A Non-Equilibrium Statistical Mechanics: Without the Assumption of Molecular Chaos

A NON-EQUILIBRIUM STATISTICAL MECHANICS Without the Assumption of Molecular Chaos

Tian-Quan

Chen

A NON-EQUILIBRIUM STATISTICAL MECHANICS Without the Assumption of Molecular Chaos

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.'•1 j ' « t v

* ***

A NON-EQUILIBRIUM STATISTICAL MECHANICS With our the Assumption of Molecular Chaos

Tian-Quan Chen Tsinghua University, PR China

^ j h World Scientific wB

New Jersey • London • Si Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

A NON-EQUILIBRIUM STATISTICAL MECHANICS Without the Assumption of Molecular Chaos Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-238-378-6

Printed in Singapore.

To the memory of my mother, who suffered many hardships for bringing up my sister and me.

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Foreword

Compared with numerous brilliant achievements in 20 th century physics, turbulence turns out to have a much earlier historical record, but remains stubbornly against understanding explanation from a basic theoretical point of view. The problem as phenomenon is so simple to expose, even without specific, scientific language. Just let fluid flow in a pipe, then we find that for low velocity or high viscosity, the flow is smooth and steady; however, if the velocity is increased or viscosity decreased, we get a transition to turbulent flow, which means the emergence of small and large vortices whirling about and chaotic streamlines interwound, and other complexities. Creating a theory to account for all these becomes a long standing difficult problem in the history of natural science. Great scientists like Heisenberg, Kolmogorov and others, possibly did lend a hand in this field, but they were not so fortunate as elsewhere, because a successful theory is still lacking up to now. Therefore, it is really encouraging for me to know that Chen's book is going to be published. I knew the author in the early days when he was a student at Peking University, he was rather famous for his mathematical talent. He undertook the vii

viii

FOREWORD

subject of turbulence by the year 1970, and has worked sincerely and persistently since then; his devotion of almost thirty years to climb high for a goal rarely frequented by predecessors give birth to the present book. 21 s t century dawned as if to be commercial in every sense, I don't expect a serious treatise like this book will soon become popular, but I do think it will be referred to by connoisseurs even after several decades. I should be considerate of writing a brief account about the content and distinguishing features of the book, but with hesitation, I decided to leave this specialist's task the author's own introduction; since my common knowledge and speculations are far from enough for bringing into the vista of the readers an even rough outline of the chief achievements.

Qian Min Professor at the Department of Mathematics, Peking University.

Preface

Statistical mechanics, founded and developed by Maxwell, Boltzmann, Gibbs, Enskog, Chapman and others, has three principal features. Firstly, it concerns itself with the relationship between the macroscopic and microscopic descriptions of the matter. Secondly, it uses the probability distributions in specifying the microscopic state of the matter. Finally, it deals with the state of the matter composed of extremely large number of small entities, e.g., molecules and, therefore, studies the asymptotic states of the aggregate of extremely large number of small entities under certain limits. Needless to say, the difficulties in attempting to build up a mathematically rigorous statistical mechanics is formidable, unless some hypotheses based on physical intuition are, explicitly or implicitly, imposed on the microscopic state, i.e., the probability distribution, of the aggregate of small entities of which the matter is composed. Usually some kinds of symmetry or mutual independence of the random variables under consideration are assumed. One of the famous hypotheses is that of molecular chaos, i.e., Boltzmann's Stosszahlansatz, put forward firstly by Boltzmann. As was pointed out by Massignon, Grad and some other people, statistical mechanics under the ix

PREFACE

X

hypothesis of molecular chaos, or something like that, will exclude the turbulence phenomena of fluid motions and make the quantization almost impossible. In order to include turbulence phenomena and to make quantization easier, Massignon constructed a theoretical framework, in which the correct Euler variables instead of the corresponding local ones in the theory of BBGKY hierarchy were introduced. He was successful in forming, exactly and correctly, the concepts of macroscopic variables in terms of the microscopical ones. Trying to study the non-equilibrium phenomena of the fluid, he was satisfied with just recalling the (Enskog-Chapman) technique used in the classical kinetic theory of gases. I think, it is the complexity of the calculation to prevent him from constructing an asymptotic technique for solving the Liouville equation.

Keeping tracks of the founders of statistical mechanics, we devote the present book to the exploration of the relationship between the microscopic and macroscopic states of matter by using the probability concepts and the techniques of asymptotic analysis (or, some perturbation schemes). Precisely speaking, a perturbation technique for solving the Liouville equation in Massignon's theoretical framework will be carried out in the present book. Of course, some hypotheses on the microscopic states have to be assumed in the process of solving the Liouville equation by perturbation method. All these hypotheses will be called the proposals of gross determinism in the present book. Gross determinism means that microscopic state uniquely determines the corresponding macroscopic state and the converse also holds in a certain sense. Proposals of gross determinism are the techniques of applying both the expression of macroscopic state in terms of the microscopic state and its inverse expression. Usually it is supposed that the (temporal and spatial) change of a macroscopic variable is far much smoother than those of the microscopic variables, of which the macroscopic variable is expressed in terms. As far as I know, all the hypotheses imposed on the microscopic

PREFACE

XI

states assumed in the present book are weaker than those, explicitly or implicitly, assumed in the classical theories of BBGKY hierarchy. As a consequence of the theory presented in the book, some new transport phenomena are discovered. The new transport phenomena discovered by the asymptotic analysis of the Liouville equation reveals the intricate relationship among the spatial and temporal variations of the density, velocity and temperature of the fluid, which was ignored by the classical kinetic theories owing to the oversimplified probabilistic models, or inappropriate asymptotic techniques (perturbation schemes), or both. I guess, the techniques presented in the book might apply to dilute and dense gases and even liquids, at least, under certain conditions. Because the mathematics of many-particle physics is too complicated, a lot of hypotheses based on physical intuition have been made for carrying out the reasoning. But, as far as I know, all the hypotheses made in the present book have been made, explicitly or implicitly, in classical fluid dynamics and thermodynamics. Any way, the techniques developed here is still immature and the works presented in the present book is only a beginning of the exploration of the intricacy of many particle physics. A great deal of difficult problems are still left open. I hope to publish the present book for throwing stones and bringing back jade (a Chinese proverb: pao zhuan yin yu), i.e., I hope my crude works may draw forth others by abler persons.

I am grateful to my teacher, Professor Qian Min at Peking university, for his discussion, criticism and instruction about the topics of non-equilibrium statistical mechanics and his kindness of writing the foreword for the book. I am indebted to Professor Wang Hong Yu at the Department of Mathematics, Yang Zhou university for his assistance in publishing the book. I really appreciate the generous financial supports from National Natural Science Foundation of China, Center of Mathematical Research and Teaching of the Ministry of Education of China and the College of Science at Tsinghua University during the long time of

Xll

PREFACE

my doing the research work on non-equilibrium statistical mechanics. It is impossible to finish the work without these supports. I am also indebted to my former colleague, scientific editor of the World Scientific Publishing Co., Dr. Lu Jitan for his patience in the cooperation during the period of preparing my manuscript. I really esteem the World Scientific Publishing Co. for its publishing the present book at the risk of economic loss. Last, but not the least, I will express my deep gratitude to my wife, whose constant assistance and encouragement, even during her suffering from leukemia, are indispensable to the accomplishment of the present book.

Tian-quan Chen at Tsinghua yuan.

Contents Foreword

vii

Preface 1

2

3

4

5

ix

Introduction

1

1.1

Historical Background

1

1.2

Outline of the Book

11

F-Functional

27

2.1

Hydrodynamic Random Fields

27

2.2

F-Functional

30

//-Functional Equation

33

3.1

Derivation of .ff-Functional Equation

33

3.2

H-Functional Equation

51

3.3

Balance Equations

54

3.4

Reformulation

64

K-Functional

69

4.1

69

Definition of liT-Functional

Some Useful Formulas

75

5.1

75

Some Useful Formulas xiii

xiv

CONTENTS 5.2

6

7

8

A Remark on H-Functional Equation

78

Turbulent Gibbs Distributions

81

6.1

Asymptotic Analysis for Liouville Equation

81

6.2

Turbulent Gibbs Distributions

93

6.3

Gibbs Mean

Euler K-Functional

109 Equation

119

7.1

Expressions of 2?2 and B3

119

7.2

Euler if-Functional Equation

136

7.3

Reformulation

140

7.4

Special Cases

144

7.5

Case of Deterministic Flows

148

Functionals and Distributions 8.1

8.2 8.3

157

-ff-Functionals and Turbulent Gibbs Distributions

157

Turbulent Gibbs Measures

164

Asymptotic Analysis

9 Local Stationary Liouville Equation

169 175

9.1

Gross Determinism

175

9.2

Temporal Part of Material Derivative of TN

184

9.3

Spatial Part of Material Derivative of TN

219

9.4

Stationary Local Liouville equation

225

10 Second Order Approximate Solutions

227

10.1 Case of Reynolds-Gibbs Distributions

227

10.2 A Poly-spherical Coordinate System

234

10.3 A Solution to the Equation (10.24)i

238

CONTENTS

xv

10.4 A Solution to the Equation (10.24)2

253

10.5 A Solution to the Equation (10.24)3

254

10.6 A Solution to the Equation (10.24)4

259

10.7 A Solution to the Equation (10.24)5

260

10.8 A Solution to the Equation (10.24)6

261

10.9 Equipartition of Energy

263

11 A Finer if-Functional Equation

271

11.1 The Expression of B2

271

11.2 The Contribution of Gi to B2

273

11.3 The Contribution of G 2 to B2

291

11.4 The Contribution of G 6 to B2

294

11.5 The Expression of B3

296

11.6 The Contribution of Gx to B3

298

11.7 The Contribution of G2 t o B 3

301

11.8 The Contribution of G 6 to B3

303

11.9 The Contribution of G 3 to B3

305

11.10 The Contribution of G 4 to B3

318

11.11 The Contribution of G 5 to B3

328

11.12 A Finer if-Functional Equation

336

12 Conclusions

339

12.1 A View on Turbulence 12.2 Features of the Finer

339 K-Functional

Equation

342

12.3 Justification of the Finer JC-Functional Equation

343

12.4 Open Problems

345

xvi

CONTENTS

A Some Facts About Spherical Harmonics

347

A.l Higher Dimensional Spherical Harmonics

347

A.2 A List of Spherical Harmonics

349

A.3 Products of Some Spherical Harmonics

369

A.4 Derivatives of Some Spherical Harmonics

402

Bibliography

407

Index

415

Chapter 1

Introduction 1.1

Historical Background

In the present book we are endeavoring to derive the functional equations governing the evolutions of fluid flows, including both the so called laminar and turbulent flows, from the first principle of non-equilibrium statistical mechanics. About one hundred and twenty years ago, based on his famous experiments on fluid flows in a pipe, O.Reynolds ([66, 67]) obtained some significant conclusions about fluid motion. In modern mathematical language, his conclusions can be summarized as follows. (1) The velocities (and the densities and the pressures) of the turbulent flows should be described by random fields. (2) Each sample of the random fields satisfies the Navier-Stokes equations. (3) The origin of turbulence is the instability of the solutions of the Navier-Stokes equations. Since then, the above view points have been accepted by most experts of fluid dynamics. The basic assumption we adopt in the present book is that a general fluid flow should be described by random fields and the so called laminar flow is only an extremely special class of fluid flows, which are described by the random fields with vanishing variances. At first glance, this proposition does not deviate far away from the first conclusion of Reynolds' theory. But it is not identical with Reynolds' theory. Reynolds assumed that a general fluid flow is, in principle, 1

2

CHAPTER 1.

INTRODUCTION

deterministic (or, laminar), which is represented by a solution to the NavierStokes equations (see, Reynolds' second conclusion). His experiments revealed some flows, called turbulent flows, are so complicated in form that we have to use a random field to describe them. Hence, according to his theory, a turbulent flow is a random flow, each sample of which is represented by a solution to the Navier-Stokes equations. The view point of the present book is that a general flow is a random field, of which the evolution should be governed by some functional equations. The exploration of the forms of these functional equations is the aim of the present book. Having assumed the randomness of the velocity (and the density and the pressure) fields of the general fluid flows, in the theory of hydrodynamic stability we ought to consider the general form of the disturbance of the basic flow to be random, even if the basic flow itself is deterministic (or, laminar). The latter conclusion is really foreign to the traditional theory of hydrodynamic stability. I think, one of the causes why only the deterministic disturbance has been taken into consideration in the classical theory of hydrodynamic stability is the fact that no equations governing the general fluid motion other than the Navier-Stokes equations, which governs the evolutions of deterministic flows, are at our disposal until now. In fact no equations governing the evolution of the random fields attributing to the general fluid flow have been derived from the first principle. Reynods astutely, but more or less arbitrarily, assumed that each sample of the random fields satisfies the Navier-Stokes equations. It is interesting to note that Reynolds' work ([66, 67]) on the statistical theory of turbulence was about twenty years earlier than Gibbs' work ([31]) on statistical mechanics of the system of particles. But Newton's laws governing the motion of the system of particles play a role far much more fundamental than that of the Navier-Stokes' equations governing the motion of fluid flows. The Navier-Stokes equations are heuristic consequences of the Newton equations under certain assumptions. Even if Gibbs'

1.1. HISTORICAL

BACKGROUND

3

averaging technique can be applied to the equations governing the motion of the particle system, the applicability of Reynolds' averaging technique to the NavierStokes is still questionable. Hence Reynolds' second conclusion must be carefully investigated. This is the reason why we have to derive the equations governing the motion of the general fluid flows from the first principle. Usually the quantity specifying the random fields is a functional, i.e., a function defined on a function space. The first task we are facing is the derivation of a functional equation governing the evolution of the functional specifying the random fields attributing to a general fluid flow. Having got the functional equation, we have to consider the stability or instability of the solutions to the functional equation governing the evolution of the random fields, which describes the motion of the general fluid flows, with respect to random disturbances. It is reasonable to imagine that under certain conditions the initially negligibly small random disturbance, measured by fluctuations or correlations, might evolve to such an extent that we can no longer ignore after a finite time interval has elapsed. Maybe this is one way of the onset of turbulence. In summary, we assumed that the general fluid flows should be described by random fields and the equations governing the evolution of the random fields should be derived carefully from the first principle.

About half century ago, D.Massignon and his collaborators ( Arnous, Bass and Massignon [3]; Massignon [56, 57, 58] ) correctly pointed out that the methods in Maxwell and Boltzmann's kinetic theory of gases ([59],[10]) and its extension to dense fluids by J.Yvon [86, 87], J.G.Kirkwood [44, 45, 46, 47, 48], M.Born, H.S.Green [11, 12, 13] (and N.N. Bogoliubov [7, 8]), i.e., the theory of BBGKY hierarchy, gave us a form of the classical statistical hydrodynamics quite different from that of statistical thermodynamics in the statistical mechanics of Gibbs [31]. D.Massignon indicated that the differences between the two mechanics were profound when we tried to quantize the theories or to study the fluctuations and

4

CHAPTER

1.

INTRODUCTION

correlations of the hydrodynamic quantities in the classical theory of turbulence. He constructed a theoretical framework of statistical hydrodynamics, which generalized Gibbs theoretical framework in his statistical mechanics. In Massignon's theoretical framework the "correct variables of Euler" were introduced instead of the corresponding local quantities in the kinetic theory and a rather detailed theory for the systems in equilibrium was developed. Precisely speaking, the correct variables of Euler at a given point are the arithmetic means of the corresponding variables of the molecules in a small neighborhood of the point in the three-dimensional physical space and the local quantities in the classical theory of the BBGKY hierarchy are the corresponding moments of a random variable attributing to the first molecule of N molecules, which obeys the marginal probability distributions of the 6N-dimensional distribution on the phase space of N molecules. Under the assumption of molecular chaos, it follows from the law of large numbers in the theory of probability that the correct variables of Euler coincide with the corresponding local quantities in the classical theory of BBGKY hierarchy, as N —> oo. If the assumption of molecular chaos does not hold, they are different in general. The moments of the random variable attributing to the first molecule of N molecules is not a legitimate candidate for representing the local quantity in fluid dynamics, i.e., the arithmetic means of the corresponding molecular variables of the molecules inside a small neighborhood of a spatial point, even if the distribution on the phase space is symmetric with respect to the group of the permutations of molecules. D.Massignon correctly elucidated the difference between two concepts, i.e., the correct Euler variables and the corresponding local ones, in statistical mechanics, which had been ignored by the experts in statistical mechanics for a long time, and constructed the equilibrium statistical mechanics on a solid foundation. Failing in constructing an asymptotic technique for the Liouville equation similar to the Enskog-Chapman expansion (

1.1. HISTORICAL

BACKGROUND

5

see, [16] and [25]) or something like that ( see, [74] ) for Boltzmann equations, Massignon did not get much progress in the theory for systems in evolution (i.e., non-equilibrium statistical mechanics). This might be the cause of the ignorance of Massignon's important work in the circle of the experts of non-equilibrium statistical mechanics for half century. Maybe the only expert on non-equilibrium statistical mechanics who has noticed Massignon's work was Yvon ([87]). But Yvon said that (Massignon's) work is too academic, therefore, useless in the theory of turbulence. On the other hand, discussing the statistical theory of turbulence developed by Reynolds, Taylor, Kolmogorov and Hopf, the late professor H.Grad stated in his last paper [35] published in 1983:

" ..., if the Navier-Stokes equations are considered to have a kinetic origin, an a priori probability has already been introduced into the n-particle dynamics (Liouville equation), and a second (Hopf) structure is redundant, incompatible, or (most likely) unduly enlarges the space. This possibility of discrepancy is compounded when (as is always necessary) approximations are introduced into one or the other probability structure".

Hence it is desirable to devise an asymptotic technique for solving the Liouville equation, if we are trying to construct a compatible molecular theory of the motions of general fluid flows (i.e., including both the laminar and turbulent flows, as called in classical fluid dynamics). About thirty years ago, V.N.Zhigulev, H.Grad, S.Tsuge, M.Lewis, T.Q.Chen and others started a study of the Boltzmann hierarchy (Zhigulev [88], Zhigulev and Tumin [89], H.Grad [34], S.Tsuge [75], S.Tsuge and K.Sagara [76], M.Lewis [54], T.Q.Chen [17], T.Q.Chen and M.E.Yuan [21], T.Q.Chen [18, 19]). The theory of the Boltzmann hierarchy is at

CHAPTER!.

6

INTRODUCTION

an intermediate stage between the kinetic theory of Maxwell and Boltzmann and the statistical mechanics of Gibbs. Although Zhigulev, Tsuge, Lewis , Chen and others successfully constructed techniques similar to Enskog-Chapman expansion, Grad's 13 moment method and Maxwellian iteration of Ikenberry and Truesdell for Boltzmann hierarchies, they got only little progresses in constructing a molecular theory of turbulence. In the last paragraph of his last paper [35] cited above, professor H.Grad criticized the theory of Boltzmann hierarchy existed until then:

"The literature in this subject ( i.e., the theory of Boltzmann hierarchy ) is quite sparse. Tsuge has considered Tollmien-Schlichting waves using a generalization of the 13-moment approach. Unfortunately, the problem is linearized, so that the inclusion of two-point correlations does not offer very much additional information.

Zhigulev and Chen have expanded the Boltzmann hierarchy using

Chapman-Enskog techniques; this basically nonlinear approach is restricted by the assumption that fluctuations are small compared to the basic flow, which is only slightly better than linearization.''''

The approximate solutions to the Boltzmann hierarchy obtained by using Enskog-Chapman technique are expanded in cumulants and the expansion starts with the local Maxwellian distributions.

The turbulent fluctuations of local

Maxwellian distributions always vanish. Therefore, Grad concluded that the approach using Chapman-Enskog technique is restricted by the assumption that (turbulent) fluctuations, which can be expressed in terms of cumulants, are small compared to the basic flow. I do not know exactly Grad's idea about the approach to the Boltzmann hierarchy. According to what stated in his last paper [35], I think, he might try to make a direct approach to the Boltzmann hierarchy without introducing any macroscopic quantities. In other words, a sequence

1.1. HISTORICAL

BACKGROUND

7

of distributions, which satisfies the Boltzmann hierarchy, on the phase space of iV-particles and its marginal subspaces instead of their moments (as used in the classical fluid dynamics) would be used in specifying the motion of fluid flows. If that were true, the approach would be formidably difficult. In 1955, C.B.Morrey, Jr. published a paper entitled " On the derivation of the equations of hydrodynamics from statistical mechanics " [61] in 1955. He declared: "This paper is also intended to be the first of a series of papers on this subject...". As far as I know, actually the paper was turned out to be Morrey's last paper on this subject. In his paper, Morrey successfully derived Euler equations. I think, Morrey would have devoted his second paper to the derivation of Navier-Stokes equations, if there had been no formidable difficulty on the way of doing that. Morrey constructed his theory directly from the Liouville equation without introducing a kinetic equation. This is one of its merits. But Morrey still studied a hierarchy and his start point is a local Gibbs distribution. Therefore Morrey's model excluded the turbulence phenomena too. In the present book, an asymptotic analysis for the Liouville equations will be worked out. The scaling in the asymptotic analysis of the present book is actually an extension of Morrey's fluid limit. But the theory developed in the present book is carried out in Massignon's theoretical framework. Hence the ingredients of the ideas of the present paper consists of Massignon's " correct Euler variables " and Morrey's fluid limit. I think, it is the most natural hydrodynamic limit along the line of the research works of Maxwell, Boltzmann, Gibbs, Enskog and Chapman. As a by-product of the asymptotic analysis for the Liouville equation worked out in the present book, some new phenomena for non-ideal fluid flows will emerge from behind the clouds produced by the hybrid product of the Boltzmann-Grad limit and EnskogChapman expansion in the classical kinetic theory. In order to remedy the fault indicated by Grad (i.e., this basically nonlinear

8

CHAPTER

1.

INTRODUCTION

(Enskog-Chapman) approach is restricted by the assumption that fluctuations are small compared to the basic flow, which is only slightly better than linearization ), about a decade ago the author tried to construct a non-equilibrium statistical mechanics directly based on Liouville equation ( Chen [19, 20] ). Being inspired by the work of E. Hopf ( Hopf [41] ) on the theory of turbulence, the author introduced a functional and obtained a functional equation governing its evolution from the Liouville equation without introducing additional assumption on the molecular distributions. But the functional introduced in [19, 20] contains far much more information than that contained in the functional Hopf introduced in [41]. In order to compress the contents of the functional solely to those which interest physicists, and get a functional equation governing its evolution, we should get the explicit form of an approximate solution to the Liouville equation. Hence we have to construct a rational perturbation scheme (or asymptotic analysis) for solving the Liouville equation. This is the very aim of the present book. Precisely speaking, a new functional H, which contains even more information than the functional introduced in [19] does, is introduced. A functional equation governing the evolution of the new functional H can be derived directly from the Liouville equation without additional restrictions on the distributions. A functional K, which is a generalization of the Hopf functional to the case of compressible flows, is defined as the restriction of the functional H to a linear subspace, generated by mass density functions, momentum density functions and energy density functions of the fluid flow, of the function space on which the H functional is denned. In order to obtain a functional equation governing the evolution of the restricted functional K, it is necessary to get an approximate, but explicit, form of the solutions to the Liouville equation. For getting it, an asymptotic technique (or perturbation scheme) similar to that of Enskog-Chapman expansion for the Boltzmann equations should be devised for the Liouville equations. The present

1.1. HISTORICAL BACKGROUND

9

book will be devoted to this task. Having obtained an approximate solution of the first order to Liouville equation, a functional equation, which is a generalization of the inviscid Hopf functional equation to the case of compressible inviscid flows, is obtained. It is not so easy, as it is thought of at first glance, to obtain an approximate solution of the second order to Liouville equation, because we are encountering the problem of solving an exceedingly large linear system of equations, of which even the existence of the solution is difficult to show. With the aid of the explicit form of the second order approximate solutions, a form of the functional equation ( precisely speaking, an infinite dimensional pseudo-differential equation ) with undetermined coefficients, which governs the evolution of the functional K, is obtained. To my surprise, in case of incompressible flows the equation thus obtained has a lot of new terms which do not appear in the Hopf functional equation. I think, the origin of the difference between the result of the present book and the classical one consists of the following facts. Firstly, the results of the present book are derived in Massignon's theoretical frame, which is profoundly different from the Maxwell-Boltzmann kinetic theoretical framework, because in the latter theoretical framework both the basic and the perturbed solutions must satisfy the assumption of molecular chaos and in the former there is no such restrictions. On the other hand, the scaling in the present book is quite different from that in the classical kinetic theory. There are two famous classical asymptotic limits: the asymptotic limit in the theory of Boltzmann equations and the fluid limit in Morrey's theory. The classical asymptotic limit in the theory of Boltzmann equations consists of two steps. The first step is the derivation of the Boltzmann equation from the Liouville equation, which is called the Boltzmann-Grad limit. The Boltzmann-Grad limit is a limit taken under the condition that the mean free path ( « l/(n . . . > v W ; . . . ; « )

= *(---;yi*),«-1y?),---,«-1yS;v?)>--->vW;...;t)

= *(--sqi"),4,),---,qS;vi"),-.-,vg;.••;*),

(1-20)

where $ is independent of K and m and q(is) = y[ s ) , q,(,) = K-'y?,

2 oo under certain specific conditions. Usually we assume that the limiting process N —> oo is taken under the following conditions: m AT = m ^ i V s - > K

(1.26)

s

and Ki < NK6 < K 2 , where K, Ki and K2 denote three positive constants.

(1.27)

20

CHAPTER

1.

INTRODUCTION

The equation (1.13) can be rewritten as follows: (S)

N.

r/-l

w / 7

+£ ti^dx? £ + ^V £ F ^

dt

JV.

i-1

£f< s >-(Z-l)f< (s)

+E

femyFi

fc=l

d dx

fc-1ax!

a Jdw|s),

F = 0.

(1.28)

Under certain plausible conditions, it can be shown (see the equations (1.23) and (1.24)) that N

- „ , (s• ;)

w

d_

at t

(

+££^

N

'

r'-1

(s)

JVs

d

a

a

|f-

AT.

E(ff-Y(xi s) ))-(/-l)(f/ s) -Y(x«))]^ y JF = 0. + ^m7(r^)I E (1.30)

For brevity, we assume that the external force field vanishes, i.e., Y(x) = 0 and the space V occupied by the N particles is of finite volume. Usually it is assumed that V is so large that the boundary effect on the behavior of N

1.2. OUTLINE OF THE BOOK

21

particles is negligible. In classical monographs (see, e.g., [28]), in order to avoid the boundary effect it is frequently to treat a system of infinitely many particles in the whole space R 3 with periodic structure in the space R 3 instead of a finite particle system, but we just treat finite particle systems with vanishing boundary effects. It is easy to see that the function of the form F(Z,*) =

(1.31) is a solution to the equation (1.30), i.e., a first order asymptotic solution to the Liouville equation, where T/v(- • •; • • •, • • •, • • •; • • •) denotes a function of hv + 1 arguments, v —

|V|/K3

being the number of cubes into which the space occupied

by the fluid is divided. Each cube corresponds to five arguments in the function T(- . . ; . . . , . . . , - . . ; . . •): the mass density of the molecules in the cube, the three components of the momentum density of the molecules in the cube and the sum of the intermolecular potential energy density and the kinetic energy density, of the molecules in the cube. We call the asymptotic solution (1.31) to the Liouville equation the density of a turbulent Gibbs distribution, or simply, a turbulent Gibbs distribution. Of course, the angular momentum density might play a role in the definition of turbulent Gibbs distribution too. But the size K of the cube is negligibly small in comparison with the macroscopic length scale, the total angular momentum of the molecules in the cube is approximately equal to the vector product of a constant vector and the total momentum. Thus the angular momentum density can be (approximately) expressed as a function in momentum density of the molecules in the cube. Hence it is redundant to include the angular momentum density of the molecules in the cube as an argument of

22

CHAPTER

1.

INTRODUCTION

T(- • • ; • • • , • • - , • • • ; • • • ) in the definition of turbulent Gibbs distributions. In classical equilibrium statistical mechanics the Gibbs (canonical) distribution is of the form Ce_/JH,

(1.32)

where H is the Hamiltonian of the system under consideration and C and f3 are constants depending on the system. If the system has a non-zero mean velocity u, the distribution (1.32) will be modified as follows: C e -/3(H-(iV m |u|

2

))

(132)

/

where N denotes the number of molecules in the system. For the sake of brevity, we shall call the distribution (1.32)' the Gibbs (canonical) distribution too. Now we consider a fluid continuum as a system of subsystems. Each subsystem corresponds to the aggregate of molecules in a small cube of the fluid continuum. We assume that the state of the molecules in each cube is a Gibbs (canonical) distribution constse-^"'-^"1!""!2), and that the subsystems are statistically independent. Then the system will be subject to a local Gibbs distribution Ce-^A(H.-(iv.m|u„P)]

s = (*i,s 2 ,*3).

(1-33)

It is evident that the local Gibbs distributions (1.33) and the convex linear combinations of several local Gibbs distributions with different parameters are special cases of the turbulent Gibbs distributions. Actually any convex linear combinations of turbulent Gibbs distributions themselves are turbulent Gibbs distributions too. The set of local Gibbs distributions (1.33) is a subset of the set of the turbulent Gibbs distributions and the latter is far much wider than the former. In accordance with the classical statistical theory of turbulence, any convex

1.2. OUTLINE OF THE BOOK

23

linear combinations of local Gibbs distributions should be included in the class of the distributions describing turbulence phenomena (of inviscid flows). Hence, among the molecular distributions, the turbulent Gibbs distributions might be the best candidate for describing turbulence phenomena (of inviscid flows). In non-equilibrium statistical mechanics without the assumption of molecular chaos the turbulent Gibbs distributions will play a role that the local Maxwellian distributions play in the classical theory of Boltzmann equations. It is evident that the turbulent Gibbs distribution of the form (1.31) depends on the subdivision of the space (1.12), i.e., on K and U, i = 1,2,3. But statistical mechanics is independent of the subdivision (1.12). Grad [33] said that statistical mechanics is a sort of asymptotic mechanics. Hence what we are interested in is not the behavior of a single turbulent Gibbs distribution, but the asymptotic behavior of a sequence (precisely speaking, a net or a filter) of turbulent Gibbs distributions. The mathematical subtleties of the theory will be touched upon in Chapters VI and VIII. In Chapter VII a functional equation, called the Euler .FiT-functional equation ( i.e.,the equation (7.19)), is derived from the //-functional equation under the assumption that the probability distribution on the phase space takes the form of a turbulent Gibbs distribution just as the Euler equations are derived from the balance equations under the condition that the probability distribution on one-molecule phase space takes the form of a local Maxwellian distribution in the classical theory of Boltzmann equations. In case of incompressible flows, the Euler K-functional equation is equivalent to the Hopf functional equation for incompressible flows in the theory of turbulence. In case of deterministic fluid flows, the Euler /iT-functional equation will be reduced to the Euler equations in classical fluid dynamics (including the equation of continuity, the momentum equations and the energy equation). Hence we can say that the Euler ff-functional

24

CHAPTER

1.

INTRODUCTION

equation is the equation governing the evolution of the statistical solutions of Euler equations, i.e., the solutions of the Euler equations with random initial data. In other words, the Euler ^-functional equation governs the motion of turbulent inviscid flows. We have seen that the approach in the first seven chapters of this paper is acceptable: it almost coincides with the statistical theory of turbulence in the classical fluid dynamics. The only difference between them is as follows. In our approach a general flow, including both the laminar and turbulent flows in classical fluid dynamics, is treated in a general theoretical framework, and the laminar (or, deterministic) flows are considered to be special cases of the general flows, i.e., those with vanishing variances. In classical fluid dynamics, a general flow is laminar (or, deterministic) and turbulence phenomena is the consequence of the instability of a basic (laminar) flows. The theoretical framework of our approach was constructed by Massignon ([58]) about half century ago. It is wider than the classical one, which is still accepted as true among most experts in the circle of fluid dynamics. According to my opinion, in Massignon's theoretical framework a large number of famous difficulties in the classical theory of transition from laminar flows to turbulent flows will be solved easily or disappear automatically. In Chapter VIII the connection between the turbulent Gibbs distribution and its corresponding fcT-functional is established by making use of the inversion formula for Fourier transform. The connection obtained in this way will be used in asymptotic techniques for the Liouville equation in deriving a finer /('-functional equation. In order to devise a scheme for solving Liouville equations, which is an analogue of the Enskog-Chapman technique for Boltzmann equations, we have to solve a local inhomogeneous stationary Liouville equation, of which the inhomogeneous term is obtained through tedious calculation in Chapter IX. In Chapter X we have got an explicit form of the solutions of the inhomogeneous station-

1.2. OUTLINE OF THE BOOK

25

ary Liouville equation obtained in the Chapter IX and, therefore, a form of the second order approximate solution of the Liouville equation (with undetermined coefficients) can be explicitly written down. In Chapter XI a functional equation governing the evolution of the AT-functional and finer than the Euler JiT-functional equation is derived with aid of the second order approximate solutions to the Liouville equation obtained in Chapter X. A lot of new transport phenomena emerge in the calculations in Massignon's theoretical framework. In Chapter XII some concluding remarks are made.

This page is intentionally left blank

Chapter 2

if-Functional 2.1

Hydrodynamic Random Fields

The 6TV-dimensional phase space for a system of N particles (molecules) is denoted by T = V w x R 3iV , where V denotes the space occupied by the N particle system. The representative point of T is Z = (xi, x 2 , • • -, x # ; v 1 ; v 2 , • • •, v ^ ) , where Xj = (XJI,XJ2,XJ3)

and Vj = (VJI,VJ2,VJ3)

denote the position and velocity of

the j " 1 particle respectively. The motion of the system of N particles is governed by equations:

n>^=f„

(2-1)

N

f,- =

]T

f(Xj-xfc)+Y(Xj),

f(x) = - ^ ' ( | x | ) = - g r a d ^ ( x ) , lxl

(2.2)

(2.3)

where m denotes the mass of the particle ( molecular mass ), f (x 7 — Xfc)= the intermolecular force of the fcth molecule exerted on the j t h molecule, tp the intermolecular potential, Y the external force field. Usually we write ip(x.) = V(l x l)i the function tp on the left hand side denoting a function of three variables and the function \p on the right hand side a function of one variable, but according to the context it will cause no confusion. Y the external force field. In order to 27

28

CHAPTER 2.

H-FUNCTIONAL

include the case of molecules with hard cores, the function ip might be supposed to be of the following form:

{

oo,

if |x| < do;

finite values,

otherwise,

(2.4) where do is the diameter of the hard core of the molecule. Usually we assume that V'(lxl) decays rapidly as |x| —> 0. Sometimes we assume that ip is of compact support and the diameter of the support is far less than K, the diameter of a fluid particle in classical fluid dynamics. An equivalent system of equations of (2.1) is (2 5)

m

~A=^

^T = v "

-

Equations (2.5) are called the Hamilton's system of equations for the system of N particles. If we introduce the Hamiltonian of the system: N

|

,2

,

N

N

N

H - E ^ + ^ E E *(l*,-**!) +5X*i), J'=l

j = lk=l;k=£j

j= l

where U denotes the potential of the external force field: Y = -grad U, the equations (2.5) can be written in a rather symmetric way: dxj _ _1_ dE_ dt m &Vj'

rfvj _ dt

1 9H m dxj

(2.6)

The equations (2.5) ( or, equivalently, (2.6) ) define a flow on the phase space T. Liouville proved that this is a volume-preserving flow on T ( H. Goldstein[32] ). In statistical mechanics we study a probability density F(Z,t),

depending on

time t, on the phase space T instead of the volume-preserving flow on T defined by the equations (2.5). The probability density F(Z,t)

is always assumed to be

symmetric with respect to any substitutions among the N particles. The basic assumption in statistical mechanics is that the flow defined by the Hamilton's

2.1. HYDRODYNAMIC

RANDOM FIELDS

29

equations is probability-preserving with respect to F(Z,t).

As a consequence

of the assumption and the volume preservation of the flow on the phase space T, Liouville derived the following equation governing the evolution of the probability density F ( Z , t ) : dF

^

8F

J^f,- dF

0

n

-at+E-r^ + Zi-wr -

,„„. (2J)

The equation (2.7) is called the Liouville equation. Being a probability density, F should be nonnegative and its integral over the phase space T should be unity. Furthermore we assume that F will decay sufficiently rapidly to zero as the total intermolecular potential tends to infinity. Precisely speaking, F should be dominated by an exponential function with a negative multiple of the total intermolecular potential energy as its exponent, in particular, F will vanish wherever the total intermolecular potential energy is infinity (i.e., whenever at least two hard cores of the molecules touch or overlap with each other). Under this assumption, the value of the ^-functional, which will be defined later, is independent of the values of the function ip(x.) in the ball |x| < do- We can modify the value of the function i/>(x) in the ball |x| < d 0 arbitrarily without changing the value of the H functional and without violating the Liouville equation. Henceforth we always assume that tp(x) will vanish whenever |x| < do- Therefore the function ip can be considered a tempered distribution in the sense of L. Schwartz and its Fourier transform %j), as a tempered distribution, is meaningful. The solutions to the Hamilton's equations are the characteristic curves of the Liouville equation. Thus solving the Liouville equation is equivalent to solving the Hamilton's equations. But the Liouville equation provides the possibilities of looking for an approximate ( or, asymptotic ) form of the solution of the physical problem, which is not easy to see from the Hamilton's equations directly. I think, this is the essence of the statistical mechanics. In order to describe the macroscopic state of the flow, we introduce the follow-

30

CHAPTER 2.

H-FUNCTIONAL

ing five random fields, called the hydrodynamic random fields of the N— particle system: N

ft(y,u,Z)

= m]T«5(xf-y)exp(-27riu-v,)Y,(xi), r

N

j = 1,2,3;

(2.8)

N

p 4 (y, u, Z) = m Y^ s(xl ~ y) exp(-27riu • v*) - ] T V(xfc - xj) + U(xt) , (2.9) fc=i

N

/95(y, u, Z) = m ] T (xk - xj) + l/(x,)

(2.16)

fc=i,fc#/

l=i

AT

P5(y,0,Z)

=m^5(xi-y),

(2.17)

N

| p ( y , 0 , Z ) = - 2 m 7 r i £ v « • J(x, - y ) .

(2.18)

The physical meanings of the right hand sides of the equations (2.15)-(2.18) are as follows. The right hand side of the equation (2.15) represents the product of the molecular mass and the external force density at y. The right hand side of the equation (2.16) represents the product of the molecular mass and the potential energy density at y. The right hand side of the equation (2.17) represents the fluid mass density at y. Finally, the right hand side of the equation (2.18) represents the product of (—27ri) and the momentum density of the fluid at y. The quantities in the equations (2.15)-(2.18) are just those the fluid dynamics concerns itself with. Hence the above five random fields contain more information than that the experts of the classical fluid dynamics are interested in. The advantages of the five random fields are as follows. Firstly, their evolution is governed by a closed functional equation for any distributions satisfying the Liouville equation. Secondly, the functional equation governing their evolution can be used to deduce equations governing the evolution of moments, even more than the ordinary five moments (e.g., Grad's thirteen moments). The following formulas for the (infinite dimensional) partial derivatives of Hfunctional are well known and will be used in the sequel: — (G)M - -2ni JF(Z,t)exp[A] Jf

mdydudZ,

(2.19)

32

CHAPTER 2. H-FUNCTIONAL

3 0 ^ ( 6 ) ^ 1 , ^ 2 ] = (-2rri) 2 jF(Z,t)exp[A]

JJ\plPkdydu

J

J

where

,„ „.

-.-UAH-*)/(«-),•

( | x , - x * | ) + £/(*)

dydu,

(3.3)2

k^i

^

at

)

= ^

/13

ff8(y, u) •

/

/" dZ F exp [A] ( - 27rmi)v,

,=1-/

gx

exp(-27riu • vi)dydu,

(3.3)3

/

f ^ ) \ u'^)^(gx~y)

exp(-27riu •

vfiYjMdydu

36

CHAPTER

3. H-FUNCTIONAL

EQUATION

N

= ^2

dZFexp[ J 4](27rmi)vj

//E0i(y>u>*)a*(yy) N

r

-27rmi V J=I

86,

•lit

ex

/ ^

3

N

= m J2 f dZ F exp[A] ff



9# 2 exp(27riy • 0 ^ — | ^ ( y > u ) > exp(-27riy • £) u ) - exp(-27riy • £)8(u) dy • du

and the convergence of the integral on the right hand side of the equation (3.6). The physical meaning of the inclusion of the last term 1 dH,

2-Tri de5 (©)

dej ,

dy-d^{y>U\

.

40

CHAPTER 3. H-FUNCTIONAL

EQUATION

in the braces on the right hand side of the equation (3.6) is that the term corresponding to the self-interaction of one molecule must be excluded, because the intermolecular potential is equal to infinity at the origin:

V>(o) = o. Actually "Pf." on the left hand side of the equation (3.6) denotes the "partie finie de Hadamard", of which the definition can be found in [69] and [38]. For the purpose of the present paper, it is sufficient to memorize the definition (3.6). Similarly we can derive the following expressions for the other four terms on the right hand side of the equation (3.3) consecutively. P r o p o s i t i o n 3.5

(°Z) \dt J

13

B I«?( 9) 27ri d6 K '

(3.7)

5

Proof

\dtj13 = J2

fdZFexp{A}(-2irmi)vi

= J2 [dZFexp[A](2irmi)vt

• ff 65(y, u ) ^ ^ — ^ exp(-27riu • v,)dydu

• ff fl6(y, u ) d S ^ ~ Y^ exp(-27riu • vf)dydu

l=i

N

= Y^ fdZFexp[A}(-2irmi)xi

• ff

" ^ ^

V^y^Ei r*i /7a^(y,u) = 22 J dZFexp[A]mJJ —^T^(

x

S(xt - y) exp(-27riu • vt)dydu

, ' ~ v)

. J2 J dZFexp[A] J J y ^ S f o

0exp(-2iriu-V|) ~Q^

d dn

y

- y) exp(-27riu • v,)dydu

3.1. DERIVATION

OF H-FUNCTIONAL

5

27Tid05

41

EQUATION

dy • 9u

(y,u)

The proof of the P r o p o s i t i o n 3.5 is completed. P r o p o s i t i o n 3.6 r

dH\

3

1 dH.^J^dej.

dYi( ,1

Proof

(*)--£/ / /

]T9j{y, u, *)<S(x, - y)exp(-27riu • v ( )—^(x ( )dydu

3

N

= f^

fdZF

exp[A}m ff £

z=i ^

- £

dZFexp[j4](-27rmi)v,

•'•' j=i

fdZFexp[A)m

9exp(-27riu-vi) dYs 0j (y, u, t)*(x, - y) L_^(y)dydu

f f (y,u,t)^(y)u.Y( y ) m dt J 21 5^5 .7=1

CHAPTER 3. H-FUNCTIONAL EQUATION

44

+- ;Lpf.y^?(0-S(©> del

uexp(27riy-0 $ ^ *i(y> u> ' ) y j ( y ) . exp(-27riy^)«(u) j=i

(3.12) Proof

at

•

]

= -4TT2 fdZj2 %Fexp[A]

/21

•/

,

= 1

Yl ^ " ( y ' u ' i ) ^ ( x ' ) ^ ( x ' ~ y) ' exp(-27riu • v,)udydu

= -4TT 2 / dZ ^ ^ f ( x ;

-x f c )Fexp[A]

" / / 5 Z ^ ( y ' u ' i ) y j ( x ' ) ( 5 ( x J - y) exp(-27riu • v;)udydu

-4TT2

/dZ^Y(x;) •>

i=i

ixp[A] / / 5 Z ^ ' ( y ' u ' * ) 5 G ' ( x 0 * ( x J ~ y)exp(-27riu • v;)udydu

AT

= -4TT2 UZJ2J2

exp(27ri^ • (x, - xfc))f(Od£ Fexp[^]

• / / ^^(y,u,i)y j (x;)(5(x i -y)exp(-27riu-v;)udydu AT •4TT^ j

dzY,n*i) i=i

3.1.

DERIVATION OF H-FUNCTIONAL EQUATION

45

Fexppl] / / y~]9j(y, u, t)Yj(xi)6(xi - y) exp(-27riu • vfiudydu

i/*fe){f(e) uexp(27riy-£)]r0i(y,u,i)y,-(y), exp(-27riy-f)«(u) 27rm1~(e)

+

2-KidH m 9^5

uE«,(y,u,%(y) }

(©) 5>(y>u>*)?i(y)ii-Y(y)

1 /" d H T ^Pf. / ^ f ( O - ^ r ( e ) u e x p ( 2 7 r i y . O ] [ > ( y . M ) ^ ( y ) . exp(-27riy-0*(u) 5

L

^=

m 305l;

1

X>(y,u,t)vj(y)u-Y(y)

The proof of the Proposition 3.10 is completed. Proposition 3.11

(f) 22 a 3 g (9) 'del 1

+

=

-imb Pf -//^ d ^ (r?)?(e)

uexp(27ri(£ + T?) • y)0 4 (y, u), exp(-2?ri£ • y)<J(u), exp(-27ri77 • y)

max . i< , ^ 3 l ^ i - «_,, l < ,«. /. ,2„

d_ dz

dy • du

dZFK" 3 m •

J

d\

v

J2

2

^

»M)

l«/

(3.28)

2

The right hand side of the equation (3.27) represents the mean of the sum of the external forces exerted on the molecules in the small cube of side length K around z. In deriving the equation (3.28), we have used Newton's third law, i.e., the function f being an odd function. Hence we have

Yl

$3/i(xi-Xfc)

!«/2

If the intermolecular force is supposed to be of short range ( i.e., the range of the intermolecular force is far much shorter than the side length K of the small cube, i.e., the size of the fluid particle ), the sum of the first term on the right hand side of the equation (3.26) and the right hand side of the equation (3.28) represents the mean of the first component of the divergence of the pressure tensor. The second term on the right hand side of the equation (3.26) represents the divergence of the mean of the first component of the quantity (u • V)u in fluid dynamics, where u denotes the velocity of the fluid particle at z. Combining the

60

CHAPTER 3. H-FUNCTIONAL EQUATION

equations (3.19), (3.25),(3.26),(3.27) and (3.28), we obtain the balance equation for the first component of the mean of the momentum of the fluid particle: fdZFn-3m dt J ' —

^ max

i_

i?f