t) = T(t,0)A (F) (2.10)



The time evolution operator T(t,t') is the solution of the equation,



- } T(t,t') = 0 (2.11)



with initial condition T(t',t') = 1. It also has the property,


(2.12)


T(t,t')T(t',t") = T(t,t").






21

Equations (2.2) and (2.9) differ because in a nonconservative

system a region of phase space does not maintain a fixed volume as it

evolves in time. This can be seen explicitly by evaluating the Jacobian

of the transformation of an element of volume in phase space as it

evolves in time,



J = {x( )J (2.13)



For a conservative system, J=1. Here J satisfies the equation,


oF
+ C }J = n (2.14)
opn


and therefore its change in time accounts for the difference in the time

evolution of the distribution function and the dynamical variables.

The average values of the dynamical variables A can then be

written



= f dF A (xl,...,xN,t)pN(xl,...,xN,0) (2.15)



The possibility of shifting the time from the distribution function onto

A will be useful in the following to display some properties of the

time-correlation functions.

At this point the dynamics of the macroscopic set {A } is deter-

mined through the time evolution of the distribution function. In this

sense Eq.(2.8) has only the meaning of a formal definition.

On a microscopic scale A fluctuates instantaneously around its

average value. The microscopic fluctuations are defined as








6A (F;t) = A (F)
(2.16)



A complete description of the system should incorporate a systematic

analysis of the fluctuations and of their space- and time-correlation

functions of any order.

The second order equal time correlation function is defined as



M (t) E <6A A ;t>



= f dr aA()6A (r)pN(r,t) (2.17)



and, in general, the k-th order correlation function is



M (t) = <6A 6A ...6A ;t> (2.18)
a"'.Va a1 C2 ak


Again, as for the average quantities, the time evolution of the equal

time correlation functions is determined through the ensemble.

The amplitude of the equal time correlation functions is a measure

of the width of the probability distribution of the fluctuations of a

variable around its average value. The fluctuations can be viewed as

the result of a stochastic force which represents explicitly the effect

of instantaneous molecular collisions not accounted for in the macro-

scopic description and averages to zero over a time long compared to the

microscopic collision time. In the language of stochastic processes the

fluctuations are then the result of the "noise" in the system and the

equal time correlation functions measure the size of this noise. It

should be pointed out however that, even if here and in the following






23

some of the terminology of stochastic processes is used, the description

presented here is entirely macroscopic and always deals with averaged

quantities.

In equilibrium, away from critical points, the probability of fluc-

tuations is Gaussian and only second order correlations are important.55

Their amplitude is proportional to kBT, where T is the temperature and

kB is the Boltzmann constant, as suggested from the fact that in the

equilibrium state spontaneous fluctuations can only be thermal in

origin. In systems driven out of equilibrium new sources of noise may

become available, in a way strongly dependent on the particular state

considered. Furthermore in a nonequilibrium state equal time

fluctuations are dynamical variables: the clear separation of static and

dynamic properties that appears natural in equilibrium does not apply to

this case.

The correlation of fluctuations at different times are measured in

terms of multitime correlation functions. The most general definition

of a two-time correlation function is



M a(tl,t2;t) = <6A (tl)6A (t2);t> (2.19)



By using Eqs.(2.8) and (2.15) to translate the time arguments, this can

also be written as



<6A (t1 )6A (t2);> = <6A (tl-t)6A (t2-t);0> (2.20)



Therefore the two-time correlation function really depends on two time

arguments and on the initial time. In the following the dependence on







24

the initial time will not be indicated explicitly, unless needed. An

average over the initial nonequilibrium ensemble will simply be written



E d f dr A(r)pN(r,0) (2.21)
N=0


The k-time correlation function is then


A A A
M(t ...t) = <6A l(t )6A 2(t2)...6A k(tk)> (2.22)



The macroscopic variables and the correlation functions of their

fluctuations around the average value have been expressed here as

nonequilibrium averages of the corresponding microscopic quantities. In

the next chapters equations describing the dynamics of the observables

here defined will be derived in two specific macroscopic conditions.



II.2 Reduced Distribution Function Formalism
A
In the special case when the dynamical variables (A } are sum of

single particle functions, two-particle functions, etc.,


N N N
A (xl,...,xN) = a (x) + b (xi,x) + ... (2.23)
i=l i=1 j=1
i j

their averages and correlation functions are conveniently represented in

terms of reduced distribution functions. Many of the physical variables

of interest in real systems, such as the microscopic mass, energy and

momentum density in fluids, have the form (2.23).

A set of microscopic phase space densities is defined as







N
fl(x ,t) = 6(x1-xi(t))
i=l

N N
f2(x1,x2,t) = I 6(x1-xi(t))6(x2-xi(t)) (2.24)
i=1 j=1
i j

etc.



The reduced distribution functions usually defined in kinetic theory,56'57


s N!
n fs(xl...,xs t) = (N! f dx ..dxN N(x ...,xNt) (2.25
s (N-s)! dXs+l1'* PN(l N):,
N>s


can be interpreted as averages of the phase space densities over the

nonequilibrium ensemble,


s^
n f (x ,...,x ,t) < (x ,...,x ,t)> (2.26)



In particular the one-particle distribution function is defined as



n fl(xl't) =



S (N-)! f dx2 ...dxN PN(x 2,...,''xNt) (2 .2)
N>1


The reduced distribution function fs represents the probability density

that, at the time t, the positions and moment of particle 1 through s

lie in a neighborhood dx ...dxs of the point {x1,...,xs1, regardless cf

the positions and moment of the other particles in the system.

The average of a variable A of the form given in Eq.(2.23) is then

written as







26

(A ;t> = n f dx a (x )fl(xl,t)



+ n2 / dxdx2b (x1,x2)f2(x ,x2,t) + ... .(2.28)



For simplicity only the expressions for the correlation functions of

variables that are sums of single particle functions will be derived

here. The extension to more general cases is straightforward.

The correlation functions of the {A } can then be expressed in

terms of the correlation functions of the fluctuations of the one-

particle phase space density around its average value, Cs, with the

result,



M a(t) = f dx dx2a (x)a (x2)C2(xlt;x2,t) (2.29)



and

Ma(tl,t2) = dx1dx2 a (xl)a(x2) C2(xl,tl;x2,t2) (2.30)



In general the k-th order correlation function is given by


k
.M (t. ...,tk) = J...f i dxia .(xi)}Ck(Xl,tl;...;xk,tk) (2.31)
1 k i=1 1


Equivalent expressions for the correlation functions of more

general variables involving also two-particle functions, etc., will

involve correlations of higher order phase space densities.

The equal time correlation functions of phase space fluctuations

introduced in Eqs.(2.29-30) are defined as









27
A A
Cs(x1,t;...;Xs,t) = <6fl(x ,t) ...6fl(xs,t)> (2.32)



where

6fl(x,t) = fl(x,t) (2.33)



It is immediately seen that these correlation functions can be expressed

in terms of the reduced distribution functions f

Similarly, the multitime correlation functions introduced in

Eq.(2.31) are given by



Cs(x1,tl;...;xs,ts) = <6fl(xl,tl)...6fl(xs,ts)> (2.34)



The one-particle distribution function and the fluctuations in the phase

space density are the objects of interest in kinetic theory and will be

analyzed in detail in Chapter IV.

The expressions given in Eqs.(2.29-31) for the averages and the

correlation functions of a general set of observables in terms of the

reduced distribution functions are obtained by carrying out a partial

ensemble average over a reduced number of particles. They provide a

connection between macroscopic observables and kinetic theory and a

formal reduction of the many-body problem to an effective one-, two-,

... s-body problem. The reduced distribution function method is

particularly useful when dealing with gases at low or moderate density.

In this limit closed kinetic equations for the lower order distribution

functions can be derived.







28

II .3 Generating Functional for Nonequilibrium

Averages and Fluctuations

To describe the dynamics of a nonequilibrium system it is

convenient to introduce a generating functional G, from which averages

and correlation functions of any order can be generated through

appropriate functional differentiation.

The same idea has already been used by Dufty58 to derive kinetic

equations for equilibrium multitime correlation functions. The advan-

tage of this formalism is mainly that once a single set of equations for

the first functional derivatives of G has been derived, the equations

for the correlation functions are simply obtained by functional

differentiation.

If (A } is the set of dynamical variables chosen to describe the

system, the corresponding generating functional is defined as



G[{k}1] = ln (2.35)



where

U[{x }] = exp Jf dt A (r,t)X (t) (2.36)



and summation over the index a is understood. The test functions {kx}

in Eq.(2.36) have no physical interpretation. The only requirement

imposed at this point is that they are sufficiently localized in time

for the integrals in Eq.(2.36) to exist.

The first functional derivative of the generating functional is

given by








A (tX) <= a (2.37)
a


When evaluated at X =0, Eq.(2.37) reduces to the nonequilibrium average
a
of A,'



a (t|X=0) = A (t) (2.38)



Similarly, by successive functional differentiation,


(2)
M (tlt2 (2) G[XJ (2.39)
ap '2 T ta 1 t I a =01


and in general,


(k)
(t V=. t 6 G[X1 (2.40)
Mal ...ak(t ,...,tk) 6xaal ) 6xk (tk a=02.40)


Here and in the following the times are chosen to be ordered according

to t > t > *** ts. This implies no loss in generality since the

dynamical variables commute in classical mechanics. As desired,

averages and correlation functions of any order can be generated from

G. The objective is now to derive a set of equations for the time

evolution of the functionals IAa(tIk)}. When evaluated at

X=0, these should reduce to the macroscopic regression laws for the

system considered. Furthermore equations for the correlation functions

of any order can be obtained by functional differentiation. In this

sense the generating functional method provides a unified description of

the system, since all information can be derived from one single set of

equations.






30

The set of macroscopic regression laws describing, at long times,

the nonequilibrium state of a many-particle system (such as the Boltz-

mann equation and the nonlinear Navier-Stokes equations which apply at

the kinetic and hydrodynamic level respectively) can be derived by

averaging the microscopic equations of motion, at least in the case of

low density gases. The nonlinear Boltzmann equation for the one-

particle distribution function is obtained by performing a low density

closure of the BBGKY hierarchy. The Chapman-Enskog expansion provides

then a systematic method for deriving the hydrodynamic equations.

Alternatively, the nonlinear hydrodynamic equations can be obtained

directly by averaging the microscopic conservation laws over a

nonequilibrium ensemble which is a formal solution of the Liouville

equation and by evaluating the constitutive equations expressing the

irreversible part of the fluxes as functionals of the thermodynamic

gradients.

The same well established methods can be applied to the generating

functional to derive a set of equations for the X-dependent functional

{A (tXh)}. Furthermore the derivation preserves the properties of the

generating functional, in the sense that the set of equations so obtain-

ed may be differentiated to derive equations for the correlation

functions. This program is carried out explicitly in Chapter III and

Chapter IV for the hydrodynamic and kinetic limit respectively. The

theory of nonequilibrium fluctuations so derived is justified in the

same well understood limits leading to the macroscopic description.

The convenience of introducing a generating functional or charac-

teristic function to describe fluctuations has often been recognized in

the literature.59 In particular Martin, Rose and Siggia60 defined a







31

time ordered generating functional, involving both the dynamical

variables of the system and a set of conjugated operators describing the

effect of small perturbation in the variables. Also, in Eq.(52) of Ref.

31, Onuki defines a characteristic function analogous to the one

proposed here, from which equal time correlation functions of phase

space fluctuations can be generated. The definition given in Eqs.

(2.35-36) is however more general and it can be applied to a larger

class of statistical mechanical systems.

The next two chapters will be dedicated to the explicit derivation

of equations for the averages and the correlation functions describing

the properties of a fluid at the hydrodynamic and kinetic level.

However, before carrying out this program in detail, it is instructive

to present the results. The macroscopic description of the dynamic

properties of a many-particle system have the same structure at both

levels of description (kinetic and hydrodynamic).

The relaxation of the macroscopic variables {A } is described by

nonlinear equations of first order in time (to be identified with the

nonlinear Navier-Stokes equations for the average densities and the non-

linear Boltzmann equation for the one-particle distribution function),



A(t) + N [{A(t)}] = 0 (2.41)



where N represents a general nonlinear functional of the (A The set

of Eqs. (2.41) has to be solved with the appropriate initial and

boundary conditions, constituting an experimentally appropriate

definition of the macroscopic nonequilibrium state considered.






32

The dynamics of the correlations of fluctuations around the non-

equilibrium state is described by a set of linear homogeneous equations,



M (tt) A L[ l(tI) ,t = 0 (2.42)
t a 1' 2 ay yp 12


where
6Na[(A(t1)}]
-L (tl) 6A. (t- (2.43)
ap 1 6A P(t


Equation (2.42) applies-as long as the separation tl-t2 is large

compared to an appropriate microscopic time, tm,-of the order of the

lifetime of the transient before the adopted macroscopic description of

the system applies. Specifically, t = z the duration of an
m c
interparticle collision, for a kinetic description, and t = tf, the
m
mean time between collisions, for a hydrodynamic description. The

linear functional Lp in Eq. (2.42) depends on the solution of the

nonlinear equations, Eqs.(2.41). In this sense the dynamic evolution of

the two-time fluctuations is entirely determined by the macroscopic

nonequilibrium state.

The correlations of fluctuations at the same time, needed as

initial conditions for the set of Eqs.(2.42), are the solutions of a set

of bilinear inhomogeneous equations,



6 '(t) + L r[A(t)}]M (t) + L [{A(t)]4 (t) = FA(t)] .

(2.44)



The specific form of the inhomogeneity F depends on the problem con-

sidered. To interpret these equations, it is instructive to compare















33

them with the corresponding equilibrium results. In this case Onsager-

Machlup's hypothesis applies and the decay of the mean values and of the

fluctuations is governed by the same set of linear equations, obtained

from Eqs.(2.42) by evaluating Lap at equilibrium,


A(tl) Ao(t )

ap 1 2 yp 1 2

where the superscript "zero" indicates the equilibrium value or that the

average has to be understood as an equilibrium average. Also the fact

that in equilibrium, in virtue of time translation invariance, the two-

time correlation function depends only on the time difference has been

indicated. The equal time fluctuations are given by equilibrium sta-

tistical mechanics as a property of the ensemble. Equations (2.44)

become then identities: the inhomogeneous term is identified with the

amplitude of the noise in the system (this appears evident by using a

Langevin description). The inhomogeneity in Eqs.(2.44) can then be

interpreted as the noise term. Indeed it originates from microscopic

excitations in the system whose lifetime is short compared to the time

scale of interest.









CHAPTER III
HYDRODYNAMICS

III.1 Generating Functional for Hydrodynamics and

Nonlinear Navier-Stokes Equations

In this chapter the generating functional formalism outlined in

Section 11.3 is applied to derive a set of equations describing macro-

scopic properties and fluctuations at the hydrodynamic level. For

simplicity, attention is limited to the case of a simple fluid whose

nonequilibrium state is adequately described by the nonlinear Navier-

Stokes equations. The method used here is easily extended to more

general systems and other nonequilibrium situations, as will be seen in

the next chapter.

In general a hydrodynamic process is one which is adequately de-

scribed in terms of the averages of the locally conserved quantities in

a many-particle system.7 This contracted description applies to time

scales large compared to a characteristic microscopic time scale, the

mean free time between collisions. The hydrodynamic equations are

derived by averaging the microscopic conservation laws and by closing

the macroscopic equations so obtained with constitutive relations for

the irreversible part of the fluxes. This procedure can be carried out

in detail by applying, for example, the methods of the response theory

and provides a precise statement of the validity of the hydrodynamic

description for processes varying on space and time scales large com-

pared to some microscopic space and time scales characteristic of the

system.6,10








35

The microscopic conserved densities for a simple fluid are the
A A
microscopic mass density p, the total energy density E, and the three
A
components of the momentum density gi, respectively given by


N
p(r) = m 6(r-q) ,
n=1 ,. .

N
g(r) = n 6(rqn
n=l

N -
g(P)= Pni6(r-qn) (3.1).
n=l


Here N is the total number of particles in a volume V and m is the mass

of the particles; r represents a point in the fluid. The function e is
n

the energy of the n-th particle: for a central pairwise additive

interaction potential, V(qnn,), with qnn' = n n n is given by,


2



n'nn

In the following, for convenience, the set of five conserved densities

will be indicated as




{ca(W)} = (p(r),^(r),gi(r)) (3.3)



or
N
-
,(r) = a (x)6(r-q) (3.)
n=1


with








^a (xn)} = (mnPn i) (3.5)



Also, here and in the following, the indices n,n',... are used to label

particles and the indices i,j,k,l,... are used to label the components

of vectors and tensors. Greek indices, a,p,... run from 1 to 5 and

label sets of hydrodynamic variables.

The microscopic conservation of mass, energy and momentum is

expressed by the set of five conservation laws,



+ a A ^ .
(rt) +- i(r,t) = 0 (3.6)



The set of Eqs.(3.6) constitutes the definition of the microscopic

fluxes Y i, explicitly given by



{Y ()} = (gi()'s,(),tij (r)) (3.7)


A A
where s. is the energy flux and t.. is the momentum flux. Explicit

expressions for the microscopic fluxes can be found for example in Ref.

7, Eqs.(4.6). It should be noted that in writing Eqs.(3.6) no external

sources have been taken into account, even if a general nonequilibrium

system in interaction with its surroundings is considered here. The

adequacy of Eqs.(3.6) as the starting point to describe properties in

the interior of the system and the possibility of incorporating the

effect of the boundaries entirely through the thermodynamic parameters

is discussed in Appendix A.

The macroscopic quantities of interest are the nonequilibrium
averages of the
averages of the I{d>},








S(r,t) = <(r~();t> (3.8



and the correlation functions of their fluctuations around the average,



Ma l. k(rl 'tl;';rk'tk) = (3.9


where

6da(r,t) = pa(r,t) <(p(r);t> (3.10


)


)


)


For a one-component- fluid the average densities are explicitly given by


(3.11)


a (r',t) + = (p(r,t),E(r,t),p(r,t)vi(rt))


where p, E and v are the average mass and total energy density and the

macroscopic flow velocity, respectively.

The macroscopic conservation laws are obtained by averaging

Eqs.(3.6) over the nonequilibrium ensemble,19 with the result,



S +r = 0 (3.12)
1


The hydrodynamic equations are obtained from Eqs.(3.12) when these are

supplemented by an equation of state, relating the thermodynamic var-

iables, and by constitutive equations for the macroscopic fluxes as

functionals of the average densities.

The intensive thermodynamic variables, {y(r,t)}, such as temper-

ature, pressure, etc., are defined in the nonequilibrium state by

requiring that the nonequilibrium average densities are the same





38

functionals of temperature, pressure and flow velocity as they are in

equilibrium.6 This is obtained by choosing



<@a(r);t> = <+ (r);t>L (3.13)



where <...;t>L indicates the average over a local equilibrium ensemble

at time t,



L = X f dr A(D)pL(t) (3.14)
N=O


The local equilibrium ensemble for a one-component fluid is given by



pL(t) = expt-qL(t) f dr y (r,t)a (r)} (3.15)


where qL(t) is a constant determined by the requirement that PL(t) is

normalized to one. The right hand side of Eq.(3.13) is an explicit

functional of the {y }: this equation constitutes the definition of the

local thermodynamic variables conjugated to the ( I}. Explicitly,

r 1 2

{y (r,t)} = (-v +1 Pv2, ', -,vi) (3.16)


-1
where vp- is the chemical potential per unit mass and P = 1/k T, where

T is the Kelvin temperature and kB is Boltzmann's constant. Equations

(3.13) do not imply a limitation to states near equilibrium or local

equilibrium. Other definitions are possible and sometimes desirable.

However, the present definition has the advantage that the functional

relationships of all nonequilibrium thermodynamic parameters are the

same as in equilibrium.






39

The main problem in closing the hydrodynamic equations is the

specification of the irreversible part of the fluxes,



S(r,t) < i(r);t>L, (3.17)



as functionals of the thermodynamic gradients. These expressions can be

evaluated to first order in the gradients by using the linear response

theory, with the result,6


ajy (r"t)
Yi(r,t) = L (rt {}) (3.18)



where L is the Onsager's matrix of transport coefficients. For times

large compared to the mean free time between collisions, after the

initial transient describing the complicated behavior of the system

before hydrodynamics applies, the time evolution of the macroscopic

state is then described by the set of nonlinear Navier-Stokes equations,

given by


S a + ijy (r,t)
r. ___ ( t;{()} + Ly
(r,t) + {E(r,t{}) + L (, ) r = 0,
i O
(3.19)



where E represents the contribution from the nonlinear Euler
a
equation. Explicitly, the nonlinear Navier-Stokes equations are



p + V(pv) = 0 (3.20a)


av
S + + (3.20b)
(-- + v'V)u + hNVv = -f.q tij (3.20b)
ij br.








at
(_ + .v 1 p ij
(- +v* (iP + ] (3.20c)
+t v i p br. r
2 j

1 2
where u is the average internal energy density, u = e 2 pv h is the

enthalpy density, h = u+p, and p the pressure. To Navier-Stokes order
*
the irreversible parts of the heat flux, qi, and the stress tensor,

tij, are given by the usual Fourier and Newton laws as



qi(r,t) = -K(p,u) r. (3.21)


vk
j(,tP) = {Ip,u)Aijkl + C(p,u)6ij6kl} r, (3.22)



where A. =6 6. + 6. 6 2 6. 6. The coefficients of thermal
ijkl ij kl ik jl 3 ij k
conductivity, <, and of shear and bulk viscosity, n and C, respectively,

are in general functions of the thermodynamic variables, as indicated.

The generating functional G for hydrodynamic averages and fluctuations

is given by Eq.(2.35), with



U[(~}] = exp f_ dt f d'r (r,t)x (r,t) (3.23)



The first functional derivatives of G are



+G (,t X) 6GlXI
6X a(r,t)

<( (r,t)U[h]>
a (3.24)



where % often will be used to indicate the set of five test functions,

{Xa}. When evaluated at X=0, Eq.(3.24) reproduces the average densities







41

given by Eq.(3.8). Similarly, higher order functional derivatives

generate the correlation functions (3.9), as shown in Eq.(2.40).

The explicit form of Eq.(3.24),


) d -Lt U[XI
aN=O ar e- []> PNO)
N=O


S< (r;t>, (3.25)



where L is the Liouville operator of the isolated system (Eq. (2.4)) and

the time evolution operator is defined in Appendix A (Eq. (A.51)),

suggests that the same statistical mechanical methods used to derive the

hydrodynamic equations from the macroscopic conservation laws can be

applied to describe the time evolution of the set of functionals

{(a (r,tl)}. In fact, the time evolution operator is the same in both

cases and the difference in the equations can be incorporated as a

modification of the initial condition for the nonequilibrium ensemble.

The initial condition becomes dependent on the test functions

a through the functional U. This program is carried out explicitly in

Appendix A, where the methods of nonlinear response theory are applied

to derive a set of equations for the functional {a (r,tX)} The

derivation is identical to the derivation of the nonlinear Navier-Stokes

equations and preserves the properties of the generating functional.

In particular, a set of h-dependent thermodynamic variables,

{ya(?,tl|)}, is defined in analogy with Eq. (3.13) by requiring,



4 (r);t>=
kL' (3.26)





42

where <...;t>kL indicates the average over a X-dependent local

equilibrium distribution functional pL(tlx), given by
SL t


pL(tx) = exp{-qL(tXl) f dr ya(r,ti k) (r)} (3.27)


The constant qL(t X) assures the normalization to one of the distribu-

tion functional (Eq. (A.7)). The functional dependence of pL(tlX) on

the set t{y(r,tlX)} is the same as indicated in Eq. (3.13) for the

case X=O. The form of the thermodynamic equations relating {Iq(r,tlk)}

and {ya(r,tlX)} is therefore preserved and is again the same as in the

equilibrium case. The comments made when imposing the equality (3.13)

apply here. The thermodynamic variables {ya(r,tlX)} depend on

the {k } only through the X-dependent densities, { (r,t l)}.

The set of generalized Navier-Stokes equations for the five func-

tionals {1 [X]} derived in Appendix A are formally identical to the

usual nonlinear Navier-Stokes equations. In a matrix form they are

given by



-- Sa(r,t 1) + {E (r-,t;{4[X]})

= ((rrt)
+ L'(rt;{c{d]}) } = y (rtlX),
Sr. Dr. ai NS
(3.28)


where the Euler matrix, E and the matrix of the transport

coefficients, L %, are respectively defined as



E (r,t;{ [k]}) i );> (3.29)
a ai XL









and

L (r,t;{cp[]}) = lim ft dt <{eL(t-)^ () [> (3.30)
tt o ai j3.3)
t>>tf


where j[X] is the total (volume integrated) projected flux defined in

Eqs. (A.23) and (A.57), and the average in Eq. (3.30) is over the

X-dependent equilibrium ensemble with local thermodynamic variables,

defined in Eq. (A.54). Both E and L are nonlinear functionals of the
a ap
densities {[( (r,tl )} and depend on the test functions {( } only through

the {~}a themselves.

Additional contributions to the irreversible fluxes, not present in

the Navier-Stokes equations, appear in Eqs. (3.28). They originate from

the X-dependence of the initial condition for the nonequilibrium
**
distribution functional. The subscript NS to y i indicates that the

right hand side of Eqs. (3.28) has to be evaluated to Navier-Stokes

order in the thermodynamic gradients and in the limit of time t long

compared to the mean free time tf, i.e.


S (r* Yir' ) lim [ y (r,t )]NS (3.31)
br. ai NS r ai I )lNS
1 t>>t I


where [A]NS indicates the result of an expansion of the function A to

second order in the gradients of the thermodynamic variables. The
**
explicit form of y is derived in Appendix A, with the result,



S(,t|) = L (3.32)

Here the nonequilibrium distribution functional has been written

Here the nonequilibrium distribution functional has been written







D(tl() (3.33)
PN(tlI) = PL(tJX)e (3.33)
with
-Lt
D(tlX) = e D (X) + D'(tlX) (3.34)



and
t + L(t-T) y (a"TX) +
D'(tlX) = .- dr J dr e-(X
S ( |

+ rai (1} (3,35)





.D o( ) = D(O i) :


Uin ] PL(0)
= In (3.36)
(U[XJ>kL PL(OIX) '


where ai and a are the projected fluxes and the normalized densities

given by Eqs. (A.23) and (A.21), respectively, when the local

equilibrium average is substituted with an average over PL(tlX). When

evaluated at X=0, the additional contributions to the irreversible

fluxes vanish identically, since D (X=0) = 0, and Eqs. (3.28) reduce to

the nonlinear Navier-Stokes equations (3.19). However, when the

generalized equations for the functionals { (r,tJX)} are considered,

these extra contributions have to be kept and will be shown to be

directly related to the strength of the noise in the system.

The Navier-Stokes limit indicated in Eq. (3.31) is consistent with

the approximations introduced in the left hand side of Eqs, (3.28), but

can only be taken after functionally differentiating y i and evaluating

the result at X=0. The evaluation of this term will be needed when







45

deriving the equations for the correlation functions, and is carried out

explicitly in Appendix B.

The only assumption introduced in the derivation of Eqs. (3.28)

regards the form of the initial state, which has been chosen to be local

equilibrium. This choice does not imply any serious restriction,

because the deviations from initial local equilibrium are expected to

decay in a time of the order of the mean free time, tf, in states

leading to hydrodynamics.3,10 These deviations are only responsible for

the initial short lived transient in the system.

Using the same notation for the functionals and the corresponding

average densities, the X-dependent Navier-Stokes equations are

explicitly given by



p[X] + V((p[I]v[I]) = 0 (3.37a)
T4


[ I v ,* v .i [ ,]
(-- + v[]. V)u[] + h[X]--v[-] = -V.q [r] -tij ] -- (3.37b)
St V1 Or

+ 1 8p[A] 1tij []
( + v[.V)vi[] = + ) (3.37)
1 j

4
where the dependence on r and t of the functionals has been omitted to

simplify the notation. The irreversible parts of the heat flux and the

stress tensor are now given by




qi[O] = -[]NS
q[] = -[ --8 +q [INS (3.38)
1

]vk[i] **
t.[] X] = -{[X]Aijkl + [x]6j } r + tij[X]NS (3.39)







46

The transport coefficients in Eqs.(3.38-39) are functionals of X

through the X-dependent densities, {( (r,tlX)}. The additional

contributions to the irreversible fluxes are defined as

** D(t^) D'(t jX)

qi (r,tlX) = kL (3.40)

= XL

t..(r,t|x) = t (r)[e- e; (3,41)



where q.(r) is the microscopic heat flux, q.(r) = s.(r) v.t. (r).

Finally it should be stressed that although the extra contribution

to the irreversible-fluxes may seem'similar to the ones suggested by

Landau and Lifschitz47 to construct hydrodynamic Langevin equations (and

in effect their physical origin is analogous), there is a basic differ-

ence in the present formulation. All the equations here are equations

for averaged variables, not stochastic equations. In this way the

description obtained is directly related to the experimental situation.



111.2 Correlation of Fluctuations

The fluctuations of interest are those of the set of microscopic

densities, {((r,t)}. In particular the second order correlation

functions are defined as



HB(rl,tl2 = <6 (rltl)6 (r2,t2)> (3.42)



Using Eqs. (3.23-25), such correlation functions can also be expressed

as the first functional derivatives of the X-dependent densities,








6
M ( tl ;r2 ,t2 ( =0} (3.43)
6s (r2,t2) a


A set of equations for the two-time correlation functions can then be

obtained by functionally differentiating the generalized Navier-Stokes

equations, derived in the proceeding section, Eqs. (3.28). In a general

nonequilibrium state no symmetry properties relating the different

correlation functions can be identified a priori. The equations for the

set of twenty-five functions M (for a=i,...,5 and 6=1,...,5) are in

general all coupled together.

By functionally differentiating Eqs. (3.28) and remembering that

the X-dependence occurs in these equations only through the {4 (r,tjk)}

(or, equivalently, the thermodynamic variables ({y(r;tlX)}) and through

the additional terms in the irreversible fluxes, yai, the following set

of equations for the correlation functions is obtained


6 + +' + 4
t IMap(rltl;' r 2 t2+ + ,o(r lt1; {}aom(rltl;r2t2
4 4


= I ,(rl',tl;r2,t2) (3.44)



where ..
E a(r,t;{(}) aLl (r,t;{0}) ay (r,t)
a aao 0
C (r't;({ ) a +
Sr. + + r.
i (r,t) Y P (r,t)

ij oy a by(r ,t)
+ L(rt;{}) ~ y (3.45)
j 84 (r,t)


and
(+ [ 6y _(r ,t, 1)
I ,rl,t l;,t ) = lim [br ] (3.46)
tl >>t li 6s (r2 t2) 2=O NS






48
**
The fluxes y.i are defined in Eq. (3.32) and [A]NS indicates, as in

Section III.1, the Navier-Stokes limit of the function A. The differ-

ential operators in Lap operate on everything to their right, including

the operand of p itself. In deriving the set of Eqs. (3.44) the

assumption that the local equilibrium correlation functions appearing in

the operator f a are localized in space has been introduced. Therefore

these equations apply only if the system is away from critical points.

In the general case the operator Lap is nonlocal and the derivatives in

Eq. (3.45) have to be substituted with functional derivatives.

The inhomogeneous term on the right hand side of Eqs. (3.44) is

evaluated in Appendix B. For t -t2 tf, it is given by


t L(t -t +T) ,
I (r;,t ;r 2,t) = lim 2 d <[e i(r )]j>0



H 0) (3H.
x po(r2,;r (3.7)
ij


where <***>0 indicates an average over a reference equilibrium ensemble

with local thermodynamic variables evaluated at the point (rl,tl) (the

definition is given by Eq. (A.54), evaluated at X=0), and is the

volume integrated flux, defined by Eq. (A.57), at X=0. Also,
H + +
G (r 2,T;rl,0) is the Green's function for the Navier-Stokes equations,

defined in Eq. (B.17).

The time correlation function in Eq. (3.47) is the correlation

function appearing in the matrix of the transport coefficients, given by

Eq. (3.30) at X=0. Its lifetime is of the order of the mean free time

tf, and therefore much shorter than the macroscopic time scales of inter-

est in hydrodynamics. Consequently the term I does not contribute to the






49

hydrodynamic equations for the two-time correlation functions if



tl>>t ,



t -t2>>tf (3.48)



In this limit Eq. (3.44) becomes




-t M ,(rltl;r2,t2)+ oa(rl'tl; t )Ma(rltl;r2,t2) =0 (3.49)



This set of coupled equations describes correlations over time scales

large compared to a mean free time, as desired in a hydrodynamic

description. The equations for the correlation of fluctuations at

different times are linear, although coupled to the solution of the

nonlinear Navier-Stokes equations through the dependence of C a on

the {4 }. The linear character of the equations is a general result and

does not restrict their applicability to small fluctuations around the

nonequilibrium state.

For the case of small fluctuations around a nonequilibrium steady

state, Eqs. (3.49) agree with the results of others.26,39-42,46 The

equations derived here represent therefore a generalization of these

results: they provide a hydrodynamic description of fluctuations, valid

to Navier-Stokes order in the gradients, for nonstationary states and

fluctuations of arbitrary amplitude. A detailed comparison with

previous work will be presented in Section III.4.

For the special case of equilibrium fluctuations, the operator

fC reduces to that of the linearized Navier-Stokes equations. The






50

equations for the fluctuations at different times are then exactly those

suggested by Onsager's assumption: the fluctuations decay according to

the macroscopic linearized regression laws governing the dynamics of a

nonequilibrium system near equilibrium.

The explicit form of Eqs. (3.49) for a one-component fluid is

conveniently written in terms of the correlation functions



Mp(rl,t ;r2't2) = <6p(rltl )6 (r2,t2)>



M I (rl,tl;r2,t) = <6u(r ,t )6( (r2,t2



Mgi rltl;r2,t2) = <6gi(rl,tl)6 (r2,t2)> (3.50)



3=1,2,...,5. It is convenient to write the hydrodynamic equations in

terms of the correlation functions MuP instead of M -, where u is

defined by


N
u(r) = En(n-m(q n))6(r-q) (3.51)
n=l


The two correlation functions are related by



u Pl ;r2t2 = M p(r ltl;r2t2) i(rlt )mgi( ,l; 2)


+2 [v(rl,tl)]2 M (rltl;r2,t2) (3.52)



A coupled set of equations for the correlation functions (3.50) is

obtained from Eqs. (3.49) by respectively setting a=p,u,gi,







S + M = 0 (3.53)
at1 pp +r i gip


S + + + M u + ( hM + hM)
Svi ri) Mup + rh li Mvi v I 3 r, r prp u U
Sli 1 ri li


S {iK (TpMpp + TuM p) + (KM + KMU ) Kr }
ri rli p pp u up p pp r

av 6v
arj r p Vijkm + jm)Mpp + p ( i +jkm + S ijm)Mup
lj 1rlm

b V k 6
2V. + M ) = 0 (3.54)
ijkm + ijkm Or Or vi.p
1wm 1 1


M + (v M + py.M
SI ip g Ir j gip 1 v p


+ M p + p) + i jk + rvi
IJ


arli (pMpP + u up) + j iqj km ij 6m) im Mk}
Ir lj IF


+ [(n A .. + 5..k )M
Br j r -m p ijkm + p i p m)pp


+ (u Ajk.. + 6.ij km)M ]} = 0, (3.55)


where

P(rl't )i p(rl,t l;r2t2) Mg p(r,,t ;r2,t2)


+ -+
v i(rl,t )M (r ,tl;r2,t2) (3.56)


The space and time arguments of the correlation functions and the

hydrodynamic variables have been omitted in Eqs. (3.53-55) to simplify

the notation. The dependence of the correlation functions is the same

indicated in Eqs. (3.50) and the hydrodynamic variables are evaluated at






52

the point (rl,tl). Also, the thermodynamic derivatives of a function

A = A(p,u) have been indicated as



A [A(P'u)]u
p p '(3.57


A [A(pu)] (3.57)
u bu p


By comparing Eqs. (3.49) with Eqs. (3.53-5), the matrix


identified as





S(r,t) =





where



up

gip



uu
giu
giu


t (rt) is


pgi

ug.

gig
ii I


pgi br. '
1


=-h
1r.
1


V.

P
1
p


(-) I + h D
ri P P


2
(C -V) D6ij]


D.j [2D.i.+
13 1 3 j


1
2


I \
br.


8 T + (IT )K
r. p or p]


+ ( 2 - ) D6.] i
3 i p


= v. + h D
i or. u
1


1 1


T T
T + (T )' ]
1


1 2
2 Dij [2Dij + (C -j)D j],j


(3.58)


+ 2[2nD ij
ij i







53
1 1 + u 2 1
= -- 2[2nD + ( -rj)D 6.
ori p p r i ij 3 ij ri p


gip 8r. p
1


---- v.v.
Or. i j



gi u or u


S8 2
2 j lD + (Cp -- ) D
br. p i + r. p 3 P


8 8 A ,1k
r. Aijkm +' ij -) r
3 m


2 uD.. + (~ -2 ) D,
J 1


where


gigj 13 jr k Or i r k Aijkm m+ ~i r p








3 1


3.59)


3.60)


D D.. = Vv. (3.6
11



The differential operators in Eqs. (3.59) act on everything to their

right, unless otherwise indicated by the presence of a parenthesis.



111.3 Equal time fluctuations

Second order equal time correlation functions, defined by setting

t1=t2 in Eq. (3.42), i.e.



M(lt;r2 t) = <6((r ,t)6 (r2,t)>, (3.6


1)


2)


ugi








54

are needed as initial conditions for the solution of Eqs. (3.49).

However, equations for the equal time correlation functions cannot be

obtained directly by functional differentiation of the equations for

{<( (rl);t> }, as was done for the equations for the correlation of

fluctuations at different times, because the functional differentiation

does not commute with the time derivative. Instead, they can be

determined from the limit,


6 .* .6,
SM(r,t;r2,t) = [lim (+P P) (r);t+
a--t p 1 ( + 12 ap 66 2 ]=0 '
2t (3.63)



where the operators P12 and P p permute rl,r2 and a, B, respectively.

By using the results of the proceeding section, (Eq. 3.44), and

observing that the e-0 limit can be taken immediately on the left hand

side of the equation, an equation for Map is obtained in the form



t Mr(rl,t;r2't) + (1+P12P ) C(rl',t1;{ )Mo(r1lt;r2,t)



= FQ(rlr2;t) (3.64)


where the operator is defined in Eq. (3.45), and



p(rlr2;t) = lim (1+P 2P a >a (l,t+E;r2,t) (3.65)



The inhomogeneous term r a is evaluated in Appendix B, with the result






55

r (r ,r2;t) = (1+P2P ) L(r ,t;{ 6(r a-r)

(3.66)

where L is the Onsager's matrix of transport coefficients.
ap
Equations (3.64), together with the set of Eqs. (3.19) and the

equations for the two time correlation functions, give a closed descrip-

tion of transport and fluctuations (up to second order correlations) in

a hydrodynamic system whose macroscopic state is described in terms of

the nonlinear Navier-Stokes equations.

Again, the interpretation of Eq. (3.64) is clarified by comparing

with the case of fluctuations around the equilibrium state. Onsager's

assumption applies then and, as seen in the proceeding section, the time

evolution of the two-time correlations is governed by the linearized

hydrodynamic equations. Similarly, Eq. (3.64), evaluated at equi-

librium, becomes



(I+P12 P ) C(r ,{( })M (r ;r 2)



= (1+P P )L' o}) 6(r( 2) (3.67)
12 ap ap ri ar 2j 1 2
ii 2j


where the superscript zero indicates the equilibrium value and 0 now

depends on r1 only through the differential operators. Equation (3.67)

is an identity or, equivalently, constitutes a statement of the

fluctuation-dissipation theorem relating equilibrium fluctuations and

transport coefficients.

The description of nonequilibrium fluctuations derived here

indicates clearly how Onsager's assumption should be modified for

systems out of equilibrium. First, it is no longer true that the






56

regression laws for the fluctuations are the same as those for the

average values. They are instead linear equations, whose coefficients

depend on the solution of the nonlinear regression laws (they can be

interpreted as a linearization about the nonequilibrium macroscopic

state, {(a(r,t)}, at each instant t). In this sense knowledge of the

average dynamics still allows determination of the dynamics of fluctu-

ations. The two sets of equations form a bilinear set that must be

solved simultaneously. Secondly, the fluctuation-dissipation theorem

must be abandoned in favor of Eqs. (3.64) and (3.66) which, for

nonstationary states, indicate that even the equal time fluctuations are

dynamical variables.

In equilibrium, rTa is determined by the left hand side of Eq.

(3.67), since the equal time fluctuations can be calculated directly in

the Gibbs ensemble. In the nonequilibrium case, however, r a must be

provided independently. Its calculation requires the analysis of the

initial transients due to microscopic degrees of freedom not

incorporated in the Navier-Stokes equations. In this sense the source

function F p can be interpreted as a noise amplitude.

The explicit form of Eqs. (3.64) for a one-component fluid can be

obtained by using Eqs. (3.58-59) for the elements of the matrix rC.

The inhomogeneous term PF (r,2r ;t) is diagonal in the labels a and
atp 1 2'
p for a and p = p,u,g.. Its explicit form is given by Eq. (3.66), with



L (r,t) = 0,



L (r,t) = lim ft d- <[eLT sir)]S>0
t>>Cf





57
ij L t (>
L (r,t) = lim dr <[e tk r)] (3.68)
gkgl t>>t ok 0 0


where s!(r) and t!.(r) are the projected energy and momentum fluxes,

given by



s!(r) = s.(r) gi(r) (3.69)
p(r,t)
and

t() = tij(r) 6ij[p(r,t) + (-) u() + () p(r)] (3.70)
13 1j 13j u 6P
p u


Equations (3.68) are the Green-Kubo expressions for the transport

coefficients, or



LJ(rCt) = 6ij.T2 (r,t)K(p,u) (3.71)



ij +
L (r,t) = KrT(r,t)[ri(p u)Aijkl + C(p,u)6j Sk] (3.72)



The transport coefficients depend on r and t through the average mass

and internal energy density.



III.4 Discussion

Several authors26'27,39-46 have recently formulated a description

of fluctuations in nonequilibrium hydrodynamic steady states which are

adequately described by the nonlinear Navier-Stokes equations. It seems

therefore appropriate to compare these approaches with the one used here

and to discuss differences and similarities.

Tremblay, Arai and Siggia26 have proposed a Langevin theory of

fluctuations in nonequilibrium steady states. They assumed that the

Landau-Lifschitz theory of hydrodynamic fluctuations can be applied not





58

only for equilibrium fluctuations (in a regime where the macroscopic

regressions laws are linear), but also for small fluctuations around the

nonequilibrium state. The equations governing the time evolution of the

fluctuations are obtained by linearizing the nonlinear macroscopic

regression laws around the nonequilibrium state. The correlations of

the random forces are assumed to have the same form as in equilibrium,

with local thermodynamic variables. This description is supported by

the work of Keizer.30 It is however phenomenological in character and

restricted to small fluctuations around the nonequilibrium state. The

results of the present formulation basically confirm the Langevin

hypothesis. There are however some important differences. First, the

equations for the fluctuations are always linear, independent of the

size of the fluctuations. Secondly, the form of the noise spectrum is

derived here on the basis of a microscopic description, that clearly

places the source of the fluctuations in the transients present in the

system before the macroscopic description applies. Furthermore the

approach used here is not stochastic, but describes the system entirely

in terms of macroscopic variables (averages and correlation functions),

as required to make a precise and immediate connection with experiments.

Microscopic or semi-microscopic formulations of the description of

fluctuations in nonequilibrium steady states, similar in spirit and

content to the present one, have been developed by Kirkpatrick, Cohen

and Dorfman (KCD),45,46 and by an MIT collaboration,39-42 as indicated

in Chapter I. The results of KCD agree with the ones obtained here,

when the latter are specialized to the case of nonequilibrium steady

states. The inhomogeneous term in their equation for the equal time

correlation functions has apparently a different form from the one in

Eq. (3.66). The correlation of interest in Ref. 46 is the deviation of




















59

the equal time correlation function from its local equilibrium value.

The noise term there originates from the action of the Euler part of the

hydrodynamic equation over the local equilibrium correlation function,

as it is shown in Ref. 41. The form given in Ref. 46 and the one

derived here, Eq. (3.66),.are however equivalent. The hydrodynamic

equations obtained by Kirkpatrick, Cohen and Dorfman are therefore

identical to the ones derived here.

Instead, the results obtained by the MIT group (and, it seems, also

the Langevin theory) differ from those obtained through the generating

functional approach because the second term in the evolution matrix ,P

[see Eq. (3.45)], containing the thermodynamic derivatives of the trans-

port coefficients, is neglected there.41 This term is often small, but

is still of first order in the fluctuations around the nonequilibrium

state, and therefore cannot be neglected even when fluctuations of small

amplitude are considered.










CHAPTER IV
KINETIC THEORY

IV.1 Generating Functional for Phase Space Fluctuations

The hydrodynamic description of nonequilibrium fluctuations

developed in the previous chapter is restricted to states adequately

described in terms of the nonlinear Navier-Stokes equations and rests on

an estimate of the lifetime of certain correlations functions.

Within the framework of kinetic theory, it is possible, at least in

the limit of low density (where a small expansion parameter is

available), to derive an exact description of nonequilibrium fluctua-

tions. Furthermore the kinetic description is more general than

hydrodynamics. It describes macroscopic processes varying on shorter

space and time scales (of the order of or smaller than the mean free

path and the mean free time between collisions) and it incorporates the

hydrodynamic limit .3,8 Also, as shown in Section 11.2, the averages and

the correlation functions of the dynamical variables of interest in the

study of a large class of systems are simply related to the reduced

distribution functions and to the correlations of phase space

fluctuations.

The generating functional method is applied here to describe phase

space nonequilibrium fluctuations. No restrictions are imposed on the

nonequilibrium state or on its stability other than the well understood

assumptions entering in the derivation of the Boltzmann equation.4,5,57

The description of fluctuations obtained is justified in the same limits

60







61

which apply to the macroscopic description of the state. In particular

no extra assumptions are introduced to evaluate the amplitude of the

"noise" in the system, which contributes to the equations for the equal

time fluctuations. Finally, in the last section, the hydrodynamic limit

is recovered. Again, the results are restricted to nonequilibrium

states in low density gases. However all of the known qualitative

dynamics of fluids in general are also exhibited by gases. In

particular, the nonlinear hydrodynamic behavior is identical.8'9

The system considered is the classical system of N interacting

point particles in a volume V introduced in Chapter II. Since the

internal degrees of freedom of the molecules are neglected, the discus-

sion is limited to monoatomic gases.

The time evolution of the distribution function of the system is

governed by the Liouville equation for a system in interaction with its

surroundings, as given in Eq.(2.2). For pairwise additive central

interatomic forces the Liouville operator of the isolated system has the

form


N N
L(x ,...,xN) = L(x.) 9(xi,xj) (4.1)
i=1 i

with +
p.*V+
i q
L (xi.)= (4.2)



and



e(x ,x.) = [ V(q .)].( + - ) (4.3)
qi q Pi j






62

Equation (4.3) applies for continuous interatomic potentials, V(qij),

with qij = qi qj The potential V is also assumed to be short

ranged (of range o) and purely repulsive. The first term in Eq.(4.1)

represents the free streaming of the particles and the second term

contains the effects of the collisions.

Kinetic theory describes phenomena whose space and time variations

occur over scales large compared to the force range, a, and the duration

of a collision, Tc. The derivation of the kinetic theory from the

microscopic N-particle dynamics rests on the assumption that, for a

particular class of nonequilibrium states, a contraction of the

description takes place over a time of the order of T For times large

compared to T all the properties of the system depend on time only

through the one-particle distribution function. This is the idea

proposed by Bogoliubov to derive the Boltzmann equation from the

Liouville equation.4 The special states to which this macroscopic

description applies are identified through the requirement that the

initial correlations decay on times of the order of the time required

for a collision.5 The kinetic theory of nonequilibrium fluctuations

derived here will be restricted to the same class of states.

As mentioned, the macroscopic variable of interest is the one-

particle distribution function, fl(x,t), defined in Eq.(2.27) as the

nonequilibrium average of the phase space density, f (x,t). The

corresponding generating functional is given by Eq.(2.35), with



U[X] = UN[X] = exp jf+ dt f x fl(xt);(x,t) (4.4)


The first functional derivative of G is then given by







6G[X]
6W(x,t) (4


and reduces to the one particle distribution function when evaluated at

6G[=O]



[(xt] =o = f1(x,t) (4.6)


To obtain a kinetic equation, it is convenient to define the

functional f (x,t [) as



n f (x,tl ) for t > t (4.7)
1 o


where to is an arbitrary parameter introduced to assist in ordering the

times obtained on functional differentiation. Again, evaluating Eq.

(4.7) at X=0 and choosing to=0, the functional fl[X] reduces, at all

times t > 0, to the one-particle distribution function, as in Eq. (4.6).

The dependence on the parameter to can be introduced explicitly by

choosing the tests functions to be nonzero only for t < t i.e.



X(x,t) = X'(x,t)8(to-t) (4.8)



where 0 is the unit step function. In this way only time correlation

functions involving time arguments smaller than t can be generated. The

ordering chosen in Eq. (2.40) can be obtained by appropriate choice of

to. The correlation functions of phase space fluctuations, defined in



*A similar approach has been used in Ref. 58 to derive kinetic
equations for multitime equilibrium fluctuations.








64

Eqs.(2.32) and (2.34), are then immediately obtained as higher

functional derivatives of G, as shown in Eqs.(2.40). In the following

the dependence on the parameter to will be indicated explicitly only

when needed.

As already done at the hydrodynamic level, it can be argued that

the detailed form of the external forces and of their interaction with

the particles of the system should not appear in the equations governing

the time evolution of the reduced distribution functions (or

functionals).657 The dependence on the surroundings can be entirely

incorporated through appropriate boundary conditions to be used when

solving the kinetic equations. This is a consequence of the fact that

the lower order reduced distribution functions are localized quantities

describing properties at a particular point in the gas. Therefore, as

long as properties in the interior of the system are considered, at a

distance from the boundaries large compared to the force range, the

Liouville equation for the isolated system can be used to describe the

time evolution of the dynamical variables. Equation (4.5) can then be

rewritten as


-Lt UfX]
n fl(x,t |) = N / dx ...dxN e U > PN(0) (4.9)



This form suggests that fl(x,tlX) can be identified as the first member

of a set of functionals defined by



nsf (x X) N! dx dx e-Lt U[,]
... t ) = dx ..dx() .
s s' (N-s)! s+1 N N(O)
Nls








65

When evaluated at X=0, fs(xi,...,xs,tI) reduces to the s-particle

distribution function defined in Eq.(2.25). The operator governing the

time evolution of the set of functionals {fsA[]} is the Liouville oper-

ator of the system and does not depend on the test function X. This

dependence only enters through the initial condition. By differentiat-

ing Eq.(4.10) with respect to time it can then be shown that the

functionals f [X] satisfy the BBGKY hierarchy. In particular, for s=1,



{-- + Lo(x1)}f(x1,tIX) = n f dx29(x1,x2)f2(xx ,tlX) (4.11)



Again, as done in the hydrodynamic case, the standard methods of sta-

tistical mechanics which allow closure of the hierarchy and derivation

of a kinetic equation for the one-particle distribution function5 can be

applied to the X-dependent functionals. By assuming that the functional

relationship between fl[X] and K, as defined in Eq.(4.7), may be

inverted, it is possible to express, at least formally, f2[\] as a func-

tional of fl[X]. A formally closed equation for fl[X] is then obtained,



{- + Lo(xl)}fl(x,tlX) = n f dx2o(x1,x2)H(x1,x2,tlfl[X]) (4.12)



The functional H is highly nonlinear but it simplifies considerably when

evaluated at X=0. However, in order to preserve the properties of the

generating functional, the evaluation of H at nonvanishing K has to be

attempted. This evaluation is carried out in the next section in the

limit of low density.






66

IV.2 Low Density Limit and Boltzmann Equation

A systematic evaluation of the functional H defined in Eq.(4.12) is

possible in the limit of low density, by using the nonequilibrium

cluster expansion developed by Cohen5 as a generalization of the

technique proposed by Mayer for equilibrium systems. In this way the

two-particle distribution function for a dilute gas can be expressed in

terms of the one-particle distribution as a power series in the reduced
3
density no When this result is substituted in the first equation of

the BBGKY hierarchy, an expression for the rate of change of the single

particle distribution function in the form of a density expansion is

obtained. The terms in the expansion depend successively on the dynamics

of clusters of two, three, etc., isolated particles. To lowest order in

the expansion parameter, corresponding to the Boltzmann limit, only

binary collisions are retained.

The cluster expansion for the functionals f [X] is formally iden-

tical to the one for the distribution functions and is outlined in

Appendix C. To lowest order in the density, the result is



f2(x1,x2,t I) = t(x ,x2)f (x1,tX)f (x2,t X)



+ R(x1,x2,tlfl[X]) (4.13)



where the streaming operator is defined as




t(x1,x2) = S_t(x ,x2)St(x1)St(x2) (4 .1)



and St(xl,...,Xs) is the operator governing the time evolution of a






67

dynamical variable in a system of s isolated particles. It is given by


tL(x ,.. .,xs)
St (x, ...,X) = e (4.15)



where L(xl,...,xs) is the s-particle Liouville operator (see Eqs.(4.1)).

In other words St(xl,...,xs) describes the dynamics of an isolated

cluster of s particles. Finally, R(xl,x2,tlfl []) is a nonlinear func-

tional of fl[k], depending on the initial correlations in the system.

Its explicit form is given in Appendix C. For times long compared to a

collision time the streaming operator $- reaches a time independent

value,57



5(xl,x2) = lim S (x1,x2) (4.16)
t>>
c

Furthermore it is shown in Appendix C that if attention is restricted to

the time evolution of initial states of the system where the particles

have only short-range correlations (i.e. the s-particle distribution

function factorizes for interparticle distances large compared to the

force range), a part of the contribution to Eq.(4.12) from the function-

al R(xl,x2,tfl [\]) vanishes for times long compared to the collision

time.

By inserting Eq.(4.13) into Eq.(4.11) and by making use of the

above results, an equation describing the time evolution of the one-

particle functional f (x1,tXl ) is obtained. The equation only describes

variations over time scales large compared to T and is given by



-- + Lo(xl)}f(x,tl) = n 7[fi[kl,fl[k]] + W(x1,t fl[k]), (4.17)







for t > t where
o


J[A,B] = dx2 K(x1,x2)A(x1)B(x2)



and

K(x1,x2) = 9(x1,x2) -5(x1,x2)



The second term on the right hand side of Eq.(4.17) is given by



W(x1,tf 1[X]) = lim n f dx29(x ,x2)St(x1,x2)
t>>f
c

x {U2[Xlx2 X]U -1[x IX]UI -[x21 ] -1



x St(x1)St(x2)f (xlt )fl(x2,t 1),


(4.18)


(4.19)


(4.20)


where U [x ,... ,x ] is the low density limit of the s-particle
s 1 s
functional analogous to the N-particle functional defined in Eq.(4.4).

It is given by



Us[Xl,...,xs ] = exp fJ' dt St(xI,...,x2) (xi,t)
i=l


where


Sexp{ [ f dt X(xS(t),t)} ,
i=l



s
x.(t) = S (x ,..,x )x (0)
1 t s )xi()


(4.21)





(4.22)


A precise justification of the use of Eq.(4.21) as the low density form

of Us is given in Appendix C.





69

By evaluating Eq.(4.17) at X=0 and observing that

Us[x1,...,xs l=0] = 1 for all values of s, it follows that



W(xl,tlfl[%=O]) = 0 (4.23)



identically. Choosing to = 0, a homogeneous equation for the one-

particle distribution function is obtained,



{ -+ Lo(x )}f1(x1,t) = n 3[fl,fl] (4.24)



The right hand side of Eq.(4.24) is a nonlinear functional of the one-

particle distribution function. When the further approximation of

neglecting the difference in position of the colliding pair is intro-

duced (this implies that the resulting equation will only describe space

variations over distances large compared to the force range), the

operator J assumes the familiar form of the nonlinear Boltzmann oper-

ator, given by

IP -P21 2 + +

J[A,B] = f dp2 i db b J d4 A(qI ,P)B(ql p
A 2qI0 m 1 i2


A(qP )B(ql' 1 (4.25)



where b is the impact parameter for the binary collision and d is the

azimuthal angle measuring the orientation of the scattering plane. The

prime over the moment indicates their values after the collision. The

nonlinear Boltzmann equation for the one-particle distribution function

is then recovered. In the following both the operators J and J will be

referred to as the nonlinear Boltzmann operator.






70

Equation (4.17) has therefore the form of a generalized Boltzmann

equation. The extra term on the right hand side of the equation is a

complicated functional of X and its behavior at long times cannot be

analyzed before setting X=0. As just shown, it does not contribute to

the dynamics of the macroscopic variables of the system. However it has

to be retained in the equation for the functional fl(x,t|X) for the

purpose of deriving equations for the correlation functions.

Equations for the correlations of phase space fluctuations are

derived in the next two sections. As in the hydrodynamic limit, the

contribution from W to the equation for the multitime correlation func-

tions vanishes on a time scale of the order of t However, when
c
correlations of equal time fluctuations are considered, this term does

not vanish, but is responsible for the appearance of an inhomogeneous

part in the equations.



IV.3 Correlation of Fluctuations

The multitima correlation functions of phase space fluctuations

defined in Eq.(2.34) are obtained by functional differentiation of

G[k], as indicated in Eq.(2.40), or, equivalently,


s
Cs(xl,tl...;xst) f(x^,tk
k=1

(s-l)f ( lt I

[65(x2 t 2) ...6(xs ,ts ) to=t2 ,=0 '



with the time ordering t > t2 > .. t Equations for the time

evolution of Cs are then obtained by functionally differentiating

Eq.(4.17) and evaluating the resulting equation at k=0.






71

It is shown in Appendix C that the inhomogeneous term

W(x1,tl X) vanishes at long times (t >> T ) if to is chosen to

satisfy t t > T This choice corresponds to the one needed to
o c
generate multitime correlation functions in a Boltzmann description. In

this sense it immediately appears that W does not contribute to the

equation for the two-time correlation function, as long as tl-t2 >c

Here this result is also explicitly shown to follow from the analysis of

the functional derivative of W. This will make evident the difference

between the unequal and the equal time cases.

The equation for the two-time correlation function (s=2) is


S6W(x ,tI If I[ ])
--a + L(xl'tllfl ) C2(xltl;x2't2 [ 6(x2,t2) to=t,k=0 '

(4.27)

where

L(xl,tlfl) = Lo(xl) n A(xl,tllf1) (4.28)

and

A(x1,t lf1) = f dx3( (x1,x3) (,x3)(1+ P13)f (x3,tl) (4.29)



By functionally differentiating Eq.(C.26), the term on the right hand

side of Eq.(4.27) can be evaluated explicitly as a functional of the

one-particle distribution, with the result,


6W(x ,tl X)
[6X(x2,t2) -=o0 = n dx3e(x1,x3)Stl(x1 ,x3){[St 2(1,x3)


St(x )S t(x )][6(xl-x2) + 6(x3-x2 ]



x St (x)St (x3)f (x1,tI)f (x3,tl) (4.30)
1 1











6W(x ,tl X)
[ 2x ) ]
6X(x2,t2)


=n fdx39(x,x3){[ St (xX3) t (x ,Xj)

Sx St -t 1-t2 + 6(B-X

1( )S (x3)[6(x1-x2) + 5(x3-x2)i
2 1 2 1


x (x x3 )f( ,t)f ,t
1


where the choice t = t2 is understood. The streaming operator

St reaches a time independent limit for t>>T as shown in Eq.(4.16).

Therefore the difference between the two streaming operators in

Eq.(4.31) vanishes if



tl >> c


t t2 >> c (4.32)



In the above limit the right hand side of Eq.(4.26) vanishes,


lim
t -t>> c
1 c


6W(x ,tI X)
[6(xt) ]t 2,=0 = 0 ,
61(x t ) ti2t,X=0


(4,33)


and the time evolution of the two-time correlation function is governed,

by a homogeneous equation given by



( + L(x1,t1f1)}C2(x ,t1;x2,t2) = 0 (4.34)
l


(4.31)






73

The conditions imposed in Eqs.(4.32) imply that Eq .(4 .34) can only be

applied to describe correlations over time scales large compared to a

collision time. Not only the times t, and t2 have to be large compared

to T but also their difference. This is the natural outcome of con-

sidering the Boltzmann limit.

As in the proceeding section, when the difference in position of

the colliding pair of particles is neglected, Eq.(4.34) becomes


--I + LB(xl'tl fl)}C2(x'tl;x2,t) = 0 ,



where,

LB(xl,tllfl) = LO(x1) nl(x1,tjf) ,


(4.35)





(4.36)


I(x1,t1f1)A(x1) = f d3 f0 db b J d4 3 [ + P(pP3)]



x fl1,p'3',tl)A(ql'P ) fl1' 3t)A(ql1p .
(4.37)



The permutation operator P(pl,p3) interchanges only the moment of the

two particles. When the one-particle distribution function in Eq.(4.37)

is evaluated at equilibrium (i.e. coincides with a Maxwell-Boltzmann dis-

tribution), the operator I reduces to the linearized Boltzmann

operator.8

The equation for the correlation function is linear. This is an

exact result (in the Boltzmann limit considered here), not the outcome

of a linearization around the state. No restrictions have been imposed





74

on the size of the fluctuations. The operator L(x,tifi) depends on tha

nonequilibrium state of the system, characterized by the distribution

function fl(x,t), which is given by the solution of the nonlinear Boltc--

mann equation. In this sense the dynamics of the fluctuations is

entirely determined by the macroscopic nonequilibrium state of the

system.

To solve Eq.(4.34) specification of the initial condition, repre-

sented by the equal time correlation function C2(x1,t;x2,t), is also

needed. The derivation of an equation for the equal time correlation is

the subject of the next section.

Finally, the method described here can be extended to evaluate

higher order multitime correlation functions. In particular the

equation for the three-time correlation function is given by



b + L(x1 ,t 1f1)}C3(x1,t ;x2,t 2;x3,t3)
{i + L(l'l l)}C3(xl 2'tx"3')



= n J dx4K(x1,x )(i + P14)C2(Xl,t ;x2,t2)C2(x4,tl;x3,t3) (4 .3)



where the operator K(x1,x4) has been defined in Eq.(4.19). The equation

for the three time correlation function is again linear, but is coupled

to the lower order correlations through the inhomogeneous term on the

right hand side of Eq.(4.38).

In general, the equation for the s-time correlation function is a

linear inhomogeneous equation. The homogeneous part nas, at any order,

the form given in Eq.(4.34) and depends therefore on the nonequilibrium

distribution function. The inhomogeneous part couples the s-th order

equation to all the lower order ones. As s increases, the equ-aicns

become therefore very complicated, but always conserve linearity.







75

Furthermore the solution of the equation for the s-th order multi-

time correlation function requires the knowledge of all the correlation

functions obtained by progressively setting tl=t2, t=t2=t3, ...,

tl =t2 = ...=ts, in Eq.(4.26). Equations describing the time evolution

of such correlation functions can be obtained by applying an appropriate

limiting procedure on the equations obtained by functional different-

iation of Eq.(4.17). The second term on the right hand side of

Eq.(4.17) will only contribute when at least two of the time arguments

in the correlation function are equal, as will be shown in the next

section.



IV.4 Equal Time Fluctuations

In order to derive equations for the correlation functions of equal

time fluctuations, defined in Eq.(2.32), a limiting procedure on the

equations for the unequal time correlation has to be used. In this way

it is possible to circumvent the problem arising from the fact that the

time derivative and the functional derivative can only be interchanged

when involving different time arguments. As in the previous section,

the calculation will be carried out explicitly for the second order

correlation function. The rate of change of C2(xl,t;x2,t) can be

written in the form,




C2(x1,t;x2,t) = lim{C2(x1,t+E;x2,t) + .C2(x2,t+e;xl,t)} (4.39)



An equation for the equal time correlation can then be obtained from the

equations for the unequal time correlations, Eq.(4.27), with the result,







76

1 + (1 + P12)L(xl,tlf)}C2(xi,t;x2,t)


6W(x ,t+E X) 6W(x 2,t+E X)
= lm[ 6x(x2,t) ]t =tI=0 6+(x ,t) t =t,X=0
EO2 o 1 0


On the left hand side of the equation the limit E-+O has been taken,

The operator L(x,tlfl) has been defined in Eqs.(4.28-29). By using

Eq.(4.31) the right hand side of the equation can be written,


r(xl,x2,t)


6W(x ,t+ I ) 6W(x2,t+ I )
S!+1([- 6(x ,t) ]t =t,X=O + L 6(x ,t) ]to=t,=0


= li+(1+P12)n f dx3 (xi,x3) S (x1,x3)[1 St(x,x3)]

Ox e(x)S1- +
x S_E(xl)SF(x3)[6(x1-X 2) + 6(x2-x3)]


(4.41)


or, taking the limit,



P(x1,x2,t) (+P12) n f dx39(x1,x ){[S(:x-x2) + 6(X3-X2)] St(x1X3



s(x1,x2 )[6(x1-x2) + 6(x-x3)]}f (xt)f(x,t) .

(4.42)


By using the form (4.3) of the interaction operator O(x1,x3), it can

easily be shown that, for an arbitrary function F(xl,x3), the followi'

identities hold:


x St+e(XX3 )fl(x1,t+c)fl(x3,t+eV ) ,





77

(1 + P12) f dx38(x1,x3)6(x2-x3)F(x1,x3) = e(x1x2)F(x1,x2) (4.43)


(1 + P12) f dx3 e(x1,x3)8(x1-x29F(X1,x3)


= 6(x1-x2) f dx3O(x1,X3)F(x1,x3)


(4.44)


By introducing the above identities and considering the limit t >> c'

Eq.(4.42) can be written in the form,



F(xl,x2,t) = n 9(x1,x2) m(x1,X2)f1(x1,t)f1(x2,t)


+ 6(x1-x2)n f dx3e(xl,x3) m(x1,x3)f 1(x1,t)fl(x3,t)


- n[A(x1,tcf 1) + A(x25t If I )]56(x 1-x 2)f1(x15t)


(4.15)


or, from Eqs.(4.18) and (4.19),


r(x1,x2,t) = n K(x1,x2)f (x1,t)fl(x2,t) + n6(x1-x2)7[f,f]1



n[A(x1,tlfI) + A(x2,tifl)]6(xl-x2)f (xt) (4.46)


The equation for the equal time correlation function is then given by


(4.47)


ja + (1+P12)L(xitlf )IC2(x1,t;x2,t)= r(= IX 0


By neglecting again the difference in position of the colliding pair,

the various terms in Eq.(4.46) can be identified with Boltzmann-like






78

operators, as defined in Eqs.(4.25) and (4.37). In this limit, the

inhomogeneity F is given by



FB(xl,x2,t) = n 6(q-q 2)K(p 2 f(xl,t)fl(x2,t)



+ n6(x1-x )J[f 1,f i



n[I(x,tlfl) + I(x2,tlfl)]6(x-x2)f (xt) (4.48)



where K(p`1,2) is the kernel of the nonlinear Boltzmann operator,

defined by



f dp2K(pl'P2 )f (X,t)f (x2,t) = J[flf ] (4.49)



The dynamics of the equal time correlation function is determined by the

same operator that governs the time evolution of the two-time correla-

tion and again depends on the solution of the nonlinear macroscopic

problem. The linear equation for the equal time correlation function

however has now a inhomogeneous term, which is also a function of

fl(xi,t), i.e. of the state of the system.

The inhonogeneity in Eq.(4.47) assumes a familiar form when the

equilibrium case is considered for comparison. Equation (4.47) reduces

then to an identity, given by



(l+P12)n[I(Pi) + I o(2)]C2 (x;x2) = o(xl,x2) (4.50)


with







79

o(x1,x2) = n[I(1) + I(02)]6(x1-x2)(p1) (4.51)



where 4(p) is the Maxwell-Boltzmann distribution and I (p) is the usual

linearized Boltzmann operator. An identity equivalent to Eq. (4.50) is

obtained when equilibrium fluctuations are described through a

linearized Boltzmann-Langevin equation.53 The right hand side of the

equation is then identified with the amplitude of the noise in the

system. In a similar way the right hand side of Eq. (4.47) can be

interpreted as a measure of the noise in the system. It is a

complicated function of the nonequilibrium state through the one-

particle distribution function. This is a reflection of the new sources

of fluctuations, besides the thermal one, present in the nonequilibrium

state. As the random force in a Langevin description, the inhomogeneous

term r originates from microscopic excitations whose lifetime is shorter

than T which sets the time scale of the macroscopic description.

An alternative, and perhaps more popular, approach to the study of

phase space fluctuations in low density gases is based on a hierarchy

method analogous to the one used to derive the Boltzmann equation. Sets

of equal time and multitime distribution functions are defined as the

nonequilibrium averages of phase space densities .3 Hierarchies of

coupled equations for these distribution functions, formally identical

to the BBGKY hierarchy, are then derived from the Liouville equation.

The correlation of phase space fluctuations of equal and different time

argument are simply related to these distribution functions. A properly

reduced density can then be used as the expansion parameter to close the

hierarchies and derive kinetic equations for the correlation functions.






80

34-37
This approach has been used by several authors 7 In particular

Ernst and Cohen37 have derived kinetic equations for the nonequilibrium

correlation functions in a gas of hard spheres at low density by
3
performing an expansion in the parameter no and by retaining, to each

order in the density, only terms involving collisions among a limited

number of particles. The results derived in Ref. 37 agree with the ones

obtained here by using the generating functional method, but the

identification of terms of different order in the density is certainly

not transparent for space scales greater than a. The reason for this

lies in the choice of the expansion parameter. Ernst and Cohen applied

the method used when deriving a kinetic equation for the one-particle

distribution function. To close the first equation of the hierarchy

information on the two-particle distribution function, f2(x1,x2,t), over

distances smaller than or of the order of the force range a is needed in
3
this case. An expansion of f2 in powers of no is therefore

appropriate. The cluster expansion in fact gives information on the

variation of the distribution functions over space and time scales of

the order of the force range a and the collision time T
c
respectively. In low density gases, however, second or higher order

correlations among particles vary appreciably also over distances and

times of the order of the mean free path Z and the mean free time tf,

both much larger than a and T A different and more systematic method

of expansion of the hierarchy can then be developed by better analyzing

the role of the various length and time scales in the problem. If the

s-ch equation of the hierarchy is scaled according to thn two character-

istic lengths over which a variation of the correlation functions is

expected, I and tf, the expansion parameter a = (n3 )1 naturally






81

appears in the equation. The terms of an expansion of the s-th order

distribution function in powers of a can then be systematically

evaluated. The condition a<<1 corresponds to a gas at low density in

the usual sense (i.e. no3 << 1, since a (no3) 2), but with a large

number of particles in a volume of the linear size of a mean free

path. This is exactly the condition under which the effect of

collisions will be important over the space and time scale considered

and will be responsible for higher order correlations. The scaling of

the BBGKY hierarchy and the evaluation of correlation functions to

zeroth and first order in a is carried out in Appendix D for a gas of

hard spheres. Again the results agree with Ref. 37 and with those

obtained by using the generating functional method.



IV.5 Hydrodynamic Limit

In the low density limit, the average conserved densities in a

fluid are defined as



(t(r,t) = f dp a(p)nf1(x,t) (4.52)



where {a (p)l are the one-particle conserved quantities, defined in Eq.

(3.5). In this section, in order to simplify the notation, x is used to

indicate also the set of variables (r,p), where r represents a point in

the fluid and p is the canonical momentum. The hydrodynamic equations

are then obtained by averaging the microscopic conservation laws, Eqs.

(3.6), over the nonequilibrium one-particle distribution function and by

supplementing the equations so obtained with constitutive equations for

the macroscopic fluxes as functionals of the average densities.




82

A set of X-dependent average densities can be defined as



((r,tlX) = f dp a^(p)nfl(x,t|h) (4.53)


where f1[X] is the one-particle functional defined in Eq. (4.7). When

evaluated at h=0 (and for to=0), Eqs. (4.53) reduce, at all times, to

the usual average densities, defined in Eq. (4.52). The functionals

[((r,tlh)} are the low density limit of the ones defined in Chapter

III.

Again, intensive X-dependent thermodynamic variables {y (r,t|X)}

are defined by requiring



ca(r,tXl) = f dp a (p)fL(x,tlh) (4.54)



where fL is a one-particle local equilibrium functional, given by



f (x,t I) = exp{-ya (,thX)a (p)} (4.55)


The local equilibrium distribution is normalized as follows,



f dx fL(x,tlh) = N,



where N is total number of particles in the system. The thermodynamic

variables {y (r,t)} are given explicitly in Eq. (3.16). The comments

made in Chapter III, when requiring the equality (3.13), apply her2.

Hydrodynamic equations for the functionals {( (r,tJl)} are simply

obtained by projecting Eq. (4.17) for f1[X] over the five conserved

densities {a( p)1 By writing
a Q('






83

fI(x,tjx) = fL(xtlx) + A(x,t Ix)


(4.56)


and substituting in Eq. (4.17), an equation for A is obtained



+ Lo(x) n AL(x,tlfL)}A(x,t|\)
bt AL(x'tifQ)IA(x't IX)


= [ + Lo(x )]fL(x,tlX) + W(x,tl) ,


(4.57)


where, for consistency with the purpose of deriving equations to Navier-

Stokes order, only terms linear in A have been retained. The operator

AL in Eq. (4.57) is the local equilibrium form of the operator defined

in Eq. (4.29), i.e.


AL(x',t fL) = f dx2e(x1,x2) 5(xl,x2)(1+Pl3)fL(x,t |)


The solution of Eq. (4.57) is given by



A(x,tlX) = T(x;t,0)A(x,OIX) + ft dt T(x;t,)W(x,T X)
0


-ft d- T(x;t~t)La + L (x)I]f(x,-rIx)


(4.58)


(4.59)


where the time evolution operator T is X-dependent and is defined as the

solution of the equation,


1 6 + L (x) n I 0
3t 0 AL'x nhXxjctlfL,>=


(4.60)


with initial condition T(x;t,t) = 1. Also, the initial deviation from






84

the local equilibrium distribution functional is given by



A(x,OIX) = fL(x,Ol) (4.61)



By projecting now the equation for fl[X] over the conserved

densities {a } and making use of Eq. (4.54), the following set of

equations for the { a[X]} is obtained



-- (r,tX) + f- dp v. a (p)f(x,tk) + -i f dp-v. a (p)A(x,tl\)
7t a br i a L Or. i a
i 1


= f dp-a ()W(x,tl ) -- (4.62)



The term on the right hand side of Eq. (4.62) can be neglected. In fact

it contains the potential part of the momentum and energy fluxes which

are of higher order in the density and therefore negligible in the Boltz-

mann limit. By inserting Eq. (4.59) for A, the set of Eqs. (4.62) beccmes



a( + r Ea(rt; [Al})


~ ft dT f dp via (p)T(x;t,T)[-O + v*V ]f (x,Tlx)
or. o ia U r L
1


= dp v.a (p)T(x;t,0)A(x,Olk)
1


t Jt dr J dp v. a (p)T(x;t,T)W(x,T X) (4.63)
1


where the Euler matrix Ei has been defined as
aX







85

E (r,t;([hX}) J dp v. a (p)f (x,tl) (4.64)



Furthermore, to lowest order in the gradients,



[- + VV fL(X,TIX) = fL(x',lX) (P) (.65)
J


where

j(p) = (i P )v a (p) (4.66"



Here P l) is the low density form of the local equilibrium projection

operator defined in Eq. (B.9) and it is given by



P()A(a) = a (p)g(r) f dp' fL(r,p',tI)a (p')A() (4.67)
t P U ap a


where

gg(~) = f dp fL(x,t |)a(p)a (p) (4.6 )


-1
and g is the (ap) element of the inverse matrix. The thermodynamic
ap
variables {y } and their gradients appearing in Eq. (4.65) can also be

expanded in a functional Taylor series around the point t=-. To Navier-

Stokes order only the first terms in these expansions have to be

retained. Substituting Eq. (4.65) into (4.63), the latter becomes


--Y((rrt)t I+)
tatIX) r a af
^ a | {-(rt;{}) +r, })


a (.69)






86

where L is the matrix of the transport coefficients f r a low density

gas, given by



L a(r,t;{}) = lim dt J dx fL(x,t X)[T (x;t,T)via(p)
t>>tf


6(r-q)] j(p) (4.70)



For convenience, the same notation that was adopted to indicate the

matrix of transport coefficient for a dense fluid is used here for the

case of a low density gas. The two differ because the coefficient of

bulk viscosity vanishes at low density. The operator T in Eq. (4.70)

is the adjoint of the operator T and is defined by



f dx a(x)T(x;t,T)b(x)f (x,T) = f dx fl(x,z)b(x)T (x;t,T)a(x) ,

(4.71)


A A
for any pair of one-particle phase functions, a and b. Equation (4.70)

can be put in a more familiar form by observing that, to Navier-Stokes
(+ t
order, the hydrodynamic variables {y (r,t)} in the operator T can be

evaluated at -=t. The time evolution operator is then given by



T (x;t,t) = exp{[v.V- nI (x,t)](t-T)} (4.72)



where
--I
nIL(x,t) = fL (x,tlX)nL(xt fL)fL(xt X) (4.73)



The time integral in Eq. (4.70) can then be performed with the result,






87

Li(r,t;()) = f dx fL(x,t|x);j (p)[v V niL(x,t)]-1



xv.a (p)6(rq-) (4.74)



The action of the derivative operator in Eq. (4.74) on the operator

IL(q,p,t) or on fL(x,tl ) generates gradients of the thermodynamic

variables {y }. To lowest order these terms can be neglected and Eq.

(4.74) can be written as



L (r,t;{}) = J dqdp fL(r,p|x) (p)[.q d- nlL(r,p,t)]



x via (p)6(r-q) (4.75)

or

L (r,t;{}) = fdp fL(r,p ) (p)[n L(r,p,t)] v ) (4.76)



which is the familiar expression for the transport coefficients in the

Boltzmann limit, derived, for example, by using the Chapman-Enskog

expansion.8 The inhomogeneous term on the right hand side of Eq. (4.69)

is



W (r,tX) = J- dp via (p)T(x;t,0)A(x,0jX)
1


ft dr f dp via (p)T(x;t,T)W(x,tI) (4.77)
1


When evaluated at X=0, the right hand side of Eq. (4.69) vanishes

identically and the left hand side reduces to the nonlinear hydrodyna'niic

equations for a low density gas.






88

The correlation functions of the fluctuations of the conserved

densities around their average value are given by


S 6c (r ,t 1x)
M(rltl;r2,t2) = f dp2 a(p2) [6( t2) (4.73)



Equations for the correlation functions are then obtained by

functionally differentiating Eq. (4.69) and projecting the resulting

equation over the conserved densities. The case of correlation of

fluctuations at different time is first considered, with the result



I Map (r tl;r2t2) + rlt a (r't;r2't2


1
= I (r tl ;r2,t2) (4.79)


where
6W (r ,tI |)
I(rltl;r 2,2 lim f dp2a (p2 x2,t2) \=0 (4.80)
tl>>t f


The operator LK is given in Eq. (3.45) and depends on the state of the

system through the solution of the nonlinear hydrodynamic equations. In

the low density limit considered here, the terms containing the bulk

viscosity and the derivative of the pressure or of the enthalpy density

with respect to the density vanish, and do not appear in L ap

The inhomogeneous term on the right hand side of Eq. (4.79) is

analyzed in Appendix E. The contribution from the second term on the

right hand side of Eq. (4.77) is shown to vanish on a time scale of the

order of the collision time, Tc. Therefore, it does not contribute to

the hydrodynamic equations for the correlation functions, which describe










89

variations over time scales large compared to the mean free time, tf.

The functional derivative of the first term in Eq. (4.77) vanishes in

the limit

lim a(ltl;r2,t2) --- 0
tl>tf (4.81)
tl-t2>t

and gives no contribution to the equation for the unequal time

correlation functions, which reduces to the same linear homogeneous

equation obtained in Section 111.2, Eq. (3.49).

An equation for the equal time correlation functions can be

obtained by using the same limiting procedure applied for example in

Section 111.3 (see Eq. (3.63)). The resulting equation is


34
M rl,t;rt) + (1 + P2P ) : (rl ,t; {4 )M (r ,t;r,t)
it UP1 2 12 ap a1 2


S (r,r ;t) (4.82)
ap 1 2


where

V (' 0 2P- ) (raiV ,t) (4.83)
i(r +,r2;t) = lim (1+P1 P )I ,t+e;r2 ,t) (4.83)



Only the contribution to I from the first term on the right hand side

of Eq. (4.77) survives on the time scale of interest, leading to the

result,
































90

The set of hydrodynamic equations derived here to describe the time

evolution of averages and fluctuations strictly applies only to a low

density gas. With this restriction in mind, they are identical to the

equations obtained in Chapter III for a dense fluid.









CHAPTER V
TAGGED-PARTICLE FLUCTUATIONS IN SHEAR FLOW

V.1 Definition of the Problem

To illustrate the application of the formalism developed in the

previous chapters, the problem of steady shearing flow is considered.

Substantial simplifications occur as the result of three conditions:

(1) low density, (2) Maxwell molecules, and (3) uniform shear flow. In

fact for Maxwell molecules the irreversible momentum flux can be

determined exactly from the nonlinear Boltzmann equation, as a nonlinear

function of the shear rate. The macroscopic conservation laws can then

be closed and solved exactly for the situation of uniform shear. All

the information on the macroscopic state of the system, which is needed

in the equations for the correlation functions, can be evaluated for

arbitrarily large shear rate, as has been shown elsewhere.19,20,22-24

In particular the velocity-velocity autocorrelation function of a tagged

particle in the fluid is evaluated here. For the particular system

chosen, this correlation function can be calculated without any

approximation. Due to the presence of the shear on the system, equal

time correlations which are zero in equilibrium are now nonvanishing,

and their amplitude increases with the shear rate.








*Haxwell (or Maxwellian-) molecules are monoatomic molecules inte acting
through a weakly repulsive potential of the form V(r) = eo(o/r) where
r is the interparticle distance.






92

The steady shear flow corresponds physically to a fluid between two

parallel plates at a fixed distance apart and in relative motion. The
4 + *
flow field, U(r), is expected to vary linearly between the plates

(except near the surfaces) and be of the form,


U.(r,t) = U + a..r. .
1 oi 13


(5.1)


The constant vector Uoi and the constant tensor aij are the velocity

the lower plate and the shear rate tensor, respectively. In the

following Uoi will be set equal to zero without loss of generality.

the geometry considered, the tensor aij has the following properties:


aii = 0,



where no summation is intended in Eq. (5.2), and


a.j.a = 0 .
1i Jk


(5.2)


(5.3)


It is also assumed

only inhomogeneity



p(r,t) = p(t)



u(r,t) = u(t)


that the externally imposed shear flow represents the

in the system, and



-q *
q (r4t) = qi(t) ,
1 i


Sij (r ,t) = t ij(t)


(5.4)


*In this chapter the macroscopic flow velocity of the fluid is indicated
with U(r,t), instead of v(r,t), as was done in Ch. III.


I







93

Therefore, the walls of the system are not maintained at a constant

temperature (the extraction of heat from the system would create an

inhomogeneity in the temperature field). Heat is produced in the system

through viscous friction and the temperature grows in time. The state

considered is therefore nonstationary.

By inserting Eqs. (5.1-4) into the hydrodynamic equations for a

simple fluid, Eqs. (3.20), these reduce to



ap(t)
t- 0 (5.5)
at


Bu(t) *
u(t) = a. .t..(t) (5.6)
at 1j ij



The mass density p is therefore constant,



p(t) = po (5.7)



The internal energy density u can be considered a function of p and of

the temperature T (or the pressure p). The hydrodynamic equations

reduce then to one single equation, given by


BT(t) *
( aij(t)t (t) (5.8)
at 13 ij


OT
where a(t) = -u) In the case of a low density gas, the ideal gas
ou p
2 -1
equation of state can be used, to obtain a = (nK where

n = Po/m is the constant number density. The temperature equation

becomes then




Full Text

PAGE 1

FLUCTUATIONS IN SYSTEMS FAR FROM EQUILIBRIUM BY MARIA CRISTIM MARCHETTI A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1982

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Digitized by the Internet Archive in 2009 with funding from University of Florida, George A. Smathers Libraries A Teresa e Piero M. http://www.archive.org/details/fluctuationsinsyOOmarc

PAGE 3

ACKl-JOWLEDGEMENTS I would like to thank Professor James W. Dufty for his constant guidance and support in the course of this work. He suggested this problem to me . It was exciting and challenging to work with someone who has such a contagious enthusiasm for physics . I have learned much from him and he has also been a good friend . My gratitude also goes to all those who have helped me, both as colleagues and as friends. In random order they are: Greg, Mark, Bonnie and Pradeep, Robert, Vijay, John, Annalisa M., Annalisa P., Martin, Carlos, Simon, Bob, Teresa, Jackie, Attilio R., and many others. Without the love and support of my parents I may not have had the courage to venture to the United States . A special thanks goes to them for having always been near me during these three years . I would like to acknowledge the support of a Fulbright-Hays Travel Grant, which allowed me to come to the University of Florida, and an Educational Award from the Rotary International for the year 1981-82. Finally, I thank Viva Benton for her quick and accurate typing of the manuscript . Ill

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TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iii ABSTRACT vi CHAPTER I INTRODUCTION I II NONEQUILIBRIUM STATISTICAL MECHANICS OF CLASSICAL SYSTEMS 17 1. The Nonequilibrium Distribution Function: Average Values and Fluctuations 17 2. Reduced Distribution Function Formalism 24 3 . Generating Functional for Nonequilibrium Averages and Fluctuations 28 III HYDRODYNAMICS 34 1 . Generating Functional for Hydrodynamics and Nonlinear Navier-Stok.es Equations 34 2 . Correlation of Fluctuations 46 3 . Equal Time Fluctuations 53 4 . Discussion .57 IV KINETIC THEORY 60 1 . Generating Functional for Phase Space Fluctuations 60 2. Low Density Limit and Boltzmann Equation 66 3 . Correlation of Fluctuations 70 4 . Equal Time Fluctuations 75 5 . Hydrodynamic Limit 81 V TAGGED-PARTICLE FLUCTUATIONS IN SHEAR FLOW 91 1. Definition of the Problem 91 2 . Transformation to the Rest Frame 97 3. Two-time Velocity Autocorrelation Function 100 4 . Equal-time Velocity Fluctuations 103 5 . Results and Discussion 105 VI DISCUSSION 113 IV

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APPENDICES A DERIVATION OF THE \-DEPENDENT NONLINEAR NAVIER-STOKES EQUATIONS 118 B EVALUATION OF THE SOURCE TERM FOR HYDRODYNAMIC FLUCTUATIONS 137 C DERIVATION OF THE \-DEPENDENT BOLTZMANN EQUATION 145 D SCALING METHOD FOR A LOW DENSITY CLOSURE OF THE HARD SPHERES BBGKY HIERARCHY 155 E HYDRODYNAMIC "NOISE" FROM KINETIC THEORY 187 F DETERMINATION OF r"? FOR MAXWELL MOLECULES 191 REFERENCES 203 BIOGRAPHICAL SKETCH 207

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Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FLUCTUATIONS IN SYSTEMS FAR FROM EQUILIBRIUM by Maria Cristina Marchetti August, 1982 Chairman: James W. Dufty Major Department: Physics A unified formulation of transport and fluctuations in a nonequilibrium fluid is described at both the kinetic and the hydrodynamic levels . The method is based on the analysis of a generating functional for the fluctuations of the set of observables of interest around their average values . It is shown that the first functional derivatives of such a generating functional obey an inhomogeneous form of the macroscopic regression laws (nonlinear Boltzmann equation or nonlinear Navier-Stokes equations) . From this equation the dynamics of multi-space and -time fluctuations may be obtained by suitable functional differentiation , In particular the equations for the second order correlation functions of fluctuations at different and equal times are obtained to illustrate the method . The dynamics of a nonequilibrium fluid is governed by three sets of coupled equations: (1) nonlinear equations for the macroscopic observables, (2) linear homogeneous equations for the two-time correlation functions, and (3) linear vi

PAGE 7

inhomogeneous equations for the equal time correlation functions . The description obtained constitutes a precise statement of the generalization to nonequilibrium states of Onsager's assumption on the regression of fluctuations . The results apply to general nonstationary nonequilibrium states, either stable or unstable. The description of nonequilibrium fluctuations obtained rests on the same theoretical basis as the macroscopic description of the system (Boltzmann kinetic theory for a low density gas and hydrodynamics for a general fluid) . Vll

PAGE 8

CHAPTER I INTRODUCTION A large class of macroscopic irreversible processes in many-body systems is adequately described in terns of relatively few (compared to the number of degrees of freedom) macroscopic variables , changing continuously in time . This description applies when the time scale of interest is large compared to a microscopic time scale characteristic of the problem and the system considered. The macroscopic variables can be identified with the quantities measured in an experiment , and their values at a time t completely characterize the state of the system at that time . Examples of such a set of observables are the hydrodynamic densities in a fluid and the concentrations describing a chemical reaction. On this macroscopic scale static and dynamic properties of the system are expected to be governed by a closed set of macroscopic laws . In particular the time dependent properties should obey a set of differential equations of first order in time . It is then a deterministic description in the sense that the equations assign fixed values to all the macroscopic variables, once the values at t=0 have been given . The completeness of a set of macroscopic observables for the description of a particular problem on a particular time scale is difficult to determine a priori and is established only by the derivation of the macroscopic equations themselves . *The complete set of observables often coincides with the set of conserved variables in the system. However nonlinear coupling between the hydrodynamic modes in a fluid can lead to the appearance of memory

PAGE 9

2 The role of statistical mechanics consists in providing a microscopic basis for the macroscopic description, defining its limit of applicability, and establishing a precise connection between the dynamics of the observables and the time evolution of the positions and momenta of all the particles in the system. A derivation of the macroscopic laws from the microscopic dynamics, governed by the Liouville equation, can be accomplished by introducing 3 the idea of several well separated time scales in nonequilibrium fluids . A system in an arbitrary nonequilibrium state is considered . On a very short time scale, of the order of or smaller than the duration of a collision, the dynamics of a system of N particles is very complex and a large number of variables (the 6N coordinates and momenta of the particles in the system or, alternatively, all the reduced distribution functions) is necessary to describe it. However, after a microscopic time, of the order of the duration of a collision, or mean collision time, X , the system relaxes to a kinetic regime, where the dynamics c may be described more simply in terms of the average density in the single particle phase space. ' Therefore, when the time scale of interest is large compared to t , a contraction of the formal description of the dynamics of the N-body problem to a description in terms of the one-particle distribution function is obtained . effects in the macroscopic equations of motion: this indicates that the chosen set of variables is not complete and needs to be enlarged (for a discussion and references on this point see Ref . 2) . *For intermolecular forces with a finite and nonvanishing range, x is defined as the ratio of the force range, a, and the mean thermal speed of the molecules, v ,-i.e. x =a/v . o c o

PAGE 10

3 At larger time scales (for gases larger than the time between collisions, or mean free time, t£ , ) a further contraction of the description takes place . The system relaxes to a state near local equilibrium and its dynamics is described in terms of an even smaller number of variables, identified with the local conserved densities, e.g., mass density, energy density, flow velocity. This is the hydrodynamic regime, which describes variations on time scales of the order, for example, of the time required by a sound wave in a fluid to cross a region of macroscopic interest . The macroscopic regression laws governing the relaxation of a system displaced from equilibrium can be derived in principle by averaging the microscopic equations of motion and introducing approximations suitable for the time scale of interest. For example, in the limit of times large compared to a collision time (i.e., the kinetic regime), the nonlinear Boltzmann equation for the one-particle distribution function C Q is obtained by a low density closure of the BBGKY hierarchy . ' The Chapman-Enskog expansion provides then a systematic method for deriving the hydrodynamic equations for the average densities in the gas, on the 8 9 longer time scale. ' Alternatively, the hydrodynamic equations can be obtained directly by averaging the microscopic conservation laws , bypassing consideration of the kinetic stage . ' The contraction of the description of the dynamics of a manyparticle system in terms of few average variables naturally introduces the concept of fluctuations . From a microscopic point of view in fact *Defined as t = l/v , where Z is the mean free path of the molecules For liquids , t^ can be of tt scales is no longer useful. For liquids , t^ can be of the order of t^ and the separation of time

PAGE 11

4 the dynamical variables fluctuate instantaneously around their average values . The physical origin of the fluctuations can be different for different systems. In general, however, the amplitude of their space and time correlations controls the adequacy of the macroscopic description and can be measured in a variety of experiments . Thus the statistical mechanics of irreversible processes in a many-particle system should also provide a framework to describe fluctuations in an arbitrary nonequilibrium state . The description of fluctuations in equilibrium systems is well established from both phenomenological considerations, and first principles calculations for some limiting cases (e.g., low density 1 2 gases) . A precise connection between the spontaneous fluctuations in the equilibrium state and the macroscopic response of the system to a small external perturbation is established by the fluctuationdissipation theorem. ^'^^ This result is concisely expressed in Onsager's assumption on the regression of fluctuations,^^ which states that spontaneous fluctuations around the equilibrium value decay in time according to the same linear laws governing the relaxation of a macroscopic state close to equilibrium. In other words the linearized regression laws apply regardless of whether the nonequilibrium condition is prepared or occurs spontaneously . The adequacy of this assumption is a consequence of the fact that the size of the equilibrium fluctuations is limited by their thermal origin. Furthermore the space correlation of the equal time fluctuations can be calculated directly from the equilibrium ensemble .

PAGE 12

5 The relationship of correlation functions and response functions to a great number of measurable properties of macroscopic systems in equilibrium is also well understood and a wide variety of experiments is available for comparison. The interpretation of Rayleigh-Brillouin light scattering experiments in fluids constitutes a well known example of the success of the equilibrium fluctuations theory . The detailed evaluation of time dependent correlation functions remains in many cases a difficult many-body problem, but the theoretical formulation can be stated clearly and is founded on the basic principles of statistical mechanics . The corresponding theoretical and experimental study of fluctuations in a nonequilibrium system is much less complete and several new problems arise, even from a purely theoretical viewpoint, in addition to the difficult computational aspects . The systems considered are in general open systems, in contact with several reservoirs, which are not in equilibrium among themselves. ' These reservoirs impose external forces and fluxes on the system, preventing it from reaching thermodynamic equilibrium and keeping it in a nonequilibrium state , which is stationary if the external forces are time independent. In contrast to the equilibrium system, which is specified by a unique thermodynamic state, a large class of nonequilibrium states must now be considered. The first problem encountered when dealing with systems out of equilibrium is therefore the adequate characterization of the particular nonequilibrium phenomenon of interest . Furthermore , the formulation of the macroscopic description of the system and the derivation of the nonlinear regression laws for a

PAGE 13

6 complete set of observables, such as the nonlinear hydrodynamic equations, presents several difficulties. At the hydrodynamic level the evaluation of the form of the macroscopic fluxes as functionals of the thermodynamic forces is required . These constitutive relationships are highly nonlinear and only a formal expression can be provided in the general case . ' An explicit evaluation has only been performed in specific model cases (shear flow ' ) or under precise limiting conditions (Burnett coefficients ') . On the other hand, in near equilibrium situations the formalism of linear response theory provides expressions for susceptibilities and transport coefficients in terms of the correlation functions of equilibrium fluctuations (Green-Kubo relations) . ' Nonlinear transport phenomena also can be related to nonequilibrium fluctuations by introducing the concept of nonlinear response functions . The understanding of the dynamics of fluctuations in the nonequilibrium state is therefore intimately related to the description of energy, momentum, mass and charge transport in systems far from equilibrium. Other characteristic nonequilibrium phenomena where fluctuations play an important role are the appearance of instabilities, bifurcations and turbulences , and the related onset of nonequilibrium phase tran• . 25 sitions . A system macroscopically displaced from equilibrium and brought outside the region of applicability of the linear response theory (into a region where the macroscopic regression laws are nonlinear) evolves through a sucession (or branch) of nonequilibrium states as the magnitude of the parameter measuring the strength of the external perturbation increases . At some critical value of this parameter the state of the system may become unstable. The system will then make a

PAGE 14

7 transition into a new branch of states . The occurrence of such instabilities may lead to a change of symmetry of the stable state of the system . In this sense the phenomenon is analogous to a phase transition in a system in thermodynamic equilibrium. Therefore, as the mechanism of equilibrium phase transition can be understood by recognizing the importance of the fluctuations near the transition point , the occurrence of bifurcations (and other nonequilibrium phase transitions) is intimately related to the fluctuations in the system at the instabilities . A large variety of physical systems shows this general behavior . The most popular hydrodynamic example is the B^nard instability, occurring in a liquid layer heated from below in the gravitational field . Other examples are found in nonlinear optics , autocatalytic reactions and biological systems . The formulation of a theory of nonequilibrium fluctuations which does not impose restrictions on the size of the fluctuations and applies near instabilities is an essential starting point for the understanding of all these nonequilibrium phenomena. However, even for the simplest case of hydrodynamic fluctuations, it is not obvious how Onsager's assumption should be generalized for such nonequilibrium states . Because of the nonlinear character of the average regression laws, the dynamics of the macroscopic observables is no longer the same as that of their fluctuations . Furthermore a nontrivial problem is the determination of the source of the fluctuations, which may no longer be simply thermal in origin and whose knowledge is required for the specification of the equal time correlation functions . There have been several recent studies of fluctuations in nonequilibrium stationary states that are similar in spirit and objectives to

PAGE 15

8 the present discussion. It therefore seems instructive to put the present work, in context by presenting a brief overview of these approaches . They can be classified in general into two broad categories: stochastic approaches, ° ^^ where the microscopic fluctuating variables are assumed to define a random process, characterized by a set of probability densities, and microscopic approaches ,^^~'^^ which attempt to derive from first principles the dynamics of nonequilibrium fluctuations . The stochastic formulations are essentially based on a generalization of the Onsager-Machlup regression hypothesis to the nonequilibrium case . The time-dependent properties of the system are governed by a set of equations for the probability densities (Fokker Planck equations) or for the stochastic variables themselves (Langevin equations). A stochastic theory of hydrodynamic fluctuations has been developed by Tremblay, Aral and Siggia^^ by assuming that the fluctuating hydrodynamic equations proposed by Landau and Lif schitz^^ '"^^ can be applied to describe not only fluctuations in the equilibrium state, but also small fluctuations around a nonequilibrium stationary state. The time evolution of the fluctuating variables is then governed by a linear Langevin equation whose coefficients depend on the nonequilibrium state of the system. The correlation of the random forces is chosen to have the same form as in equilibrium, except with local thermodynamic variables for the nonequilibrium state. The characteristic feature of the Langevin theory is the separation of the time variation into a slowly varying and a rapidly varying part . The basic assumption is that the correlations of the fast variables are determined by a local equilibrium ensemble . This Langevin theory has been successfully

PAGE 16

9 applied by several authors ° » ^~ to the evaluation of the scattering of light from a fluid with a temperature gradient and/or in shear flow. However the limitation on the size of the fluctuations clearly restricts its applicability to regions far from instabilities and critical points . Furthermore in Ref . 26 the example of a case where the theory fails is given: electrons scattering off impurities to which a potential difference is applied . Presumably this is a case when the local equilibrium assumption for the fast variables does not apply. A formal theory of nonequilibrium fluctuations and nonlinear irreversible processes has been developed by Grabert , Graham and Green. ' Their approach generalizes to the nonlinear regime the functional integral expression for the transition probability between two macroscopic states introduced by Onsager and Machlup ' as an extension to the time dependent domain of Boltzmann's relationship between entropy and probability . The basic assumption of the theory is that the probability of a given fluctuation from one state to another is measured by the minimum increase in action associated with the change among the two states considered . This minimum principle provides the possibility of constructing a path integral expression for the conditional probability which appears as a natural generalization of the Onsager-Machlup functional . However this minimum principle is purely an assumption and has not been justified on any microscopic basis . Only a posteriori can it be seen that the results obtained agree with those derived through different formulations . In particular the conditional *It should be pointed out that the results of Kef . 50 for the Brillouin scattering in temperature gradient are not in agreement with the results of the other calculations . '

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10 probability can be shown to satisfy a Fokker-Planck equation . This equation has also been applied by Grabert^^ to evaluate the spectrum for light scattering from a fluid in a temperature gradient . His results agree with those derived by other methods . Within the framework of this generalized Onsager-Machlup theory, 25 Graham has also analyzed in detail problems associated with the stability and the breaking of symmetry in nonequilibrium stationary states . Several authors^^"^^ have formulated studies of nonequilibrium fluctuations based on a master equation in an appropriate stochastic space. With the aim of generalizing Landau-Lif shitz 's fluctuating hydrodynamics to the nonlinear region, Keizer^^ has rewritten the microscopic conservation laws in a fluid in the form of master equations by describing the fluid in terms of elementary molecular processes . His results support the Langevin theory proposed in Ref . 26 . Coarse grained master equations for inhomogeneous systems have been assumed as the starting point to describe phase space fluctuations by Onuki^^ and 32 Ueyama. For systems near equilibrium the linearized BoltzmannLangevin equation is recovered. Furthermore by using the ChapmanEnskog expansion method fluctuating hydrodynamics can be derived. Finally, van Kampen^^ has applied the method of expansion in the size of the system to the solution of master equations for several problems, from the Boltzmann equation to the rate equations governing chemical reactions . The principal limitations of the master equation approaches are the necessity of assuming (instead of deriving from first *This is in effect a generalization of an analogous master equation proposed by Logan and Kac"*'^ for homogeneous systems.

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11 principles) the basic equation, and the strong dependence of this equation on the model considered. Finally, a generalization to the nonlinear regime of the linearized CO o o Boltzmann-Langevin equation-^ has been proposed to describe nonequilibrium phase space fluctuations in low density gases . Tliis method is similar in spirit to the fluctuating hydrodynamics of Ref . 26, since the correlation of the random forces is again assumed to have a local equilibrium form. The microscopic studies are based on several different techniques . First of all the methods of the kinetic theory of gases have been used ~ to describe phase space fluctuations . To make any progress with the formal theory the limitation of low density has in general to be imposed. This restriction, however, allows the introduction of controlled approximations. Furthermore, the kinetic theory is more general than the hydrodynamic description since it describes fluctuations on shorter space and time scales and it incorporates the latter . In particular Ernst and Cohen have applied the methods of the nonequilibrium cluster expansion to derive kinetic equations for the equal and unequal time two-point correlations of phase space fluctuations for a gas of hard spheres . The meaning and adequacy of their expansion will be discussed in Chapter IV and Appendix D, where an alternative method, leading to equivalent results » is presented. An alternative microscopic theory of fluctuations in nonequilibrium steady states has been developed by an M.I.T. group .-^""'^ For states not too far from equilibrium (up to second order in the parameters measuring the deviations from equilibrium) they apply the method of nonlinear response theory to express nonequilibrium averages and

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12 correlation functions in terms of higher order equilibrium correlation functions . These correlation functions are then evaluated (in the hydrodynamic case) by assuming a separation of time scales in the system, implying that the correlations involving the dissipative fluxes decay on a time scale shorter than the macroscopic time scale of interest . Their results are also generalized to nonequilibrium stationary states arbitrarily far from equilibrium by employing the projection operator techniques . They conclude that in general a simple generalization of the equilibrium fluctuation-dissipation theorem to a local equilibrium form does not hold . The formalism used however obscures many of the physical hypotheses entering the derivation. Furthermore the separation in fast and slow variables is not well founded for states arbitrarily far from equilibrium. Finally Kirkpatrick, Cohen and Dorfman^" " developed a hydrodynamic theory of nonequilibrium fluctuations in stationary states based on the use of projection operator techniques and on the KadanoffSwift mode-coupling theory for the evaluation of the equal time correlation functions . They also have applied their description of hydrodynamic fluctuations to the problem of light scattering from a fluid subject to a temperature gradient, for both the cases of small and large gradient . The formulations briefly described here suffer from several limitations . Their applicability is often restricted to nonequilibrium stationary states and to small fluctuations around the states . The latter condition implies that the system has to be far from instabilities and critical points . A precise connection between the stochastic and the microscopic approaches, as is possible for the case

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13 of equilibrium fluctuations, has not been established. In the stochastic methods it is often not clear how the macroscopic information on the nonequilibrium state, as specified in an experiment, enters the problem. In general there is a need for developing an exact treatment of nonequilibrium systems whose validity extends to both the kinetic and hydrodynamic regimes and that establishes a precise connection between the theoretical and the experimental description. The objective of the present work is to provide a unified formulation of the description of nonequilibrium fluctuations founded on a microscopic basis and constituting a precise statement of the generalization of Onsager's assumption on the regression of fluctuations to arbitrary nonequilibrium states . The problems mentioned above are addressed explicitly. An open system in contact with external reservoirs is considered . The reservoirs prepare or maintain the system in a general nonequilibrium state . The results derived apply to nonstationary states, either stable or unstable. The method rests on the definition of a generating functional whose first functional derivatives are directly related to the complete set of macroscopic nonequilibrium observables relevant to the problem considered . Higher order functional differentiation generates the correlation functions of the fluctuations of the chosen dynamical variables around their average values . The first advantage of this approach is that it makes it possible to discuss the problems of average behavior and fluctuations in a unified context . Characterizing the nonequilibrium state amounts to providing a statistical mechanical derivation of the macroscopic regression laws . This can be accomplished both at the

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14 kinetic and hydrodynamic level in such a way that the properties of the generating functional are preserved . A set of nonlinear equations for the first functional derivatives is first obtained: these equations have the same form as the macroscopic regression laws, the only difference being the appearance of additional contributions arising from transient effects in the system whose lifetime is short compared to the time scale of interest . By functionally differentiating these equations, equations for the correlation functions are obtained . After defining precisely in Chapter II the class of problems considered and the general form of the associated generating functional, the specific cases of hydrodynamic and kinetic regimes are analyzed in Chapter III and Chapter IV, respectively. The appropriate macroscopic variables for the hydrodynamic description are the five average conserved densities . For simplicity attention is limited to a simple fluid whose nonequilibrium state is adequately described by the nonlinear Navier-Stokes equations. The set of equations for the first functional derivatives can be derived by using the nonlinear response theory, as shown in Appendix A. The equations are an inhomogeneous form of the nonlinear Navier-Stokes equations . Additional contributions appear as extra terms in the irreversible heat and momentum fluxes . By functionally differentiating these generalized Navier-Stokes equations, a set of linear equations for the correlation functions is obtained. The additional contributions to the irreversible fluxes survive only in the equations for the equal time correlation functions, derived by a limiting procedure on the equations for the multitime fluctuations.

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15 A similar procedure is carried through at the kinetic level in Chapter IV. Here the macroscopic variable of interest is the oneparticle distribution function, interpreted as the average of the microscopic phase space density . A low density kinetic equation for the first functional derivative of the associated generating functional is derived in Appendix C by following closely the nonequilibrium cluster expansion used to obtain the Boltzmann equation from the BBGKY hierarchy . Again the equation obtained differs from the nonlinear Boltzmann equation by the presence of extra short-lived terms that only contribute to the equations for the equal time correlations . The equations for the correlation functions are obtained by functional differentiation of this generalized Boltzmann equation . At both levels of description, kinetic and hydrodynamic , the same structure is obtained: nonlinear equations for the averages and linear equations for the correlation functions . The coefficients in the linear equations depend on the solution of the nonlinear problem: the dynamics of the fluctuations is then entirely governed by the nonequilibrium state. The amplitude of the noise, or source of fluctuations, enters in the form of an inhoraogeneous term in the equations for the equal time correlation functions and is derived without introducing any assumptions other than the ones entering in the derivation of the macroscopic equations . The formulation presented here starts from the Liouville equation for the system, but all the equations considered for the macroscopic description involve only averaged quantities , either observables or correlation functions . The information on the nonequilibrium state enters through boundary and initial conditions for such macroscopic variables: these are the parameters controlled in an experiment.

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16 The linear character of the equations for the correlation functions appears as a general result in the present discussion, not as the outcome of a linearization around the nonequilibrium state, applicable only when the fluctuations are small in size . In this sense the results obtained here agree with the conclusions of the M.I.T. group: the dynamics of the fluctuations is, as in equilibrium, determined by the state of the system in a form that naturally generalizes Onsager's regression hypothesis . As an application of the general description of nonequilibrium fluctuations obtained by the generating functional method, the velocityvelocity autocorrelation function of a tagged particle in a fluid in shear flow is evaluated in Chapter V, for arbitrarily large shear rate . Substantial simplifications occur in the problem as the result of three conditions: (1) low density, (2) uniform shear rate and (3) Maxwell molecules . In fact it is well known that the irreversible part of the stress tensor may be evaluated exactly from the nonlinear Boltzmann equation for a gas of Maxwell molecules in uniform shear flow. Consequently, the hydrodynamic problem can be solved exactly to all orders in the shear rate .^''"'^^ Here the velocity-velocity correlation function is evaluated by solving the appropriate set of coupled kinetic equations . The velocity correlations do not decay to zero at long times , since the asymptotic velocity of the particle is equal to the velocity of the fluid at the position of the particle . Nonvanishing position velocity correlations are then present in the fluid at large t. Also, due to the presence of the shear in the fluid, equal time correlation functions which are zero in equilibrium are now nonvanishing .

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CHAPTER II NONEQUILIBRIUM STATISTICAL MECHANICS OF CLASSICAL SYSTEMS 11,1 The Nonequllibrlum Distribution Function; Average Values and Fluctuations To prepare or maintain a system in a nonequllibrlum state external forces have to be introduced in general; nonequllibrlum statistical mechanics involves the description of a system in interaction with its surroundings in addition to specified Initial conditions. ' The external forces are assumed to have the character of reservoirs, in the sense that they can be specified Independently of the state of the system. In the case of a fluid, for example, the most general situation is represented by Interaction with particle, energy and momentum reservoirs. In classical mechanics a system of N interacting particles enclosed in a volume V is described in terras of the canonical coordinates, q , . . .q , and the canonical momenta, p , . . .p , of the particles. The state of the system is represented by a point in the 6N-dimensional space of the canonical coordinates . In the language of statistical mechanics the state of the system at the time t is described by the N-particle distribution function, Pj^(x^, ...,Xj^,t) = pj^(x^(-t) , ...,x^(-t)) , (2.1) where x. = (q.,p.) . Here x.(t) are the canonical coordinates of the 1-th particle as evolved from their initial value x.(0) ^ x., according to the dynamics of the system interacting with its surroundings . The 17

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18 distribution function p represents the probability that the system lies in a neighborhood dr = dx . . .dx of the phase space point r = (x ,...,x ) at time t. The time evolution of the distribution function is governed by the Liouville equation for a system in interaction with its surroundings, ' d ^^ fe + ^}pN + Pn-T = ° ' (2.2) ^Pn where here and in the following summation over repeated indices is understood. The operator C in Eq.(2.2) is £ = L(x^,...,Xj^) + ?^ . A_ , (2.3) ^Pn where L is the Liouville operator for the isolated system, defined as LpN = {p^.H} • (2.4) Here {•••} indicates the classical Poisson bracket and H is the Hamiltonian of the system. ?^ is an external force representing the effect of the reservoirs on the n-th particle of the system: it depends in principle on all the degrees of freedom of the system and reservoirs and on the details of their interaction. The force f is in general n ^ nonconservative and time dependent, and can be assumed to be applied everywhere at the boundary of the system. This assumption does not constitute a serious restriction since it corresponds to most experimental situations of interest.

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19 Equation (2.2) can be integrated formally when supplemented by an initial condition at t=0, p^(x^, ...,Xj^,0) = Pj^ q(x^(0) , ...,Xj^(0)) . (2.5) In general p will be assumed to represent an arbitrary nonequilibrium state. Also, the distribution function p,, is normalized at all times. I I dx ...dx p (x ,...,x ,t) = 1 . (2.6) N=0 IN .< i As already mentioned in Chapter I, a macroscopic description of a many-body system only involves a set of relatively few (compared to the number of microscopic degrees of freedom) conveniently chosen variables . The appropriate set of observables depends on the system considered and on the macroscopic space and time scales of interest: the one-particle distribution function is the object to consider in the kinetic theory of gases and the five average conserved densities are the appropriate se; of variables to describe hydrodynamics in a simple fluid . In any case the choice is guided by the requirement that the macroscopic description is closed on some time scale when given in terras of the variables considered . A discussion of this point can be found for example in Ref . 1 . Within the formalism of statistical mechanics the macroscopic measurable quantities are defined as ensemble averages of corresponding microscopic dynamical variables, {a (x , . . . ,x ) } , ^ , functions of the phase space coordinates , he

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20 ^a'^^^ = <\^''r""\^''^> ' (2.7) where <...;t> indicates an average over the nonequilibriura distribution function at time t, = / dr A^(x^,...,x^)p^(x^,....Xj^,t) (2.8) oo and j dT = I / dx . . .dx . Here and in the following a caret N=0 indicates the microscopic variable as opposed to the averaged one . An alternative expression of Eq.(2.8) is obtained by observing that the time evolution of the dynamical variables (a } is governed by the equation of motion, ^feJ V^'^) = 0' (2.'^) where the operator has been defined in Eq.(2.3). Equation (2.9) can be formally integrated with the result, A^Cr.t) = T(t,o)I^(r) . (2.10) The time evolution operator T(t,t') is the solution of the equation. {^} T(t.t') = , (2.11) '.d.th initial condition T(t',t') = 1. It also has the property, T(t,t')T(f,t") = T(t,t") . (2.12)

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21 Equations (2.2) and (2.9) differ because in a nonconservative system a region of phase space does not maintain a fixed volume as it evolves in time . This can be seen explicitly by evaluating the Jacobian of the transformation of an element of volume in phase space as it evolves in time , T = &li>^iCO)iJ • (2.13) For a conservative system, J=l . Here J satisfies the equation, and therefore its change in time accounts for the difference in the time evolution of the distribution function and the dynamical variables . The average values of the dynamical variables A can then be written A A = / dr A^(x, ,...,Xj^,t)pj^(x^,...,Xj^,0) . (2.15) The possibility of shifting the time from the distribution function onto A will be useful in the following to display some properties of the time-correlation functions . At this point the dynamics of the macroscopic set {a } is determined through the time evolution of the distribution function. In this sense Eq.(2.8) has only the meaning of a formal definition. On a microscopic scale A fluctuates instantaneously around its a average value . The microscopic fluctuations are defined as

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22 5A (r;t) = A (D . (2.16) a a a A complete description of the system should incorporate a systematic analysis of the fluctuations and of their spaceand time-correlation functions of any order . The second order equal time correlation function is defined as M^p(t) . <6A^5Ap;t> A A = / dr 6A^(r)6Ap(r)pjj(r,t) , (2.17) and, in general, the k-th order correlation function is M (t) = <6A 6A ...6A ;t> . (2.18) «l---\ "l "2 "k Again, as for the average quantities, the time evolution of the equal time correlation functions is determined through the ensemble . The amplitude of the equal time correlation functions is a measure of the width of the probability distribution of the fluctuations of a variable around its average value. The fluctuations can be viewed as the result of a stochastic force which represents explicitly the effect of instantaneous molecular collisions not accounted for in the macroscopic description and averages to zero over a time long compared to the microscopic collision time. In the language of stochastic processes the fluctuations are then the result of the "noise" in the system and the equal time correlation functions measure the size of this noise . It should be pointed out however that, even if here and in the following

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23 some of the terminology of stochastic processes is used, the description presented here is entirely macroscopic and always deals with averaged quantities . In equilibrium, away from critical points, the probability of fluctuations is Gaussian and only second order correlations are important . Their amplitude is proportional to k^T, where T is the temperature and kg is the Boltzmann constant, as suggested from the fact that in the equilibrium state spontaneous fluctuations can only be thermal in origin. In systems driven out of equilibrium new sources of noise may become available, in a way strongly dependent on the particular state considered . Furthermore in a nonequilibrium state equal time fluctuations are dynamical variables: the clear separation of static and dynamic properties that appears natural in equilibrium does not apply to this case . The correlation of fluctuations at different times are measured in terms of multitime correlation functions . The most general definition of a two-time correlation function is M^p(t^,t2;t) = <6A^(tp5Ap(t2);t> . (2.19) By using Eqs.(2.8) and (2.15) to translate the time arguments, this can also be written as <6A^(t,)6Ag(t2);t> = <6A_^(t^-t)6Ap(t2-t);0> . (2.20) Therefore the two-time correlation function really depends on two time arguments and on the initial time . In the following the dependence on

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24 the initial tine will not be indicated explicitly, unless needed. An average over the initial nonequilibrium ensemble will simply be written a I / dr A(r)p„(r,o) . (2. 21) N=0 The k-time correlation function is then M (t, ,...t, ) = <6A (t,)6A (t„)...6A (t, )> . (2.22) a, ...a, 1 k' a, 1 a„ 2 a, k.' Ik 12k The macroscopic variables and the correlation functions of their fluctuations around the average value have been expressed here as nonequilibrium averages of the corresponding microscopic quantities . In the next chapters equations describing the dynamics of the observables here defined will be derived in two specific macroscopic conditions . II .2 Reduced Distribution Function Formalism In the special case when the dynamical variables {a } are sum of single particle functions, two-particle functions, etc., N ^ N N . A (x ...,x ) = I a (x ) + I I b (x ,x ) + ... , (2.23) i=l i=l 3 = 1"-^-" their averages and correlation functions are conveniently represented in terms of reduced distribution functions . Many of the physical variables of interest in real systems, such as the microscopic mass, energy and momentum density in fluids, have the form (2.23). A set of nicroscopic phase space densities is defined as

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25 N f^(x^,t) = I 5(x^-x^(t)) i=l N N f (x ,x ,t) = I I 6|x -X (t))6[x -X (t)) (2.24) i=l j = l ^ ^ ^ " i^2 etc . The reduced distribution functions usually defined in kinetic theory ,^^ '^'' Sr / .X V N! n f (x s ^....,x^,t) = I j^^ j dx^_^^...dx^^ pj^j(x^.....Xj^.t) (2.25) N>s can be interpreted as averages of the phase space densities over the nonequilibriura ensemble, n f^(x^, ...,Xg ,t) = . (2.26) In particular the one-particle distribution function is defined as n f^(x^,t) = " ^ (N-1)! •' "ix^ ...dXj^ pj^(x^,x2,...,x.^.,t) . (2.2:) The reduced distribution function f represents the probability density that, at the time t, the positions and momenta of particle 1 through s lie in a neighborhood dx,...dx of the point {x ,...,x }, regardless of the positions and momenta of the other particles in the sytem. The average of a variable A of the form given in Eq.(2.23) is then written as

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26
= n / dx.a (x, )f,(x, ,t) a ^ 1 a 1 1 1 2 " + n J dx dx b (x ,x )f (x ,x^ ,t) + ... . (2.28) For simplicity only the expressions for the correlation functions of variables that are sums of single particle functions will be derived here . The extension to more general cases is straightforward . The correlation functions of the (a } can then be expressed in terms of the correlation functions of the fluctuations of the oneparticle phase space density around its average value, C , with the result, M^p(t) = / dx^dx2a^(x^)ap(x2)C2(x^,t;x2,t) , (2.29) and M ^pCt^.t^) = / dx^dx^ a^(x^)ap(x2) C2(x^ ,t^ ;x2 .t^) . (2.30) In general the k-th order correlation function is given by \, ...a^'^l'-'V = /..-/.n {dx.a^.(x.)}Cj^(x^,t^:...;x^.t^) . (2.31) 1 K. 1=1 1 Equivalent expressions for the correlation functions of more general variables involving also two-particle functions, etc., will involve correlations of higher order phase space densities . The equal time correlation functions of phase space fluctuations introduced in Eqs .(2 .29-30) are defined as

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27 Cg(x^,t; ...;x^,t) = <6f^(x^,t) ...5f^(Xg,t)> , (2.32) where 6f^(x,t) = f^(x,t) . (2.33) It is immediately seen that these correlation functions can be expressed in terms of the reduced distribution functions f . Similarly, the multitime correlation functions introduced in Eq.(2.31) are given by Cg(x^,t^;...;Xg,t^) = <6f^(x^,t^)...5f^(x^,tg)> . (2.34) The one-particle distribution function and the fluctuations in the phase space density are the objects of interest in kinetic theory and will be analyzed in detail in Chapter IV. The expressions given in Eqs .(2 .29-31) for the averages and the correlation functions of a general set of observables in terms of the reduced distribution functions are obtained by carrying out a partial ensemble average over a reduced number of particles . They provide a connection between macroscopic observables and kinetic theory and a formal reduction of the many-body problem to an effective one-, two-, ... s-body problem. The reduced distribution function method is particularly useful when dealing with gases at low or moderate density. In this limit closed kinetic equations for Che lower order distribution functions can be derived .

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28 II .3 Generating Functional for Nonequilibriu m Averages and Fluctuations To describe the dynamics of a nonequilibrium system it is convenient to introduce a generating functional G, from which averages and correlation functions of any order can be generated through appropriate functional differentiation. The same idea has already been used by Dufty^S to derive kinetic equations for equilibrium multitime correlation functions . The advantage of this formalism is mainly that once a single set of equations for the first functional derivatives of G has been derived, the equations for the correlation functions are simply obtained by functional differentiation . If {a^} is the set of dynamical variables chosen to describe the system, the corresponding generating functional is defined as G[{?^J] = ln . (2.35) where ^[{\]] = exp j'Zdt I^(r,t)X^(t) , (2.36) and summation over the index a is understood. The test functions {x in Eq.(2.36) have no physical interpretation. The only requirement Imposed at this point is that they are sufficiently localized in time for the integrals in Eq.(2.36) to exist. The first functional derivative of the generating functional is given by a'

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29 A (t \) a (2.37) When evaluated at \ =0, Eq.(2.37) reduces to the nonequilibrium average of A , a A (tl\=0) = A (t) . (2.38) Similarly, by successive functional differentiation. % t > ••• t . This implies no loss in generality since the dynamical variables commute in classical mechanics. As desired, averages and correlation functions of any order can be generated from G. The objective is now to derive a set of equations for the time evolution of the functionals (A (t|\)}. When evaluated at \=0, these should reduce to the macroscopic regression laws for the system considered. Furthermore equations for the correlation functions of any order can be obtained by functional differentiation. In this sense the generating functional method provides a unified description of the system, since all information can be derived from one single set of equations .

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30 The set of macroscopic regression laws describing, at long times, the nonequilibrium state of a many-particle system (such as the Boltzmann equation and the nonlinear Navier-Stokes equations which apply at the kinetic and hydrodynamic level respectively) can be derived by averaging the microscopic equations of motion, at least in the case of low density gases . The nonlinear Boltzmann equation for the oneparticle distribution function is obtained by performing a low density closure of the BBGKY hierarchy. The Chapman-Enskog expansion provides then a systematic method for deriving the hydrodynamic equations. Alternatively, the nonlinear hydrodynamic equations can be obtained directly by averaging the microscopic conservation laws over a nonequilibrium ensemble which is a formal solution of the Liouville equation and by evaluating the constitutive equations expressing the irreversible part of the fluxes as functionals of the thermodynamic gradients . The same well established methods can be applied to the generating functional to derive a set of equations for the ^-dependent functionals {A^(t|\)}. Furthermore the derivation preserves the properties of the generating functional, in the sense that the set of equations so obtained may be differentiated to derive equations for the correlation functions. This program is carried out explicitly in Chapter III and Chapter IV for the hydrodynamic and kinetic limit respectively. The theory of nonequilibrium fluctuations so derived is justified in the same well understood limits leading to the macroscopic description. The convenience of introducing a generating functional or characteristic function to describe fluctuations has often been recognized in the literature .59 in particular Martin, Rose and Siggia^*^ defined a

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31 time ordered generating functional, involving both the dynamical variables of the system and a set of conjugated operators describing the effect of small perturbation in the variables. Also, in Eq.(52) of Ref . 31, Onuki defines a characteristic function analogous to the one proposed here, from which equal time correlation functions of phase space fluctuations can be generated . The definition given in Eqs . (2.35-36) is however more general and it can be applied to a larger class of statistical mechanical systems . The next two chapters will be dedicated to the explicit derivation of equations for the averages and the correlation functions describing the properties of a fluid at the hydrodynamic and kinetic level . However, before carrying out this program in detail, it is instructive to present the results . The macroscopic description of the dynamic properties of a many-particle system have the same structure at both levels of description (kinetic and hydrodynamic) . The relaxation of the macroscopic variables JA } is described by nonlinear equations of first order in time (to be identified with the nonlinear Navier-Stokes equations for the average densities and the nonlinear Boltzmann equation for the one-particle distribution function) , I^A^(t) +Nj{A(t)}] = , (2.41) where N represents a general nonlinear functional of the (A } . The set of Eqs. (2.41) has to be solved with the appropriate initial and boundary conditions, constituting an experimeniially appropriate definition of the macroscopic nonequilibrium state considered .

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32 The dynamics of the correlations of fluctuations around the nonequilibrium state is described by a set of linear homogeneous equations. Ir7%(^'h> + ^aynA(S)}]V'l''2) = • <2-^2^ where 6N [{A(t )}] -V^^ = 6Vt ) • (2.43) p i Equation (2.42) applies as long as the separation t,-t2 is largecompared to an appropriate microscopic time, t_, of the order of the lifetime of the transient before the adopted macroscopic description of the system applies. Specifically, t = t , the duration of an m c interparticle collision, for a kinetic description, and t t^ , the m f mean time between collisions, for a hydrodynamic description. The linear functional L in Eq . (2.42) depends on the solution of the nonlinear equations, Eqs.(2.41). In this sense the dynamic evolution of the two-time fluctuations is entirely determined by the macroscopic nonequilibrium state . .. The correlations of fluctuations at the same time, needed as initial conditions for the set of Eqs.(2.42), are the solutions of a set of bilinear inhomogeneous equations , ar\p(^) + L,,[(A(t)}]M^p(t) + L^p[{A(t)}]M^^(t) = r^p[{A(t)}] , (2.44) The specific form of the Inhomogeneity T _ depends on the problem considered . To interpret these equations, it is instructive to compare

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33 them with the corresponding equilibrium results . In this case OnsagerMachlup's hypothesis applies and the decay of the mean values and of the fluctuations is governed by the same set of linear equations, obtained from Eqs.(2.42) by evaluating L at equilibrium, . A°(t,) A°(t,) where the superscript "zero" indicates the equilibrium value or that the average has to be understood as an equilibrium average . Also the fact that in equilibrium, in virtue of time translation Invariance, the twotime correlation function depends only on the time difference has been indicated . The equal time fluctuations are given by equilibrium statistical mechanics as a property of the ensemble. Equations (2.44) become then identities: the inhomogeneous term is identified with the amplitude of the noise in the system (this appears evident by using a Langevin description). The inhomogeneity in Eqs.(2.44) can then be interpreted as the noise ter.n. Indeed it originates from microscopic excitations in the system whose lifetime is short compared to the time scale of interest.

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CHAPTER III HYDRODYNAMICS III.l Generating Functional for Hydrodynamics and Nonlinear Navier-Stokes Equations In this chapter the generating functional formalism outlined in Section II .3 is applied to derive a set of equations describing macroscopic properties and fluctuations at the hydrodynamic level . For simplicity, attention is limited to the case of a simple fluid whose nonequilibrium state is adequately described by the nonlinear NavierStokes equations . The method used here is easily extended to more general systems and other nonequilibrium situations , as will be seen in the next chapter . In general a hydrodynamic process is one which is adequately described in terms of the averages of the locally conserved quantities in a many-particle system. This contracted description applies to time scales large compared to a characteristic microscopic time scale, the mean free time between collisions . The hydrodynamic equations are derived by averaging the microscopic conservation laws and by closing the macroscopic equations so obtained with constitutive relations for the irreversible part of the fluxes . This procedure can be carried out in detail by applying, for example, the methods of the response theory and provides a precise statement of the validity of the hydrodynamic description for processes varying on space and time scales large compared to some microscopic space and time scales characteristic of the system .^»^'^ 34

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35 The microscopic conserved densities for a simple fluid are the microscopic mass density p, the total energy density e, and the three components of the momentum density g. , respectively given by N p(r) = I m 5(r-q ) , .n=l «.-a'iu..c..j..i< N n= 1 t ; 'I'v^.V^OVili;..,. n=l (3.1). Here N is the total number of particles in a volume V and m is the mass of the particles; r represents a point in the fluid. The function e is the energy of the n-th particle: for a central pairwise additive interaction potential, '^(^nn'^' "^^h q^^ , = jq^ q^ , | , £^ is given by, 2 nn' pN n' = l (3.2) nil n'?^n In the following, for convenience, the set of five conserved densities will be indicated as {(|;^(?)| = (p(?),£(?),g.(?)) , (3.3) or i (?) = I a (X )6(?-q" ) , n=l a n (3.i) with

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36 Also, here and in the following, the indices n,n',... are used to label particles and the indices i,j,k.,l,... are used to label the components of vectors and tensors. Greek indices, a,p,... , run from 1 to 5 and label sets of hydrodynamic variables . The microscopic conservation of mass, energy and momentum is expressed by the set of five conservation laws. X The set of Eqs.(3.6) constitutes the definition of the microscopic fluxes Y • » explicitly given by {^C?)} = (g.(?),s.(?),t.j(?)) , (3.7) where s. is the energy flux and t.. is the momentum flux. Explicit expressions for the microscopic fluxes can be found for example in Ref . 7, Eqs.(4.6). It should be noted that in writing Eqs.(3.6) no external sources have been taken into account , even if a general nonequilibrium system in interaction with its surroundings is considered here. The adequacy of Eqs.(3.6) as the starting point to describe properties in the interior of the system and the possibility of incorporating the effect of the boundarias entirely through the thermodynamic parameters is discussed in Appendix A. The macroscopic quantities of interest are the nonequilibrium averages of the (cj; },

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37 4'^(r,C) = <(|.^(?);t> , (3.3) and the correlation functions of their fluctuations around the average. \...a^(^l'^---'\'V = <^ia/^'S^-"^V'^'\)> ' (^-^^ where 64; (?,t) = 4; (r,t) <4; (r);t> . (3.10) For a one -componentfluid the average densities are explicitly given by {cl;^(?,t)} = (p(?,t),£(?,t),p(?,t)v^(?,t)) , (3.11) where p, e and v are the average mass and total energy density and the macroscopic flow velocity, respectively. The macroscopic conservation laws are obtained by averaging Eqs.(3.6) over the nonequilibrium ensemble, with the result, |^ +|_<;^.(?);t> = . (3.12) 1 The hydrodynamic equations are obtained from Eqs.(3.12) when these are supplemented by an equation of state, relating the thermodynamic variables, and by constitutive equations for the macroscopic fluxes as f unctionals of the average densities . The intensive thermodynamic variables, {y (r,t)}, such as temperature, pressure, etc., are defined in the nonequilibrium state by requiring that the nonequilibrium average densities are the same

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38 functionals of temperature, pressure and flow velocity as they are in equilibrium. This is obtained by choosing = , (3.13) where <...;t>T indicates the average over a local equilibrium ensemble at time t , ^ = ^ / dr A(r)p^(t) . (3.14) N=0 The local equilibrium ensemble for a one-component fluid is given by Pj^(t) = exp[-q^(t) / d? y^(?,t)(J.^(?)} , (3.15) where qL(t) is a constant determined by the requirement that p-(t) is normalized to one. The right hand side of Eq.(3.13) is an explicit functional of the {y }: this equation constitutes the definition of the local thermodynamic variables conjugated to the [(\) }. Explicitly, (y^tC^.t)} = (-V +1 pv^, p, -pv^) , (3.16) where vp is the chemical potential per unit mass and 6 = 1/k T, where T is the Kelvin temperature and kg is Boltzmann's constant. Equations (3.13) do not imply a limitation to states near equilibrium or local equilibrium. Other definitions are possible and sometimes desirable. However, the present definition has the advantage that the functional relationships of all nonequilibrium thermodynamic parameters are the same as in equilibrium .

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39 The main problem in closing the hydrodynamic equations is the specification of the irreversible part of the fluxes. \i^^'^> " <'^ai(^)'^> L' (3.17) as functionals of the thermodynamic gradients . These expressions can be evaluated to first order in the gradients by using the linear response theory, with the result, * > ii ^ , ayg^^'*^) where L 7^ is the Onsager's matrix of transport coefficients. For times large compared to the mean free time between collisions , after the initial transient describing the complicated behavior of the system before hydrodynamics applies, the time evolution of the macroscopic state is then described by the set of nonlinear Navier-Stok.es equations, given by (3.19) where E represents the contribution from the nonlinear Euler a equation. Explicitly, the nonlinear Navier-Stokes equations are 1^ p + V.(pv) = (3.20a) 9v. (^ + ^^)u + hV-v = -^.rt. . ^ (3 .20b) *-5t ^ ^ ij ar.

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40 * At" r|_+ ;^.^)v. = -1 r|P_ + -JJ. ) , (3.20c) ^5t ^ X p ^5r. dr. ^ 1 2 where u is the average internal energy density, u = e -rpv , h is the enthalpy density, h = u+p, and p the pressure. To Navier-Stokes order the irreversible parts of the heat flux, q., and the stress tensor, * t . . , are given by the usual Fourier and Newton laws as q*(?,t) = -K(p,u) |I(3.21) 1 A dv t..(?,t) = {ti(p,u)A..j^j^ + C(p,u)6.^5j^^} -^ , (3.22) 2 where A..,, = 6..6, , + 6.,6., --^^ 6..6, ,. The coefficients of thermal ijkl ij kl ik jl 3 ij kl conductivity, < , and of shear and bulk viscosity, ri and C> respectively, are in general functions of the thermodynamic variables, as indicated. The generating functional G for hydrodynamic averages and fluctuations is given by Eq.(2.35), with U[{^„}] = exp /^ dt J dr J (?,t)\ (?,t) . (3.23) The first functional derivatives of G are (i)^(r,t|\) = " 6\^(?,t) <4>^(?,t)U[\]> (3.24) where X often will be used to indicate the set of five test functions, (\ }. When evaluated at \=0, Eq.(3.24) reproduces the average densities

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41 given by Eq.(3.8). Similarly, higher order functional derivatives generate the correlation functions (3.9), as shown in Eq.(2.40). The explicit form of Eq.(3.24), „(?.t|».J^/dr;„c?)e-"4i|^p,(0) <* (?);t> , (3.25) a A. where L is the Liouville operator of the isolated system (Eq . (2.4)) and the time evolution operator is defined in Appendix A (Eq. (A.51)), suggests that the same statistical mechanical methods used to derive the hydrodynamic equations from the macroscopic conservation laws can be applied to describe the time evolution of the set of functionals {(\) (r,t|X)}. In fact, the time evolution operator is the same in both cases and the difference in the equations can be incorporated as a modification of the initial condition for the nonequilibrium ensemble . The initial condition becomes dependent on the test functions X through the functional U. This program is carried out explicitly in Appendix A, where the methods of nonlinear response theory are applied to derive a set of equations for the functionals {^ = «i;^(?);t>^^, (3.26)

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42 where <»«»;t> indicates the average over a \-dependent local equilibrium distribution functional p (t|\), given by p (t|\) = exp{-q (t|\) / dr y (?,t|X)(l; (?)} . (3.27) The constant qj^(t|\) assures the normalization to one of the distribution functional (Eq . (A. 7)). The functional dependence of p (t|\) on Li the set (y^(r,t|x)} is the same as indicated in Eq . (3.13) for the case X=0. The form of the thermodynamic equations relating {(\, (r,t|X)} and {y (r,t|X)} is therefore preserved and is again the same as in the equilibrium case. The comments made when imposing the equality (3.13) apply here. The thermodynamic variables {y (r,t|X)} depend on ^^^ (^(j} °"ly through the X-dependent densities, {4; (r,t|\)}. The set of generalized Navier-Stokes equations for the five functionals {cl-^lX]} derived in Appendix A are formally identical to the usual nonlinear Navier-Stokes equations . In a matrix form they are given by |-c^^(?,t|X) +|_{Ej(?,t;{4.[X]}) i ii.-> V 9yR(r,t|\) J 1 (3.28) where the Euler matrix, E , and the matrix of the transport coefficients, L^^ , are respectively defined as ap E^(?,t;{c^[X]}) H ^^ , (3.29)

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43 and L^J(?,t;{4.[X]}) = lim jl dt <{e^^'"^^Y„i(r)]5p.[X]>o^, (3.30) where i^. [X] is the total (voW integrated) projected flux defined in Eqs . (A. 23) and (A. 57), and the average in Eq . (3-30) is over the X-dependent equilibrium ensemble with local thermodynamic variables, defined in Eq . (A .54). Both E and L „ are nonlinear functionals of the a ap densities [(\) (r,t|\)} and depend on the test functions {\ } only through the {(i) ] themselves. Additional contributions to the irreversible fluxes, not present in the Navier-Stokes equations, appear in Eqs. (3.28). They originate from the X-dependence of the initial condition for the nonequilibrium ** distribution functional. The subscript NS to y • indicates that the 'ai right hand side of Eqs. (3.28) has to be evaluated to Navier-Stokes order in the thermodynamic gradients and in the limit of time t long compared to the mean free time t^, i.e. Y!?(^.t^)M<; = li[^ Y:-^, . (3.32) Here the nonequilibrium distribution functional has been written

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with and 44 Pj^(t|\) = p^(t|\)e°^^l^^ , (3.33) D(t|\) = e ^"^ D^(X) + D'(t|\) , (3.34) D'(t|M = jl dT J d? e-^(^-^){.^^2 ^.%^(?\X) . ._ . i * 4. ._----.5y .(Cj'tlX) + ^ \(r\x)} , (3.35) i D (X) E D(0 a) o where . and £ are the projected fluxes and the normalized densities given by Eqs . (A .23) and (A. 21), respectively, when the local equilibrium average is substituted with an average over p (t|?\.). When JLi evaluated at \=0, the additional contributions to the irreversible fluxes vanish identically, since D (\=0) = 0, and Eqs. (3.28) reduce to the nonlinear Navier-Stokes equations (3.19). However, when the generalized equations for the functionals {4; (r,t|A)} are considered, these extra contributions have to be kept and will be shown to be directly related to the strength of the noise in the system. The Navier-Stokes limit indicated in Eq . (3.31) is consistent with the approximations introduced in the left hand side of Eqs. (3.28), but can only be taken after functionally differentiating y . and evaluating the result at ,\=0 . The evaluation of this term will be needed when

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45 deriving the equations for the correlation functions, and is carried out explicitly in Appendix B. The only assumption introduced in the derivation of Eqs . (3.28) regards the form of the initial state, which has been chosen to be local equilibrium. This choice does not imply any serious restriction, because the deviations from initial local equilibrium are expected to decay in a time of the order of the mean free time, t^ , in states •3 1 n leading to hydrodynamics. ' These deviations are only responsible for the initial short lived transient in the system. Using the same notation for the functionals and the corresponding average densities , the \-dependent Navier-Stokes equations are explicitly given by 1^ p[>^] + V.(p[\]v[\]) = , (3.37a) (|_+ v[\].v)u[\] + h[X]^.v[\] = -V.q*[\] -t*^[X] ^-1^-1, (3.37b) c . J * ^ i J where the dependence on r and t of the functionals has been omitted to simplify the notation. The irreversible parts of the heat flux and the stress tensor are now given by q*[M = -<[M-|^+qrtMi,3 , (3.38) * , , av [X] ^^ t..[X] = -{^[MA,.^, + aM6,.\,}^— + t..[X]^3 . (3.39)

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46 The transport coefficients in Eqs .( 3 .38-39) are functionals of X through the ^-dependent densities, {cj; (r,t|\)}. The additional contributions to the irreversible fluxes are defined as ,;*(?,t|.) = 4(?)[e°(^l^' a-'-Ci^'liO^^ , (3.40) t;;(f.tU) = ^^ , (3.41) where q. (r) is the microscopic heat flux, q.(r) = s.(r) v.t..(r). Finally it should be stressed that although the extra contribution to the irreversiblefluxes may seem similar to the ones suggested by Landau and Lifschitz to construct hydrodynamic Langevin equations (and in effect their physical origin is analogous), there is a basic difference in the present formulation . All the equations here are equations for averaged variables, not stochastic equations. In this way the description obtained is directly related to the experimental situation. Ill .2 Correlation of Fluctuations The fluctuations of interest are those of the set of microscopic densities, (fjj (r,t)}. In particular the second order correlation functions are defined as Using Eqs. (3.23-25), such correlation functions can also be expressed as the first functional derivatives of the ^-dependent densities.

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47 (3.43) A set of equations for the two-time correlation functions can then be obtained by functionally differentiating Che generalized Navier-Stokes equations, derived in the preceeding section, Eqs . (3.28). In a general nonequilibrium state no symmetry properties relating the different correlation functions can be identified a priori . The equations for the set of twenty-five functions M (for a=l,...,5 and p=l,...,5) are in general all coupled together . By functionally differentiating Eqs . (3 .28) and remembering that the ^-dependence occurs in these equations only through the {'4^ (r,t|\)} (or, equivalently, the thermodynamic variables (y (r;t|\)}) and through the additional terms in the irreversible fluxes, y ., the following set of equations for the correlation functions is obtained ar;\8^^'^l'"^'^2> ^ ^aa^^^'^l'^V^%(^^'h'^^'^2) ^ap(^l'h'^2''2^ ' (3.44) where V^'^'^V) =lr-i' 5E\?,t;{cl;}) bLl{{r,t;{}) Qy (?,t) aa i 9'^p(r,t) + L^J(?,t;{(i.}) a4^g(r,t) a ^0^''^^ aa 5r i a4'p(r,t) } . dr. (3.45) and V^''l'^''2'^2^ lim ** -> 6Y^(r^.t.|X) ar, t^»t^ li 5X (r^.t^) J \=0 NS (3.46)

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48 The fluxes y ^^ are defined in Eq . (3.32) and [H]^^ indicates, as in Section III .1 , the Navier-Stokes limit of the function A. The differential operators in C operate on everything to their right, including the operand of C itself. In deriving the set of Eqs . (3.44) the assumption that the local equilibrium correlation functions appearing in the operator C are localized in space has been introduced . Therefore these equations apply only if the system is away from critical points . In the general case the operator 1^ „ is nonlocal and the derivatives in Eq . (3.45) have to be substituted with functional derivatives. The inhomogeneous term on the right hand side of Eqs . (3 .44) is evaluated in Appendix B. For t^-t2 » t^, it is given by ,^ -> .^2 r L(t -t +T) . tj»t^ 'lF7:^Pa^^2'^'^l'°) ' (3.47) where <"'>q indicates an average over a reference equilibrium ensemble with local thermodynamic variables evaluated at the point (r ,t ) (the definition is given by Eq . (A. 54), evaluated at \=0) , and i is the oj volume integrated flux, defined by Eq . (A. 57), at X=0 . Also, ^pa^^2'"'^l'^^ ^^ ^^^ Green's function for the Navier-Stokes equations, defined in £q . (B.17). The time correlation function in Eq . (3.47) is the correlation function appearing in the matrix of the transport coefficients, given by Eq. (3.30) at \=0 . Its lifetime is of the order of the mean free time tf, and therefore much shorter than the macroscopic time scales of interest in hydrodynamics. Consequently the term I „ does not contribute to the ap

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49 hydrodynamic equations for the two-time correlation functions if S»^f • t^-t^»t^ . . (3.48) In this limit Eq . (3.44) becomes " This set of coupled equations describes correlations over time scales large compared to a mean free time, as desired in a hydrodynamic decription . The equations for the correlation of fluctuations at different times are linear, although coupled to the solution of the nonlinear Navier-Stokes equations through the dependence of il on the {<\> } . The linear character of the equations is a general result and does not restrict their applicability to small fluctuations around the nonequilibrium state. For the case of small fluctuations around a nonequilibrium steady state, Eqs. (3.49) agree with the results of others .2*^' ^^"''^ ''^^ The equations derived here represent therefore a generalization of these results: they provide a hydrodynamic description of fluctuations, valid to Navier-Stokes order in the gradients, for nonstationary states and fluctuations of arbitrary amplitude. A detailed comparison with previous work will be presented in Section III .4 . For the special case of equilibrium fluctuations, t'.ie operator C reduces to that of the linearized Navier-Stokes equations . The

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50 equations for the fluctuations at different times are then exactly those suggested by Onsager's assumption: the fluctuations decay according to the macroscopic linearized regression laws governing the dynamics of a nonequilibrium system near equilibrium. The explicit form of Eqs . (3.49) for a one-component fluid is conveniently written in terms of the correlation functions %p(^'S'^2''2) = <6p(?^,tp6(l.p(?2.t2)> . , ->• -^ \s^^l''^r'^2''^2^ " <6u(r^,tp64; (r2,t2> , ^g.p^^l'^l'^2'*'2^ = <6g^(r^,t^)5<\>^(v^,t^)> , (3,50) P=l,2,...,5. It is convenient to write the hydrodynamic equations in terms of the correlation functions M „ instead of M „ , where u is up eP' defined by ^ ^ N u(r) = I e^[p^-mv(q^))6(?-q ) . (3.51) n=l The two correlation functions are related by %(^1'^'^2'^2) = 'W^r'rh^^l^ ^i(^l'^>\p(^l'^'^2''^2^ + l[v(?^,tp]2M^^(?^,t^;?2,t2) . (3.52) A coupled set of equations for the correlation functions (3.5C) is obtained from Eqs. (3.49) by respectively setting a=p,u,g..

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51 I— M „ + I M ^ = , (3.53) (I— + V. I ) M „ + h I M , + M „ l^i— + (J^lfh M „ + h M J {< I (T M „ + T M „] + (k M „ + K M ^1 1^ } ^ 5r,^ ^ P p3 u uB-' ^ P pP u up-' Sr, . ' dr li ^^li P pP u up^ ^ P PP u up^ ar^^ 5v . 5v, ^^ ^r-^ {(ti A. ., +C5. .5, )M„ + (tiA. ., +C6. .6, )M.} dr 5r ^^ 'p ijkm p ij Km pp *u ijkm u ij ttm' up' ^ ' 11 km 11 km Sr, 5r, . v.p -' -" Im li i'^ I — M „+| fv.M „ + pv.M ^ dr^^ ^^p pP ^u up^ 5r ^^ ' ijkm ^ ij"km^ 5r^^ v^^p' dr ^or L V ip i-jk-m ^p ij^m-' pp + (n A... + C 6..6v )M o]} = 0. (3.55) *• u ijkm u ij km'^ up-"^ where P^^l'S^^.p^^l'^l'^'l'^Z^ "g.p(^'-l'^2'^2^ -^(^l''^l>V'^''l''^2'^2) • ^'•''' The space and time arguments of the correlation functions and the hydrodynamic variables have been omitted in Eqs . (3.53-55) to simplify the notation. The dependence of the correlation functions is the same indicated in Eqs. (3.50) and the hydrodynamic variables are evaluated at

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52 the point (r ,t ) . Also, the thermodynamic derivatives of a function A = A(p,u) have been indicated as ^ r aA(p,u) -| rdA(p,u) u '-Su ]p(3.57 By comparing Eqs . (3.49) with Eqs . (3.53-5), the matrix iAv ,t) is ap identified as -%^^^'^ up \ SiP uu -C . pgi "Si '^i^j / (3.58) where "-pg . 5r . ' C = -h|_-^ up dr^ p (|L.) _i. + h D ^ar^^ P P dr p ^5r -' p-i 2 ij ^ ' ij '^ 3'^ ij J dr. ^ or. p ^or. ' p + 2[2nD + (C -|ti)D6.. a _x_ dr. p ' J *= V. uu dT _u Jh D — r — i or.. ' u dr. '-'^ 5r . *u " ^-drV-'^uT + l-s — ]< 11 'rtr •' n -I -iD..[2,D.. +(C-|,)D6..],

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53 5 1 , 1 rQu C = h^l + 1 (^) 2[2tiD.. + (C -4ti)D6..] 1—1 , ug 5r p p ^dr^ ^ ij 3 " ij J 5r p i. = I— P + 2 I— Ti D. . + -I— ( C ^Ti ) D g^p 5r^ ^p dr^ 'p ij 5r^ ^p 3 'p V. V. fr) A. ., + C6..6, ) -S ^ . * iikm 11 Km-^ 5r p dr. i i 5r . ^ ' ijkm ij Km-' 5r p J J m C =|_p + 2I— t;D. . +1— (C -|ti)d, s . u or . u or . \i 1 1 dr . ^ u 3 u '^ ^11 J -^ 1 a __ . a __ . a r _. . >.. c >, a i (3.59) C = 6. . -^ V, + I— V. + -^ (nA. ., + C6., 6. 1 -^ , g.g. 11 dr. k 5r . x 9r, ^ iikm ik jm'' 5r p ^i^j -^k J k-* -'m'^ where , av . av . D.. = 4 {-^ + ^) , (3.60) 11 2 "dr . 5r . ^ and DSD.. = V«v . (3.61) 11 The differential operators in Eqs . (3.59) act on everything to their right, unless otherwise indicated by the presence of a parenthesisIII .3 Equal time fluctuations Second order equal time correlation functions , defined by setting t,=t2 in Eq . (3.42), i.e. M^p(?^,t;?2.t) = <6i^(?.,t)66p(?2.0>, (3-62)

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54 are needed as initial conditions for the solution of Eqs . (3.49). However, equations for the equal time correlation functions cannot be obtained directly by functional differentiation of the equations for {<(jj (r );t> }, as was done for the equations for the correlation of fluctuations at different times, because the functional differentiation does not commute with the time derivative. Instead, they can be determined from the limit, £^0 ^ 6\ (r ,t) P (3.63) where the operators P,2 ^^^ P o permute r ,r and a, |3 , respectively. By using the results of the preceeding section, (Eq. 3.44), and observing that the e^ limit can be taken immediately on the left hand side of the equation, an equation for M is obtained in the form (Xp lr%(^l'^'^2''^) + <^-^^12^ap> ^aa^^^'S'^^>)%(^l''^'^2'^) = ^aB^^'^Z'^^ ' (3-6^> where the operator C is defined in Eq . (3.45), and e->-0 '^ ^ The inhomogeneous term r . is evaluated in Appendix 3, with the result cxp

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55 (3.66) where L { is the Onsager's matrix of transport coef f iciencs . Equations (3.64), together with the set of Eqs . (3.19) and the equations for the two time correlation functions, give a closed description of transport and fluctuations (up to second order correlations) in a hydrodynamic system whose macroscopic state is described in terms of the nonlinear Navier-Stokes equations . Again, the interpretation of Eq . (3.64) is clarified by comparing with the case of fluctuations around the equilibrium state. Onsager's assumption applies then and, as seen in the preceeding section, the time evolution of the two-time correlations is governed by the linearized hydrodynamic equations. Similarly, Eq . (3.64), evaluated at equilibrium, becomes = ^^+^2^ap) I77757'^h-^2^ > <3-^^> li 2j where the superscript zero indicates the equilibrium value and C „ now depends on r only through the differential operators. Equation (3-67) is an identity or, equivalently , constitutes a statement of the fluctuation-dissipation theorem relating equilibrium fluctuations and transport coefficients . The description of nonequilibrium fluctuations derived here indicates clearly how Onsager's assumption should be modified for systems out of equilibrium. First, it is no longer true that the

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56 regression laws for the fluctuations are the same as those for the average values. They are instead linear equations, whose coefficients depend on the solution of the nonlinear regression laws (they can be interpreted as a linearization about the nonequilibrium macroscopic state, {i\> (r,t)}, at each instant t) . In this sense knowledge of the average dynamics still allows determination of the dynamics of fluctuations . The two sets of equations form a bilinear set that must be solved simultaneously. Secondly, the fluctuation-dissipation theorem must be abandoned in favor of Eqs . (3.64) and (3.66) which, for nonstationary states, indicate that even the equal ciae fluctuations are dynamical variables . In equilibrium, F „ is determined by the left hand side of Eq . (3.67), since the equal time fluctuations can be calculated directly in the Gibbs ensemble. In the nonequilibrium case, however, T „ must be provided independently. Its calculation requires the analysis of the initial transients due to microscopic degrees of freedom not incorporated in the Navier-Stokes equations . In this sense the source function T „ can be interpreted as a noise amplitude . ap The explicit form of Eqs. (3.64) for a one-component fluid can be obtained by using Eqs. (3.58-59) for the elements of the matrix C „. The inhomogeneous term F „(r,,r -t) is diagonal in the labels a and dp 1 z p for a and p = p,u,g. . Its explicit form is given by Eq . (3.66), with L;j(r,t) = 0, L^J(?,t) = lim Jl dT <[e'-^ S|(?)]S > , t»t.-L J u

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57 ^k^l t»t, ° lie ji u where s!(r) and t!.(r) are the projected energy and momentum fluxes, given by s:(r) = s (r) -:ii±-L^g (r) , (3.69) p(r.t) i^^N " /^\ h(r,t) '^ .^. i;(r) = s^. (r) g,(r) , and 4j(r) = t^.(r) 6.^[p(?,t) + (|£) n(T) + (-|E) p(r)] . (3.70) Equations (3.68) are the Green-Kubo expressions for the transport coefficients , or L^hht) = 6,.K^T^?,t)<(p,u) , (3.71) UU IJ o (?,t) = KgT(?,t)[Ti(p^u)A,^^j^ + C(p,u)6^^6j^^] . (3.72) The transport coefficients depend on r and t through the average mass and internal energy density. in .4 Discussion Several authors '' have recently formulated a description of fluctuations in nonequilibrium hydrodynamic steady states which are adequately described by the nonlinear Navier-Stokes equations . It seems therefore appropriate to compare these approaches with the one used here and to discuss differences and similarities . Trembiay, Aral and Siggia have proposed a Langevin theory of fluctuations in nonequilibrium steady states . They assumed that the Landau-Lif schitz theory of hydrodynamic fluctuations can be applied not

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58 only for equilibrium fluctuations (in a regime where the macroscopic regressions laws are linear) , but also for small fluctuations around the nonequilibrium state . The equations governing the time evolution of the fluctuations are obtained by linearizing the nonlinear macroscopic regression laws around the nonequilibrium state . The correlations of the random forces are assumed to have the same form as in equilibrium, with local thermodynamic variables . This description is supported by 30 the work of Keizer . It is however phenomenological in character and restricted to small fluctuations around the nonequilibrium state . The results of the present formulation basically confirm the Langevin hypothesis. There are however some important differences. First, the equations for the fluctuations are always linear, independent of the size of the fluctuations. Secondly, the form of the noise spectrum is derived here on the basis of a microscopic description, that clearly places the source of the fluctuations in the transients present in the system before the macroscopic description applies. Furthermore the approach used here is not stochastic, but describes the system entirely in terms of macroscopic variables (averages and correlation functions), as required to make a precise and immediate connection with experiments . Microscopic or semi -microscopic formulations of the description of fluctuations in nonequilibrium steady states, similar in spirit and content to the present one, have been developed by Kirkpatrick, Cohen and Dorfman (KCD) , ' and by an MIT collaboration ,^^~'^^ as indicated in Chapter I. The results of KCD agree with the ones obtained here, when the latter are specialized to the case of nonequilibrium steady states . The inhoraogeneous term in their equation for the equal time correlation functions has apparently a different form from the one in Eq . (3 .66) . The correlation of interest in Ref . 46 is the deviation of

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59 the equal time correlation function from its local equilibrium value , The noise term there originates from the action of the Euler part of che hydrodynamic equation over the local equilibrium correlation function, as it is shown in Ref . 41 , The form given in Ref . 46 and the one derived here, Eq . (3. 66),. are however equivalent. The hydrodynamic equations obtained by Kirkpatrick, Cohen and Dorfman are therefore identical to the ones derived here . Instead, the results obtained by the MIT group (and, it seems, also the Langevin theory) differ from those obtained through the generating functional approach because the second term in the evolution matrix C [see Eq . (3.45)], containing the thermodynamic derivatives of the transport coefficients, is neglected there. This term is often small, but is still of first order in the fluctuations around the nonequilibrium state, and therefore cannot be neglected even when fluctuations of small amplitude are considered .

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CHAPTER IV KINETIC THEORY IV. 1 Generating Functional for Phase Space Fluctuations The hydrodynamic description of nonequilibrium fluctuations developed in the previous chapter is restricted to states adequately described in terms of the nonlinear Navier-Stokes equations and rests on an estimate of the lifetime of certain correlations functions . Within the framework of kinetic theory, it is possible, at least in the limit of low density (where a small expansion parameter is available) , to derive an exact description of nonequilibrium fluctuations . Furthermore the kinetic description is more general than hydrodynamics . It describes macroscopic processes varying on shorter space and time scales (of the order of or smaller than the mean free path and the mean free time between collisions) and it incorporates the hydrodynamic limit. ' Also, as shown in Section II .2, the averages and the correlation functions of the dynamical variables of interest in the study of a large class of systems are simply related to the reduced distribution functions and to the correlations of phase space fluctuations . The generating functional method is applied here to describe phase space nonequilibrium fluctuations. No restrictions are imposed on the nonequilibrium state or on its stability other than the well understood assumptions entering in the derivation of the Boltzmann equation .^'^ »^' The description of fluctuations obtained is justified in the same limits 60

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61 which apply to the macroscopic description of the state. In particular no extra assumptions are introduced to evaluate the amplitude of the "noise" in the system, which contributes to the equations for the equal time fluctuations. Finally, in the last section, the hydrodynamic limit is recovered. Again, the results are restricted to nonequilibrium states in low density gases . However all of the known qualitative dynamics of fluids in general are also exhibited by gases . In Q Q particular, the nonlinear hydrodynamic behavior is identical. ' The system considered is the classical system of N interacting point particles in a volume V introduced in Chapter II . Since the internal degrees of freedom of the molecules are neglected, the discussion is limited to monoatomic gases . The time evolution of the distribution function of the system is governed by the Liouville equation for a system in interaction with its surroundings, as given in Eq.(2.2). For pairwise additive central interatomic forces the Liouville operator of the isolated system has the form N N L(x^,...,x^) = I L^(x.) I e(x^,x ) , (4.1) i=l i
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62 Equation (4.3) applies for continuous interatomc potentials, V(q..), with q.j = l^i 'Ij I • The potential V is also assumed to be short ranged (of range a) and purely repulsive. The first term in Eq.(4.1) represents the free streaming of the particles and the second term contains the effects of the collisions. Kinetic theory describes phenomena whose space and time variations occur over scales large compared to the force range, a, and the duratio of a collision, x^ . The derivation of the kinetic theory from the microscopic N-particle dynamics rests on the assumption that, for a particular class of nonequilibrium states, a contraction of the description takes place over a time of the order of x^ . For times larg. compared to x^ all the properties of the system depend on time only through the one-particle distribution function. This is the idea proposed by Bogoliubov to derive the Boltzmann equation from the Liouville equation.^ The special states to which this macroscopic description applies are identified through the requirement that the initial correlations decay on times of the order of the time required for a collision .5 The kinetic theory of nonequilibrium f lactations derived here will be restricted to the same class of states. As mentioned, the macroscopic variable of interest is the oneParticle distribution function, f^(x,t), defined in Eq.(2.27) as the nonequilibrium average of the phase space density, f^(x,t). The corresponding generating functional is given by Eq.(2.35), with U[X] HU^IX] =exp/::dt/dxf^(x,t)X(x,t) . (4.,^ The first functional derivative of G is then given by

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63 6\(x,t) ' ^^-^^ and reduces to the one particle distribution function when evaluated at X=0, To obtain a kinetic equation, it is convenient to define the functional f (x,t|\) as n f^(x,t|\) H . for t > t^ , (4.7) where t is an arbitrary parameter introduced to assist in ordering the times obtained on functional differentiation. Again, evaluating Eq . (4.7) at \=0 and choosing t =0, the functional f.[X] reduces, at all times t > 0, to the one-particle distribution function, as in Eq . (4.6). The dependence on the parameter t can be introduced explicitly by choosing the tests functions to be nonzero only for t < t , i .e . \(x,t) = \'(x,t)e(t -t) , (4.8) where 9 is the unit step function. In this way only time correlation functions involving time arguments smaller than t can be generated . The ordering chosen in Eq . (2.40) can be obtained by appropriate choice of t . The correlation functions of phase space fluctuations, defined in *A similar approach has been used in Ref . 58 to derive kinetic equations for multitime equilibrium fluctuations .

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64 Eqs.(2.32) and (2.34), are then immediately obtained as higher functional derivatives of G, as shown in Eqs.(2.40). In the following the dependence on the parameter t will be indicated explicitly only when needed . As already done at the hydrodynamic level, it can be argued that the detailed form of the external forces and of their interaction with the particles of the system should not appear in the equations governing the time evolution of the reduced distribution functions (or functionals) . ' The dependence on the surroundings can be entirely incorporated through appropriate boundary conditions to be used when solving the kinetic equations . Tills is a consequence of the fact that the lower order reduced distribution functions are localized quantities describing properties at a particular point in the gas. Therefore, as long as properties in the interior of the system are considered, at a distance from the boundaries large compared to the force range, the Liouville equation for the isolated system can be used to describe the time evolution of the dynamical variables. Equation (4.5) can then be rewritten as CD n f^(x.t|X) = J^ N / dx^..,dx^ e-^^ ^^ p^(0) . (4.9) This form suggests that f (x,t!\) can be identified as the first mftniber of a set of functionals defined by n^f^
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65 When evaluated at X=0 , f (x ,...,x ,t|\) reduces to the s-particle distribution function defined in Eq.(2.25). The operator governing the time evolution of the set of functionals (f [X]} is the Liouville operator of the system and does not depend on the test function X. This dependence only enters through the initial condition. By differentiating Eq.(4.10) with respect to time it can then be shown that the functionals f [X] satisfy the BBGKY hierarchy. In particular, for s=l, {^5t + L^(x^)}f^(x^,t|\) = n J dx29(x^,X2)f2(x^,X2,t|\) . (4.11) Again, as done in the hydrodynamic case, the standard methods of statistical mechanics which allow closure of the hierarchy and derivation of a kinetic equation for the one-particle distribution function can be applied to the \-dependent functionals . By assuming that the functional relationship between f [\] and X, as defined in Eq.(4.7), may be inverted, it is possible to express, at least formally, f«[\] as a functional of f [\] . A formally closed equation for f , [\] is then obtained, {^+ L^(x^)}f^(x^,t|\) = n J dx2e(x^,X2)H[x^,X2,t|f^[\]) . (4.12) The functional H is highly nonlinear but it simplifies considerably when evaluated at \=0 . However, in order to preserve the properties of the generating functional, the evaluation of H at nonvanishing X has to be attempted . This evaluation is carried out in the next section in the limit of low density .

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66 IV .2 Low Density Limit and Boltzmann Equation A systematic evaluation of the functional H defined in Eq.(4.12) is possible in the limit of low density, by using the nonequilibrium cluster expansion developed by Cohen as a generalization of the technique proposed by Mayer for equilibrium systems . In this way the two-particle distribution function for a dilute gas can be expressed in terms of the one-particle distribution as a power series in the reduced 3 density no . When this result is substituted in the first equation of the BBGKY hierarchy, an expression for the rate of change of the single particle distribution function in the form of a density expansion is obtained . The terms in the expansion depend successively on the dynamics of clusters of two, three, etc., isolated particles. To lowest order in the expansion parameter, corresponding to the Boltzmann limit, only binary collisions are retained . The cluster expansion for the functionals f [X] is formally identical to the one for the distribution functions and is outlined in Appendix C. To lowest order in the density, the result is f2(x^,X2,t|X) = ^(x,,X2)f j^(x^,t|\)f ^(x^.tlX) + R(x^,X2,t|f^[\]) , (4.13) where the streaming operator is defined as |.(x^,X2) = S_^(x^,x^)S^ix^)S^i:i^) (4.14; and Sj.(xp . . .,Xg) is the operator governing the time evolution of a

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67 dynamical variable in a system of s isolated particles . It is given by tL(x , . . .,x ) Sj.(x^, ...,Xg) = e ^ , (4.15) where L(xj^ , . . . ,Xg) is the s-particle Liouville operator (see Eqs.(4.1)) In other words Sj.(xp . . ..x^) describes the dynamics of an isolated cluster of s particles. Finally, R(x ,x ,t |f [\] ) is a nonlinear functional of f.[\], depending on the initial correlations in the system. Its explicit form is given in Appendix C . For times long compared to a collision time the streaming operator 5 reaches a time independent value , ^^(x^.x^) = lim ^^.(x^.x^) . (4.16) t>>i; c Furthermore it is shown in Appendix C that if attention is restricted to the time evolution of initial states of the system where the particles have only short-range correlations (i.e. the s-particle distribution function factorizes for interparticle distances large compared to the force range), a part of the contribution to Eq.(4.12) from the functional R(x ,x ,t|f [\]) vanishes for times long compared to the collision time . By inserting Eq.(4.13) into Eq.(4.11) and by making use of the above results, an equation describing the time evolution of the oneparticle functional f (x ,t|\) is obtained. The equation only describes variations over time scales large compared to t and is given by f— ^dt + L^(x^)}f^(x,t|\) = n J[f^[\],f,[\]] + W(x^,t|f^[?'v]), (4.17)

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68 for t > t , where o J[A,B] = / dx^ Ux^,x^)A(xpB{x.p (4.13) and K(x^,x^) = eCx^.x^) SJx^.x^) . (4.19) The second term on the right hand side of Eq.(4.17) is given by VJ(x^,t|f^[A]) = lim n / dx2e(x^,X2)S_|.(x^,X2) t>>T C X {u°[x^,X2|\]U°~^[xJ\]U° ^[x^lX] -1} X S^(xps^(x2)f^(x^,t|\)f^(x2.t|\), (4.20) where U [x, ,...,x Ix] is the low density limit of the s-particle sis' -^ ^ functional analogous to the N-particle functional defined in Eq.(4.4). It is given by s U°[x^,...,Xg|\] = exp J_^ dt S^(x^,...,X2) ^ ^^^i'^) i=l s = exp{ I j_^ dt \(x^(t),t)} , (4.21) i=l where x^(t) = S^(x^,...,Xg)x.(0) . (4.22) A precise justification of the use of Eq.(4.21) as the low density form of U is given in Appendix C .

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69 By evaluating Eq.(4.17) at X=0 and observing that U [x , . ..,x |\=0] = 1 for all values of s, it follows that W(x^,t|f^[\=0]) = , (4.23) identically. Choosing t = 0, a homogeneous equation for the oneparticle distribution function is obtained, {|^+ L^(x^)}f^(x^,t) = n J[f^,f J . (4.24) The right hand side of Eq.(4.24) is a nonlinear functional of the oneparticle distribution function. When the further approximation of neglecting the difference in position of the colliding pair is introduced (this implies that the resulting equation will only describe space variations over distances large compared to the force range) , the operator J assumes the familiar form of the nonlinear Boltzmann operator, given by J[A,B] = J dp^ /q db b jI"" d0 \ ^ {A(q^,ppB(q^,pp .-* -> . , ->• -* A(q^,p^)B(q^,P2)} , (4.25) where b is the impact parameter for the binary collision and ({) is the azimuthal angle measuring the orientation of the scattering plane . The prime over the momenta indicates their values after the collision . Tha nonlinear Boltzmann equation for the one-particle distribution function is then recovered . In the following both the operators J and J will be referred to as the nonlinear Boltzmann operator .

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70 Equation (4.17) has therefore the form of a generalized EoltZEann equation. The extra terra on Che right hand side of the equation is a complicated functional of \ and its behavior at long times cannot be analyzed before setting \=0 . As just shown, it does not contribute to the dynamics of the macroscopic variables of the system. However it has to be retained in the equation for the functional f (x,t|\) for the purpose of deriving equations for the correlation functions . Equations for the correlations of phase space fluctuations are derived in the next two sections. As in the hydrodynamic limit, the contribution from W to the equation for the multitime correlation functions vanishes on a time scale of the order of i . However, when c correlations of equal time fluctuations are considered, this term does not vanish, but is responsible for the appearance of an inhomogeneous part in the equations . IV .3 Correlation of Fluctuations The multitime correlation functions of phase space fluctuations defined in Eq.(2.34) are obtained by functional differentiation of G[\], as indicated in Eq.(2.40), or, equivalently , C^(x^,t^;...;x^,t^) . < n 6f (x.^,tj^)> k.= l 6^"-^)f^(x^,tjM " Uux„,t„)...6\(x ,t )^t =t^,\=0 ' (4.26) LI s s o 2 Lme with the tine ordering t, > t„ > ••• t . Equations for the tic 12 s evolution of C^ are then obtained by functionally differentiating Eq.(4.17) and evaluating the resulting equation at \=0 .

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71 It is shown in Appendix C that the inhomogeneous term W(x,,t,|A.) vanishes at long times (t, » t ) if t„ is chosen to i. 1 i c o satisfy t t > x . This choice corresponds to the one needed to o c generate multitime correlation functions in a Boltzmann description. In this sense it immediately appears that W does not contribute to the equation for the two-time correlation function, as long as t,-t„>T . 12c Here this result is also explicitly shown to follow from the analysis of the functional derivative of W. This will make evident the difference between the unequal and the equal time cases . The equation for the two-time correlation function (s=2) is g 6W(x^,tJf J\]) ^ar;;-" ux^.tjfp} c^cx^.t^jx^.t^) = [ — 6^(^^^t^) — lt^=t2,x=o ' (4.27) where L(x^,tjfp = L^(Xj) n A(x^,tjfp (4.28) and A(x^,tjfp = / dx2e(x^,X2) Jx^,x^)(HP^2)f^(x2,tp . (4.29) By functionally differentiating Eq.(C.26), the term on the right hand side of Eq.(4.27) can be evaluated explicitly as a functional of the one-particle distribution, with the result, 5W(x^,tJX) t^ToT^TTp— \=0 = " ^ ^^3®^^'''3^'-t/^'^3)^ts,^(x^.X3) Sj. (x^)S^ (X3)][6(x^-X2) + 6(X3-X2)]} X S^ (x^)S^ (X2)f^(x^,t^)f^(x3,tp (4.30)

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72 or, 6W(x,,t |\) X S^ _^ (x )S (X )[6(x -X ) + 6(x -X,)]} 2 12 1-^ g^ (x^,X3)f ,(x^,tpf^(x3,tp , (4.31) where the choice t = t2 is understood . The streaming operator S reaches a time independent limit for t»x , as shown in Eq.(4.i6) t c Therefore the difference between the two streaming operators in Eq.(4.31) vanishes if S » \ ' t, t>> T . (4.32) 12c In the above limit the right hand side of Eq.(4.26) vanishes, 6W(x t |\) lim [^_L_^_] ,0 = 0. (4.33) t,»-c '6X(x2,t2) ^t^=t2,X=0 1 c S-^^2»^c and the time evolution of the two-time correlation function is governe.! by a homogeneous equation given by (|^+ L(x^.tJf^)}C2(x^,t.;x2.t2) = . (4,34) '1

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73 The conditions imposed in Eqs.(4.32) imply that Eq.(4.34) can only be applied to describe correlations over time scales large compared to a collision time. Not only the times t, and t2 have to be large compared to T , but also their difference . This is the natural outcome of conc sidering the Boltzmann limit . As in the preceeding section, when the difference in position of the colliding pair of particles is neglected, Eq.(4.3A) becomes t§t-+ S^^l'^I^^JS^^I'h'"''2'''2^ = ° ' ^^'^'^ where, Lg(x^,tjfp = Lq(xP nl(x^,t Jfp , (4.36) and I(x^,tJfpA(xp = / dp3 Q db b /^^ d^ "^^^^' [1 + P(p^,P3)] In ,, |PrP3' X {f^(q^,P3,t^)A(q^,p') f^(q^,P3,tpA(q^,p^)} . (4.37) The permutation operator P(p,,Po) interchanges only the momenta of the two particles. When the one-particle distribution function in Eq.(4.37) is evaluated at equilibrium (i.e. coincides with a Maxwell-3oltzmann distribution) , the operator I reduces to the linearized Boltzmann Q operator . The equation for the correlation function is linear . This is an exact result (in the Boltzmann limit considered here), not the outcome of a linearization around the state. No restrictions have been Imposed

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74 on the size of the fluctuations. The operator L(x,t|fp depends on tha nonequilibrium state of the system, characterized by the distribution function f,(x,t), which is given by the solution of the nonlinear Boltz-mann equation . In this sense the dynamics of the fluctuations is entirely determined by the macroscopic nonequilibrium state of the system. ' To solve Eq.(4.34) specification of the initial condition, represented by the equal time correlation function C2(x, , t ;x2 ,t ) , is also needed . The derivation of an equation for the equal time correlation is the subject of the next section. Finally, the method described here can be extended to evaluate higher order multitime correlation functions . In particular the equation for the three-time correlation function is given by = n / dx^K(x^,x^)(i + V^^)C^(x^,t^;x^,t^)C.^{x^,t^;x^,t^) , (4.3S) where the operator K(x,,x,) has been defined in Eq.(4.19). The eauation 1 4 for the three time correlation function is again linear, but is coupled to the lower order correlations through the inhomogeneous term on the right hand side of Eq.(4.38). In general, the equation for the s-tirae correlation function is a linear inhomogeneous equation. The homogeneous part nas, at any order, the form given in Eq.(4.34) and depends therefore on the nonequilibrium distribution function. The inhomogeneous part couples the s-th order equation to all the lower order ones. As s increases, the equazicns become therefore very complicated, but alv/ays conserve linearity.

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75 Furthermore the solution of the equation for the s-th order multitime correlation function requires the knowledge of all the correlation functions obtained by progressively setting t,=t2, t,=t2=to, ..,, t. =t2 = ...=t , in Eq.(4.26). Equations describing the time evolution of such correlation functions can be obtained by applying an appropriate limiting procedure on the equations obtained by functional differentiation of Eq.(4.17). The second term on the right hand side of Eq.(4.17) will only contribute when at least two of the time arguments in the correlation function are equal, as will be shown in the next section. IV .4 Equal Time Fluctuations In order to derive equations for the correlation functions of equal time fluctuations, defined in Eq.(2.32), a limiting procedure on the equations for the unequal time correlation has to be used . In this way it is possible to circumvent the problem arising from the fact that the time derivative and the functional derivative can only be interchanged when involving different time arguments. As in the previous section, the calculation will be carried out explicitly for the second order correlation function. The rate of change of C^Cxi , t ;x2.t) can be written in the form, |^2(-i.t;x2,t) = l±^_^{^^i.^,t+e;.^,t) + l^^^x^ '^"^^'^I '^^^ ' ^' '''^ E-K) An equation for the equal time correlation can then be obtained from the equations for the unequal time correlations, Eq.^,4.27), with the result.

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76 6W(x, ,t-l-e|\) 6W(x„,t+£|\) On the left hand side of the equation the limit £->0 has been taken. The operator L(x,t|fj^) has been defined in Eqs .(4 .28-29) . By using Eq.(4.31) the right hand side of the equation can be written, 6W(x^,t+E|\) 6W(x2,t+£|\) r(x^,x,,t) H li^^{[ 6Ux^,t) It =tA=0 ^ ^ 0X(x,.t) 3t =t,X=0^ E"^ Z O i O = lim_^(l+Pj^2)" / dx^Q(x^,x^) S_(x^,x )[l gj.(x ,x )] £->-0 X S_^(x^)S_Jx2)[6(x^-X2) + 6(X2-X3)] S^^^(x^,X2)f^(x^,t+c)f^(x3,t+e) , (4.41) or, taking the limit, r( x^.x^.t) ^ (l+P^2^ "" ^ dx3e(x^,X3){[5(x^-X2) + SCx^-x^)] S^(x^ ,X3) Sj.(x^,X2)[5(x^-X2) + 5(X2-X3)]}f,(x^,t)f^(x3,t) . (4.42) By using the form (4.3) of the interaction operator 9(x ,x.), it can easily be shown that, for an arbitrary function F(x^,X3), '-^'^ following identities hold:

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77 (1 + P^^) / dx^e(x^,X2)6(x2-X2)F(x^,x^) = e(x^ ,X2)F(x^ .x^) (4.43; and (1 + P^^) / dx3 e(x^,X3)6(x^-X2)F(x^,X3) = &(x^-x^) j dx3e(x^,X2)F(x^,X2) . (4.44) By introducing the above identities and considering the limit t » t , Eq.(4.42) can be written in the form. r(x^,X2,t) = n e(x^,X2) SJx^,X2)f^(x^,t)f^(x2,t) + 6(x^-X2)n / dx3e(x^,X2) SJx^ ,X2)f ^ (x^ ,t)f ^ (x^ ,t) n[A(x^,t|fp + A(x2,t|fp]6(x^-X2)f^(x^,t) , (4.-''5) or, from Eqs.(4.18) and (4.19), r(x^,X2,t) = n K(x^,X2)f^(x^,t)f^(x2,t) + n6(x^-X2) J[f ^ ,f ^] n[A(x^,t|fp + A(x2,t|f p]6(x^-X2)f^(x^,t) . (4.46) The equation for the equal time correlation function is then given by {-^+(l+?^pL(,x^,t\f^)}C^(x^,t;x^,t)=T(x^,x^,t) . (4.47) By neglecting again the difference in position of the colliding pair, the various terms in Eq.(4.46) can be identified with Boltzmann-like

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78 operators, as defined in Eqs.(4.25) and (4.37). In this limit, the Inhomogeneity F is given by T^(x^,x^,t) = n 6(q^-q2)K(p^,P2)f j^(x^,t)f ^(x^.t) + n6(x^-X2)J[f^,f^] n[l(x^,t!fp + I(x2,t|fp]6(x^-X2)f^(x^,t) , (4.46) where K(p ,p„) is the kernel of the nonlinear Boltzmann operator, defined by dP2K(p^,P2)f^(x^,t)f^(x2,t) = J[f^,f^] . (4.49) The dynamics of the equal time correlation function is determined by the same operator that governs the time evolution of the two-time correlation and again depends on the solution of the nonlinear macroscopic problem. The linear equation for the equal time correlation function however has now a inhomogeneous term, which is also a function of fi(Xj,t), i.e. of the state of the system. The inhomogeneity in Eq,(4.47) assumes a familiar form v/hen the equilibrium case is considered for comparison. Equation (4.47) reduces then to an identity, given by (l+?^2>n[l^(p.) + I^(P2)]4°^^^l'^2^ = ^o^^'l'^2^ ' ^"^'^^^ with

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79 r^Cx^.x^) = n[l^(pp + I^(p2)]6(x^-X2)
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80 This approach has been used by several authors .-' In particular Ernst and Cohen have derived kinetic equations for the nonequilibrium correlation functions in a gas of hard spheres at low density by 3 performing an expansion in the parameter no and by retaining, to each order in the density, only terms involving collisions among a limited number of particles . The results derived in Ref . 37 agree with the ones obtained here by using the generating functional method, but the identification of terms of different order in the density is certainly not transparent for space scales greater than a. The reason for this lies in the choice of the expansion parameter . Ernst and Cohen applied the method used when deriving a kinetic equation for the one-particle distribution function. To close the first equation of the hierarchy information on the two-particle distribution function, f (x ,x ,t), over distances smaller than or of the order of the force range a is needed in 3 this case. An expansion of f2 in pov;ers of na is therefore appropriate. The cluster expansion in fact gives information on the variation of the distribution functions over space and time scales of the order of the force range a and the collision time i c respectively. In low density gases, however, second or higher order correlations among particles vary appreciably also over distances and times of the order of the mean free path Z and the mean free time tf, both much larger than a and x . A different and more systematic method of expansion of the hierarchy can then be developed by better analyzing the role of the various length and time scales in the problem. If the s-th equation of the hierarchy is scaled according to the. two characteristic lengths over which a variation of the correlation functions is 3 -1 expected, S. and t^ , the expansion parameter a = (ni ) naturally

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81 appears in the equation. The terms of an expansion of the s-th order distribution function in powers of a can then be systematically evaluated . The condition a«l corresponds to a gas at low density in 3 3 2 the usual sense (i.e. no « 1, since a = (no ) ), but with a large number of particles in a volume of the linear size of a mean free path. This is exactly the condition under which the effect of collisions will be important over the space and time scale considered and will be responsible for higher order correlations . The scaling of the BBGKY hierarchy and the evaluation of correlation functions to zeroth and first order in a is carried out in Appendix D for a gas of hard spheres . Again the results agree with Ref . 37 and with those obtained by using the generating functional method . IV .5 Hydrodynamic Limit In the low density limit, the average conserved densities in a fluid are defined as (|;^(?,t) = / dp a^(p)nf^(x,t) , (4.52) where (a (p)} are the one-particle conserved quantities, defined in Eq . (3.5). In this section, in order to simplify the notation, x is used to indicate also the set of variables (r,p), where r represents a point in the fluid and p is the canonical momentum. The hydrodynamic equations are then obtained by averaging the microscopic conservation laws , Eqs . (3.6), over the nonequilibrium one-particle distribution function and by supplementing the equations so obtained with constitutive equations for the macroscopic fluxes as functionals of the average densities.

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82 A set of ^-dependent average densities can be defined as -> .-> 4; (r,t|\) = / dp a^(p)nf^(x,t|\) (4.53) where f,[\] is the one-particle functional defined in Eq . (4.7). When evaluated at X=0 (and for tQ=0) , Eqs . (4.53) reduce, at all times, to the usual average densities, defined in Eq . (4.52). The functionals {cl; (r,t|\)} are the low density limit of the ones defined in Chapter III . ''"' Again, intensive \-dependent thermodynamic variables (y (r,t|X)} are defined by requiring 6 (?,t|\) = / dp a^(p)f^(x,t|\) , (4.54) CX ex Xj where fj is a one-particle local equilibrium functional, given by fj^(x,t|\) = exp{-y^(?,t|\)a^(p)} . (4.55) The local equilibriui3 distribution is normalized as follows , / dx fj^(x,t|\) = N, where N is total number of particles in the system. The thermodynamic variables {y (r,t)} are given explicitly in Eq . (3.16). The comments made in Chapter III, when requiring the equality (3.13), apply hers. Hydrodynamic equations for the functionals {(Jj (r,t|A.)} are simply obtained by projecting Eq . (4.17) for f,[A.] over the five conserved densities {a (p) 1 . By writing

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83 f (x,t|\) = f^(x,t|\) + A(x,t|\) (4.56) and substituting in Eq . (4.17), an equation for A is obtained {|^+ L^(x) n Aj^(x,tif^)}A(x,t|X) = [|r+ L.(^i)]fT(^'ti^) + W(x,t|\) , (4.57) 'ot o i " L where, for consistency with the purpose of deriving equations to NavierStokes order, only terms linear in A have been retained. The operator Ain Eq . (4.57) is the local equilibrium form of the operator defined in Eq . (4.29) , i.e. Ajx^,t|f^) = / dx2e(x^,X2) SJx^,X2)(l+P^3)fL(x^,tiX) . (4.58) The solution of Eq . (4.57) is given by A(x,t|\) = T(x;t,0)A(x,0|\) + /^ dt T(x;t ,i:)W(x ,t| \) jl dx T(x:t,T)[|:^+ L^(x)]f^(x,xK) , (4.59) where the time evolution operator T is \-dependent and is defined as the solucion of the equation, {^+ L^(x) n Aj^(x,t|f^)} T(x;t,x) = , (4.60) with initial condition T(x;t,t) = 1. Also, the initial deviation from

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84 the local equilibrium distribution functional is given by A(x,0|X) =— i^___ _ f^(x,0|\) . (A.61) By projecting now the equation for f.[\] over the conserved densities {a } and making use of Eq . (4.54), the following set of equations for the {c(; [X]} is obtained ^(l.^(?,t|\) +-^pJ dp V. a^(p)fj^(x,t|X) +-^^ / dp V. a^(p)A(x,t|\) i i . = / dp-a^(p)W(x,t|\) . ^-^ (4.62) The term on the right hand side of Eq . (4.62) can be neglected. In fact it contains the potential part of the raomentum and energy fluxes which are of higher order in the density and therefore negligible in the Eoltzmann limit. By inserting Eq . (4.59) for A, the set of Eqs , (4.62) becomes |_4.^(?,t|X)+|_E^j?.t;(>[M}) i " dFT ^o "^^ J" "^P ''i^a^P^'^^^-'^'^^tl^ + v.V^]fj^(x,T|\) = --gp/ dp v^a^(p)T(x;t,0)A(x,0|X) Ip/q d-r / dp v^ a^(p)T(x;t,T)W(x,T|X) , (4.63) i where the Euler matrix E has been defined as a.

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85 E\?,t;{c!;[\]}) = / dp V a (p)f (x,t|X) . (4.64) Furthermore, to lowest order in the gradients, [-^+ v.V^]f^(x,x|\) = fj^(x,T|\) P ^^ pj(P) = (i P^^Vj ap(p) . (4.66) Here P£ ^ is the low density forn of the local equilibrium projection operator defined in Eq . (B.9) and it is given by P^^^A(p) = a^(P)g^p(?) J dp' f^(?,p',t|\)ap(p')A(p') , (4.67) where g^p(?) = / dp f^(x,t|x)a^(p)ap(p) , (4.63) and g is the (ap) element of the inverse matrix. The thermodynamic P variables {y } and their gradients appearing in Eq . (4.65) can also be expaiided in a functional Taylor series around the point t = -r . To NaviarStokes order only the first terms in these expansions have to be retained. Substituting Eq . (4.65) into (4.63), the latter becomes 1 ^ J W^(?,t|\) , (4.69)

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86 where L , is the matrix of the transport coefficients f:>r a low density gas, given by L^J(?,t;{ci.}) = lim ll dt / dx f (x ,t | \) [T^x;t ,T)v,a^(p) ap ^^^^^ o L la . ->• ^. T^ 6(r-q)]4)p^(p) , (4.70) For convenience, the same notation that was adopted to indicate the matrix of transport coefficient for a dense fluid is used here for the case of a low density gas . Tlie two differ because the coefficient of bulk viscosity vanishes at low density. The operator T in Eq . (4.70) is the adjoint of the operator T and is defined by / dx a(x)T(x;t,T)b(x)f^(x,T) = / dx f ^(x ,T)b(x)T^(x;t ,T)a(x) , (4.71) A A for any pair of one-particle phase functions, a and b. Equation (4.70) can be put in a more familiar form by observing that, to Navier-Stokes order, the hydrodynamic variables {y (r,t)} in the operator T can be evaluated at T=t . The time evolution operator is then given by T (x;t,T) = exp{[v.V;^ nT (x ,t ) ]( t-x)} , (4.72) q Li where nl (x,t) = f/(x,t|\)nA,(x,t|f^ )f,(x,t|\) . (4.73) ij i-i Li L u The time integral in Eq . (4.70) can then be performed with the result.

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87 L^J(?,t;{4>}) = / dx fj^(x,t|\Hp^(p)[v.V^ nl^(x,t)]"^ X v^a^(p)6(r-q) . (4.74) The action of the derivative operator in Eq . (4.74) on the operator I (q,p,t) or on f (x,t|A.) generates gradients of the thernodynamic Li Ju variables {y } • To lowest order these terms can be neglected and Sq . (4.74) can be written as L^Jp(?,t;{4,}} = J dqdp fL(?,PK)?pj(P)[v-V^ nl^(?,p,t)] ^ X v.a (p)6(r-q) , (4.75) 1 a or L^Jp(r,t;{4.}) = /dp f^(?,p|>O0p.(p)[n l^(?,p,t)] Va^(p) , (4.76) which is the familiar expression for the transport coefficients in the Boltzmann limit, derived, for example, by using the Chapman-Enskog expansion. The inhomogeneous term on the right hand side of Eq . (4-69) is W^(?,t|\) = |_ J dp v^a^(p)T(x;t,0)A(x,0|X) i 4— (^ dx / dp v.a (p)T(x;t,x)W(x,T|\) . (4.77) or . -^ o la 1 When evaluated at X=0, the right hand side of Eq . (4.59) vanishes identically and the left hand side reduces to the nonlinear hydrodynauiic equations for a low density gas.

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88 The correlation functions of the fluctuations of the conserved densities around their average value are given by 64) (r ,t |\) Z Z A.— U Equations for the correlation functions are then obtained by functionally differentiating Eq . (4.69) and projecting the resulting equation over the conserved densities . The case of correlation of fluctuations at different time is first considered, with the result = ^ap(^'^'^2'^2) ' (^-'^^ where ^ -> 6W (? t |\) ^ap(^l'^'^2'^2) = 1%^ / '^P2^p(P2> Ux(x^.t^) ^^=0 ' ^' ''^^ The operator £ is given in Eq . (3.45) and depends on the state of the system through the solution of the nonlinear hydrodynamic equations. In the low density limit considered here, the terras containing the bulk viscosity and the derivative of the pressure or of the enthalpy density with respect to the density vanish, and do not appear in C . The inhomogeneous term on the right hand side of Eq . (4.79) is analyzed in Appendix E. The contribution from the second term on the right hand side of Eq . (4.77) is shown to vanish on a time scale of the order of the collision time, x . Therefore, it does not contribute to the hydrodynamic equations for the correlation functions, which describe

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89 variations over time scales large coapared to the mean free time, t^ . The functional derivative of the first term in Eq . (4.77) vanishes in the limit ti>tf (4.81) ^-^2>'^f and gives no contribution to the equation for the unequal time correlation functions, which reduces to the same linear homogeneous equation obtained in Section III. 2, Eq . (3.49). An equation for the equal time correlation functions can be obtained by using the same limiting procedure applied for example in Section III .3 (see Eq . (3.63)). The resulting equation is wv^v^'h''^ + (1 + p^^v ^aa(^^'^'t^^>v^^'^'^^2'^) = r^B^^i'^2'''^ ' ^^-^^^ where Only the contribution to I _ from the first terra on the right hand side of £q . (4.77) survives on the time scale of interest, leading to the result , r^p(?^,?2;t) = (1 + P^^P^p) 1^ Lj^p(^,t;(,}) 1^ 6(rW . (4.84) were L 7, is given by Eq . (4.76).

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90 The set of hydrodynaraic equations derived here to describe the time evolution of averages and fluctuations strictly applies only to a low density gas. With this restriction in mind, they are identical to the equations obtained in Chapter III for a dense fluid .

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CHAPTER V TAGGED-PARTICLE FLUCTUATIONS IN SHEAR FLOW V.l Definition of the Problem To illustrate the application of the formalism developed in the previous chapters, the problem of steady shearing flow is considered. Substantial simplifications occur as the result of three conditions: (1) low density, (2) Maxwell molecules, and (3) uniform shear flow. In fact for Maxwell molecules the irreversible momentum flux can be determined exactly from the nonlinear Boltzmann equation, as a nonlinear function of the shear rate . The macroscopic conservation laws can then be closed and solved exactly for the situation of uniform shear . All the information on the macroscopic state of the system, which is needed in the equations for the correlation functions, can be evaluated for IQ OO 00 0/ arbitrarily large shear rate, as has been shown elsewhere. ' ' In particular the velocity-velocity autocorrelation function of a tagged particle in the fluid is evaluated here. For the particular system chosen, this correlation function can be calculated without any approximation. Due to the presence of the shear on the system, equal time correlations which are zero in equilibrium are now nonvanishing, and their amplitude increases with the shear rate . *Maxwell (or Maxwellian-) molecules are monoatomic molecules interacting through a weakly repulsive potential of the form V(r) = e (a/r) , where r is the interparticle distance. 91

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92 The steady shear flow corresponds physically to a fluid between two parallel plates at a fixed distance apart and in relative motion. The flow field, ij(r),* is expected to vary linearly between the plates (except near the surfaces) and be of the form. U.(r,t) = U . + a. .r. . (^'-^ i^ ' -^ oi ij J The constant vector U^^ and the constant tensor a^^ are the velocity of the lower plate and the shear rate tensor, respectively. In the following U . will be set equal to zero without loss of generality. For the geometry considered, the tensor a^^ has the following properties: a = (5.2) ^ii ^' where no summation is intended in Eq . (5.2), and a..a.^ = . ' (5.3) It is also assumed that the externally imposed shear flow represents the only inhomogeneity in the system, and p(?,t) = p(t), q.(r,t) = q.(t) , u(?,t)=u(t), tj^^^'"^) = ^IjC'^ • ^^'^^ *In this chapter the macroscopic flow velocity of the fluid is indicated with U(r,t), instead of v(r,t), as was done in Ch. III.

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93 Therefore, the walls of the system are not maintained at a constant temperature (the extraction of heat from the system would create an inhomogeneity in the temperature field) . Heat is produced in the system through viscous friction and the temperature grows in time . The state considered is therefore nonstationary . By inserting Eqs . (5.1-4) into the hydrodynamic equations for a simple fluid, Eqs. (3.20), these reduce to ^&^-^)-=0. (5.5) at ^"(t) ^ _3..t*.(t) . (5.6) at ij ij The mass density p is therefore constant. p(t) = p^ . (5.7) The internal energy density u can be considered a function of p and of the temperature T (or the pressure p) . The hydrodynamic equations reduce then to one single equation, given by |Iil)-= a..a(t)t* (t) , (5.8) where a(t) = frr— 1 . In the case of a low density gas, the ideal gas ^ou^p 2 -1 equation of state can be used, to obtain a = -^ ^'^'^r^ where n = p /m is the constant number density. The temperature equation o becomes then

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94 Mii.-|(„V-'ayt*.(0 . (3.9) The system is prepared at t=0 in a state of local equilibrium with a temperature T(0) = T and a uniform flow field of the form given in Eq . (5.1). External forces will also be present at t>0 to maintain the desired flow field. Consequently t..(t=0) = 0, and the initial state of the system is described by the local equilibrium ensemble. pjO) = exp{-qj^(0) + v(0)N p(0)H'} , (5.10) where q,(0) is the normalization constant defined in Eq . (A .7), and Li H' = H({p^ mU(q^)}) , (5.11) where H is the Hamiltonian of the system. The choice of initial local equilibrium is equivalent to the assumption that the effect of the external forces can be incorporated entirely through boundary conditions on the thermodynamic variables . At low density, the state of the fluid (or bath) is described entirely in terms of the one-particle distribution function, f,(xj^,t), whose time evolution is governed by the nonlinear Boltzmann equation. i|f + % • \K^\''^ =--J[^'fb] ' ^5 -'2) where J is the nonlinear Boltzmann operator defined in Eq . (4.21). The suffix b is used to distinguish the variables of the bath from the variables of the tagged particle. Consistent with the preparation of

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95 the system described above, the initial condition for Eq . (5.12) is "^(^'0) = ^L^^'O) = 't'^Po'^'b^ ^5-13) where , (5.15) where the average is over the initial ensemble of the system. The time evolution of f-j, is governed by a linear equation, ^^^^'%-^hL^h^^V^^^^ =0 ' (5-^6) where ^BL'^b'^T^"'^' ^ ^ "'b ^o "" " •'o "''''' 'b' ^"b^^'"'b''"^T Im[f>,JfT(^.t) = / dv. r db b /2^ d*|v-vj{fjq,v' t)f r^,v',t) f,(q,v. ,t)f„(q,v,t)} . (5.17) b"^' b' ' T *In this chapter the velocity of the particle, instead of its momentum is used as a phase space variable, i .e . x = (q,v) .

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96 The operator defined in Eq . (5.16) reduces to the Boltzmann-Lorentz operator when f, is a Maxwell-Boltzmann distribution . Here the operator It>t will be referred to as the nonequilibrium Boltzmann-Lorentz oL operator . The tagged particle is assumed to be initially in local equilibrium, and localized at the origin, i.e. f^(x,0) =-^ 6(q)(^(p^,v') , (5.18) -y -> -> -> where v' = v U(q) . The correlation function of interest here is the velocity-velocity correlation function of the tagged particle, defined as (t > 0) G^(t,T) = ]> . (5.19) As shown in Section II .2 , G..(t,T) can be written in terms of the correlation function of the fluctuations of the tagged particle phase space density. G^(t,x) = / dx^dx^ v^^v^.C^^ix^,t+x;x^,x) , (5.20) where C2(x^,t+T;x2,T) = <6(x^-x^(t+x))[6(x2-x^(i;))-<6(x2-x^(t))>]> . (5.21) T As shown in Section IV .3, the time evolution of C2 is governed by a linear equation, given by

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97 ^h^^l'%^"^3L^^bl}^^^l''^+^'^2''^^ = °' (5.22) with initial condition C^(x^,t;x^,^) = &(x^-x^)f^ix^,x) f^(x^,T)f^(x2,T) , (5.23) since the equal time two-point correlation is zero by definition for a tagged particle . The dynamics of the average properties and the fluctuations of the tagged particle is described by the set of three coupled equations (5.12), (5.16) and (5.22), to be solved with the initial conditions (5.13), (5.18) and (5.23), respectively. V .2 Transformation to the Rest Frame The form of the initial distribution functions for the bath and the tagged particle, Eqs . (5.13) and (5.18), indicates that the dependence on the externally imposed flow field can be transposed from the initial conditions into the kinetic equations by introducing a pseudo-Galilean transformation (or transformation to the rest frame), defined by q! = q. a. .q .t = A. .(t)q. , v! = V. a. .q . , (5.24) 1 1 iJ J where the time-dependent tensor A..(t) is A..(t) = 5,. a..t . (5.25)

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98 For a flow field satisfying the properties (5.2-3), this transformation is easily inverted, since [A"^t)].^ = A^jC-t) , (5.26) with the result q. = q! + a. .q:t = A. .(-t)q! , V. = v! + a. . q'. . (5 .2?) 1 1 ij J A function F, which depends on the phase variables q and v, F = F(q,v) , transforms as follows F(q,v) = F({A^^(-t)qj}, {v| + \j ^ j ' 1 ) = F'(q',v') . (5.28) It can immediately be seen that this change of variables does not change the form of the binary collision operators in the kinetic equations obtained in the previous sections . The transformed equations are then given by: (1) for the bath distribution function, ^i-" V^Ji(^)i--%i-jii-l^b(^>^) ="'Jtf;'^bJ ' (5-2^) bj -J bj (2) for the tagged particle distribution function.

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99 ^If + Vji(^>lrvjii:-"'BLt^;nf;(x.t) = , (5. 30) (3) for the phase space correlation function, tfe-^ ^li^ji^'^) fer^li^ji I7-^Vt^bH^z'^'^i'^+^'^z'^^ = • (5.31) The prime on the transformed variables has been suppressed to simplify the notation. The initial conditions for Eqs . (5.29-31) are now f^Cv^.O) =
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100 fjl (5.35) or G'f'f(t,T) = R^.^'.ityT) + a.-R^'fCt.T) + a.^R^.'i(t,x) ij xj ikTcj J 1 il + ^ik^-ji^i^t.T) , (5.36) where R.. (for a and p = q,v) are the position and velocity correlations in the rest frame, defined as Rl^(t,x) = j dx^dx^a^^?^xl'(^^,t+TyA^,x) . (5.37) The velocity autocorrelation function will be evaluated in the next two rt R sections by solving a set of equations for the R... V .3 Two-time Velocity Autocorrelation Function The asymptotic (t large compared to a microscopic collision time) behavior of the two-time velocity autocorrelation function G. (t,-c), defined in Eq . (5.19), can be analyzed immediately, before evaluating w G. . itself for all times. In fact, it is expected that, at large t, the velocity of the tagged particle will be equal to the velocity of the fluid at the position of the particle, i.e. lim = lim a . (5.38) t->co -J t-H^ By using,

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101 qj^(t+T) = q^{x) + /^ ds Vj^(s+t) , (5.39) the long t limit of the correlation function is given by, Therefore, the velocity autocorrelation function does not decay to zero at long times . This is a result of the presence of the shear in the system and of the nonstationary nature of the state considered. At vv large t, G.. reaches an asymptotic limit, which depends on correlations between position and velocity of the particle that are created by the presence of the shear . For the purpose of evaluating the velocity correlation function at all times t and x it is not necessary to solve the kinetic equations themselves. Instead, it is more convenient to project the kinetic equations over the one-particle dynamical variables of interest (the components of q and v) , and obtain equations for the correlation function. In this way the Maxwellian form of the interparticle potential will lead to substantial simplifications . A set of equations for the two-time correlation functions in the CtPi rest frame, R..(t,i;), is immediately obtained by projecting Eq . (5.31) onto v^ and q • . For Maxwell molecules the velocity moments of the Boltzmann-Lorentz operator can be calculated exactly (see Appendix F) and simply generate eigenvalues of the operator itself . The resulting equations are

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102 and ^If"^ ^Kj(t,T) + a^j^I^^^(t,T) = 0, (5.41) |^Rj°^(t,T) A.j^(t)R^J(t.T) = 0, (5.42) for a = q,v, where v is the collision frequency given in Eqs . (F. 10-11). Equations (5.41-42) can be integrated immediately, with the result -V t R^J(t,T) = e A.j^(t)R^J(0,T) , (5.43) and R where i?^'^) = R,^;(0.x) -^i[6^^ T^,(t)Kj(0.x) . (5.44) T..(t) = 6..e ^ +^ [1 (l+v^t)e '^ ] . (5.45) Substituting Eqs. (5.43-44) into Eq . (5.36), the velocity autocorrelation function is then given by G^(t.x) = T.^(t)[R;:;(0,.) +a.XJ(0,x)] + ^ik^Ik^O'^) + ^ik^jl^^O'-^) • <5.46) It can easily be shown that the asymptotic value G^(°=,t) given in Eq . (5.40) is identical to the one obtained by taking the limit v t » 1 in Eq. (5.46) . In principle, all four equal time correlation functions r"^(0,t), ij for a = v,q and p = v,q, have to be evaluated. However, it is shown in the next section that Eq . (5.46) can be reexpressed entirely in terms of the velocity autocorrelation function in the rest frame, r^^(o i) ij *

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103 V .4 Equal Time Velocity Fluctua tions The equal time (t=0) velocity autocorrelation function is given by G^(0,t) = = / dx v^v f^(x,T) . (5.47) Equations for the equal time correlation functions in the rest frame are then obtained by projecting Eq . (5.30) for f,^ over the components of v and q. Again, the collision integrals can be evaluated in the case of Maxwell molecules (Appendix F) , and the resulting equations are •|^rJ](0,t) A.j^(-c)R^](0,T) A^i,(^)R^i(0,x) = 0, (5.48) ^fc+ ^K-(0'^) -'^jk^^^^^Q'^) + ^ik^](°''^) = ° ' (^-^^^ (-^ + V, + v„lR'''f(0,x) + a., R?'y(0,T) + a., R^'f(0,T) ^dx 1 2-^ ij lie Tcj ' ' jk kj T^2^ij^(0'^) = hi^^^ ' (5.50) ^PJ VV where R, , is the trace of the tensor R. . , and I.,(t) = -^ t..(T) + 6,.^ P(t) . (5.51) The correlation functions R.. and R.: enter Eq . (5.46) only in combination with the shear rate tensor. Making use of the properties (5.2-3) and of Eqs . (5.48) and (5.49) it is then shown that G^^(t,T) can

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104 be expressed entirely in terms of the single equal time correlation function R..(0,t), with the result, + /o ds[a.^^T.^(x-sHa.^T.^^(x-s)]R;^(0.s) + a. jT.^(t) 5. J /^^ ds e '^ ' ' A^^(x-s)r;;^(0 ,s) . (5.52) The problem is now reduced to the one of solving Eq . (5.50) . The inhomogeneous term I..(t) on the right hand side of Eq . (5.50) depends on the hydrodynamic state of the fluid through the pressure (or temperature, T(i;) = (nK^) p(t;)) and the irreversible stress tensor. D k For the case of Maxwell molecules a closed equation for t. . has been 1 Q* obtained elsewhere, with the result. (^+ 2v-)t. . + a., t., + a., t ., + nK„T(a. . + a.. ) ^5t 2^ ij ik jk jk ik B ij ji -!^ij\i\i = <^ • (^-53) Equations (5.50) and (5.53), together with the heating equation (5.9), are a closed set of equations that can be solved exactly for a *The equation for t.. is also immediately obtained by projecting the nonlinear Boltzmann equation for f^ over v. v. . '"^ ^ b 1 J **In Ref s . 19 and 20 the nonlinear viscosity v\ and the viscometric functions have been calculated as nonlinear functions of the shear rate. Also, the rate of increase of the temperature in the system has been evaluated .

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105 specified flow field. The solution is carried out in the second part of Appendix F. The results are presented in the next section. V .5 Results and Discussion The coupled set of Eqs . (5.9), (5.50) and (5.53) is solved in the second part of Appendix F for the following choice of the flow field, a. . = a6. 6. . (5 .54) The Laplace transforms of the temperature field and of the equal time correlation functions (see Eq . (F.21)) have three poles in the complex z-plane , one real and two complex conjugate, corresponding to excitations with different lifetimes . The real part of the complex poles is always negative, for either the temperature or the correlation functions . These poles are responsible for contributions that decay exponentially in a time of the order of v, or v„ (both v and v„ are of the order of the mean free time,t£), and should therefore be considered as initial transients in the system. Here only the properties of the system at times well separated from the initial time are considered (x » t ). Such terms are therefore neglected. The real pole of the temperature field is positive and corresponds to a contribution which grows exponentially in time at a rate that increases with the shear rate. Explicitly, in terms of the reduced quantities introduced in Appendix F, i.e. T*(t*) = T(v/t*)/T , (5.55) 1 o

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106 * * with t = t/v , the temperature field at t » 1 is given by T (t ) = H(x)exp[t 2v X (x/2)] , (5.56) * * * * where v = v„/v , x = a /v = a/v„, with a = a/v , and H(x) = JL±.pimii±^JA^ . (5.57) 1 + 4\ (x/2) + 3X (x/2) * The function \ (x) is defined in Eq . (F.37). It is a highly nonlinear and monotically increasing function of x. The temperature as a function of time is shown in Fig . 1 for several values of the shear rate . The expression (5 .56) applies with no restrictions on the size of the shear rate . In contrast, the real pole of the equal time correlation functions changes sign as a function of the shear rate . As discussed in Appendix F, for T >> 1 its contribution is important only at very large shear rate, i.e. x > 3.3, when the pole is positive and leads to a term which grows in time in the correlation functions , For shear rate such that X < 3 .3 this term decays to zero in a time of the order of the mean free time . The time dependence of the equal time correlation functions for k T » 1 is therefore the same as that of the temperature, given in Eq . (5.56), and can be scaled out by simply normalizing the correlations with the temperature at the time considered . These normalized correlation functions are then constants, independent of time. The form of these constants is indicated in eqs . (F. 44-45): they are highly nonlinear functions of the shear rate . At nonzero values of t, another effect entirely due to the presence of the shear in the system arises . The tensor q^ (t %) ^^ ^° longer ij

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is -k * Fig. 1 Reduced temperature T (t )/H(x) as a function of t = v t, for X = .5, 1.0, 1.5, 2.0 (Eq. (5.56)).

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108 4. 3. _ 2, _

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109 symmetric in i and j . The asymmetries depend on time and are complicated functions of the shear rate . Their amplitude increases with the shear rate. As an example, define, A (t,T) = G^^(t,T) G^(t,x) . (5.58) xy xy yx From Eq . (5.52), with the flow field chosen in Eq . (5.54), -v.t -v.(t:-s) A (t,T) = a(l e ^ ) r ds e R^(0,s) xy •'o yy -V t + ^ [1 (1 +v^t)e ]R^(0,T) . (5.59) As discussed before, for v t » 1 and x < 3.3 the time evolution of the equal time correlation function is entirely governed by the same growing exponential as controls the heating in the system, i.e. by the second term of Eq . (F .42) . By inserting this expression in the right hand side of Eq . (5.59), the time integration can be performed, with the result. —V t A * , . / N fo *., 1 . 1 + V \ (x/2) A (t.-t) = {2a (1 e ) ^ '— ^ 1 + 2v \ (x/2) a v,t e ^ }r'^(0,t) , (5.60) 1 J yy or, defining the corresponding reduced quantity, -1 * -1 J. J. A (v,t,v, T*) A* (t*) E "y ,\,, / , (5.61) "' ' ' " Vm the asymmetry in the xy components is given by

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110 A* (t*) = y (x.v*){2a*(l e'^^) ' ^ ^^^ j^/^) ""^ Vy ^ ,^ 2v\*(x/2) * * — t* , -ate I , (5.62) * where t = t/v. , and y is given in Eq . (F .46) . The expression (5.62) applies at all times t , with the restriction x » 1 and a < 3.3. The behavior of A /v as a function of t is shown in Fig. 2 for several xy yy ^ values of the shear rate .

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o

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112

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CHAPTER VI DISCUSSION A unified description of transport and fluctuations in systems macroscopically displaced from thermal equilibrium has been formulated . The general structure of the theory obtained has been identified in Section II .3 and only a brief summary is given here . The class of nonequilibrium states considered is quite general . It includes situations where the important variations occur on both kinetic and hydrodynamic length and time scales , for either stationary or nonstationary conditions . No limitations on the stability of the solution of the nonlinear macroscopic regression laws (hydrodynamic equations or Boltzmann equation) are imposed. In summary, it is shown that a complete theory of the dynamical behavior of a nonequilibrium fluid requires three coupled sets of equations: (1) equations for the average values, which are in general nonlinear, (2) equations for the correlation of the fluctuations around the average values at different times, (3) equations for the correlation of fluctuations at the same time. The generating functional method used here allows the derivation of these three sets of equations from one single equation and states a precise connection between the dynamics of the macroscopic observables and that of the correlation functions (see Eqs . (2.42-43)). The description of nonequilibrium fluctuations so obtained is macroscopic in character and applies under the same conditions that set the adequacy of the macroscopic regression laws. This, together with the generality of 113

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114 the nonequilibrium state considered, mostly differentiates the present formulation from others with similar objectives . Another important result is the linearity of the equations for the correlation functions . No restrictions are imposed in the present work on the size of the nonequilibrium fluctuations. The linearity of the equations is therefore an exact result , not the outcome of a linearization around the nonequilibrium state . The point of view adopted here is to describe the state of the system entirely in terms of macroscopic quantities, either observables or correlation functions . The information on the nonequilibrium state enters in two ways. First, they are introduced as boundary conditions on the macroscopic observables when solving the nonlinear regression laws: these are the quantities that are controlled in an experiment. Secondly, they are contained in the noise terra, which constitutes the inhomogeneous part of the equations for the equal time correlation functions (see Eq . (2.44)). The evaluation of this term requires a microscopic analysis of the transients present in the system at times shorter than those at which the macroscopic description applies . The details of the analysis depend on the time scale of interest, and no general prescription for the calculation of the noise has been identified . Here only equations for the correlation functions have been considered . No attempt has been made to construct the full probability distribution of fluctuations or a closed form of the nonequilibrium ensemble of the system. This is in agreement with the spirit of the present discussion, whose objective is to present a theoretical description that can be put in direct correlation with the experimental

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115 one. However, it would be very useful to derive a prescription to determine the noise spectrum in a nonequilibrium system, analogous to the relationship between the equal time fluctuations and the entropy in 27 a system in equilibrium. The work of Grabert , Graham and Green constitutes an attempt in this direction. However, their results rest entirely on the assumption that the macroscopic deterministic regression laws of the system correspond to a minimum of a properly defined timedependent action functional . The justification of such an assumption is not clear . A limitation of the present work regards the class of hydrodynamic nonequilibrium states considered. In fact, while the kinetic theory of fluctuations is quite general (it imposes no restrictions on the size of the displacement of the system from equilibrium), the hydrodynamic theory is limited to macroscopic states which are adequately described by the nonlinear Navier-Stokes equations . In other words the nonequilibrium state is one which is close to local equilibrium, in the sense that only terms to second order in the gradients of the thermodynamic variables are retained. The description of systems with large thermodynamic gradients is an open problem, even at the level of deriving the macroscopic regression laws . It requires in fact the evaluation of the constitutive equations for nonlinear transport, expressing the irreversible fluxes in a fluid as nonlinear functionals of the thermodynamic forces . It is straightforward to calculate the first corrections to the Navier-Stokes result ' (Burnett coefficients), but the solution of the general problem represents a very difficult task. An example of a model system where the evaluation of 23 the nonlinear irreversible fluxes is possible is the low density gas of Maxwell molecules studied in Chapter V.

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116 Another problem which has not been addressed hare is the determination of the boundary conditions for the correlation functions which are needed when solving the equations . Such boundary conditions could be formulated on a macroscopic scale and then checked at low density through kinetic theory calculations . Out of equilibrium the walls can play an important role since the system is prepared or maintained in the desired state through the application of external forces at the boundary. Finite size effects have for example been identified as responsible for a large part of the discrepancies between * 61 the theoretical calculations and the experimental measurements of the asymmetry in the Brillouin scattering from a fluid with an applied temperature gradient . The application of the present formalism to a concrete problem is in general a difficult computational task. It requires first of all the solution of the nonlinear equations describing the macroscopic state considered with specified initial and/or boundary conditions . Even this is possible only in a few cases . There are very few experiments to check the description of nonequilibrium fluctuations . Primarily these have been Brillouin scattering experiments from mechanically or thermally driven fluids, where the presence in the system of long range correlations that vanish in equilibrium has been identified . There is a need for proposing new situations where nonequilibrium fluctuations could be probed and that are still simple enough for a theoretical description. ^Several different methods have been applied to this problem, See Refs. 26, 28, 33, 38, 40, 43-46, 49, 50.

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117 Finally, as discussed in Chapter I, nonequilibrium fluctuations can be very important in hydrodynamic flows near an instability or a bifurcation point, since they are a possible mechanism to amplify the spontaneous thermal fluctuations and determine the transition of the 25 system to a new branch. Another class of related phenomena where fluctuations play an important role is constituted by nonequilibrium phase transitions . The three sets of hydrodynamic equations obtained in Chapter III are local in space, as a result of an expansion of the local equilibrium distribution function around a reference equilibrium ensemble (Eq . (A. 53)) and of the assumption that some local equilibrium correlation functions are short ranged. Consequently, they do not apply in this form for fluids near critical points or instabilities. However, the same procedure can be followed to derive a nonlocal form of the three sets of hydrodynamic equations , suitable to describe nonequilibrium effects in systems near a phase transition.

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APPENDIX A DERIVATION OF THE \-DE PENDENT NONLINEAR NAVIERSTOKES EQUATIONS The methods of nonlinear response theory are applied here to the first functional derivatives of the generating functional for hydrodynamic fluctuations to derive the set of generalized Navier-Stokes equations given in Section III.l, Eqs . (3.28). The derivation is essentially identical to the derivation of the nonlinear Navier-Stokes equations, except for the treatment of the initial conditions and the definition of the parameters in the local equilibrium ensemble . As described in Section II. 1, the system considered is prepared and maintained in a general nonequilibrium state through the action of external particle, energy and momentum reservoirs, whose thermodynamic parameters are controllable . The time evolution of the distribution function and of the dynamical variables of the system is then governed by Eqs. (2.2) and (2.9) respectively. The first functional derivatives of the generating functional G, defined by Eqs. (2.35) and (3.23), are then explicitly given by 4.^(?,t|x) = J^ / dr[T(t,o);^(?)] ^^ p^(o) = J/^^ V^>'^^'°) Tgrny^N^o) ' ^""'^ where the time evolution operator T( t , t ' ) has been defined in Eq.(2.11) and T (t,t') is the solution of the equation, 118

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119 (j^ + -^) T'(t,t') = , (A.2) J pi J. with -r— = -T— + C, and initial condition T (t,t') = 1. In other words, d t ot T (t,t') is the operator that governs the time evolution of the nonequilibrium distribution function; Pj,(0) is the initial nonequilibrium ensemble of the system. Equation (A.l) can be reinterpreted as the average of (\> (r) o.er a \-dependent nonequilibrium distribution functional, defined as 4. (?,t|\) = I / dr (i.(r)p(t|\) =<(!.(?) ;t> . (A.3) The time evolution of p (t|X) is governed by the same equation describing the time evolution of the distribution function, Pj^(t|\) = T^t,t')pj^(0|X) . (A.4) The \-dependence of the functional p (t|X) is entirely placed in the special initial condition to be used when solving Eq . (A.4), Pn(oI^) =W]yPN(o) • ^^-'^ The system considered is contained in a finite volume V. The external nonconservative forces are localized at the boundaries and short ranged. The objective here is to derive a set of hydrodynamic equations describing the behavior of the conserved densities in the interior of the system, where the effect of the external forces does not depend on the details of the interaction. The resulting description

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120 will therefore apply in a subvolume Q, differing from the volume V of the system by a boundary layer whose width is determined by the range of the external forces . Since the aim here is to obtain a hydrodynamic description, it is convenient to represent the solution, Eq.(A.4), in terms of the deviation from a local equilibrium functional, p (t|\), defined as Li Pj^(t|\) = exp{-qj^(t|X) /^ d? y^(?,t|\)4.^(?)} , (A. 6) where q (t|\) is the normalization constant determined by the condition I j dr p, (t|\) = 1 . (A.7) N=0 The parameters {y [X]} are defined by requiring the equality (3.26), rewritten here for completeness , indicates an average over the local equilibrium functional , = I / dr A(r)p,(t|X) . (A.9) ^^ N=0 ^ Therefore the {y [\]] in Eq . (A. 6) depend on the {X } through the set of functionals {(\) (r,t|A.)}. The form of the functional relationship , r a ,-> between the {y (r,t|X)} and the (([> (r,t|\)} is the same as the one defined in Eq . (3.13) for the thermodynamic variables, which is recovered by setting {\ = O} .

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121 The microscopic fluxes associated with the conserved densities 4; (r) are defined by ^V^) = -|F:^ai^^") ' (^-i^) where L is the Liouville operator for the isolated system. Since the \ dependence enters only through the initial condition, as indicated in Eq.(A.5), and through the thermodynamic variables, in the sense specified in Eqs . (A .6-8), the derivation of a formal solution (A. 4), suitable for obtaining the hydrodynamic equations, is essentially identical to the case of \=0 more usually considered . Therefore the X dependence will be suppressed in the following to simplify the notation and will be introduced only at the end of the derivation. Setting, Pj^(t) = pjt)e°^^^ (A. 11) in Eq . (A .4), an equation for D(t) is obtained, given by a? ^D(t) =-^ln pjt) --^, (A.12) ^Pn where ZT 1" Pl(^) = If ^l(^> + /q '^ ^ [y,ir,tH^(h] . (A.13) Making use of the requirement that p, (t) be normalized to one and letting a tilde over a phase function denote its deviation from the local equilibrium average ,

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122 A(r;t) = A(0 ^ , (A. 14) Eq. (A. 13) can be rewritten in the form IF 1" PJ^) = ^Q ^^^^^^ET^ ^K^'-''^ ^ y^(?.t)[L;^(r) + f . V^ I (?)]} . (A. 15) n pa " n The time dependence on the left hand side of Eq . (A. 14) will sometimes be left implicit to simplify the notation. Using Eqs . (A. 10) and integrating by parts this becomes ^ ay (?,t) ^ ^ By^(?,t) . , (A. 16) where S is the boundary surface of the volume Q considered . To proceed it is convenient to eliminate the time derivative of y in favor of its spatial gradients . This can be done by using the a definition of the fy } and the macroscopic conservation laws, obtained ' a by averaging the microscopic conservation laws . In the volume Q, the latter are given by dt or. ' 1 since no external sources are present inside Q. The macroscopic laws

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123 may then be written 1 1 * where use has been made of Eqs.(3.13): y • are the irreversible fluxes, ai ' defined as ^ai^^'"^) = ^ . (A.19) Since <6 ;t>^ and ^ are specified functionals of the (y } , these macroscopic laws can also be expressed as equations for the {y ] (see a' for example Ref . 18) with the result, ?s > ^ -i ^ ^ 5y (?',t) ^ -1 > ^ aylCr-.t) where Scxp^^'^'> '= <^a^^)^p(^')''>L ' 4p^^'^'> =L ' l,ih = /^ d?' g;^ (?.?')y?') , (A.21) and g „ indicates the (aS) element of the inverse matrix. The time dependence of g and K has been left implicit . Also surface terms have been neglected for points far from the boundary in the manipulations

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124 leading to Eq . (A. 20), due to the short range character of the correlation functions in Eqs . (A. 21). In Appendix D of Ref . 18 the explicit form of Eqs. (A. 20-21) for the case of a one-component fluid is also given. Substitution of Eq . (A. 20) into Eq . (A. 16) gives 1 1 /s/S.y^(?.t);^.(?) + /^ dr^ y^(?,t)t^^. ^^_l^(h . (A. 22) Pn with \i(?;t) = Y^i(r;t) / d?' 4^p(?';t) k^^(?' ,?) • (A.23) Because of the short range character of the forces F and the definition '=' n of the volume Q, the integrand of the last term on the right hand side of Eq . (A. 22) is zero over the region of integration. This term can therefore be neglected and Eq . (A. 12) becomes ay (r,t) 5y .(r,t) 1 X (A.24) where S(t) = V-> • ?^ + (^ dS.y (?,t)Y .(?) . (A. 25) p n ' b^ 1 a ai n '^ The boundary values of the thermodynamic variables y appearing in the second term on the right hand side of Eq.(A.25) are at this point arbitrary. They can be specified to satisfy S(t) = 0, or,

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125 \K = k^^h \^^'^)^^i(^) • (A.26) In this way the effect of the external forces is entirely incorporated through the boundary conditions on the thermodynamic variables, as desired for a macroscopic description of the nonequilibrium state . The same result can be obtained by following the argument proposed by McLennan in Ref . 6. In order for the Liouville equation to admit a solution representing equilibrium between the system and the reservoirs, the forces must satisfy the condition S=0 in equilibrium. It is therefore sufficient to require that this condition holds for the nonequilibrium state as well to recover the result given in Eq . (A. 26). This choice does not represent the most general situation, but is consistent with the experimentally interesting case in which the interaction is entirely specified through the y . Equation (A. 24) for D(t) becomes then , by (r,t) 5y .(r,t) 1 i whose solution is given by. t tt ^ &y„(r,i;) D(t) = T'(t,t^)D(t^) + Jl dx T^t.x) Vr{-^^^ ^cd^'^'^^ o i 9Y„i(r,-t) ^ ^ + ^_ C^C-^;^)} • (A.28) Here D(t^) represents the initial deviation from local equilibrium and will be important when deriving the generalized hydrodynamic equations for the functionals 4; (r,t|\) . The time evolution operator,

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126 T (t,t'), has been defined in Eq . (A. 2). By substituting Eq . (A. 28) into Eq . (A. 11), the formal solution of Liouville equation is obtained in the form, p^(t) = p^(t)exp{T^t,t^)D(t^) + jl dT T^t.-c) /^dr[-^^-^ $^.(?;t) o i i An important class of nonequilibrium states is constituted by nonequilibrium stationary states. A stationary solution is defined by pi t-t ->-oo o The limit may be implicitly incorporated by imposing the initial condition in the infinite past. The solution given in Eq . (A. 28) becomes then D(t) = lim T^t,t^)D(t^) + jl^ dT T^t.x) /^ d?{ ^"^J'' ^^,(?;x) O "* + — ^ y^;^)} • (A.31) i For initial equilibrium, the first terra on the right hand side of Eq . (A.31) vanishes. This choice of the initial condition restricts the applicability of the stationary solution to the case Ix =0| (the initial condition given in Eq . (A. 5) cannot be used). In fact when deriving equations describing fluctuations in nonequilibrium steady states by using the generating functional method, the condition of stationarity

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127 has to be imposed only after functionally differentiating the generalized (X-dependent) Navier-Stokes equations and evaluating the resulting equations at {x = O} . In order to lead to a stationary solution, the external forces must be asymptotically constant, or, F (t) = F for t finite, n n ' and, consequently, T (t,T) = T (t t) for t-T finite The stationary solution is then given by Pm = Pre , (A.32) with D = jO dx t\x) jdr {— ^ $ .(?) +_f— ^ (^)} . (A.33) ' -'° •' Q ' dr. ^ai 5r. a •' 1 1 A comment on the stationary solution may be made regarding the treatment dy ct of in Eq . (A. 16). In the derivation of the stationary solution of the Liouville equation presented here , the time derivative of y was first eliminated in terms of the space derivatives, and then it was assumed that the (y } are independent of t, at large t. A different °^ ay result would have been obtained if — — had been set equal to zero at the ot beginning, in Eq . (A. 16). The first procedure is the correct one, because the y in the solution are integrated over an infinite time a interval and in general, for finite t, y (r,T) y (r,-o°) is finite.

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128 The time dependence of the (y } may therefore be neglected only if the phase functions in the integrand vanish sufficiently rapidly as x -y -^ . This is not the case for the phase functions y .(r) and cj; (r) , appearing in Eq.(A.16). However the term T (t.x)^ .(r) may be expected to vanish rapidly as x -> -°° because the <^ . are orthogonal to the conserved densities fcL ] , <$^.(?)4;p(?');t>^ = . (A. 34) The term containing T (t,T) | (r) causes no problem, because it may be accounted for as a modification of the time development operator on ({) . to represent the dynamics in the subspace orthogonal to the {(j; } , as shown in Ref . 18 (see Eqs.(2.29) and (2.30) of Ref . 18). The above evaluation of the formal solution (A. 29) also applies for the special initial condition (A. 5). Furthermore, the definitions (A. 6) and (A. 8) imply that the \-dependence of p (t|\) occurs only through the initial condition and the hydrodynamic variables {y^(r,t|A.)} . Making this ^-dependence explicit, Eqs . (A. 11) and (A. 28) become P^Ctl^) = PL(t|\)e°^^l^^ , (A.35) and D(t|\) = T^(t,0)D^(\) + D'(ti\) , (A.36) with

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129 dy^(r,T|\) D'(t|\) = jl dx T^t.T) J^ d? {-^ ^^i(r;T|X) i + ^ 5^(r;TU)} , (A.37) i and D (\) = D(0|\) The initial time has been chosen at t =0 . The densities 5 (r;t|\) and the fluxes ^ .(r;t|\) are identical to the ones defined in Eqs . (A. 21) and (A. 23), when the local equilibrium average is substituted with an average over the functional p (t|\). The irreversible fluxes Li * -» I Y .(r,t|X) are given by Y*i(?.t|M = ^ ^L . (A.39) When evaluated at X.=0, Eq . (A. 35) reduces to Eq . (A. 29). In this case the initial condition term D (X=0) represents the deviation of the initial state from local equilibrium, and is usually neglected when deriving hydrodynamics, at \=0 , because it is expected to decay on a time scale of the order of a mean free time for states leading to hydrodynamic behavior . This deviation is only responsible for the initial transients in the system, occurring before hydrodynamics applies . This is not the case for the X. -dependent functional, D (X). However the same restriction can still be imposed on the class of initial states considered, by choosing

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130 9^(0) = pjO) . (A.40) The initial condition D (\), becomes then and vanishes identically when evaluated at X=0 . The generalized (^-dependent ) hydrodynaraic equations are obtained by averaging the microscopic conservation laws, £qs • (A. 17), over the nonequilibrium ensemble given by Eqs . (A. 35-38), i.e. 1 1 and then closing Eqs. (A. 42) with constitutive equations for the irreversible fluxes . The local equilibrium averages on the left hand side of Eq . (A .42) can be evaluated explicitly. From the definition of the local equilibrium functional, p (t|\), it follows that the functional relationship between the \-dependent fluxes, , and the ^.-dependent cci A.L densities, <'i'j^(r) ; t> , is the same as in the case X=0 . In particular for a simple fluid the local equilibrium averages are found to be {^} = (p[X], elX], p[\]v.[\]) (A.43) {^^} = (p[\]v.[/.],(h[\] +1 p[\]v2[\])v.[\], p[\]6. + p[\]v.[\]v.[\]) . (A.44)

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131 The r and t dependence of the functionals has been omitted to simplify the notation. When evaluated at \=0, the functionals on the right hand side of Eqs . (A .43-44) are identified with the thermodynamic variables: p, e and h are the mass, energy and enthalpy density, respectively, v is the macroscopic flow velocity and p is the pressure . The functional relationship between the A.-dependent thermodynamic variables {y (r,t|X)}, defined in Eq.(A.8), and the \-dependent densities, {4; (r,t|X)}, is the same as in equilibrium. This is merely a convenient definition and does not imply any restriction on the nonequilibrium state considered. The second term on the left hand side of Eq . (A. 42) is then written lF:
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132 and When Eq.(A.46) is evaluated at \=0, the additional contributions v . ai vanish identically, since D(t|0) = D'(t|0), and the usual formal expression for the irreversible fluxes in a nonequilibrium fluid is recovered , Y*.(?.t) = ^ (A.49) where D'(t) = D'(t|0) (see for example Ref . 17, Eq . (54)). The additional contributions are entirely due to the \-dependence of the initial condition of the nonequilibrium distribution functional and have to be kept when deriving equations for the \-dependent densities, {ci;^(?,t|\)} . For simplicity, only the case of a nonequilibrium system whose macroscopic state is completely described in terms of the nonlinear Navier-Stokes equations is considered here. The two terms in Eq . (A .46) need then to be evaluated only to second order in the gradients of the hydrodynamic variables. However, the additional fluxes v can be 'ai evaluated to Navier-Stokes order only after performing the functional differentiation and evaluating the result at \=0 . The Navier-Stokes part of the firs term on the right hand side of Eq . (A. 46) is evaluated here. It appears from Eq . (A. 37) that D'(t|X) is already at least of first order in the gradients. Therefore, to second order in the

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133 gradients , ^ai^^'^I^^H = /> / d?'<[T(t,x);^.(?)r*e,(r'U);tXL-^ ayo(r',T) J (A .50) where the operator T(t,T;), adjoint of T (t.x) has been defined in Eq . (2.11). Before proceeding further, it is useful to point out that the dynamics of average quantities inside the volume Q should be independent on the external short ranged forces . For the purpose of evaluating properties away from the boundaries, it can therefore be assumed that the time evolution of a dynamical variable A is described by the Liouville operator for the isolated system. A(r,t) = e^*^ A(r) . (A.51) The adjoint of the evolution operator e , in the sense defined in Eq . (A.l), is simply (e '') = e~ . Equation (A. 50) can then be written, 4(?.c|M„ . /^d,/„d?. !Vi;^ <[e«-'';^,(?)]L.(f. |W;0 J M' " '». (A .52) The irreversible fluxes v .(r,t A.),, are then evaluated to Navier-Stokes 'ai ' H order by expanding the thermodynamic variables , y , in a Taylor series around the reference point (r,t). To first order in the gradients, the local equilibrium distribution functional can be written ay (r,t|\) Pl^*^!^^ = Peq(^'t|^)(l ^5?: V^' ^^i-'^i^'^a^^' ' ^^ ^ ' ^^'"^

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134 where and H are the total (volume integrated) densities, \ = k ^^ '^a^^) • (A. 55) The constant q (r,t|\) takes account of the normalization to unity of the distribution functional. When evaluated at \=0 the distribution function defined in Eq . (A.5A) reduces to a reference equilibrium ensemble with local thermodynamic parameters. To Navier-Stokes order, only the first term in the expansion (A. 53) has to be kept, with the result V^^'^I^^H = ''C^' '^ /o '^ 0.' (^•^^> where %[^] = /q d? *pj(?^). (A .57) Here *^***>q, indicates an average over the equilibrium ensemble defined in Eq. (A .54) ,
ox = I JdrAp (?,t|\) . (A.58) N=0 ' The r and t dependence of the averaged quantity through the thermodynamic variables has been left implicit to simplify the notation. Finally, hydrodynamics describes variations over time scales large

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135 compared to the mean free time t = X/v (where I is the mean free path and Vq the thermal velocity of the particles). In this limit, Eq . (A .46) can be written as Y*i(r.t|X) = Y**(r,t|x)^3 + L^^^^'"' ^^^ ^^ ^ ^ If" y a^^ ^^^ ' (^-59) J where L -^ is the matrix of the transport coefficients, defined as L^J(?,t;{4.[X]}) = lim ll dx <[e^^''"^) V^^^ ]^6i ^ ^^>0X' ^^'^^^ t»t f ** -> and the additional fluxes y .(r,t \).,„ are defined as 'ai ' NS 1 t»t^ 1 Here [A]j^g indicates the result of an expansion of the function A to second order in the gradients of the thermodynamic variables . The generalized Navier-Stok.es equations for the set of functionals [>\i (r,t|\)} are then given by |^4^^(?,t|X) +|_Ej(?,t;{4.[M}) i + -S— L o(r,t;{4;[\]})^ y„(r,t \) = -^y -(r.t X)^,„. dr. aB > >-t-'. j • ' 5^ ' g ' 5r. ai NS 1 j 1 (A.62) Both the Euler matrix and the matrix of transport coefficients in Eq.(A.62) are functions of r and t and functionals of the (x } only through the average X-dependent densities, {4; (r,t|X)}. The explicit

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136 form of Eqs . (A. 62) for a one-component fluid is given in Section III .1 . When evaluated at X=0, the right hand side of Eqs. (A. 62) vanishes identically, since y .(r,t|0) = 0, and the Navier-Stok.es equations for the conserved densities are recovered . ** -y The additional contributions y .(r,t \) will be evaluated to Navier-Stokes order only after performing the functional differentiation and evaluating the result at X=0 . A detailed analysis of these terms is given in Appendix B, where they are shown to determine the amplitude of the noise in the system.

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APPENDIX B EVALUATION OF THE SOURCE TERM FOR HYDRODYNAMIC FLUCTUATIONS The \-dependent densities, {(jj (r,t|\)}, defined in Eq . (3.24), have been shown to satisfy the set of nonlinear hydrodynamic equations (3.28). These equations differ from the usual nonlinear Navier-Stokes equations only by the presence of additional contributions to the irreversible fluxes , given by the Navier-Stokes limit of .>V,X)=^^. (B.„ Where D(t|>^) and D'(t|\) are defined in Eqs . (A .36-38). These terms vanish at \=0, when the nonlinear Navier-Stokes equations are recovered. However, their functional derivative contributes to the equations for the correlation functions through the inhomogeneous term I (see Eq . ccp (3.44) and Eq . (3.64-65)), given by 6y .(r, ,t, |\) I t^»t^ li ^^0(^2' 2^ ' As indicated in Sections III .2 and III .3, the right hand side of Eq . (B .2) has to be evaluated to Navier-Stokes order for consistency with the approximations introduced in the left hand side of Eqs. (3.44) and (3.64). The objective here is to evaluate I „ and to show that it only gives a nonvanishing contribution to the equation for the equal time correlation functions, on the time scale of interest.

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138 By carrying out the functional differentiation and making use of the fact that D(t|0) = D'(t|0) = D'(t), 6Y**(r .t |\) I ^ D'(t ) 6^U) 6\ (r t ) |\=0 6\ (r t ) ^ ^ ^ P ^ 2 (B.3) The nonequilibrium distribution function at time t, is given by Eqs (A. 4) and (A. 28) for initial local equilibrium, or p^(t^) = T^t^,0)pj^(0) D'(t ) = Pl('^i)^ • (B.4) Making use of the properties of T ' , which acts as a point operator,' i .e . ;T^(t,T)A][T^t,T)B] = T^t,T)A B , (B.5) and of Eq. (A. 38) for D (\), I ^ can be written as o ap -<[x][ ° ^^^,p:(0U) ].^o^°>."^ (B.6) where A,-L *This property follows from the fact that the exponential of T^ is a linear combination of differential operators .

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139 Also, in writing Eq . (B .6) , the definition of T(t,T) as the adjoint of T (t,-u), as given in Eq . (A.l), has been used. The functional differentiation in Eq . (B.6) can be carried out explicitly. Using the fact that the local equilibrium functional pj^(t|\) depends on \ only through the densities {(\i (r,t|\)}, the result is J dr |^(r;0)^ , (B.8) ~ . -> where the normalized densities i (r) are defined in Eq . (A. 21). It is a convenient to define the time dependent local equilibrium projection operator onto the set (l,4; }, given by P^A = 1 . ^ + / d? C^(?;t) 4^(r;t)A;t>^ . (B .9) This definition agrees with the one used, for example, by Kawasaki and Gunton in Ref . 2. Inserting Eq . (B.8) into Eq . (B.7) and using the definition of P^., Eq . (B.6) becomes SY**(?,,t |X) I -^ ^ ^ ^ 777—-— I ^ = <[T(S'0>V^^l)]QoV-2'^2) = 0>L • (^-^0) where Q = 1 P . t t As done in Appendix A, it can now be argued that the dynamics of average quantities at points in the interior of the system is independent from the boundary forces. The evolution operator T(t|,0)

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140 can then be replaced with the corresponding evolution operator for the isolated system (see Eq , (A. 51)), with the result, ^V^^r^l^) I . Lt SX (r^,t^) |\=0 = ^t^ ^ V(^)^Vb^^2''2^'°>L • ^^-^^^ The time evolution of the dynamical variable 4; (r,t) is governed by the P equation , 1^ ip(?,t) Li^(?,t) = . (B.12) By applying the projection operator on the left of Eq . (B.12), an inhomogeneous equation for Q ((; (r,t) is obtained, o fj {|f%^}%Y^'^^ = Qo^Vp^^"'^) ' (^-^3) whose solution is ^ t Q L(t-x) . Vp^'''^^ = Jo d^ e ^o^^o'^'p^'''''^ ' ^^'^^^ since Q cj; (r,0) = by definition. Substituting into Eq . (B.ll), I o p ap can be written as V^l''l''^''2) = ^^^ t^F/o' ^^ / ^^' xe° 2 ^j(r-;0);0>,|^ P ; M ,s> J 6<6 (r');0> ° ^ (B.15) where

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141 ^0^.(?;O) H Q^y^.(?) = y Ar;0) J dP % (P ',0). , (B.16) UJ (X CC Oj Li i .e . (^ . is the flux already defined in Eq . (A. 23), now projected at t=0. No approximations have been introduced at this point . To evaluate I , the right hand side of Eq . (B.15) now has to be expanded to second order in the gradients . The correlation function on the right hand side of Eq . (B.15) is peaked at t = t -t : the functional derivative in the integrand can then be evaluated at this value of the time argument . If the difference t,-t2 is large compared to a mean free path, t^, the functional derivative is identified with the Green's function of the hydrodynamic equations, defined by 6 = / dP G"p(?,t;?',0)6 , (B.17) with the result (B.18) The Green's function G can be expanded in a Taylor series in the gradients around the point r' = r . To lowest order Eq . (B.18) becomes

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142 ^|F-^pa^^2'^-^2'^l'°) • (^-15) The identity e =e -[dxe^ ^PLe (B.20) has been used to show that, to lowest order in the gradients, the projected evolution operator on the right hand side of Eq . (B.18) can be substituted with the usual evolution operator. Also, the local equilibrium ensemble in Eq . (B.18) has been expanded around the reference equilibrium ensemble p^ (r ,t ), defined by Eq . (A. 54), at X=0 , as required to Navier-Stokes order. The correlation function on the right hand side of Eq . (B.19) is the correlation function entering the expression for the transport coefficients, Eq . (3.30). Its lifetime is of the order of the mean free 2 time, tf . The inhomogeneous term 1 „ therefore vanishes if ^ » tf tjt^ » t^, (B.21) or, explicitly. t^»t^ ap i i z z S-^2»^f

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143 and does not contribute to the equations for the unequal time correlation functions . The contribution from the inhomogeneity I to the equations for (Xp the equal time correlation functions is given by where e^O^ ap z ^^^^^ or^^ o ai xe° , (?.;0);0>,|^ J J \^ . (B.24) The correlation function on the right hand side of Eq . (B.24) is now peaked at t=0 . The functional derivative can then be evaluated at -c=0 and reduces to a spatial 6-f unction, with the result, 2 r -(?, ,?„;t) = (1+P,„P ^)[^-\ j^ dT<{e^^Y •(?,)} aB 12 12 aB '-5r,.Qr„. -^ o • ' ca I ' li 2j *p.(r2;0);0>J^^ . (B.25) To proceed, it should be observed that, in the solution of the hydrodynamic equations for the correlation functions, F „ will always appear in the combination , ^^['^^2 0^'^'l'^'^2'^2''-^)rap(^^I'^2'^) ' ^'-2^)

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144 where K is the two-point hydrodynamic Green's function for the equation considered. By integrating by parts twice, the double derivative in Eq . (B.25) can be moved over the Green's function. The latter can be expanded in a Taylor series in the gradients around the points r^ and r^ . To Navier-Stokes order the result is [^p^ K«J U„ (l+P^,P„p) Si d.<[e'-f„J$p.>^ , (B.27) where, to the desired order in the gradients, the local equilibrium average has been substituted with an average over the reference equilibrium ensemble p^^(?^,t). Also, Eq . (B.20) has been used to show that the difference between the projected and the usual evolution operator generates terms of higher order in the gradients. Finally, in Eq. (B.27) r^^ is the volume integrated flux. A convenient form °^ ^ap' "'^'^'^ -^^""^^ ^° ^^^ result indicated in Eq . (B.27), is then given by r,p(r^,r^;t) = lim^( 1.P^^P^^)__£__,( >^^ (B.28) or. ^2 r ,p(r,,r2;t) = (1+P^2^^p) ^p^ 6(?^-?pLj^J(?^,t;{^}), (B.29) where L^^ is the matrix of the transport coefficients, given by Eq . (3.30) for A.=0.

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APPENDIX C DERIVATION OF THE \-DEPENDENT BOLTZMANN EQUATION The objective here is to evaluate the functional H(x ,x ,t|\), defined in Eq.(4.12), to lowest order in the density, deriving a closed kinetic equation for the one-particle reduced distribution functional f^(x,t|\). From the definition of the reduced distribution functionals, Eq.(4.10), it appears that the ^-dependence can be incorporated entirely in the initial condition, while the time evolution operator is the same as that appearing in the definition of the usual reduced distribution functions . A nonequilibrium cluster expansion, analogous to the one used by Cohen in deriving the nonlinear Boltzraann equation, can then be performed to express f„[X] as a power series in the density in terms of f [\] . This is done by eliminating the test function X between the two functionals . To simplify the notation, the s-particle functional f^[X] is written as 2 I V N! 1 II n f (X,,...,X ,t X) = ) -y-ri r-r... ^ I dx , , . . .dx, J ,,(x , , • . • ,X .. , t \ ) , s 1 ' s ' ^.t". (N-s)! N!Q ^ s+1 N N 1 N ' (C.l) for t > t , where o F^(x^,...,x^,t|X)= S_Jx^,...,Xj^)U^[X]pj^(0) , (C.2) 145

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146 and pjj(O) is the initial distribution function for the system of N particles, without normalization in the grand canonical ensemble. The normalization constant Q is defined by requiring. ^ N!Q ^ '^''i**-'^''n^N^''i'--*'''n''''^^ " ^^^'^^ The reduced distribution functionals are normalized as follows, / dx^...dXgfg(x^, ...,Xg,t|\) = V^, (C.4) where V is the volume of the system. In order to define the cluster expansion, the set of functions F (x , . . .,x, ,t |\) , for k = 1,2,...,N, is chosen as the set of basic functions in terms of which the distribution functions of the system will be expressed. The dynamics of F, is the dynamics of a cluster of k isolated particles . To derive an expansion for the oneand two-particle distribution functions, two sets of cluster functions, xC^, |x„ , • • . ,x, |X) and X(x ,x„ |x„ , . . . ,x, I \) have to be introduced (the time dependence of the cluster functions has been left implicit) . They are defined by F^(x^, ...,Xj^,t |\) = x(x JX)Fj^_^(x2, ...,Xj^,t |\) k + I X(x^ |x^ |\)F^_2(X2, •..,x^_^ ,x^_^^ , ...,Xj^,t|\) k " .IX X(x^|x.,x |\)F (X ,...,x. x ...,x X ^^,...,x^;t|M + ... , (C.5)

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147 and F^(x^ , ...,x^,t|\) = xCXj^jX^ |\)F|^_2(x^, ...,Xj^,t|\) i=3 T«»»» (C«6) The functions x can be expressed in terms of the F by writing explicitly Eqs.(C.5-6) for s=l,2,...,N and then inverting them, with the result. X(xJ\) = F^(x^,t|X) , xCxJx^lX) = F^Cx^.x^.tl?.) F^(x^,t|X)F^(x2,t|X) , etc. , (C.7) and xCx^.x^lX) = F2(x^,X2,t |X) , X(x^,X2|x2|X) = F^(x^,X2,x^,t |X) F^Cx ^ .x^.t | X)F ^(x2,t | X) , etc. . (C.8) It appears from Eqs.(C.7-8) that the functions x have cluster properties, i.e. vanish when the F factorize to a product of lower order functions . They are associated with the motion of isolated groups of molecules in phase space . The cluster expansion provides a means for

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148 systematically treating the distribution functions of a many-particle system in terms of those of small groups of isolated particles . As the density of the system considered increases, correlations among a larger number of particles will become important. Substitution of Eqs.(C.5) and (C.6), for k=N, in Eq.(C.l) for s=l and s=2, respectively, gives n f^(x^,t|\) = x(xJ/\) + I -(jr[yr / dx2...dx^x(xjx2,...,x^|\). (C.9) and 2 n f^(x^,x^,t\X) = xCx^.x^K) "*" ^ (A-2)! ^ dx^...dx^x(x^.X2^^3""'^Al^^ * (C.IO) In this way the oneand two-particle functionals have been written as expansions in terms of clusters of an increasing number of particles. This is therefore an expansion in the density. The cluster functions x depend on the test function X through the initial condition for the functions F, . The functional relationship between X and f [X] is given by Eq.(C.l) for s=l or, equivalently, by Eq.(C.9). The latter expression can be used to invert the relationship in the limit of low density. Eq.(C.9) has to be solved for X in terms of f [X] and this relationship is then used in Eq.(C.lO) to eliminate X in favor of f [X] . However, since the distribution functions always depend on X only through the operator U^^fX], it will be sufficient to invert Eq.(C.9) in terms of such an operator .

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149 To lowest order in the density only clusters involving the smallest number of particles contribute. In this limit the desired functional relationship is given by n f^(x^,t|\) = S_j.(xpu°[xJ\]n f^(x^,0) , (C.ll) where a subscript or superscript zero indicates, here and in the rest of this Appendix, the low density limit. U [x ,...,x \X] is an s-particle version of the operator U [\] , defined in Eq.(4.4), and it is given by Ugfx^ , ...,Xg IX] = exp /_^dt / dx \(x,t)S^(x^ , . . ,x^ ) I 6(x-x^), i=l (C.12) where the test function X has the form chosen in Eq.(4.8). The low density limit U is formally given by s U°[x^,...,x^|X] = U^[x^,...,x^|X^] , (C.13) where X denotes the first term in the expansion. \[n.f^(x^,t|\)] = X[0,f^] + 0(n) . (C.14) Use of Eq.(C.ll) in Eq.(C.lO) gives the low density expression for the functional H in terms of f,[X], H (x ,x ,t|X) = S5 (x ,x )f (x ,t|X)f, (x ,t|X) + R(x ,x ,t|X), ol^ tl211 12 Iz (C.15) where S (x ,x ) is the two-particle streaming operator defined in

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150 Eq.(4.14) and U°[xJ\]U°[x2|\]f^(x^,0)f^(x2,0)} . (C.16) The first term on the right hand side of Eq.(C.15) has the form desired to obtain the Boltzmann expression for the rate of change of the oneparticle distribution function due to collisions . The second term is a highly nonlinear funcional of f,[X] through the solution, \ , of Eq.(C.ll) and depends on the initial correlations in the system. To analyze this term, the case corresponding to X=0 is first considered. By setting \=0 in Eq.(C.16), R becomes R(x^,X2,t|0) = S_^ix^,x^){f^{x^,x^,0) f ^(x^,0)f ^(X2,0)}. (C.17) This term has to be inserted in the integral on the right hand side of the first equation of the BBGKY hierarchy, Eq.(4.11), where it is premultiplifed by 9(x ,x ), which restricts the range of integration to interparticle distances smaller than or of the order of the force range. For purely repulsive, short ranged potentials the operator S_ (x,,X2) brings two particles initially closed together far apart (i .e . to relative distances larger than the force range) in a time of the order of 1 . Therefore the contribution from R(x,,X2,t|0) vanishes in a time of the order of t for all the initial nonequilibrium states where the c ^ s-particle distribution function factorizes for interparticle separations large compared to the force range. This is the restriction

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151 usually imposed on the class of initial states when deriving the Boltzmann equation. Explicitly, for initial conditions leading to a state that is adequately described by the Boltzmann equation, the vanishing of the contribution from initial correlations is expressed by lim / dx^ e(x^,X2)R(x ,x ,t|0) = . (C.18) t>>T; c For nonzero values of \, however, R is a complicated functional of the initial condition and will contribute to the kinetic equation for f [\] . It is convenient to rewrite Eq.(C.16) as the sum of two terms, R(x^,X2,t|\) = S_^(x^,x^)U°^[x^,x^\X]{t.^i^^,K^,0) f ^ (x^ ,0)f ^ (x^ ,0) } + S_^(x^,X2){U°[x^,X2l>.]U°"^[xJ\]U°"^X2|M l} X U°[xJ\]U°[x2U]f^(x^,0)f^(x2,0) . (C.19) By applying the group properties of the streaming operator, the first term on the right hand side of Eq.(C.19) can be written in the form, {S_(.(x^,X2)U°[x^,X2|\]}S_^(x^,X2)[f2(x^,X2,0) f ^ (x^ ,0)f ^ (x^ ,0) ] . (C.20) When substituted in the first equation of the hierarchy this term gives no contribution in the limit t >> x for the same class of initial c states leading to Eq.(C.18). The long time contribution of R(x ,x„,t|\) is then given by

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152 lim R(x ,x ,t|\) = lim S_ (x ,x„){U2[Xj^,X2|X]U°" [xJ\]U°~ [x^l?^]-!} t»T t»-c ^ c c X U°[x^|\]U°[x2|X]f^(x^,0)f^(x2,0) . (C.21) Using the properties of the streaming operators and Eqs . (C. 12-13) for the low density Ug operators, Eq . (C.21) can be rewritten as t lim R(x^,X2,t|\) = lira (exp(/_° dx [^t-T^^r^2'' "^t;^^1'^2^] t»x t»x c c X [\(x^,T;t) +K(x2,T;t)]} l) X S_j.(x^,X2)U°[xJ\]U°[x2i\]f ^(x^,0)f^(x2,0), (C.22) where the choice (4.8) of the test function has been inserted explicitly (t is a parameter to be chosen when functionally differentiating such that t > t ) and o \(x.,T;t) E S ^(x.)\'(x. ,t) . (C.23) 1 X-t 1 1 It has previously been pointed out (Eq . (4.16)) that the streaming operator S (x ,x ) reaches a time-independent asymptotic limit for t>T . Therefore the argument of the exponential in Eq . (C.22) vanishes if the parameter t^ is chosen to satisfy t-t >t; . As seen in o o c Section IV .3 this choice is appropriate when deriving equations for unequal time correlation functions. For consistency with the Boltzmann limit considered here, the appropriate multitime correlation functions are in fact those whose time arguments are separated by at least a

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153 collision time. Therefore, it already appears from the present analysis that the inhomogeneous term R(x ,x ,t|X) will give no contribution to the equations for the unequal time correlation functions . The same result is restated in Section IV .3 by showing explicitly that the functional derivative of R generates terms that vanish in this case . However, when deriving equations for the equal time correlation function, the parameter t^ has to be chosen arbitrarily close to t, and the difference between the two streaming operators in Eq . (C.22) does not vanish. The asymptotic form of Eq . (C.22) can only be evaluated after functionally differentiating and setting \=0 . In Section IV .4 this term is shown to contribute to the equations for the equal time correlation functions. In general, it has to be kept when deriving a kinetic equation for f.[\] to be used to generate equations for the correlation functions . Finally, substituting these results into Eq.(4.12), an equation describing the time evolution of the functional f (x ,t|x.) for timevariations large compared to x and t > t is obtained. |A_ ^+LQ(x^)}f^(x^,t|x) = n / dx^Q(K^,x^) 5Jx^,Kpf^(x^,t\X)f^(x^,t\X) + W(x^,t|f^[\]) , (C.24) with

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154 W(x^,t|f^[\]) = lim n/dx2e(x^,X2)S_^(x^.X2){(U°[x^,X2|\]U° ^ [x J\]U°"^ [x^ | \]-l) c X U°[xJ\]U°[x2|\]f^(x^,0)f^(x2,0)} . (C.25) Again, the dependence of Eqs . (C .24-25) on t^ has been left implicit. Equation (C.25) can also be rewritten in a form that will be useful in the following, given by W(x^,t|f^[\]) = lim n/dx2e(x^,X2)S_^(x^,X2){u°[x^,X2|MU° ^ [x J MU°"^ [x^ | \] l} c X S^(x^)S^(x2)f^(x^,t|\)f^(x2,t|M . (C,26)

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APPENDIX D SCALING METHOD FOR A LOW DENSITY CLOSURE OF THE HARD SPHERES BBGKY HIERARCHY Introduction An alternative approach to the study of the kinetic theory of fluctuations in low density gases is based on the derivation of hierarchies of equations for equal time and multitime reduced distribution functions . These equations are obtained from the Liouville equation and are formally identical to the equations of the BBGKY hierarchy. The availability of the density as an expansion parameter provides then the possibility of performing controlled approximations to obtain closed equations for the correlation functions . Several authors have adopted this approach to formulate a description of nonequilibrium phase space fluctuations . In particular 37 Ernst and Cohen have treated the case of a gas of hard spheres by using the method of the nonequilibrium cluster expansion to derive kinetic equations for multitime and equal time correlation functions . 3 This amounts to performing an expansion in pov^ers of no (where n is the number density and a is the force range), selecting, at each order, only collisions involving a limited number of particles . The lower order terms in a cluster expansion describe correlations over distances of the order of the force range, a. This constitutes an adequate approximation for the two-particle distribution function when closing the first equation of the hierarchy, since the presence of the force on the right hand side of the equation restricts the integration to interparticle 155

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156 distances smaller than the force range. The resulting kinetic equation describes the behavior of the one-particle distribution function over space and time scales as small as a and the collision time t , respecc tively . However the higher order distribution functions are expected to present appreciable variations also over scales of the order of the mean free path 1 and the mean free time t^ between collisions, in a low density gas much larger than a and t . In other words additional c correlations among the particles in the gas are important after the particles have undergone an appreciable number of collisions . The mean free path and the mean free time are the two characteristic scales in the problem. The kinetic equations for higher order distribution functions of interest here describe variations over lengths larger than or of the order of Z and t^ . In particular, the BBGKY hierarchy for a gas of hard spheres is considered here. The choice of the hard sphere potential simplifies the treatment, because the collision time is zero. At the end of this Appendix it will be indicated how the results for a continuous potential can be recovered by identifying the appropriate collision operator. When the s-th equation of the hard spheres hierarchy is scaled according to the two characteristic lengths of the problem, Z and t^ , the natural expansion parameter in the equation is identified with 3 -1 a = (nl ) . The condition a <<" 1 represents a gas at low density in the usual sense (i.e. with no « 1, since a = (na^)^), but with a large *The collision time x^ is defined as x =a/v , where v^ is a the thermal speed. For continuous -potential it represents the duration of a collision, or interaction time. In the case of the hard sphere potential this identification is not possible, since the interaction time is zero, but the definition of x used here is still meaningful.

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157 number of particles in a volume of the linear size of a mean free path . Only when the latter condition is satisfied, will the collisions have the effect of correlating different particles over distances of the order of Jl. By expanding in powers of a the s-th equation of the hierarchy, a systematic approximation for the s-particle distribution function in terms of one-, two-, ..., etc. particle distribution functions is obtained . By imposing the requirement that this approximated form constitutes, to each order in a, a solution to the entire hierarchy, kinetic equations for the lower order correlation functions are obtained . In particular kinetic equations for the second order two-time and equal time correlation functions are explicitly derived here and are found to be in agreement with those obtained in Ref . 37 by the cluster expansion method. However, the present formulation clarifies the approximations introduced and provides a systematic prescription to derive kinetic equations for higher order correlations . Finally, the results also agree with those derived in Chapter IV by applying the generating functional method . The advantage of the latter method is evident, since there all the desired information can be derived by a single kinetic equation. BBGKY Hierarchy for Hard Spheres The dynamics of a system of particles interacting via a discontinuous potential is difficult to describe, e.g. in terms of Newton's equations of motion, since the force is infinite at the points of discontinuity of the potential . To circumvent this problem a formulation

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158 of the dynamics which avoids explicit dependence on the force is required . Let A(x ,...,x ) be an arbitrary N-particle phase function. The time development operator is defined by A ^ A(x^,...,Xj^,±t) E A(x^(±t) , ...,Xj^(±t)) e A(x^, .. .,Xj^) , (D.l) where t > 0. The Liouville operator L is the generator of time translations. For continuous potentials it is given by Eqs . (4.1-3), rewritten here for convenience, L = i L (x ) I e(x ,x ) , (D.2) i=l ° i
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159 ( V^ for r . . < a I V^ij'V M I^^'^^l-^ij^ for a-|.r.. < a + | (D.5) for r^. > a + ^ From Eq. (D.5) it follows that where the limits have to be taken in the order indicated . In the following, to simplify the notation, the dependence on the parameter V in the potential defined in Eq . (D.5) will be omitted. The generator of time translations for discontinuous potentials, L, (the + and sign refers respectively to forward and backward streaming), may be defined from Eq . (D.l) as A 7\ ^ L^A(x^, ...,Xj^) = ± lim_|_ lim — A(x^ , . . .,Xj^,±t ) t->-0 eO V >«> o xt L(,x ,...,Xj,)<^ = lim_^ lim L(x , . . .,Xj^)e A(x , . . .,Xj^) , t>0 eX) V HK= (D.7) o or, formally, L = lim lim L(£)e**^^*-^^ . (D.8) t->0 e^O V ^oo o The e-dependence enters in the Liouville operator through the potential defined in Eq . (D.5) and has been indicated explicitly in Eq . (D.8).

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160 The limits in Eq . (D.8) have to be taken in the order indicated, i.e. the exponential cannot be simply set equal to one because the e^O limit generates singularities in L(e). The time evolution of a dynamical variable is then given by A I A A(x^ , ...,x^,±t) = S_^^(x^ , ...,Xj^)A(x^ , ...,x^) , (D.9) where the N-particle streaming operator S has been defined as S^^(x^,...,x,p = exp{±tL_j_(x^,..,Xj^)} . (D.IO) The scalar product of two N-particle phase variables A and B is defined as their equilibrium average, A A
^= / dr p^(r)A(r)B(r) , (d.ii) where p (r) is the N-particle equilibrium distribution function. The adjoint, K , of an operator K is defined by / dr p^(r)B(r)[K A(r)] = / dr[KVQ(r)B(r)]A(r) . (d.i2) In particular, setting l]^ = L_ , (D.13) the adjoints of the streaming operators are

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,+ 1 1 -r^ [\ ] = s _^. l&l [s:j^ = s^ . (D.14) where S_^^(x^,...,Xj^) = exp{±t L_j_(x^,...,x^)} . (D.15) The barred streaming operators are explicitly defined by / dr p^(r)B(r) [s*^. md] = / dr[s!^ p^(r)B(r)]A(r) . (d.i6) They translate in time a phase space point according to S_^ X. = X. (±t) . +t 1 1 ' The definition of the adjoint given in Eq . (D.12) is the convenient one for the purpose of the present calculation . However it implies that an operator and its adjoint have different domains and it requires special attention when dealing with the streaming operators for hard spheres . As extensively discussed in Ref . 62, hard sphere molecules are not allowed to overlap and the streaming operator is not defined for overlapping configurations. The distribution function p (r) gives a vanishing weight to initially overlapping configurations, but the combination S_ p is not well defined. The derivation presented here, +t '^o t» however, deals with a potential with a finite height V , and therefore does not present this problem. Care will be required when the limit V ^^ is taken . The barred pseudo-Liouville operators L_ are formally

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162 defined in a way analogous to Eq . (D.8) as L = lim lim L(e) e ' ^ , (D.17) t^O e->0 V ->^ o where the operator in Eq . (D.17) is understood to act on the distribution function, as in Eq . (D.12). By using Eq . (D.2), Eq . (B.17) can be rewritten explicitly as L = lim lin, { ^ L (X )e^^^(^)I 9(x . ,x .; e)e^^L(e) j ^ ^^ ^^^ t->0 e^O i=l i°° o In the limit e-K) the streaming operator e^ generates a discontinuous function of the momenta of the particles, but a continuous function of their positions . The limits can therefore be taken immediately in the first term on the right hand side of Eq . (D.18), since the operator Lq(x) only contains derivatives with respect to the space variables . The second term, however, involves gradients with respect to the momenta and generates singularities in the regions common to both ?^ and exp[+tL(£)]. Equation (D.18) becomes then r ^ +tL(x ,x. ;£) ^+ = ^ I ^o^^^i^ ~ ^^^+ ^^"^ e(x x.;e)e ^ } i=l t^O EM) ^ ^ V ^^ o N N _ = { I L^(^i) ^ I T (X X )} , (D.19) i=l i
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163 and L(x^,x.) is the two-particle Liouville operator. Therefore the effective interactions for discontinuous potentials are still pairwise additive . This conclusion can be reached by performing a binary collision expansion of the corresponding N-particle scattering matrix. In order to determine the specific form of the collision operator defined in Eq . (D.20) a simple two-particle problem can therefore be considered. The scalar product of two real phase functions A(x ,x ) and B(x ,x ) (assumed for later convenience to be continuous functions of |r |) in the two-particle phase space is defined, in analogy with Eq . (D.ll), as
2 J dx^dx^ p^Cx^.x^) A(x^,X2)B(x^,X2) , (D.21) where p^Cx^.x^) = f^(ppf^(p2)e~P^^''^ . (D.22) Here f (p) is the Maxwell-Boltzmann distribution and r E r = |q -q | . The operators T_ are defined by their action on the combination p B , i .e . o A A J dx^dx^ A T_ B p^Cx^.x^) +tL(x^,X2;e) . = ± lim lim / dx dx A(x ,x )9(x ,x„ ;e)e B(x ,x„) t->0 e^O V ->oo o = ±lim lim M_(t,£) . (D.23) t->0 e^O "^ V ->-oo o To evaluate the matrix element defined in Eq . (D.23), the case of back-

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164 ward streaming will be considered explicitly. By introducing center of mass and relative coordinates and momenta, -> ->• ->• -> s.lO j.!ilZi the matrix element M can be written as A "tLf X X * r ^ M (t,£) = / d^d? / dSdg A(x, ,x^)r^>V (r)] . ^^ ^'2' i z 'r e g X p^(x^,X2;e)B(x^,X2) , (D.24) where the two-particle Liouville operator is intended as a function of relative and center of mass coordinates. ^ • V| g • V^ L(x^,X2;s) = ^— + -_[v^ V^(r)] • ^> . (D.25) r r± g The gradient of the potential V, is nonzero only in a small spherical region l^ = {a e < r < a + e} . The r-integration in Eq . (D.24) can therefore be restricted to this region. Furthermore using the identity, -tL(x^,x ) -tL(x,,x„) Eq . (D.24) can be rewritten as

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165 [_(t,£) = / d^ d^ dg fo(P,)fo(P2) / dr I(x^.X2)([-^+ L^(x^) + L^Cx^)] e -tLCx^.x^;^) -pV^(r) >< e e BCx^.x^)} . (D.27) Let r be a unit vector in the direction of r and dQ = sin6d9d({) denote -> 9 the angular part of the r-integration . Since r^A is a continuous function of r, Eq . (D.27) can be approximated by M_(t,£) = J d^ d^ di fo(Pi)fo(P2) J dQ a^ A(R,ot;^,|) ^. ?.C H -tL(x ,x ;e) -pv(r) ^ r (• O+E .g«rdr Iz £ ^, . ,t X {/ dr ^2 — -r— e e B(x,,x„) ^•' 0-e m dr L 1 2 -i ?. ^'h ^4.c-tL(x ,x ;e) -pV (r). + fe +-ir J W^ ^ ^ BCx^.x^)} , (D.28) where g has been chosen in the z direction. The last term in Eq . (D.28) vanishes in the limit e->-0 because the integrand in the r-integration is finite . The other term on the right hand side will in general give a nonzero contribution because singularities may arise from the derivative of the discontinuous potential. Carrying out the r-integration, Eq . (D.28) becomes then lim M_(t,e) = / d^ d^ d| fo(Pi)fo(P2) / ^Q a^ A(ft,ar;5,g) ^ -pv _^, . -pV _ . X (lia[e ^ e ^^ b] limfe ^ e """^ b] } , (D.29) where "lim" indicates the three limits, as in Eq . (D.23), and

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166 [f 1 = f (r=a±e) . The second term on the right hand side of Eq . (D.29) vanishes in the limits indicated, since limfe 1 = . The scattered momenta for collisions originated at a+e can be evaluated explicitly by distinguishing the cases of particle initially on the positive (r«g > 0) or negative (r«g < 0) hemisphere, with the result, lim[e e '''^ B] . ^ =e "^ ^ B(R,0r ;d,g ' ) , (D.30) (a.g)>0 a+e lim[e e b] . ^ = e B(R,ar;G,g) , (D.31) (a«g)<0 a+e where g' is the scattered relative velocity in a binary hard spheres collision. g' = g 2r(r • g) b^2 g • (D-32) By inserting these results into Eq , (D.29) and noting that exp[-pVjjg(a_|_)] =1, Eq. (D.29) becomes lim M_(t,e) = / d^ d^ d| / dQ A(S,S; ar ,g) a^ ^ [e(r.g)b^2 e(-r.g)] p^ (x^,X2)B(5,S;ar,g) , (D.33) where 9(x) is the Heaviside step function. Finally, by performing a change of variable in the angular integration over the negative hemisphere and reinserting the r-integration through the introduction of

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167 6-functions, Eq . (D.33) can be written 2 r.| ,,:,t^ lim M_(t,e) = / dx^dx^ A(x^,X2){a ^ e(r -g) [ 6(r-a)b^ HS ^ 6(r+a)]p^ (x^,X2)B(x^,X2)} . (D.34) Comparison with Eq . (D.23) gives 2 r T_(x^,X2) = o^ ^ e(r.|)[6(r-a)b^2 " ^(^+°^] 2 = a / . da -^^ [6(?-aa)b 6(?+aa)]. (D.35) (a.g)>0 "^ ^^ Here da = sinGdGdcj), where 9 is the scattering angle in the binary collision and (}) is the azimuthal angle which determines the plane of the scattering. The spatial 6-functions in Eq . (D.35) take account of the fact that a collision takes place only where the particles are at contact . A similar derivation can be carried out for the operator T , with the result. T (x ,x ) = J d^^ [6(? aa)b,„ 6(r + aa)] . (D.36) (a«g)<0 The barred N-particle pseudo-Liouville operators are then given by Eq . (D.19) with the form (D.35) or (D.36) for the binary collision operators . An expression for the unbarred pseudo-Liouville operators defined in Eq . (D.8) can now be obtained by using the definition (D.13). It is

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168 immediately shown that [L^(x^) + L^Cx^)]^ = -[L^(x^) + L^Cx^)] . (D.37) Furthermore, making use of the invariance of the scattering cross section for direct and restituting collisions, the adjoints of the collision operators are found to be [t_]^ e T , (D.38) where T^(x. ,x^) = +a2 ; da BC+a-g.^ ) --il 6(t.aa)[b.^ l] , (D.39; -> * -y and g^^ ^ Pi ~ P-j • -^^ operators L^ are then given by N N L^(x^,...,x^,) = I L^(x.) ± I T (x X ) . (D.40) i=i i = . (D.42) Since the collision operators do not act on the spatial variables and

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169 the kinetic energy of a pair is conserved in a binary collision, it immediately follows that (T_) *= (T_)^ = T^ . (D.43) The adjoint of the free streaming part however is now given by -> [L^(x.) +L^(x.)]*= -[L^(x.) +L^(x.)] -Iii-!ii6(r..-a). (D.44) By combining these results it immediately follows that (L_^) = -L_, (D.45) and consequently, for t>0. + + (S^) = S_^ . (D.46) The last property implies the time reversal invariance of the equilibrium average, since ^ = <[s:^ a]b>^ . (D.47) From the definition of the barred streaming operators it follows that they govern the time evolution of the distribution function of the system. The Liouville equation for the nonequilibrium distribution function, p^^.(x , . . . ,x ,t) of a hard sphere gas is then given by

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170 {5F'*"'^+^''i'-*--%)}Pn^''i"--'^N'''^ = °' (D.48) where the or + sign has to be chosen depending on whether t > or t < 0, respectively. A hierarchy of equations for the one-time reduced distribution functions, defined in Eqs . (2.25), can be derived by performing successive partial averages over the Liouville equation (D.48). However, when considering a gas of hard spheres, a more convenient method consists of first deriving a hierarchy of equations for the phase space densities fg, defined in Eqs. (2.24). The derivation presented here follows closely Ref. 37. Equations for the nonequilibrium distribution functions, given by n^fg(x^,...,x^,t) = , (D.49) are then obtained by averaging this set of microscopic equations. Here, as in Section II. 2, the same notation is used for both the phase space coordinates and the field coordinates . The time evolution of the phase-space densities is governed by the pseudo-streaming operator defined in Eq . (D.IO), i.e., tL . f^(x^....,x^,t) = e fg(x^,...,x^) , (D.50) where the + or sign has to be chosen depending on whether t > or t < 0. By differentiating Eq . (D.50) with respect to time for s=l,2,..., a hierarchy of equations for the functions f is obtained. s The s-th equation of the hierarchy is given by

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171 '-^ JJ '^"s+l V'^i'^s+P ^^s+l^'^l'-'-'^s+l'^) ' (°-51) 1=1 where the upper and lower sign has to be chosen for t>0 and t<0 respectively. To derive Eq . (D.51) the presence of 5-funcions in the definition of the phase space densities has been used, together with the property of the collision operators given in Eq . (D.43), to transform the operators in the equation to operators acting on the field variables Instead of the phase space variables. The hierarchy given by Eq.(D.51) for s=l,2,3,... is known as the Klimontovich hierarchy. In particular, since f^Cx^.x^.t) = f^(x^,t)f^(x2,t) 6(x^-X2)f^(x^,t) , (D.52) the first equation of the hierarchy can be written as a closed nonlinear equation for the one-particle phase space density, the Klimontovich equation, given by ^If ^ L^(^i)}f\(^l't) = ± / dx^ T_(x^,X2)[f\(x^,t)f ^(x^.t) 6(x^-X2)f^(x^,t)] . (D.53) The BBGKY hierarchy for the one-time reduced distribution functions f^ is immediately obtained by averaging Eq . (D.51) over the initial nonequilibrium ensemble and by using Eq . (D.49). The s-th equation of the BBGKY hierarchy for hard spheres is given by

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172 x=l where the sign has to be chosen as indicated before . The introduction of the phase space densities allows the definition of a set of multitime distribution functions as n^fCx^.tJ-.-lx^.t^) = < n f^(x^,t^)> . (D.55) k=l Hierarchies of equations for the multitime distribution functions can again be obtained from the ICimontovich hierarchy. In particular the case of the two-time distribution is considered. For s=2 Eq . (D.55) can be rewritten explicitly in the form, y t'L (t-f)L . n^f^^^(x^,t|x|,t') = 5; J drpj^(0)e * f^(x',0)e *f^(x^,0), (D.56) where the + or sign has to be chosen in L, depending on whether the corresponding time difference is positive or negative . A hierarchy of equations for the two-time distribution functions is then obtained by the Klimontovich hierarchy for t ->• t-t', with the result , = ±nl I dx^^^ T_(x.,x )f^ (X ..,x t|x;.f), (D.57) 1=1 '

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173 for s = 1,2,3,..., where n^^-^f^ ^(x^,...,Xg,t|x|,f ) = , (D.58) and the upper or lower sign refers to t >t ' or t < t ' respectively. Hierarchies for three or more times distribution functions are derived in a similar way . Low Density Expansion In the preceeding section the dynamics of phase space fluctuations in a nonequilibrium gas is described in terms of infinite hierarchies of coupled equations . In the limit of low density approximate closed equations can be derived for the lower order correlation functions by performing an expansion in powers of a properly chosen reduced density . A standard way of identifying the appropriate expansion parameter consists of scaling the equations according to the length and time scales characteristic of the problem. The one-time BBGKY hierarchy is first considered, as given by Eq . (D.54) for s = 1,2,... . For simplicity, the case of forward streaming (t > 0) is analyzed. Reduced variables, scaled according to the characteristic length and time scales of the problem r and t , are defined as r* = r/r , t* = t/t , p* = p/mv , (D.59) o o o where v is the thermal velocity of the particles and t^^ = r^/v^ . A dimensionless form of the binary collision operator is also introduced.

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T_*(x. ,x^) = ^ " _ T_(x. ,x.) , (D.60) where 3 r o

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175 P = Ti/I nj^r . (D.&5) o The chosen scaling parameters r and t must be of the order of the lengths over which the properties to be described have appreciable variations . For r =a and t =t , the parameters a and B become o o c a = Ti/2 , p = n/2ina^) . (D.66) 3 The reduced density no is the appropriate expansion parameter when deriving kinetic equations to describe variations over space and time scales of the order of a and x , however, as discussed in the introc duction, the interest here is not in two-particle or higher order correlation functions that change appreciably over such short scales . By choosing instead r = A and t = t. = Jl/v , where the mean free ^ o o f o path I is given by X = [n/l na^Y^ , (D.67) the parameters a and |3 become a = (nP) ^ , p = 1 . (D.68) This choice is appropriate to derive kinetic equations describing the desired space and time correlations in a low density gas . The condition a « 1 represents a low density gas outside of the Knudsen regime (the parameter K in (D.61) is now given by K = a/ I, i.e. is identified with the Knudsen number ).° In this limit the particles

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176 undergo several collisions over a length Z and a time t^, resulting in nonvanishing correlations over such ranges . Kinetic equations for the two-particle and higher order correlation functions are then obtained by a perturbative solution of the s-th equation of the hierarchy, using a as the expansion parameter . The s-particle distribution function is expanded in powers of a, fg(x ,...,x t) = y a^f^^'^x ...,x t) . (D.69) n=0 By substituting Eq . (D.69) into Eq . (D.63) (with the choice of the parameters given in Eq . (D.68)), a set of coupled equations for f^"^ is s obtained. To zeroth order in a, the equation is given by ^h\l\(-M"'^^i—s^^^ " J^ ^ ^^s+l ^-^^i'^s+l^^s+l^^l'-'-'^s+l'^) = ° ' (^-^O) and to k-th order in a, with k > 1, s i = i fe^ .1 \^-i^K (-r-'-^s'^^ .1 /^-s+l^-(^'-s+l>fs+i(-l"--'-s+l'^) s = I T_(x ,x )f "^\x ,...,x ,t) . (D.71) i
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177 f^°^(x^,...,x^,t) = n f^^°^(x.,t) . (D.72) i=l Substitution of Eq . (D.72) in Eq . (D.70) shows that this form provides a solution to the entire hierarchy, to zeroth order in a, if f^ \x,t) satisfies the equation, ^If+ ^^^l)J^l°''^^l'') = /dx2f^(x^,X2)f^^°^(x^,t)f^^°\x2,t) . (D.73) By translating Eq . (D.73) back into the original variables and by observing that, since the distances of interest are large compared to the force range, the difference in the positions of the colliding particles can be neglected and the binary collision opertor can be approximated by a point operator, T_(x^,X2) "" ^(^i-^2^'^^Pl'P2^ = Hv^^)a^ /. ^ d^ [-^][h^^ 1] , (D.74) Eq . (D.73) is immediately recognized as the nonlinear Boltzmann equation for hard spheres . The equation for the first order correction of f is obtained by setting k=l in Eq . (D.71) and substituting the zeroth order solution in the right hand side, with the result, {-|+ ); L (x.)}f*^^\x,,...,x ,t) ^5t .^, oi^s 1 s 1 = 1 I ! dx^^i ^(-i'-s+l)fs+l(-l"-"^s+l'^) 1=1 = 1 {n f;°\x, ,t)} T* (x.,x.)f5°)(x.,t)f5'^\x.,t) . i
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178 The form of Eq . (D.75) suggests the solution, s s f^^\x^,...,Xg,t) = I n f^^°\x t)f[^^(x.,t) i=l j=l J s s + I n fr'^x, ,t)g^/^(x.,x.,t) . i
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179 order term in an expansion in powers of a of the two-particle cluster function , g2(x^,X2,t) = f^Cx^.x^.t) f ^(x^,t)f ^(x^.t) . (D.81) In analogy with Eq . (D.62) a dimensionless form of the cluster function is introduced, *,* * *^ , -6s . * * -k ^l^^l'^l'^ ) = (°iv^) S2^'^o^^o^l '^o™^o^'^ ''^o^ ^ • (D.82) The expansion of g in powers of a is then given by g2^yi^,-^^,t) = I a"g^"\x^,X2,t) . (D.83) n=0 From the definition, Eq . (D.81), and from Eq . (D.72) for s = l and s=2 , follows that g^ -"(x^.x^.t) = . (D.84) Furthermore g (x ,x ) is by definition the two-particle function introduced in Eq . (D.76), whose time evolution is governed by Eq . (D.78). Having made this connection, it is easily shown that, when the reduced variables are eliminated in Eq . (D.78) in favor of the original ones and the approximation given in Eq . (D.74) for the binary collision operator is used, the equation for g^ (x ,x„,t), the first nonvanishing contribution to the two-particle cluster function, agrees with Eq . (3.3) of Ref . 37 .

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180 The procedure outlined here can be easily carried further to evaluate the next term in the expansion of i„ . This will involve threes particle correlations together with corrections to the oneand twoparticle distribution functions. In this way, kinetic equations for progressively higher order correlations are consistently obtained . Correlation Functions In order to directly compare the results derived here with those of Ref . 37 and of Chapter IV, it is convenient to consider the multi-time and equal time correlation functions as defined in Eqs . (2.34) and (2.32), instead of the corresponding distribution functions. In analogy with eq . (D.62), a dimensionless form of the correlation functions is defined as * * * A A mv 3 T O^S C^(x ,t • . ..;x ,t ) = ( ) C (r mvx,,t t-...;r mvx ,t t ) , s 1 1 s s ^ n -' s o o 1 o 1 o o s o s (D.85) where this definition applies to both the cases of equal and different time arguments . An expansion of the correlation function in powers of a is then written as * " (n) ^(x^.t^;...;x^,t^) = I a^ C^ ^(x^,t^;...;x^,t^) . (D.86) In this section equations for the time evolution of the lowest order terms in the expansion of the second order correlation functions are obtained .

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181 The equal time second order cumulant is first considered. By definition it can be written as C2(x^,t;x2,t) = g^Cx^.x^.t) + a6(x^-X2)f^(x^,t) . (D.87) An equation for C is therefore immediately obtained from the equations for the one-time reduced distribution functions derived in the prei/ious section. Since the two-particle cluster function vanishes to zeroth order in a, Eq . (D.87) shows explicitly that C^°^(x^,t;x2,t) = , (D.88) and in nonequilibrium situations both terms on the right hand side of Eq . (D.87) contribute to the lowest order in the expansion parameter. This point has caused confusion in the literature since some authors have argued that only the second term had to be kept to lowest order in the density. Ernst and Cohen correctly estimated the relative order of the two terms. The present scaling method, however, provides a more explicit statement of the result . By using Eqs . (D.73), (D.78), and (D.87), the kinetic equation for * the lowest order nonvanishing term in the expansion of C is immediately obtained. {|^+ (i+p^2^^*^''i'*^^^4^'*^''i''''''2^^ = ^*^''r''2'^^ ' ^°'^^^ with

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182 * — (0), r (x^.x^.t) = [a (x^.t) + A (x.^,t)]&(x^-x^)£Yhx^,t) _A (0)., .^^(0), (x^-x^) / dx3 T_(x^,X3)f'|^^x^,t)f^^^^x3,t) — * ( 0^ en's Again, reexpressing Eq . (D.90) in terms of the original unsealed variables and functions, the low density equation for the second order equal time correlation is found to be fe *" (l+Pi2^^^''l'''^JS^''l'^'''2'^^ " rCx^.x^.t) , (D.91) with L(x^,t) = L^(x^) nA(x^,t) , (D.92) and TCx^.x^^t) = -n[A(x^,t) + A(x2,t)]6(x^-X2)f^(x,t) + 6(x^-x^) / dx^ T_(x^,X3)f^(x^,t)f^(x3,t) + T_(x^,X2)fj^(x^,t)f^(x2,t) . (D.93) Here, A(x^,t) is simply given by A (x ,t) with T_ replaced by T_ . The one particle distribution function is the solution of the nonlinear Boltzmann equation. ^5t + L^(^l)}fl(x^,t) = n J dx^ T_(x^,X2)f^(x^,t)f^(x2,c) , (D.94)

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183 Equation (D.91) agrees with Eq . (3.5a) of Ref . 37 (the factors of n in Eq . (D.91) are due to the different normalization of the distribution functions) . The unequal time correlation functions are directly related to the multitime distribution functions defined in the previous section, Eq . (D.55). In particular, for s=2 this relationship is simply given by C*(x^,t;x|,t') = f*^^(x^,t|x|,f) f*(x^,t)f*(x^,t') , (D.95) where the dimensionless form of the distribution functions f , is s,l defined as in Eq . (D.62) (a factor of (mv ) is needed to scale ^s,l)The hierarchy of equations for the two-time distribution functions, as given by Eq . (D.57), for s=l,2,3,..., therefore has to be analyzed. This can be done in the same way described in the previous section, by scaling Eq . (D.57) with r = Jl and t = t^ , with the result. r6 1=1 a I T_ (x ,x )f (x ,...,x ,t|x:,t') i • (°-^^^ 1 = 1 Again the distribution functions are written as a power series in a, °° f* ,(x,,...,x^,t|x ',f) = I a" f^''J(x,,...,x ,t|x;,t'). (D.97) n=0

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184 By proceeding as before the lowest order terns in the expansion of f s, 1 can be easily evaluated . To zeroth order in a the result is given by (D.98) i=l and, to first order, f^|J(Xj,,...,x^,t|xJ,t') = f[^\x|,t') n f[°\x.,t) + y f^,°>(x.,t.)n fS°)(x^.t)f(i\x^,t) S S + I nf5°\x t)c^/\x.,t;x;,t') i=i j=i ^ J ^ ^ ^ + 1 nf(^o\x^,t)f[0)(x' t')g(l)(x^,x^.t), i
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185 the original unsealed variables and functions (the result of this transformation is simply the substitution of the operator L (x,t) with the operator L(x,t) as given in Eq . (D.92)) reduces to Eq . (3.9) of Ref . 37 and to the equation derived in Chapter IV by using the generating functional method, Eq . (4.34). The derivation presented here has the advantage of interpreting the physical meaning of the appropriate expansion parameter and of the approximation introduced . As a consequence of the scaling of the hierarchy, a clear identification of the order of the various terms in every equation follows. Therefore, the method outlined provides a systematic and controlled way of obtaining kinetic equations for higher order correlations. However, it is easy to see that the lowest order nonvanishing term in the expansion (D.86) of the s-th order equal time correlation function is the coefficient of a . This implies that every time the lowest order approximation for the next order cumulant is desired, it is necessary to derive an extra term in the expansion of the solution of the hierarchy. Therefore, the method, even though straightforward in principle, becomes quite lengthy when third or higher order correlations are desired. The advantage of the formalism developed in Chapter IV, based on the introduction of a generating functional for phase space fluctuations , is then evident . There all the information desired can be derived from a single kinetic equation, Eq . (4.17). The analysis presented in Chapter IV also applies to systems with continuous purely repulsive interparticle potentials . The resu!' ts here are however identical if che following correspondence between binary collision operators is made.

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186 T (x. ,x. ) f-^ e(x. ,x. ) S (x. ,x. ) . (D.lOl) The fact that the right hand side of Eq . (D.lOl) represents the long time (compared to a collision time) limit of the binary collision operator for continuous potentials can be derived as a rigorous result .^^

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APPENDIX E HYDRODYNAMIC "NOISE" FROM KINETIC THEORY The contribution to the equations for the hydrodynamic correlation functions arising from functional differentiation of the inhomogeneous term W (r,t|?0 ii^ Eq . (4.77) is analyzed here. For convenience, W is rewritten as the sum of two parts, w^(?,t|;o = \i^J-\v,t\\) + w^2\?,t|x) , (E.i) with a (T,t\X) = -|_ / dp v.a^(p)T(x;t,0)A(x,0|X) , (H.2) and V)^\r,t\\) = |_ /^ dx J dp V a (p)T(x;t,T)W(x,Ti\) . (E.3: 1 The functional derivative of the second term on the right hand side of Eq . (E.I) will be shown to vanish on a time scale of the order of the collision time, t . Consequently it does not contribute to the hydrodynamic equations for equal nor unequal time correlations functions. The first term, W^ , is instead responsible for the inhomogeneous part in the equations for the equal time correlation functions . ( '') The contribution from the term W^"'^ is first considered. Its a functional derivative is given by 187

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(2) ^ , ^^^ X T(x^;t^,T)9(x^,X2){[S^_^ (x,,X3) " ^^(x ^ ,X3) ]s° .^(x,.X3) X [6(x^-X2) + aCx^-x^)]} S^(x^,X3)f .(x^,T)f^(x3,T) , (E.4) where S^(x^ x^) = S (x )S (x ), and the parameter t^ has to be chosen to assure the proper order of the times. Since t > x > t > t , and only the case when t,-t > x is considered here, the appropriate choice is the one that restricts the integration tot, >TJ^t +t. Both the 1 2 c streaming operators in square brackets can then be replaced with their asymptotic limit, S^, since % > i and t t > t over the entire range of integration, and the integrand in Eq . (E.4) vanishes. Similarly, the contribution from W '^ to the equation for the equal a time correlation function is given by ^ . ^ 6W^^^(r,,t+e|\) lim, (l+P,^? J f dp^a (p )[ — 1 rv c^ where P-^2 interchanges r and r and 6W^2^(?^,t+£|x) t— \\(x,,t) \=0 = W;T C''^'^ 6(^-to)0(t^-t)J dPidx3V^.a^(p^) X T (x,;t+e,x)9(x^,X3){[5^_^(x^,X3) 5^0^^ .-3) ]S°_^(x^ .X3) X [6(x^-X2) + 6(X3-X2)]} 5^(x^,X3)f^(x^,-r)f^(x3,t) . (£.6) An appropriate choice of the parameter t^ is now t =t . The range of the o o ^

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189 time integration in Eq . (E.6) is then restricted to t+s > t > t, i.e. vanishes when the limit £-»0 is taken . Since the integrand is well behaved over the time interval of interest, W^ ^ gives no contribution to the equations for the equal time correlation functions . The contribution from the first term in Eq . (E.l) is given by X [C^ix^,0',x^,t^) f^(:c^,0)a^,(ppg^J(?^) j dp a^(P)C^(r^,P ,0;:i^,t^)} , (E.7) where C2 is the two-time correlation function of phase space fluctuations, defined by Eq . (4.26) for s = 2 . Its time evolution is governed by Eq . (4.34) for the variable X2 (since t2 > 0), whose solution is C2(x^,0;x2,t2) = T(x2;t2 ,0)C2(x^ .O-.x^.O) , (E.8) with C2(x^,0;x2,0) = 5(x^-X2)f ^^(x^ ,0) . (E,9) In writing Eq . (E.9) use has been made of the fact that only states such that the initial correlations in the system factorize in products of one-particle distribution functions are considered. Furchermore, the system has been assumed to be in local equilibrium at t=0 . Substituting, Eqs . (E.8-9) into (E.7), this becomes

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6W^'^(r,,t,|X) ^ dP2^VP2>^ °^6;v(x' t\ ];.=0 = ItT / ^^ fL(x,0)[T^x;r^,0) 2' 2' " "m S(r^-q)v.a^(p)](l-py'')[T^x;t2,0)5(?2-q)ag(p)] , (E.IC) where T is. the adjoint of the operator T, as defined in Eq . (4.71). The hydrodynamic part, up to Navier-Stok.es order, now has to be extracted from-Eq.. (E .10) . Tliis can be done by following closely the method used in Appendix B. The same result is then obtained, i.e. ^ . ^ 6W^^\? t |\) li.^^ / dp2a^(pp[_3^^^_^-^_]^^^ -> , (E.U) ^-^2»^f and t>>t, f> ^'"'l^^^p) / ^P2-p^P2)[ V.(x;,t) U where L^ is the matrix of transport coefficients for low a density gas, defined in Eq . (4.76) .

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APPENDIX F DETERMINATION OF R°^: FOR MAXWELL MOLECULES Equations for the Rest Frame Correlation Functions In the first part of this Appendix Eqs. (5.41-42) and Eqs . (5.4850) are obtained. Consider first R. .(t,T), defined by Eq . (5.37) for the case a=v , and in general given by R^jCt.T) = / dx^dx^v^^a^^C^^' ix^,t+T;x^,x) , (F.l) T where a„. = q„., v„., and C.' satisfies the equation 2j ^2j' 2j' 2 ^If-^ ^i^ji(^> fcr^i-ji fer"^BLff;ncJ'(Xi.t+x;x2,x) = 0, (F.2) with I [f'] given by Eq . (5.17). By projecting Eq . (F.2) onto v,, and JJL b Ai vet a„., an equation for R. . follows 2j ij •^R^j(t,T) + a^j^R^"(t,T) = n / dx^dx^v^^a^. Ig^lf ^^^7* ^^1 '•^''"'^''^2'^^ (F.3) voc In general, Eq . (F.3) is not a closed equation for R. . because of the term on the right hand side. However, for Maxwell molecules the right hand side can be evaluated, with the result, T va n / dx^dx2v^.a2jIg^[f^]C2'(x^,t+T;x2,T) = -v^Pj^"(t,T) . (F.4) 191

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192 The proof of Eq . (F.4) follows closely the one presented in Ref . 19. The symmetry properties of the collision operator can be used to write n / dx^dx2V^^a2jl3L[f^]C2'(xi.t+T;x2,T) ,^ „T. = n / dx^a / dx^dVj^C2'(x^,t-!-T;x2,-c)f^(v^,t) X /°db b|v^-v^| /^^ d^fv^.-v^.] , (F.5) where v. denotes the velocity after the collision. Defining the center of mass and the relative velocities. G =^(v^ + v^) g = v^ Vj^, (F.6) Eq . (F .5) becomes n / dx^dx2V^.a2jIgL[f(,]C2'(Xi,t+T;x2,T) = n / dx^a J dx^dv^C2'(x^,t+T;x2,T)f.^(v,^,t) r2Ti ,, 1 X /^ db b g /^" d4.y [i. g.] . (F.7) Now let g. = g.cos9 a.g sin6, where a is a unit vector orthogonal 111°' ^ -> .2% to g. Then, since J dA a. = 0, •'o 1

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193 T n / dx^dx^v^.a Ig^ff -jC^'Cx^.t+Tjx^.T) 2j X Tt g 1°° db b g(cose-l) . (F.8) For Maxwell molecules, J db b g h(9) is independent of the velocity, for an arbitrary function h. Thus n / dx^dx2V^.a2.l3L[f^]C2'(Xi.t+x;x2,T) ^1 / d-i'^V^^^ii-V^^y^b^^'^^S'^^i'^+^'^'-z'^) = -v^R^?(t,T) . (F.9) where V = mi; /" db b g(l-cos8) , (F .10) or V, = Tina^(e /m)^^^ 1.19 . (F.ll) 1 o The equation (5.42) for R?.(t,i;) can be derived immediately in a similar way, by observing that 1 is an eigenf unction of the Boltzmann-Lorentz operator with zero eigenvalue . The equations (5.48-50) for the equal time correlation functions, R"?(t) = R^^(0,t) = J dx a.0^f^(x,t) , (F.12)

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194 T where fl satisfies the same equation as C ' , are obtained in a similar w way. In particular, the equation for R. .(t) is 5 „w at (F.13) For Maxwell molecules, the right hand side of Eq . (F.13) can be evaluated as shown before. Using f d
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195 v^ = -^ u na^(e^/m)^'^ 1.23 . . (F.17) The cross terms in Eq . (F.15) vanish, and n / dx v.v.I^^[f.]f.(x.t) = (v^+V2)R;;(t) ^^6..Rl^it) ^""2 * "l + -^t..(t) +/5..p(t) , (F.18) where the pressure tensor of the fluid is defined as * r -* P.. = J dx.m V, .V, .f '(v. ,t) ij •' b bi bj b b' = t*j + 6^^ P . (F.19) Substituting Eq . (F.18) in the right hand side of Eq . (F.13), Eq . (5.50) is immediately obtained . w Evaluation of R_(0,t) The second part of this Appendix is dedicated to the solution of Eq . (5.50) for the following choice of flow field. a. . = a6, 6. . (F.20) ij ix jy The Laplace transform of a function A(t) is defined A(z) = f" dt e ^^A(t) . (F.21) 'o

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196 In terms of the transformed quantities the set of Eqs . (5.50) is than given by (z + v^ .V2)R^(z) -. 2a R^^(z) j v^R^.Cz) = I^(z) -. ^ (+ v^ + V2)R (z) + aR (z) = ! (z) . xy yy xy (z + V + V )R (z) + aR (z) =1 (z) , 1 2 xz vz xz yz xz (z + V + v.)R (z) iV-R, , (z) = 1 (2) + -5-^ i ^yy 32 kk' yy m (z + v^ + V2)Ry^(z) = I^^(z) , K_ T ^^ -^ -^1 -^ ^2)^zz^^) -i ^2\k<^> = \z^^> +-|-^ > (F,22) and 3K (z + v^R^j^Cz) 42aRy^(z) =_2 (,^x(z) + T^) , (F .23) where T^ is the initial temperature of the fluid, and R..(t) H R^(0,t) = R^.(t) . (F.24) These equations have to be supplemented Vv-ith equations for the components of the stress tensor,

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197 (z + 2v-)t = at 2 XX 3 xy (z + 2v-)t = (z + 2v„)t* = 4 at* , 2 yy ^ 2 XX 3 xy ' (z + 2v^)t = -at anK„T(z) 2 xy yy ^ ' k A t = t = , (F .25) xz yz V ^/ and the heating equation, znKgT(z) = nkgT^ | at* (z) . (F.26) The components R and R do not have any interesting behavior. The other components are given by \x^^^ = °-l i[(^+^+^2^^ "^ 2a2](v^T(z) + T^) 2 „ a -(z+vj(z+v +v„) + [|(z+vp + v^ 2 .^2v^ ](v,-vp(zT(z) T^)} . R (z) = D — f-(z+v,+vJ^(v,T(z)+T ) + [— -^ xy ' m ^ ^ 1 2 1 '^ o^ '-z+2v~h (z+v^+V2)](2+v^)(v^-V2)(zT(z) T^)} , S 2 ^ (z+v^Xz+v^+v^) V^'^ = °^ {(-^v^+v^) (v^T(z) + T^) + [v, ^^^^ , (v^-v^XzTCz) T^)} ,

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198 ^B rr.,_ .2 . „ 2 \k^^^ = ^ i" f[3(z+v^+V2)^ + 2a-](v^T(z) + T^) R (z) = R (z) , . (F.27) zz yy \^ ••<-' / where D = [(z+v^Cz+v.+v^)^ -| a^v^]"^ . (F.23} The set of Eqs . (F .25) and (F.26) can be solved in terms of the temperature, with the result (z-f2v )^ + 2a^/3 T(z) = T ^— . (F.29) ° z(z+2v2)^ I a^(2v2) The correlation functions and the temperature field have three poles, corresponding to the roots of their denominators . The roots of the third order polynomial D, defined in Eq . (F .28) , are z^Ca.v^.v^) = XCa.v^) v^ , z^Ca.v^.v^) = -[i^Ca.v^) + v^ + vj + i _|sinh[lc:(^) J , /3 2 z^Ca.v^.v^) = z^Ca.v^.v^) , (F .30) where the dagger indicates here the complex conjugate, and ^(a.v^) =3 ^2^^"^^'! ''^T-^^ ' CF.3i)

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199 with 2 «(^) = 1 + 9 ^ • (F.32) Similarly the poles of the temperature field are given by T z^Ca.v^) = z^(a,0,2v2) ' (?-33) for i=l,2,3. The real part of the two complex conjugate roots (for both the correlation functions and the temperature) is always negative, and leads to an exponential decay of the function characterized by a lifetime shorter than v (or v^ ). The relaxation times v~ and v~ are of the order of the mean free time between collision of the particles in the system. This exponentially decaying behavior represents therefore an initial transient in the system and can be neglected if the time T in the two-time correlation function, G. (t,T), is chosen to satisfy t >> t^ . T The real pole of the temperature field, z, , is always positive and leads to an exponential growth of the temperature in time, with a rate which increases with the shear: this is the m.anif estation of the viscous heating in the system. The real pole of the correlation functions, z, , changes sign as a function of the shear rate. For < a/v„ < 3 .3 it is negative and therefore contributes only to the initial transients in the system. At larger values of the shear rate it becomes positive and it has to be retained. Therefore, as long as the shear rate is not too large (a/v^ < 3.3), all the time dependence of the equal time correlation functions is determined by that of the temperature field .

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200 It is convenient to define reduced quantities as T*(t*) = T(v/t*)/T , (F.34) ic "k "k z^(a ,v ) = z^(a,v^,V2)/^i' (f-35) with, * / * * , * a = a/v^ , V = v^/v , t = v^t , X = a /v = a/v . * The real pole z is then written as "k "k ic "k "k z^(a ,v ) = V \ (x) 1 , (F .36) where ,*/ s 4 . ,2ra(x)i \ (x) =-j smh [-—-] . (F.37) At long times (t » 1), the reduced temperature field is then given by T (t ) = H(x,v )exp[t (2^(2v ) + l] , (F .38) * where H(x,v ) is the residue of the right hand side of Eq . (F .29) at the pole z^ , i.e ., ^ Y^^ilv ) + 1 + 2v j^ + 2a •-,/3 [z (2v ) z (2v )j[z (2v ) z' (2v )]

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201 The a dependence of the poles in £q . (F .39) has not been indicated to simplify the notation. Similarly, reduced correlation functions are defined as a O By neglecting the contribution from the two complex poles, which only survive at very short times, at all shear rates, the reduced correlation functions are given by * . *, r * A. . * * , ^r" ^1^^ '^ )(^ "^) * * -k * * * R.j(T ) = [v A (x)A.^(x,v ) + B.^(x,v )]JJ ds e ^ T (s) * A * * * 2 (a ,v ) ^ + A^.(x,v )T (t;) + e B..(x,v ) . (F.41) The coefficients A. and B. are linear combinations of the residues of the functions on the right hand side of Eqs . (F .27) at the pole z = z , . Their explicit form is not given here . By inserting Eq . (F .38) into Eq . (F .36) , the time integration can be carried out, with the result R*.(x*) = b(l>(x,v*)e-^*t^*^*(-^)-^ +b^^^(x,v*)e^*2^*^*^^'^2) (F,42) where • ^ ^ ,,v . B. . + V X' (x)A. . b). (x,v ) = B. . H . * , -' '-' 1 + z^(2v ) z^(v ) A A /„v . B. . + V \ (x)A. . b f (x,v*) = H{A + ^—^^-} . (F.A3) ^ ^J 1 + z^(2v ) z^(v )

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202 * The expression (F ,42) applies for x » 1 and all values of the shear rate. If only shear rates such that a < 3 .3 are considered, the exponent in the first term on the right hand side of the equation is negative and this term can be neglected for t » 1 . In this case the time dependence of the correlation function is the same as that of the temperature and can be scaled out by normalizing the correlation function with the temperature, i ,e , •k A ^ii^^ ^ * * * -^A — = Y,-^(x,v ) , for T » 1, a < 3 ,3 T (X ) "-J (F .44) where ^ ^ ^ B. . + V X (x)A. . Yi.(x,v ) = A +^^ — ^ ^r-ir • (F-^5) ^J 1 + z^(2v ) z^(v ) In particular the explicit form of y is needed in Section V .5 , This yy is given by Yvv^-.v*) = {[1 + X*(x)]2 + 2(l-v*)Ax/2)[l _ _A^x2ll±>^x)J_ J ^ ^^ V \ (x) + 2v 1 X {[l + 4\*(x) + 3\*^(x)][l + 2v*\*(x/2) v*\*(x)]}~^ , (F.46)

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REFERENCES 1. M.S. Green, J. Chem . Phys . ^O , 1281 (1952); Tl_, 398 (1954). 2. K. Kawasaki, Ann. of Phys . _6j_, 1 (1970). 3. D.N. Zubarev, Nonequilibriuni Statistical Thermodynamics (Consultants Bureau, New York, 1974). In Chapter IV an extensive list of references on the definition of the nonequilibrium statistical operator can be found . 4. N.N. Bogoliubov, in Studies in Statistical Mechanics , vol. I, eds . J. de Boer and G .E . Uhlenbeck (North-Holland Pub. Co., Amsterdam, 1962) . 5. E.G.D. Cohen, Physica _28 , 1025 (1962); _28, 1045 (1962); 28, 1060 (1962). 6. JJ^. McLennan, Adv. Chem. Phys . Vol. V, ed . I. Prigogine (Wiley, New York, 1963) . 7. D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (W.A. Benjamin, Inc., hlassachusetts , 1975). 8. J.H. Ferziger and H .G . Kaper, Mathematical Theory of Transport Processes in Gases (North-Holland Pub . Co., Amsterdam, 1972). 9. S. Chapman and T. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, Cambridge, England, 1952). 10. K. Kawasaki and J .D . Gunton, Phys. Rev . A8_, 2048 (1973). 11. S .R . de Groot and P. Mazur , Nonequilibrium Thermodynamics, (North Holland, Amsterdam, 1962) . 12. J .L . Lebowitz, J .K . Percus and J. Sykes , Phys. Rev . 183 , 487 (1969). 13. H.B. Callen and T Ji . Welton, Phys. Rav . _83_, 34 (1951)14. R. Kubo and K. Tomita, J. Phys. Soc . Japan _9_, 388 (1954); R. Kubo , J. Phys. Soc. Japan _12_, 570 (1957). 15. L. Onsager, Phys. Rev . _37_, 405 (1931); 2§.» 2265 (1931): L. Cnsager and S. Machlup, Phys. Rev . _9j_, 1505 (1953); S. i-iachlup and L. Onsager, Phys. Rev. ^, 1512 (1953). 203

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204 16. P .C . Martin, in Many Body Physics , eds . C. de Witt and R. Balian (Gordon and Breach, Inc., New York, 1968). 17. R.D. Mountain, Rev. Mod. Phys . _38_, 205 (1966); V. Ghaem-Maghami and A.D. May, Phys . Rev. A22_, 698 (1980). 18. C.K. Wong, J .A. McLennan, M. Lindenfeld and J .W . Dufty, J. Chem. Phys . _68^, 1563 (1978). 19. J.W. Dufty and M.J. Lindelfeld, J. Stat. Phys . ^, 259 (1979). 20. J.W. Dufty and M.C. Marchetti, J. Chem. Phys . _71» 422 (1981). 21. M.H. Ernst, B. Cichocki, J.R. Dorfman, J. Sharma and H. van Beijeren, J. Stat. Phys . _18 , 237 (1978). 22. R. Zwanzig, J. Chem. Phys . _7j_, 4-^16 (1979). 23. M.C. Marchetti and J.W. Dufty, Int. J. Quant. Chem. (1982). 24. S. Hess, Phys. Rev. A _25_» 614 (1982); N. Herdegen and S. Hess, preprint (1982) . 25. R. Graham, Springer Tracts in ttod . Phys., Vol. 66, 1 (1973); in Fluctuations, Instabilities and Phase Transitions , ed . T . Riste (Plenum Press, New York, 1975). 26. A. -M.S. Tremblay, M. Aral and E .D . Siggia, Phys. Rev. A23 , 1451 (1981). 27. H. Grabert and M.S. Green, Phys. Rev. A _19_, 1747 (1979); H. Grabert, R. Graham and M.S. Green, Phys. Rev. A _2j_, 2136 (1980). 28. H. Grabert, J. Stat. Phys . _26_, 113 (1981). 29. N.G. van Kampen, Phys. Lett. 50A , 237 (1974); Adv. Chem. Phys., Vol. XXXIV, 245 (1976) . 30. J. Keizer, Phys. Fluids _2i. 198 (1978). 31. A. Onuki, J. Stat. Phys . _18 , 475 (1978). 32. H. Ueyama, J. Stat. Phys . ^, 1 (1980). 33. T. Kirkpatrick, E .G .D , Cohen and J.R. Dorfman, Phys. Rev. Lett. 42, 862 (1979). 34. F.L, Hinton, Phys. Fluids _13_' S57 (1970). 35. V. Gantsevich, V .L . Gurevich and R. Katilius, Sov . Phys. JETP 32, 291 (1971) . 36. S. Tsugd and K. Sagara, J. Stat. Phys. 12, 403 (1975).

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205 37. M.H. Ernst and E.G.D. Cohen, J. Stat. Phys . _25^, 153 (1981). An extensive list of references on the kinetic theory of nonequilibrium fluctuations can be found here . 38. J.W. Dufty, in Spectral Line Shapes , ed . B . Wende (W . de Gruyter it Co ., Berlin, 1981) . 39. I. Procaccia, D. Ronis and I. Oppenheim, Phys . Rev. Lett. 42, 287 (1979). 40. I. Procaccia, D. Ronis, H .A . Collins, J. Ross and I. Oppenheim, Phys. Rev. A J_9_, 1290 (1979); D. Ronis, I. Procaccia and I. Oppenheim, Phys. Rev. A J^, 1307 (1979), 1324 (1979). 41. I. Procaccia, D. Ronis and I. Oppenheim, Phys. Rev. A 20, 2533 (1979); D. Ronis, I. Procaccia and J. Machta, Phys. ReTT A 22, 714 (1980). 42. J. Machta, I. Oppenheim and I. Procaccia, Phys. Rev. A 22, 2809 (1980). 43. T. Kirkpatrick, E.G.D. Cohen and J .R . Dorf man , Phys. Rev. Lett. 44, 472 (1980). 44. T.R. Kirkpatrick and E.G.D. Cohen, Phys. Lett . 78A , 350 (1980). 45. T.R. Kirkpatrick, Ph.D. Thesis, The Rockefeller University, New York, 1981. 46. T.R. Kirkpatrick, E.G.D. Cohen and J.R. Dorf man, Phys. Rev. A, August 1982, to be published. 47. L.D. Landau and E .M . Lifschitz, Fluid Mechanics, Chap. XVII (Pergamon Press, New York, 1959)~I 48. R.F. Fox and G .E . Uhlenbeck, Phys. Fluids JJ, 1893 (1970). 49. D. Ronis and S. Putterman, Phys. Rev. A _22 , 773 (1980). 50. G. van der Zwan, D. Bedeaux and P. Mazur , Phys. Lett . 75A , 370 (1980). 51. C. Tremblay and A.-M.S. Tremblay, Phys. Rev. A ^, 1692 (1982). 52. J. Logan and M. Kac, Phys. Rev. A _13_, 458 (1976). 53. M. Bixon and R. Zwanzig, Phys. Rev . 187 , 267 (1979); R.F. Fox and G.E. Uhlenbeck, Phys. Fluids _11, 288TTl970) . 54. E. Ikenberry and C. Truesdell, J. Rat. Mech . Anal . _5, 55 (1956). 55. L.D. Landau and E.M. Lifschitz, Statistical Mechanics, Chap. XII (Pergamon Press, New York, 1958).

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206 56. P. R^sibois and M. de Leaner, Classical Kinetic Theory of Fluids , Chap VII (J. Wiley & Sons, Inc., New York, 1977) . 57. J.R. Dorfman and H. van Beijeren in Statistical Itechanics B, ed . B.T. Berne (Plenum Press, New York, 1977) . Ihis article constitutes a recent review on the kinetic theory of gases . 58. J.W. Dufty, Phys . Rev. A J_3 , 2299 (1976). 59. M. Lax, Rev. Mod. Phys . 28_. 359 (1966). 60. P.C, Martin, E .D . Siggia and H.A. Rose, Phys. Rev. A _8 , 423 (1973), 61D. Beysens, Y. Garrabos and G. Zalczer, Phys. Pv.ev . Lett. 45, 403 (1980). 62. M.H. Ernst, J.R. Dorf man , W.R. Hoegy and J.M.J, van Leeuwen , Physica 45, 127 (1969) .

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BIOGIIAPHICAL SKETCH Maria Cristina Marchetti was bom on December 11, 1955, in Pavia , Italy. There she received her primary and secondary education. In November 1974 she entered the University of Pavia, Italy, where she received the degree of "Laurea" in Physics in July 1978. In 1979 she was awarded a Fulbright-Hays Travel Grant and in March of the same year she began graduate studies in Physics at the University of Florida. During her graduate career she was given a 1980 Phi Kappa Phi Scholarship Award and a 1982 Sigma Xi Graduate Research Award . She is a member of the American Physical Society. 907

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. U^v L J^mes Wi Duf rofessor of tnQh Ph>Wi hair man cs I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. c?W>?ut>. y Oa. Thomas L . Bailey Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy . T"! di.. /^^.,./L Arthur A. Broyles Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a disser/aiciojf) for the degree of Doctor of Philosophy . Charles F. Hooper, Jr Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy . i '•-OTvv^' -t^ f. U^. Kwan Y. Chen Professor of Astronomy

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This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate Council , and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1982 Dean for Graduate Studies and Research


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