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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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<72)) is held positively constant. The result under the BoltzmannGrad limit is valid only for dilute gases. If you are trying to get macroscopic
10
CHAPTER
1.
INTRODUCTION
equations governing the motion of fluids from the Boltzmann equation, a second limiting procedure, the so called Enskog-Chapman technique, has to be made. The latter is an asymptotic limit taken under the condition that the mean free path (w l/(n<72)) shrinks to zero. These two limits are logically inconsistent, at least, formally. Usually one believes that the results of the above iterated asymptotic limits are valid for cases of positive, but very small mean free path, although we lack a rigorous theory to show its truth and even lack its precise formulation. Morrey's fluid limit is a simple (i.e., not iterated) asymptotic limit taken under the condition that ncr3 is held positively constant. Morrey succeeded in deriving the Euler equations under a rather strong condition, but failed in the derivation of the Navier-Stokes equations. Grad pointed out [35], "(Under Morrey's fluid limit), the mean free path shrinks to zero, as do the transport coefficients; thus the limit is a macroscopic, ideal fluid that presumably satisfies the Euler equations ". It should be noted that Grad's argument is somewhat paradoxical. Although the Boltzmann-Grad limit is taken under the condition that the mean free path tends to a positive constant, but the Enskog-Chapman aymptotic limit, under which the transport phenomena emerge, is taken under the condition that the mean free path shrinks to zero. Under the fluid limit devised in the present book, which is an analogue of Morrey's in Massignon's theoretical framework, new phenomenon different from the classical transport phenomena emerge. This is a result of the asymptotic analysis of the Liouville equation under the fluid limit worked out in the present book. In the present book, the new terms of the functional equation corresponding to the second term in the asymptotic expansion of the solution to the Liouville equation relate to heat energy, or temperature, and its spatial derivatives.
1.2. OUTLINE OF THE BOOK
1.2
11
Outline of the Book
The present book is organized as follows: In Chapter II, an .ff-functional, which generalizes the iJ-functional introduced in [19], is denned as the characteristic functional of the five random fields: N
Pj{y,\i,Z)
= m ^ ( 5 ( x ; - y ) e x p ( - 2 7 r i u • vj)lj(xi),
j = 1,2,3,
(1.1)
^(xfc-xj) + r/(xj)
(1.2)
i=i
N
p1
Pi ( y , u , Z ) = m ^ < 5 ( x / - y ) e x p ( - 2 7 r i u - v i ) i=i
N
^Z k=i,k&
N
p 5 (y, u, Z) = m ^
5(xi - y) exp(-27riu • vj),
(1.3)
(=i
where x^ = (£fci,£A:2,£fc3) and vjt = (vki,Vk2,Vk3) denote the position vector and velocity vector of the kth particle respectively, V(xfe
—x
i ) denotes the inter-
molecular potential between the fcth particle and Zth particle, Y = (Yi, Yb,!^) the external force field and U(x) its potential. It is easy to see that N
p 5 (y,0,Z) = m ^ 5 ( x , - y ) i=i
and
^ 5 ( 9u U ' Z ) ( y '°' Z ) = - 2 7 r i m £ V ^ X ' - y ) i=i
The right hand side of the first equation represents the fluid mass density at the point y and that of the second equation represents the (—27ri) multiple of the fluid momentum density at the point y. They are the two quantities the experts of the fluid dynamics concern themselves with. The quantity ps(y, u, Z) contains far much more information than the above two quantities, i.e., the fluid mass density and the fluid momentum density, do. Therefore the equation governing the evolution of the former is far much more complicated than the latter. The reason why we introduced the quantity ps(y, u, Z) into non-equilibrium statistical mechanics is that its evolution is governed by a closed equation, which can be
12
CHAPTER 1.
INTRODUCTION
deduced solely from the Liouville equation without any additional assumption on the distributions, but there are no closed equations governing the evolutions of the fluid mass density and the fluid momentum density, unless some additional restrictions are imposed on the solutions to the Liouville equation. Because the total energy density and the external force field play great roles in Hamilton mechanics, besides the quantity ps(y, u, Z), the quantities Pi(y, u, Z), i = 1,2, 3,4 have to be introduced into non-equilibrium statistical mechanics too. Having introduced the five random fields pi(y,u, Z),i = 1,2,3,4,5, we can write the //-functional as follows: H(0;t)
= H(e1,e2,e3,e4,65;t)
= f dZF(Z;t)exp[A},
(1.4)
where A = -2m
11^2ej(y,u,t)Pj(y,u,Z)dydu, •'•'
(1.5)
j=i
and 6 = (01,02,03,04,05)-
(1-6)
The experts of statistical physics concern themselves only with a few moments of the molecular velocity distribution and those of fluid dynamics only a few moments of the fluid particle velocity distribution in the theory of turbulence. They encounter two closure problems in a similar, but actually quite different, way (see, [35]). In the theory of BBGKY hierarchy (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]), various techniques were devised to deal with the closure problem in statistical mechanics. In 1952 E.Hopf [41] introduced the characteristic functional of the fluid particle velocity distribution to overcome the closure difficulty in the theory of turbulence. We are facing two closure difficulties simultaneously. Hence the //-functional (1.4), which is the characteristic functional of the characteristic functions of the molecular velocities at y, the characteristic functions of the molecular velocities
1.2. OUTLINE OF THE BOOK
13
with the total energy at y as its power and the characteristic functions of the molecular velocities with the external force at y as its power, should be introduced into the non-equilibrium statistical mechanics. In Chapter III, an iJ-functional equation, (i.e., the equation (3.16)), which governs the evolution of the if-functional, is derived from the Liouville equation without any additional restrictions. Hence the ^-functional equation (3.16) governs the evolutions of any i7-functionals corresponding to general solutions to the Liouville equation. It plays the role of the balance equations in the theory of Boltzmann equation, which are satisfied by the moments of the general solutions to the Boltzmann equation. The difference between them is that the iJ-functional equation is a closed equation, but the balance equations are not. In the classical theory of Boltzmann equations, in order to get Euler equations we have to restrict the general solutions of the Boltzmann equation to local Maxwellian distributions, which are asymptotic solutions of the Boltzmann equations. A parallel approach to asymptotically solving the Liouville equation will be carried out in the Chapters IV-VII. Because the experts of the classical fluid dynamics concern themselves only with mass density field, velocity field and energy density field, in Chapter IV, a .fir-functional, which is a restriction of the if-functional to the linear subspace spanned by the mass density field, velocity field and energy field, is introduced as follows:
K(a,b,c;t)=
f dZF(Z,t)exp
i - 2-Tri /
+b(v)m • £ U - ^1*2 W , _ y) + 1=1
V
m
a ( y ) m ^ < 5 ( x j - y)
c(y) (™ V
J
£ | Vl ,2j (x| _ y)
* L=l
N 1 N M l + - ^ ^ # l - x f c | ) i ( x 1 - y ) + ^[/(xl)5(x,-y) dy
l=i
fc#i
l=i
/J
J
14
CHAPTER
1.
= H(0i,02,03,04A ;t),
INTRODUCTION
(1.7)
where ej=-biws(v)t
,
j = W ;
(18)
c(yWu) m
b(y) a*(u) c(y) 05 = a(y)8(u) + ^ -^ - ^ A2 u * ( u ) . 2ni
'
du
8TT
(1.10)
The JC-functional thus defined contains all the information the experts of the classical theory of turbulence concern themselves with. At the end of Chapter IV, the connections between the derivatives of Hfunctional and those of AT-functional, which is a restriction of the iJ-functional, are shown. These connections will be used in the derivation of a functional equation, which the /^-functional for inviscid flows should satisfy. In Chapter V, a class of distributions, called the turbulent Gibbs distributions, is introduced as a class of asymptotic solutions to the Liouville equation. The turbulent Gibbs distributions are generalizations of the convex linear combinations of local Gibbs canonical ensembles. We assume that the intermolecular force is of short range. We can divide the space occupied by the fluid into a large number of cubes of the same size and with their faces in parallel with one of the coordinate planes. The cubes are assumed to be so small that the arithmetic means of the molecular variables of the molecules in each cube will be used as the corresponding macroscopic variables of the fluid particles in fluid dynamics and the aggregate of the molecules in the cube (or , for brevity, the cube) plays the role of a fluid particle in fluid dynamics. On the other hand, the sides of the cubes are assumed to be far much longer than the range of the intermolecular force so that the pairs
1.2. OUTLINE OF THE BOOK
15
of the interacting molecules, which are situated in different neighboring cubes, occupy a negligibly small fraction among all the pairs of interacting molecules. Hence the total intermolecular potential energy between two molecules in different cubes is negligibly small in comparison with the total sum of intermolecular potential energy between two molecules in the same cube. In the classical fluid dynamics or thermodynamics, the internal energy for a fluid particle is defined as the sum of the heat energy of the fluid particle (i.e., the cube representing it) and the total intermolecular potential energy among the molecules inside the particle (i.e., the cube representing it). We usually assume that the internal energy is an extensive property of the particle, i.e., the additivity of the internal energy. In other words, it has been implicitly assumed that the total intermolecular potential energy between two molecules in two different neighboring fluid particles (i.e., the cube) is negligibly small in comparison with the total intermolecular potential energy among the molecules inside the same particle (i.e., the cube). Therefore the assumption previously made about the subdivision of the space into a large number of cubes is consistent with the classical thermodynamics and fluid dynamics. In the sequel, we assume that the space occupied by the fluid has been subdivided into a large number of cubes in the manner stated above. Having done that, we can write down the Liouville equation in the following way: dF s xjECs
L
s x;€C3
S
XieC8
t?,3
xtecs
Xfc6ct
where s = (si,S2,S3) is a triple integer-index, C s are small cubes into which the space occupied by the fluid is subdivided: Cs = { y = (2/i,y2,2/3);(si-l/2)K + Zi
16
CHAPTER 1.
INTRODUCTION
K denoting the side length of the small cube and 0 < k < K, i = 1,2,3. The right hand side of the equation (1.11) is negligibly small. Precisely speaking, it is plausible to assume that
E E =£ E E * - » ) • « E k E '<*-*<• S
XjGCs
t;is
Xfe6Ct
In order to get an asymptotic solution to the equation (1.11), we would like to neglect the right hand side of the equation (1.11) and rewrite the equation thus obtained in the following way: d
dx ( s ) N.
-(/-i)
d
dx (s)
i-i
+ f^m^(l-l)l[^{ E E^-c-itf (<•)
9w'(•)
^F = 0,
(1.13)
where JVS denotes the number of the molecules in the cube Cs, x\(s)' the position of the Ith molecule in the cube C s , fj
the force exerted on the Ith molecule in
the cube Cs and w| 8 ' is defined as follows:
/ „,(•) \
/ v(s) \
(s)
wN , l (•)
= Av ( 5 ) ,
wW =
(1.14)
Ns2
w
(s) V13
(•)
\ w^
I
\ u (s)
/
where v} 8 ' = (v^ , v^ , w(j') denotes the velocity of the Zth molecule in the cube
(1.15)
17
1.2. OUTLINE OF THE BOOK and Ss = 1
1
1
1
1
0
0
0
-2
0
0
-3 \/l2
0
1
-i
V5
V2
1
1
V5 1 \/l2
l Vl2
1 Vl2
V(iv.-2)(7\r.-i)
y/(Nm-2)(N.-l) -(N.-l) y/(N.-l)N.
1 y/(N.-l)Nm
I
V
y/(N.-l)N.
)
(1.16)
We introduce a set of new variables y( s ) as follows:
x<s> JV.l
x<s>
W12
,(»> -
=A
12
= Ax< s ».
(1.17)
x<s> JV.2
x<s>
tf
13
13
The side length n of the cube (i.e., the size of the fluid particle in the classical fluid dynamics) is an infinitesimal (in comparison with the macroscopic length scale) and the intermolecular potential ip is assumed to depend on the parameter K and the molecular mass m in the following way: y,(|x|)=m~f
|Xi
(1-18)!
The external force Y is assumed to be of the form: Y(x) = mT(x),
(1-18)2
18
CHAPTER!.
INTRODUCTION
where 5 and T are functions independent of n and m. Moreover we assume that the distribution function F depends on K in such a way that dF(Z-t) dy\a)
=
1 «
f0r/>2.
(1.19)
The equation (1.19) will be satisfied if F is of the form: F(Z;t) = F ( . . . ; y [ " ) , y ? ) , - - - , y g ; v W > . . . > 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
(1.21)
C.B.Morrey, Jr. ([61]) has proposed assumptions similar to (1.17), (1.20) and (1.21). But Morrey has not introduced the subdivision of the space occupied by the fluid into a large number of cubes and the variables y, , I = 1, • • • ,iVs, in his theoretical framework. In Morrey's theory, as well as in all the theories of BBGKY hierarchy, the first variable xi among all the variables xj, I = 1, • • •, N plays the role y\
in the theoretical framework in the present book, i.e., the
Massignon's theoretical framework. The latter represents the position of the mass center of the molecules in the cube Cs, i.e., the position of the fluid particle represented by the cube Cs. Hence Massignon's is far much better than those in the theory of BBGKY hierarchy in describing the reality. I think, ease in making calculations is the only cause why the local quantities were introduced in the theory of BBGKY hierarchy in a way inconvenient in describing the reality. Hence Massignon's correction should be considered reasonable. Of course, the complexity of calculations is the inevitable price for the reasonability.
1.2. OUTLINE OF THE BOOK
19
We should point out that the condition (1.19) is wider than (1.20). The asymptotic limit taken in the present book satisfies the condition (1.19), usually not (1.20). It is easy to see that the terms containing the spatial derivatives of F on the left hand side of the equation (1.13) can be written as follows: N
'
(s)
N.
a
w
—^ .w, Z / i r a . . (ai ) ^ Z ^
<•)
Ei-(M) dxd (s)
ti v ^ ax|-> U VtTW L ti d4 N.
(•)
E-i i=i
"yi fly!" j
^=w<^+fS 3yi
»
1=2
dF d
(8)-
(1.22)
y\
Owing to (1.19), the first term on the right hand side of the equation (1.22) is negligibly small in comparison with the second term. Precisely speaking, w
(<0 dF (»)
O(l)
(1.23)
dy'i
and Nm
E 1=2
(S) dF dy{*)
= K-M1)-
(1.24)
On the other hand, we have
f(x) = - | ^ ( | x | ) = - ^ ( V S ) ( M ) .
(1.25)
As was pointed out by Grad ([33]), statistical mechanics concerns itself with the asymptotic behavior of the iV-particle system as N —> 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-<'-^]^H
+ ^Em v ^ i j i
(1.29)
Neglecting the higher order infinitesimal W.
a
k 8i
(»)
a
+££5fe • ^^VKa4"l
in the equation (1.28), we have the following approximate form of the equation (1.28), which will be used to get the first order asymptotic solution to the Liouville equation: Ns
(<0
i-i
E d4 S ) £s 1.£1=2VU^Wnt^dx] W,
a-i)
a ax{'>
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 <5(xf - y) exp(-27riu • vj).
(2.10)
i=i
2.2
^-Functional
Definition 2.1 The characteristic functional of the above five random fields (2.8), (2.9) and (2.10) is called the if-functional (hydrodynamic functional) for the probability distribution F(Z;t)
on the phase space T = *VN x R 3JV and
denoted by H(Q;t) = H(e1,e2,e3,e4,95;t)
= fdZF(Z;t)exp[A],
(2.11)
where 6 = (0i, 02,03,04,0s)
(2.12)i
A = -27ri ff Y^Ojiy, u,t) P j (y, u, Z)dydu.
(2.12)2
and j=i
In the sequel, we always assume that the forms of dependence of 9j, j = 1, • • •, 5 on t are always as follows: 0 i (y,u,i) = t0i(y,u), 0 fc (y,u,t) = 0k(y,u),
1 = 1,2,3;
(2.13)
k = 4,5.
(2.14)
2.2. H-FUNCTIONAL
31
It is easy to see the following special values of the above five random fields and the derivative of the random field p5 with respect to u at (y, 0, Z): N Pj(y,0,Z)
= mYlS(xi-y)Yj(xi),
j = 1,2,3;
(2.15)
i=i N
ri N (y,0,Z) = m ^ < 5 ( x i - y ) - J ] i>(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
etc. Occasionally we shall use some formulas expressing integrals ( or antiderivatives ) of the if-functional like (2.19) and (2.20). We will not derive them until they are needed. The functional equation governing the evolution of the Hfunctional will be derived in the following section.
Chapter 3
i7-Functional Equation 3.1 Derivation of //-Functional Equation According to the definition of tf-functional (2.11), (2.12)i, (2.12) 2 , (2.13), and (2.14), it is easy to see that the H-functional depends on t through both
F(Z,t)
and 0j(y,u, t)(j = 1,2,3) in the expression (2.11). Thus we have dH = IdZ— dt
3
exp[A] - 27ri / dZFexp[A]
ff ] T ^(y,u)p,-(y,u, Z)dydu. 3= 1
(3.1) Making use of the Liouville equation (2.7) and the formulas (2.19) and (2.20) for partial derivatives of the if-functional, we have the following: Proposition 3.1
The temporal derivative of the H-functional corresponding
to the distribution density F, which satisfies the Liouville equation (2.7), is of the following form
dH (dH\ ^ (dH\ dH _ (dl£\ (2E\ + (dH\ (dH>
where
,„ „.
-.-UAH-*)/(«-),•
<32)
{™)r-jMp.S£,MAl
(3.2),
33
CHAPTER 3. H-FUNCTIONAL
34
EQUATION
and 'c>H\
dH
0j(y,u)K}(y)
(3.2)3
Integrating by parts, we obtain the following expression for the quantity in the equation (3.2): P r o p o s i t i o n 3.2
(fX-Z-S-S-^-tCf),,. <-> where (6H \ dt / 1 1
e
i&>u'*)^^~
/ / £
\dt)12
dZFexp[i4](-27rmi)vj l=1J
exp(-27riu • vfiYjWdydii,
• / Y 0 4 (y, u ) 8 < 5 ( x ^
= J2[dZFexp[A](-2mni)v,
1
y)
(3.3)!
exp(-27riu• v,)
N
- 5 > ( | 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 ^ ) \ <" / 14
= -2m7ri]T
[dZFexp[A]vi
j _ i •>
J J E ^ ( y ' u ' * ) ^ X ( ~ y) e x P(- 2 7 r i u • vi)-Q^(xi)dydu,
(3.3)
3.1. DERIVATION
OF H-FUNCTIONAL
^ )
II
= -2m7ri f"
= -2m7ri££ '16
^
Remark
fdZF
exp{A}vt
0 4 (y, u)6(xi - y) exp(-27riu • v*) — ( x / ) d y d u ,
( ^ )
• Jl
35
EQUATION
j=
1 k^i
(3.3)5
fdZFeM^i J
e4(y, u)5(xfc - y) exp(-27riu • v f )
^^
*">dydu,
(3.3) fl
In deriving the equation (3.3), we have assumed that the probability
density F approaches zero so rapidly as |vj| or |x(| (I = 1,2,..., N) tends to infinity that the boundary terms arising from integration by parts will vanish and we have used the easily verified fact that d 3-x.i
N
N
GEE *(i*i-*i))=£|£(i*« \
j=i
Xfcl).
k=i
Making use of the formulas (2.19) and (2.20), etc, for partial derivatives of the iJ-functional and integrating by parts repeatedly, we obtain the following expression for ( ^ r ) n : P r o p o s i t i o n 3.3
V at y n
(3.4)
2it\debK '
Proof dH
at
) -±J
dZ F exp[A](-27rmi)v i
/ll
l=1J
3
/ / E Wy>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
<9u
= - m ^ y
dZFexp[A]vi
(y, u, t)6(xi - y) exp(-27riu • vj)Y^(y)dydu
T[dy
N
P(-27riu • v,)Yj(x,)4ydu
r
rr dZFexp[A}jJ
j i ( y , u , t)5(x, - y)
exp(-27riu • vj) )5^(y)dydu
3
^
i=i
£
g2^ ^—|^(y,u,i)J(xi-y)exp(-27riu-v,)yj(y)dydu
i=i
2m 86$
' ^
<-]3= !
dy • du
The proof of the P r o p o s i t i o n 3.3 is completed. The quantity ( ^ ) i 2 can be transformed in a way similar to, but a little more tedious than that of the derivation of (3.4). P r o p o s i t i o n 3.4 \ & J12
}
=W^ iOT
2
dO ( e ) exp(27riy • Ojj—4^(y,
u ) , exp(-27riy • £)6(u)
3.1. DERIVATION
OF H-FUNCTIONAL
1
37
EQUATION
dH
'/^
(3.5)
U
MW1RL<*>"\
where V>(£) denotes the Fourier transform of the function ip(x) — ip{\x\): VK£) = / dx^(x)exp(-27rix-^) and 2 pf. y"de^(o 47r12m a // (©)
dO2 exp(2vriy • 0 g — 7 ^ ( y » u ) - exp(-2?riy • £)<5(u)
t^i
d2H
exp(27riy • 0 » 47r2m d0 2 (©)
. g u ( y > u ) - exp(-27riy • £)<5(u)
1 dff a^ 27ri 96»5 (©) 0y • dxi(y
,u)]|.
(3.6)
Proof TV
£
y dZ Fexp[^](-27rmi)v i • J J 0 4 (y, u ) ^ '
1
J2
~
y )
exp(-27riu • v,)
W
dydu
I dZ Fexp[A] 2TT miv, • / / " 0 4 (y, u)
1
x
(
*^
y)
exp(-27riu • v,)
*
-X>(|x,-x f c |) + tf(x,) dydu k±l
-27rmi /" dZ Fexp[i4] ff | ^ ( y , u) • j ^ U,«(xj - y) exp(-27riu • v,)
38
CHAPTER 3. H-FUNCTIONAL EQUATION
1 N "l\ - ^ V ( | x j - x f c | ) + £/(xO jdydu k*l
N
m JdZFexplA] Jj
^(y,u) • £
/
(<5(x, - y ) A (exp(-27riu • v<))
JV
5>(|x,-x fe |) + t/(xO
= —m j dZ F exp[A] J J ^ - ^ (y, u) £
dydu
( 5(xj - y) exp(-27riu • v.
N
1
-X>(|x ( -x fc |) + t/(xO
\dydu
k*i
= -f/,ZF«pM/^_(y,„) w
N ( \ 5 x x J ] ( ( ' ~ y)exp(-27riu • vz) ^ V(|xj - xfc|) jdyd
1 +
dH
2 ^ ^
,^
( 9 )
^
^
dZ Fexp[A] y ^ 7 ^ ( y - u) £
x
E/^) ex p( 27ri ^-( x '- x *))^) d y du +
^
^(x, - y) exp(-27riu • v,)
^iw5(e) uMwk
"\
3.1. DERIVATION
OF H-FUNCTIONAL
m "2
39
EQUATION
[<%$(£) [dZFexp[A]
N f d62 4 L , (y, u) exp(27ri£ • y) ] T <S(xj - y) exp(-2?riu • vfidydu
N f x / exp(-27ri£ • y)S(u) ^ J
fc
S(xk - y) exp(-27riu • vk)dydu
—1
+ --(e) 27Ti9fl6
e
l/«*0{*=w< >
'"<*>«$=<*">
9# 2 exp(27riy • 0 ^ — | ^ ( y > u ) > exp(-27riy • £)<5(u)
1 dH del (y,u) 27ri d0 5 (©) dy • du
1
•]}
+
8fi2
^ / ^
2 ^
( G )
2
50 exp(27riy • g) g . g ipf. _/'^€)iS(e) '47r m d0 2
2
1
+
Sff/^ 2^^(0)
u
( y ' u ) ' exp(-27riy-^)5(u)
u
^)^k^ )
The proof of the Proposition 3.4 is completed. Remark It is not difficult to show divergence of the integral 1
/
d2H
df^(0 47r2m BO2
d# 2 (9) exp(27riy • 0 a.. 4 Q ..(y> 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 <£
90.,(y,u,£) 9u
^£^-5(xl-y)exp(-2mu.vl)^(y)dydu 9y
-—(e)
27ri905V ' L J = I The proof of the P r o p o s i t i o n 3.6 is completed. P r o p o s i t i o n 3.7
\dt J1B
2m 905
9y
(3.9)
9u
Proof 3ff\
N
Y2 / dZFexp[A](-2mm)vr
94(y,u)6(xl-y)exp(-2mu-v,)
—
(xi)dydu
CHAPTER 3. H-FUNCTIONAL
42
^2 J dZFexp[A]m
ff 0 4 (y, u)<5(x, - y)
J2 IdZFexp[A]m ff
9exp(-27riu • vj) du
EQUATION
dU (y)dydu dy
d9i(y,u) ., . , „ . \&U, . , , ^ '- • 8{xi - y) exp(-27riu • vi) — (y)dydu
1
dH
(B\
dU_ d9i dy du
The proof of the Proposition 3.7 is completed. Proposition 3.8 \dtJi6
W** (0 *-w (e)
27rm
0 4 (y,u)exp(-27ri£-y), exp(27ri£ • y) — (u) (3.10)
Proof (SjM
=-2m7ri
^2
fdZFexp[A]vi
l
a
S,(y, u)6(xk - y)«p(-2»iu • v k ) ^ ^ - ^ « i y < t a
" 5= M * {W<e)0 (y, u) exp(-27ri£ • y ) , exp(27ri£ • y) — (u) 4
+ d 2 tf,
2^-0)
d94 8u (y,u)
85 0 4 (y, u) exp(-27ri£ • y ) , exp(27ri£ • y ) g ^ ( u )
•L^-h^-w^
The proopf of the Proposition 3.8 is completed.
3.1.
DERIVATION
OF H-FUNCTIONAL
EQUATION
43
A transformation similar to that used in the derivation of the equation (3.3) will yield the following expression for (7^)2: Proposition 3.9
U). "/*£=• 3^*4-gU),-
(311)
where N
dH ) = -4TT 2 / dZ £ f ^ e x p [ ^ ] • [f ] T Ojfr, u, t)Yj(Xl)5(Xl dt / 2 1 J ,=1 •/•/ J = i
x exp(—27riu • v/)udydu, afp
dt
,
,22
X
j
(3.11)i
AT
-27r 2 / dZ^fiFexp[A}-
^
- y)
^2
f 0 4 (y,u)<5(x/-y)exp(-27riu-v ; )udydu
/ exp(27ri77 • (xj - xk))4>{v)dil,
(3.11)2
= -47T2 / d Z £ f,Fexp[.4] • / / 0 4 ( y , u)*(x, - y)
x exp(-27riu • v;)[/(y)udydu,
(3.11)3
and — J
= -4TT 2 / d Z ^ f j F e x p [ i 4 ] - / / 0 5 (y,u)<S(x|-y)exp(-27riu-v ( )udydu. (3.11)4
The methods of transformations used in the proofs of the P r o p o s i t i o n s 3.33.8 will yield the following expressions for the (7^)2*, 1 < k < 4. Proposition 3.10 dH\
2mdH
r
3
= - — «sr(©) 5>(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)<5(u)
t
rPTT
u • Y(y)(9 4 (y, u) exp(27ri7? • y ) , exp(-27ri7? • y)<5(u) (3.13)
where
¥t.jjdtdn${rj)l{£)
46
CHAPTER 3. H-FUNCTIONAL
EQUATION
d3H
uexp(2?ri(£ + rj) • y)0 4 (y, u), exp(-27ri£ • y ) J ( u ) , exp(-27ri7j • y)<S(u) 'del (e)
4mi337ri d3H
uexp(27ri(£ + 77)-y)0 4 (y,u), exp(-27rif-y)<5(u), exp(-27ri?7-y)<5(u) del (e)
+2m7ri(^S(e) m2 dOl +
W(e)
uexp(27rif • y)0 4 (y, u ) , exp(-2ni£ • y)<5(u)
uexp(27ri7? • y)0 4 (y, u ) , exp(-27ri7/ • y)(5(u)
air
+(2m7r)2 — ( O ) u0 4 (y,u) and o 2 rx
u • Y(y)0 4 (y, u) exp(27ri7j • y ) , exp(-27ri7? • y)<5(u)
/
= / dr)i>(r))i
d2H
u • Y(y)0 4 (y, u) exp(27rir? • y ) , exp(-27ri?7 • y)<5(u) del (e)
+2
™W e )
u.Y(y)fl 4 (y,u)
Proof dH\
at j 22
2TT2 j dZ^TfiFeyip[A}-
^2
/ / 0 4 (y,u)<$(x ( -y)exp(-27riu-vi)udydu
/ exp(27ri77 • (x ( - Kk))ip{T])dri
3.1.
DERIVATION
OF H-FUNCTIONAL
= -2TT 2 [dzJT
\J2
EQUATION
47
/exp(27ri£- (x, -Xfc))?(Od£ + Y(x,)]Fexp[i4]
l\ ^ 4 ( y , u ) u < 5 ( x ; - y ) e x p ( - 2 7 r i u - v z ) ^ ^ /
exp(2Trir)-(xi-xk))ip(r))dr)dydu
i = i fc#i
2TT2 / 7 d^dr]ip(ri){(C) •
f dZFexp[A] j ^ / / 0 4 ( y , u ) e x p ( - 2 7 r i u • v,)u
AT
AT
x5(x; — y) exp(27ri(£ + 77) • x;)dydu Yjexp(—27ri£ • x^) 2__]exp(-27rir7 • Xj) fe=i
j=i
N
— I dZ Fexp[A] 2~] 11 <My, u)uJ(x; — y) exp(—27riu • v;) exp(27riry • xi)dydu
N
x y^exp(—2mr) • Xj)
/ dZFexp[A}^2
/ / ^4(y,u)ui5(x( - y)exp(-27riu • v;)exp(2-7ri£ • xt)dydu
N
x^]exp(-27ri£-xfe)
+
/ d Z F e x p [ y l ] ^ / / 04(y,u)u<5(xi - y)exp(-27riu • v/)dydu
r
2TT2 /d7?^(77)
AT
fdZFexp[A] ^
/70 4 (y,u)u<5( X i - y ) • Y(y) exp(-27riu• v,)
CHAPTER 3. H-FUNCTIONAL EQUATION
48
N
x exp(2iriT) • xi)dydu ^
exp(-27rirj • Xj)
0 4 (y, u)u6(xi - y) • Y(y) exp(-27riu • vt)dydu
^JJdtdnfajf®
4m37ri
(d*H
'{del
(e)
uexp(27ri(£ + 7?)-y)0 4 (y,u), exp(-27ri£-y)<5(u), exp(-27ri77-y)S(u)
+2m7ri( - ^ - ( 0 ) uexp(27ri£ • y)0 4 (y, u ) , exp(-27ri£ • y)5(u) -(
+ ^
uexp(27rir? • y ) 0 4 ( y , u ) , exp(-27rir? • y)<5(u)
+ ( 2 m 7 r ) 2 ^ - ( e ) u0 4 (y,u)
l
u • Y(y)0 4 (y, u) exp(27ri77 • y ) , + 2m-22 JdV^V){^(Q) dej
exp(-27rir; • y)S(u)
+ 27rmi—(0)|u-Y(y)04(y,u)
7 0
^db*-//** ^ d3H
'del
(6) uexp(27ri(f + rf) • y)0 4 (y, u), exp(-27ri£ • y)<5(u), exp(-27rir? • y)tf(u)
3.1.
DERIVATION
1
f
-~
OF H-FUNCTIONAL
f)^H
+
EQUATION
49
u • Y(y)0 4 (y, u) exp(27ri77 • y ) , exp(-27ri?7 • y)S(u)
The proof of the Proposition 3.11 is completed. Proposition 3.12
V dt J23 1
f
—
fP H
uexp(27riy • f)#4(y, u)t/(y, u ) , exp(-27riy • £)6(\i)
(3.14)
u-Y(y)04(y,u)[/(y) Proof fdH
V dtat J 23 = -AT:2 jdZ'^rfjFexp^]
= -4TT 2 I dZ^2
• //0 4 (y,u)<5(x, - y)exp(-27riu • v,)l/(y)udydu
\Y^
Iexp(27ri£ • (x ( - xfe))f«)de
xFexp[A] • / / 0 4 (y, u)5(xi - y) exp(-27riu •
4TT2 fdZ^2Y{yn)Fexp[A}
^/* r «»{w (e)
•
vt)U(y)udydu
ffeA(y,u)S(xi-y)exp(-2iria-vi)U(y)udydu
uexp(27riy • £)04(y, u)U(y, u ) , exp(-27riy • Q5(u)
50
CHAPTER 3. H-FUNCTIONAL
+ 2^—0)
u0 4 (y,u)C/(y)
^Pf./«?tt)-|£<e)
EQUATION
u-Y(y)04(y,u)t/(y)
uexp(27riy-0^4(y,u)C/(y,u), exp(-2?riy-£)(5(u)
m ae5y
'
u-Y(y)04(y,u)[/(y)
The proof of the P r o p o s i t i o n 3.12 is completed. P r o p o s i t i o n 3.13
\dtJ24 1
f
~
dzH
uexp(2?riy • £)#5(y, u ) , exp(-27riy • £,)6(u)
^Pf./ief(0-w(e)
m 90 5
u-Y(y)fl5(y,u)
(3.15)
Proof 4TT2 J dZ^2
f/Fexp[yl] • / 7 6>5(y, u)<S(xi - y) exp(-27riu • v ( )udydu
4TT2 [dzJT
f J ] /"exp(27ri£ • (x, - x , ) ) f ( 0 ^
;Fexp[A] • / / 0 5 (y, u)<5(xj - y) exp(-27riu • vt)udydu
N
-4TT 2 /"dZ J2 Y(x ( )Fexp[^] • ff 0 6 (y, u)<5(x, - y) exp(-27riu • v,)udydu
3.2. H-FUNCTIONAL
51
EQUATION
^/«'(0-{^(e) aei
uexp(27riy • 0^s(y. u ) , exp(-27riy • £)6(u)
27ri dif u-Y(y)05(y,u) ~m~d^ (©)
+ 2^11(9) uMy,u) 1
f
-^
FPU
U ^ P f . / ^ f ( « ) w ( e ) uexp(27riy • £)0s(y,")» exp(-27riy • £)
27ri d i f
( 9 ) u.Y(y)fl 6 (y,u)
m 96>5V
The proof of the Proposition 3.13 is completed.
3.2
iY-Functional Equation
Combining the Propositions 3.1-3.13, we obtain a functional equation governing the evolution of the if-functional — if-functional equation. T h e o r e m 3.1 If the distribution F on the phase space R 6JV satisfies the Liouville equation, then the if-functional corresponding to this distribution satisfies the following functional equation—If-functional equation: dH —— =Wi+W2+
(3.16)!
TU3,
where Wi
-—(e) 2TH
de5 OH
K
'
^
8%
(y,u,t)r,(y)
1 dH
3=1
7= 1
(9) ^ ^ ( y , u ) y , ( y )
d6i
,^
2m dH (9) m 895
dYj, J
S^(y.u.*)n-(y)«-Y(y) j=i
52
CHAPTER 3. H-FUNCTIONAL
1
+^ P f .
EQUATION
f d H J d e ? ( O - ^ r ( e ) uexp(27ri^y) £ * ; ( y , u , ^ ( y ) , e x p ( - 2 7 r i £ - y ) < S ( u ) J'=I
(3.16)2 ^2
+i
Q2Q
~nJ<xmig-M ml
e x p ( 2 7 r i y - 0 ^ — ^ ( y , u ) , exp(-27riy-£)<5(u)
p t
exp(-27ri£ • y)0 4 (y, u ) , exp(27ri£ • y ) ^ H
+
•/««*«>« H< e >
P f . / / ^ ^ ) f ( 0 - ^ 3 ^ | ( B ) uexp(27ri(£ + 77) • y)0 4 (y, u ) ,
exp(-27ri£ • y)<S(u), exp(-27rir7 • y)<5(u)
+
ipt>*>w'si
+ Pf rfe?(
^ -/
u • Y(y) exp(27ri77 • y)0 4 (y, u ) , exp(-27ri?7 • y)<5(u)
°-w ( e )
uexp(2iri£ • y)0 4 (y, u ) [ / ( y ) , exp(-27ri£ • y)<J(u)
2m dH ( 6 ) u-Y(y)0 4 (y,u)E/(y) m d# 5
„ a2a4 ,
;
1
+
27T1 96>5
dH
m dU_ &h
2^W^)
dy
du
(3.16)a
and 1
o
f
rfiH
,-ffpf./««0-W(e)
+
2^i^
( e )
9^5 (y,u) dy • du
uexp(27ri£ • y)0 5 (y, u ) , exp(-2?ri£ • y)tf(u)
2m dH
(©) fl5(y,u)u-Y(y)
(3.16)4
3.2. H-FUNCTIONAL
Corollary 3.1
53
EQUATION
In case of null external force field, i.e., Y(y) = 0, U(y) = 0,
we have wx = 0,
^ = sbPf-/"«««»H<e» +
24 pf -/ d ^^-w (e)
exp(27riy-g)
exp(—27ri£ • y)S(u),
f
frTJ
^ ( y , u ) , exp(-27riy£)<*(u)
exp(-27ri£ • y)0 4 (y, u ) , exp(27ri£ • y ) ^ ( u )
+p f . / / ^ ^ ) f ( o - ^ | | ( e )
1
(3.16) 5
uexp(2?ri(£ + 77) • y)0 4 (y, u ) ,
exp(—2nir] • y)8(\i)
u • Y(y) exp(27ri7y • y)0 4 (y, u ) , exp(-27ri7? • y)<S(u)
+^ P f . / ^ ^ ) w ( e )
+ Pf
i -/ d e ? ( °-w ( e )
uexp(27ri£ • y)0 4 (y, u)C/(y), exp(-2vri£ • y)J(u) (3.16) 6
and 1
—
f
,(
ffiT-f
rf"-/* °-W(0) +
uexp(27ri£ • y)0 5 (y, u ) , exp(-27ri£ • y)S(u)
1
dH,^
2 ^
{ 0 )
d% (y,u) dy • du
(3.16),
54
CHAPTER 3. H-FUNCTIONAL
EQUATION
It is easy to see the following properties of the expressions w\, w2, and tj73): W!=0,
if 9j =Q,j = 1.2.3;
Q72 = 0,
if 64 = 0;
073 = 0, if 0 5 = 0 .
3.3
Balance Equations
The ^-functional equation holds for J7-functionals corresponding to any distributions F(Z, t) satisfying the Liouville equation. The five balance equations in classical kinetic theory (see, e.g., Truesdell and Muncaster [74]) hold for mass density, momentum density and internal energy density fields arising from any distributions satisfying the Boltzmann equation. In non-equilibrium statistical mechanics without the assumption of molecular chaos the H-functional equation plays the role the balance equations in classical kinetic theory do. But the Hfunctional equation (3.16) is formally better than the five balance equations in classical kinetic theory in that it is a closed equation for H-functional and the balance equations are not a closed system of equations for the five moments. The form of #-functional equation is awfully clumsy. If we take account of the fact that the form of the five balance equations (especially, the energy balance equation) in the classical kinetic theory is clumsy too, the length of the iJ-functional equation, from which the five balance equations can be deduced, is acceptable. The iJ-functional contains more information than that the physicists are interested in. In order to concentrate our efforts upon the information the physicists are concerned, a functional, called X-functional, will be introduced in the next section. Before introducing the if-functional and deriving various X-functional equations governing the evolution of the if-functional, we shall derive some balance
3.3. BALANCE
55
EQUATIONS
equations in classical kinetic theory as consequences of the //-functional equation to illustrate the physical significance of the //-functional equation. Inserting 0 = (0,0,0,0,0s) into the //-functional equation (3.16), we obtain the following functional equation: 8H
~dt
a2e5
1 dH = W3 =
{e)
2riW5
dy • du
d2H
+^Pf./dCf(0 del (e)
(y,u)
2m dH (9) 0 5 ( y , u ) u . Y ( y ) m 69$
uexp(27ri£ • y)0 5 (y, u), exp(-27ri£ • y)S(u)
(3.17)
Differentiating both sides of the equation (3.17) with respect to 0 5 , we obtain the following equation: d2H
d2H.„.
1
27Ti d2H
m dOl
(©) 0 5 ( y , u ) u . Y ( y ) , ^
ra3//
+ ^Pf./de?(o-{ d9\
+
3205 (y,u),
W<e)
1
2^3// m d0 5
dH
,^
d2tp 9y • du (y,u)
^u-Y(y)
uexp(2?ri£ • y)0 5 (y, u), exp(-27ri£ - y)<5(u), y>
uexp(27ri£ • y)v3,exp(-27ri^ • y)5(u)
(3.18)
Setting 05 = 0 in the equation (3.18), we have the following equation: d2H
/n.r
,
1
dH,^ 9y • du (y.u)
+
i Pf /^)-w (e)
^ 9 K) VU-Y(y) m d95 '
uexp(27ri£ • y)y>,exp(—27ri£ • y)<5(u)
(3.19)
The equation (3.19) is a corollary of the equation (3.17), which in turn a corollary of (3.16). As it will be shown later, the balance equations of mass,
56
CHAPTER 3. H-FUNCTIONAL
EQUATION
momentum and kinetic energy in classical fluid dynamics are corollaries of the equation (3.19). A large number of terms on the right hand side of the equation (3.16) contained in the expressions vo\ + xn-i, which make the equation (3.16)i awfully clumsy, are irrelevant to the balances of mass, momentum and kinetic energy. If we set the function if in the equation (3.19) to be K-35(U),
if \yt - Zi\ <
for i=l,2,3;
K/2,
(3.20) otherwise, where z = (zi, 22,Z3) denotes a fixed point in R 3 and K a positive number so small that the aggregate of the molecules in a cube of side length K can be considered a fluid particle in fluid dynamics, but so large that the aggregate still consists of a large number (e.g., > 108) of molecules. It is worth noting that
(3.21)
Taking account of G = (0,0,0,0,0) and using the formulas for derivatives of the if-functional (2.14), we have the following equations: d2H
,
n
dtd65
N
(Q)[
J
- z),
1=1
and
--(e) where
(y u)
k-au ' J
~ K3 *(x) = {
lo.
d
N
f
3
= 2m-— • / dZFK~ mS^wMxi
if max
1
- z),
\xi\ < f; (3.22)
otherwise.
We call 6(x) the macroscopic delta function of size K. The above results can be summarized in the following
3.3. BALANCE
EQUATIONS
57
Theorem 3.2 If the distribution F is a solution to the Liouville equation, then we have | - / dZFn-3m
] T ?(xj - z), + ^ • / dZFK-3m
J^ v/<5(x, - z) = 0.
(3.23)
It is easy to see that the integrals on the left hand side of the equations (3.23) are the mean fluid mass density and the mean fluid momentum density at (the macroscopic point) z respectively. Therefore the equation (3.23) is a generalization of the classical equation of continuity. It should be stressed that the fluid mass density K _ 3 m^ i = 1 <5(x( - z) and the fluid momentum density / $ " 3 m ^ J = 1 V(J(x/ — z) are random fields, which Massignon called the "correct variables of Euler". The mean fluid mass density and the mean fluid momentum density are their mathematical expectations. In case of laminar flows, of which all the fluid variables are deterministic, the random fields coincide with their mathematical expectations. So the equation (3.21) is a generalization of the classical equation of continuity, which describes the mass conservation of the laminar flows. If the assumption of molecular chaos holds, in virtue of the law of large numbers for the arithmetic means of independent random variables in probability theory, the fluid variables must be deterministic. In non-equilibrium statistical mechanics without the assumption of molecular chaos, the fluid variables, which are the arithmetic means of a large number of exchangeable random variables, are random in general. Because statistical theory of turbulence concerns itself with the random variables, e.g., the velocity, attached to fluid particles, the nonequilibrium statistical mechanics without the assumption of molecular chaos is the most convenient theoretical frame for describing turbulence phenomena. If we set the function ip in the equation (3.19) to be KT3(27r)-2dUl<5(u),
if \Vi -
Zi\
< K / 2 , for i = 1,2,3; (3.24)
otherwise,
58
CHAPTER 3. H-FUNCTIONAL
EQUATION
we shall have the following expressions for the terms in the equation (3.19), d2H dtdOi
-27ri- a^ ( 5ev ) ;
az
(e)M = | / d Z F « - 3 m 0 1
/
/ dZFn~3m.
(3.25)
(^-iv(z)- 1
a
l<, <3Ni-^l<"/
(dZFK-*mN{z)-1
^2
ujivj
i
Y
\
- ~
">
max . i< , ^ 3 l ^ i - «_,, l < ,«. /. ,2„
d_ dz
dy • du
dZFK" 3 m •
J
d\
v
J2
2
^
»M)
l<™*\Xki-Zi\
max i
iv'^/o
'
'i
E
l<J<3k/H-2i|<«/2
u
Y,
v
*>
(3-26)
where N(z) denotes the number of molecules in the small cube around z of the side length K.
5 1
^s|*H-«i|<«/2
1 f a H \ ^ j P f . y d f f ( 0 • - ^ - ( 9 ) |uexp(27ri£ • yV,exp(-27ri£ • y)J(u)
= ^ 2 /d£f(£)-47r 2 m 2 f dZFexp[A] f dy m J J ASEslw-nKK/a
x
E
/
•*
duuex
P ( 2 7 r i £ • y) ( - «
dy
du exp(-27rif • y)S(u) *
3 2?r
(
29
)
«i
^ Kk
S x
( k ~ y) exp(-27riu • vfc)
3.3. BALANCE
EQUATIONS
= fdZFK~3
59
Yl
E
/i(^-x fc ).
2
islssl*"-**'^"/ i<*"sl*«-**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)
!<,"., l*li-Zi|
+
=
J2
S
/i(xi-xfe)
£
£
/i(x/-xfc)
E
/i(x(-xfc).
53 i<~3l*«-*l<"/2
1^3|xfci-Zi|>«/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<3l*"-z*l<"/2
x (v, - AT(z)-1 \
v
Yl max i _ i l<™ a\*ki-Zi\
+ ^ - f dZFK^mNiz)-1 max
i_
•/
*
£ max
J2
_ t ^.. m
= /dZF«-3
+ fdZFK-3
v
Yl
J
i_
*
„ /o
J-
max
Yjfo) _ i ^ _ /rt
Y
£
l
1
Vfe
i_
/i(xi-Xk).
^3|lfci-Zi|>K/2
The balance equations for other components of the mean of the momentum of the fluid particle can be derived in a similar way. Theorem 3.3 If the distribution F is a solution to the Liouville equation, then we have the following balance equations for momentum / dZFK~3m
J
+^-fdZFK-3m J
Y
«
(vij-Niz)-1
Y max
v
max i „ _ i -._ in 1 " " 3 | i « - 2 i l < « / 2
i
_ |
in \v
x<*<3l^*-^l<«/2
v
Y max
i
>v)
™ t-'^./o
.^ji.H-ziKit/a
'
3.3. BALANCE
EQUATIONS
61
max
\
r
_ i v „ / o
i
i
Vfe J
max
_ i j . . . /rt
i_
= fdZFn-3
+ fdZFK~3
max
J2 max
i_
Y .. \ ^
i
i^sl^-^K"/2
Yl _ i J * ._ /<•»
JZS*\*H-*i\
max
i_
•
i?,a<3N^-Zil<«/2
J2
•/
«/
max
i<"sl*«-^l<"/2
^
m
/,(x/-x fc ), j = 1,2,3. (3.29) _ i^ _
m
T.<~*\Xki-Zi\>Kl2
If we set the function
{
-m/(87r 2 K 3 )A u <5(u),
if \y{ - zt\ < K/2 for i = 1,2,3;
0,
otherwise,
(3.30)
the terms on both sides of the equation (3.19) will have the following expressions respectively: | ^ ( 0 ) k ] = - 2 7 r i m ^ fdZFK-3
27ri a
' dy • du (y.u)
2- 1 m| V / | 2 ,
Yl
= m27ri— • fdZFn-3
(3.31)
v
^ max
•/
_ i ^
I._
'lv'
,n
i
(3.32)
- ^ | f ( © ) ^ u • Y(y)] = -27rim f dZFn'3 **
^
£ max
i
v, • Y(x,), (3.33) i J-
/«
i
62
CHAPTER
-ipf./«?«)• 0(6) = -27rim I dZFK~3
3. H-FUNCTIONAL
EQUATION
uexp(27ri£ • y)<^,exp(—27ri£ • y)S(u)
^
^ v j • f(x, - x fe ).
(3.34)
Combining the equations (3.19),(3.31),(3.32),(3.33) and (3.34), we obtain a balance equation for the mean of the kinetic energy. The balance equation for the potential energy can be derived as follows: differentiating both sides of the if-functional equation (3.16)i with respect to #4 and inserting 9 = (0,0,0,0,0)
(3.35)
and
{
KT 3 (5(U),
if \yi - Zi\ <
0,
Otherwise
K/2
for i = 1,2,3, (3.36)
into the equation thus obtained, we get the following equation: d2H , _ r , dm-2£, /ir -5s M dtdO^ (©)bl ' = 86,
• ~ • |pf. | d ^ ( O i n ^ 0 ( e ) [ e x p ( 2 7 r i e • y)<W,exp(-27ri£ • y)*(u)]|
-sis-^ e ^^ + s i ^ e > ^ - ^ +
i f 2m^Pf' J d ^
m
d^H ' - ^ 2 - ( 0 ) [ e x P ( - 2 7 r i £ ' y)v,exp(27rie • y)du5(u)}
= mTri— • / dZFK~3 Z
]T
Yl v i ^ ( x j - xfc)
1<™3\xii-Zi\<*/2W
3.3. BALANCE
63
EQUATIONS
+2m7ri— • / dZFn~3 Z
E
VJ£/(XJ)
,< m K3l^-^l<«/ 2
-2m7ri / dZFtC3
v
E
''
-2m7ri f dZFn'3
E
E
i<™3|zii-*il<*/2
v
I T ^ y
a
' • ^ ( x / - x fc ).
(3.37)
Mi
Since
a2g
( e ^ = - ^ l / d Z F ( m a x E "<*> V
+
^2
E
l
E
V>(xfe-Xi)),
z
i
(3.38)
/
the equation (3.37) is the balance equation for the mean of the potential energy. Adding both sides of the balance equations for the mean kinetic energy and mean potential energy and taking account of the fact that the right hand sides of the equations (3.33) and (3.34) and the third and fourth terms of the right hand side of the equation (3.37) will be cancelled out after addition, we obtain the following theorem for the balance of energy: Theorem 3.4 If the distribution F is a solution to the Liouville equation, we have the following form of the balance equation of total energy: 8_ fdZFK~3l dt
\
E
E
]T 2- 1 m|vi| M
^(**-*)+
E
u
<*n
64
CHAPTER 3. H-FUNCTIONAL EQUATION
\,??;,i*n-*ii<"/3
+
Y.
X>/V>(xi - xfc) + 2
5Z
v,^(xj)J=0.
(3.39)
We have seen that the classical balance equations are the consequences of the ff-functional
equation. Of course, we can get more balance equations for the
means of higher moments, e.g., Grad's thirteen moments. Usually the system of balance equations is not closed, but the J?-functional equation is a closed (functional) equation governing the evolution of the if-functional. The means of moments are the values of the H-functional and its derivatives at 0 = (0,0,0,0,0). Since the means of moments are of physical significance in an obvious way, they interested the experts of fluid dynamics a long while ago. The physical significance of the //-functional is rather confusing and the functional equation governing its evolution is much more complicated ( and cumbersome ) than the equations for means of moments. But the //-functionals have the advantage that they exhibit the panorama of turbulence phenomena and the means of moments only a part.
3.4
Reformulation
The form of the equation (3.16)i is really clumsy. In order to get a (formally) simpler form of it we would like to use the language of topological tensor product [37] to express the higher order derivatives of the //-functional in a (formally) compact form. Because the compact form has the same content as the original form (3.16)i, it is useless in actual computations. It will not be used in the computations in the sequel.
3.4. REFORMULATION
65
A second order derivative of the H-functional with respect to 65 at 0 is a bilinear functional. A bilinear functional on two linear topological spaces can be extended to be a linear functional on their topological tensor product. So the second order derivative of the i7-functional at 0 can be considered a linear functional on the topological tensor product of two linear topological spaces. In this paper we will not be concerned about the mathematical theory of topological tensor product of linear topological spaces. Formally the extended second order derivative of the .//-functional at 0 for sufficiently smooth function y> = ¥>(yi,y 2 ,ui,u 2 ) is of the following form: d2H,
del
•(Q)[
III!
dyidy2duid\i2
IdZFexp[A]
u i , u 2 )/9 5 (yi, u i , Z)p 5 (y 2 , u 2 , Z).
(3.40)
But the trouble we are encountering is that the function
v(yi5y2,ui,u2) = V(yi -y2Mui.u 2 ),
(3.41)
where ip is the intermolecular potential ( sometimes, intermolecular force ). The function <^(yi,y 2 ,Ui,u 2 ) of the form (3.41) is not smooth enough( even discontinuous ), because it tends to infinity as |yi — y 2 | tends to zero. If we assume that the distribution function F(Z, t) is dominated by an exponential function with a negative multiple of the total intermolecular potential energy as its exponent, the following limit exists: lim
d2H
(3 42)
J£ - •S0 & a f l r ^ '
-
where
v(yi,y2,ui,u 2 ),
if|yi-y2|>e;
0,
otherwise.
e (yi,y2,uiu 2 ) = {
(3.43)
66
CHAPTER
3. H-FUNCTIONAL
EQUATION
It is easy to see that the function <£ e (yi,y2,uiU2) defined in (3.43) can be expanded as a double Fourier integral. It makes the possibility of using the method of topological tensor product in treating the second order derivative of the Hfunctional. In the sequel we frequently consider the second order derivative of the Hfunctional at 9 as a linear functional on the topological tensor product and its value at the function
w
(e)M=
rfiH
-w ( e ) b e ]
(3 44)
-
The above techniques apply to higher order derivatives of the .//-functional too.
Having made the above conventions for higher order derivatives of the H-
functional, we can simplify the forms of many terms on the right hand side of the //-functional equation (3.16). For example, it is easy to show that r
1 fl2
IT
exp(27riy • O ^ ^ C Y .
u
) - exp(-27riy • £)6(u)
d2H (3.45) (6) V»(yi - y 2 > « — — ( y i . u i ) < * ( u 2 ) 8mir2 del a y i • ux Using the formulas similar to the formula (3.42), we can reformulate the Theorem 3.1 as follows: Theorem 3.1' If the distribution F on the phase space R 6JV is a solution to the Liouville equation, the //-functional corresponding to this distribution satisfies the following functional equation—//-functional equation:
l»=w5ma]
+
Miem
+
Mb]'
where 1 d 2m 8y
/J^d9i, \ ^ duyj'
, „ , . TTdOir t)Ys(y) + U-^(ywtu) >''"'• du '
, 86;5 , + -^(y,u) ' du
(3 46)
-
3.4.
67
REFORMULATION
+27riJ^(y,u)Y j (y)
4T^
+ ^m- u • Y(y) ( J 2 *i(y,«, *)>i(y) + «4(y, u)tf(y) + 05(y, u)
(3.47)
-myi-y2)^-(yi,ui)J«(u2)-^-(yi-y2)e4(yi,ui)-^-(ua) 0 = 87r2m 9y r
3
1 dtp (yi-y2)-ui<J(u 2 ) Y^ ^ ( y i , u i , *)^(yi)+«4(yi, u i ) t / ( y i ) + 0 5 ( y i , ui) m25yi j=i
+
2m2^yi-y2^Ul'
Y
(yi'ui)04(yi'ui)<J(U2)'
(3.48)
and
7 = -r-^-VKyi - y2)f(yi - y2)ui0 4 (yi, ui)<5(u2)<J(u3). 4m'5 7r
(3.49)
The equations (3.16)i and (3.46) are different forms of the same if-functional equation.
This page is intentionally left blank
Chapter 4
K- Functional 4.1
Definition of if-Functional
In classical fluid dynamics the quantities which concern physicists mostly are mass density, velocity and internal energy per unit mass (see, e.g., Truesdell and Muncaster [74]). It is natural to introduce the following K-functional to meet physicists' interest: Definition 4.1 The if-functional for the probability distribution F(Z, t) is defined as follows: K(a,b,c;t)=
+b(v)m • E
j dZF(Z,t)exp
(v« -
1 N + - ^ ^ W x , - x
tJ
^-)^i
i - 2vri J
a(y)m]T<5(x; - y)
- y) + c(y) ( f £
|v,| 2 *(x, - y)
N f e
M l \dy \,
| ) ^ - y ) + ^[/(x^(xi-y) i=i
i=i k^i
where a(y),b(y) and c(y) are functions defined on V.
It is easy to show the following 69
/J
J
(4.1)
70
CHAPTER 4.
K-FUNCTIONAL
Proposition 4.1 The if-functional for the distribution F(Z, t) is a restriction of the if-functional for the same distribution to the linear subspace composed of the vectors of the following form: (#1,#2,03,04,#5) = (©,#4, #5)
\
b ( y ) i * ( u ) c(y)J(u) b(y) , a(y)d(u) + — m m 27ri
dS(u) c(y) — - - —jr2 Au«5(u)V du &w
where © = (0i,0 2 ,0 3 ) =
b(y)t*(u) m
Precisely speaking, the iC-functional can be expressed in terms of if-functional as follows: K(a,h,c;t)
= H(e1,92,93,9i,05;t),
(4.2)
where
e)
= =hmm,
j = w
,
( 4.3)
* = 2W£W,
(4 . 4)
^wo+m.aM-fM^
(4,,
Five moments corresponding to mass density, momentum density and internal energy density play very special roles in the classical theory of Boltzmann equations. The kernel of the linearized Boltzmann integral operator is just the subspace generated by them. In the theory of Liouville equations, five functions corresponding to mass density, momentum density and total energy density and the composite functions of them belong to the kernel of the Liouville operator. Of course angular momentum density belongs to the kernel too. The reason why we exclude the angular momentum density in the definition of K functional is
4.1. DEFINITION
OF K-FUNCTIONAL
71
that in a small neighborhood of x ( precisely speaking, the size of the neighborhood is that of the fluid particle in classical fluid dynamics ) the total angular momentum density is nearly equal to the cross product of a constant vector and the total momentum density, therefore, a function of the total momentum density. We do not know whether there are other functions belonging to the kernel of the Liouville operator. In analytical dynamics (see, e.g., [39] or [82]), there are some results due to Bruns, Poincare and Painleve relating to the problem at hand. As far as I know, the problem of characterizing the kernel of the Liouville operator is still open. Of course, we can take functions outside the kernel of the Liouville operator ( e.g., 13 functions corresponding to Grad's 13 moments) to define other functional. But we will not do that in the present paper. Usually, statistical mechanics concerns itself with the Hamilton system of TVparticles satisfying the condition mN = K,
(4.6)
where K denotes a constant. It is straightforward to show the following Proposition 4.2
If the Hamilton system satisfies the condition (4.6), then
the if-functional is of the following form: K(a,b,c\t)
x jdZF(Z,t)
+b(y)m • £
— exp I — 27rKi / a(y)dy
exp J - 2vri f
U ( y ) - f a(y)dy\m
( v , - ^ ) * ( x , - y) + c(y) ( f £
^ S ( x , - y)
|v«| a *(x, - y)
N 1 N \1 "\ + oEEV'(|x/-xfc|)<5(xi-y) + ^C/(xi)5(x(-y) dy\,
(=1 kjtl
l=i
(4.1)'
CHAPTER 4.
72
K-FUNCTIONAL
where a(y),b(y) and c(y) are functions denned on V. Corollary 4.1 Under the condition (4.6), in order to determine the Kfunctional it is sufficient to know the values of the AT-functional on the hyperplane
ja(y)dy = 0.
(4.7)
As an immediate consequence of the Definition 4.1 we have the following propositions and their corollaries. Proposition 4.3 dK — (o,b,c;t)[p(y)]
= -27rmi f dZ F ( Z , t) exp I -2ni
+b(y)m • j ^ L
i = l k^l
- ^
^
a(y)m ^
W < - y) + c(y) (f
i=l
/J
= -27rmi / dZ F(Z, t) exp[4] / dyp(y) ^
£
5(xt - y)
|v,|a*(x« - y)
J ^
;=
1
*(x, - y).
(4.8)
Corollary 4.2 N
^-(a,b,c;t)[6(x-y)\
= -27rmi JdZF(Z,t)exp[A]
J'
(4.9)
where <S(x) denotes the function defined in the equation (3.22). Proposition 4.4 — (a,b,c;t)[q(y)]
= -27rmi /" dZ F ( Z , t) exp[A] J dyq(y) • £
(v, - ^ ^ )
* ( * - y).
(4.10)
4.1.
DEFINITION
OF
73
K-FUNCTIONAL
Corollary 4-3 — (a,b,c;«)[<J(x-y)]
N
-27rmi fdZF(Z,t)exp[A]
N
f dy j S ( x - * ) E (vu -
^
^ ^ l \
1=1 ^
i=i
J
t
(4.11) '
for i = 1,2,3. Proposition 4.5 dK — (a,b,c;t)[u(y)] TV
/
N
-2m I dZ F(Z, t) exv[A} f dyu(y) jr *(X, - y) (j J
J
1=1
Iv< l ^ x ' " ^
L=l
^
N
+\ E
£
N
E W l * - Xfc|)*(x, - y) + £
(=1 k±l
£/(x,)<5(x, - y ) ) .
1=1
(4.12)
'
Corollary 4.4 dK ~ — (a,b,c;t)[<5(x-y)] N
,
5 x
N
dZ F(Z, t) exp[A] y ^ E ( ~ <) ( f E lv< 2 N
x
N
+^EE^i <- *i) + E t W ) ' (=1 k^l
x
x
1=1
(4-13)
'
The formulas for higher order derivatives of the functional K(a, b , c) can be obtained in a similar way. In order to obtain a functional equation governing the evolution of the Kfunctional from the ^-functional equation, we should study the connection between derivatives of the H-functional and those of the corresponding K-functional and then seek an asymptotic solution to the Liouville equation. The next chapter is devoted to the former task and the chapter 6 to the latter one.
This page is intentionally left blank
Chapter 5
Some Useful Formulas 5.1
Some Useful Formulas
Making use of the expression of K-functional in terms of the corresponding Hfunctional (4.2)-(4.5), we obtain the following formulas expressing the partial derivatives of if-functionals in terms of those of the corresponding ff-functionals. In the following formulas the 0 =(0i,#2,#3,#4,#5) in the H-functional and the a, b, c in the corresponding if-functional always satisfies the equations (4.3)-(4.5). It is easy to prove the following relations between the derivatives of if—functional and those of its corresponding K—functional. P r o p o s i t i o n 5.1 acr r ~\ _ dK air dH (a,h ,C)[(T}. (9) a(y)<J(u) d65 da
(5.1)
Since BJ-f
dK. 9b(a
1 d5(u)~ 2ni du -
^§>> -%«»
1 dS(u)' 2iri du
a •
1 d5(uy 2?ri du 75
j"=i
°jtS(u)~ m L
3
>
[ m
J(U)
dK 'Y-crt' (a,b >c) da m
,
(5.2)
CHAPTER 5. SOME USEFUL
76
FORMULAS
we have Proposition 5.2 dH
d5(u) du
(©)
ae5
„ .9K,
^
= 2m—(a,
xr
,
dK
,
n
^
b , c)[- a) + 2m—(a,
s Y-at
b , c)
(5.3)
m
Since dK 8H — (a,b,c)[a] = — (9)
•£« 1
/"
8TT
rP'H
2
8TT
2
A u <5(u)
•
•£«
A u <5(u)
>
m
£/<7<5(u)
exp(27ri£ • y)cr(y)<5(u), exp(-27ri£ • y)<5(u) (5.4)
we have Proposition 5.3 dH 80.
•(e)
= -8n2 — (a,b,c)[a} + 8n2 —
aAu5(u)
(a,h,c)
Ua m
2m
+ m^ Pf.y"dg^(0|^(o,b,c)
exp(27ri£ • y ) a ( y ) , exp(-27ri£ • y)
(5.5)
Similarly we can derive the following formulas: Proposition 5.4 d2H,
del = -
2
4?r
4 7 r
d2K TS^T dbdb
92K
9b^(a'blC)
( 9 ) o\ • V u <5(u), cr2 • V U
2
K b, c) [
r
y-ffif
-
C T 2
'm -^-
d2K — 9b9a
(a, b, c)
_47r2_(a,b,c)
Y-a2t <^i
m
' Y • (7i*
Y • a2t
m
m
, (5.6)
5.1.
SOME USEFUL
FORMULAS
77
Proposition 5.5 d3H Q93 (©) \°1 • VU5(U) , (72 • VU5(U) , (T3 • Vu<5(u)
•8^i(^r(«,b,c)
+
a3*: , .
ab^ ' '
d3K
+
, r
(a b c):
,
u x
+
+ 7r^7^-(a,b,c) da2db
Y-<7 3 *'
m
m
m
'-^- +
r '
Y-a2* m
d3^ / L ^9b(a'b'
Y •CT3t
Y-cr 2 *
a3/r
Y-a3t
i
0 3 , Cl ,
9b^:(a'b'c)
Y-o-x*
N
+
2
Y-aif
(a,b,c) :
C2 i 0"3 ,
9b da
a 3 *:
m Y • (7j * Y • (72 *
(a, b , c)
da2db
, -0-3
m
Y-a2t
Y-a3t
c )
m
•
m
Y • (71 t
m
m
•cr2
+
ab3" ( a '
, c ) : <7l , <72 ) (73
}'
(5.7)
P r o p o s i t i o n 5.6 d2H
del
(6)
CTAU<5(U)
92AT
, r • V u <5(u)
a 2 A:
Y-Tt
167rJi
^>>b-c) dadc
m
T , (7
+
+167r3i—(a,b,e)
47T
1
r
~
r d3#
a 2 i<: b c ^:K . ) dadb
167ri
C/(7
Y-rt
m
m
£/CT
m
exp(27ri£ • y)o-(y), exp(-27ri£ • y ) , • r
md 3 A!" T Y •Tt + -fa3-(a> b > c ) exp(27ri$ • y ) a ( y ) , exp(-27ri£ • y ) , — ^ ~ } ,
(5-8)
Proposition 5.7 d2H
del d2K
= 27ri
(a b c)
ab^ ' '
(9) (7 • V u 5 ( u ) , r J
o
a,T
-
d 2 K
,
X- N
+ 27ri—-2-(a,b,c)
Y-trt m
(5.9)
78
CHAPTER
5.2
5. SOME USEFUL
FORMULAS
A Remark on //-Functional Equation
Making use of the equations (5.1),(5.3),(5.5),(5.6),(5.7),(5.8) and (5.9), all terms on the right hand side of the .ff-functional equation (3.16) can be expressed in terms of the partial derivatives of the corresponding /("-functional ( under the condition that (4.3)-(4.5) are satisfied ), with the only exception of the term of the following form, d295 1 dH (0) 27ri 865 <3y • 9uJ
——(e)
EE
SSe) 27ri dd$
dbjjy) d26(u) 1 dxjk dujduk]
da(y) dy
dS(u) du
1 dH dc(y) • VuAuJ(u) (9) 3 16n id95 dy
(5.10)
B\ + B2 + B3,
where A u and V u denote the Laplacian and the gradient operators with respect to u respectively. On account of (5.3), we have
da(y) dy
8K.
,
, *Y m
da(y) dy
(5.11)
The function space consisting of the functions on the 6-dimensional space R^ x R „ can be considered a module over the ring of the functions on the 3-dimensional space R^. Because d%. 8(a) and V u A u <J(u) do not belong to the submodule generated by 5(u), V u 5(u) and Au<5(u), B2 and £3 cannot be expressed in terms of the derivatives of the corresponding AT-functional.
We will summarize the
results thus obtained in the following Theorem 5.1
Under the condition that (4.3)-(4.5) are satisfied, all terms
on the right hand side of the if-functional equation (3.16) can be expressed in terms of the partial derivatives of the corresponding ^-functional, with the only
5.2. A REMARK
ON H-FUNCTIONAL
EQUATION
79
exception of the term of the following form,
--(e)
d295 dy • du
B\ + B2 + B3,
(5.10)'
where 1 =
dK, , ~db^a'
, '
,C
da(y) dy
d K
, u N + -£-(«, b,c)
*-^f<°> R, = -
s^dbjiy) i = i fc=i
9yfc
tY m
dajyY dy J'
a2<5(u) dujduk
1 dH 9c(y) • VuAu<5(u) 167r 3 i9^ (e) 9y
(5.11)'
(5.12)
(5.13)
and A u and V u denote the Laplacean and the gradient with respect to u respectively. Among three terms B\,Bi
and S3, B\ can be expressed in terms of the
derivatives of the corresponding A"-functional and B2, B3 cannot.
Therefore the functional equation obtained from the if-functional equation by using the connections between the partial derivatives of the if-functional and those of the corresponding /iT-functional is not a closed equation for K-functional. In the classical theory of Boltzmann equations, the balance equations for the five moments are not closed, because the pressure tensor field and the energy flux vector field cannot be expressed in terms of the five moments (i.e., the mass density, momentum density and energy density). It is easy to see that d\.Uk<5(u) and VuAu<5(u) correspond to the pressure tensor field and the energy flux vector field respectively. Hence we encounter the closure problem similar to that in the classical theory of Boltzmann equations. In order to solve the closure problem, we have to restrict our consideration to a special class of distributions on the phase space T just as we have to restrict the solutions of the Boltzmann equations to the class of local Maxwellian distributions in classical theory of Boltzmann equations. In the next section the special class of distributions on T, called the turbulent Gibbs distributions, will be introduced.
80
CHAPTER 5. SOME USEFUL It should be stressed that the coefficients of d^.UkS(u)
FORMULAS
in Bj, and V u A u £ ( u )
in B3 are dVkbj(y) and dyc(y) respectively. They are similar to what happened in classical theory of Boltzmann equations. Hence the functional equation corresponding to the first order asymptotic solution of the Liouville equation will coincide with the Hopf functional equation for inviscid flows. But the Liouville operator is different from the linearized Boltzmann integral operator in an important aspect so that the second order asymptotic solution to the Liouville equation has an important feature different from that to the Boltzmann equation. The second order approximate form of the if-functional equation is completely different from the Hopf functional equation. This is what we shall work out in the following sections.
Chapter 6
Turbulent Gibbs Distributions 6.1
Asymptotic Analysis for Liouville Equation
In order to derive a closed functional equation governing the evolution of the Kfunctional, we have to restrict the probability distributions on the phase space of N particles to a class of asymptotic solutions to the Liouville equation under certain limiting processes. We assume that the intermolecular force is of short range. We can subdivide the 3-dimensional physical space occupied by the fluid into a large number of disjoint cubes of the same size with planes in parallel with the coordinate planes. The sides of the cubes are assumed to be so small that the macroscopic variables of the fluid, which are themselves random variables, can be regarded as nearly (functionally) independent of the position vectors inside each cube. On the other hand the sides of the cubes are assumed to be far much larger than the range of the intermolecular force so that the total intermolecular potential energy between two molecules in different neighboring cubes is negligibly small in comparison with the total intermolecular potential energy of all pairs of molecules in the same cube. Actually, the cubes, into which the space occupied by the fluid is subdivided, 81
82
CHAPTERS.
TURBULENT
GIBBS
DISTRIBUTIONS
play the roles of fluid particles in fluid dynamics and thermodynamics. In fluid dynamics or thermodynamics, the internal energy for a fluid particle is defined as the sum of the heat energy inside the fluid particle and the total intermolecular potential energy among the molecules inside the fluid particle. We usually assume that the internal energy is an extensive property of the particle, i.e., the internal energy of the larger particle composed of two ( or more ) neighboring particles is the sum of the energies of the two ( or more ) neighboring particles. Thus it has been implicitly assumed that the total intermolecular potential energy between the two molecules in two different neighboring fluid particles is negligibly small in comparison with the total intermolecular potential energy among the molecules inside the same fluid particle in classical fluid dynamics and thermodynamics. The assumption previously made about the manner of the subdivision of the space into a large number of disjoint cubes is consistent with what the experts of the fluid dynamics and thermodynamics have done for a long while. So there would be no fluid dynamics or thermodynamics if the assumption had not hold. In the sequel, we assume that the space occupied by the fluid has been subdivided into a large number of disjoint cubes in the way stated above. Having done that, the Liouville equation can be written as follows:
s xj€C s
s xi€C,
S
where s = {si,S2,sz)
L
Xj6Ca
(or t = (ti,t2,h))
J
xkec,
xt€Ct
denotes a triple integer-index, Cs (for
each triple integer indix s) denotes the small cube with the side length K: Cs = {y = (yi, Ife.lfe); ( * - 1/2)K
(s< + l/2)«,for i = 1,2,3.}, (6.2)
6.1. ASYMPTOTIC
ANALYSIS
FOR LIOUVILLE EQUATION
c = (ci,02,03) being a constant 3-dimensional vector.
83
Of course, it is more
convenient to denote the cube by Ci . For the sake of brevity we shall always assume c = 0 and omit the index (c), unless it is necessary to distinguish one subdivision from the others. For fixed 3-dimensional vector c = (01,02,03), all the cubes C s form a system of disjoint cubes, into which the space occupied by the fluid is divided. It is easy to see that the last term on the right hand side of the equation (6.1)
3
*k£ct
xj€Cs
can be written as
E E„ im 9iv i' E **.-*>. S
X16C.
It£C, l»-t| = I
where s - t = max Is< —t»|, 1<»<3
because the range of the intermolecular force is far much shorter than the side length K of the cube and the intermolecular force between two molecules in two different cubes C s and Ct with |s — 1 | > 1 vanishes. The reason for separating the term
S
Xl6Cs
"fc£C7 t
in the Liouville equation from the term
E E i f E f(x(-xfc)+Y(x,) s
x,€C.
xfc€C»
OF dvi
(6.4)
is based on the following two considerations. Firstly, during a very short time interval, the molecules inside a cube (or, a fluid particle) approximately constitute a Hamilton system. And secondly, the expression (6.3) is negligibly small in comparison with the expression (6.4).
84
CHAPTER 6. TURBULENT
GIBBS
DISTRIBUTIONS
Because we shall deal with the distribution densities that have discontinuous jumps on the boundaries of the cube cylinders Ilt=i ^
x
^-3N
m tne
phase space
R 6 N , as was shown in [69], the spatial derivatives of the distribution density with discontinuous jumps on the boundaries of the cube cylinders Yli=i Cs ; x R 3 N will be separated into two parts: d dx-k
d d-x-k
+
A Ax f c '
where the derivative ^ - on the left hand side denotes the derivative in the sense of Laurent Schwartz' distribution theory, the derivative with a square bracket [3^-] on the right hand side the ordinary derivative of the density within the cube cylinders and the last term ^ - on the right hand side a measure with the boundaries of the cube cylinders as its support, which exhibits the discontinuous jumps on the boundaries of the cube cylinders (see, [69], p.44). The precise expressions of -£^ operating on a special class of functions ( called turbulent Gibbs distributions ) will be specified later on. Because the variation of a macroscopic variable between two neighboring cubes is far much smaller than that of the corresponding microscopic variable, it is reasonable to assume that the term -£- is an infinitesimal in comparison with the term [gf^]- Some readers might suspect the validity of the proposition -£^- «
[gf-], since a derivative of a discontinuous
function will take the value 00 at its discontinuities. The plausibility of the above assumption can be explained as follows. What we mean by saying ^ - < < [^-] is that the outcomes of the operators on both sides of the above inequality operating on a special class of functions (called turbulent Gibbs distributions) is subject to the above inequality. In the sequel, for the sake of brevity, we shall use the notation ^ - to denote the derivative in ordinary (not Laurent Schwartz') sense ( within the cube cylinders ) instead of the clumsy notation [ ^ - ] . No confusion will occur, because the derivative in the sense of Laurent Schwartz' distribution theory will not be used in the sequel unless it is specified explicitly.
6.1. ASYMPTOTIC
ANALYSIS
FOR LIOUVILLE
EQUATION
85
Another explanation, which is easier to be understood than the previous one, of the decomposition •£- = [^-] + ^ - will de deferred to the paragraph immediately after the definition of the concept turbulent Gibbs distribution. In order to get an asymptotic solution to the Liouville equation, we would like to rewrite the Liouville equation (6.1) in the following form: »
N.
i-i-i
d
d
£ £ tZVtrwi £ \-tid4
(i-i)
,<•>
N.
a*«
i-i
E f * s) -(/-i)f/
(s)
1=2
SF ~dt
my(n^)U£i
N.
££=T mdwi S
&W
fx
(s)
x
(s)
dF
N
^
AF
£ ( < *) E E ^ a ( S ) - f-" L v ' Ax('
X,€C.
xt£C, I*.
S
(= 1
V
S
UX
-l
1= 1
'
(6.5) where Ns denotes the number of the molecules in the cube C s , xf of the Ith molecule in the cube Ca, fj
the position
the sum of the forces of the molecules in
the cube Cs exerted on the Zth molecule in the cube Cs, the expression
£ f (x' _ x*) JcfcgCt
in the second term on the right hand side of the equation the sum of the forces of the molecules outside the cube Ca exerted on the Ith molecule in the cube Ca. And w ( (s) = («£> ,w$,w$)
the vectors defined as follows: '
%1
'
, „iV , N„l 12
,(•> =
= Av ( s >,
= AS 13
\ <>.)
\
/
(6.6)
86
CHAPTER 6. TURBULENT
GIBBS
DISTRIBUTIONS
where v} 8 ' = (% ,«}* ,Vf8 ) denotes the velocity vector of the Ith molecule inside the cube Ca and A. = I
0
Bs
0
I ,
(6.7)
Ba = 1
1
1
1
VTT;
VK
VA^
VNl
i
-1
0
0
0
0
0
-3 VT2
0
i
0
V2 1
1
-2
V6
V6
V6
1
1
1
Vl2
vT2
VT2
v /(W.-2)(iV.-l)
\
1 y/(N.-l)N.
^(^.-2X^.-1) 1 y/(Nm-l)N.
4r
-(Af.-l) y/(Nm-l)N.
\
)
(6.8) The benefit of writing the equation (6.1) in the form of (6.5) is that the (s)
quantity -J== denotes the mean velocity of the molecules in the cube C s , but the quantities w} 8 ' ,1 >2 are linear combinations of the relative velocities of the molecules in the cube C s with respect to the center of gravity of the molecules in the cube C s , and conversely, any relative velocity of the molecule in the cube C s with respect to the center of gravity of the molecules in the cube Cs can be represented as a linear combination of the quantities w^ ,Z > 2. The relative motion of the molecules in the cube Cs with respect to the center of gravity of the molecules in the cube Ca represents the heat motion of the molecules in the cube. w<"> The quantity -jk^ represents the velocity of the fluid particle in fluid dynamics, i.e., a macroscopic variable, and the sum of the squares of the quantities wj , Z > 2 is proportional to the heat energy of the fluid particle. Hence the reason why the
6.1. ASYMPTOTIC
ANALYSIS
FOR LIOUVILLE
EQUATION
87
orthogonal transformation (6.6) is introduced is that it will separate the variables representing the macroscopic motion of the fluid particle from those representing the heat motion inside the fluid particle. We introduce a set of new variables y^s^ as follows:
„(•) y
N,l „(•) »12
»
= Ax'(•)
= Aa y(8) y"(«)
Now we assume that the side length K of the cube is an infinitesimal and the intermolecular potential ip depends on K and the molecular mass m in the following way:
lK|x|)=ms(§[).
(6.9)
Therefore the intermolecular force is of the form:
a
f W - - 5 « H > = -5
(6.10)
Statistical mechanics concerns itself with the asymptotic behavior of the system of N particles as N —> oo. Usually we assume that the limiting process N —> oo is taken under the following conditions: m AT = m ] T j V . - » K
(6.11)
and Ki <
NK6
< K2,
where K, Ki and K2 denote three positive constants.
(6.12)
88
CHAPTER
6. TURBULENT
GIBBS
DISTRIBUTIONS
Of course we can make a more general assumption: V(|x|) = m «
Q
-
2
H ^ , a > l
(6.9)'
and Ki < Nn3a < K 2 ,
(6.12)'
where K, Ki and K2 denote three positive constants. For the sake of definiteness, we shall restrict our discussion to the special case (6.9) and (6.12). The general case can be treated in a similar way. It should be emphasized that there are two kinds of quantities in statistical mechanics. One is called the kind of macroscopic quantities, of which the variations inside each cube Cs are negligibly small. Macroscopic quantities can be regarded as constants inside each cube C s . Or, in van Kampen's [78] terminology, they are slow variables. The other is called the kind of microscopic quantities, i.e., van Kampen's fast variables, which are rapidly oscillating inside each cube C s . All macroscopic quantities we are interested in are (nearly) constant inside the cube. Therefore, the macroscopic variables may be regarded to depend on the cube and to be independent of the points inside the cube. They can be regarded as functions in cubes, i.e., with the cube as its independent variable. It should be emphasized that most of them are random variables with respect to a specific probability distribution on the cylinder, i.e., they are functions in w^ , CJ[ , w2 , W3 , w£ . Although we frequently regard the macroscopic quantities to be constants inside a cube C s , but actually they are variables inside the cube Cs. What we mean by saying that is that the variations of the macroscopic quantities is of the order O(K).
Besides, we always assume that the spatial derivatives of the macroscopic
quantities are of order 0(1) and approximately equal to the corresponding difference quotients of their (constant) values inside the neighboring cubes, e.g., ^ - ( x ) « — M s + ei) - w(s - ei)] = O(l),
6.1. ASYMPTOTIC
ANALYSIS
FOR LIOUVILLE EQUATION
89
where w(x) is a macroscopic quantity, x £ Ca and e\ = (1,0,0). In the sequel, the only microscopic quantities we are interested in are the quantities expressed in terms of the quantities V(l x i — x jl) — niKH(|Xj—XJ|/K 2 ) and their derivatives. The occurrence of an infinitesimal denominator K 2 in the expression of the independent variables of the function 5 exhibits the rapid oscillation of the quantity inside the cube with the side length K —* 0. Precisely speaking, the spatial derivatives of
I/J
is usually of the order
0(K_1).
Statistical mechanics concerns itself with
the connections between the microscopic and macroscopic quantities by using the method of averaging the microscopic quantities with respect to specific probability distributions. It is plausible to assume that
s x/€C»
x f c ec,
N.
(-1
-«">?(g^[£*,-<,-i>«t4*)* <-) because the molecules in the cube C s interacting with the molecules outside the cube Cs must be situated in the boundary layer with thickness of order K2 of the cube C s and the ratio of the volume of the boundary layer to that the cube is of order O(K). The external force Y is of the form: Y(x) = mT(x),
(6.14)
where S and T are independent of K and m. Moreover we assume that the distribution function F has the property that dF
/ 1 \
^=°U> For example, if the distribution F depends on K in the following way:
F(Z;0 = *(---;y| s ) I ^ 1 y^ s ) ,---,«- 1 yS;v«,...,vS;...;^
(615)
90
CHAPTER 6. TURBULENT
GIBBS
DISTRIBUTIONS
= *(--Sqi8US\---,q^v^---,v#;.••;*),
(6.15)!
where $ is independent of K and m, (»)
(s)
qi =y\ , and qj'^K"1^,
2
the assumption (6.15) will be saisfied. So does it, if we propose that F is of the form: F(Z;0 = $i(---;yiS),«5(«-2y(s),---,«-2y^);vis),---,vW;..-;*)
= $i(---;p?),G;vi,),--.|vW;.-.;t),
(6.15)a
where $ i and 5 are functions independent of K and m, and Pi
s)
= y [ s ) , p | ' ) = «- 2 y}" ) , 2 < i < J V 8 ,
G = ng(K-2y{2\- • •.K'VJV]) = ^ ( P ^ V • ->P$)Actually, the turbulent Gibbs distribution obtained later on is of the form (6.15)3. C.B.Morrey, Jr. [61] has proposed assumptions similar to (6.9), (6.11), (6.14) and (6.15)i. The differences between Morrey's and ours are as follows. Firstly, we „(•>
have used 7^=, which is the position vector of the mass center of the molecules in the cube C 8 , instead of the position vector of the first molecule among the molecules in the whole space, as was used by Morrey and the authors of the theory of BBGKY hierarchy. I think, what we have done is better in modelling
6.1. ASYMPTOTIC
ANALYSIS
FOR LIOUVILLE EQUATION
91
reality than that Money and the authors of the theory of BBGKY hierarchy did, because the candidate for the position of the fluid particle ought to be the position of the mass center of the molecules inside the cube (i.e., the fluid particle), not the position of the first molecule among the molecules in the whole space. Even if it is assumed that the distribution is symmetric with respect to the molecules, both programmes will yield different results, unless the assumption of molecular chaos holds. The technique of subdivision of the space occupied by the fluid into a large number of disjoint cubes provides us with the possibility of using the mass center of the molecules in the cube instead of the first of the molecules in the whole space. Secondly, Morrey assumed that:
^(|x|) = m~'(M),
(6.9)"
which coincides with the assumption (6.9)' with a = 1. The assumption (6.9)' with a > 1 means that the range of the intermolecular force will shrinks to zero much faster than the size of the fluid particle. Morrey's assumption (6.9)", which coincides with the special case a = 1 of the assumption (6.9)', only means that the range of the intermolecular force will shrinks to zero and does not mention the limiting behavior of the ratio of the range of intermolecular force to the size of the fluid particle. I think, (6.9)' with a > 1 is better than Morrey's (6.9)" in modelling the reality, because the number of molecules in a fluid particle is tremendously large. The limit studied in the present paper will be the limit under the conditions (6.9), (6.11), (6.12) and (6.15) (or (6.9)', (6.11), (6.12)' and (6.15) with a > 1). In a formal derivation of the Boltzmann equation from the Liouville equation, which has been made rigorous by Lanford (see, e.g., [51], [15] or [70]) with partial success, Grad [33] has proposed a limit under the conditions:
^(|x|) = K 2 E'(i^Y
(6.16)
92
CHAPTER
6. TURBULENT
GIBBS
DISTRIBUTIONS
mN = const.,
(6.17)
NK2 = const..
(6.18)
and
Usually we call the latter the (Boltzmann-)Grad limit. With the aid of (6.17) and (6.18), (6.16) can be written as follows:
x|)=mS"C^y
(6.16)'
The Boltzmann-Grad limit is good for describing the dilute gases. The limit (6.9), (6.11), (6.12) and (6.15) used in the present paper is different from those of Morrey and Grad in that it has distinguished between two length scales: the range of intermolecular force and the size of the fluid particle. Various ways of describing the above asymptotic limits have been devised by different authors. For example, van Kampen [78] has used the concepts of fast and slow variables to explain the Enskog-Chapman technique as a method of elimination of fast variables. Muncaster [74] introduced the method of stretched field and the concept of gross determinism to get the results identical to those of the Enskog-Chapman technique. Before Muncaster did that, a more special form of Muncaster's idea had been elucidated by Grad in [33]. Some authors used the multi-scale techniques to derive the fluid dynamic equations from the Boltzmann equation (see, e.g., [49]). All these ideas will play the roles of backgrounds for inspiring us to construct the methods used in the present paper. It is easy to verify the following three equations: ZL
W.
E
(s) W
i
r'-l
a
V
°
dx ( s ) fe=i ax fc
„»
,.
air
(s)
Q
n
i\
d
dx ( s ) ax J
T
P
a$ OVi
N. a;* "• -19*1V^ dG
,_
(s)
V7 /
(sh
6.2. TURBULENT
GIBBS
93
DISTRIBUTIONS
and f(x) = - J ^ ( | x | ) = - m K - 2 ( V H ) ( * - 2 | x | ) . Bearing in mind that the expressions 1 dF m9v
N
-
f x x
„,(•> wis;
ES XE E ^^' ("V'" *)' """ EE V^v^ l6C.^
8F
N
AF
axf"'• 5> U" Ax<
»*« and the temporal derivative of F are of order O(l) and the left hand side of the equation (6.5) is of order 0 ( / t _ 1 ) , we have the following local homogeneous stationary Liouville equation governing the first order asymptotic solution of the Liouville equation: N.
(s)
d
•<-i
s
*• 1=2
Vi^-Wi
fc=i
9x
9x| s )
!i
N.
+ f^m^l^TYl E
E(^-Y(X«))-(Z-1)(II«-Y(X<'>))].^}F = 0 (6.19)
6.2
Turbulent Gibbs Distributions
Definition 6.1
A turbulent Gibbs distribution on the phase space T is a prob-
ability distribution with the density of the following form:
F(Z,t)=TN( ...;4m\U[m) ,J^
,Ui'\J^]...;t)
^ - f - ^ ^ E ^ ^ f m E l ^ t ^ E *
94
CHAPTER 6. TURBULENT GIBBS DISTRIBUTIONS
is subdivided and is)
mNs
w
(6-2°)i
o = —r N.
("l
, W2
, C/g
«? > =2J?(*E|v I W | a +
; - - m 3 2^Vi
'
(0.20)2
^(|x|s)-x«|))
E
(6.20)3
denote the mass density, momentum density, energy density of the fluid flow in the cube C s respectively. Sometimes, for brevity, we simply call the function T^ a turbulent Gibbs distribution. For brevity, we introduce the notations ir ' — (UJ0 ,ui1 ,u)2 ,w3 ,u>4 )
(o.2U)4
fi = (---;^ ( s ) ;---),
(6-20)5
and
then the turbulent Gibbs distribution T/v can be written as follows: TN(Sl;t)
= TN(-
• • ; n<->; . . . ; t)
=
TN(.
• •; ^
s )
, u,<s), ^
s )
, ^
s )
, u , < s ) ; • • •;
t).
(6.20)6
It should be noted that any probability density on the phase space F is a (measurable) function on T. For any given system {WQ , s € Z 3 } , there exists a unique subset T, <«> s 6 Z 3 , of the phase space T of the following form: T
{J0s\seZ3}
=
R
{u.< 8 ) ,s6Z3}
X R
'
where N R
{o,<"\s€Z3} =
U
HCs
6.2. TURBULENT
GIBBS DISTRIBUTIONS
95
For any fixed system {CJQ \ s € Z 3 } , the function
J- NV • • , k'o
i
w
l
> u2
>
w
3
>
w
4
' '
'
l
)
in {u>[a', u>2 i V3 , W4 , s e Z 3 } denotes a function denned on the set T , ( . )
s€Z3i,
which is the restriction of the turbulent Gibbs distribution T/v on the set T, <•) „,™31. A remark about the concept of turbulent Gibbs distribution is in order: Remark
In Definition 6.1, the cube Ca, which is defined in the equation
(6.2), could be replaced with rectangular parallelepiped Pa of different sizes. In most cases treated in the present book, we will confine ourselves to the turbulent Gibbs distributions with the cubes of the same sizes, as stated in the Definition 6.1, since it is easier to deal with and it is enough for our purpose. But there are some exceptional cases, e.g., those in Chapters 8 and 10, a more general concept of the turbulent Gibbs distributions, i.e., those with rectangular parallelepiped P3 of different sizes, is in need. Having defined the turbulent Gibbs distribution, the reason for replacing the differential operator d/chc}*' in the Liouville operator, operating on the turbulent Gibbs distribution, with the operator (d/cbc-
+ A/Ax^ s ) can be explained as
follows. When x^- goes from the cube C s to one of its neighboring cubes CCT, where a = (ffi,cr2,
t \
1=1
V
1=1
1=1 k^l
'
will undergo a ( discontinuous ) jump. Hence the rate of change of the turbulent Gibbs distribution with respect to spatial variables consists of two parts. The first part is expressed in term of the partial derivatives with respect to spatial variables, which represents the rate of change inside the cube. The second part
CHAPTER
96
6. TURBULENT
GIBBS
DISTRIBUTIONS
represents the rate of change of the turbulent Gibbs distribution as one of the spatial variables goes from one cube to one of its neighboring cubes. Precisely speaking, we have to replace the differential operator d/dxj operator with the operator (d/dx.j
in the Liouville
+ A/Ax^- ), of which the first term d/dxr*'
represents the rate of change inside the cube C s (or, in the notation of Laurent Schwartz, it is equal to [d/dXj ]) and the second one A/Ax^
represents the
rate of change between neighboring cubes. The precise expression of the second term A/Ax^-
will be specified later on, (see, the Third Proposal of Gross
Determinism ). Usually we assume that the second term (A/Ax^- )T/v, which represents macroscopic variation of the fluid flow, will be far much smaller than the first term (d/dx.* )T/v. In other words, the macroscopic change of T/v between two neighboring cubes is far much smoother than the microscopic change inside a cube. This is the very reason why a discrete version of the multiple scales in perturbation methods should be used in solving the Liouville equation. The physical significance of the assumption is that the dependence of the quantity, including the turbulent Gibbs distributions, on macroscopic variables is far much smoother than its dependence on microscopic variables. Some remarks about the turbulent Gibbs distribution thus defined are in order. The function T;v(ft;t) in (6.20) is regarded as a composite function, i.e., it depends on Z through £1 = (• • • ;wj s ,Wj ,^2 , w 3 i w 4 i ' ' •)• Since the total mass of the particle system has been assumed to be constant (see, (6.17)): mN = K, the transformation Zi —> U
— (•••,U0
,Wj
,U/2
,U)3
,U4
,•••)
defined in the equations (6.20)i, (6.20)2 and (6.20) 3 maps the phase space V ^ x R 3Ar into the affine subspace {il : S
s
WQ = K / K 3 } of the 5^-dimensional linear
6.2. TURBULENT
GIBBS DISTRIBUTIONS
97
space {fi = (• • •; ^ s ) , w[ s) , u^, w?*, wj s) ;•••)}• Hence the turbulent Gibbs distribution Tjv(fl;i), as a function in Q. for a fixed t, i.e., a function defined on the space {£2; ft 6 R 5 " } , is undetermined outside the hyperplane
s
When we consider the turbulent Gibbs distribution T/v(ft;£) to be a function on the space R 5 ", e.g., as a factor of the integrand of an integral over the space R 5 ", we should use 5(
V°
< s j E ^ _ - K - J jTtf(n;t) instead of TN(Q;t). Both TN(fl;t) and - ^ 3 )TAr(n;t) are called the turbulent Gibbs distributions. It will
cause no confusion. The values of the function Tjv, as a function in ft, is undetermined outside the hyperplane K 3 ^ S JVS = K. Its values outside the hyperplane are irrelevant for our study. For the sake of convenience we always assume that T/v = 0 outside the hyperplane, unless stated contrarily. When the system
{LJQ
} is fixed with
=
K35ZSWO
K,
tne
turbulent Gibbs
distribution T/v(ft;t), considered as a function in Z, denotes the probability dis3
(») /
tribution on the poly-cylinders of the form ([\s C£ u°
m
) x R 3 i v . Therefore, the
system {LJQ } is a parameter specifying the poly-cylinder to which the turbulent Gibbs distribution is restricted. The turbulent Gibbs distributions are distributions determined by the mass density, momentum density, energy density of the system of molecules inside the cubes Cs, which play the role of fluid particles in fluid dynamics. Why can we use such distributions in specifying the behavior of a fluid flow? I think, the reason why we can do that is as follows: fluid dynamics concerns itself with fluid density, fluid particle velocity and fluid temperature (or , pressure), i.e., the local quantities of the N particle system. Therefore, all these local quantities play great roles in fluid dynamics. On the other hand, in the classical kinetic theory
98
CHAPTER
6. TURBULENT
GIBBS
DISTRIBUTIONS
of gases, i.e., the theory of Boltzmann equations, the distribution / ( x , v , t) describing the behavior of gases specifies a distribution of molecular velocities at x and x represents the position of fluid particle, i.e., the position of the cube Ca- In other words, the classical kinetic theory does not distinguish the positions of the molecules in the same fluid particle. Actually in the heuristic derivation of the Boltzmann equation we do not distinguish the positions of two colliding molecules and do not distinguish the positions of the same molecule before and after the collision, (see, e.g., [33]). Hence, using the turbulent Gibbs distribution in specifying the behavior of the fluid is in accordance with the classical kinetic theory of gases. Finally, the statistical mechanics established by Gibbs concerns itself with the equilibrium phenomena. Gibbs (micro-canonical or canonical) ensembles are independent of x. In the theoretical frame of the present paper it is independent of the cube Cs. The non-equilibrium statistical mechanics without the assumption of molecular chaos studied in the present paper might be considered to be a natural synthesis of the Maxwell-Boltzmann kinetic theory and the Gibbs equilibrium statistical mechanics. Usually we assume that V > > K 3 and the boundary effect of the space occupied by the fluid on the behavior of the fluid is negligibly small. Each cube corresponds to five arguments: the mass density, the three components of momentum density and the density of the sum of the total intermolecular potential energy and the total kinetic energy of the molecules in the cube. The notations WQ > ui
i w 2 » ^3
an(
^ w4
denote the five densities
in the cube Ca respectively. It should be emphasized that for each fixed system of integers {Na}, the values of the turbulent Gibbs distribution function T/v(- • •) denote the restriction of a probability density on the phase space V 3 N x R 3Ar to the union of the sets of the form (V3N
u
x F] s C^m), i.e., the set
(v3jv*rK)'
6.2. TURBULENT
GIBBS DISTRIBUTIONS
99
where V denotes the space occupied by the fluid. Ignoring the order of the factors in the cartesian product, we regard each set V37V x n,-=i C*j w * t n ^»
=
S j = i ^ss,-
to be a set of the form (V3N x [ ] s C*m). It is easy to see that there are N\/ ]JS Ns\ sets of the form V3N
x n , C s w '.
The quantity (s) _ mATs Wn
is a constant inside the cube cylinder V 3 i V x n , = i ^
and when we take the system
of variables x j , • • •, XJV. ; w i , • • •, V/N. as new system of independent variables for specifying the state of the system of particles inside the cube C s , the components of the vector , (.)
(.)
(.)v _ W ^ _ , < • ) _ K
m
r* v W
K
i=i
are independent of the variables (s) X
and the quantities f^
l
(s) i • • • > XJV.'
(s) W
2
(s) > ' ' ' »
W
JV. »
- Y(xj. s) ), k = 1, • • •, N3 in the second term on the left
hand side of the equation (6.19) are equal to the components of minus gradient of the total intermolecular potential energy in the cube Cs with respect to xjj.8':
fl s) -Y(xi s) ) = -grad,,
£ 1<J,
*(xW-xM) 1
Recalling that the partial derivative -^f-jy in the equation (6.19) denotes the derivative in the ordinary sense inside the cube Cs (i.e., not in the sense of Laurent Schwartz' distribution theory), we can verify that the distribution density (6.20) satisfies the asymptotic form of the Liouville equation (6.19). We call the asymptotic solution of the form (6.20) to the Liouville equation (the density of) a turbulent Gibbs distribution and the probability with density (6.20) a turbulent Gibbs probability.
100
CHAPTER
6. TURBULENT
GIBBS
DISTRIBUTIONS
In classical equilibrium statistical mechanics the Gibbs (canonical) distribution is Ce-"H,
(6.21)
where H denotes the Hamiltonian of the system under consideration, ft a constant inversely proportional to the absolute temperature of the particle system and C the normalization constant. If the velocities of the molecules in the system has a common non-zero mean velocity u and the system relative to the inertial reference system of the constant velocity u is subject to the Gibbs canonical distribution (6.21), then the system is subject to the distribution: Ce-^(H+fi^|u|
where q =
(LJ\,U2,
2
-«3u-q))
w 3 ) T , Uj = ^ £ J L i vtj.
In the sequel, we call the above distribution a generalized Gibbs canonical distribution. Now we consider a fluid continuum as a system of subsystems. Each subsystem corresponds to the aggregate of molecules in a small cube C s of the fluid continuum. We assume that the subsystem consisting of the molecules in each cube is subject to a generalized Gibbs canonical distribution with parameters depending on the cube C s : const s . e -A(H.+=^|u.| a -« s u..q.) ) where q s = (wj ,a4 , w 3 )T> uf
(6 22)
= jfs J2iLi vi* • Furthermore, we assume that
subsystems are statistically independent of each other, then the whole system will be subject to a local Gibbs canonical distribution C e x p f - ^ / 3 S ( H S + —— | u s | 2 - K 3 u s - q s ) j ,
s=
(s1,s2,s3),
where C denotes the normalization constant: C =
[
dZe -E>(
H +i:i
»
T 1 l u 'l 2 - K3u '- c u)
-l
(6.22)'
6.2. TURBULENT
GIBBS
DISTRIBUTIONS
101
-l
N.
-n[(^)7*--(-fgg^-^))]
(6.22)"
It is easy to see that mJV 8 |
fdZCexp( -J^/3S(H;
^
2a> 0
,=i
l2
.
3
i
'
*5«
(6.23)
= 7i + r2> where -A(Ht + ^ | u t | a -
...
71 =
w
/
dV
3 K
ut.
q t
e
-i^1
x / dVw e
t
*(|x{ t ) -xj, t , |))'
fc
*<
:(Ht + ^ i | U t | 2 _ K 3 U t . q t _ l
£
Vdx^-x^l))
fc#l
3ATt 2/St'
To
[/
dV
w
(6.23)!
- A ( H . + = ? * | u . | » - « s u « - « l . - j £ i < f c . l < N t *(|x, ( t ) -x< t >|))'
e
X / cfV ( t ) e
**'
-/5t(H t + 2 ^ t | u , | 2 - K 3 u t . q t - i X_; i < f e , <<wt V d x ^ - x ^ ' l ) )
**'
102
CHAPTER 6. TURBULENT
XI - ^ l (t) ! - ^ |
U t
|
2
GIBBS
DISTRIBUTIONS
+ K 3 u t .q t N
2w,
3, ,(t)
K IV,
I
dV(t)
e
*5"
x / dVwe
qt
*#'
3V^
ut
(6.23)a
2ft '
Summarizing the results in the equations (6.23), (6.23)i and (6.23)i, we get the new physical significance of the parameter /?s: T h e o r e m 6.1
If the system of particles is subject to the local Gibbs canon-
ical distribution (6.22)', the mean of the heat energy in the cube C t is of the form:
JdZC exp ( - £> S (H S + ^ K l
0
2
- «3us • q.))
i< f c . ' < ^ t
j=l
2/?tV
VN~t)
(6.23)'
In other words, we have the following physical significance of the parameter
A: Corollary 6.1
Under the condition stated in the Theorem 6.1, we have
6.2. TURBULENT
GIBBS
103
DISTRIBUTIONS
|dZexp(-£/3a(Hs + ^|us|2-K3us.qs))
-1
(6.23)" ZU}n
\
Kk.
KN.
'
Since 1 + 1/y/Nt « 1, the equation (6.23)" can be (approximately) rewritten in the following way, Corollary 6.2 1
jt «
Under the condition stated in the Theorem 6.1, we have
2mC f ,„ —W) J dZexP
(
v-^ n
/TT
s ,,2
( - X , &( H s + — lUsl ~ K
x /( t) _Ek^!)!_^_ E ZK \
mN
2t^
3 Us q
\\
' *)J
^ . ^ v
!
(6.23r
/
MI
Theorem 6.2 Let a system of N particles be subject to a turbulent Gibbs distribution Tjv (6.20) and suppose that when the positions of the particles in the system be given, the velocities of the particles are statistically independent, i.e., N
,i<j
(6-24)
i=l then the system should be subject to a local Gibbs distribution (6.22)'. Proof Suppose the positions {xi, • • • , x # } of all the particles in the system and the velocities {v\, • • •, vj, • • •
,VN',J
^ I ^ k} of the particles except the j t h
and fcth in the cube C s are given, the conditional probability distribution C(VJ, vjt) of the velocities of the j
t h
and A;th particles should be of the following form:
Cfo.vfc) = /(v,- +v f c , | V i | 2 + |v fc | 2 ),
104
CHAPTER
6. TURBULENT
GIBBS
DISTRIBUTIONS
where / is a function denned on R 4 . According to the the assumption made in the Theorem 6.2, it also has the following form: C(vj,vk)
=
g(vj)g(vk),
where g is a function defined on R 3 . According to the famous Boltzmann-Gronwall Theorem on Summational Invariants in the classical kinetic theory of gases (see, e.g., [74], p.72, or[14]), the logarithm of g is of the form: l n 5 ( v ) = a | v | 2 + , u - v + /?, where a and /? are two scalars, and \i a vector independent of v respectively. In other words, g is of the form: <7(v) = a e - 6 l v - u l 2 , where a, b are scalars, and u vector independent of v respectively. Hence we have C(VJ.V*)
= A exp ( - b(\Vj\2 + |vfe|2) + c •
(VJ
+ v f c )J,
(6.25)
where A, b are two scalars, and c a vector independent of Vj and v^. Generally speaking, the scalars A, b and the vector c depend on the velocities of the particles other than the j t h and the fcth particles in the cube C s and those in the other cubes. Of course, they depend on the system {WQ , s € Z 3 } too. By virtue of the symmetry with respect to N particles of the distributions on the phase space, it is easy to see that b and c are independent of all v^ ,1
T*(fl; *) = A. exp ( - bs £ ^
where Aa,b3,ca
i=i
N, (s) 2
|v, | + c s • J ] v, i=i
v (s)
J, '
are independent of all v | , 1 < I < Ns. Of course they depend
on the system of numbers {WQ , Vt}.
6.2. TURBULENT
GIBBS
105
DISTRIBUTIONS
Being a turbulent Gibbs distribution, T^(fi;i) is of the form: TN(tt;t)
2«3
= Cs exp ( - —b,ur
ci
N.
+ cs
where C s , 6S and c s are independent of 7S-S\ Since the turbulent Gibbs distribution satisfies the equation (6.24), it should be of the form: TN(Sl;t)
=Ciexp
2K d
m
S
S
1= 1
-• '
= C 2 exp(-K 3 £/? s U s ) -£I>^H), ^
s
L
s j=\
-I'
where (3a, u s = (u s i,u S 2,w s _ ? ), Ci and C2 denote constants. This is exactly a distribution of the form (6.22)'. The proof of the theorem is completed.
It is evident that the local Gibbs canonical distributions (6.22)' and any (continuous) convex linear combinations of local Gibbs distributions 3
/
daC(a) exp
3
K X>,ff(u4 -£Usj,Xs)) s
s)
^
j=l
(6.26)
'
are turbulent Gibbs distributions. Turbulent Gibbs distribution (6.20), which was defined to be an asymptotic solution to the Liouville equation, is a natural generalization of the (continuous) convex linear combinations of local Gibbs distributions (6.26). O. Reynolds [66, 67] proposed that the turbulent flows are random solutions of the basic equations in fluid dynamics (e.g., Navier-Stokes equations or Euler equations) with random initial data. It is well known that a solution of the Euler equations corresponds to a local Gibbs canonical ensemble. If we take Euler equations as the basic equation governing the motion of fluid flows, the Reynolds' proposal is equivalent to the assumption that the molecular distributions of the turbulent flows are (continuous) convex linear combinations
106
CHAPTER 6. TURBULENT
GIBBS
DISTRIBUTIONS
of local Gibbs canonical ensembles (6.26), i.e., a special case of turbulent distribution (6.20). This is the very reason why the distribution (6.20) is called a "turbulent Gibbs distribution".
Generalizing de Finetti's result, Hewitt and Savage ([40], [73], [22]) have proved that in an infinite cartesian product of a measure space any symmetric probability distribution can be expressed as a ( continuous ) convex linear combination of symmetric independent distributions. Combining Hewitt-Savage's result with the T h e o r e m 6.2, we might be close to getting a proof of the conjecture that any turbulent Gibbs distribution is a ( continuous ) convex linear combination of local Gibbs canonical ensembles. Of course, the obstacle in the way of proving the conjecture is the fact that we are dealing with a limit (in a certain sense) of a sequence (or net) of finite cartesian products of a measure space. Although Hewitt and Savage had some results about the symmetric measure on a finite cartesian product of a measure space, I am not sure whether the conjecture is correct. If it were not, the class of turbulent Gibbs distributions would be substantially wider than the class of convex linear combinations of local Gibbs canonical distributions. Even if it were true, the (continuous) linear convex combinations of the distributions perturbed from a local Gibbs canonical ensemble might be different from the distributions perturbed from the corresponding (continuous) linear convex combinations of the Gibbs distributions. In other words, even if the first order approximate solutions to the Liouville equation, i.e., the turbulent Gibbs distributions, were the linear convex combinations of the local Gibbs distributions, which is used in the classical fluid dynamics, but the second order perturbed solutions to the Liouville equation might be different from the linear combinations of the distributions perturbed from the local Gibbs distributions, which is used in the classical fluid dynamics. Hence it is reasonable to conjecture that the statistical solutions of the Navier-Stokes equations, as is still used in the classical fluid dy-
6.2. TURBULENT
GIBBS DISTRIBUTIONS
107
namics, might not be a good candidate for describing all turbulence phenomena, of which the most natural descriptions would be the solutions corresponding to the turbulent Gibbs distributions and the distributions perturbed from them. Now we put forward Assumption A
The turbulent Gibbs distribution of the form (6.20) corre-
sponds to the first order approximate description of fluid motions, including both laminar (i.e., deterministic ) and turbulent ( i.e., random ) motions. Definition 6.2 The special cases of the turbulent Gibbs distributions of the form (6.26), which are (continuous) convex linear combinations of local Gibbs canonical distributions, will be called the Reynolds-Gibbs distributions.
Although we do not know whether each turbulent Gibbs distribution is a Reynolds-Gibbs distribution, but it might be expected that the turbulent Gibbs distribution for N large enough could be approximated by a Reynolds-Gibbs distribution in a certain sense. The coefficient C(cr) in the equation (6.26) denote a constant with the parameter a satisfying the following normalization condition: J dJc(
2TT
II
dXexp " /
= 1. s
l
J
(6.26)i
)
We should make four remarks on the concept of turbulent Gibbs distribution. Firstly, since statistical mechanics concerns itself with the asymptotic behavior of a system of many particles as the number of particles tends to infinity. In fact the Assumption A states that the turbulent Gibbs distribution (6.20) is a nice statistical description of the system of particles as the number of particles is large enough. What we are interested in is not a single turbulent Gibbs distribution,
108
CHAPTER
6. TURBULENT
GIBBS
DISTRIBUTIONS
but a sequence (more precisely, a net or a filter) of turbulent Gibbs distributions, for which the sequence (or net) of the corresponding K functionals converges in a sense specified later. Such a sequence (or, net) will be called a turbulent Gibbs measure for the system of molecules. The study of the details on the connections between the turbulent Gibbs distributions and their corresponding K functionals will be deferred to section VIII. Secondly, we assume that the turbulent Gibbs distribution decays to zero rapidly, as one of the quantities (i.e., for a certain s), which represents the density of the internal energy of the molecules in the cube Cs, N.
Ev«v (i
(S)
1=1
tends to infinity. Precisely speaking, usually we assume that the turbulent Gibbs distribution should be dominated by a constant multiple of a local Gibbs canonical ensemble. Thirdly, any integral of the motion of the particle system is a solution to the asymptotic form of Liouville equation (6.19) and could have been taken as an argument of T in the definition of the turbulent Gibbs distribution.
But
we have not included the total angular momentum inside the cube Ca as an argument of T in the definition of turbulent Gibbs distributions, because the size of the cube is so (macroscopically) small that the total angular momentum approximately equals the vector product of a (nearly) constant vector and the total momentum, therefore, a function in total momentum.
Of course, there
might be other interesting integrals of motion, but the results of Bruns, Painleve and Poincare (see, e.g., [39] or[82]) show us that the integrals of motion, which have some algebraic or analytic properties, are functions in total mass, total momentum, total energy and total angular momentum. The later can be ignored for a cube of (macroscopically) small size.
6.3. GIBBS MEAN
109
Finally, it is worth noting that under the assumption A, i.e., the distribution on the phase space VN
x H3N is of the form of a turbulent Gibbs distribution
(6.20), the total intermolecular potential energy of the molecules in the cube Cs is not a constant inside the product of the cubes f l s ^ s - * t s behavior is quite complicated inside C^ x R 3JV . Hence we ought to regard the total intermolecular potential energy in the cube Ca as a microscopic (or, in van Kampen's terminology, fast) variable. But the average of the total intermolecular potential energy in the cube Ca with respect to a certain measure, which will be specified later on, is really a macroscopic (or, in van Kampen's terminology, slow) variable. Seeking "a finer if-functional equation", we frequently need to "slowly" differentiate the total intermolecular potential energy in the cube C s with respect to spatial variables, i.e., to compute the difference quotient of the intermolecular potential energies in the neighboring cubes. It is nonsense to "slowly" differentiate a fast variable. Then we have to use the average of the total intermolecular potential energy in the cube C s with respect to a certain measure instead of the total intermolecular potential energy in the cube C s . It seems uncomfortable to do that, but now I can do nothing out of the way. Hence it is necessary to illustrate the average in great detail.
6.3
Gibbs Mean
In the kinetic theory of dilute gases, the total intermolecular potential energy is assumed to be negligibly small. But in statistical mechanics of general fluids (even dense gases), the total intermolecular potential energy cannot be considered to be negligible. On the contrary, it will play an important role in deciding the properties of the fluids. Usually it causes a great trouble in theoretical treatment. In the present section an important concept, called Gibbs mean, will be introduced. It plays the role of the density of the difference between the internal energy and
110
CHAPTER 6. TURBULENT
GIBBS
DISTRIBUTIONS
the heat energy of a fluid particle in thermodynamics or hydrodynamics. The total intermolecular potential energy of the molecules inside the cube Ca
^EEMs)-xIs)) 1=1 k?i
is a function on (CS)N'.
Usually it fluctuates drastically on (CS)N".
For exam-
ple, it may take the value infinity when one molecule approaches the other. But the drastic fluctuation will occur very rarely with respect to a turbulent Gibbs distribution. That is the reason why the concept of the intermolecular potential energy of a fluid particle, which corresponds to a cube Ca, can be used in the classical fluid dynamics and thermodynamics. Although we have not got a rigorous proof of the above assertion, it is plausible to assume that the total intermolecular potential energy of the molecules inside a cube Ca can be replaced with the its mean with respect to the turbulent Gibbs distribution under consideration. So we introduce the concept Gibbs mean. In order to calculate the mathematical expectations of random variables with respect to the turbulent Gibbs distributions, which will be encountered frequently in the sequel, we have to introduce a new system of curvilinear coordinate variables in the phase space ~VN x R3N.
As a first step, we will introduce a poly-spherical
coordinates in the subspace C^s x H3N' of the phase space V w x R 3iV spanned by the 6./Vs-dimensional vectors representing the totality of the random velocities of all the molecules inside the cube Ca, i.e., the totality of the relative velocities of the molecules inside the cube Cs with respect to the mass center of the molecules in the cube C s , which represents the position of the fluid particle composed of the molecules inside the cube Cs. It has been remarked that the random velocities of all the molecules inside the cube C s are represented by the linear combinations of the vectors W2, • • •, w^r.. The curvilinear coordinates, which we are to use and will use frequently in the sequel, are defined as follows:
6.3. GIBBS
111
MEAN
w
21
r
—
s m
(s)
rst
s l n
V2
•
(s)
s
¥>1
•
m
(a)
"AT,-2 1 '
•
-
"S
l n
/i(s)
•
s i n
"21
„(s)
u;^' = r w sin
W
(s)
AT.I
= r
(s)
Cal •
(s)
•
(s)
simp2 ' siny>J (si
(s)
•
fsl
.
(s) N.2 = (s)
r
(si
(s)
•
Sm<
P2
•
(s)
(s)
COSl
Pl
.
n(s)
(s) Cst (s) t/;^ 3 ' = r w COS <^2
(s)
(si
=rWcos
•
/i(s)
.
0(B)
2
2 i 2
„(a)
a(a)
• • • sin022 coso\2 ,
,
sin 6jy _ 2 3 • • • sin ^ 3 sin #13 ,
u;23 = rv ' cos tp2 '
1^.3
/i( s )
•
/i(s) COS^_
(s)
(a)
^11 >
sin 6N _2 2 • • • sin 622 sin 8\2 ,
W32 — r w siny?2 cos(/?j' s\ndyN'_2
W
n(s)
cos
/i(s)
(s)
(s)
'
c o s ^ _ 2 x,
^22 = r w sin
"ll
s i n
^ - 2 , 3 •- " s
i n S
23
c o s
^W '
COS0 (»)
(s)
¥,2
TV.-2,3 '
(6.27)!
0 < v4 s) <
TT/2;
0 < ^2s) <
(6.27) 2
TT/2;
3(8) 9(8) 0 < flj;' < 2TT, fc = 1,2,3; 0 < 0™ < vr, j = 2, • • •,Ns - 2, fc = 1,2,3.
(6.27) 3
(si
where w\-, i = 2,- • • ,NB; j = 1,2,3 denote the variables defined in the equation (6.6) for the cube Cs and r N,
rW
3
1/2
EEKf)
(*h2
(=2 j = i
2£s m
26,s,kin m
"&s,pot
m
-11/2
25.s,/ieat
1/2
m (6.27)4
£ s the total energy of the molecules in the cube C s , £s,fcm the kinetic energy of the macroscopic motion of the fluid particle composing of the molecules in the
112
CHAPTER 6. TURBULENT
GIBBS
DISTRIBUTIONS
cube C s , £a,Pot the total intermolecular potential energy of the molecules in Ca and £atheat the heat energy of the fluid in C s respectively:
£> = ? E I>?) 2 + \ E EM S) - *,(s)) = *J;\ (=1j=i
i=l
3
f
.. -
m
(6.28)
k^l
3
n) 2
rU )
- "
i=i
3
/, ,(sh2 ( } V ^' j=l w0
(6.29)
^4EEMS)-^),
(6.30)
(=1 fe^i N, £s,/ieat — ^ s
£s,fcin
1=2
,(»)
3
-IE
3
^s,pot — mx—>\—», „ / J / X^g
f,/ , ,A( »s ;)\\22
T
00 w,
2K3
(sh2 )
j=l
AT.
(6-27)1 J=I fc,ti
It is easy to verify the following Proposition 6.1
The absolute value of the Jacobian of the transformation
of variables inside the poly-cylinder C^":
x « , v<s> - x< s \ £m, «,<•>,«,«, w$, $ w , e< s) , e 2 s ) , ei s )
(e.si)
IS
d(x{*\
• x(s)-v(s) •
v w(sh ~l
9(xi,»,-)xi;;;f.)wW$W,e[,)18i,)10i,)) ( r (s))3JV.-5
m
3
s i n 2 " - 3 4 S ) c o s " - 2 ^ 2 S) s i n " - 2 ^ s ) c o s " - 2 <s) n i=l
JV a -2
]][ sin'" 1 0 « . i=2
(6.32)
Or, equivalently, the absolute value of the Jacobian of the transformation of variables:
x<->, v<s> -> x<s>, J*], 4S), 4 S ) , 4 S ) , $(s>, e<s) , e 2 s ) , e<s)
(6.3i)'
6.3. GIBBS is
MEAN
113
m",
. x ( s ). v < s)
(s)
£)(„(*)
v (8 h
„ ( • ) . , . ( » ) , ,(«) . ,00 , ,(s)
ft(s)
ff,(s)
(111)5/2(0,^)3/2
ft(s)
ft(sh
i = i »=2
(6.32)' where ,(•)
2£a
2£.s,kin
m
m
,3,» 3 / , >W ) \ 22 2«"a, W «33E=1K ) K
r
1/2 ^&s, pot
2£s,heat
m
m
VN«
E E t X W W(»)' - * „(»h T>
(s)
m
1/2
(6.27)2
m
mw.
In other words, r^
1/2
denotes the square root of the double heat energy per unit
mass inside the cube Ca.
When the distribution F(Z, i) on the phase space WN x R 3Ar takes the form of a turbulent Gibbs distribution
F(Z,t)=TN(Q;t),
(6.20)'
the average of the total intermolecular potential energy of the molecules inside the cube Cs
\
£
v>(ixis)-x[s>D
l
-\%"'-
with respect to the turbulent Gibbs distribution (6.20)' is
fdZF(Z,t)\ J
J2 V-(|xit)-x[t)|)= fdZTN{^t)\ l
J
Y. Z
l
V^-x^l)
l
-?[n(/eS.«wL*^/w..,*^-*ffi);
114
CHAPTER 6. TURBULENT GIBBS DISTRIBUTIONS
l£ ^ElX^-xh 1=1 k^l
^("^•^"'.•'^("Eiwi-'^EEM-'-xi"!));-^) V
\
i=l
1=1 k^il
'
)
- E iDb(n/c„.^")(n/R,^")^i; E «4,,-!") n y T
K 3 JV!
J
v nf-
9_f(AT.-l)
/-oo
r( .,,(s) ,,(•) ,.(•) ,.(•) ,.(»).
27T2 W '
1;
K
E 3 =A ,VViV s !r(|(7V s -l))m
I
J
s LK ^
2
.,
<W S) <W S) <W S) <W S)
;
(=i fe^(
(6.33)
where £s,int denotes the internal energy of the particles inside the cube Cs, i.e. 3
(6.34) 3=1
(s )
_ miV s
6.3. GIBBS
MEAN
115
. (.> (.) (Wj
mV]VsW<8)
(.),
, W 2 ,U> 3 J -
-5
.
w _ m | w ' s ) | 2 + m ( r ( 8 ) ) 2 + S_& s » * ^ 4 S ) -*l 8) )
W4
and "+
(a, \ 0,
if a > 0; if a < 0.
Hence it is natural to introduce a kind of mean of the total intermolecular potential energy of the molecules inside the cube CB in the following Definition 6.3 The Gibbs mean * ( t , w ^ . w ^ . w ^ . u / ^ . w ^ ) of the total intermolecular potential energy density inside the cube Ct at the point £1^ = (WQ ,U)[' ,U4 ,(^3 ,o>4 ) is denned as follows: w(t,u; 0 ,iv1 ,u2 ,UJ3 ,w 4 J AT,
• /C t
V
i=l
/
fc?tJ
+
1=1 kjtl
(6.35) the expression Aft
(2«- 3 f Mnt -K- 3 ^^V(xi t) -x[ t) )) ^
i=i
fc#(
2
' +
will be called the Gibbs power inside the cube Cs at the point o(») , ,(*) , ,(*) , ,(*) , ,(*h s r -— 1, (LJ,<*) 0 ,UJ1 ,u2 , w 3 , w 4 ; and the integral Gibbs power, denoted by
G(tfa,«,Wit),4t)fWW,a;W) = G(tinW)>
(6.36)
116
CHAPTER 6. TURBULENT
GIBBS
DISTRIBUTIONS
is defined to be Nt
3Af t -5
G(t,^))=/ jvd X«(l2^-5:5:V(xi t) -x| t )))
2
.
(6.37)
It is easy to see that both the Gibbs mean * ( t , WQ ,U>I ,^2 , ^ 3 ,014 ') and the integral Gibbs power G(t,tjQ ,w[ ,u4 ,t<4 ,1^4) depend on Cl^ through the dependence on the functions £t-nj and U>Q , therefore sometimes we would like to write 9{t,u,g\UV,Jf\u;g\Jf>) = *(t > ^, 4nt ,4 t) ) l
(6.38)
G(t,U^\J^\LJ^,J^,J^)
(6.39)
and = G(t,€t>inUuj^).
But it should be stressed that the quantity f — K3 U) (») t-t.int — "•
1
3 3
(^))2n
2 - ^ (8) 3 = 1 <4'
depends on M}"', i = 0,1,2,3,4 too. Therefore, 9 (.)G(t,£t,int,Wo ) does not represent the partial derivative of G(t,£t,int,k>o ) with respect to U>Q for fixed £t,int, but for fixed u\ , i = 1,2,3,4. It is intuitively reasonable to assume that the Gibbs mean *(t,£ t ,mi,Wo ) depends only on UQ . But we do not need the assumption for the time being. Now we can reformulate the equation (6.33) as the following T h e o r e m 6.3 The mean of the total intermolecular potential energy inside the cube C t with respect the turbulent Gibbs distribution F(Z;t) — T/v(fi; t) is 'dZF{Z,t)\ / •
£
Vdx^-x^l)
l
27r2^» VK 2
N
E. -=
N s
NaT(UNa-l))mSi^1N^\
6.3. GIBBS MEAN
117
: J] (J (W'W'W'W"* ^{^(fij^GCt,^^,^)*^,^,^,^) (6.40)
This page is intentionally left blank
Chapter 7
Euler X-Functional Equation 7.1
Expressions of Bi and B3
In this chapter we shall derive an approximate form of the if-functional equation, called Euler AT-functional equation, which holds under the condition that the distribution F(Z, t) on the phase space is of the form of a turbulent Gibbs distribution (6.20). In order to do that we have to obtain expressions for the quantities B2 and B3 in the equation (5.10) in terms of partial derivatives of the /{"-functional under the condition that the distribution F(Z, t) takes the form of a turbulent Gibbs distribution, i.e., F(Z, t) = Xjv(fi;£)The quantities Bi in (5.12) and B3 in (5.13) are of the following forms
*
2
1 8H
=
- 4 ^
r
e
3
3
(7.1)
>
and 1
B
>=
9H,^ \dc(y) • VuAu<5(u) - l e ^ r W 9 ) dy
(7.2)
respectively. If b is so slowly varying with respect to y that it and its derivatives almost take constant values inside each cube C s , i.e., inside each fluid particle, we shall write b(s) instead of b ( x ^ ) and db/dsi(s) 119
instead of db/dx\s
( x ^ ) for
120
CHAPTER
7. EULER K-FUNCTIONAL
EQUATION
any x' s ^ € C s . Actually, it is equivalent to assuming that the quantities b(x(s>)-b(s) and db/&r| s ) (x ( s ) ) -
db/dsi(s)
are infinitesimals of order K, for any x ^ € C s . The above conditions will be fulfilled, if the function b and its derivatives of orders one and two are bounded on V. The physical meaning of the above conditions about b is that the measurement in fluid dynamics will not be made finer than K. It is in accordance with the meaning of fluid particles in fluid dynamics. Having made the assumption, we have the following expression for B 2 :
B2 = ^
J dZTN(n;t)exp[A}
3
3
TV
-27rmi f dZTN(Q; t) exp[,4] £ fl E
N A ,'A*
^ T ^ K " ,
3
3
s n = l ? = lfc=l
3 A J**
3
„,
s j=lk=l°8k
where A = {si,---,s f e ,---,SAr}, N
AX = [ I ( ^ X R 3 ) fc=l
"xnk
N. n=l
7.1. EXPRESSIONS
OF B2 AND B3
121
and X W,
n=
l,2,---,N.
denotes the position of the n t h molecule x; belonging to the cube Ca, i.e., {x n , n = 1.2, •••, Na} is a rearrangement of the molecules in the cube Cs, and Na the number of molecules belonging to the cube C s :
Sometimes we would like to write
s
Here we have identified two Cartesian products Y\k=i(CSk x R 3 ) and Y\k=i(Ctk R 3 ) if Ylj=i ^s,s3 = X^7=i ^s,t3 for
au s
x
- I n the sequel we use the same notation
I"Is(Cs x R 3 ) ^ " to denote them. Of course, there are N\/ \\s Ns\ A's corresponding to the same ris(Cs
x
R-3)^*-
We have introduced the following notations in the above argument:
This convention will apply to any slowly varying functions, i.e., the functions, which are nearly constant inside each cube C s . It should be stressed that the center of the cube Cs is KS. Maybe a more appropriate convention is as follows: dbj
dbj ,
(s).
For the sake of brevity, we would like to use s instead of KS in g-*-(-)Of course, the number Na of the molecules belonging to the cube Ca depends on the sequence A = (si,S2, • • -,s^v). A more convenient notation for Na is Na '. In order to avoid making notations too heavy, we would like to use the simpler notations Na instead of the more precise but heavier ones. This convention will apply to other notations too, e.g., the notation x^ is used instead of x ^ ' .
122
CHAPTER
7. EULER K-FUNCTIONAL
EQUATION
Again we shall use the matrices Aa and Ba introduced in the last chapter:
(7.4)
Aa where Ba 1
1
7TC
7K
i
72 i
V6
-i
71 i
75
1
1
1
0
0
0
-2
0
0 0
V6
1
I
1
-3
Vl2
vT2
Vl2
7l2
y/(Nm-2){Nm-l)
y/(Nm-2)(Nm-l) 1
1 y/(N.-l)Nm
-(iV.-l)
V(W.-l)Af.
/ (7.4)i
It is easy to see that both AB and Bs are orthogonal matrices. We introduce new vectors / «,«
/ «J? \
\
Af.l
12
w« =
Av(s>.
=A (•)
iu JV.2 u;(•>
V <>, /
N,2
\ <>. /
The quantity B2 hi (7.3) can be expressed in the following form:
(7.5)
7.1. EXPRESSIONS OF B2 AND B3
123
3 A
-27rmi E
jAx
3
S j=l
N
fe=l
fc
i(s)_^^|W)„g.„<;
A dZ7V (SI; t) exp[A] E
3
9sfc
V
7=lfc=l
AT.
Q t
• ^ ^ • E E ^ M E ^ j=l
1=1
fe#j
*
(»)
1=2
(7.6)
B21 + B22 + B23,
where B21
= -2«ni£ /
divb(s)V J g ) - J s )
dZT fi
^( =') exP^ E E E ( | N - **^T^)
(7.7)
B22 = - 2 7 r m i E / A
dZTN(Sl;t)exp[A]
T
ddivb(s)
iXM |
(s)2
(7.8)
J
*>
and 3 B23
-27rmiE/ J
N
dZTJV(n;t)exp[^EEElrWl)
A *»
s
OSk
j=ik?j
tl,
&)u'S)-
(7
"9)
1=2
In deriving the equation (7.6), we have used the following equation: AT.
dZT w (fi;t)exp[yl]EK 8) ) 2
/
/
dZTN(Sl;t)exp[A}^2\wls)\2,
(7.10)
124
CHAPTER
7. EULER K-FUNCTIONAL
EQUATION
which follows from the elementary equation /
|w,| a dS,
wldS=\f
where dS denotes the area element of the 3-dimensional unit sphere. Now we are going to derive the expressions for B21, #22 arid B23 in terms of derivatives of if-functional under the condition that F(Z,,t) = T/v(fi;i), consecutively. B21 = 3 X
JA
*
3 k
a j = l fc=l \
3
3
1-
-27riWA dZTw(fi;Oexp^]EEE A ^
*
j=lk=l
'
'dbj(s)
( dsk L^
a
N.
N.
(a)
l=\
f=l
Na
divb(s) djfc i
I
\C ..^^//^^E^-rVpf-^u.v!-.,^ S
AT.
"//
m
xc(y)a*(u) E \C,\
duk
^
<^(y _ x ( ) exp(-27ri u • Vj
3
3
f-
'c%
= - ^ E / ^^(^Oexp^EEE \C4 Z1T1
A J*»
x
j = lk=l
a
L
)dyd\i
divb(s)
^
\
m6{y x|S))exp( 2?riu vS)) ydu
IIlcf^t'
"
ttlcfi^rt'
mS{y
" ''"
"
X/(s))exp(_27riu vsVyrfu
''
7.1. EXPRESSIONS
x
125
OF B2 AND B3
(// l£r6{u) ^m5{y ~ x's))exp(_27riu • v l 8) ) d y du )
•iMx:{(f>-^) fl277
o
/
dz
* ( y - ^ . « ( y - ^
-Qg2^i^2,03,e4,e5
+ z*(y - rj)8(u))
= f*n±±{( ^0fy0 0
divbM
— **
'
<*(y - » ? ) . <*(y - » ? )
Hy - v), ^ * ( y - * )
dbjda
92AT +
96^ (a
+
^(y-7?)'b'c)
d2K + -Q^-(a + zS(y - r]),b,c)
s(y-v),^^-s(y-v)
m
m
])}•
(7.11)
where C s denotes the cube with the index s and |C S | the volume of the cube Cs,
xc(y)
1i ,
I'
y € c,, (7.12)! y£Cs.
and *(y)
xco(y) ICol '
(7-12)2
126
CHAPTER
7. EULER K-FUNCTIONAL
EQUATION
Co being the cube with center 0. We call 6 the macroscopic Dirac 6 function. If b is a function which varies so slowly that b takes almost constant value inside each cube C 8 , then
/ '
6(y)<S(x-y)dy«6(x).
In deriving the formula (7.11), we have used an elementary formula for the integral ( or antiderivative ) of iZ-functional, which can be easily verified by using the definition of ff-functional: 3
3
^ £ f^ dZTsV; t) eMA] £ £ £ [|C| (|i(s) - ^ f ^ , ) 2m x
//^fE^-!!Vp(-2-.vf>)^u
x//^^£m*(y-xl<'>)«xp(-5hri«.v«)4ydu
-i
s{u)
( / / lcT
^
mS{y
~
x,(s)) exp(_27ri u v (8) d du
' i ) y )
~±h±±{(&>-*?*<*) ^£ioodz^-(e1,e2,e3,e4,e5
+ zS(y-r,)5(n))
S(y-r,)^L,S(y-r,)^.
}.
In the sequel similar formulas for antiderivatives of the if-functional will be used frequently. It follows from the P r o p o s i t i o n 5.3 that
*•--£<•>
div b(y) 12TT 2
-A u <S(u)
7.1. EXPRESSIONS
2dK
i
1 + ^2^
2dK (a, b , c) ^ d i v b C y ) 3 da m
K ^ divb(y)
f P f
127
OF B2 AND B3
d2K
~-
- /
rfC^(0-^r(o.b,c)
exp(27ri£ • y ) d i v b ( y ) , exp(-27ri£ • y) (7.13)
In deriving the equations (7.11) and (7.13) we have used the assumption that the distribution F(Z, t) is of a form of turbulent Gibbs distribution only in deriving the equation (7.10), since the turbulent Gibbs distribution is independent of the random molecular velocities. For a distribution perturbed from the turbulent Gibbs, (7.10) does not hold in general, but we can use the equipartition of heat energy (see the section 10.9) to avoid using the independence of the turbulent Gibbs distribution from the random molecular velocities. Hence the equations (7.11) and (7.13) hold for any distributions F(Z,t)
perturbed from the turbulent
Gibbs distributions. In order to derive an expression for B23 in terms of the K-functional we have to use the assumption that the distribution F(Z, t) is of the form of a turbulent Gibbs distribution. On account of the fact that the integrand in the integral on the right hand side of (7.9) is an odd function with respect to w/ (I > 2), we have £ • 23
(7.14)
0.
Summarizing the results of the above argument, we have the following P r o p o s i t i o n 7.1
If the distribution is a turbulent Gibbs distribution, B2 is
of the form: o
o
*-/*g£(^)-^^) div b(77)
3=
dz
d2K (a + dbjdbk
/" { j—100
k.
z6{y-T]),b,c) <5(y - v), Hy - v)
CHAPTER
128
7. EULER K-FUNCTIONAL
+
^k{a+z~s{y-^h'c)
+
d2K dhd-a{a
+ -Q^2Ka +
+
5(y-v),^^~S(y-r}) m
Z 6{y
' -^h'c)
z8(y-TJ),h,c)
m
-^— Hy - v), —^~ 6 (y - v) 2dK (a,b,c) ^ d i v M y ) 3~da m
2dK + 3 ,^ ( a , b^, c^) div b(y)
+
EQUATION
eiiPf-/^^)l^(a'b'c)
exp(27ri£ • y ) d i v b ( y ) , exp(-27ri£ • y) . (7.15)
In order to derive an expression for B3 (see (7.2)) in terms of partial derivatives of X-functional under the condition that F(Z,t)
= T/v(fi;t), we need a formula
8
connecting a cubic form of the vectors v j ' and a cubic form of the vectors wj in (7.5): L e m m a 7.1 N.
3
}
2w\s)
J £1=1 i ^ W = «#•# + 4" + £ -TW & Ns
( 7 - 16 )
3=1
where 7V(»)
/(s)_
f
z^i
(i-2)|wW|»mW
f><">|a + - ^ | w « | a ,
(7.16)!
m
f
f
|w|->|^W
*
*
wf'.wf^'
i=3
(7.16)a
,
)
1
jU=y i« -* " Z=2 ^
(8)12
w1
(7.16)3
129
7.1. EXPRESSIONS OF B2 AND B3 Proof N.
N.
s) 2
,
r
,
,
N.
Ei^ i ^ = E 1=1
1=1
N,
(s)
j=l+l
(s)
«E
-
(s)
2
(l-l)wf'.w{s>
(s)
V ,
>/20
i\9|
^Vs / I
J^^z. 1
(s)iO (s)
vI'-'lVilH + "It y-izJ|w«)i2_ £ /=2 'v^O-i)
«# A *
A7.
A/.
|w(s)|2
I
N.
^
( | _ 1 \
w
;=2
N
'
N
iw^l2^
« . w ( s > .„<•)
1
WB
fc—1
i
-
j_ v v ' - ' i_(.in..w l i I fei fztliVW^) w
w
AT.
AL
(s)
(s)
(s)
2
=+ EE — NsVJU^T)
^)VHk-i)
hjiti VNsj(j-i)i(i-i) 2f,(l-
w
«.J-l(|.i)|wWtf
( s ) i 2 (s)
+1=1 E Ei^+i E kit+i Hi ~ «•
>
E /k(fc-i) Ars^r^T) + E stfjv iv
^ -i
N
v^^i) ^ l V ^ F i )
JV.
i
(1-pwW-wj"
(t - I X
(s) i
I ( s ) i 2 (s) _ V - _l_..W,a.>) V^^ q - l ) | ,(")ia..,W wna<
»i»J JJV, Bs
*
o
•
(»)
1
+2
v^
| (s)(2
iV.
i=i j = i + i
AT.
N.
(s)
(s)
(s)
tijitikiiiVNsJti-mk-i)
l)w<»> • wf' uff , n ft d - DwW • WW u , " iVs^I^l) /NJ
' 11 t w W ™,(s>,,/s>
*• ^
fl-lW(,).w(!)i»w
CHAPTER 7. EULER K-FUNCTIONAL EQUATION
130
+2
»• »• «-l)w(»>.w<%(*>
* *
S,Si
l=2j=l+lk
iyfflF^ JV.
£ i£lVWi-i)*(*-i)i(i-i) (8)
(»)
(»)
fe <+i
W A . H + 1 ViO' - D*(* - Did -1) AT. AT.-l
N.
N.
+ 2V* Y" V"
"
{s)
V !w(s)l2 -
- -^—w
£ 1=1
AT.
—
(s)
wl
3V3V:
(s)
J
N
r
+2
W
V"
I
|w 2
W
(.)
(s) W
fc
(s)l2
* '
mi
w
(s)
"
W
« U*iW}3 WH ) + 4_^(«) S?g((^,.-!=C!!) VK N.
fc-1
N.
w(s)
_,,(s),„(s)
JV. , ,
„,,
(s)|2
—
W V lw<8>|2
?_lw«| V > - V ( / - 2 )' W ' S ) I 2 ^ ) 3vW
i=i
^
|
(
<
^
-
^
)
(s)
7.1. EXPRESSIONS
131
OF B2 AND B3
3
2«,< 8)
-M+^+E^Mg Using the above identity we can derive an expression for B3 in terms of the partial derivatives of /^-functional as follows: Proposition 7.2
If the distribution is a turbulent Gibbs distribution, B3 is
of the form:
B3= d
a+ y r?) b c)
l/ %^7_l{SI( ^ - ' ' d2K
*(y - v), <J(y - v)
Yt —S(y - v), <5(y - v)
d2K (a + z5(y dadb
-T]),b,c) US(y - v). % - V)
d2K (a + z5(y - rf), b , c) I U5{y - rj), ™ £ ( y - *?)] } da2
+ 12m2:
*{
d3K (a + z6(y-ri), da2dh'
d3K + -fa3-(a + zS(y-il)>
b
aptJd"^Ld*h9K) b , c) j ( y - r / ) e x p ( 2 ? r i ^ - y ) , exp(-27ri£-y),
S(y-rj)
\Yt» c) <5(y-r/)exp(27rif-y),exp(-27ri£-y), — S(y-Tj)
-lId^MJLdziL
dzo
E { ^ ( a + (^i+22)%-r/),b,c)&"(y-r?),^(y-7,))^%-r?)
132
CHAPTER
+ 2
d^d-a{a
+ {Zl +
7. EULER K-FUNCTIONAL
Yt~ <*(y - v). <*(y - v). —<*(y - v) m
Z2) 5{y
' -^h^
d3 K + Qiftfa (« + (*i + «2>*(y - f),b,c) <5(y -»?), S(y -
+
da2db^a
+
^Zl
~d^db{'a
YiTI)
, —6(y
- v)
~ ^'b'^ ^(y-»?),^(y-»/),%-»/)
+ Z2 Y
^
a 3 A: + 2
EQUATION
+ ( z i + 22)<J(y
+ ~gbdtf(a + (2i + z^(y
~ ^
b c)
'
Yt Ft —<5(y - r?), -^-<*(y - V), S(y - T))
- v),b, c) *(y - » / ) , <*(y - * ? ) , *(y - v)
}•
Proof I
Ba =
dZTN(£l;t)exp[A]
x / / d y d u - ^ ( y ) - V u A u <5(u) ^
=
(5(xf - y)exp(-27riu • vj)
^ydZrAr(n;t)exp[^]5]^(xj)(-87r3i)-vj|vJ|2
-imri£ /
dZrJv(n;t)exp[^X;E£(8)-vl')|v}')|2
A ^
s
1=1
a S
3
= —m7ri W A •/A^ 9,.,(s),
3^
(s)|2
r
dZT„(ft;*)exp[A]£]T!^(S) s i=l °Si
N.
fe
..
„x
(8),
vFi)
(s)|2
3 ^
L Ws
JV.
(s),
(s),2
UMi VJiF^)
(7.17)
7.1. EXPRESSIONS
N.
133
OF B2 AND B3
N.
(.) /
(s)
(eU
N.
j
«E E ""J"; " ' ^ E ^ ' E ^ S ? J=2j=!+1
V J U - l )
2
V^Vs
(•) V ^ /'c
+
M\2
^
J=2
Ml2
W:
su.,( s )|2
^
ou„(s>|2>
-—E^^mo-PWEiw -f ( g U r - w ) -m7ri
ac,, 5 dzrjv^Oexp^iX;^^) 3m2
£ |
x/ / ^
X
/ / Iclf
^
A
•mTri^y
m
£
|c.| ^
*(X - x| S) ) e x p ( - 2 r i « • v<8Vy<*u
u<*(u)m ] T «5(y - x| s ) ) exp(-27ri u •
dc. . dZTiV(fi;t)exp[^]^^(s)
A ""»*
x
i
(-87r 3 i)
2
v\s))dydu
1
|C S
31X13 8 ^ 1 ( ^ ) 2 |C.|
// i £ r ^ r m p{y - x<(s))exp(-27riu • ^s))dydu
x E (/ / ^
^ » £ *(y - *«) «xp(-2*i «• v « ) ^ ) ' 5i 487T 3
x^-(61,62,63,e4,65
K< f . dc
+ z~5(y - r,)S(u))\S(y
dz
- i/)V„J(u), <5(y - r?)Au<5(u)
CHAPTER
134
7. EULER K-FUNCTIONAL
•^ I d^iv) f.jzi
EQUATION
JLdz2
&H X Y, -573-(01'02,03,04,05 + (Zi + Z2)6(y - l,)*(u)) 3= 1
S{y - ?7)Vu(5(u), S(y - T))-— , 6{y - 77) dui duj d2K (a + zd(y - 77), b , c) dbdc
=lh>L{ d2K
Yt m
% - v), Hy - n)
<*(y - v)»<*(y - v)
d2K (a + z6(y - r?), b , c) U6(y - rj), S(y - 77) dadb d2K (a + zS(y da2
-r)),b,c)
^(y-^.^y-r,)] J m
ipiId^Ldzh^
+ 12m27n x
a 7 b c {^i2^:( da2dby +^(y- ?)' ' )
+
Hy -V)exp(27ri£•y),
exp(-27rif • y ) , 8(y- 77)
^3-{a+zS{y-r]),b,c) 5(y-77)exp(27ri^-y), exp(-27ri^-y),
1 f
dc
f°
f°
Yt-
'
—5(y-r))
7.1. EXPRESSIONS
:
M"^ (o
135
OF B2 AND B3
+ (zi + z2)8{y -
ri),b,c)
m
m
m
Yt <5(y - v), *(y - » ? ) , - = - * ( y - » ? ) m
a3/vr
Yt+ -Qtffo(a + ( z i + ^ ( y - r?),b,c) *(y -»?), Hy - v). —<5(y -»»)
a3K + g a 2 g b ( a + (21 + *2)<$(y - r?),b,c)
^ ( y _ ^ ) , ^ ( y
Yt 93A" + ^ 2 ^ - ( a + ( z i + - ^ ( y - v), b , c) —s(y 2
_,),%-»,)
Yt - 77), -^-*(y -»?). *(y - v)
9s K + gbQtf(a + ( 2 i + z 2 ) % - »/),b,c) <*(y - v), Hy - v), 6(y
-„]}.
In deriving the above equation we have repeatedly used the formulas for antiderivatives (even the higher order antiderivatives) of the iJ-functional and the fact that -m7ri
W A
^{l-2) \?\^\\^ E v^T) W
1=3
S)
+ -^IH
XMM?
dZTw(n;t)exp[A]£;£;|i(s) JA
a i=l
*
OSi
+E E ^ p i + 2 E £ • VJiF^ *
«#>|w}'>ro£ ^
^(w^-wf)
+ ^ » « E (rf) 2 - ^ ) ] = o, (7.18)
which follows immediately from the fact that F(Z) = T/v(fi;i) and the facts that the first three terms in the square bracket is an odd function in w 2 , • • •, w)y and
CHAPTER
136
the fourth term odd function in w^
7. EULER K-FUNCTIONAL
EQUATION
and the vanishing of the integral of the last
term is due to the elementary integral identity (7.10). It should be stressed that in case of distributions other than turbulent Gibbs distributions the value of the above integral would not vanish. And it is the only change we have to make in the computation of B3 for deriving a finer X-functional equation, which holds for a distribution different from the turbulent Gibbs distribution.
7.2
Euler if-Functional Equation
Making use of the H-functional equation (3.16) and the equations (5.1),(5.3),(5.5)(5.11), (7.7),(7.11),(7.15),(7.16) and (7.17), we obtain the following functional equation governing the evolution of the X-functional: T h e o r e m 7.1
If the distribution F(Z,t)
distributions, i.e., F(Z,t)
is of a form of turbulent Gibbs
= T^(fl;t), then the evolution of the K-functional is
governed by the following functional equation, called Euler X-functional equation: dK.
at
(7.19)!
(a,b,c)=tfi+tf2+tf3,
where dK
K ^ vi9 = —( (a,b,c)
dK
E— k=i
fe=l
abkK
dbk'
\t a,b,c
T-^
da
dbj
—\Yi—Jm ^—' oykK L j=i
m
U3^
2dK. , . + - ^ - ( a , b , c ) divb
1
f
+
9y
dzK
^ ( a , b , c ) AY- — m dy dK p (a,b,c) da m-
3
(7.19)2 3
^ ^ d b j . dyk j=ik=i
•EESw
"
dK (a,b,c) da
3
3
j = i fe=i
WkbjYk
2dK (a,b,c) —divb m 3 da
b(y) exp(27ri£ • y ) , exp(-27ri£ • y)
7.2. EULER K-FUNCTIONAL
EQUATION
137
+/*gt(gw-^) f-jz[Bh{a
+
z6(y-T)),b,c)
dK
Hy - v), -^-*(y - v)
d2K dbrf-a{a +
+
Hy - v), Hy - v)
Z S{y T]) h c)
~ - >>
Yt~ Hy - v). -=-*(y - v) m
d2K -^-*(y - v). ~ < * ( y -*?)
^ d 2 tf. •^pf./de^)^(0>b,
C
)
div b(y) exp(2?ri£ • y ) , exp(-2vri£ • y) , (7.19)3
^3 =
5 / , 9c„ + d (??)
sy % +
-g^(a,b,c)
U_dc_ mdy
«7
+ -£-<«, b,c) m 2
/-0
, f a2AT , ?/ d (a+zs{y • y_ i0O H ^- c
d2K d^d-c{a
Yt +
d2K (a + dadb d2K (a + da2
Z S{y
~ -^h'c)
m
z8(y-r)),b,c)
z6(y-v),^,c)
-
9c dy
, x ^ b < c )Hy - n), Hy - v)
x
*(y - v), Hy - v)
us(y - v), Hy - v) -
Yt-
U6{y - v). —*(y - v) m
138
CHAPTER
7. EULER K-FUNCTIONAL
+ 12m2 7T1 J'dljjjil) • J
EQUATION
dzPf.Jdtfc)
{ d3K (a + z5(y - rj), b , c) exp(27ri f • y)<j(y - rj), exp(-27ri £ • y ) , 6(y - vj) \da2db[
Ytz6{y-ri),b,c) exp(27ri£ • y)<J(y - rj) , exp(-27rif • y ) , —<5(y - rj) m
1)
+ -Q^-(a +
4/^1^71^/1^ ^ \ ^ { a +
{zi+z2)8{y-rj),h,c)\^5{y-ri)^8{y-rj),^8{y-rj)
d3K
+
YtHy - rj), 8{y - r{), -^$(y
WfoL^
d3K + 2da2Qb
+
(^
+ Z2 Y
^
Yt~ ^ ' ^ ^ <*(y - v). <*(y - v), ~6(y
{a + (Zl + z2)Z{y - Tf),h,c)
d3K
Yt -
~ *))
Yt -
~s(y - v), ~^6(y - v), Hy - v) ^s(y-v),^s(y-v)J(y-v)
d3K + QbQtf(a + ( z i + Z^)5(y - v)ib- c)
i ±ptjdtfc)< + 27rm
- v)
d2K (a, b , c) dbda
*(y - v). hy - v) > *(y - v)
exp(27ri£ • y) dc (y), exp(-27ri£-y) m dy
7.2. EULER K-FUNCTIONAL
EQUATION
139
d2K (o,b,c) f ^ ^ ( y ) - Y ) e x p ( 2 7 r i f : - y ) , e x p ( - 2 7 r i ^ y ) 27rm da2 ""' ' ' \_\rri* # y " " J
+•
Proof
(7.19)4
In deriving the equations (7.19)i, (7.19)2, (7.19)3 and (7.19)4, we have
used the following facts: If we set -bj(y)tS(u)
Bi
c(y)S(u)_ J = ! , - ' , "5,
-»
m
#4 =
m
into the equation (3.16)2 and (3.16)3, then we come to the following three conclusions: 3
1
dH
,^
rfia
dYi
dej(
E^.M>if +§>
5>(y,u)Yj(y)
(3.16)2
J=I
and 1
Q2
rx
- 2 =! §Pf./^(0^ w (e) " • / •
+ pf
i - /««« 4#<e> 1
+
dH
2W
exp(27riy-0
92ft ^ (y, u), exp(-27riy-0<J(u)
exp(-2?ric; • y)0 4 (y, u ) , exp(27ricl • y ) ^ ( u )
,^ 7 - — — (y,u) 0 ) E dy • du
+*>[*•£]•
<3I6
»'
where wx and VJI are the quantities denned in the equations (3.16)2 and (3.16) 3 . (2). The sum of the following three expressions dH
5>(y,u)Y}(y) oe5 (©) L i = i
1 dH 8U d04 2m dH fl (y,u)u-Y(y) 27ri90 5 v(©)y [ d y d u j ' m 90 5(©) 6
140
CHAPTER
which are terms in the expressions
f>
•
7. EULER K-FUNCTIONAL respectively, vanishes:
WI,W2,VJ3
3
-i
1
£*;(y,u)Y;-(y)
EQUATION
dH,^ dU_ 804 dy du
J=I
2m dH (©) 0 5 ( y , u ) u - Y ( y ) m 90 5
0.
(3). We come to the equation exp(-27ri£ • y)0 4 (y, u ) , exp(27ri£ • y ) ^ ( u )
del
+
;W* ?tt) 'W (e) uexp(27ri£ • y)0 (y, u ) , exp(-27ri£ • y)<J(u) 5
1
/"
frK
b(y) exp(27ri£ • y ) , exp(-27ri£ • y)
of which the two terms on the left hand side are terms in the expressions tJ72 and W3 respectively and the term on the right hand side is a term in the expression 02-
7.3
Reformulation
If we use the convention made at the end of the section III ( see (3.40),(3.42), (3.43) and (3.44) ), the equation (7.19)i can be written in a (formally ) simpler form. Theorem 7 . 1 '
If the distribution F(Z,t)
distributions, i.e., F(Z,t)
is of a form of turbulent Gibbs
= Tjv(fi;i), then the Euler K— functional equation
can be reformulated as follows: dK, , . dK . , . . . dK, , . r , dK, , . . . — (o,b,c) = — ( o , b , c ) [ a i ] + — ( a , b , c ) - [a2] + — ( a , b , c ) [ a 3 J
+
_(a1b,c)[ft]
+
^(a,b>c)[fc]
7.3.
REFORMULATION
r
141
3
3 ,
+ /«»££( J
i=ife=i
d2K (a + J—ioo I dbjdbk
x /
dzl
v
£<,>-^) z8(y-r)),b,c) <*(y - v)»*(y - v)
d2K
*(y - v). -^-*(y - v)
+
d2K dblj-a{a
Yt+
Z 6{y T ) h C)
~ - >> >
d K
t(y - v). ~^s(y
-r-*(y - v). - £ - % -»?) m
+
SK»-/H
a2 A: Sbdc
m
(o + z J ( y - 7 j ) , b , c ) *(y - »7). *(y - v)
d2K
Yt m
d2K (a + dadb d2K -j^(a
+
z8(y-r)),b,c)
*(y - f ) . Hy - v)
us(y - v), Hy - v)
' ~ YtzS(y-rj),b,c) US(y - rj), —(5(y - rj)
+ 12m2 «»-j
{
- v)
*%<*)• [jbj**®
d3K _ £ ( a + , * ( y - , / ) > b , c ) exp(27ri£ • y)S(y - ??),exp(-27ri^ • y), 6(y - rj)
142
CHAPTER
7. EULER K-FUNCTIONAL
+ 7te5-(a + 2
EQUATION
Yty), —<5(y - rj) m
1)
-lId^{v)JLdzilLdz2 :
S { 0 ( a + ( * + z ^ y -1)>b-c) [^ 5 > -*>) - ^ > -»»)> ^ ( y ~ ^ a3 i^-
<*(y - » ? ) . <*(y - v), —<*(y - v) m 9 s A" Yi~ + QiMfa(a + ( 2 i + * 2 ) % - »?),b,c) <5(y - v), Hy - v), —<*(y - v) m d3K
Yt~ 2-^-^-(a+(z1+z2)S(y-ri),b,c) —6(y m
+
d3K + Q-2Q^(a + (Zl + Z 2)^(y - »?), b , C)
Yt~ - rj), - M ( y - 1?), S(y - rj) m
2£j(y_,,),^(y-,,),%_,,) m
m
d3K + ^ ^ 2 ( a + ( z i + z 2)^(y -»/), b , c) <*(y - »?). * ( y - » ? ) . * ( y - » ? )
][
(7.20)
where <*i =
--
rrr
!<•>->
a2
„ tU„ dc t „ 9a 2f/ , Y + —Y •— + - ¥ • — - — V - b , m ay m ay 3m
may V
,
aa ay
as=3(V-b),
1/ 9c may'
.„„,, (7.21)
(7.22)
(7.23)
7.3.
143
REFORMULATION
* = 4 ^ ( y i - y 2 ) ^ ( y i ) •Y(yi) + toki^1 -y2)v •b(yi) +
^2
dm^0r( y l - y 2 ) b ( y l ) ' =
4 ^ ^
y i
-
y 2 )
^
( y i )
(7 24)
-
(? 25)
-
-
On the right hand side of the equation (7.20), we have simplified the forms of a part of the terms on the right hand side of the equation (7.19)i. The other part of the terms (including the last four integrals ) are kept the same forms as in (7.19)i. These four integrals are outcomes of ( infinite dimensional ) pseudo-differential operators operated on the if-functional. Since this article is devoted to the formal calculation of a perturbation method, we will keep the above forms for the four terms for the time being. The definition of infinite dimensional pseudo-differential operators and its application in formulating a finer if-functional equation will be deferred to the Chapter XI. Because the range of the intermolecular potential ( or, of the intermolecular force ) is far much shorter than the side length n of the cube C s and the function b(y) is almost constant inside the cube C s , we have | ^ - ( y i - y 2 )b(yi) « \§^(yi
~ y2)[b(yi) + b(y 2 )]
o r - ( y i - y2)b(yi) - -x— (y 2 - yi)b(y 2 ) ctyi dy 2 therefore, ^T(a,b,c)
-(yi - y 2 ) b ( y i ) 27rim2 dy\
0.
The equations (7.24) and (7.25) will be simplified as follows:
144
Pl
CHAPTER
EQUATION
= 4 ^ ( y i ~ y2)|^yi) • Y(yi) + ^ b ^ y i " y2>V • b ^ (7-26) ft
7.4
7. EULER K-FUNCTIONAL
47rm2
(7.27)
V>(yi-y2)g—(yi)-
Special Cases
Theorem 7.2
In case of null external force field ( i.e., Y = 0,U = 0 ) the
Euler AT-functional equation governing the evolution of the .ST-functional will be reduced to the following simpler form: (7.19);
— ( a , b , c ) = i ? 1 + ^ 2 + ^3, where 1 =
^2
3
/
(7-19)i
divb
96,
divbfo)
>
x
j=ik=i
d2K
o /
^
'
3^(a'b'c)
3
r
+
=
da dy
,c
~d~b^
% dz
au
a
Z(
b
- » ? ) . *(y - v)
c
M. ( + *(y ~ *?)' ' )
-6^Pf-/de^)^(°'b'C)
divb(y) exp(27ri£ • y ) , exp(-27ri£ • y) (7-19)3
5 f dc f $3 = 3 / dri—(ri) • /
d2
d K 0J^(a
+
~ ^
Y
\~ ^'b' ^ ^Y ~ ^ ' ^
y
~ ^
145
7.4. SPECIAL CASES
dT]
+ 12m2 7T1 l
%{r,)f-- teW.fdtfc)
exp(27ri£ • y)<5(y - 77), exp(-27ri£ • y ) , 5(y - rj)
x
d^db{a+zs{y-r')^c)
d {v) dzi dz2
-lI ^ L lL 3
d3K Hy - v), <*(y - v), <*(y - v)
j=i
1
i
f
+
i
f)2K
exp(27ri £ • y) dc (y), exp(-27rif • y) m dy (7.19)i
27rm dhda
It is easy to see that the last terms in the expressions of $2 (7.19)3 and $3 (7.19)4 can be expressed as 2
T.~ d K ^pf.|de&o|£(«,b, 2 C) 67rm
d2K da2
67rm2
divb(y)exp(27ri£ • y ) , exp(-27ri£ • y)
T^(yi - y 2 ) d i v b ( y i )
(7.28)
and 1
/
\
d K
82K dhda
exp(27ri £ • y) dc (y),exp(-27ri£-y) m dy
i^2^(yi-y2)^(yi)
(7.29)
CHAPTER
146
7. EULER K-FUNCTIONAL
EQUATION
respectively. Here we have made use of the convention made at the end of the Chapter III. In case of incompressible flows, i.e., NB being independent of s, setting a = 0, c = 0 and div b = 0 into the equation (7.20) will yield the following form of the JiT-functional equation:
dK (a,b,c) dt
j*n±±^)f_J>^
+ *(y-^c)*(y-»?).<5(y-'7) • (7.30)
The right hand side of (7.30) can be transformed as follows:
httgf j
0
ioo
IK
j=ik=i
dz
d2K ,,, (a + zS(y - 77), b , c) <*(y - v), Hy - v) objdbk
j
3
x
JA
A
27TK5i m
JVs
N,
i=i
P =i
dZTN(£l;t)exp[A\ •"*>•
'sr^\r^\r^ Ns bj(s + ek) - bj(s) J2i=i vij Z^Z^Z^ Ks K N s j = l fe=l
r
N.
1
S J = I fc=iuyk
*
27rmiy" /
x
3 „,
3
J2P=i vPk Ns
s
3
s
rw,((s)),.,( w (s))
6 s
K
3
s_ efc
,.,({s~"ek'uj ), L ,( uj
s_efc
)
E / dzrw(n; t) exptA] E E E i( ) - V — S T
(7.31) where efe = (^ifc , <52fe, 63k)•
7.4. SPECIAL CASES
147
Now we obtain the following expression for the right hand side of the equation (7.30) 3
2-KQX J
dZTN(Q; t) exp[A] J b(y) • £
fe^(y)dy,
(7.32)
where w»
1 V
^3
(!)
«
(S)
,
>.
It is easy to see that /x represents the velocity of the fluid particle composed of the molecules inside the cube Ca. It should be stressed that /z is a random vector representing the velocity of the fluid particle in a turbulent flow. In the derivation of the expression (7.32) we have used the condition of incompressibility: p = K~3mNa
K~3mN8-ek.
—
Thus we obtain the following Theorem 7.3 (Hopf [41])
In case of null external force field the Euler K-
functional equation governing the evolution of the A'-functional for an incompressible flow is simplified as follows:
~(a,b,c)
=2ni J dZTN(^t)eMA]
J b(y) •J2^^-(y)dy,
(7.33)
where b(y) satisfies the condition divb = 0. Because we are using the concepts and the notations of the differential calculus in infinite dimensional space and Hopf ([41]) used Volterra's functional calculus, the forms of the equations in the present paper are different from those in Hopf's. But it is easy to see that (7.33) coincides with the Hopf functional equation for turbulent inviscid incompressible flows: 9$ dt
/ - dWa l 8\ dxa
(see, Hopf [41] or Monin and Yaglom [60]). This is the very reason why we call the equation (7.19)i with (7.19) 2 , (7.19)3, (7.19)4 or its equivalent form (7.20)
CHAPTER
148
7. EULER K-FUNCTIONAL
with (7.21),(7.22),(7.23),(7.26) and (7.27) (7.19)2,(7.19)3, (7.19)4
for t h e c a s e o f n u l 1
EQUATION
(and its special form (7.19)i with external force) Euler /f-functional
equation.
7.5
Case of Deterministic Flows
If the mass density field, the momentum density field and the field of total energy per unit mass are deterministic (i.e., the so called laminar flow in classical fluid dynamics), the differential form of the turbulent Gibbs distribution, considered as a function defined on the space ~VN x R 3iV , is of the form
F(Z,t) = C g , , ^ . ) ^ I ] <*(4S)-P(s)) A m
'm
S
S(u}f)-p(s)t,j(S))6(J;)-e(s)p(s))
J —1
(7.34) where p(s),p(s)p,j(s)
and p(s)e(s) are five deterministic quantities, which repre-
sent the mass density, the components of the momentum density and the energy density of the fluid particle represented by the cube C s , respectively. And the normalization constant (m) 5 / 2 (4 s ) ) 3 / 2 r((3AT 8 - 3)/2)7Vs!
n
M Y |.2«( 9JV -+ 9 )/ 2 7r( 3 ".- 3 )/ 2 G(s,£(s)p(s) -
lp(s)\fi(s)\^,^s))
.
(7.35)
In the language of differential forms (see, e.g., [55]), the differential form
c5
«>r ,0 n M s ) - POO) n s(^s) - P(S)N(S))SUS) - <*)/*.*))] dz J
m
j=i
m
is the pull-back of the differential form 3
K>y „(., m
^)~p{s))X[8{^-p{s)N{s))5{4)-e{s)p{s)) dQ,
II ' m
S
**
J= l
under the transformation of variables L —> 12 — (•• - , i Z v ' , • • • ) — ( . " ' i ^ O
>W1
>w2
>W3
>W4
1
)•
7.5. CASE OF DETERMINISTIC
149
FLOWS
Under the assumption (7.34), the X-functional will be of the following form:
K(a,b,c) = cJdZl[
S^eapi.
Q ( m £ |v, (8) | 2 +
exp ^ - 27ri /
£
( f[ s(m£
v^ - ^^(s)*3) J
V(|xi s) - xj8>|)) -
a(y)m ] P J(x/ - y) + b(y)m • ^
p(s))e(s)K^
vz<S(x( - y)
dy}
(7.36)
If the functions a(y), b(y) and c(y) are so slowly varying that they can be considered constants inside each cube C s , the If-functional will be of the following form: K(a, b , c; t) — exp< - 2 7 r i ^
al(s)ps + b ( s ) • /J,sps + C(s)£s/9S
4
(7.37)
Inserting a = S(y — x ) , b = 0, c = 0 and (7.37) into the Euler If-functional equation (7.19)'x, we have -27riexp(-27rip(x,0)^7 = -27riexp(-27rip(x,t)) / up—^-
= 2mexp(-2irip(x,t))
'-dy
— • (p/*)(x).
Hence we obtain the equation of continuity. T h e o r e m 7.4
Under the assumption (7.34), we have the equation of conti-
nuity: |
+
| - . ( p M ) ( x ) = 0.
(7.38)
150
CHAPTER
7. EULER K-FUNCTIONAL
EQUATION
It should be stressed that in the equation (7.37) s denotes a triple integer-index (i.e., a discrete valued vector ), but in the equation (7.38 ) x represents a continuously varying vector. In the sequel, we frequently make the exchange between the discrete variables and the corresponding continuous variables arbitrarily without explicitly mentioning the condition for the validity of the exchange. An more rigorous formulation of the problem will be postponed to section VIII. Inserting a = 0, b = (<5(y - x),0,0), c = 0 and (7.37) into the Euler Kfunctional equation (7.19)j, we have
— (a,b,c) = -27riexp{-27ri/3(x)^ 1 (x)}
IdK (a, b , c) div b(y) = ^ S~dc
',
(7.39)
exp{-27rip(x)Mx)}^(x),
(7-40)
^ P f . / ^ ^ ) ^ r ( a , b , c ) exp(27ri£ • y ) d i v b ( y ) , exp(-27ri£ • y)
67rm2
^ e x p ( - 2 ^ ( 8 ) / ! ! (s))
^ AT
^
VLU^W-^;]^-
'
m/
•\S'1s'AT
T. 1<1
nfdx^
d X ( f f )
^
4r-*r)
/
(3JV„-5)/2>
7.5. CASE OF DETERMINISTIC
3 (p(
151
FLOWS
>2
xi
w
(3AT f f -5)/2-i - 1
*n(^w* - ty - £ *( "-" i
= - ^ e x p f - 27ri/9(s)/x(s)j
A x { * ( s - ei, e(s - ei)p{s - e{) - -p(s - ei)|ji(s - ei) 2' W..(•-«! 0 )
- t f f s + ei,e(s + ei)p(s + ei) - | p ( s + ei)|/i(s + e i ) | 2 , 4 s + e i ) J I
47ri
d\t
= —^-exp{-2mp{x)m(x.)} — (x, w 0 (x), a;i(x), w 2 (x), w 3 (x), w 4 (x)), (7.41) where, for brevity, we have made the following convention for notations: /Ws>....= J
/
N
dx{*} • • • dxff
• • •.
J(C.) °
The convention will always apply in the sequel. According to the Definition 6.1, the Gibbs mean of the total intermolecular potential energy density inside the cube Cs is W(_S,
/«<•>
£
UJQ , W 1
*(x«-x}->)(n-
l
2K J
,W2
/Wn-
,W3
,U!4
)
S)
(S)
(3AT,-5)/2
x; ^(4 -*, ) l
£ l
v>(xLs)-x!*>)
(3AT s -5)/2-i - 1
(7.42)
152
CHAPTER
7. EULER K-FUNCTIONAL
EQUATION
where
s)
a-i^) (s)
( 2w}"'
mNa
/>
(7.43)
/
Elementary calculation yields that 3 J
3 x
i=ife=i
£(,,--^>,
• )
a 2 is
*(y - » ? ) . *(y - v)
3
3
s j=lfe=l^a2/fc
A •'A*
£J
*
N.
(=1
(=1
iVs
/
fi
A
. TV.
/
I
12 \
= 2 7 r i e x p { - 2 7 r i p / i 1 } ^ — ( / i 1 | t f c / 9 ) - ^ e x p { - 2 7 r i W l } — f ^ | L j . (7.44) Inserting a = c = 0,b = (0,5(y - x),0) and a = c = 0,b = (0,0,6(y - x)) into the equation (7.28) respectively, we shall obtain the equations similar to the equations (7.39)-(7.44), but with X2 and X3 instead of Xi respectively. Combining all these equations, we obtain the Euler equations in classical fluid dynamics. T h e o r e m 7.5
dt
Under the assumption (7.34), we have the Euler equations:
k (PM) + Yl dx »Z-(W P) + * Srad, * - * ! - * k
T h e o r e m 7.5'
= 0.
(7.45)
fe=i
Under the assumption (7.34), the Euler equations can be
reformulated as follows:
dt
8
+ ^ fc=i
-5—iWkp) dxk
+ gradp = 0,
(7.45')
where V
3 V
2
p
(7.46)
7.5. CASE OF DETERMINISTIC
FLOWS
153
The equation (7.45)' is of the very form of Euler equations in the classical fluid dynamics. By the way, the equation (7.46) is an equation of state for general fluids. Of course, a more definite form of the equation of state will be obtained only after getting an explicit form of the Gibbs mean in a cube in terms of the mass density in the same cube and the form of the intermolecular potential energy tp. Getting a precise form of the total intermolecular potential energy is a difficult mathematical problem. Although we called the equation (7.19)! ( or its equivalent form (7.20) ) the Euler /^-functional equation, the equation (7.45), which is a special form of the functional equation (7.19)i in case of deterministic flows, is quite different from the classical Euler equations in the following point. There are six unknown quantities in the classical Euler equations: mass density, three components of velocity, temperature ( heat energy ) and pressure. Because there are only five equations in the Euler system of equations, we have to use the equation of state to reduce the number of unknown quantities in the equations to five. The equation of state used in classical fluid dynamics is an equation based on experiments. The Euler if-functional equation governs the evolution of a functional of five random fields. We do not need the equation of state to eliminate any quantities. The equation of state is implied in the Euler /sT-functional equation. In Euler if-functional equation the intermolecular potential tp plays explicitly a role. In classical Euler equations it is implied in the equation of state. Inserting a = 0, b = 0 and c = <5>(y — x) into the Euler if-functional equation (7.19)i and taking (7.18), (7.34) and (7.37) into account, we have
— (a,b,c) = _ 2 7 r i e x p { - 2 7 r i £ p } - ^ ,
(7.47)
154
CHAPTER
lnf
[ ,,7,^[
i d2K . , 27rm dbda
7. EULER K-FUNCTIONAL
.
[exp(27ri£-y) dc, m dy
= 2 / d ? ^ ( 0 s - ^ e x p { - 2 7 r i £ p } / p(y2)ti(y2)
.
EQUATION
, „ .,
exp(27rig-y 2 ) dJ (y 2 - x)dy 2 m dy2
/ P(yi)exp(-27ri^-y 1 )dy 1 + 27Tiexp{-27rU/o} / p(y)fi(y) — — ( V
= 27riexp{-27ri£p} — •(/J(X)#(X,
,
dy\
w 0 (x), u>i(x), w 2 (x), w 3 (x), u>4(x)) J, (7.48)
and Q
r
o21^-
/»0
/
+ Ultra2
i/^'/W^'
f d3# x
r
S ^ 2 g b ( a + zS(y ~ v),b,c)
.
exp(27ri^ • y)<5(y - rj), exp(-27ri£ • y ) , 5(y - r?)
Q2 !£•
+ 2 7 r m i ^ ^ ( a + z<5(y - 77), b , c) * ( y - » ? ) . * ( y - »7)
1 f
x
dc
f°
f°
J2 gbdtf(a + (Zl+Z2^y ~ 7?)>b'c) p(y - f) - *(y - *?) - % - *?)
-mm Y ] /
dZF(Z,t)exp[j4]
7.5. CASE OF DETERMINISTIC
*£>
N.
Dw
A')(. \3VNSfrl
27riexp{—2m ep}
155
FLOWS
dx
W 12
3VK
f5 P a,heat c
Ml n
|w[s)|2)'
+ £,s,kin
(7.49)
where
(=Ei-n>)
'a,heat
(7.50)
and m
c
i
I
(7.51)
£»,kin = y l W l l
denote the heat energy and the kinetic energy of the fluid particle represented by the cube Cs respectively. It would be recalled that the distribution F(Z, t) in the second last line is of the form in the equation (7.34). We introduce the heat function per unit mass ( enthalpy ) as ^
5 gg.fteot
P
3
(7.52)
p
Combining the equations (7.28), (7.47)-(7.52), we obtain the energy equation as follows. Theorem 7.6
Under the assumption (7.34), we have the following energy
equation: d_ (ep) + div dt
MgM 2 + h)
0.
(7.53)
The equations (7.38),(7.45') and (7.53) are the equations governing the motion of ideal fluid in classical fluid dynamics. The Euler lf-functional equation governs the motion of ideal fluid in our theory. Of course we could have derived the equations (7.38),(7.45') and (7.53) from the balanced equations formulated at the end of the section III, which were derived from the i?-functional equation, by taking account of the fact that the distribution F(Z, t) is of the form of the special turbulent Gibbs distribution (7.34). In order to illustrate the significance
156
CHAPTER
7. EULER K-FUNCTIONAL
EQUATION
of the Euler /^-functional equation, we have derived them directly from the Euler ^-functional equation. The system of Euler equations in classical fluid dynamics is a special case of the Euler JsT-functional equation corresponding to the special turbulent Gibbs distribution (7.34). In describing fluid motions there are no a priori reasons for favoring the special turbulent Gibbs distribution (7.34) among the general turbulent Gibbs distributions (6.11). In Massignon's theoretical frame, the Euler iiT-functional equation, not the system of Euler equations, is the basic equation governing the evolution of fluid flows. In other words, the general fluid flows themselves are turbulent (i.e., not deterministic). A laminar flow (usually appeared in the laboratory) is a very special form of general flows, of which the correlations and variances vanish. Precisely speaking, they are negligibly small. In case of the Reynolds number exceeding the critical Reynolds number, the correlations and the variances as well as the (deterministic) disturbances will increase to infinity exponentially, at least according to the linearized equations governing their evolutions and the turbulence phenomena, i.e., the randomness of the flow, prevails. Therefore, in Massignon's theoretical framework, a turbulent flow is not necessary to be a deterministic chaos. We have to change our attitude in attempting to find out the ways of "the transition from laminar to turbulent". Turbulence occurs, when the randomness of the flow (which, in general, exists for every flow and is specified by the correlations and variances) grows to such an extent that it cannot be neglected in the problem under consideration.
Chapter 8
Functionals and Distributions 8.1
if-Functionals and Turbulent Gibbs Distributions
In the remainder of the paper, for brevity, we consider only the case of null external force field. We assume that the distribution F(Z, t) is a turbulent Gibbs distribution:
F(Z,t)
= TN(Q;t)
\
= TN(- • . - u ; ^ , ^ , ^ , ^ , ^ ; • • •)
'—1
'—1
kjtll
'
(6.20)' In order to emphasize the dependence of the if-functional on the turbulent Gibbs distribution T/v in the equation (6.20)', we shall use KN instead of K in the remainder of the paper. It is of the following form ( see (4.1)):
KN(a,b,c;t) = fdZTN(il;t)expl J
-2ni K
f ^
a(y)m£6{x { - y) + b(y)m • ^ v ^ x , - y) ^
1=1
157
1=1
158
CHAPTER 8. FUNCTIONALS
a
+ c(y) f ? E N * ( x , - y ) \ z 1=1
Yl
I
dZT
AND
DISTRIBUTIONS
+ 5f;5]^(|x I -x fc |)«J(x l -y) > )l4y} z
i=i M I
/J
J
t) exp i - 2 7 r i ^ a(s)mATs + b(s)mAr s /i(s)+c(s)miV s £(s)
N(^;
1
-
}
T
3
exp ( - 2nnH£ a(s)W<s) + V ^ ( s ) ^ s ) + c(s)<4 s) j } , •> s L JJ j=1
= J2 I dZTN(Q;t) A J**
(8.1) where Na denotes the number of particles (molecules) in the cube Cs, (therefore, mJVs the total mass of the particles in the cube C s ), i \
* f \^
(»A
1/ (»)
(s)
(s)x
mArs
the average velocity of the molecules in the cube Ca, ( therefore, mNsn(s)
the
total momentum of the particles in the cube C8 ), and Nm
-, N.
x
3=1
(s)
x
)
3=1 kjtj
)
• ? - x ? ) ) = p^ .
(8.3)
the total energy (=kinetic energy 4- potential energy) of the system of the particles per unit mass in the cube C s , (therefore, mNae(s) the total energy of the system of the particles in the cube C s ). It is worth making a remark: for the special turbulent Gibbs distribution (7.34), Na: /i(s) and e(s) are independent of Z and, therefore, deterministic, but Ns, /i(s) and e(s) defined here depend on Z and, therefore, are random. The formulas (4.9), (4.11) and (4.13) can be reformulated as follows. — {a,b,c;t)[S(x-y)}
-27riy] /
= —(a,b,c;t)
dZTN(n;t)exp[A)Jo\
'
(4.9)'
8.1. K-FUNCTIONALS
AND TURBULENT
oK. .,~. — (a,b,c;t)[S(x-y)]
= -27ri
..
uK. =
GIBBS
DISTRIBUTIONS
. —(a,b,c;t)
159
K-
[dZTN(n\t)exp[A]u?\
(4.11)'
for i = 1,2,3, and — (a,b,c;t)[<S(x-y)] = — ( a , b , c ; t ) «t,.
-27ri I
dZTNifytfexplA]^.
(4.13)'
In the equations (4.13)' and (4.13)' we have, for brevity, assumed that Y = 0. The velocities of the molecules inside the cube is represented by a 3iVs-dimensional vector (vj , • • •, v ^ ) . In order to single out the mean velocity of the molecules inside the cube C s , which will play a prominent role in the further study, it is convenient to make use of the orthogonal transformation in the 37Vs-dimensional space:
(s)
w N.l w («) »
Av(s),
(8.4)
(s)
w w <•) (S)
\
N,3
\
/
N.3 I
where •As —
Bs 0 0
0 Bs 0
0 0 B
(8.5)
160
CHAPTER 8. FUNCTIONALS
AND
DISTRIBUTIONS
and 1
1
1
1
1
1
-l
0
0
0
-2
0
0
-3 \/l2
0
1
1
V(Af.-2)(W.-l)
V/(JV.-2)(AT,-1)
0
V2 1
l
V6
V6
1 %/l2
B.
1 Vl2
1
Vl2
1 y/(N.-l)N.
\
-(Af.-l) y/(Nm-l)Nm
1 y/(Nm-l)N.
)
(8.6)
It is easy to see that the mean velocity of the molecules inside the cube Cs is proportional to w | s ' = (w^,Wi2',
Wi3'), i.e., the mean velocity of the molecules
inside the cube C s is represented by the 3NS- dimensional vector (w[ , 0, • • •, 0). we have KN{a,b,c;
{n
exp
t) « V
f TT ( f
dX&
dw<s) f
f
dw<s) • • •
dw#)
,(s) (•) s; + 6 (s)a;^ ,(s) ; + c(s) s) » ; + 6 2 (s)w 2«d7ri a(s)u>£> + &i(s)w}' 2 3
vi )'
XJJV(.---,WQ
>W1
'W2
>W3
>W4
I
-
(8.7)
" ' ) ^
where (s) _ mATs
, (.)
,<•> (.), _ mVTCwr
(.) w
(.^1 >^2 > 3 J —
, _ m | w [ s ^ + m(rW) 2 + £ & wi
(6.20)1
2K 3
Z3
(6.20)2
'
s
E ^ ^» - x v(^ h)
(6.20)&
8.1. K-FUNCTIONALS
AND TURBULENT
GIBBS
DISTRIBUTIONS
161
In order to get a form, more amenable to the further study, of the right hand side of the equation (8.7), a system of poly-spherical coordinates will be introduced in the subspace of the random velocities of the molecules, i.e., the relative velocities of the molecules with respect to the mass center of the system of molecules in the cube Ca. Obviously, the subspace is generated by the vectors w\ ',i = 2, • • •, Ns. Firstly, we introduce the length of the random velocities in the cube C s , which is a 3(Na — 1)- dimensional vector, as r N-
,-(*)
L
(s),2
£|w< t=2
A poly-spherical coordinates in the subspace spanned by the random velocities can be written as follows: (
(s)
(s) •
(s)
(s)
(B)
•
(s)
•
(s)
•
(s)
• n(s)
.
As)
o(s)
•
/i(s)
.
a(s)
u>2i = r w smip2 ' sini/?! s m ^ _ 2 1 • • • sin0 21 ; s i n ^ , w^i = r w smip2
(s)
W
N.l
=
r(
(s)
(s\
ys>
w22 = r (s)
u>32 = r
(s) N.2 (s)
W
' Slnf2 •
=
•
(s)
•
Sm<
Pl
(s)
/i(s)
COS^_21 , (s)
(s)
sin (f2
• /i(s)
• n(s)
(s)
(a) • (s) ' Sln(f2 (a) (»)
w23' = rw cos
N).3=rl-a)cOS'P2
• /i(s)
.
sin
• n(s)
„(s)
^12 > n(s) (s)
cos
r(
w
/i(s)
sint/)/_ 2 x • • • sin0 2 i cost^i ,
sin
la) •
w
(s)
•
(a)
s\nip\
(•)
(s)
COS(
Pl
n(a) N.-2,2
cose
(8.8)
>
S i n e<
' • ' S i n 9Zi
Sin 6
• • • S i n e<23
N.-2,3
NI-2,3
Sin 0
U
C0S
'
^
>
(») cos#JN.-2,3-
Having introduced the poly-spherical coordinates, we can express the Kfunctional KN as follows:
CHAPTER 8. FUNCTIONALS
162 Theorem 8.1
AND
DISTRIBUTIONS
The K-functional KN(O, b , c;t) corresponding to the turbu-
lent Gibbs distribution T„( -,Ss) »,,("> ,, ( s ) ,,<") ,, ( s ) w •LNK---,U0 i ' w 2 ' w 3 1^4
-t\
r--,t)
can be expressed as follows: KN(a,b,c;t)=
^
^ , exp f - 27ri^a(s)mAT s j
)(n/K,-i->){n
27TK J V -- 1 )
Lr(|(iv.-i))
x /"°° dr<8> (r< s >) w -- 4 J J e x p [ - 2*37ri(a(s)a><s) + ] T ^ s j w j ' * + Jo L V s J=1
.,.(«) , ,(•) , .(») , .(») , , ( s ) .
vT„/
.,\
c(s)jA /.
(8.9)
Proof The equation (8.9) is a consequence of a change of variables. In deriving the above expression for the AT-functional K^, we have used the simple facts that the number of the hypercubes n»=i £»< with J2i=1 Sa Si = Ns for all s is exactly jV!/ f ] s Na\ and the area of the (3iVs — 4)-dimensional unit sphere is 27r f
(iv.-i)/r(|(iVs _
1})
Introducing the following quantities, which are the mass density, momentum density and energy density of the system of particles in the cube C s , respectively: mATs
(s)
W,0
(8)
w.
~
m y/Kwfl,
(8.10)
«3
j = 1,2,3,
(8.11)
8.1. K-FUNCTIONALS
AND TURBULENT
<4S) = ^ ( m | w < s > | *
+
GIBBS
DISTRIBUTIONS
m(r<°>)2 + E E ^ ( x £ 8 > - * , W ) ) .
163
(8.12)
have r rW
JV
«
(s)|2
j
I T
Ei< i=2
-v/m
2w (»)
>3
(mlw^l^ + ^ E ^ r - ^ ) )
2 K3,.,(») V
^
/ ,(sh
EtiKs s ; ) 22
i
,( )
1=1
ky&l
'
N.
EEw-h
(8.13)
1 = 1 kytl
Inserting the expression (8.13) for r^
in the right hand side of the equation
(8.9) for the expression if-functional KN and taking account of (6.11), we obtain the following Theorem 8.1' Under the condition (6.11), the expression of the X-functional KN in terms of the turbulent Gibbs distribution TV is of the form: V
3N
3V
9JV „,!<-_,
.
= 2 ^ 7 T — - i ^3NK ^ r ( ^ a )
KN(a(s),b(s),c(s);t)
m 2
-
E X f
In"
3
)
0
,(•) ) j r f « " n ° + m \ r / ' 3 K 3 u ; ! ' - 3 m \
dLJ^dUi^dUi^dUi^
TN{Q;t)
J J j e x p ( - 27r«3i a ( s ) 4 s ) + E & i ( s V J s ) + c ( s ) ^ s ) ^ G ( s , 5 s , i n t , 4 s ) ) | 11 (8.14)
By virtue of the inversion formula for Fourier transforms, we have the following
164
CHAPTER 8. FUNCTIONALS
AND
DISTRIBUTIONS
Theorem 8.2 Under the condition (6.11), we have the following expression of the turbulent Gibbs distribution T/v in terms of its corresponding AT-functional
3N
TN(Sl;t)
V
EA'\
31*
3V
12V
9N
n : t.KN(a(s),
b(s), c(s); t) exp
, V
,i^,
n
(4 s ) )^(^^)r(^^) G{S,€atint,U)0
)
/ 2 m da(s) f db( s)dc(s)
UTTKH
^
a ( s ) 4 s ) + J2 W^f
+^
^
) }• (8.15)
8.2
Turbulent Gibbs Measures
A s s u m p t i o n B For any Euler (or, inviscid) fluid flow, there is a net (or, filter) of turbulent Gibbs distributions {T/v} such that the corresponding net (or, filter) of .ftT-functionals {K^}
converges to a if-functional K:
K(a,b,c)
= lim prJV_+00.fi:Ar(ajv,bAr,cAr),
where fliv(s) = -o / a(x)dx, K Jc.
bN(s)
= — /
b(x)dx,
CJV(S) = - ^ /
c(x)dx,
and lim pr^^oo denotes the projective limit ([50]).
8.2. TURBULENT
GIBBS
165
MEASURES
The mapping (a,b,c) —>
(aN,bN,cN)
in the Assumption B is a projection from the infinite dimensional linear space composing of the vectors (a(x), b(x), c(x)) to the finite linear space composing of the vectors (ajv(s),bAr(s),CAr(s)). -phe former space is called the projective limit of the (net of the) latter spaces [50] and the ivT-functional K the projective limit of the /f-functionals
K^.
The A s s u m p t i o n B may be restated as the Definition of Euler fluid flows. Hydrodynamics does not concern itself with the general Hamilton systems of TV-particles. The hydrodynamics of inviscid flows only concerns itself with the Hamilton systems of TV-particles satisfying the Assumption B. A remark about the concept of Euler fluid flows is on order: Remark In classical fluid dynamics, the sizes of fluid particles have never been specified definitely. Usually they were crudely described as "macroscopically infinitesimal" and "microscopically infinitely large". Hence the sizes of the cubes or rectangular parallelepiped, which represent the fluid particles in fluid dynamics, in the Assumption B are not specified definitely either. In the sequel we assumed that, for each turbulent Gibbs distribution Tjv in the Assumption B, the corresponding system of cubes (or rectangular parallelepiped) can be made finer not less than one tenth or coarser not more than ten multiple, and the turbulent Gibbs distribution corresponding to the new system of cubes (or rectangular parallelepiped) T'N will be very close to T/v. For brevity, in the remainder of the present book we frequently use a(s), b(s) and c(s) instead of aiv(s),bw(s) and
CJV(S)
respectively.
Now we are going to introduce an important concept — turbulent Gibbs measure, which plays the fundamental role in the classical statistical theory of turbulence.
166
CHAPTER 8. FUNCTIONALS Definition 8.1
AND
DISTRIBUTIONS
The functional measure on the infinite dimensional linear
space consisting of the vector- valued functions (uo{&), wi(cr), w2(
OT(fi(
3N
3V
9 I(2$n™-%*K T~ -^r(*±*)TN(n-,t)
x lim pr N — ao J
3N_
m 2
G(s,€atint,u:0 (sh)
n [(^))lr(^^)r(^z-)
n[dwo ( s ) dwi ( s ) dw 2 ( s ) dw3 ( 8 ) ^4 ( s ) ] >• (8.16)
will be called the turbulent Gibbs measure corresponding to the fluid flow under consideration, where the limiting process "lim pr^^oo" in the equation denotes a projective limit as N —• 00 and it is assumed that the net of turbulent Gibbs distributions {T/v} is so chosen that the above projective limit exists.
The present book will be devoted to the formal computations in an asymptotic analysis of the Liouville equations. The study of the subtleties of the theory of infinite-dimensional integrals is outside of the scope of the present book. Therefore we will not be involved in the detailed study of the concept of projective limits of measures. Even the specification of the vector-valued functions (UJO(CF)
, wi(cr), ^2(0"),
W3(CT) ,
^ ( a ) ) , the elements of the function space on which
the functional measure Tjv is defined, has not been made. The basic assumption in the statistical theory of turbulence is the existence of the turbulent Gibbs measure. What we have done here is to formulate the turbulent Gibbs measure in the terminology of the non-equilibrium statistical mechanics without the assumption of molecular chaos. In other words, the introduction of the second stochastic structure, as was done in the classical statistical
8.2. TURBULENT GIBBS MEASURES
167
theory of turbulence, is really redundant in the theoretical frame of the present paper. Now we can reformulate the equation (8.14) as follows: T h e o r e m 8.3 The Jf-functional K(a,b, c;t) corresponding to the turbulent Gibbs measure DT(w 0 (cr), wi(
= /
T>T{u}0{a),u)i{a),u)2{(r),u}3{a),w4(
xexp ( -27ri / dcr o(cr)wo(<7) + ^6 J (cr)a;j(or)+c(a)w4(
),
(8.14)"
where fT>T(uJo((T),uji(a),u!2(&),W3(cr),uj4(cr); t)(- • •) denotes the functional integral with respect to the turbulent Gibbs measure 2?T (see, e.g., [42], [24], [30] and [77]).
We introduce the following: Definition 8.2 The signed if-measure VK. on the infinite dimensional space consisting of the vector-valued functions (a(a),b(a),c(cr))
is defined as follows:
PK(a(<7),b(a),c(<7);t)
= limpr jv ^ oo K"^"m^A'Af(aAr(s),bAf(s) ) CAr(s);i)n da 7v (s)dbjy (s)dcjv (s) (8.17)
Having defined the signed If-measure 2?K we have the following
168
CHAPTER 8. FUNCTIONALS
T h e o r e m 8.4
AND
DISTRIBUTIONS
Under the condition (6.11) , we have the following inversion
formula for the turbulent Gibbs measure VT in terms of the signed if-measure VK:
VT(U0{(T)
, WI(
= lim p r ^ J
lay
v -w-T
x exp ( 27r«3i ^
Jlidu^dw^du;^^^}
dap/ (s)dbjv (S)
/
,u4(a);t)
{K
N
(aN (s), bN (s), cN (s); t)
(•)
^ s
= Do;o(cr)X»a;i(CT)2?W2(o-)I>W3(a-)I»a;4(cr) / i 2?K(a(a),b(a),c(o-);i)
exp(27ri/
O(CT)W0(CT) + ^6j(«r)u;^(
H.
(8.18)
P r o o f It follows from the equation (8.15) that V
3N__3V
9N
12V
2^7T^-"^«;~_"^-r(^±21) 3N i V
m
2 +
^
n (4 s ¥r(^^)r(^£^) G(s, £s,int,U0 )
%
II
daN(s)dbN(s)dcN(s)
<
TN(n,;t)
KN{aN(s),hN(s),cN(s);t)
x exp (27TK3i £ a^(s)4s) + ^6^(s)u;js)+cw(sVis) H . \ L ,-^j J/ J s
(8.15)
8.3.
ASYMPTOTIC
169
ANALYSIS
It is easy to see that the equation (8.18) is a consequence of the equations (8.15)!, (8.16), (8.17) and the following fact about the Dirac delta function in infinite dimensional function space: s( f u>0{a)do - K J = l i m p r - A r ^ f « 3 ] T ] 4 s ) - K J
lim J>TN-
.y/V/* The equation (8.18) is the inversion formula for the infinite dimensional Fourier transform (8.14)". As far as I know (see, e.g., [42], [24], [30] and [77]), an inversion formula for general infinite dimensional Fourier transforms is unavailable. Because the infinite dimensional measures encountered in the present paper are always the projective limits of finite dimensional measures, the inversion formulas for the Fourier transforms encountered in the present paper are immediate consequences of the inversion formulas for the corresponding finite dimensional Fourier transforms. In the next section they will be used frequently.
8.3
Asymptotic Analysis
As was pointed out by H. Grad ([33]), statistical mechanics is an asymptotic mechanics. Precisely speaking, a general fluid flow is not specified by a single turbulent Gibbs distribution, but by a turbulent Gibbs measure, i.e., a net of turbulent Gibbs distributions for which the projective limit on the right hand side of the equation (8.16) exists. In other words, a general inviscid fluid flow is specified by a turbulent Gibbs measure defined in the equation (8.16) or, equivalently, by a /f-functional defined in (8.14)'. The former was done by O.Reynolds, ( and then by G.I.Taylor and A.N.Kolmogorov ), the latter by E.Hopf in the classical statistical theory of turbulence, respectively. What has been done in the present paper is the exploration of the possibility of describing the classical statistical theory
170
CHAPTER 8. FUNCTIONALS
AND
DISTRIBUTIONS
of turbulence in terms of the non-equilibrium statistical mechanics without the assumption of molecular chaos. Making use of Stirling asymptotic formula for T function (see, [83] or [2])
r(o) = Vw'-se-o
+
1 12a
+
1 288a2
+
°m
(8.19)
|Q| < 1,
(8.20)
and the elementary formula (l+a)*(Q>=expU(a)^(-l)^+1^J, we have the following lemmas about the T function: Lemma 8.1 K+m m
)-H^)
m
K+m /
„,
A
(8.21)
e—™-fl + 0(m) j .
This is an immediate consequence of the equation (8.19). Lemma 8.2
p.,/|A£+m\ Af+n1'1 *3u,(°> + Y
=
— ( — ) V^\K3U>^J
«3^s)
1 -
exp
12« 3 ci s )
+ 0 -2"
(8.22)
Proof. According to the equation (8.19) we have K3W£S) + m
m *348) + f 1
m
K3UJ^S) + m
exp
m
1+
m
+
o(HL
3
UK U>LS)
Making use of the equation (8.20) and Taylor expansion for the exponential function, we have K3W£S) + m
m
8.3.
ASYMPTOTIC ANALYSIS
171
*3"^
- 3 , ,( s ) , m
= — ( — )
x exp
1 /
exp
V
m
3 «£!±a. \ -
m
T^UF)
m
3
/ exp exp I -
l
)
112/s 2 K 3wi" ^
/L
1+
~ 12K 3 W< S)
m
yi(
V«V
m'
exp
4 ( K 3 4 ' ) + m) 2
+
V
/ ^,3,8,00 ' + m\
UJ,
.3„(.)
r+*
m 12K348)
V^TT
+ o(£)]•
Lemma 8.3 /3/t3h^8J-3m\
V 3 («)
-
1
f
2m
-l
2m J
4m
2H
exp
//Q.,3,.,00' 3 ^ > U V
2m
1
;L
__47m_
/rf\
36«3O;W
U V
.
(8.23)
Proof. By virtue of the equations (8.19) and (8.20), it is easy to see that «3,»_
'/3/t34s) - 3 m \ l - 1 . V 2m JJ
x exp
1 / 2m V _ 3 s) 72^V3K 4 -3m;
/3« 3 4 s) -3m\r
m 3
18K 4
S)
- 18m
+
<%}
3«3ul">-4m
'3K3^s)-4m
1 / 2m ^V3K348)
exp
2m
2
y ^ (~1) J+1 (
U
3
m_ \K^
XO.J
172
CHAPTER 8. FUNCTIONALS
GOH^t^E /
2m
»«3-f)-«-
.
2
/3
\
3
s>
x e x p/3K ujk
">
-3m\
3
18K U{S)
+
+0
*6 ) .
^/m2\\
m 5m
m
(—k—J
DISTRIBUTIONS
m 18/t 3 ti s ) - 18m
l + Ol
1
AND
°(5
3K3U^*>-4III
V 2m /L
^F\3«34'V
36«3^8)
V«V.
Corollary 8.1 Under the conditions (6.11) and (6.12), we have VT(u)0(a) , WI(CT) , w 2 (cr), w3(cr), w4(
= Vw( 3/V + l
2~
x lim pr
n
Proof.
31V
5V
. M i l
, 1 .
3N
i „
,„
,
^•7r~~^+2(KH-m)Af+te3r-1rJV(n;t) 3W
2V
3V
I
TM
3V
3
<»)\
25m
d _
2V
Ju)0{a)d
^ g(s.g..int,^r) ]Jdwo ( s W ( s W s W s W s ) ] 0
K )
2m
(8.24)
It is an immediate consequence of the lemmas 8.1—8.3 and the
equation (8.16). The Theorem 8.1' can be reformulated as follows, Theorem 8.1" Under the conditions (6.11) and (6.12), we have the following expression of the K-functional KN in terms of its corresponding turbulent Gibbs distribution T^:
8.3.
ASYMPTOTIC
173
ANALYSIS
3JV
KN(a(s),b(s),c{s);t)
n
E
1V,1
3jV
5V I 1
2 2 ^T+^TT 2 3^3-+2(K + m ) w + 2 e ~ 3JV
18K3^S) 3
18K (4
S)
2V
^Iv
"9V"
x
3V i 1
25m
iMfi;*)
+1
)^^
x n { exP ( - 27r«3i fl(s)4s) + E 6 i( 8 ) w ? } + C(S)W4S) W , £ a , i n U 4 S ) ) } }• (8.25) Proof It is an immediate consequence of the lemmas 8.1—8.3 and the equation (8.14). The Theorem 8.2 can be reformulated as follows, Theorem 8.2' Under the conditions (6.11) and (6.12) and assuming that m is sufficiently small, we have the following expression of the turbulent Gibbs distribution T/v in terms of its corresponding /^-functional > w o , ux
J
AT(.
«<5K3 V U{"\ £"•<>'
n
K
2
3N + 1
2V
2
^(N
rKs;)
, UJ2
i ' it J K, ,.,,,1 1K-SV
+ l) w + 27r
18K3^S) +
18G(s,59,int,4s))
xS.K N (a(s),b(s),c(s);t)exp
(l^h
, w3 ,u>4 , •••
£
2
K^: ,t)
2KJ
,i
2^3-+5m
N_SV
3„
~
i^e"-1
25m
|J
/ da(s)db(s)dc(s)
a(s)a;<s) + £ 6 , ( s ) ^ s ) + c(s)u,<s) ) } . (8.26)
Proof It is an immediate consequence of the lemmas 8.1-8.3 and the equation (8.15).
This page is intentionally left blank
Chapter 9
Local Stationary Liouville Equation 9.1
Gross Determinism
The turbulent Gibbs distribution
F(Z,t)
=
^(--^^^Ev^^(m|;|v«|» \
I — J.
+
I— J.
f;S^|x?)-x{'>|));..s*) I — i. / C ^ l
/
(9.1) is a function discontinuous on the boundaries of the cube-cylinders C^* x R 3Ar x R in the space F x R = V ^ x R3Af x R. We have elucidated the meaning of the decomposition of the spatial derivatives of the turbulent Gibbs distribution (9.1) into a smooth part and a singular part. (See, the arguments immediately after the equations (6.4) and (6.20)). A similar argument applies to the decomposition of the temporal derivative of the turbulent Gibbs distribution (9.1). If we subdivide the time axis into a large number of small time intervals, inside each of the small time interval the turbulent Gibbs distribution is independent of t.
But
as t goes from a time interval to one of its neighboring interval the turbulent Gibbs distribution will undergo a ( discontinuous ) jump (i.e., the turbulent Gibbs 175
CHAPTER 9. LOCAL STATIONARY
176
LIOUVILLE
EQUATION
distribution Tyv merely depends on the time intervals, but is independent of the moments inside each time interval). The differential operator d/dt (in the sense of Laurent Schwartz' theory of distributions) in the Liouville equation must be replaced with the operator (d/dt + A/At),
of which the first term, that represents
the ordinary differential operator inside each time interval, vanishes as it operates on a turbulent Gibbs distribution T/y: dTN
at
(9.2)
= 0,
and the second one represents the rate of the change as t goes from a time interval to one of its neighboring time intervals. Hence the differential operator d/dt in Liouville equation should be replaced with the operator A/At
in the process of
seeking an asymptotic solution to the Liouville equation perturbed from a turbulent Gibbs distribution. Having done that, the approximate Liouville equation (6.1) will be of the following form: N.
A_ At
)
d dw\(s)
,»
>F = 0, (9.3)
where, and henceforth, — ^ denotes the usual differential operator inside the cube C s . Making use of the orthogonal transformation (7.6), we have the following form of the equation (9.3): ,(•> Hi. /
*
A
(s) +
N.
w
(s)
Ax, ( s )
r'-l
+ £ tlVO^W N.
+ f^m^r^TTi £
E
94s)
it=i •
+ Ax1(•)
(1-1)
d dx (•)
+ Ax'(s)
i-i
Etf'-a-irt(»> •fe=i
9w|(s)
>F = 0.
(9.3)'
9.1. GROSS
177
DETERMINISM
The asymptotic analysis used in the section VI to derive the equation (6.19) can be generalized as follows. First Proposal of Gross Determinism
The following local (inhomoge-
neous) stationary Liouville equation will be used in a scheme of iteration to obtain approximate solutions to the Liouville equation: . . (s)
N.
rf-1
1)1
d
d
y;-^-(i-i)-4 dx.\ l s)
f-1
N.
+ 1=2 £ my/(I
- 1)1
£ f f -(<-l)f/(») fc=l
aw ( ( s ) J J
i At + ? i S ' ^ ' Ux<s) + Ax}->). N.
W
(»)
+ Ey/iTW
Ax (s)
(=2
N.
(s)
• s+Z E3
"n-1
N.
,, + I > w
Ax«
n—1)
(9.4)
the right hand side of the equation representing the macroscopic variation of F n _ i . The First Proposal of Gross Determinism means that the approximate solutions to the Liouville equation we are interested in are those of which the dependence on macroscopic temporal and spatial variables is much smoother than that on microscopic temporal and spatial variables. So the microscopic and macroscopic scales play quite different roles and have intricate relationship in the process of determining the functional equation governing the fluid motion. This is the reason why we call the iteration a Proposal of Gross Determinism. The term Gross Determinism was used by Muncaster (see, Truesdell and Muncaster [74] and Muncaster [62]) in his reformulation of the techniques of
178
CHAPTER 9. LOCAL STATIONARY
LIOUVILLE
EQUATION
the Enskog-Chapman expansion. The First Proposal of Gross Determinism plays the role of the method of the stretched field in Muncaster's reformulation of the Enskog-Chapman technique for solving the Boltzmann equation. Because Massignon's theoretical framework is quite different from that in the theory of Boltzmann equation, the First Proposal of Gross Determinism is quite different from the stretched field method in scaling and its result is different from that of the method of the stretched field too. According to my opinion, the First Proposal of Gross Determinism is more natural, and therefore better, than the stretched field method. In order to make our iteration scheme feasible, other Proposals of Gross Determinism will be made later on. If Fo = 0, then Fi = Tpi (i.e., the turbulent Gibbs distribution ) satisfies (9.4). F\ = T/v is the distribution corresponding to the general inviscid flows, including both laminar (i.e., deterministic) and turbulent (i.e., random) ones. What we are interested in is F%- In the classical kinetic theory of gases, the sum of the first two terms in Enskog-Chapman expansion is the distribution corresponding to the flow satisfying the Navier-Stokes equations. It is natural to expect that F2 were the distributions corresponding to the general viscous flows in the classical fluid dynamics. But the calculation carried out in the sequel will show us that it is not the case. The special case of the First Proposal of Gross Determinism is to solve the following equation N.
(•)
r
i-l
&4S)
+E^™fi:# ) -('-^ ) k=l
9x|s'J
3w«
?iSlyfe 'bxf+ Axf)
9.1. GROSS
DETERMINISM
179
The remainder of the present paper is devoted to getting an approximate solution i*2 of the equation (9.4)'. The iteration scheme (9.4) (or its special case (9.4)') is different from that in the classical kinetic theory of gases. The latter consists of two asymptotic limits. The first asymptotic limit is the so called Boltzmann-Grad limit, that is used to derive the Boltzmann equation from the Liouville equation. In the BoltzmannGrad limit the mean free path is assumed to tend to a positive limit. The second one is used to derive Euler or Navier-Stokes equations from the Boltzmann equation, in which the mean free path is assumed to diminish to zero. The modes of scaling in both asymptotic limits are inconsistent, at least, formally. The iteration scheme used in the present book consists of only one asymptotic limit. A new length scale K, which represents the size of the fluid particle, has been introduced in the definition of the iteration scheme in the present book. In the classical kinetic theory the size of the particle plays no role. There both the fluid particle and the molecule are regarded as a points of zero volume. The spatial and temporal rates of change of the distribution function have been subdivided into two parts, one of which for the macroscopic rate of change and the other for microscopic one. Furthermore we assumed that the macroscopic change is far much smoother than the microscopic one. Finally, the iteration scheme (9.4) in the present book is carried out in the iV-particle phase space, i.e., in Massignon's theoretical framework, but the second asymptotic limit in the classical kinetic theory of gases is carried out in the one-particle phase space. Hence the outcome
180
CHAPTER
9. LOCAL STATIONARY
LIOUVILLE
EQUATION
of the iteration scheme (9.4) is quite different from that in the classical kinetic theory of gases. The latter is the constitutive relations for deriving the Euler or Navier-Stokes equations in classical hydrodynamics, but the outcome of the iteration scheme (9.4) is a constitutive relations for deriving a closed functional equation governing the evolution of the /("-functional, which, as it will be shown later on, is different from the Hopf functional equation or its generalization for compressible flows. In case of deterministic flows, the new constitutive relations thus obtained will be reduced to a system of constitutive relations different from those for deriving the Navier-Stokes equations. The iteration scheme (9.4) is close to Morrey's limit (see, [61]) in scaling, but slightly different in some other aspects. Roughly speaking, the perturbation scheme (9.4) is an analogy of Morrey's limit in Massignon's theoretical framework, in which Massignon has introduced "correct Euler variables" instead of the traditional (or, incorrect) Euler variables in the classical theory of BBGKY hierarchy. Maybe we can call the iteration scheme (9.4) the correct Morrey's limit. I think, it is the mathematically simplest hydrodynamic limit and will be expected to provide us for a nice model describing the phenomena of the fluid motions we see in our environment. The outcome of the correct Morrey's limit is quite different from that of the Enskog-Chapman technique for the Boltzmann equations, as will be shown in the remainder of the present paper. Now we are going to deduce an approximate form of the solutions to the equation (9.4) in case of n = 2, i.e., the equation (9.4)'. Making use of the definition of the turbulent Gibbs distributions and taking account of the fact that dip(xk — x ()/^ x fc
= —
di>(xk — xi)/9x.i,
which is a math-
ematical formulation of the 3 r d Newton's Law in mechanics, we can easily verify that
N
*
(s)
KIT"'
yAl.^LS=0. U^°
d*\
(9.5)
9.1.
GROSS
DETERMINISM
181
Hence among the terms on the right hand side of the equation (9.4)' what we should seek are merely the expressions of the following two terms:
ATN
(9.6)
At and r Ns
(s)
.
TVs
/'-1
(s)
N.
A
Y
(s) AT/y .(»)'
EE »
A
(=i
(9.7)
— i
According to Theorem 8.2, the turbulent Gibbs distribution TN can be expressed as the outcome of the linear operator $ operating on the if-functional KN: TN = *{KN), where $(KN)
(9.8)
denotes the expression on the right hand side of (8.15). For any
functional JN(a,h,c)
in a,h,c,
the functional <&(JJV) in WQ ,w^',LJ2
,uj^',UJ±
is defined as follows: 8^ \-* m
*(JAO 2
n
( w
(s)
/
J.
(B) 0
3JV , V Km T2 + Z5
' m
S,T¥-»,6¥-*TV T K( ±^ B)
3
,) 3 a) _L ^^ _ +± m . )s rr r( 3 K 4 - 3 m ^ 2m / da(s) / G(s,£Stint,u)0 )
) | rK(
dh(s)dc(s)
a(s)4S) + X!M8)"^"*
+ c s w
( ) 4 (8.15)'
Under the conditions (6.11) and (6.12), $(J;v) has the following approximate form:
3^-^«3"+^ ^» °
2
2
^(iV +1)^+5^
2
i^+2m
^?ei—x
182
CHAPTER 9. LOCAL STATIONARY
n
rrf)-"*"1
LIOUVILLE
EQUATION
18K 3 4 8 ) + 25m / da(s)db(s)dc(s)
18G(s,£ Mnt ,4 S) )
x | j j V ( a ( s ) , b ( s ) , c ( s ) ; t ) e x p UKKH
a(s)^s)
£
+ ^ ^ ( s j w j ' * + c(s)a,<s)] \ j . (8.26)'
The local Maxwellian distribution can be expressed in terms of mass density, momentum density and energy density in a rather straightforward way. The equation (8.15)' or (8.26)', which expresses the turbulent Gibbs distribution TN in terms of its corresponding AT-functional KN, is its analogue in the non-equilibrium statistical mechanics without the assumption of molecular chaos, i.e., in Massignon's theoretical framework. Of course, the latter is quite different from the former both in form and in essence. The if-functional Kp/ satisfies the Euler iiT-functional equation (7.19)^ with (7.19)2, (7.19)3, (^19)4, i-e> dKff/dt
can be expressed in terms of derivatives of
the if-functional KN as follows:
dKN dt
=
(9.9)
Z(KN),
where H(KN) denotes the expression on the right hand side of (7.19)i. ( Henceforth, for brevity, we always restrict ourselves to the case of null external force): H(KN) = 1?! + tf2 + 0 3 ,
(7.19)'/
where dKN 0i = - ^ g - ( a , b , c )
^2
=
da
2dKN divb 3"9c"(a'b'c)
(7.19)2
9.1.
GROSS
+
x
183
DETERMINISM
f° dz dz
J-ioo
ht±{g 92RN
dbjdbk
M
- ^ 5
(a + zS(y - 77), b , c)
j k
)
<5(y - v). *(y - v)
^ P f . / d ^ ( 0 ^ ( a , b , c ) divb(y)exp(27ri£ • y ) , exp(-27ri£ • y) ,
67rni2
(7-19)2
^=l!d^{r')-Ljzi^ia+z~s{y-r,)^c) + 12m2 X
da2db^a
+ Z
d
vJ "^L ^h^
^ Y ~ ^ ' b ' C ) I e x P( 2 7 r i £ ' y)^(y
1 f
X
d
J2 g^(a
<*(y - v). <*(y - v)
+ {zi +
dc
f°
_
f ) ' exp(-2?ri£ • y ) , S(y - rj)
f°
z2)S(y-v),^,c) 6{y - v), &(y - v). *(y - v)
3=1
i ^pf.Jdtfc) + 27rm
d2K N (a,b,c) dbda
exp(27ri £ • y) dc (y), exp(-27ri£-y) m dy (7-19)i'
It is easy to see that $1 or $2 o r $3 w iH vanish, as a(y) or b(y) or c(y) vanishes respectively. Inspired by the idea used in the classical Enskog-Chapman technique for the Boltzmann equation, which was called Enskog's juggling owing to its strange form
184
CHAPTER
9. LOCAL STATIONARY
LIOUVILLE
EQUATION
and reformulated in a rather mathematically natural way by Grad [33], and then generalized as a method of stretched fields with gross determinism by Muncaster (see, Truesdell and Muncaster [74] and Muncaster [62] ), we make a proposal for solving (9.4)' as follows: Second Proposal of Gross Determinism
In getting the right hand side
of the equation (9.4)', we propose that A7V
*(£(#*))•
At
(9-10)
The formal similarity between the form of the equation (9.10) and the proposal used in Muncaster's theory ([74] and [62]) is obvious, that is the reason of the use of the term gross determinism. But the relationship between the turbulent Gibbs distributions and their corresponding Jif-functionals is quite different from that between the local Maxwellian distributions and its five principal momenta, it should be noted that the consequences of the equation (9.10) is quite different from those of the Muncaster's gross determinism in the classical theory of Boltzmann equations.
9.2
Temporal P a r t of Material Derivative of 7V
Making use of the equation (8.14) in the Theorem 8.1' and the elementary formulas:
ex+e « ex(l+e),
(9.11)
and dyj
dyf
2K
we have dKN
V
JJV
JV
QN
, T/-
i „
da
2^7T—-3^«¥r(^±2i)
dy.
m 2
3Af
,
9.2. TEMPORAL PART OF MATERIAL DERIVATIVE OF TN
x
185
s) s) s) s) /<W <W <W <W |(-2™ i)n (w(8))3r(,^)r(3^p^J 3
£
H | exp [ - 2™ 3 i a ( s ) 4 s ) + ^ 6 > ) u ; j s ) + c ( s ) a 4 s )
)G(s,£.,int)4'))
9a
x7M^)££uf-gi(t) t
j=i
23r7r^-£*,s^r(£±*) 3Af
m 2
K3
E
E."O8)=K
n
*>
S
JdWs>aWs)<Ws)<Ws) U«0
)aF(
EX>r4Mn exp (- 2 ™ 3
T(
" V^ 1
?
V
2m
^/
a(s)4 s) + £ ^ (s)wiS) + C(S)W4S) i=i
)}
l[G(s,es,int:J0s))TN({l;t) V
3JV_J5V
3N
2^7r~_^«;iTLr(^±2l) m
3N 2
E
I II /<WsWsWsWs)
f ] exp f - 2™ 3 i a ( s ) 4 S ) + £ ^ ( s ) ^ s ) + s \ L j=i
x V V . f ' A a wfrN(n;«)TT
G
C(S)LJ^
} -I /
,( s h
(-^'"o ) (9.13)
186
CHAPTER
9. LOCAL STATIONARY
LIOUVILLE
EQUATION
where the outcome of the differential operator -^- on the function .A(t) is defined as d
„ A{t +
ej)-A(t-ej)
It should be noted that d (t)7V(ft;£) depends on t , although Tff(Q;t)
is inde-
pendent of t. This operator and those of similar forms will be used frequently in the sequel. Here and henceforth the differential quotients and their corresponding difference quotients will be used for the same quantity alternatively without mentioning the conditions under which the alternation is justified. Since the quantity ! — K3 W.(•)
P
3( ^ 2 ,
2 ^= 1 j=l
,..(•) WA
depends on w\ , i — 0,1,2,3,4,. The integral Gibbs power G(s,£s,int,Vo)
1S a
function in LJ] , i = 0,1,2,3,4, too. Therefore, it is worth noting that the partial derivative d <.)G(s,£ 8 ,int,^o ) in the equation (9.13), and in all the equations in the remainder of the paper, should be understood to be the total derivative of G(s,£atint,cjQ
) with respect to UIQ for fixed u>^ ,i = 1,2,3,4, i.e.,
9 <.>G(s,£ s , int ,t4 8) ) = [d <.>G(s,£S)int,w£s))] + 9 f . i „ l G(s,5 S i i n t ,w^ ) )S>>£,,<„*, where [dj.)G(s, £s,mt,^o
)] denotes the partial derivative of G(s, £atint, U)Q) with
respect to UQ for fixed £a,intAccording to (8.15)' and (9.13) we have #(tfi) w
m
2 +7T
2^^-^«--^r( K+m m
n
1
/
~\ f
da(s) / db(s)dc(s)
~2m
«s ,.,(•) ^-\
n i J V
m
•^-'*
m
«. L •/
rfVrfi^jrfii^1) V 3N
m 2
9.2. TEMPORAL PART OF MATERIAL DERIVATIVE OF TN
x j H exp ( - 2™ 3 i a(s)4 s ) + £
187
^•(sjwj"* + c(s)^ s )
)
^ g ^ " ^ " [T"(";i)?tf))JrjSfe')r7i-'lI-3-)]} x e x p ^ i r n 3 ! ^ ofsjuf'+ £ V")"'}'1 + cMu?'
(4")^r(^g^)r(2fs^=^)
n
<-*(s, £s,int,W 0 )
EE4
(t)
t
j=i
G(s,£sMt,uJoa))
X
W^
m
/^/a
0
' m
)1
- (-r)^(^^)r(^£^)
(9.14)
0,
(9.15)
-•• •••
Since 9 „
a^r we have
(9.16)
*o»i)« - ( £ + ^ + c +«)*„.y „(.)K, Z—/s ° '
where
t
j=i
j
t j=i
(9.16)!
a
r, = Tjv(fi;t)J] (j
(. s ) c -s,int! w o /J
t j=l
J
s
(9.16)a , 33 , »
{ = TN(Cl;t)l[
r/ic
a C + m \ r / 3 « 3 a ; ^ - 3m\
188
CHAPTER 9. LOCAL STATIONARY
LIOUVILLE
EQUATION
d t
J=l
J
S
K3Ug +m^j-,,3K3u^'' — 3m m / V 2m
^
(9.16)3
)J
and 1
1
(t> (t+ej) TN{n.t)t j2j2^ Wn j=i L^o
v= _
, ,(*-ej) W,
a t
(9.16)4
.<*>
J
j=i
It is easy to see that
tl = TN{n;t)Y[
t
t
— 2/t
n
2K
i=i
= TN(fi;t)EE
G(s,£Stint,u;0 (»h)
G(S, £a,int,W0
)
G(t + e i t A + e i , i n t , 4 t + e i ) ) G ( t - c ^ f t - ^ . i n t . w i *
Ci)
)
i=\L
I O (t + e ^ - C ( t - e j ) I w \ uo o /
3
G ( t + e j , 5 t + e j , m t ) W0
)
° )
= TN(Sl;t) 3
a,™
*££tr t
9 (t+e^jG^+ejj^t+ej.mtj^O
(t+e,-)-. ' )
G(t+e
j=i
d
.rr,,!')^ T*(n;t)^4 t j=i
3
) G(t — ej,£t-ejtint,^o
(t)>
^O^C^'^t.'nt'^O )
')
(t)>
G(t,£t,int) w o ) (9.17)
9.2. TEMPORAL
PART OF MATERIAL
DERIVATIVE
Making use of the asymptotic formula for dlnT(z)/dz
OF TN
189
= T'(z)/T(z)
(see, e.g.,
[2]):
r'(z) r(z)
lnz
-hH^
(9.18)
we have •j
^,( K3U)Q + m
m
J
V
(s)
,
" v m; + m x
M - l
^ ^ ln(K3a;o + m) — lnm — + o'( 33 m 2 2(«3^s)+m) ' V V((//tt c4iss))++m m)) 2 /
(9.19)
and ^f3K3cj^a) - 3 m 2m
-l
3/c3wis) - 3mN 2m
=ln(|)+ln(«»a,«-m)-lnm-—p^ -+o( ) • (9.20) ( f \^/ 3(/t3u>o - m) \(/t 3 a;o - m ) 2 / By virtue of the above two asymptotic formulas, it is easy to see that , j Xt+ej) ln(K,s3w^ > + m) + t
j=\
+ ln(K34t_ej)+m)
m
+ 3- ( K 3 4 t + e i )
3/t 3
2(K34t_ei)+m)
+ ln(K34t_ei)-m)- m)
+ 2m
^
2(«34t+e^+m)
l n ( K 3 4 t + 6 j ) - m)
3 ( K 3 4 t _ e j ) ~ m).
K2TN(n-,t) 2m
zp?{l(H«r^M^)+l(^-^
+0 1 ^
190
CHAPTER
9. LOCAL STATIONARY
LIOUVILLE
EQUATION
Here and henceforward, as in (9.13), the spatial difference quotient and the corresponding spatial derivative for macroscopic quantities will always be used alternatively. Summarizing the results in the equations (9.16),(9.16)i,(9.16)4, (9.17) and (9.21), we have the following Proposition 9.1
-^zA-)^^)Md^)TN{m
*(*) * +TN(Sl;t)
t
j
j=i
djJ)G(t,£t,int,Uo
(t)\
i
) 5K3
G(t,ft,i„t,4t))
2m
(9.22)
ln( W W)
It is worth noting that both the domain of integration and the integrand of the expression G(t,£ttint,W0
=4
dx(t)
K
)
AW-si.
Ej-i^)2 ,(*>
1=1 kjil
' +
depend on Nt (or, equivalently, onwj ). Therefore the form of dependence of the function G(t,£t,int,^>o
) on Nt (or, equivalently, on wj ) is rather complicated.
We shall return to the problem later on (see the proof of L e m m a 9.8).
Making use of (8.14), we have 2 dK divb 3 ~dc
2^+17r^-^+1K^+2r(K±H) 3m 2
9.2. TEMPORAL
><
PART OF MATERIAL
E a
B) K
« ZA =
n[/
DERIVATIVE
OF TN
191
exp(—27rK3ia(s)u>o )
(n l 8
da; 1 (s) da;2 (s) dw3 (s) dw4 (s)
(l>i 8) I> (s + ^
8
e^-b^s-ej))
j= l
J ] [exp ( - 2 7 r « 3 i [ ^ 6 i ( s ) ^ s ) + c ( S ) 4 8 ) ] ) G ( s , f s , i n t , 4 s ) )
3N
V
3V
9JV
,
TN(n;t)
,v,
3m 2 exp(—27TK3i O(S)UJQ )
EX"-* l *
?[/
{(E^DW'0)
<W S) <W 8) <W S) <W S)
exp ^-27r«3i J ]
^(sV^+cfs)^
V
3 N _ W
)}
nG(s^s,int,4S)) ^ v ^ ; 0
9JV *•
m
'
3m 2
E
?[/
n L(4
du/i ( " ) dw 2 (,) <W" ) <W" )
exp(-27TK3ia(s)wQ ) s)
)*r(^^)r(2^^)
exp f -27r/t 3 i ^ ^
J ^ 6j (s)a;j s) +c(s)w, s
<-j=l
I
192
CHAPTER 9. LOCAL STATIONARY
E
LIOUVILLE
EQUATION
W S)
4 XX<- + '<> -flw<—i>> ) { I I G ( S ' ^,int, 4 S ) ) IV(Q; t ) } (9.23)
Making use of the inversion formula for the Fourier transform and the equation (8.15)' we have the following L e m m a 9.1 25
V3 3
t
j=i
dc
L
3
J/
f) r
9 ( t>G(t,&,*„*,a;^)
(9.24)
J
Now we are going to compute the second term of the expression (7.19)3 °f ^2Firstly, it follows from the equation (8.14) that V
82K,N (a + dbjdbk
31\
QN
JV
.ISK
^
zS(y-ri),b,c) S{y - rj), S(y - 77)
m
'
3JV
m 2
E 3
S)
- E.4 =K
l
n
(wW)lr(^±=)r(2^£=2=)
s
nf/ <W <W <W <W (n s)
s)
s)
x exp ( - 2 7 r « 3 i ^ (a(s) + z ^ ^ \ 4
s)
s )
^ ( S , ^s,intT^o
+ J2 bj(s)^s) + c(s)u ,00
x(-4,V)Tw(O;0EE^XdtC'('')}}Hence we have
^
7= lfc = l
V
'*
)
(9.25)
9.2. TEMPORAL PART OF MATERIAL DERIVATIVE OF TN
0
l
* / -ioo 3
/
^
fl2 d'K N db^h{a
dZ
+
zS{y r ) h c)
-' > > >
Hy - v), Hy - v) V
3
(
c(-4*V)
vVk
>
£
3
^
X><->=4"0 - ^ 3 -
)
J]
/
193
) J_ioo
3N_3V
gv
m—
* 1
k s L w
2
( o ) T(—^—)r(—^—)J
xc iT,)x cM x Y[ I /<Ws)<Ws)<Ws)<Ws) TN(n;t)^^Jv > 6
H | exp ( - 2™ 3 i (a(s) + * ^ ^ ) ^
x G ( s , fs.mtj^o
a )
bj(s)W<s) + C(s)u,<s)
+£
)ff
V
3N
3V
gjv
,v . V
3JV
m 2
53
I /
dzexpf
-27ri0^xc8(r/)w^s))
» 3 E,^= K
nL
(w(s))fr(«!^)r(3^_^)j
(-47r J K b )
m
'
)
194
CHAPTER 9. LOCAL STATIONARY LIOUVILLE EQUATION
x ft | exp ( - 2™3i a(s)48> + £ 6;(s)u;<s> + c(sV<s) ^G(s,5 s , i n t ,4 s ) )|j 1
V
, ,
3JV
3V
,,
,ir
ON
,
3JV
m 2
><
n
E 3
r ^
3
J<Ws)<Ws)<Ws)<Ws) L(4-))lr(^g±=)r(2^g=2=)J
TN(Sl;t)
/ 96^ ^ divb(>y)t \ E s S t "j-<•),.(*), M Xc.(??)xct (v) v 8 ^ 3 '*/ E S 4 S) XC.W
i=lfc=l
n { e x p ( - 2 ™ 3 i a(s)u>ia) +J2bj(s)^3)
V
, i
31V
3V
i i
+0(8)^
ow
, ^
W,£ 8 , i n t ,u,( s ) )}
,
3JV
m 2
E 3
3
r /
Z E E (ft S
X
/<Ws><Ws)<Ws><Ws>
(n L 4.))fr(^±=)r(»=!^=»=).
K2TN(Q;t)
(
,(s+efc)-6j(s-efc
EUiM^ + ei)-b,(s-e,)]s )]-
(s), ,(s)
\ < H "i* /
(.)
j=l fe = l
I I { eXP ( -
27rK3i
" ( ^ ^ + E ^ ( ^ + C(S)W4S) ) G(8, 5s,int, 4 S ) ) }
V
3JV
3V
9V
, i^ ,
2^7T — - ^ K ¥ r ( ^ ± 2 ! ) 3Af
m 2
9.2. TEMPORAL PART OF MATERIAL DERIVATIVE OF TN
«3T
/<W s) <W s) <W 8) <W 8)
£ (n [ "<'>=K <•
(
s
^)
) i r (
^^
) r (
o 3
J ] { exp ( - 2 ™ i
195
3^
±
n
)
-i
8
a ( s V < > + £ 6i(.)«i')+C(.Vi
,)
3
) } £
3
£
£
:{^^< t) -^£|-^ (t) ||r N (f2;t)nG(s,5 s , int) 4 s) )}||.
(t) L w.
(9.26)
Therefore, 3
$
3
(/*±S (£«-*?**) (0+ (y ,?),b c)
*ls *' ~ '
% - */). % -*?)
3N . V
M i p V
m Z^B 0 ' m 3« 3V 9JV
n
"""^
12V
C ( S , Sa^nt,
£
^E^^ (-2™ 3 i£ ^
2
. ,^ .
2m
{
exp
,.(.) K m
11
V V
2^7T
3JV
2
3V
^
UJ0 )
9JV K
-
r (
K ± m )
3JV
m 2
/dA 1 (u) dA 2 (u) rfA 3 (u) rfA 4 (u)
n
(u)X3T,/K3A(|U)+m^<,3K3A(i")_3m
V" U A r ) ^ ( ^ ^ ) r (
a(u)A( u) +£6 j (u)Aj u) + C (u)A: (») j=i
2m
I
3
3
rx^xW k(t)
196
CHAPTER 9. LOCAL STATIONARY LIOUVILLE EQUATION
(9
6jk
E J ; ^ } {r*(A; *) II G(u> fu,int(A), A
/
r
x exp I 27TK3i J2 ^
3
L
s
j=i
/rfb(s)dc(s) "STZJ.^O
J
'm
--L
-.
a(s)u>ia) + E fc^sju/j"* + c(s)w<s)
G(s, £ s ,inti w o )
dAi ( s ) dA 2 ( s W s W s )
]((?[/
exp ( - 2™3i E [JX-JA?*+ c ( s ) A i 8) l) E E E :
W .(*)
{ ^ t y - %E^ a Ar}{ T ^( A ; t )Il G ( s '^nt(A) ) ^ s) )J r 3
x exp (27TK3i E
E
h
i i8)^
3
3 V 1 , .<•) K FT Z ^ s O • m
ELj^fo^Mn^Wo
, X
9
\dsk
(Jjfc 9 (8)
—
»V
3
+ C(S)W4S
3
^uP j "k
u
EEE
) s i = l fe=l L
O»
w
E ^ 9 ^ » ) ( T ^(^ *) II G(s,£S:int,^s))
Summarizing the above results we obtain the following L e m m a 9.2 We have 3
$ \
x
r° /
7-ioo
dz
J
j=lfe=l ,dvk
8*KN au »u ( a + OOj^Ofe
3
zS
(y ~ 7 ?)' b ' c )
% - v), % - v)
9.2. TEMPORAL
PART OF MATERIAL
> =
'
DERIVATIVE
^
d 5jk Wkd»? ~ T
- S K • T J*\ & 1*, 1^ 1*.— (t) w •
io
t j=ik=i
9 ^(t)G(t,£ M nt,w£,(*)') +TN(n ; * ) ( 3*fc
OF TN
s,k
G(t,£ tl< nt,4 t) )
3
3
3
o
197
v
Tjv(n;t)
a w ( t ) G(t,£ t ,int,wj )
a
j^S'i' S ^'
GG(t,£t,int,J^)
(*
). ' (9.27)
Now we are going to compute the last term of the expression (7.19)3 °f $2According to the equation (8.14), we have d2K»
^Pf./^(0^(a,b,C) 67rm2
div b(y) exp(27ri £ • y ) , exp(-27ri £ • y)
i2^r+17r~~^r+1K;~
+
^3T(K±ni) *• m '
3m 2
E
n L(4s¥r(^±f£)r(^£^)J J<W s ) <W s ) <W s ) <W s )
n | e x P r - 2 7 T « 3 i a(s)4s) + ^6,(sVJs)+c(s)a;is)
W,£s,int,4s))j
(9.28)
Hence we have $
(-6^ pf 7 rf ^ (0 ^ (a ' b ' c) V
i)«3v
15V
(•> K 47ri m ^ K " ^ 3 "
n
/
n
div b(y) exp(27ri $ • y ) , exp(-27ri £ • y)
(4s¥r(^C±^)r(2^£^)' G(S, t s . m t i ^ o )
da(s) / db(s)dc(s)
198
CHAPTER 9. LOCAL STATIONARY LIOUVILLE EQUATION
n
E
/dA 1 ( , ) dAa ( ' ) dA3 ( , ) dA 4 ( ' )
(^>)ir(^£±=)r(2^£=2=)
x M ] { e x P ( - 2 ™ 3 i a ( s ) A o° + £ 6 i ( s ) A 5 S ) + C(S)A4S> )G(s,5 s ,int(A), A ^ ) |
cr,(A;t) £ *(t,A, i n t (A), A<<>) £
^t + ^ ^ t - e j )
i=i
3 x exp ^27T« i £
<5«»v,,,<»)
IT 2 ^ . "o • S
9 (t +Cj > —
I
)
2 L
\ m ) C{s,£atint,UJQ
l
\
(*h)
2S
1
da s
( ) / dM8)^8)
I1 / s
3
(w0
TT
X
nr
I"/ >Mrr* 3u 'o' >+m '>rY 3K3a 'o* ) ~ 3m V
K 47rim^'K"^"
5m
J V (A;t)nG(s,4,int(A),A(
s)
)^
^ ^ ( A ) ^ )
d^(t-Cj)
-47TK4i
JJexpfft
exp ^27TK3i £
27r«; 3 i
a(s)A< s) +£6>)A< 8) + C (s)Ai s)
^
U(s)4s) + £ b^uf
+ c(s)^ s ) )
9.2. TEMPORAL
PART OF MATERIAL
DERIVATIVE
2<$ £ 3 ^ m Z-^a
OF TN
199
(.) K ° ' m
3n s G(s,f s , int ,4 8) ) X J2 *(*. £t,int, 4^) E ^ A ( t ) T^(fi; *) II G(S' ^,mt, 4 S) ) t
j=l
1
'
L
s
Summarizing the above result, we have L e m m a 9.3 $
(-6^ p f 7^ ( 0 ^da( a , b , c )
divb(y)exp(27rif-y), exp(-27ri£-y)
2
m /
JfS
0
])
' m
(t)>
9 (t)G(t,ft,mt,^ ; )l
d
t)
x£*(t,^»U )E
9 (,)Tw(fi;0 + T w ( f i ; 0 - ^ G(t,£t,intj^>o
(t)s
)
(9.29)
According to the equation (7.19)3
an<
^
tne tne
L e m m a s 9.1-3, we have the
following Proposition 9.2 28 3
$(tf2)
" EAB)>K
E {(%r - *(t,ft,«-«,«r)) E -frd^TsMt) o
+Tiv(n;i)
3
3
(t)
(t)
200
CHAPTER 9. LOCAL STATIONARY LIOUVILLE EQUATION
3^
(£t,int + \ K3
(t)N dJt)G(t,Sttint,^>)
d
*(t,ft,mt,W, 9=
(9.30)
G(t,£ttint,uj0 (th)
•>
1
Now we are going to compute $(#3). 5
f
J
d
dc
,
^
f
j
d K
N,
-st
s
,
,
3/ %^7_ i o o ^ab^ ( a + z<5(y - 7?) ' b ' c) V
3N
3V
9N
<*(y - » / ) . 5(y - v)
, v ,„ ,
5 2^7rT-5 = T/ S *f t r(£±!n) 3m 2
n
E ^
n{exP(
U
,
27TK3i
/(Ws)<Ws)<Ws)<Ws)
^
S
^
("(B^ + D i ^ + ^ f
£s))}
C(s, f s ,i„(, W(
i=i
(t) 3
xTN(fi;t)(-7r«;2i)^
^ ^ ( t + eO-cft-ei)]^ w
o
(=1
5 2^7r^-^«Tr(g±B) 31V 3 m2
E
n
/<Ws)<Ws)<Ws)<Ws) (4"))*r(^±E)r(2^2=)
J] { exp C - 2™3i (a(s))4s) + £ 6j(s)a;js) + c(s)4s) ) j
E +
% E w J ( t ) ^ K r PM«;*) I I G(s- £*,intAs))} Lw n
;—1
'
s
(9.31)
9.2. TEMPORAL PART OF MATERIAL DERIVATIVE OF TN
201
Hence we have $
5
h>L
^ ( a + ZHY - V), b, c) <5(y - v), Hy - v)
dz
5 x -T- ^ TT " o » « i V nW K m « 3 K " 3 I I
n
/
da(s) / db(s)dc(s)
/dAi(s)rfA2(sWsWs)
{ s {n x
(^•>)tr(^=)r(2=^=2=)
n {exp ( - 2™3i [(«( s ))4 s ) +E^^+ c ( s ) A i s ) ])}
*E
" \ E AiW LAo i=i
^ 9 A<*> PMA; *) I
I G(s> fMnt(A), A<s))]
x exp [27TK3i 5 3 a ( s ) 4 s ) + E M 8 M " } + ^ ^ s L =1 55, 3 v 1 ,,(s) K 3[lsG(s,£Stint,u}0
(s)
)
3
^
) } J/ J
~
E -feE^'V^'^^^n^8^.-"^)] u
0
i ~4
{=1
Summarizing the result thus obtained, we have Lemma 9.4
<5/*>/:
•I i/<(">• jd'TBEC + Jfr-**')
E
5<5^3 \ p m / ^ff
(s) KW4'
(.) 0
» m
3a;,(<0
<j(y - *?). <*(y - v)
202
CHAPTER
^1 = 1
9. LOCAL STATIONARY
"d"Sl » L
EQUATION
dj.)G(s,eatint,u)y) (*h.
dj.yTN(Q;t)+TN(n;t)
l
LIOUVILLE
"
G(s, £a,intt<^Q
(9.32)
)
The second term on the right hand side of the equation (7.19)4
can
be com-
puted as follows.
12m2 d3KN,
-,
xJ^-L""-/*9®
. ,
,
exp(2?ri£ • y)<5(y - 77), exp(-27ri£ • y ) , S(y - rj)
5i 2 ^ + 3 7 r ^ - ^ + 1 / c 2 f 3N 12 m 2
E *3V
n
W<->=K
(n s
^
+ 3
r(^)
/<W s) <W s) <W s) <W 8)
(4 s ¥r(^£^)r(^^)
I exp ( - 2™ 3 i a ( s ) 4 s ) + £
6;(s)u;<s) + c ( s ) ^ s ) ) G ( s , £ s , i n l , w<">) J
3
1 ZK
t
1=1
( t )
l l
(9.33)
^o J J
By virtue of (8.15)' and (9.33), we have $
&KN X
d^db{a
,
5, +
h^L*"-/*9®
5 12m27ri J
drj
. . . exp(27ri£ • y)5(y - 77), exp(-27ri£ • y ) , S(y - 77) - ''
Z5{y V) h c)
5 . V 12V J 3 « o « 3 ^ (.) £m -27rim^ 3 "K"^ 3_ m Z^s 0 ' 3
9.2. TEMPORAL
PART OF MATERIAL
DERIVATIVE
n •#135!^=)^ G(s,£s,int,^o
/
(«h )
OF TN
da(s) / db(s)dc(s)
/M (s) <W s) <W s) <W s) 8
{ s {n (4
¥r(^^)r(^£^i)
J J { exp ( - 27TK3i a(s)w<s) + £ 6,(s)u;j s) + c(s)u4 s) ) G ( s , £ s , i n t ,
(t)
3
l xTs(n;t)—^(t,Sttint,LJ^)
11
£ > ( t + ei) - c(t - e , ) ] \ \
3 s) s) C S S) x exp ( 2 ™ i £ [ a ( s ) 4 + £ > ( s ) u , < + ( V4 1 ) } \ L \ J) s j=1
56. 3 v * , ,
K
^(t.fc.int,^)
3n,G(s,fs,int,^S)) 3
,,(*)
£
9a;,(t) TAr(fi;t)JjG(8>£;,int,4"))
Summarizing the result thus obtained yields the following L e m m a 9.5 r0
$
X
| ^
( a
+
25
12m 7ri JdnjfiW-J. dzPi. j dZfc) 2
"(y-7?)'b'c)
exp(27ri£ • y)<5(y - 77), exp(-27ri£ • y ) , S(y
%
m
V
Z-rfa
a.'*) K * ( S ) ^ S , i n t , W 0 S ) O
'
m
204
CHAPTER
9. LOCAL STATIONARY
3
u,r» & di.}TN(ri;t) ti^3)ds<
*£
+
LIOUVILLE
EQUATION
0 ( S )G(s,5 S ] i n t ,^ s ) ) TN(n;t)-
(9.34)
G(s,£atint,Ld0
)
The third term on the right hand side of the equation (7.19)4 *s of the following form:
1 f
3
x
dc
f°
f°
d3K
H 5baS(a + (zi + Z2)5{y ~v),h'c)
Ky - v)> Hy - v). *(y - v)
2^+17r^-^+1K2f+3ir(^) 3m 2
E
n{exP(
E
, ,(»>_
27TK3i
K
n
/<W s) <W s) <W s) cW s) (4s))^(^±^)r(3^£^)J
a(a)wP +J2bj(s)uf
G(s,£, s,int
+C(8)UJ.
,4 S) )}
j=\
3
3
/ , ,<*) \ 2 1 1
a
(9.35) t j = l 1=1
'
xw
0
'
J J
Hence we have
K~ IId^{r,y Ldzi
Ldz
Hy - v), Hy - v), Hy - v) j=l
J
9.2. TEMPORAL PART OF MATERIAL DERIVATIVE OF TN
2^j£v>
(.)
m / j
n
B
205
j<m^i
0 ' m 12V a
(4 s ¥r(^^)r(^£^^ /'± da(s)
/ dh(s)dc(s)
G ( s , S a ^ n t , u } 0(*)\)
/<Ws)<Ws)<Ws)<Ws)
£ (n
( w
( s )
3
)
^ ^
r (
) r (
3 ^
z
3 »
)
>ZA'}=K
J ] | exp f - 2™ 3 i a(s)4s)
+£
bj (s)^s)
+ c( S ) W i s) \ G(s, £a,int, ^ s ) ) |
J=I
3
t
x exp f 2™ 3 i Y^
3
(t) v 2-
j= l i= l
vw
'
0
7
.
a ( s ) 4 s ) + E fyfaV?* + c ( s ) ^ s )
)
j=i
ui(-') &
• ^ V m / .n
0
» m
3 [Is ^ ( s i ^ , i n t i
x
3
3
(t)
/ a ) ( t ) \ 2 /)
T
w
0 )
f
G s
s)
1
~
EEE^ (^y) |;^r[n{ ( >^nt,4 )}7M^)
Summarizing the result thus obtained yields the Lemma 9.6
K-Udv^yLdziLdz2 3
x
d3K
T
E absi 1= 1
3
(a+(21+Z2) 5{y 7?) b c) (y
(y
1\
(y v)
~ ~ ' ' r ~^' ^ "^' ^ ~ ) <-
i '
206
CHAPTER
9. LOCAL STATIONARY
3
E.
m Z_>«
9sj
0
LIOUVILLE
,
3
EQUATION
(s) N 2
' m
a uW r w (fl;t) + rAr(n;t)
; (•)\ d w (.)G(s,£ s ,int,w5 )
G(s, £s,int,W0
(9.36)
)
The last term on the right hand side of the equation (7.19)4
c a n De
computed
as follows,
pt
)
exp(27ri £ • y) dc (y), exp(-27ri£-y) m dy
b c
5 /«« i^<°' ' >
2^7r^-^K^+6r(^s) 3JV, o +-4
m 2
E.
{n
E "o
JM(s)<Ws)<Ws)<Ws) ( w
( s )
) f r (
^ ^
) r (
3 ^
z
3
E )
—&
3
3
J ] { exp C - 27TK i a ( s ) 4 I \ L s 3
(t)
8)
+ £ j
M 8 ) " ^ + C(S)W4S) ) J /
= 1
I "j
Q
x E * ( t ' ^ < , 4 t ) ) E ^ a r 9 u ' " [ I I G ( s ' ^ t , 4 s ) ^ ( f i ; f ) ] I (9-37) Hence we have $
exp(27ri£ • y) 9c (y), exp(-27rif-y) m dy )
fl b c Qpf-/**<0=S£< 27rm dbda ' ' >
S„3 y-> m Z->B
(.) K 0
' m
n8G(s,£,,i„t,4s)) 3
w , .w wL d
xj;^^,^)^-^^,,, fe=l
W
0
Tw(f2;t)nG(s,5s,in„4s))
9.2. TEMPORAL
PART OF MATERIAL
DERIVATIVE
OF TN
207
Summarizing the result thus obtained yields the following L e m m a 9.7 $
O - J' «%&iL%&+™ '
3
(t)
»E
*
m / , ^H
0
exp(27ri £ • y) dc (y), exp(-27ri£-y) m dy )
' m
a
(*) k=\w0wA"'
5
OT at* fe
Combining the equation (7.19)4
wrt
h
tne
(9.38)
L e m m a s 9.4-7, we obtain the fol-
lowing Proposition 9.3
E. "0
, ,(»)
4(0 3 ) «
-
3
K
• m y
3
, ,(»)
^
^
I
^ s
34
s
)+2^^-*(s,£8(int)a;(s)))
, .<•)
1=1
^0
(.)G(s,£Siint,oj^3))
d dsi
8
( .,r w (n;t)+TAr(fi;t)
(9.39) G ( s , £ S j j n t , w0 )
Combining the equations (7.19)" and (9.10) with the Propositions 9.1-9.3, we obtain the following Theorem 9.1 The expression ^fATN
is of the form
{ATN\
At
\ At J1
(ATN\
1 A \ At^ A '
(9.40)
where
N
7
x
t
I. j = l
•>
j=l
fc=l
w
0
K
208
CHAPTER
9. LOCAL STATIONARY
t)
+f ( ^ r -*(t,£t.«n*,4 )) E ^ 3
LIOUVILLE
EQUATION
t ) T i v ( f i ; t}
, ,(t)
+1^+2^ '-&• -2*(t,£t,intAt))) E n t r l r ^ ^ C n ; * ) , (9.40),
"" -r„(n;.)E(E"l"- " " "' ' """ " (m8 i G
tX't'i3
+: 3 +
£-t,int —L3
, .(*) r
^ , . ( t ) ( =1 W0
l T ,/,
~ V(t,£t,int,Wo
(tK )
^ + 2 ^ - $ ^
i
3
fl
£f<»i
£ ,
C^o^t,^,^)
, £t,int,<^0
* r(,w
2i = i^
a
S >
G(t,€ttint,u>^)
(t)N
G(t,£, in i><"o )
9
5 3
- ^
m
9
; (*)> )
w<*)G(t,5t,tnt,^ 4
))
9
'<
G ^ , ^ , ^ )
WI/*M
- * ' ln(wo aT
(9.40)2
; )f-
In order to simplify the form of the expression (^55^)2 defined in the equation (9.40)2, w e need the following L e m m a s . L e m m a 9.8 du(*)G{t,£ttint,U0 G(t,£t,int^o
(t)>
)
(*h
)
+ ln
3^.3
Ms
£t,kin - ^ ( t . g t . m t ^ o ' O „(t)\^,»r«,»(tAli.trfl)>0 ~ 2m F K1\T,(* F £t,int — « W^t,C.t,mt,W0 J
3~
* ( t , St,inu 4 4 ) ) ) + In 2 + 2 In K
(9.41)
9.2. TEMPORAL PART OF MATERIAL DERIVATIVE OF TN
209
where f 1, if €t,int - « 3 *(t,£t tint ,w£') > 0; |\ 0 0, > otherwige otl
X£ t ,^ ot >0-X £tint _ K 3* (t , £ , i „ t ,„(.) ) > 0
In order to prove the Lemma 9.8 and carry out the computation in the sequel, we have to introduce Third Proposal of Gross Determinism When the expressions
j)
' ec sr^c*., t+Cj
£ tfyi*-^-*{*> + «,) yk<«-
ec t + e .
and
£ l
^Ix^-x^l) l
do not appear under differential operators in the processes of solving the equation (9.4)', we always assume that the following approximate equations
(9.42)!
- ^ 9 { t - eh£t-ejAnUJtei))
« ^
^ ( y r e i ) - x,(t) + ««,•),
£ yfc
ec t + e j (9.42)2
and ^(t.^nt.a;^)*^
£
^(Ix^-xfl)
l
hold. Now we return to the Proof of the Lemma 9.8 Recalling that G( S ,,f s , i n t l 4 s ) )= /
dX« 25Mnt-«-3
£
E^4S)-x|s))j
(9.42)3
210
CHAPTER
9. LOCAL STATIONARY
LIOUVILLE
The outcome of d (.) operating on G(s,£atint,u>Q)
dJt)G(t,£t!int,Jf))
EQUATION
is composed of two parts:
= I + II,
where
3
s;
K
(S)N2
/
2a;(»)
dX<">du(.)
3K3^(*^-5II
u,<s,/m
E-=1K ) -"-8 E EM S) -*, (S) ) W,(•)
and H = d..w (») 3
dX ( s ) •/c."SK3^'*'-
(») / „
3
s)
s)
«- E EM -x[ ))
"(*,« -
1=1
"0
fc#f
fc#J
' +
Using the Third Proposal of Gross Determinism we shall get the following approximate expression for J through a routine computation: I ~ ~—G{t,£ ttint ,u) Q )
£t,kin ~
K
^(ti^t,int)W0 )
——
(t)TX*. h ..,>o + In
2(%^-*(t,ft, i n t ,4 t ) )
The computation of II can be carried out as follows. We are facing an integral over a domain with variable dimension and have to compute the derivative of the integral with respect to the variable dimension. Subdividing each side of the cube C^' into n equal parts, we have the following approximate expression for an integral: r
™3W*
/
\
3N
°
3K 3 W ( S )
where M(A) denotes the mean of the function over the cube C^*: n3N*
9.2. TEMPORAL
PART OF MATERIAL
DERIVATIVE
OF TN
211
It is plausible to assume that M(A) is (approximately) independent of the domain of integration, hence we have d ,<•>
Jc?'
dX
(s)
n"
A(X^)*du
3Ar
-'
/-
\- >3AT.-,
3^3
— \nn K m
,<•>
— ln/c/ m
7cAf.
3*3u^B> m
MM)
dX<sU(X(s>),
which completes the proof of the equation (9.41). Some words on the Third Proposal of the Gross Determinism are in order. Since the turbulent Gibbs distribution (9.1)' is merely an approximate specification of the statistical distribution describing the state of the iV-particle system. Near the boundaries of the cube-cylinders C^' x R3Ns,
where the turbulent Gibbs
distribution (9.1)' behaves discontinuously, the discrepancy between the reality and the turbulent Gibbs distribution (6.20) is particularly large. Moreover, the total intermolecular potential energy density ^ s S z = i £fc=si V^Xfe ~~ xi
) inside
the cube Cs is a microscopic quantity, i.e., a quantity depending on the position variable X^8' of the point inside the poly-cube (CS)N" and its behavior near the boundary is rather confusing. In classical thermodynamics and hydrodynamics (even in statistical theory of turbulence), it is always assumed that the concept of the temperature of a fluid particle is meaningful. The temperature is a quantity in proportional to the heat energy, which is the difference between the internal energy and the intermolecular potential energy of the molecules inside the fluid particle, i.e., the cube Cs.
If the Third Proposal of Gross Determinism
were invalid, the concept of temperature of the fluid particle would be meaningless. Hence it is compulsory to take the strategy of making the substitution of the Gibbs mean * ( s , £s,int,^o density -^ Yli=i 12kM ^(xfc
—x
) for the total intermolecular potential energy l ) inside the cube C s and the substitution of
the corresponding part of the Gibbs mean for the total intermolecular potential
212
CHAPTER 9. LOCAL STATIONARY
LIOUVILLE
EQUATION
with a fixed molecule. It should be noted that the expression Yli=i vi" ^ f H ^ *) would appear in the inhomogeneous terms in the local inhomogeneous stationary Liouville equation, which will be used in seeking a second order approximate solution to the Liouville equation. In seeking the form of the inhomogeneous terms and the coefficients of the local inhomogeneous stationary Liouville equation, the above substitutions would not cause significant error for solving the local inhomogeneous stationary Liouville equation. But it should be noted that the above substitutions for the expressions of the solution to the local inhomogeneous stationary Liouville equation will not be used, because their derivatives will appear in the equation, that might cause significant errors. The above argument is not mathematically rigorous. I do not know how to make it rigorous. Maybe the use of the substitutions is one of the restrictions on the iV-particle system for which the theory of the present book can apply. The following two Lemmas can be verified through routine computations. Lemma 9.9 djvGfaStjnt,^)
3/c3
WW
(9.43) Lemma 9.10 0 u <.>G(t,&,i„t l u4 t) )
3K3wu (t)
1
0
(9.44)
Lemma 9.11
£'^jzC £-( n„3
3
t) d d w (t)G(t,£ t ,int,^o (t))
' i G(t,€t,mUJ^)
m
£t,kin
£t,int
Q rft^nt
2 m•E-ST-?'^ 3
(^-nt^m^))2^
r °U
3/C3
"2m
E t
(t) V"> j=l
(t) ~^S
1 t
t
*V > t,inti<*>0
,,(t)
U_ )
.T./j. f
3
"*(t, £,*>*, W^)' ,,(t)
, ,(*)Y
9.2. TEMPORAL PART OF MATERIAL DERIVATIVE OF TN
OK
3
3
v—> T—> v—>
r
(t), -
£*,i
(t) "
5
jkUQ
$*-*(t,£,*»«,<"o)
t j=l k=l
d dtj
„(t) Lu>0 J
213
(9.45)
Proof According to Lemma 9.8, (t) d ^w^^^'^')
V V U
2^2^i 3 t
fit. J
,-,,. c (t)\ G(t,£Mnt,w^)
OT
i=1
3 ^ v ^ ^ (t)_9_ £t,fcm - K 3 ^ ( t , g t , i n t , W ^ ) J C>\ *3vl>f+ ^ n(t)/«.,i»l-« *(tA,™.,<')>0 2m 2-f^-'fa'i at< ^
=
+ m
3 K 3
V ^ V ^
(t)
I g t ' i n f _VW+ * ( t ,*\£ . n. t,.,(*)' ,^)
+ln2 + 21nK
M
r£t,kin
- ^^(t.gt.int,^)
d
*• ^fe" ^^S
+
£ES>M t^j=l
+
J
(t)N
^t,int - ^ ^ ( t . f t . i n t , ^ ) ^
9
£ES>?° 2m V f e ' 3«3
££2m V ft ~
(t) j
3K
3
[£,fci»-K 3 *(t, £,<„«, W^)]
t)T oTJ^t.tnt — « ^ ( t j f t . i n t i ^ o(t)-v)] 3 £t,int-K *(t,£t,intAt))dt*
£t,kin — £t,int 3
$_ t)
(£t,int ~ « *(t, £ M „t, 4 ))2
^
(t)\ [ £ t , i n t - K d * ( t , &,«„«, 0 # ' ) ]
3
2 m +^2.2^} Vfe '
3V 2m
(t)N
£t,int-« 3 *(t > £t,int,W? ) )^ £t,kin 3
,St
£t,in
[£t,fcin - « * ( t , f t , i n t , ^ ')]
d
A,
214
CHAPTER 9. LOCAL STATIONARY LIOUVILLE EQUATION
3 «3
3
(t)
hg
+ 2^2^. t
(%Fi-*(t,ft,<„t,4t)))2
j=i
3K3 +
in
^-f^ ' £Mpi _ " ^ 7 t * .
^-^ (t) ^ ?
(t) J
" j£
3/c3 ^
~^s
?r-
)
(t)
a
y^Stsnt,^) (t) U),
r ^
^ri-*(t,£t,<„t>4t))3*j
ft.fcin
(t) L w,
9tj
1
3K 3
ft,,
a
(% -*(t,^,
K-3
(^ri-*(t,^,i„t,u;
!£
(t)
-t,fcin
(t)x 9f.
(t) v-» (*) "T^12-+ ^(t,^t,mt ; ^o ) - 2 - ^ -
+£E4"E fe - '
Mt^intiuP)]
a
V\^..(*)
2m ^
dtj
£t,int
(t)
L W,
^
^
"2m EV E £ T^J ' T (%*!£^-*(t,ft,tat,4t)))a«?) ^ 3 ^
V
f
(t) ^
+ ^(t.gt.int,^) ~ 2 % ^ ^(t.gt.in*,^)
^
£Lti- a..(t)
3K 3
2m
£t,kin 3
£t,i *
d / l ^ ° A^ 3 (Etjju _ ^ , . - . , J * ) ^ a t ,
E^'E^'T^ 3K 3
2m
9
Ei" ^"EPL- f' x^-r £r- -* (^t , ^ , i „ t , 4 t ) ) ^
£t
^ i - $ ( t , g M „ t , g , ^(t)N )
^(t,^,^) (t)
w,
1
9.2.
TEMPORAL
PART
Q
OF MATERIAL
3
3
3
OK
^-™v ^ - ~ v ^ ~ ™ \
2m
1—I2^I2-^I
t
DERIVATIVE
OF TN
r, .(*)
>(t), . ( t ) j Wk
, U £t,int
(t) •w,0 J
^{t,£t>inUJ^)dtj
j=ife=i
215
-^ «<*> &,£>
3K
^ V f t ^ 9i'3/t 3 2m
Et w
(t)
3J "K3
3
V J j=l
V^
^t.fcin.
t}
^t,tnt
~^
^~
(-£3
W(t,t t ,int,U> 0
(t)
j-
(t) V ^
r f t^ - v f ( t , 5 M !L (t) w, ;j"wtJ £\
n t
, ^(th )
9
(t) u.
2m
3
3
+^:EEE t
u
^j
k
lt)7 ~ ^ .%pi-¥(t,&,iBt,a;W)
j=\k=l
fcW
(t) o
9 at,-
r, .(t)-i Lw 0
J
In proving t h e last equality, we have used a summation by parts. T h e proof of t h e L e m m a 9 . 1 1 is completed. L e m m a 9.12 3
3K3
m
3
3
3
3
EEE t
j = l A:=l
3
+—EEE-F—
(t)N
^
9 £t.
^(t.ft.int.a;^)
L
) \ 22 , A' ,(t) /,w A( tl M l j ) uk
r
W n J M
( t )
£t,i
(t)\ %r*-tt(t, £,<„«, off')
r U fi (%*• - ^(t.ft.int.w^)) 2
9tfc
I-
(t) (9.46)
Proof
According t o t h e L e m m a 9.9,
3
3
VW l ^ l ^ l ^ t j=lk=l
^W j k w
(t) 0
a a^t,^,^) \Qt V
fc
,,(t)N r ( , F "(.tiCt.int.Wo J
216
CHAPTER
3
3«a m
+
t
LIOUVILLE
, ,(t), ,(t)
3
j=ifc=i
("fW? t]
w (t)
EQUATION
du>( t ) \e^L_nt£tinuU)(t))J
d (£t,int
4
)
, ,.
mk
(t)A
c
(^-*(t,^,«„ t ,^>))a^ 3
t
9. LOCAL STATIONARY
3
j=ife=i
(*h f * $ * - *(t, £*,**,"£*') ft
t) 2 t)
a
+M ) 4 -
(t)
w,
The proof of the Lemma 9.12 is completed. Lemma 9.13 (t)y
£
1 JL, 9 ^(t)G(t,£:t,mt,Wol;)
&t.in
t)
*(t,£t,intA )
2/c ^KT ^ m
(t)
+
w,
V-5—
(t) ^-> a
J
La>0
E
00 N
^-«(t,A,»j,<')
(t)_9_
(t)
•*(t,^,i„t,wiv;)^t
w,
(9.47) Proof
According to Lemma 9.9, Oh
El 2K 3
m
£
£,«
-^(t.^nt.W^)
^ - ^ t , ^ , , ^ )
a ^9t
9
- £t,i„. _ V I y / -
(t)
£•
(t)v
9.2. TEMPORAL
2K 3
m
PART OF MATERIAL
(t)
EE t
dt
j=l
OF TN
217
(t)
w.
(^-*(t,£t,«n«,o;S t ) ))
t)
j
%sr-*(t,£tlin«,4 )^ 2K 3 V-^
(t) v-^
t
.7 = 1
,(»)
2 K 3 -T-^
+
DERIVATIVE
•E-f
7 7
d
(jj
(t) ,(t)
3
_9_
(t)\
r^t
^-fr(t,gt,int,g,n (t)
w,
The proof of the Lemma 9.13 is completed. Lemma 9.14 1 t
3
,,(t)
3w(*)+2(^-
3 ^ (*> i=i ww o
(t) \ *(t,^,i„ t ,«r)
d
u,it'>G(t^£t^t,^ot))
d \dU
G(t,£ t , iBt ,4 t) )
3 ^ + 2 r % ^ - *(t, 5Mnt, 4 4 ) ))-.
~EY,^
f£t,in
(^-*(t,5
t f=i
,(t)> 2
M n t )
^))
%rt-*(t,£t,
(9.48)
(t)
at.
w,
Proof According to the Lemma 9.10, 1
3
(t)N
M(t)
9 ^>)G(t,5 t ,mt,Wo ) , ( * ) •
•* Z^ (t)
K3
3
,,(t)
( £t,int
m 2 - Z - , w(t) 3 ^ + 2 \ t z=i o
a 9
( -SrES> t ;=i
K3
_
4
^(t.ft.int.W^)
(t)
W,
*'^-*(t,£t,i„t,4t))
(£i^i_*(t,5Mnt,w(t)))2
218
CHAPTER 9. LOCAL STATIONARY
£-t,int
lT,/.
LIOUVILLE
EQUATION
(t)\
c
dtt
3_ , ,(t) ^ f
+ 2( % i - ^ t ^ t . i n t , ^ ) ) £t,i
(t)N
^F i -*(t,A,
m
a*;
r^J** + 2( % S r i - * ( t , £ t , i „ t , 4 t ) ) ) - |
^EE-W m t
9w,(t)
(f ££%,int ^ i _ ^ ( t , 5 t i i n t ) W W(t)\) )
(=1
ft
M ^ - * ( t , £ M n t , <(*)\n ')
'9t/
(t)
w.
The proof of the Lemma 9.14 is completed. Lemma 9.15 a; 2m .j = i
t
Proof
(t)
(9.49)
(t)
^
t
•w,
j=i
Evident.
Combining the equation (9.40)2 in the Theorem 9.1 with the Lemmas 9.119.15, we have the following Theorem 9.2 The expression (^5^)2 is of the form
(*)) a*. 3
3
+EEE t
j=i
k=i
(t)
^,,
£t,fcin_
t)
*(t,ft, i „t,4 ) _ ft.in
+ 3-SjkWo
5 . J (£^i_^(t)5Mnt)W(t)))2^
E^E-?
(*>\^,..(*>.
9
u
k
dtj Lw0 J
St.i
'•£?•-9{t,£t,int,wj>')
(tK
(t)
u.
(9.50)
9.3. SPATIAL PART OF MATERIAL
DERIVATIVE
OF TN
219
Proof By virtue of the equation (9.40)2 of the Theorem 9.1 and the Lemmas 9.11-9.15, we have 3
o r.r.,. ^
(^),~™«>{?£^£ t
Wn
j=ifc=i
where Atj,Btjk
J
t
n
(t)
w,
}•
(t>
'at*
0=1
,(t)
tt(t,£M»«.<"o )'
w.
and Ct,j denote three coefficients, which are the sums of the
corresponding coefficients on the right hand sides of the equations (9.44)-(9.48) of the Lemmas 9.11-9.15. Precisely speaking, they are of the following forms respectively. A
3
_
« 3 , . (t)..(t)
2m ° i 3/f3 Bt,jk
= —
(9.51)
Zp-Vit^A*)' (t)
^m
(9.52)
+ ?Sjkub 3
£t,i
^-^(t,£tMucj^)
and 3£MH2.
3K 3
^ - 2 m ^ ^
+
£L^
(^-»(tl^t">W?))> (t) 3UJ
J
m
(^!£-*(t,^,int,wSt)))2 %=
2m
o
i
(
(9.53)
£^_*(t)£Mnt;U;(t)))2-
The proof of the Theorem 9.2 is completed.
9.3
Spatial Part of Material Derivative of TjN
The turbulent Gibbs distribution T^ is of the form:
^(••^^^E^^^f-E^i^EE^ix^-x^i)),..; \
1=1
V
i=l
1=1 kM
'
(9.1)'
220
CHAPTER
9. LOCAL STATIONARY
in which the expression Yli=i ^2kM ^(l x k able X<s) of the point inside (Ca)N-.
Ai xx/ '
v
—x
j
LIOUVILLE
EQUATION
I) depends on the position vari-
The meaning of the expression
1=1 kM
'
is rather confusing. Maybe the following definition for it is the most reasonable:
^xi
K =
1
V
1=1 k^l
'
S x ( « ) e c t ^iyit+ei) 1
j - K " E x (*) € C t ^(yfc „ 0,
~ x i ( t ) - «ej)> t_ei)
x (t)
- ;
J
if s = t + ej;
+ «e_,-), if s = t - ey, otherwise.
Having made the above specification of the expression N. ^xl
X
(=1 kjtl
and taking account of the Third Proposal of Gross Determinism, we have the following equations N. ^xi
x
1=1 k&
ft)^(t,St,int,oJo'),.<*>
2K;"
0,
if s = t + e j or t - e,-;
(9.54)
otherwise.
Now we are ready to get the expression for the spatial part of the material derivative of Tjv. Theorem 9.3
The spatial part of the material derivative of the turbulent
Gibbs distribution TN(Q.;t) (6.20) is of the form
E t
v-^ x "(t)6 C t
(t)
ATjy
3
v ^ / ^—v ^-* fc=1 v
* x^ec.
(t)
ATN
Ax{*>/*
(9.55)
9.3. SPATIAL PART OF MATERIAL
DERIVATIVE
OF TN
221
where
(E E v,<" .f2&) =E{E^£viw*«> 3 3
i - <..<4) 9
JSP d
+E E
^ ^ t y ^ t ) + *« £ ^ ^ r ? M ^ )
E
+:
3= 1
d J? a 11 _a w r ) r J v ( n;t) + -| E y ^a < i ( , ) r w (n;t)j|, (9.55)!
(E E ^-j$i) x}*'€C,
-E ^EEE(•£> -8' -^w)^vjw°!«» j = l fc=l J=2
^ ( Z - 2 ) | w(t)|2„„(t) HHl %/^ 3 T)
m ^ f
+2K
3
E{-E
N
<
N
< w^-w^u^
i=2j=i+i
VJU
X
;
2
Nt_
Nt,
1=2 j=l + l
| w ( * ) 12^(1)
VJU^T)
*
v ^ t ,=2
^
^i/%-
+ ^|( ( ^-J=CE)} :
^--^JI*.*™
(9.55)a
and
(?£*-^),-E£tf6C.
(*)N
*(t, £,<„«, hff') (t)
W,
(9.55)3
Proof
Recalling that the turbulent Gibbs distribution is of the form:
F(Z,t) = TN(Q;t) = TN(- • -,up,u,['\4B\J'\Jf>;
• • •)
222
CHAPTER
9. LOCAL STATIONARY
LIOUVILLE
=r " ( - " ^ ' S 5 > . ^ ( m £ N a + E ^
'—1
'—1
EQUATION
*(l**-*.l));-;t),
M'l
'
(9.1)' and taking account of the fact that the total intermolecular potential energy in the cube Ct+ej will get an increment £
(t+^o yfc
particle at x[
~ xi
Ke
i)
as
a
sCt+ej
goes from the cube C t to the cube Ct+ej and a similar form of the
increment as a particle at x(( V^
E
VKyL
Y^
( t)
goes from the cube Ct to the cube Ct-ej, we have
ATN(£l;t)
_ ^
v-
(t) J m d
J
j=X
-dJt-Cj)TN(Q;t)
^ytei)
]T
~ x((t) + «e,-)}.
Taking account of the equations (7.16), (7.16)i, (7.16)2, (7.16)3, the elementary identity Aft
Aft
V « ( t ) w ( t ) - w{t)w{t) / , vik vu - wik wij
+ V™"',,)"' + Z^ ^ * " ^ 1=1
(=1
_ #yj m
"k Wn(t)
^ yhjfStMt 2
_
xll(,
,(t) A _,_ ^
/„
+ b-T-RF + E(. ^^m 3 V K 5 " -- *(t,ft.*-,^') ' m< ° )) J + ^vV (fe, y« P
and the Third Proposal of Gross Determinism, we have Y-
y*
2^
2^
w
(t (t))
i
ATN{Nn,i) (n;t) isi •
A(t)
3
_ y^ ^ u
-2^ i
(t) (t)
dP a or9*?
.T,,(Q-t\
N{
''
9.3. SPATIAL PART OF MATERIAL
7= 1 K = l
DERIVATIVE
223
OF TN
"'O
2 3
+
,(»)
(t)\
+ ; 5-V-5*(t,£ t , i B t ,u;^)-2 3
n^/ciTl
E^^w)
3 Aft
+S E E £ (»% • "V - sfw2) i;9^™') j=lfc=l/=2
3
2/t ;
Af»
r
E{-E i=l
*•
i=3
V^T^l)
+EE E v^^U^Mm -5J.-.i,rJV(n;t)
VJCT 7 !)
i=2 j = ( + l
Y, o&r*-^-***) ^y{tej)-4t]
J2
+ ^)\-
(9-56)
It follows from the Third Proposal of the Gross Determinism that 3
E E *
J=1
E «8}4^{»,^>2Wn;*) 4
x<*>6Ct
4
£ t+
V>(y r^-x^-^)
yi ^ect + .,.
(«-«,•>,„ J ect-e,
y fc
^
224
CHAPTER 9. LOCAL STATIONARY
(t)TJV(O, t )
EE^^»^^€ t
EQUATION
(*M
V v(t) —9
—W
LIOUVILLE
^(t^t.int,^)
dtj L
j=i
^ (t)
WW
,(*h (9.57) Inserting the expression (9.57) into the equation (9.56), we have
E E W-^^-t^E^w*') 4
Ax
x{*>€C,
3
3
3=1
,,(t), ,(t)
+ EE^^<*<> 7= 1 K= l
2 +
+
3
^ 0
^-^t.ft.in,,^) j=i
3
1
3oJ
3
- ^ 3 - - 2 V ( t , t t , i n t , W 0 ) + ^3
4
9flwt
p
, ,(tK ,
+SEEEK>-»S> K°
i=l *•
2g
^
^
i=l ""0
--^•|w,|
)
-d Cfc
j=lfc=l (=2
1=3
,(t)
t,int
1=2 i=I+l
wTN(Q;t)
i
v^F7!)
9.4. STATIONARY
LOCAL LIOUVILLE
225
EQUATION
^{t,etMUw^) t
(t)
3
V,
(t)
ATN\
j=i
-tCr, E vi xJ"€C,
The proof of the Theorem 9.3 is completed
9.4
Stationary Local Liouville Equation
Theorem 9.4
The inhomogeneous stationary local Liouville equation deter-
mining the second order approximate solution to the Liouville equation is of the form N,
(s)
z
l
U^f^ ^)
+
Z_* fc=l
a 9x
(s) fc
('
l
>a
9x
(s) <
S=v^[& r - ( '- 1)f '1'i^h^'^ 2 ^ 3 ' <9 ' 58) —
( ^
" T m-J V V
;
8
"«'"'"
2m
*(t,^,i„t,4 t ) ) ,(t)
3
+EEE t
)
i ) ( t ) , >(t)
3 3
3
M
j=ik=i
*,* ^ - ^ t . ^ n t . W ^ )
1 ^t,tin
ft,.
+
-E^T-JW - ' ( ^ -* ^t,^,^,^))^,-
(t)
~&jkUo
3
3*i
LW0
J
%F i -*(t,^,i„t,a;i t ) ) (t)
w,
(9.58)!
226
CHAPTER
9. LOCAL STATIONARY
V
(
3
t
„
LIOUVILLE
^X,
xl"€C,
/
iVt
3
p|w(|2)
m ~ ~73 2 ^ 1 A ^ A - A ^ wfi) • w$ t
EQUATION
I. j = i k=i i -
d
«<*> d
i f f ft(* + i)|w,(t)|at#>
d
AAwf'-iwl^'^^W)
x-a^Tsi^t)
,
(9.58)2
and W3
( E E vp>-^ V
4t
„
^X,
= -EE^>?M^)A t
Proof
-AT+E
/ 3
'*(t, £,*„«, W^) ,(t)
3
j=i
(9.58)3
By virtue of the equations (9.40) and (9.55), we have
E v<•^M = E ( ^ r ) . + E ( E E vJ*' -Axf>A ^)
* x^gc.
xi ,J 6C t
It follows from the equations (9.40)i and (9.52)i that
K
' i
v
t
_
x!"6G
^x;
7
i
The Theorem 9.4 is a consequence of the equations (9.59) and (9.60).
(9.59)
Chapter 10 Second Order Approximate Solutions 10.1
Case of Reynolds-Gibbs Distributions
The present chapter will be devoted to seeking the second order perturbed solution to the Liouville equation under certain conditions. In other words, we are trying to solve the equation (9.4)'. On account of the equation (9.5), (9.6), (9.7), (9.8) and the Theorem 9.4, we are trying to solve the following equation under certain conditions: Ns
(s)
^vtr^wi
s
(»)
*> 1=2
N.
dx (»)
i-i
+ f^m^J^lYl E
L
Eir-a-itf
(s)
fc=l
-£yh
= (Wn + Wl2 + W13) + (W21 + W22) + W3,
(10.1)
where Wn, •••, W22 and W3 are defined in the following equations, respectively: Wu 3
3K 3
2m'
t
3
j=ik=i
, >(t), >(t) £t,i„:
^-V(t,£t,int,U>Z')
(t)n
(t).
+
2^-fc^p
d (u),,(t)
dtAwPJ' (10.1)!
227
228
CHAPTER
10. SECOND ORDER APPROXIMATE
SOLUTIONS
W:12 t
,( ),.,W/£LM S*h W T. - ^ W\\2dt, ) a /^-ttftAint.gff')
2m Tjv(fi;i)2^2j
£t.
(t>
w,
(10.1)2
wr
3K'
r
(t)>
t ,(t), .(t)
n
» = -2sr ^ ;')EE^r t 1 = 1 —^3
; c"'« \
*(,t,tt,in(,W0
U>0
(10.1)a
^E E EU'-^-'fw)
W 21
*t
1<J,*:<3 i<^o t ^ s 1=2 ;=•>
^
'
w (t) a
a
dt k a u — -fc—d<*>T i M ^ t ) + J? N(n;t) dtk
t
l
i=l 1=2
^ w{*> • ( w ^ u f f + 2wW i f f )
(f + l)|wW|» W W
v^3^)
V7(J^T)
5 / ^ ( t , g^int, faff*) t
L e m m a 10.1
J v
j=l
(10.1)4
w
t)
(io.i) 5
(10.1)e
0
If the turbulent Gibbs distribution T/v(f2,£) is a Reynolds-
Gibbs distribution (6.26)' and the temperature field of the fluid flow is not too high, i.e., the probability that the temperature field is very high is negligibly small, (hence, the thermal fluctuations of the mean velocities and mean kinetic energies of the molecules in all of the cubes Cs are small), then
d,t)TN(n;t)
3K 3
(jj^TNJfyt)
(10.2)
10.1.
CASE OF REYNOLDS-GIBBS
229
DISTRIBUTIONS
and 3K3
d,t)Tif(Q;t)Ks Proof
ui^'TN(Q;t)
(10.3)
Being a Reynolds-Gibbs distribution, T/v(£2;£) is of the form: 3
TN(Q;t) = /
3
dcrC(a)exp
W S)
K ^2 &*<" ( 4 - E
Ua
(6.26)'
i>
Hence we have = K 3 I dtrC(
dJt)TN(n-t)
s )
-^u
s j
,
f f
a;]
s )
J
(10.4) and a u (.,TAr(n;t) = - K 3 / dcC{o)$t,a
exp
- «3 ^
&,»• [ ^ s ) - ^
us
j t
^ A (10.5)
According to the equations (6.22)' and (6.22)", we have the following approximate expression for 75^, 1
2m
A, a
to™
JdZexp
Jdz\exp
ft)
f - ^/?S)(T(HS + ^ | u s , C T | 2 -
f - ^
L,-=iK- ) W 2 W.(*)
A, f f (H. + ^ | u s , f f | 2 -
1
5^3 2K :
K3US,<7
K 3 U S , CT
• qs)J
• q,))
5 : *(!*<*> - * j ? | )
(10.6)
On the other hand, the equation (6.26)' can be rewritten as follows,
TN(Q;t) = J dalC(a)
x exp L
-E^(fEiv|s)i2-mtEUs,^+iEEMs)-x|s)))l} S
V
1= 1
j = l 1= 1
k=llyik
'1'
230
CHAPTER
I
10. SECOND ORDER APPROXIMATE
m
daC(a) exp
SOLUTIONS
E A* (E X>S - u° i. *)2 - "• X> * *)a)
-*EA..£5>(*i*
>h
(8)
fe=i /^fe
= / daD(a, X) exp [ - ™ £
/?,,„ ( £
f > < " > " «• i, * ) 2 )
(10.7)
where D(<7,X)
C{a)exp
j E A.- (*. ]T> ,,.)2) - i E A.. E E *(*(») 4 S) ) A:
S
^
7= 1
'
S
fc=l
l=tk
(10.7)! According to the equation (10.7), for given fc,s and X, the aggregate of the random variables Vi'k, 1 < I < Ns is subject to a distribution, which is equal to a linear convex combination of spherically symmetric higher-dimensional normal distributions. Each spherically symmetric higher-dimensional normal distribution is a cartesian product of the same one-dimensional normal distributions. In other words, the random variables subject to the spherically symmetric higherdimensional normal distributions are independent random variables with identical distribution. If the parameters j3a
(s)
i
L N,
U),(•)
N
>
E«
(10.8)
Us k, c
(=i
holds in the probability measure with density C exp
"-^EAX^-E*.^)' L
s
V
j=i
J
10.1. CASE OF REYNOLDS-GIBBS
231
DISTRIBUTIONS
Precisely speaking, for a macroscopic infinitesimal e > 0, the probability (with the above density) of the event that
(•) w.
-
Us
k,a
>e
is negligibly small. Hence any integral of the integrand with a factor of the form (•)
W:k
,(s)
-
" s k, a
exp
K 3 £/V("i S) -i>^ S) ) s
i=l
^
(10.9)!
'
is negligibly small. By virtue of the equation (10.6), we can prove the following assertion in a similar way. Any integral of the integrand with a factor of the form 2m (Stjnt I _ \b(+ y,. F...... F ,.^\
flt,<7
,(t) \ 3UJ,
x exp
-l
«P))
*3
\
(10.9)2 L
s
^
j=i
'
is negligibly small. Taking account of the assertions (10.9)i and (10.9)2, we can reformulate the equations (10.4) and (10.5) as the equations (10.3) and (10.2) respectively. The proof of the Lemma 10.1 is completed. Corollary 10.1
Under the condition stated in the Lemma 10.1, we have W21 « 0.
(10.10)
Corollary 10.2 Under the condition stated in the Lemma 10.1, we have W13 + W3 « 0. Theorem 10.1
(10.11)
If the turbulent Gibbs distribution T/v(fi,t) is a Reynolds-
Gibbs distribution (6.26)' and the temperature field of the fluid flow is not too
232
CHAPTER 10. SECOND ORDER APPROXIMATE
SOLUTIONS
high, i.e., the probability that the temperature field is very high is negligibly small, (hence, the thermal fluctuations of the mean velocities of the molecules in all the cubes C s and the temperature are small), then the local inhomogeneous stationary Liouville equation for determining the second order approximate solutions to the Liouville equation is of the form N
*
(*)
'"
I"'"1
a
^
d
a d
-(/-I)
ax<*>j
+
£W(Z-1)Z
= Wu + Wl2 + W22 3
3K3
3
^m" 7Mft;*)£££
3w3 "2m
w
3
2S
u
k
St.i
(t).
^-*(t,£Mn^n
t
j=lk=l
3
, • (*), .(t)^£t,»l.t
£^tin\
+
jkUp
Q / £t,int
(t)
u
k
, T . / i £•
\
,(*)\\
1 m v-^V^ t
i=l
xAaw«rN(n;t). Proof
(10.12)
This is a consequence of the equation (10.1) and the corollaries 10.1
and 10.4.
It follows from the L e m m a 10.1 that the equation (10.12) can be rewritten as follows:
10.1. CASE OF REYNOLDS-GIBBS
233
DISTRIBUTIONS
T h e o r e m 10.2 If the turbulent Gibbs distribution T/v(fi,£) is a ReynoldsGibbs distribution (6.26)' and the temperature field of the fluid flow is not too high, i.e., the probability that the temperature field is very high is negligibly small, (hence, the thermal fluctuations of the mean velocities of the molecules in all the cubes Ca and the temperature are small), then the local inhomogeneous stationary Liouville equation for determining the second order approximate solutions to the Liouville equation is of the form Nt
rl-l
(t)
d
ft£>
d
dx?\
l-l
Aft
1=2
-(1-1)
my/{l - 1)/
£#>-(*-!#(*) it=i
aw}"}' (t)i
2m
t
j=l
^(t.^.int.W^)
k=l
+
2^jfe^o
dtAJVJ
.(*)
Tl^f^-^t.ft.int,^))2^ (t)
3
w, t
i=l
*
V
^ ? ^ -^(t.ft.int.W^)
f>(/ + l)|wW|»WW * « w j ' M w j ^ f f l + ^ W ) •^ . /rm~i\ 2L~i 2—, ^W^T) ' f^jrl VW 3 !) l=2
(10.12)'
Since Reynolds put forward that the turbulent flows were random solutions to the Navier-Stokes equations with random initial data, it is natural to assume that the turbulent inviscid flows are random solutions of the Euler equations with random data. Hence it is natural to assume that the molecular distribution on the iV-particle phase space corresponding to the turbulent inviscid flows is a Reynolds-Gibbs distribution. According to the argument immediately after the
CHAPTER
234
10. SECOND ORDER APPROXIMATE
SOLUTIONS
Definition 6.2, although the above assumption has not been rigorously justified, but it is very likely to be true, at least, under certain reasonable conditions. Hence, in the remaining part of the book, only the JV-particle system obeying the Reynolds-Gibbs distributions will be considered.
10.2
A Poly-spherical Coordinate System
Now we are trying to get an approximate form of the solution to the equation (10.21) with under-determined coefficients, using the classical orthogonal expansion in higher dimensional spherical harmonics. We shall introduce a polyspherical coordinate system in the subspace of the phase space spanned by the random velocities of the molecules in the cube Cs, i.e., the relative velocities of the molecules with respect to the mass center of the molecules in the cube C s , which represents the position of the fluid particle: (s)
(s)
.
(s>
•
(si
.
(s)
.
n(s)
.
/)(s)
a(s)
•
w2{ = r^s> sin
w
w^i = r
(s)
W
N 1= (s)
w22
(si
a(s)
"11 1
a( s )
• • -sinflji- c o s t ^ ' ,
/i(s)
cos6*^ _ 2 x ,
(s)
n(s)
•
•
/i(s)
•
sin
fl(s)
"12 >
=
r
(s\
•
(s)
Smi
f2
s
= rr(K s >' cos
(s)
K N.3=
r
q(s)
• ( * )
= rr(v s)' cos y?(») 2
i> w.33
(s)
(s)
•
s
W.i23)
•
s\n
1
/](s) Sln
» 2 cos sin
(s)
W
(s)
' sva.6N'_2
s
w.32
N,2
•
sin<^ 2
•
= r("> sin ip2 ' cos
(<0
W
(s)
•
sin<£>2 ' smip\
fsl
r
(s)
.
(si W
(s)
«*¥>2
(s)
COSi
Pl
!
n(.s)
COse
N.-2,2
»
sin Off _ 2 3 • • • sin3 (») # 2 3 ; sin3 (s) #13 > sin 03(s)^ _ 2 3 • • • sin9(<0 023 cos9(8) 0\3 ,
(s)
COS0 AT -2,3 : s
(10.13)
10.2. A POLY-SPHERICAL where w^, v
COORDINATE
235
SYSTEM
i = 1, • • •, Na; j = 1,2,3 denote the variables denned as follows:
/ «,« \
( *i? \
U,(8) 12
12
w<«> =
Av ( 8 ) ,
— «/is N.2
JV S 2
13
13
?//S)
1
J
L
/
Bs
«^*s —
0
V o
?;(s)
0 Bs 0
(10.14)
-
0 0 B
(10.15)
and 1 ,/7vT
1 ,/JvT
1 ./XT
l
l
I
-1
V2
V2
0
0
0
0
0
-3 •/12
0
V(W.-2)(AT.-1)
0
1
1
-(JV.-l)
y/(N.-l)Nm
V(W.-1)N.
/
6s =
1
1
-2
V6
V6
V6
1
1
1
>/l2
vT2
vT2
^/(iV.-2)(iV.-l)
V
(10.16) and JV. r(s)
3
=
(=2 j = l
1/2
2£s m
1/2 ^^•s,/ci7i
m
^s,pot
m
1/2 ^£-s,heat
m (10.17)
^8> ^s.feini ^s,Pot and £s,heat denoting the total energy of the molecules, the kinetic energy of the macroscopic motion of the fluid particle composing of the molecules,
236
CHAPTER 10. SECOND ORDER APPROXIMATE
SOLUTIONS
the total intermolecular potential energy of the molecules and the heat energy of the fluid particle composing of the molecules, in the cube C s , respectively:
£
» = f E i > ? ) 2 + \ E E M s ) - *,w) - *?J;\ £.,«n = f E ^ f f ) 2 '
(10.18) (10-19)
J'=l 1
JVs
£..,»t = ? E E ^(4 S) - *ls)), 1=1 k^l N. £s,heat =£s-
Sa,kin ~
(10.20)
3
£
s,pot = "o" E E^tf 2 1=2 j=l
^ ^
(10.21)
In the sequel we would like to introduce the following quantities as a new set of independent variables in the phase space for the system of molecules in the cube Ca:
£a, « , « , « , « , «,<•>, ^ , e<->, e< s ) , e< s ) , and x<s>
(10.22)
as a new set of independent variables in the phase space for the system of molecules in the cube C s instead of the ordinary variables V ^ , X^s^ in the phase space, where €s = K 3 W < ' \
$<S>
= ( ^ <^s)) and 9< s) = (0<'\ • • •, 0#_ 2 ,i)> i = 1,2,3. It
is a consequence of routine calculations that the Jacobian of the transformation from the ordinary variables in the phase space into the new variables (10.22) is
d(4'\ ,x{g;yM, ,vg) a(xi'),..-,xW;£.,wi,)l$c))e(1'),eW,eW) I (s)\3N - 5
=
(r
}
'
/
s i n 2 " ' - 3 v4 s) ( cos ip2s) sin ^ \
\ N.-2
cos ^<s) J '
3 JV9-2
f [ [ J sin*"1 6>g>, j=\
i=2
(10.23) where r^ is denned in the equation (10.17).
237
10.2. A POLY-SPHERICAL COORDINATE SYSTEM
In order t o seek a solution t o t h e equation (10.12), we t r y t o solve t h e following six equations consecutively:
(
Nt
r'-1
(*)
r)
r)
+ 1 -Tff-n, [£> - (i - wH • A W ' m
l=2
V(l-1)l\-k=i 3
3K3
-1
3
w(t)a;(t)
dvf\>) /w(t)\
d
(10-24)l
-hh
2m
N
*
r'-1
,.,(*)
-^ dx VlSv^^)l £ fe=l A:
(t)
(/-I)
«*<«,
OX
1
t
r
* L*
E{E
<• 1=2
t
„,<*) W
v^T)7
fc=i Z-l ^
a dx(t)
K w
o
7
(i-i)
ax:(*)
I
•}*-3
3
T,v(fi;0 53 51 2m
, •(*), , ( t ) £ t , t i n
f
° '
^
K
(t)
/ ft.int
JW+
C-
,,(*)•»'
,00 (10.24)3
238
CHAPTER
10. SECOND ORDER APPROXIMATE
SOLUTIONS
i-i
Nt
(t) + ^£m ^ J - l J U ^EtfMi-itf I
- ~2m^(n,*)2,1,
(£
(4)
9w,
*S
(t)
_ _ ^ 5 t i n ( ) w ( t ) ) ) 2 « , (,
^t,
J• (10.24)4
(t)
C ^t
ri-l
Q
5
JVt
i
-(«-D
i-i
9
(t) + i =E2 mVO-l)U f cE= 1t f ' - d - i t f
(t), ,(t)
3/^ 2m
9
iF|s»
/^-^(t^nt.W^) (t) UJ,
rWt
w, ( t )
r'-1
*
a£> JV,
-(/-i)
(-1
(t) + ^Em V ? 3 ! ^ Etf'-tf-Df/ fc=l q
9
ax}",
3
,
(t)v2
i
a )
)• (10.24)B
a
ax
(t)
—1F( aw}*> J
6 ) 1
/^-^(t^t,^,^)
2a
r^(% -*(t,^,<„t,^ )) **
,(t)
.
10.3
(10.24)6
A Solution to the Equation (10.24)]
Firstly we are trying to get the solutions to the equation (10.24) i. It is reasonable (t)
to assume that the expression ^r(-fjy) is of the following form:
>D
r^xciMn
,
(10.25)
10.3.
A SOLUTION TO THE EQUATION (10.24)i
239
where C[j and C\j denote the right half and the left half of the cube C t with respect to the j t h coordinate, respectively. Precisely speaking, the set CI consists of the points y = (y 1,2/2,2/3) satisfying the inequalities: (si -
1/2)K
(si + 1/2)K, for i # j ,
and SjK
(SJ
+
1/2)K;
and the set C\ consists of the points y = (2/1,2/2,2/3) satisfying the inequalities: (si - 1/2)K
< 2/i - Ci < (si + 1/2)K,
for i ^
j,
and (SJ
—
< yj — Cj < SjK.
1/2)K
Let Trrx(th
(
Xc
lM^
Xc*,(x$) v
V EiJ'1 Xc* (x, 0
E i = i Xc:,. (x{ ) /
.,.-.„r;-
(t)
^r'(X • " i) -- , „ : -
\
rM. •
<»>^
2 7 ( X ^ ) = ( ^ ' ( X " ) ) , • • •, 75"" r (X(")) = r j ( X ( t ) ) S t T ,
(10.27)!
3j(X ( t ») = ( ^ ' ' ( X O ) , • • • , 7 f ' ' ' ( X ^ ) ) = rj(XW)B t T ,
(10.27)2
Furthermore,
and
where JBJF denotes the transpose of the matrix Bt defined in the equation (10.16). The equation (10.25) can be reformulated as (t) «> // o ; (t) \
~AJ*)
= 2[ 7 7 ( x ( t ) )
«
-^ ( x ( t ) ) 1 *» l ) '
(10 25)/
-
240
CHAPTER 10. SECOND ORDER APPROXIMATE
SOLUTIONS
where / <>
w
\
r(t)j9fc(»(t))=l
(t)
w. w
(t)
/0(tK
^
(t)
(10.28)
^W!!l=i
(t)
(t)N
Since the first column of the matrix Bj is of the form
and the row vectors rj(X<*)) and
T](X(*>)
are of the forms (10.26)i and (10.26)2
respectively, the first components of the row vectors 7?(XW)(= 7^ 1 (X( t ')(=
TJ^X^BJF")
and
should be equal. Hence we have the following
T){X.^)BJ)
P r o p o s i t i o n 10.1
The expression
^©"« is independent of w^.
P7PC< , ,?(xM f>
""
"* '
(10.25)"
Precisely speaking, it is a homogeneous polynomial of
order one in variables w^, w^, • • •, w^
k
with functions in spatial variables X ^
as coefficients, i.e., d (' w
di
(^)=f[^7(XW)-7?(XW)]wW N -1
t t 2r(*)/3 fc($( )) ^ r
JZl_ , ' £ [^'r(X(t)) - W l S j , , (6^),
(10.29)
where sin<^2 sin^i , &(*(t))
iffe = l;
(t) „<*) cos <^i sin y>2 , if k = 2;
cos y?2 ,
if fc = 3,
(10.30)
241
20.3. A SOLUTION TO THE EQUATION (10.24)i and Nt r(t)
= L
3
1/2
2
2 £ t _ 2£t,fcm _ 2£t,pot
EEK?)
m
m
1/2
2£t, heat
1/2
m
1=2 j=l
(10.17)'
It is easy to show the following Proposition 10.2
t
I
The equation (10.24)i can be reformulated as follows:
^i=laxij
1=1 j=\ Nt
3
,i-\
fl { i)fS
±.(An-±
- [ AV-) r ( t ) \ '•»' a
"w \
(*)
, p(t) ^' o
acos
c 5
3
"
(t)
(t)
9 Qr(t)
D ij
Nt-2 3 | V^ V^r(t) (t) """ Z ^ Z ^ npij
acosip2'
3
l+l,j/
(t)
^7w^m '- 'i
ax
!
n=1
( t ) o / j 5 ( t ) \ ATt-1
d
(1)
>F±
ocoso. ( t ) ^ _ T - ; , i r i f ( t ) M - i . (@(t)^
^ " ( " ^ t l j =El fc=lE E ^ S T £ K"(x<'))-7:»(x«)]HfJ,(6i'») (=1
(10.31)
where (t) OTD.V
0, • 2
i f j = 3; (t) • (t)
- cosip\
(t) .
j-iN -
I*- 2t 8in*g>cos«£> l i 2 ,
(t) .
(t)-i-rjV t -2
sin
.
„(t)
i f j = 2; ^(t)
sinfyy cos6\_ l 1,
T=i 2sinfl}3) c o s ^ g , (t) .
= <
(t)
(t) . k
(t) .
(t) T-rAft-2
- cos
.
if j = 1,
if j = 3;
(t)T-rArt-2 •
- cos if 2 sin
.r .
(10.32)
n(t)
A(t)
.r .
„
i f
2
^ C Ocos B ^0 - ;1 j, 2 . , if i j == 2; ; sin 0\ 2 x 2
.
/,(t)
Sln
fyi
^(t) cos
^-l.i.
.r .
lf
..
J = 1>
(10-33)
242
CHAPTER 10. SECOND ORDER APPROXIMATE
cw
_ c w (Q(Q e<*> e ( t ) t ^ i _
SOLUTIONS
wo«>sgnp
9wy
0,
iip^r,
0,
ifp = j , i > n + 2;
- c o s ^ c o s ^ n ^ s i n ^ / xCcos^.^n^sin^n^+iSin^)-1,
if p = j , t < n;
•sin-^^cos^^n^sin^n^+i
8 1 1 1
^)"1'
* P = j ,i = « + 1 , (10.34) (t)
^($(t))n^r2sin^,
ifi = 2; (10.35)
(t)
ft-(* )n£:isin9S««fti.
if
2
*> '
1
aj =
(10.36)
y/(Nt - l)(l + 1)1 and /3 fc ($ (t) ) as denned in (10.30). In deriving the equation, we have used the following approximate equation
^ i - *(t,5 t , m t ,4 t } ) * ^ ( r ( t ) ) 2 ,
(10.37)
which is a consequence of the T h i r d P r o p o s a l of Gross D e t e r m i n i s m . In order to get approximate solutions to the equation (10.24)i, it is more convenient to introduce two new sets of independent variables in the phase space of the system of molecules in the cube Cs r « , «&>, « # , « # , $<s>, 9 < s ) , e ^ s ) , 9 < s ) , G<*> and Y<s>
(10.38)
10.3.
A SOLUTION TO THE EQUATION
243
(10.24)i
and £. , u&> , « , « , w$ , d>(*> , 0 W , 9 2 s ) , e W , G<s> and Y<s>
(10.38)'
instead of that in (10.22), where G<s> =
V ^ ' v(s) (10.38)]
7VS
and w _ U,&B Y--(») = ( y 2 V - - , yOON £),
(10.38)2
/Etl/^-^-i)//;^ («•)
- i - l r(s)
y<
f(8)
E;=f/^-«-i)/i,
\AF^)
(10.38)3
VwSV
vsi=i//3s)-(i-i)/i3s)/ The Jacobian of the transformation from the ordinary variables in the phase space into the new variables (10.38)' is «
(<0
• x(s)-v(s)
• v(s))
a(G( s ),Y( s );5 s ,w[ s) ,$w,ei s) ,e 2 s) ,e' s) ) xsin2^-V2S)(cos^2s)sin^s)cos^s)V
(s)\3N.-5 a(G(s),y(s)) n - l ( r W)
9(X<S))
fl II ^"^i?-
m
t10-38)*
The new variables (GWjWj^GW.yW-.yW) may be called the mechanical coordinates of the system of the particles inside the cube C s , because they describe the mechanical interactions among the particles inside the cube Cs. It should be noted that the map X(s)-+(G(s),Y(s)) is not bijective. If we disregard the boundary effects, the exchange of two blocks of particles will not change the new variables ( G ' s ' , Y^s^), at least, approximately.
C H A P T E R 10. SECOND ORDER APPROXIMATE
244
SOLUTIONS
Although the above map is not a bijection, the formula for change of variables still holds, (see, e.g., [52] and [53]). Since the picture of the total intermolecular potential energy is too complicated and the present book will not endeavor to specify the coefficients of the second order approximate solutions of the Liiouville equation, we will not be involved in the mathematical sophistication of this difficult problem. An obstacle in the way of the exploration of the non-equilibrium phenomena of many particle system is the meagreness of the knowledge about the behavior of the total intermolecular potential energy of the molecules inside the cube Cs. Theoretically, we are most interested in the asymptotic behavior of the total intermolecular potential energy of the molecules inside the cube Ca in the neighborhood of its critical points. In order to solve the problem it is crucial to know the asymptotic behavior of the Jacobian
«fx(°)')
> especially, near the critical
points. As far as I know, there is meagre knowledge about this Jacobian. This might be relevant to the difficult problems of sphere packing (see, e.g.,[23]). Sometimes the new variable r^
is more convenient than the variable LJ^ in (s)
(s)
fs)
fs}
( \
use, the turbulent Gibbs distribution in new variables UJQ ,OJ\ ,&), ,LJ\ , r w is of the form: &N(---,U0
,UJ1
,LJ2
,LU3
,r
w
, • • •) — 1N(-
• • ,<^o
>wi
'w2
>w3
'w4
>'••)•
(10.39) It should be noted that for given U>Q J w i > <*4
an
d w3 >
we
nave
For convenience of the discussion in the sequel, we shall introduce the following Definition 10.1 The Liouville operator for the JV-particle system is defined to be
^ = Ef (t) E 1 E/'i(* (t) )««^( e ? , )(E^ (t) t
(.
1=1 j = l
V
i=l
°xij
l-
d
a-(t) ox l+l,j
10.3. A SOLUTION TO THE EQUATION
Aft
' (t) a o r w
-,
i /wt) r
\
3
a
,i-\
„(t)
OCOSy>x
245
(10.24)i
a
OCOS
| Vfr(t) n=l p=l
g
V
a COS 6/„ p '/ J ^
(10.40) The equation (10.31) can be rewritten as follows:
^
EE E ^ P
EV<*«>-WW?')(10.31)'
It is easy to show the following two lemmas. L e m m a 10.2
Any functions, which are independent of X( s ',$( s ) and 6 ^ ,
belong to the kernel of the differential operator C. Lemma 10.3
Let / and g be two functions in independent variables (10.38).
If / is independent of X<s>, $<s> and e<s>, then C(fg) = fC(g).
(10.41)
Furthermore, we have L e m m a 10.4
A necessary condition for solvability of the equation Cf = g
is J ghdZ =0 for any functions h, which are independent of X' s ^, $ ' s ' and Q^s\ Proof
This is a consequence of the Lemma 10.2 and the following easily
verified identity
246
CHAPTER 10. SECOND ORDER APPROXIMATE
SOLUTIONS
Jcf-gdZ = -jf-CgdZ. Since the components of the angular momentum belong to the kernel of the Liouville operator C too. Hence the condition stated in the L e m m a 10.4 cannot be sufficient in general. I do not know whether the necessary condition stated in the Lemma 10.4 is also a sufficient condition under certain asymptotic limiting process. Anyway, it seems to be really sufficient for solving the linear equations arising from Ritz-Galerkin techniques. Proposition 10.3
The equation CF{( i )
m /
t ,-=!*=!
r(t).
LW
/=i (10.31)"
satisfies the necessary condition of solvability stated in the Lemma 10.4. Proof
It is a consequence of the Proposition 10.1 and the orthogonality
of spherical harmonics Ei£(©i) on the unit sphere (see, e.g., Appendix A). It should be stressed that the quantity r(t)
N
E i w i " i ^ j | 3 r a t,int
, ,/
c
, ,(*)
*T( t+, £ t , i n t , W ^ )
is of dual characters. On the one hand, it is a quantity only depending on the random velocities w^,2 < k < Nt, and on the other hand, it can be regarded to be independent of the random velocities Wfe,2 < k < Nt, i.e., only depending on u- , 0 < i < 4. This duality is due to the Third Proposal of Gross Determinism. In order to get an expansion of the solution to the equation (10.24) i, we need some orthonormal systems of functions.
10.3.
A SOLUTION TO THE EQUATION (10.24)x
247
Firstly, we introduce an orthonormal system of functions defined on the interval [0,oo) with the power
p1(rW)=(r(t>)3Ar«-2/'(nfEnd^i') n ^Hr^f^-4 J
3
V s L ,,(•) t=l
-I s#t ••
3 ,.(t) (t)
.
5
-EEf^ ^--^'^'^,^,^;---)), j=\
k=l ( w 0 ) '
(10-42),
'
The orthonormal system Li(r^),
n = 0,1, • • • can be obtained by Schmidt
orthogonalization process. Usually we assume that L\ (r^) is a constant. It is worth mentioning that the orthonormal system L" (r'*'), n = 0,1, • • • depends on the turbulent Gibbs distribution JN\--
•,^o
>W1 ' W 2 '
w
3 i
r
i
, ,
' ) -
J
W V " i
u
0
>W1 > w 2 ' ^ 3 i W 4
i ' ' V-
This is one of the causes of the nonlinearity of the finer /^-functional equation we are going to seek. Secondly we introduce an orthonormal system of functions
{YE($^)}
on the
part of 2-dimensional unit sphere S 2 in the first octant with the power sin 2 "'" 4 V2 S) c o s " - 2 ¥#> s i n " ' " 2 ip™ c o s " ' " 2
(10.43)
or, equivalently, on the square [0,7r/2] x [0,7r/2] with the power s i n 2 " - 3 v4 s) c o s " ' " 2 ^
s i n " - 2 <*>.
(10.43)'
It is easy to see that
y*(*<*>) =
p
where l^ (v?i , y>2
2 [ y 2 f ( ^ ) , ^ ) ) + y 27 ^(^,^))] / •
2Nt-i
ysin
'
(s)
M- 9
v 2 cos" 1
2
(*) • Af t -2
(t)
{wM)
JV.-9 (*)
sin * ^ ' c o s ™ ' ¥>i
) denotes the spherical harmonics on 2-dimensional unit sphere
with indices p and q (see, e.g., [29]). Precisely speaking, Y?(
248
CHAPTER
10. SECOND ORDER APPROXIMATE
SOLUTIONS
where P^{y) denotes the associated Legendre function:
Usually we may replace the index E in the function YE($^
with the double
index (i,fc), 0 < k < I. In the sequel we shall use the (higher dimensional) spherical harmonics E ^ e O , etc,
(10.45)
which was defined in the A p p e n d i x A or [2], [4], [5], [79], [80]. It is worth noting that the spherical harmonics Yffliyr* , <J>2 ) o n two dimensional unit sphere defined above is not simply a special case of the higher dimensional spherical harmonics E£,(©i), because their normalization constants are different. Finally, we need an orthonormal system of functions
{<7J(Y(*))}
on Y ' 4 ' -
space R 3 W t ~ 3 with respect to the power ^(GW.YW)]'
a(x(*>)
.
1
'
It should be stressed that the orthonormal system of functions
{SJ(Y^)}
depends on total mass mNt of the molecules inside the cube Ct. Hence, a more convenient notation for gj(Y^)
rather use gj(Y^)
than
is gj(Y^\mNt).
But, for brevity, we would
gj(Y^\mNt).
The orthonormal system of functions {gj(Y^)}
can be obtained, for exam-
ple, by Schmidt orthogonalization technique. We always assumed that the first element of the the system is a constant: 3 o ( Y ( t ) ) — constant. An alternative of obtaining the orthonormal system of functions {gj(Y^)} to express the function gj(Y^)
as a product
gj(YW) = Jp(YW)
^GW.YW) />j(Y ( t ) ), d(X<*>)
is
10.3.
A SOLUTION
TO THE EQUATION (10.24)i
249
where {/ij(Y^)} denotes any well known orthonormal system of functions with respect to the power ^ ( Y ^ ) , e.g., the system of Hermite functions. It is easy to show that the orthonormal system of functions {gj(Y^)} independent of G ^ .
is
In order to get an exact form of the finer /^-functional
equation it is necessary to get approximate, but explicit, expressions of the functions {gj(Y^)}.
It should be done for given intermolecular potential tp(x - y).
The accomplishment of this difficult task is beyond the scope of the present book. Having introduced the above orthonormal systems of functions, it is natural to assume that the solution F x ' to the equation (10.24)i is a series of the following form:
^ ^ ^ ( . . . i ^ U l M M V V - o E E E E £ ""'"'* t
j=l
k=l n = 0 B J F G H
* ' >•<*•'
><^L JF GH(G (t) )il n \- (t) )3 J (YW)y £ ($( t ))4(©i t) )s 9 G (e( t) )^(er), (10.46) where the underdetermined coefficients V'tjfcBjFGH
can
De
obtained (approxi-
l
mately) by Galerkin-Ritz technique. All the coefficients ' PtjkEJFGVi(G'^), with the only exceptions V'tjitoooooC^'^)' ^tjiooooo (G
a r e un
i°, u e ly determined. The coefficients
) c a n take arbitrary values, because the turbulent Gibbs distribu-
tions, i.e., the functions depending only on w0 ,UJ[ , w2 ><<4 ,u>4 , are solutions to the homgeneous stationary Liouville equation. In the sequel, we frequently abbreviate iptjkEJFGHi^*^)
to
^tjkEJFGU-
Taking account of the equation (10.31)', we have the following Proposition 10.4 For all t , l < k,j < 3, the coefficients V'tjfcBjFGH^* 0 ) in the equation (10.46) satisfy the following equations
£ (t) ( £ BJFGH 0
^SEjFGH4B)(r(t))w(YW)yk(*W)s/(GW)3£(eW)^(eW)
250
CHAPTER
10. SECOND ORDER APPROXIMATE
SOLUTIONS
Nt-1
= -^y/3,($(t>) £ [7}<(XV) - T;^)}-^^),
(10.47)
i=i
where cm
= r(») "
^ i=l
^
/- *
&r ( t )
dx{t)
ox
OX
iq
+
l+l,q
(
S^s(l^- '-^')
(t) (t) [^iq
d ~
, g(t) (t) ' "n
Q
dcos(p\'
Nt-2
9
3
.+ V V r (t) (t) ^ ^_^ Z-f
9cos<^'
a
«P»9 9 COS 0 „ p
n =i P =i
(10.48)
Sometimes we would like to write the operator £(*) in the following way:
*(*) = rct) "g ± mW) aiEk (e?)) £ ± * » +£ -* £ *8} 1=1 j = l
u=2v=l
BW^ + ( t i) Vf «Je^ *J fl-.(t) OT
r r
\
9
°yuv
i=2 Vl\l
Nt-2 3 (t) + B.»J (t)
O COS lf\'
(t)
+ EE4
(t) ' A ^ Z ^ -npij OCOSif^' n=l p=l
Q
*•) j=l
a
(t) , OCOsOnp/.
(10.49) where ®uvlj
-(t^-'s®-)^-"-1'^
(10.50)
Usually we use the following notation 7 / ' r ( G ( t ) , YW) = 7?,r(X<*>), T]'\GW,YW)
= ^'(X(t)).
(10.51)
It is worth mentioning that in the differential operator (10.49) there are no derivatives with respect to G^). Hence, in the equation (10.47), the variable G ^ plays merely the role of a parameter.
10.3.
A SOLUTION TO THE EQUATION
(10.24)!
251
Given I and j , the quantities buvij form a matrix, which is determined by the form of intermolecular interactions among the molecules in the cube C t . This matrix will play a great role in the calculation of the coefficients ^tjfcBJFGH^^)Expanding the left hand sides of the equation (10.47), we have the following form of the equation (10.47):
£(t) ( £
^S£JFGHi(r)(-(t))pJ(Y(t))^($(t))4(e^)^G(e«)^(eit»)
EJFGH 0
Nt-1
r
Nt
( t )
3 oo
£ £ £ £
£
6utf,/%(*«)a I S} I l (eW)#> tjfcBJFGH
1=1 u = 2 » = l n = 0 £ J F G H
xLW(r(t))yE($(t))4(eit>)^G(eW)sJI(e^) Nt
dy£v
9^{t))
3 oo
+£££ £ -—*?< i=2 g = l n=0 E J F G H V H 4
x
%jkEJFGHYE\9
ATt
3
)9J\*
X
J
)"GlH2
)^F\PI
J-HVH3
)
o"7i)
oo
,(n) £££ £ ^i i U„,(t)*1™ ^^')^^') i=2 < j = l n = 0 £ J F G H V H '
t)
t,
^
t,
x4(ei )s'G(ei )^(ei )(4
5
l) Q
acosi^j Aft
3
ATt-2
+Bir-Azv)^($w)
(t) '
»i
Q
(t)
ocos<^2
3
+ £ £ £ £ EE-7T=TW2)^J«5H»(Y«) I = 2 i ( = l n = 0 £ J F G H m = l p = l V H ' *•)
xL<">(rW)y B (*M)C« mp
a ^dcos0™p
sF(e(t))sG(e
252
CHAPTER
= ^ * (
$ ( t )
)
10. SECOND ORDER APPROXIMATE
SOLUTIONS
[ ^ ' ( G W , YW) - ^ ! ( G ( ' ) , Y ( ' ) ) ] ^ ( e f ) .
E
In order to compute the unknown coefficients V't?fc£jFGH>
we
(10.47)'
should compare
the coefficients of the orthogonal expansions of the both sides of the equation (10.47)'. In doing that, we shall use the formulas for the spherical harmonics obtained in the Appendix A and get approximate, but explicit, expressions for the orthonormal system {gj(Y^)}.
Carrying out all these calculations is a great
and difficult task because of the huge number ( « 1020) of the molecules in the cube Ct . Any successful computations must be done after a drastic simplification based on physical intuition has been made. This difficult, but interesting, task will be postponed to a forthcoming publication. It should be stressed that the coefficients V'tjfcEJFGH
m
ight depend on JVt and £t,mt- The mathematical form
of the dependence is crucial in determining the form of the finer If-functional equation we are going to seek. Although we have not got the precise form of the dependence of the coefficients ^tjfcEJFGH of the function F[
on
-^t
an<
^ £t,%nt, the expansion (10.46)
will give us some hints about the form of a finer K-functional
equation, which is valid, at least, under certain conditions, e.g., in case of nearly incompressible flows of almost uniform internal energy. It is easy to show the following Proposition 10.5 The outcome of the Liouville operator L (10.40) operating on a product of spherical harmonics Ili=i —G (®i )
w
^ n Si=i3i
=
°dd is a
linear combination of the products of spherical harmonics rii=i —H (®i )
wn n
^
Yli=i hi = even, and vice versa. Because the right hand sides of the equations (10.47) and (10.47)' are linear combinations of spherical harmonics of order one, hence all terms on the right hand side of the equation (10.46) containing the product of spherical harmonics H ^ ( e * t ) ) S ^ ( 0 ^ t ) ) S j i ( e ^ ) ) with / + g + h = odd should vanish. We have got the
253
10.4. A SOLUTION TO THE EQUATION (10.24)2 following
Theorem 10.3 The approximate solution (10.46) to the equation (10.24)j is of the following form *cV2cA I
pd)
K
.,.(*)
i>N(---,aJ0
(t)
,w1
(t)
(t)
( t
, w 2 ,w3 ,rK
,3/2
).
x
',-••)
EEEE E t
j = lfe = l n = 0
EJFGH
/ + g + fi = e v e n
/ >(t)/ >(t) W^,* V
"
(10.46)' It should be stressed that the coefficients V'tjfcEJFGH^^) depends on the ,W 2 >W3
turbulent Gibbs distribution
SN(---;WQ,W[
depends on 5jv(- • •; w o ^ l
>w2 >w3 J7"^*'; • • •) m
a
> r ^ > ' ' ')• Hence the F[ '
nonlinear way.
It is worth mentioning that the Proposition 10.5 is a simple property of the Liouville operator C, which has no corresponding role for the Boltzmann (or, linearized Boltzmann) integral operator. In other words, the property of the Liouville operator shown in the Proposition 10.5 has disappeared after making the Boltzmann-Grad limit under the assumption of (initial) molecular chaos. This is one of the causes for the difference between the outcomes of the perturbation methods for the Liouville equation and the Boltzmann equation.
10.4
A Solution t o t h e E q u a t i o n (10.24)2
Now we are going to solve the equation N
(2)
CFi
*
,„(*)
{ E y/U^Tji JVt
1=2
Nt
l-l
E A (t)- C - D fc=l
9 x
fc
8 axj'>
l-l
d
^mVTTTl)lL^ fc
lF(2) (t)
(t)
>™^ZM%)
254
CHAPTER
T (n-,t)£ -v/m N
10. SECOND ORDER APPROXIMATE
SOLUTIONS
AT.-l
t} (
V4 r *>A(*W) £ [7*'(x<*>) - ^ ( x w ^ ^ c e W ) . (=1
(10.24)^ An argument in parallel with that in the preceding section will yield the following Theorem 10.4 An approximate solution to the equation (10.24)2
ls
°^
tne
following form
p (2)
r~ _ VK c /
.,,(t) ,,(t)
Vm
(t)
(t) (t).
^
r t
°° "
n=0
V^
i ,/,,(')L(t))2
EJFGH / + g+h=e«en
x^)JFGH(G(t))5J(Y(t>)4")(r(t))^($(t))4(©it))S9G(e2t))HjI(eit)),
(10.52)
where {Z/j ( r ^ ) , n = 0,1. • • •} denotes an orthonormal system of polynomials on the interval [0,oo) with the power 3 dr(s)(r.(s))3iV.-4
S
L
w(.)i=l
J
s/t
(10.42)a
X V ^ S N O " ; ^ . ^ . ^ . ^ . ^ ) ; - . - ) ) .
10.5
A Solution t o the Equation (10.24)3
The equation (10.24)3 is as follows:
t
K
JVt
+
X
1=1 j = l
.,
3
/i-1
i=l
x
ax
ij
(t)_0_ D. O" g r (t)
OX
l+\,i
10.5. A SOLUTION TO THE EQUATION (10.24)3
i
1
r(t) w
d
(A
r
o
d
i RW
a (») OCOSI^i'
*J Q (t) OCOS<^2
3
255
i V" STr(t) L^i Z^, n=l p=l
,,(*>,,(*)> ,
d
npij
(3)
>*i
(t) O COS &np
a
S f%ir* - * ( t , £,<»«, 4**)
2m^^oEE(%r_^(ti5t)mt)4t)))2^
(t) U>,
(10.24)^ According to the Third Proposal of Gross Determinism, the following approximate equality holds JVt
St,i
ffi1 - ^ ( t , g t , t n t , ^ Q,(*h) ^ 1 V ^ I (*) I2 (t) ~ 27Vt ^ | W J
E
w,
JV,
1
Ei
27Vt
-JVt _.(t)|2 -3 = 1 Vj I
,(»)|2
iV t
i=2
(10.53)
Hence we have ^t,int
9
3«i
(t)>
/^-tf(t,£(t)iint ,utfO w,
,(*) s^iivSTxc^xr) -JVt
i„(t)|2.
V« „ < * ) „ , „ ^ ( * ) M 2 I JE ^V^^I
,(t)
JVt
[ E ^ i X ^ C x, (^t ) \) ]
E^iXC^)
i:;:iivriaxcjl(xi,,) + iE;: 1 vrx c i 1 (^)i a -JVt
|„(t)|2.
(*)M2
E&XCJX')]
^ 3 =l^!,lXJ >
r
i
**
LE^lXCJ^x) ' ) , = !
r
JV,
_if^
v-JVt
*TNt
Nt
i
LE£,
-JVt „ ( t ) 2^«=i v « Xc;,(x„W\j
vv ( t ) i
x.. (xi") t t (t)N
xc-J*?') N
A I-* \r * ^lEuiiXcj^xliO
vv
(t)
i
/ (t)N
(t) T=W '
*
(t)>
Xci,(^)
v ( t ) v , fx ( t ) , l
\ / j - 1 (t) =—-wr '
VI
(*h
XclS^')
EuilXc'(Xu')
1
/jVt
^
JV,
—/ fc=j+i
(t)
WL
256
CHAPTER
10. SECOND ORDER APPROXIMATE
'u-l
W
, v^^Vt
(t)\ (*h
l
SOLUTIONS
/
Itu A*)\
'}
J E „ i i X Q , ( x i(t)N )
Iff
Nt
n—T
Xci,(^)
•-
-lV^n>Wi
yy
Af,
(t)
(th
E Z^ciS^')
-JA'-A1},
(10.54)!
where Nt
(*h Xq,(xr)
^ = 2 ^ *„=iXc t ',(xi ; ) v^Af, /
V^T
(t)
vJVt
wt
v^T
_ ^ ^ ^
,(t)
(t)-,
br<W^> ; (t)\ EC:ixci,(xsr )
Aft (t)|2
>ATt
j=2
(10.54)2 and N
*
v
, Cv ( t ) , l
fiZZiXciXxX') 2^„=i(—^-w«
(t)s
^tf=T. w
Aft (t)
vT > + fc=jf £ r^V^3!)*
+,Lfc=«+i ^ (fc _ 1)fc w fc )Xcj,(.x„ J (t)N
ESliXcl.Wn
,(*>
Aft
1 -ALEI
W
(t),2
J=2
(10.54)3 Hence we have P r o p o s i t i o n 10.6
For any given t and 1 < I < 3, the quantity
,(t)
10.5. A SOLUTION TO THE EQUATION (10.24)3
257
on the right hand side of the equation (10.24)'3 is the product of the quantity (*),u ,«<
3
-w«
-TN(n-,t)
2m
' * *
' '(£^_*(t)£Mnt,wW))2'
depending on wj ' , 0 < j < 4, a homogeneous polynomial of order two in variables w\ki 1 < j < Nt, 1 < A; < 3 with coefficients depending on spatial variables X ^ ) . Furthermore, as JVi
AT,
Y^Xcli^) ^ °°
a n d
u=l
YtXcJx^) ~* °o,
u=l
the following two integrals are infinitesimals. Precisely speaking,
r y dz^Tvi^TT-c—^^V-^ >z^z^{si^_n^£tint^)))2 2m .(*),.,(*)*•. . . !,„(*) |2
(10.55)i and 3
o
/
2 m
t
(*)
(*)
^ ( ^ - ^ ( t . ^ i n t , ^ ) )
2
W , , W p , |vwv ( t ) | 2 in| 2 I
?^-^^)/<^I#^
int.aO2 (10.55)2
Proof
The first conclusion of the Proposition 10.6 follows immediately
from the equation (10.54). Now we are going to prove the equation (10.55)i. Let Nt
n(X) = X)xc i l (xJ l t ) ), u=l
Since the integral on the left hand side of the equation (10.55)i is independent of the arrangement of the order of the molecules inside the cube Ct, it does not lose generality to suppose that Ynr (x!M) = I *' x-Cti ^u ) ^ o,
lf l
~U ~ n ( X ) ; if n(X) + 1 < u < Nt.
258
CHAPTER
10. SECOND ORDER APPROXIMATE
SOLUTIONS
An elementary calculation yields that JVt
VJ^T
w
(*)
vi " '
=1 E«ii xcj, (xl. 0
v/(fc-l)fc 1
1
AT, Nt
x
n(X)
n ( X ) V5=T (t) v n ( X ) y^TI w Zvu=l ^ « - 2^fc=2 ~~7/T
(t)
fc
r (t)
J
v7
V
_
k=j+i
V(fc-i)fc
..,._, "(X) viVt 2^fc=n'v-\-Li 1 '
w W
(t)
2
W+ vn^ij* *
n(X)
— V|w (t) | 2 4-—!— JVt
"*
AT,
V^EI W (*)
AA Z ^ l
+-
(t)
— V|w (t >| 2
(t)\ ) u=lXC,',(Xu
, |w (t) | 2 + -—Y\
w
VJ=T. ( t ) V7" w '
V
n(X) £
3=1
n(X)
+H £+ I V I M *
„<*)
Taking account of the facts that for all k > 2, o
3
, ,(t), Mr.
|w(t)|2
T^(%i-*(t,^,int,^t)))2
/
and forfcx^ /c2, / dZ^-TN{n;t)YY
o
'
*•**" *i
fr
0,
we have (10.55)i. The equation (10.55)2 can be proved in a similar way. As an immediate consequence of the Proposition 10.6, we have the following Proposition 10.7 The equation r p(3) -
o
6
3
T to- i\ X^ V ^
(t) ( t ) r
w
°
w
i
t kin
'
At
T^ -w r^A-^-n^^inu^'))
20.6.
A SOLUTION TO THE EQUATION
259
(10.24)4
approximately satisfies the necessary condition of solvability asserted in the Lemma 10.4. An argument in parallel with those in preceding sections will yield the following T h e o r e m 10.5
An approximate solution to the equation (10.24)3 is of the
following form C
F (3) =
/
(*)
(*)
00
(*)
It)
\
SN{---;ux0',u)\',wy,ujy,r^>;---)
3
00
y^y^^y
K
t
L(n)(r(t))
y
( = 1 n=0
EJFGH f + g + h = odd
(10.56) where {Z/3 ( r ^ ) , n = 0,1. • • •} denotes an orthonormal system of polynomials on the interval [0,oo) with the power
*>3(rW)=(rwr-B / (n [ E E W I n W^ (r
J
^
s
L
,
(
.
Js j
) i = 1
t
W(s)N3iVs-4 )
L
x^ t ) W j ( t >^ i f c i n 5 A r (. • • ; « i , ) 1 « i % i ' ) , ^ ) , r ( , ) i •••))•
10.6
(10.42)3
A Solution t o t h e E q u a t i o n (10.24)4
The equation (10.24)4 is as follows:
Wt
+E^
3
E E^-(-D4 >/*(*-!)i=J V ^
-(*)
V
(t)
d
i(t) , J
/ i - l
P^c^t);
acos(/?i
Nt-2
D
3
t) ^'r^y+EE^ acosv?^ n = 1 p=1 , J
(
(t)_9_
o
nptj o /](t) 0 COS 0no
^
(4)
260
CHAPTER 10. SECOND ORDER APPROXIMATE
SOLUTIONS
^ off^gft,^,^) 9 f^-n^stMtA^v
2m
t
i=i
{—^s
W(t,£.Mnt,a;0
,(t)
))*VH\
(10.24)4 An argument in parallel with that in the preceding section will yield the following An approximate solution to the equation (10.24)4 *s °f t n e
Theorem 10.6 following form c
I
(*)
FW_SN(---;w0
(*)
(*)
(*)
(t)
K
3 oo
L^ n ) (r( t ) )y E (* ( t ) )
t
,(t), ,(*)« X(
\
,w, ,u>2 ,u3 , r u ; . . . )
( = 1 71=0
EJFGH
,(*h
HZEJFGHV
(10.57) where {L 4 n '(r^)), n = 0,1. • • •} denotes an orthonormal system of polynomials on the interval [0,oo) with the power
TVrW)=(rwr.-B /• r n r ^ n ^ H n
dr(.)(r(.))3AT.-4
x4t)Wj(t)*(t,£t,i„t,4t))5w(...;^,),a;i,),u;W,a;^,rW; •••)).
10.7
(10.42)
A Solution t o t h e E q u a t i o n (10.24)5
The equation (10.24)5 is as follows:
t
l
JVt
+
v
1=1 j=l
1
3
/ » - l
(t) i=l "^ij
»„.(*> 9x i+i,j
(t) 9 £>.1] Q (t) r
10.8. A SOLUTION TO THE EQUATION (10.24)6
Nt-2 +
+^(t) ( K
C7C0S(^j
ij
a
dC0SV?2
,(t)
n=l p=l
ft)
(5)
(*h
£t,i
,(t)
t int,u^)9ti
—^~
>n
OCOSttnp
8 f^gr* ~*(t, £,*>«. " o )
= -SrW)EE^r ^(t,S , i=i
3
(t) + 2 - , 2_^ npij
3K
t
261
(10.24)5 An argument in parallel with those in the preceding two sections will yield the following T h e o r e m 10.7
An approximate solution to the equation (10.24)5 is of the
following form 2
^-^ t
n=0
EJFGH
(=1
(10.58) where L 5 " ' ( r ^ ) , n = 0,1. • • • denotes an orthonormal system of polynomials on the interval [0,oo) with the power
%(r(t))=(r«)^-3 / (n [ E I W ' 1 n *,,(t), ,(t)
10.8
-,,(s)
,,(s)
,,(s)
,,(s)
dr(S){r(s))3Ns-4
r-( s )-
(10.42)B
^
A Solution t o t h e E q u a t i o n (10.24)6
The equation (10.24)6 is as follows:
£F ( 6 ) =
j2 L(t) "g 1 J2mW) t
1
(=1 j=i
aiEk{Q?) r J2 ^ i=i = 1 OXij
OX
l+l,j-
262
CHAPTER
10. SECOND ORDER APPROXIMATE
Nt
3
,i-\
Nt-2
+r-
d
D,ij(t)
+
,(t)
(t)
\
acos^i
+ B.
Qr(t)
9
(6) >^1
+ n =El p =E * 0 COS $ l
a COS ^>2(t)
3
o
3
SOLUTIONS
a / r %^-^(t,^,
(. ,(*)\2
rfcr(%Ft-*(t,^„t,4t)))25'*
.<*>
• (10.24)^ According to the Proposition 10.6, the quantity Q
/%t-*(t,ft,int,^t))\ (t)
%
w.
is a homogeneous polynomial of order two in wj(t) , 2 < / < JVt. Hence the quantity £t,int
9 f ^ dU
(t)>
-«, < Q ,(t)
(10.59) i=2 j = i
is a homogeneous polynomial of order five in W; ,2 < I < Nt.
It follows from
the theory of spherical harmonics (see, e.g., Appendix A,[79]) that the quantity (10.59) is of the form
£
(r(t))5
cFGH4(e«)^G(e^)E^(e|t))
(10.60)
FGH / + g + h=l,3,or5
An argument in parallel with those in preceding sections will yield the following Theorem 10.8 following form
An approximate solution to the equation (10.24)g is of the
10.9. EQUIPARTITION
263
OF ENERGY
Ff = ^(..,4%fUM t) ,r (t) ;-)EE t
£
n=0
EJFGH
(^(t))2
x 4 " » ( r ( t ) ) r £ ( $ ( t ) ) ^ J F G H ( G ( t > ) g J ( Y ( t ) ) 4 ( e ^ ) H « G ( e ( t ) ) ^ ( e i t ) ) . (10.61) where {Lg ( r ^ ) , n = 0,1. • • •} denotes an orthonormal system of polynomials on the interval [0,oo) with the power
j
3
p6(r<*>)=(rwr- / (n [ E I W I n J
V
L <.)i=i
s
dr(.)(r(.))3W.-4
-I s #t
x^^ArO-.j^.^.^.^.rW;---)).
10.9
(10.42)6
Equipartition of Energy
According to the Laplace asymptotic theory of integrals (see, e.g., [84]), if the minimum value of / occurs at (a:o,3/o), the following asymptotic equation holds as A —> oo : / /
g(x,y)exp[-\f(x,y)]dxdy
2TT
g(x0, y0)[detf"(x0,
y0)]
1/2
exp{-A/(x 0 , y0)},
where det/"(a;o, j/o) denotes the value of the Hessian of the function / at (XQ, J/O)Substituting
1
(
• 2
(t)
(t) .
(t)
(1
— log I sin
264
CHAPTER
10. SECOND ORDER APPROXIMATE
SOLUTIONS
for g, / and A in the above asymptotic equation respectively, we have the following approximate equation for the integral:
Jo
= f '
Jo
2
f/2
d^dip?
( s i n 2 " ' - 3 ¥><*> c o s " - 2 ^
s i n " ' " 2 *,<*> c o s * " 2 ^*>
^ / s i n 2 * " 4 ¥><*> c o s ^ - 2 „<*> s i n ^ - 2 ^
cos*-* ^
7
=2 r/2 r'^p^w^w? (^)^))+Y2fH^\^))} Jo
Jo
x ^/sin2^-2 ^
c o s * - a v4 4 ) s i n W t - 2 c o s * - '
,/TT . /2 / , arcsin W 3(3JVt-2)/4jVt-'yk4' V 34TT
^
Y2f ( ^ , arcsin ^ | j +F 2 ( 2* f | , arcsin ^ |
where / denotes an arbitrary function of two variables. Hence we have P r o p o s i t i o n 10.8
The function
y B ( * ( t ) ) sin 2Aft ~ 3 ¥#> c o s ^ - 2 <4*> s i n " - 2
^
= 2 [ y 2 f ( ^ , ^ ) + y 27 2 '=(^ I ^)] x ^sin2"-2 ^
c o s * " * <,<*> s i n " ' - 2 ^
COB*-2
^
on the the 2-dimensional square [0,7r/2] X [0, TT/2] approximately equals the following constant multiple of the delta function: YE(*W) s i n 2 " - 3 tp™ c o s " - 2 <,<*> s i n " - 2
^
10.9. EQUIPARTITION
OF ENERGY
265
2[y 2 f(^%^^ 2 7 2 V 1 t) ,v4 t) )] /•
2N.-2
(t)
jv - 9
(t)
• AT.-2
(t)
N.-1
(*)
xysin^"* "v?2 ' c o s ™ ' - ^ 'sin * tp\' c o s ^ ' - ^ i
47T
3(3^-2)74^
Y ^ [ - , arcsin \ / - ) + Y2l 2k ( ^-, arcsin \j ^
xs( ip™ - TT/4, tp1^ - arcsin y/2/3 j .
(10.62)
Therefore, the factors Y^S^*)) on the right hands sides of the equations (10.46)', (10.52), (10,56), (10.57), (10.58) and (10.61) can be replaced with a constant multiple of a delta function and all the series on the right hands sides of the equations (10.46)', (10.52), (10,56), (10.57), (10.58) and (10.61) represent probability densities with the submanifold $'*' = (i , <£2 ) = ( T / 4 , arcsin \/2/3) as their supports. Precisely speaking, we have Theorem 10.9 If we consider the phase space of the molecules inside the cube Ct to be the measure space with the coordinates
st, «&>, «,<*>, „,«, * w , e « , e 2 tJ , e « , and x « and the measure density „(t)\3/Vt-5
3
w
«-2
m j = l i=2
an approximate solution to the inhomogeneous stationary Liouville equation (10.12)' on this measure space is of the following form:
F = SN(.---u,^,J^,^\^\r^;-..)J2Gi, i=0
(10.63)
CHAPTER
266
10. SECOND ORDER APPROXIMATE
SOLUTIONS
where G
4?r
EE
'(
(*)
„IA
,„(*)
3(3jy t l2)/4jv/ (^i*' - * / 4 ' ^
° = E
'
N
- arcsin y/2/SJ
Y^ ( ^ , arcsin ^ | J + y 2 u 2 " ^ , arcsin ^ |
(10.63)o
«=0 u=0
G
3
3
i = V ^ E 3 (3^2)/4^(^ ( i t) - */4> <*>? ~ arcsin V^)
00
00
u
EEE E EE j=lfc=ln=0
JFGH
r
xM ^ • ./2\ Y£ - , arcsin A / -
, v-ivf* + KT T, arcsin W -
U—Ou=0
/ + g + h=enen
(t), ,(t)
x wi J 'w
7 ^^S-^H(G (t) )ir(r (t) ) 9J (Y«)s^(e^)E« G (e^)^(e^), (^o ) 3 / 2
(10.63)i G
* = y ^ E
3(3^2)74 ^ ( ^ i t } - * / 4 - ^ - arcsin
E E EE
n=0
JFGH
(rW)2An
3
( 7 ' a r c s i n \ / 1 ) + y 2« 2 " ( 7 , arcsin
«=0t)=O
(n) ^tuuJFGH
G
^
^ 3
( G W ) 5 J ( Y W ) L W ( r ( t ) ) 4 ( e i t ) ) S ^ G ( e 2 t ) ) S j I ( 0 i t ) ) , (10.63)2
HI «—-v
3 = ^ L
47T
/
(t)
3 (3JV t -2)/4^^i
(t)
- ^ ' ^
1—r-\
- arcsin^2/3J
00
EE E EE 1=1 n = 0
JFGH
f+g+h=odd
U=0D=0
-2u 1 F ^ - , arcsin W - + K 2u
n
I ,. ),
arcsin
10.9. EQUIPARTITION
267
OF ENERGY
,(t)
t ) t) s Ot.ki ><^i JFG H(G (t) )^^5 J (Y( ))Lr (r^)4(ei )H G (e^)^(0^) )
(10.63)a
t
^
*E E EE j=l
JFGH
'
Y
iu
n=0
Y2u-2v ~-1 - , arcsin
7 , arcsin \ -)+
u=Ot)=0
f+g + h = odd
X
M:iutaFGHO
J5J( Y
J
"FlUl
^)
J-GI*^
J-H(,W3 J> (10.63)4
G
* = ^ E
3(3iv!!2)/4 ^
- '/4. ^
- arcsin
y/2/3)
E E EEEk(j.--v1)+y-"(i'arcsinv1 f + g + h = odd
^i JF GH(G (t) )5 J (Y (t) )4 n) (r( t ))4(©i t) )S G (e«)^(Gi t) ),
(t)„,(t) „(n)
xr("'u
(10.63)s Ge
= " E
><E E n=0
JFGH
3 (3iv t -2)/4^(y
EE
r2vl
y2-
? t)
i ~ */*, ^
7T _
.
_ /2\
- arcsin ^/2/3J
-._2„/7T I A '
- a r c s i n \ - ) +Y1 2u
«=0i;=0
x4t)(r(t))2^lJFGH(G(t))ffJ(Y(t))4">(r(t))4(eW)=G(e«)^(ef)I (10.63)fl
268
CHAPTER 10. SECOND ORDER APPROXIMATE
SOLUTIONS
with the following restraints: /
= /fll8ooo(G(t>)dG« = /
= / < & o o ( G w ) d G < * > = / e [ 8 o o o ( G ( t , ) d G W = 0. It should be noted that in the expression (10.63) (10.63)i - (10.63)6, the solution to the equation (10.12)' is expressed in terms of the independent variables r ( t ) v(*)-,.(*> , , ( t ) , , ( t ) , , ( t ) *(*) A ( t ) A ( t ) ft(t The corresponding probability density is its product with the following function
n{
a(GW,YW) 3(X(*))
-'(rl^-^^jj2^.!, ml m11
sin
j= l j=2
It is worth mentioning the following point. The fact that the function rikf.M
ft)
2*7,„(t> ,„(*^
YtfW^+Y^W^') :^sin2^"2 ^
cos*"2 ^
s i n ^ " 2 ( ^ cos"*"*
^
approximately equals a constant multiple of the delta function lJ ,(*) » <5( <^ - n/4, tp%> - arcsin ^ 2 / 3 j ,
as was shown in the P r o p o s i t i o n 10.8, is a non-equilibrium version of the theorem of equipartition of heat energy. When
(10.64)
10.9. EQUIPARTITION
OF ENERGY
269
In fact, ft ($<*>) = sini/4*0 sin(p^ = v ^ V ^
= A/VS.
and /33($(t))= 0 0 8 ^ = T V S . Hence we have the following Theorem 10.10 (Equipartition of Heat Energy)
If the probability dis-
tribution F is the sum of a turbulent Gibbs distribution and those on the right hand sides of the equations (10.46)", (10.52), (10,56), (10.57), (10.58), (10.60) and (10.61), then
IK')11 = B^) 2 = IKM j=2
j=2
( 10 - 65 )
j=2
i.e., the equipartition of heat energy holds. Proof By virtue of the equation (10.64), we have ( r ( t ) A ( $ ( t ) ) ) 2 = ( r (t) / 3 2 ($(t)))2 = ( r (t) / 3 3 ($(t) )) 2
=
The equation (10.65) follows from the equation (10.66).
V_L
(1Q
g6)
This page is intentionally left blank
Chapter 11
A Finer if-Functional Equation 11.1
T h e Expression of B2
In chapter 7, the Euler K functional equation (7.19)i, (7.19)2, (7.19) 3 and (7.19)4 and its simplified form (7.19)i, (7.19)'2, (7.19)^and(7.19)^ for the case of null external force were derived under the assumption that the probability distribution on the phase space of iV-molecules takes the form of a turbulent Gibbs distributions (6.20). In the last chapter we got a distribution, which is an approximate solution to the inhomogeneous stationary Liouville equation perturbed from a turbulent Gibbs distribution. The perturbed solution F is better than the turbulent Gibbs distribution T in that it, at least partially, takes account of the effect of the rate of variation of the distribution with respect to the macroscopic temporal and spatial variables. In the classical theory of Boltzmann equations, an approximate solution to Boltzmann equations perturbed from a local Maxwellian distribution will display some transport phenomena, which is ignored in the Euler equations. It is reasonable to expect that the new distribution will generate a .R'-functional equation, which is finer than the Euler K functional equation in that it will display something, which is ignored in the Euler K functional equation. Thus we 271
272
CHAPTER 11. A FINER K-FUNCTIONAL
EQUATION
call it a finer K functional equation. But the result obtained by the perturbation method for Liouville equation displays not only the classical transport phenomena obtained by the perturbation method for Boltzmann equation , but also some new ones, which reveals the intricate relationship among the variations of the density, velocity and temperature of the fluid, including dilute and dense gases and even liquids. The discovery of the new transport phenomena is the triumph of the perturbation method for the Liouville equation. In chapter 7, we obtained the expressions for B^ and B3 under the assumption that the distribution F takes the form of a turbulent Gibbs distribution T/v. If we replace the turbulent Gibbs distribution T/v with the distribution F in (10.63), which is an approximate solution to the Liouville equation perturbed from the original turbulent Gibbs distribution T, the form of the expressions for B2 and £?3 will be changed. Firstly we shall consider the case of B?.. We have proved that B2 = B21 + B22 + B23, and that among the expressions for B21, B22 and B23 obtained in chapter 7, those for B21 and B22 are valid for any distributions satisfying the Liouville equation and only the fact that B23 = 0 is derived under the assumption that F takes the form of a turbulent Gibbs distribution. (It should be noted that the equation (7.10) and the independence of the turbulent Gibbs distribution relative to $' s ^ have been used in the derivation of the expressions for S 2 i and B22- Although the distribution F in (10.63) depends on $( s ) in principle, but the equipartition of heat energy, shown in the last section of the last chapter, can be used to make a detour to bypass the obstacle due to the dependence of F on $ ' s ' ). Thus what we should do is the derivation of an expression of B23 for the distribution F of the form (10.63). According to (7.10), we have the following expression for B23 corresponding to the subdivision R 3 = [jCa:
11.2. THE CONTRIBUTION
273
OF Gx TO B2
3
B23 = - 27 rmi Y,[
N
Ah
'
E E E fdbir (, s)N ^E » «(s)w
dZF(Z,t)exp[4
(s)
lk
= -2*mi£ / dZF(Z>*)«xp[A]X;EE('-("))aStW
£* ( ^y^(eS-)H8,,(ey).
JV.-l X
(n,)
J=l
When we substitute the right hand side of the equation (10.63) for F into the right hand side of the equation (11.1), the quantities Gz,Gt and G 5 will make no contribution to £23-
11.2
The Contribution of d to B2
Substituting the expression ^
E
3
3
- , i ( s ) , , ( s ) , ,(s) , ; ( s ) r(s>i w 0 > w l ' ^ 2 >W3 > r i
3 (3Aft-a)/4
5
(yi t) -
7r 4
/ '^
-^ V
- arcsin v 7 ^ ^
00
EEE E EE j=lfc=ln=0
JFGH
U=QV=0
y22u"( - , a r c s i n ^ / - J + y 2 ; 2 u l - , a r c s i n ^ L
/ + S + h —even
x
7^^^^«H(G(t))4B)(r(t))ffj(YW)4(eit))s«G(eW)sfe(e) (^o ) 3 / 2
for F ( Z , i ) in the equation (11.1) and changing d Z ^ into d Z ( t ' , where dZ(t) and the volume element dZ^
are defined in the following equations:
dZ<*> = sin 2 "'" 3 „<*> ( c o s ^ s i n ^
cos^Y^dZ^
274
CHAPTER 11. A FINER K-FUNCTIONAL EQUATION
s i n 2 " - 3 v4 4) (cos¥><*> sin*,<*> c
^)\
o s
Nt 2
~
-Q^
2
^ ^
^t)
xde^de^de^d^dz^,
(11.2)1
ax(t) dz« = dY^JdGt^d^dw^d^^Jde^de^de^ -a(Y(*),GW) 3
(t)^3Af,-5 x
i!_i
^t-2
1 TT sin*„s„<-i/i(t) nTT n ^ m
,=: i=
(11.2)2
2
and
"2<" - ^ " > j G ( " ^ - ' " i t > a ( Y ^ , G , . > ) (
L •
<"•»>.
and taking account of the equation (10.62), we have the contribution of G\ to B2 in the following form: B?
=-27rmi^^ / \
t
J
dZ^ \{dZ^\[^SN{. ^
„=4t
V
••
•,J*\J*\J*),J*\r^--•
p=l q=l n=0
EE
•)
m
^u ( ^ . a r c s i n y 3 J + YiuV f 4 - arcsin ^ -
JFGH
(W0 ) '
npowJFGHl^
/
u=0u=0
xL^)(rW)5j(Y(t))exp[A]H^(eit))^G(e^)E^(eW)5:5:5:(r(s))2g(s) s
2^ (=1
AT - 1 s
i=i fe^j
^ N , l u j J-N 1 l (J fc J
275
11.2. THE CONTRIBUTION OF Gx TO B2
\ A
t* JiM ** A
u„^J 4t .
V
m
3 u
\
EE
/
p
3
=lg=ln=0
(t) (t) lW0 i
J
L< n) (r( t ))exp[yl](r( t )) 2
) + Y 2 ; 2u I - , arcsin
Y% ( £ , arcsin Jf
oo
«=0 v=0
EEE^^(<^ j = l fc^J (=1
= -87r2m1/V/2iVV^ A
y
t
3 (3A/,+2)/4
w/8)/i(s)/,(s)
Jvr''wo
EE
r2W 7T
1 ) / 2
,3
y d Z iu ^it / Lr((iv•t-i)/2)J t -
/ , W r-W-
> w i >w2 >w3 » r
-
2 7 r W
^
,
3 3 oo ^ P \ ^ , i ^ \ ^
z
(t) (t) Wp
' " • • ) /I,H / ,^ / ,T/ K ^ 7' )i 3 ' 3 p=lq
._ , / 2 \ , ^ _ « /
7 r
*2« ( ^ > arcsin y - J + Y 22„" I - , arcsin
g
B=0»=0L
EE^Ui*(G(t))4B)(r(t))flj(YW)exp[^](rW)2|i(t) at* j = i fc^j
27TK7/2
3mV2 _
(t) 3
,
3
x s i n 2 " - 3 ^(cos¥#>
sin„<*> cos ^ V "
2
iV t -2
£ yB($<*>) J ] J ] s i n ^ 1 #g j = l i=2
276
CHAPTER 11. A FINER K-FUNCTIONAL EQUATION
x
a(Y«C-»)(r'''r"'s"('-^»,'"-'^'"3->.^-v--)i:gJ(^') 3
3
oo
(t)
(t)
3
* E E E fwfc
„,
E E V&^(G(<>)L<*V<>) exp[A](r(*))*(t)
P:
' 3x^/2
x
3
3
A " ^
oo
(t)
t
(t)
3
£
(t)
J
n)
2
„,
EEEf^EE«^(G )4 (^)(^) ||(t) p = l g = l n = 0 (,W0
)
k
j=lkjtj
27T K 1 /2 :
3m1/2
3
- fdZTN(Z,t) */
/ " d x e x p [ y l ] ^ y B ( $ o ( x ) ) ^ 3 j ( Y ( x ;)) */
»p
1
3
x E E l ^ 0 EE^—(x)^0)(-w)(Kx))2^(x), do) where YE(<J>0(x)) = ^£($0 ) denotes the value of the function YE defined in (10.44) at the point $ 0 (x) = ^ „=o 2^v=o
= (7r/4,arcsins/2/3), E = (u,v),
£
B
=
a n d
i>Jpquvjk(x) = Tpjiquvjkix)'
Nt
1 I'jWtvjkW = ] v ^ n
r
Vjpquvljk(K)
~
<
( n - 4 )l
E
^Siuvljk W'
VfiivJN.N.ofr).
if j = 1, * = 2 or j = 2, k = 1;
i f j = l,fc = 3 o r j = 3,fc = l;
I VfiLjON,N, « - if j = 3,fc= 2 or j = 2,fc= 3.
(11 - 4 )2
(11.4)3
11.2. THE CONTRIBUTION
OF d
277
TO B2
In the formulation of the equation (11.3), we have made the convention that the discrete variable t and the continuous variable x always be used alternatively. Of course we always assume that x 6 Ct and limw,- ' = Wi(x),limG^ = x and J • • • dx. = l i m ^2t • • •
K3.
In deriving the equation (11.2), we have made use of the orthogonality of the system of spherical harmonics (see, e.g., Appendix A or [79], [80], [4]) and the orthogonality of the system of polynomials {L^ ( r ^ ) , n = 0,1, • • •} with respect to the power (10.42)i. The equation (11.3) can be shorten as follows: B{21] « f
dZTN(Z,t)exp{A]
/*£E^£B*w>"i^M.
<->
where 2-KK7'2
BSbM
3m 3 / 2
- ^ y £ ( * o ( x ) ) ]Tsj(Yo(x))Vj P 9 O T J fe(x)4 0 ) ,
Y 0 (x) = l i m Y ( x ( t ) )
(11.6)!
(11-6)2
r(t) (for the significance of XQ see the Proposition 11.1), and
i ? (x) = l i m v / | r « .
(11.6)3
Recalling that ATt r(t)
=
Ei
W
(t)|2 1=1
»=2
-3
I, , ( * ) \ 2
w,
Nt 1=1
k^l
1/2
k^l
278
CHAPTER
M" -
11. A FINER K-FUNCTIONAL
E
3
/
(t>N9
EQUATION
-il/2
(8.13)'
(t)
w.
we have
(x) EtiK x O M ' .(*)
(X)N2
i?(x)
Since Lj '(r^) L\ '(r^)
- 2 *
x,w,,(x)
(x)N2
E k M T . .f^ x0 ) (x)
2w,
1/2
• (11-6)3
is independent of r^*', we have simply written L\ ' instead of
in the above equation.
In order to write the equation (11.5) in a compact form, we introduce a five dimensional vector valued function fi(t) defined on the set {C t } and its continuous version f2(x) as follows: fi(t)
, ,(t)
(t)
(t)
(t)
(t)
(11.7)!
ui0 ,u;1 ,UJ2 , w 3 , w 4
fi(x) = (w 0 (x),wi(x),u;2(x),W3(x),a;4(x)).
(11-7)2
In the sequel we just introduce the continuous version of a quantity and omit the description of its discrete version. Qijfc(0)(x) denotes a quantity defined as follows:
Qufc(n)(x) = ±±
^gy^Wx)) 2 BS,(x)
3 3 V^ V^
^2-i
p—1 q=l
(x)
M*' -
E i = i K( Xx) )N ,(x)
2
Ulp(x.)wq(x)
(W0v(X))V2 "
v
(x) 2* x , w r -
(X)N2
TLii»n - . w (x) 2w,(x)
0
BiW ^fc(x).
(11.8)
For a fixed x, Qi3-fc(fi)(x) is a functional in five dimensional vector valued function Q(x). Since 0(x) is a function in Z, Qijfc(fl)(x) can be considered to be a composite function in Z and x.
11.2. THE CONTRIBUTION
OF Gx TO B2
279
The equation (11.5) can be reformulated as follows:
j = l k^j
The integral / d Z T ^ ( Z , t ) • • • in the equations (11.5) and (11.9) ought to be replaced with an infinite dimensional integral (see, e.g., Definition 8.1). For brevity, we will just use the finite dimensional integral /dZTV(Z,£) • • • in the sequel, that is enough for the purpose of the present book. In order to get an expression for B2
in terms of the if-functional, we have
to introduce the concept of infinite dimensional pseudo-differential operator as follows. Definition 11.1
We introduce the notations:
D
D =
i =1 2 D =
- = i& * h^' > >*> * hl-
Q(fi)(x) denotes a function in Q, and x: Q(ft)(x) = Q(wo(x),wi(x),w 2 (x),W3(x),W4(x)), where wo(x),wi(x),a;2(x) ) a;3(x),W4(x) denote the functions in x for the given point Z in the 6iV-dimensional phase space. We define the infinite dimensional pseudo-differential operator Q(D) = Q(£>0,D6l,£>62,£>63,£>c) as follows: Q(r>)K{a,h,c)(x)
= fdZexpiJ
=
Q(Da,Dbl,Db2,Db3,Dc)K(a,b,c)(X)
2™ 3 i ^2
y.
s
a
1
( s ) 4 S ) + Yl M s ) w i S ) + C(S)W4S) 1 J J i=1
xTN(Z,t)Q(4t),4t),4t),4t),rW)
(1110)
280
CHAPTER
= / dZ exp | - 2m I
11. A FINER K-FUNCTIONAL
EQUATION
o(y)w 0 (y) + £)&i(y) w »(y) + c(y)w 4 (y)
xTff(Z,t)Q(wo(x),Wi(x),W2(x),W3(t),W4(x))
= /"dZexp[i4]r A r(Z > t)Q(wo(x) > «i(x),W2(x),W3(t),W4(x)).
(11.11)
In order to guarantee the existence of the pseudo-differential operator Q(D) the functional Q(w 0 (x), wi (x), w 2 (x), w 3 (t), w 4 (x)) should satisfy some mathematical conditions just as in the theory of finite dimensional pseudo-differential operator (see, e.g., [72]). But we would not be involved in the mathematical sophistications of the theory of infinite dimensional pseudodifferential operators. Henceforth the function Q(wo(x),wi(x),w 2 (x),a;3(t),u;4(x)) is called the symbol of the pseudo-differential operator Q(Da,Dbl,Db„Db3,Dc). It should be noted that the outcome of a differential operator on a X-functional is a multilinear functional for given (a, b , c). But the outcome a pseudo-differential operator on the .^-functional cannot be denned as a certain functional, because the theory of fractional functional is unavailable. The quantity Q(D)K(a, b, c)(t) defined in the Definition 11.1 is a function in t for given (a,b,c), which can be formally written as follows: Q(D)K(a, b , c)(x) = Q(D)K(a, b, c) ( J J ®<5(x - y ) ) .
11.2. THE CONTRIBUTION
OF Gx TO B2
281
But we would use the simple form rather than the clumsy one. It is easy to see that the simple form is in accordance with the classical concept of partial derivative, when Q is a polynomial and the functional is a function in several variables. Having introduced the (infinite dimensional) pseudo-differential operators, we can get an expression of the quantity (11.5) in terms of K functional as follows. Theorem 11.1
The contribution of G\ to B2 is
B ]
2
/dx§i(x)QiJ-fc(D)K(aIb,c)(x),
= E E j = l kytj
J
(11.12)
k
where Qijjt(fl)(x) denotes the quantity defined in the equation (11.8). In order to get an explicit form of the pseudo-differential operator Qijfc(D), we have to specify the form of the Gibbs mean ^(t,£t,int,^o
)• Of course, the
specifications of the Gibbs mean ^ ( t , St,int, ^o ) should be deduced from the form of the intermolecular potential energy. But it is not an easy mathematical task to do the derivation rigorously. We shall put forward the following assumption, of which the plausibility will be explained later on. Assumption C
It is assumed that the Gibbs mean is of the form
^ ( t . f t . i n * , ^ ) = K ( 4 4 ) ) \ A > 1, K > 0,
(11.13)
for fixed m, K and N. The constants K and A depend on the fluids. Now we are going to explain the reason for the plausibility of the A s s u m p t i o n C. If the fluid is a dilute gas, the equation holds for K = 0. If the fluid is a dense gas or liquid, the Gibbs mean is of the form
- 5? f L <*«(»*. - EE«4" -!")): 1=1 k^kl
282
CHAPTER
11. A FINER K-FUNCTIONAL
Mt^S
Nt
JC
t
*
^
(=1 k&
' +
EQUATION
Nt
1=1 k^l
At first glance, we would tempt to apply the Laplace asymptotic theory of integrals (see, e.g., [84]) to the expression (6.35)'. But it must be cautious in doing so, because the dimension of the integration domain of the expression (6.35)' depends on 7Vt too. Hence the Laplace asymptotic theory of integrals cannot be applied to the expression directly. But a simple estimation of the integrals occurred in the expression (6.35)' will shed some light on the problem. P r o p o s i t i o n 11.1 point in {Ct)Nt
Let e be a positive number, XQ = (XQJ, • • •, XQ N ) be a
such that
iEE^-^-^^g^EE^i"-x<\ (ii,4) ( = 1 k^l
1=1 k^l
where X'*' = (xj* , • • •, x ^ ) and C be a positive number such that for any satisfying the inequalities x j f - X $ | < C K 2 , k=
(11.15)
l,---,Nt,
we have Nt
££>(x< t >-xj t) )<2 K 3 a +
(11.16)
1=1 k^l
and lim
1
^ ' -,(*) x [ „, (' )* )< 2 ( ! J a +
ATt—oo K°
l/JVt = OO,
(11.17) where fi denotes the 3A^t-dimensional Lebesgue measure. Then we have the following asymptotic equation (as Nt —*• oo) Nt Nt
, E(Ct)'
1=1 kj=l
283
11.2. THE CONTRIBUTION OF d TO B2 Proof
We have (th *(t, £,*»*> <"o)
L/x(t)K--E
1 2^3
V
•'^
1=1
Nt
\T
J JV t — a
2K 3
3N -S 3F*-i-i
N
t
t}
+ f
dXW f2€ttint - £ £ V(x< - x « ) )
• ^
^
1=1
JVt
"'Di
\
1=1
y
fc/i
k^i
x
2
'
+
^
+
AT.
(=i
fc^i
Nt
+• ' ^
(11.19) V
i=i
fc^i
' +
(=1 fe/i
where Di = {xW e (Ct)N< : f ; ^ VCx^ - x((t)) < 2*3a + e ] "-
i=i
fc#i
(11.20),
-1
and D2 = ( x W € (Ct)"< : E 1=1
By virtue of (11.18), we have JVt
1=1
fc^i
E k^l
^
- x«(t)) > 2« 3 a + e}.
(11. 20)2
284
CHAPTER
11. A FINER K-FUNCTIONAL
EQUATION
Therefore, it is clear that
2
f ««(2s tMt - E E *(*{? - xh)
JDl
V
1=1 kM
>2K3a(2£t,int-2K3a-^
'
' +
EE^
-4
1=1 kjtl
/ ^ { x ^ :E E ^ x f - x ^ )
< 2K3a+|j). (11.21)
On the other hand, it is easy to see that Nt
\ i^pi
2
[ dxw (25Mnt - E E M ' - ^)) < 2K3Nt£tlint(2£t,int
Nt
E E^(4 4) - x,(t))
- 2n3a - e )
1
^.
(11.22)
Hence we have ft
(t)
3Nt-5
r «<*> (2*^-££>(*«-*, )) fc#J
' +
2
ft
EE ^ ( x ^ - x . W )
1=1 k ^ i
/ ««( 2 £ Mnt - E E * < & - ^ ) ) ^ E £ ^
j D i
^
1=1 *!#J
<
' +
i = i *;^
-x;
-1
2K3iVt£t>int f2£tti^-2K3a-, 2 3JV 2K aNt\ ( C K ) ' V2^t,mt - 2« 3 a 3
-1
.
(11.23)
Using Stirling formula 7Vt! « s/2-KNt
(^\
and the condition (11.17), we have the following estimate for the right hand side of the equation (11.23) 2£t,intK?Nt 3 2K aNt\(CK2)m*
( 2£t,int - 2K3a - e' \2£t,int ~ 2« 3 a
3N.-S
11.2.
THE CONTRIBUTION
285
OF Gx TO B2
[^^.({x^gg^-*!•>> <^° + f}) 3JVt-5
6&t,intK
2£t,mt - 2K 3 Q - e
e
2K3ay/&fN\N?t(CK2)3N*
- 2K3« - §
\2€t,int
xw
<
2
*{
^TV-
By virtue of the equation (6.12), we have K3N*
1
1 2 3JV
N?* ( C K )
3N
'
N
N * (TCK)6^ '
N?* (CK) *
Summarizing, we have
• ^
^
2£t,int
<
3
\
i=l k^i
' +
A
/
e 3
\
~5
Aft
1=1 kM
Aft
6
2K a v 2iiVt VC iVK /
\(CK2)3
3JVt
Aft
* / 2ct.in*, ^fct.int — - 2 ^ «K"
\2£t,int - 2n3a 1/Nt)
i ,(*) „(*h ^({ x ( t ) : EE^ x "-xrx2« L
*"
V
*•
••an
( = 1 k^l
-Nt
Taking account of (11.17), we have Nt •'^
^
i=i k^i
t ^ * +
Aft i=i *;#;
= 0 (7 rfx(t»(2^-EEMt)-xf))) '2 £$>(*^ - x f " ) V-70!
^
1 = 1 k#'
' +
i=l fc#l
•
(11.24)
CHAPTER 11. A FINER K-FUNCTIONAL
286
EQUATION
Using an argument similar to that in proving the equation (11.24), we get the following equation Nt
L^
v
,D
*
(
j
i=i w N
+ 3Wt-5 >
jD «<*> (2£tMt - £ £ tK*i? - «,(t)))
J•
(H-25)
According to (11.19), we obtain the following approximate equation: 3AT t -5
JVt
* ( t , £ M n t , (th ^ ) = 2^3
N
t
3JVt-5
t)
/ dx( )(2ft,mt-x;^v(xi -xf')) « a
i = T T
2
^
EEv'^'-x^) (i+od))
Aft
min
V V W * * - x, (t) ).
(11.26)
The proof of the P r o p o s i t i o n 11.1 is completed.
By virtue of the symmetry of the total intermolecular potential energy of the molecules inside the cube Ct with respect to the permutations of the molecules the outcome of the permutation of the Nt molecules on each minimum point is still a minimum point. This is the reason for the occurrence of the factor ^-j in the square bracket on the left hand side of the equation (11.17). If the equation (6.9) about the asymptotic behavior of the intermolecular potential energy ip holds, the existence of the constant C asserted in the assumption of the P r o p o s i t i o n 11.1 will be guaranteed. I do not know any simple form of the conditions imposed on the form of the intermolecular potential energy ip, which is necessary and sufficient for the validity of the equation (11.17). It might be a difficult mathematical
11.2. THE CONTRIBUTION
OF d
TO B2
287
problem, because the graph of the total intermolecular potential energy of the molecules inside the cube C t is really a complicate hyper-surface in (3Nt + 1) dimensional space and what we are interested in is its asymptotic behavior as iVt —» co. I think, the problem might be relevant to, but more difficult than, the problem of sphere packing ([23]). I guess, the condition (11.17) would hold for dilute gases and highly dense fluids, especially, liquids. The former is specified by the condition: TVK4 —> C > 0 and the latter NK6 —» oo. And also it is difficult to find out the minimum points of the total intermolecular potential energy of the molecules inside the cube Ct • At the minimum points the derivatives of the total intermolecular potential energy, i.e., the components of the intermolecular forces exerted on molecules, should vanish. A heuristic argument will inspire one to conjecture that the positions of the molecules inside the cube Ct will have an approximate periodic and symmetric structure, i.e., like the structure of the sites of the vertices of a crystal. Hence for the configuration of the minimum total intermolecular potential energy inside the cube C t the — 1/3
distance between two neighboring molecules is proportional to Nt
. Assuming
that, for the distance |x — y| between two molecules belonging to a small interval around the distance between two neighboring molecules, the intermolecular potential energy is approximately of the form ^(x-y)»Co|x-y|-" we have ^ ( t . f t . i n t , ^ ) « CATt • N?/3 = C 1 ( 4 t ) ) 1 + " / 3 . This is the reason of the plausibility of the A s s u m p t i o n C. Under the Assumption C, the quantity Qijk(D)K(a,
b,c)(t) can be ex-
pressed in terms of derivatives or fractional derivatives of the /^-functional as follows.
288
CHAPTER 11. A FINER K-FUNCTIONAL EQUATION
3 D
2
= E E
/
(11.12)/
where 3
'<^HW,n,v»2n(l)
Q.^Xx) = E E fgp^(««) 2 B^(x) 3
3
EE ElMx))2
2w w
ojp(x.)ujq(x)
TLi
29(xuv
B^fc(x)
(11.8)'
and (!)
O
x
_
27T«; 7 / 2 i
= - ^ T E y ^ ( $ o W ) E^(Yo(x))^jw.„J-fc(x)L(10),
)
£
(11.6)i
j
On the other hand, we have 3
Qy*(fi)(x) = £
Qg fc (n)(x),
(11.27)
where
0»w = «E i ^ a BS.w.
<"•»>.
Qi>>W = - E E ^ g H E(".(x))2B™k(x),
(11.27)
3
3
Qg!(fl)(x) = - 2 K ^ ^ W p ( x H ( x ) ( W „ ( x ) ) ^ ^ B « f e ( x ) .
(11.27)3
p=lg=l
Hence
B
3
3
^ = ^EEE/^WQ^( D )^K b ' c )( x ) dx ' j=ifc^j t=i
(n-28)
11.2. THE CONTRIBUTION
OF Gx TO B2
289
where Qg f e (D)tf(a,b,c)(x) = JdZTN(Z,t)exp[A}Q^k(n)(^),
i = 1,2,3.
(11.29)
In order to give the specifications of the three terms on the right hand side of the above equation we need the concepts of the fractional derivatives (or integrals) of the /f-functional. Definition 11.2
If the if-functional is given as follows: K(a,b,c)=
f
dZTN{Z,t)exp[A],
then the fractional derivatives of the K-functional is defined in the following way: DlDl\DllDllDhcK(a,h,c)(t)
= |dZTiV(Z,0exp^](4ty(^t))^(4t))^(4t))^(a;(t))/1 where Da,D^,
Db2,Db3,Dc
(1L30)
are differntial operators defined in (11.10).
It is easy to see that the above concept of fractional derivatives is a generalization of the derivative of integral order. Having introduced the concept of fractional derivatives of the /^-functional, we can easily show the following expressions for the three terms on the right hand side of the equation (11.29).
Qg(D)K(a,b,c)(x)
p=lg 3
3
= ^ECkW^'X^^^^cjfx), p=lg=l
Q$(D)tf(a,b>c)(x)
(11.31)
CHAPTER 11. A FINER K-FUNCTIONAL
290
w p (x)w g (x)
- / a =PM T„(Z, ,) £ £ ^ 0 3
3
EQUATION
£<«•«>*£»W
3
B
fl 3/2 2 fl b c x pqjk » EEE S^) (^) ^^/( ' ' )( ) p=l q=\ t = l
(n-32)
and Q>(3) £i(D)X(a,b,c)(x) 3
3
= -2K /*dZTN(Z,Oexp^]^^ci;p(x)a;(?(x)(Wo(x))A-5B^(x) P=I 9=i
3
=~
2K
3
^ r > W~2D D K{a, b, c)(x). E E B»(!) SS*«^~ bp bq p
(11.33)
=ig=i
Summarizing, we have Theorem 11.2
The contribution of G\ to B
3
„ ^,
3 3
"f-ZE/^wESCw i = i fc^j
p
2D-1/2DbpDbDc
=ig=i
3
-Y,D-3/2(Dbi)2DbpDbq
- 2KD*- i D 6 p £> 6 , tf(a,b,c)(x)dx,
(11.34)
i=l
For brevity, we introduce the following formal derivative Dh of the functional K. Definition 11.3 The (formal) derivative of the K-functional with respect to heat energy is defined as 1 3 £> fc A'(a,b,c)(x) = D c A ' ( a , b , c ) ( x ) - - 5 ] Z ? - 1 D g I A - ( a , b , C ) ( x ) - K i 3 ^ ( a , b , c ) ( x ) . Z
i=i
(11.35) The formal differentiation Dh is called the derivative with respect to heat energy, because it is related to the quantity
11.3. THE CONTRIBUTION
W)
2
=
2 w 4 (x) -
OF G2 TO B2
E-=iO*(x))2 2w 0 (x)
- *
291
x,w4(x)-
SU(<*(*))2 , ^ o ( x ) j 2w 0 (x)
which is proportional to the heat energy in the cube CtThe Theorem 11.2 can be reformulated in a compact manner as follows. Theorem 11.2' The contribution of G\ to B2 can be written as follows: 3
, „,
3
3
= | E E /£wEEB?iw^1/2^cw.b,C)(x)dx.
B?
j=lk*jj
°Xk
p=l,=l
(11.34)'
11.3
The Contribution of G2 to B2
Substituting the expression y—s&NvtUo
E
3 (3iv t -2)/4
E E EE n=0
JFGH
u=0
<5
,u1 ,w2 ,w 3
,rvVv
( v i 4 ) - */*> V>2] ~ arcsin y/2/3
Y£(
^arcsin ^/f )
7T + F2;2"L-,
.
/2'
-,arcsin,,
v=Q
«^^LTOH(G(t))w(YW)4B)(r(t))sJ(ei*>)s&(eW)s&(eW) for F(Z,t) and changing dZ ( t ) into dZ ( t ) in the equation (11.1), we have the contribution of G2 to B2 as follows. .(»).
•)
292
CHAPTER
11. A FINER
: 3 ( 5 5 5 ^ 2 ) 7 4 * M ^ - TT/4, ^
K-FUNCTIONAL
EQUATION
- arcsin y/2J?>\
*E E EEk(^--vf)+r-ufi'arcsinV/I) f + gJFOH + h = even
„=0
„=0„=0 L
\
4
»0/
\4
V •*/
>)E^(e2t))H^(e3t))exp[^]
'EIE(
•
J'=I
t
6
fc/i
i=i
K
2 7 r (Ar.-i)/2
r((ivt -1)/2)
•'Ax
S ? H
3 oo u |y 2 2 ^( | , a r c s i n ^
EE
u=0u=0
)+K ^ ^arcsmy-
L
^ E E E E %^ flJ (Y(*))L 2 ° ) (r«) exp[^]||(t) 27r-v/m«;i A
t •'A*
u=0t)=0
x W ^ E E E ^ ^ ^ ^ ^ ^ W ^ i ^dtw J
2•K^/mi '
3K5/2
k
j=ifc/j
. .(•) V / dZ5N(---;4S ) ,w,1.(») ,UJ
A •'^
2
, .(•)
,r^;.--)Jdx^(KJ
,UJ3
293
11.3. THE CONTRIBUTION OF G2 TO B2
OO
4
x(Kx)) £
U
6
E W o W ) E
r\i
E E ^ W * ^ ) ^
1
" ^ )
exp[^]^-(x), (11.36)
where 6^fe(x)=^%-fc(x),
(11.37)!
$ U W - AT^T E &U*(*>.
(11-37)2
*
1=2
^N.N.oto.
if i = 1, * = 2 or j = 2,fc= 1;
W,
if J = 1,fc= 3 or j = 3, * = 1;
^j«vO'fc( x ) ~ " ^ " J N . O N ,
I £ " J O N , N , ( x ),
if
(11.37)3
0 = 3, k = 2 or j = 2, fc = 3.
Hence we have 3
/dZTw(ZI0/rfx(i^(x))4^^^x)exp[^]^^^-(x)Bffc)(x),
£#>
(11.38) where „ B
S
)(X)
7/2 . oo
u
Nt-1 $
= - f n V EE^( o(x))E E «=0»=0
J
W(x)9J(Y0(x))L2°\
1=1
(11.39) Using the arguments in proving the Theorem 11.1 and Theorem 11.2, we have the following two theorems. Theorem 11.3
The contribution of G2 to B2 is 3
B 2)
2
= E E
= E E
[dx7T-(x)
/dx^-(x)Q2jfc(D)K(a,b,c)(x)
[dZTN(Z,t)exp[A]Q2jk(n)(x),
(11.40)
294
CHAPTER
11. A FINER K-FUNCTIONAL
EQUATION
where Q2j/;(£J)(x) denotes a function in £2 denned as follows: Q;« fc (n)(x) = Bg ) (x)(i^(x)) 4 ^^^(x),
(11.41)
(2)
The quantity B\
can be expressed in terms of i^-functional as follows.
Theorem 11.4
The contribution of G
( ]
/dXBg)(x)^(x)JD^y2^(abc)(x)
B2 = 4 £ E
11.4
( n
42)
The Contribution of G^ to B2
Substituting the expression ^ ( • • • ; 4 S ) ^ i S ) ^ 2 S ) ^ i S ) ^ ( s ) ; - - - ) ^ ^ (t) ^ " I Vi
47T
• 3 (3JV t -2)/4
E
E
n=0
(t) /4> V2
7r lA
_
. nni arcsin y 2/3
E E k ( ^ - - \ / I ) + ^ ( ^ a r c s i n y | ) '
JPGH
u=Oi)=0
x4t)(r-(t))2^JFGH(G(t))9J(Y(t))4"'(r(t))4(eit))s«G(e2t))^(eW) for F(Z,t) and changing dZ^
into dZ^) in the equation (11.1), we have the
contribution of GQ to Bi as follows. ZN{---,W0
A
t
^
47T i ) 7^-m5^
E E EE Y% n =
0
JFGH f + g + h=even
u
=0i;=0
,OJ1
,W2
,W3
,rw,---j
s^t
- TT/4, „<*> - arcsin y ^ j
-,arcsinJ-)+
Y2'u2v[
'-,arcsinA/-
11.4. THE CONTRIBUTION OF G6 TO B2
295
x4t)(r(t))2^JFGH(G(t))5J(Y(t))4n)(r(t))s/(e
s
j=\ k^j
o_2m:
r
Q I dZ
Z.Z. 3 (3iV t+ 2)/4 J
2?r(iVt-l)/2
|r((jv t -1)/2)
a
1=1
3
'
ll
E E E E n=0
J
. , , ( s ) , ,( s ) , ,( s ) . ,( s ) „.(s).
d Z
k - ( ^ arcsin ^ / f )
+
^ - ( ^ , arcsin ^ / f ) "
u=Ov=0
x^^r^)4 £ £
flSU^^CYWj^CrW)
exp[ A t ) k
j = i k&
2-KK3 [
1
f dz,2N\-
TT
3
,W0
,U)X
,U)2
, W 3 ,rK
',•••)
A t ' " *
*E ^ ) ' ^ )
4
r
2TCK'
y (t) EE *(* ) u=0»=0
E EftJ-«ifc(G(t))w(YW)40)exp[A]|i(t) r-
00
u
))4
3m 2
x
\
«
u=0t>=0
Q7
E E E ^«^(x)pj(Y 0 (x))4 0 ) exp[J4]--^(x), J
(11.43)
k
j=ikjtj
where Q3uvjk(x)
=
1
Q},tvjk(x),
(11-44)!
Nt
(11.44)2 1=2
296
CHAPTER
&juvijk(x)
11. A FINER K-FUNCTIONAL
el"JN,N,O(X)'
if J = 1, *: = 2 or j = 2, fc = 1;
ei'&N.ON, W .
if J = 1, * = 3 or j = 3,fc= 1;
I ei^ioN.N, W .
if J = 3,fc= 2 or j = 2,fc= 3.
— '
EQUATION
(11.44)3
Hence we have 3
S<
3)
JdZTN(Z,t)Jdx(a;o(x))2(/i(x))4^^exp[^]H(x)Bffe)(x),
«
(11.45) where B
27TK5i ^
S>(X) = ~ ^ r
E
^
E ^ ( * o ( x ) ) J ] eJ O »i*(x)sj(Y 0 (x))4 0 ) .
u=0»=0
(11.46)
J
Using the arguments in proving the T h e o r e m 11.1, T h e o r e m 11.2, T h e o r e m 11.3 and T h e o r e m 11.4, we have the following two theorems. T h e o r e m 11.5
The contribution of G§ to JB2 is
3 B
2
3)
fdZ
= E E
/dxTw(Z,i)exp[J4]|^(x)Q3,fc(fi)(x),
(11.47)
where Q3jfc(ft)(x) denotes a functional in Q for any given x denned as follows: Q3,-fc(fi)(t) = Bf fc ) (x)( Wo (x)) 2 (i?(x)) 4 . T h e o r e m 11.6 B
™ =
11.5
4
(11.48)
The contribution of G§ to B
E E
[dxB$(x&x)D2aDlK(a,b,c)(x).
(11.49)
The Expression of B3
In chapter 7 we proved the P r o p o s i t i o n 7.2, which holds under the assumption that the distribution F(Z, t) is a turbulent Gibbs distribution. Examining the
11.5. THE EXPRESSION
OF B3
297
whole proof, we conclude that the above assumption has never been used except for showing the following equation: 3
dZTN(Q; t) exp[A) £ £ ±g-(s) -ni7ri £ /
- 7= IH S) 5>iW } + ~ ^ £ (rf) 2 - ^ ) ] = o, (7.18)' It follows from the equipartition of energy that the last term of the integral (7.18)' will approximately vanish for large Na, even if the turbulent Gibbs distribution T/v is replaced with the distribution (10.63), i.e.,
A J&*
L
k=0
. . JL // JV
2
x
..
w
s
i=l
°Si
IS | 9 \ I,.»i2>
I
, , J
7m !?E((-ir ) - V-)
= 0.
Now it is clear that T h e o r e m 11.7
If the turbulent Gibbs distribution TN is replaced with the
distribution (10.63), the additional terms to the expression B3, i.e., the integral (7.18)' will be separated into two parts, i.e., B3 = B31+B32,
(11.50)
where £31 =
A
JA
*
fe=0
s
i=l
aSi
298
CHAPTER
N
' (1
yM*
N
9U„«lwW|i» w
~ z)wu
\ i
I
+
y'
'
11. A FINER K-FUNCTIONAL
N
>
y^
(s) w w
ji
| ww ( s ) l 2 l i
I
N +QST
N
'
EQUATION
™j i ( s ) ( wl ( s ) - wj ( s ) n
-
V*
^
'
(11.50)! and dZ52v(---;a;^)wit),^t),4t),r(t);...)
B32 = - m 7 r i V /
N,
x£
G, exp[A] J ] £
fc=0
s
£(s)
i=l
- ^ £ *
J V s
«,«
£
Z=2
^
^
(11.50)2
j#<
It is clear that the expressions in the square brackets in B31 and B32 are homogeneous polynomials of orders three and two in {wj, / > 2} respectively.
11.6
The Contribution of Gx to B3
Substituting the expression 3(3^2)/4^1t) - * l ^
£ £ t 3
3
-
arCSln
V2/3
^
00
00
u
EEE E
EE
j=lfe=ln=0
U=0D=0
(t)
]
JFGH
\r1v
2«
•» 9
•K
V + Y'2u 9J [
7,arcsin
-,arcsinW-
(t)
for Gi, setting G; = 0,1 > 2 and changing dZ^
into dZ^) in the equation
(11.50)2, we have the contribution of G\ to B3 as follows. R
(i) _
R
(i)
-m7ri
:
EE/
^3(3^-2)74^ ( ^
fi(t)n^(-;4,|.« - */4' ^
-
arC5in
(s)
1
w(s) w(s) iw2
V^J
'
w
3
r^---A ' '
i
/
11.6. THE CONTRIBUTION
3
3
OF Gx TO B3
299
oo
EEE E EE j=lk=ln=0
JFCH
-2v
Y£ I - , arcsin \j - ) + Y^v
( - , arcsin A/ -
U=0i;=0
f+g + h = even
(t)
(t)
(t) n) (t) t t x^^^l„ ; 3/2JFG H(G )^i (r )5 J (Y( ))4(ei ))s« G (e^)^(ef)exp[^]
K )
s
i=l
LV
*
a
1=2
m^i
—VKm7ri A
t
•"*>•
s^tt
4ir • 3(37V t -2)/4
3
3
oo
oo
u
EEEEEE j=ljt=ln=0
J
,(*) - TT/4, p™ ,(*) - arcsin y/2/Z 1 5 (ip™
V2f
2u I ,, >arcsin
W - ) + Y"2„
7T 7,
arcsin
«=OD=0
(t)
(t)
iVt
K ') 3 / 2 (=2 (t) V~* V ^ ^ m i=l m^i
ac u c
(r )2 ( ) ! "' ^'!f "" (^._,(ef'))'(Ht,,.,(eg))^
/i\
W
Q
= -EE t
A
^
5?H
2 7 r (AT.-l)/2 4v / ran7r 2 i 3(3JVt-2)/4 Lr((ATs - l)/2)
3
3
oo
oo
u
-EEEEEE j=lit=ln=0
J
u=0u=O
Y% ( 7 , arcsin \/^)+
Y2~u2v I - , arcsin
300
CHAPTER
11. A FINER K-FUNCTIONAL
i(t)
i }**)/
3
(*)
EQUATION
»
^ ^ ( G ' t f ^ t e f Y ' V i i i E E ^ f (t) V 0
2K7/2_:
/•
31^/2
3
A
3
•/A*
t
n
7/2 "7T1
*»
2 K ' '3V2" 3m / .
oo
oo
«
j = l f e = l n = 0 J u=0i>=0
( G ( t
/»
3
^
^ 3
3
OO
U
dz / ^ ( z , f ) E E E E E w x ) ) •*
^
3
(t) (t) (t)
xE E ^ ^ ^ - -
x
J
i = l j n ^ i "'0
/
|(r«)a
» i T ...— n »•—f\ J-."—i = lfc=l J u=0t>=0
^(xK(x)o;m(x) ^r(0)„ , , ^ _ l J ^ ^ ^ ^ W ^ ^ m ( x ) L i % J ( Y 0 ( x ) )ve x p [ A M] g : ( x )
W
x))», (11.51)
x
where V'j/fcutnm ( ) denotes the function denned in the equations (11.4)i, (11.4)2 and (11.4) 3 . Using the arguments in proving the Theorems 11.1-6, we have the following two theorems. Theorem 11.8 The contribution of G\ to B3 (or B32) is 3
3
3
B(31} = B$ = E E E E
= E E
E
E
/
d x
^ W
/rfx^(x)Q 4 , f e i m (D)X(a,b,c)(x)
/ dZTN(Z,t)exp[A}Q4jkim(n)(x),
(H-52)
where Q4jfcim(fi)(x) denotes a functional in fi for any given t defined as follows:
Q*wn)(x) = y ' w ^ i L w . and BL' i m (x) denotes the function defined in the equation (11.6).
(n-53)
11.7. THE CONTRIBUTION Theorem 11.9
301
OF G2 TO B3
The contribution of G6 to B2 is
B? = *£> = 2 ± ± ± £ /dxBSL(x)£(x) /c=l j = l i = l m ^ i
xDhD-3/2DbjDbkDbmK(a,
11.7
(11.54)
b, c)(x).
T h e C o n t r i b u t i o n of Go t o J53
Substituting the expression
S E
E
E
n=0
JFGH
4 It
g
(» > i t ) - V 4 ' ^
- arcsin
y/ljl
EEk(i^ r csin^) + y-(^ arcsin
f+g + h=e-ven
l'
3 (3i^-2)/4
u=0i;=0
L
x
7
v
'
for G2, setting Gi = 0,1 ^ 2 and changing dZ^) into d Z ^ ' in the equation (11.50)2, we have the contribution of G2 to B3 as follows. (2) _
D(2)
B?> = B.32
-m7Tl
x^
E ^ 2 ) 7 4 *( ^
E E E E n=0 J 3
dz^Y[^{s)SN(---As
2 /
[ ^ (I
- */4, ^
)
(s) ,w1
- arcsin
(s) ,w2
(s)
,
^ i
-csin ^ f ) + Y ^ ( J , arcsin ^ | '
u=0ti=0 Nt
, (t)\2
E E E ^7=CSU(6(t))^CYW)4»>(r<')) exp[A] i = l j / i ;=2
\/CJQ
(S)
c^V'V")
302
CHAPTER 11. A FINER K-FUNCTIONAL
dc . .
(t)
,w. 2
2(^,_1(ef)) -iy(^1_i(ef))
A
(rW) 2 A($ ( t ) )ffj($ ( t ) ) JVt-1
dZV]JdZMSN(..-;uJis\J*\4s\J3s\rV-,...)
•£ / ,/ii
2
EQUATION
>'
s^t
27r(W,-l)/2 8m3/27r2i Z ^ 3(3Nt-2)/4K5/2 |_r((JVt - l)/2)
v X
OO
OO
U
r
EEEE n=0 J
x
2
*/ ^ ;„ l \ , v-lvl* — , arcsin ,, y 22i:(-,arcsin A /-)+r 2u 4 V 3
tt=0u=0
A_^(r(t))4 ,1 dc (t, n UW WL 7 w j^ fl c ( ;GA<") L ) 9J JC( O Y^«J)V4 W ' ( r ( t )^)Oe Ix P [ J 4 ] ^ ( t ) t) 3 2 tJUOTj ^ y , [UJn fc?^ (4 r) / /Z
6 T
°i
i==1
OO
2\/mK7ri A •/Ax
3 Z—JZ-^I
^
I , (tt;) \ l1//22
K )
' 3m 3 / 2 /
E E
t n=0 J
U
u=0ti=0
,.(*)/ „(t)\4
Wj (x)(i?(x)) (^(x))i/a
dc_ 'dti
^tJuvi0^
»
2K 7 / 2 7Ti
X
OO
OO
U
J Tu =^OT u =^0Y H ^ x ) ) dZTN(Z,t) / d x ^
dc
6^(x)ffJ(Y0(x))L2°>exp[A]g(x),
(11.55)
where £j u „jj(x) denotes the function denned in the equations (11.37)i, (11.37)2 and (11.37)3. Using the arguments in proving the Theorems 11.1-6 and Theorem 11.8-9, we have the following two theorems.
11.8. THE CONTRIBUTION Theorem 11.10
OF G6 TO B3
303
The contribution of G2 to B3 (or B 3 2 ) is
BP = £ $ = ^ ^ y , d x ^ ( x ) Q 5 i j - ( D ) A - ( a ) b , c ) ( x ) i=l j^i
= E E
/
d x
| ^ ( x ) /dZrAr(Z)t)exp[^]Q5y(n)(x),
(11.56)
where Q5<J-(fi)(x) denotes a functional in ft for any given t denned as follows:
and BJ- (t) denotes the function defined in the equation (11.39). Theorem 11.11
The contribution of Gi to B3 is
3
B32)
= B$ = 4J2Y1 f d^\X)^)DlD^DbjK(a,b,c)(X).
11.8
(11.58)
T h e C o n t r i b u t i o n of G6 t o J53
Substituting the expression
KE
3(3^t-2)/4
*E E EE n=0
JPGH
J
( ^ l t ) - */*, V24) -
7T
.
/2\
arcsin
V^j
,,_,„/)r
- , arcsin J - I + F2„
( 4 -arcsl- u 3
u=0v=0
x4tV(t))2^JFGH(G(t))5J(YW)4">(r(t))4(e«)s«G(e«)^(eit») for G 6 , setting G; = 0, / ^ 6 and changing dZ ( t ) into
CHAPTER 11. A FINER K-FUNCTIONAL EQUATION
304
x
x
^E3(3^2)/4^(^it)-7r/4^2t)-arcsinv^73)4t)(rW)4exp[^]
E E E E n=0
J
k"(f,arcsin j | ) + y 2 - ^ ( j , a r c s i n ^ )
W ^ V ^ )
«=Oi)=0
3
(t)
t=l j ^ i
„
Nt »
WQ
,
1=2
2m7ri 3K
t
A
n=0
J
Ja
*
Syit
u=Ov=0
2 7 r (Ar.-D/2 4n 3 (3JV,-2)/4 |_r((JV t - l ) / 2 )
3
o#V t ) )W ( t ) )4'V t ) )exp[4
, .(t) a or he
N,
OT
,=1 « « ^
1 "tjuvlij
i
JVt
X
( = 2
oo
2K87ri
«
rfZTw(Z,*)X:EEE^($(t))(i?(t))4^(Y(t))i60)exp[A] " 3m 2 / t J u=0 u=0
EE-r^^wE^U^)^ 9 6m
5
•
/•
/•
J
oo
J
Ji
u
„=n„=n = 0 v=0
u
dc ^^WjWwoW-g^Wgjvvijix),
x exp[A] i=l
j^i
(0) 6
(11.59)
11.9. THE CONTRIBUTION
OF G3 TO Bz
305
where Q3UVij{x) denotes the function denned in the equations (11.44)i, (11.44)2 and (11.44)3. Hence we have 3
3)
B< * /"dZT N (Z,i)EE/ dxw o( x H( x )(^ x )) 4ex P^^( x ) B l 3) ( x )' i=l
j^i
(11.60) where BJ- (x) denotes the function defined in the equation (11.46). Using the arguments in the preceding sections, we have the following two theorems. Theorem 11.12 The contribution of GQ to B$ is fi 3)
3
= E
E fdZTN(Z,t)
/dxexp[.4]|^(x)Q6i,(fi)(x),
i - i &iJ
(11.61)
OXi
J
where Q6ij(^)(x) denotes a functional in fi for any given x defined as follows: Q « i ( n ) ( x ) = Bg ) (x)o;o(x)a; j (x)(il(x)) 4 .
(11.62)
T h e o r e m 11.13 The contribution of GQ to B2 is B
11.9
33) = 4 ^ ^ /
d x B
l
3 )
(
x
)^(
x
)^^^^(a'
b
'c)(
x
(n-63)
)-
The Contribution of G3 t o B3
Substituting the expression m Y-^ 47r / (t) (t\ ,——\ ^ i JL, 3 ( 3 ^ - 2 ) 7 4 ^ 1 - */*>
3 X 'E
oo
E
1=1 n=0
E
E E k(^,-sin/I)
JFGH
f + g + h=odd
„ = 0v=0
L
V
'
'
^-(J.arcsinyf) ^
*
'
(t)cXr
?t(uuJFGHl
,J
^
(t)
& H * J-^3 V
J-F<, W 1 y-GV W 2 ) - H ( H 3 )
306
CHAPTER
11. A FINER K-FUNCTIONAL
EQUATION
for G 3 , setting G; = 0,/ ^ 3 and changing dZ ( t ) into dZ ( t ) in the equation (11.50)i, we have the contribution of G3 to B3 as follows. (4) _ „(4) « -niTri B£> = B£> "^ —111/11 E / X
X
^
i^
E / /
\
dZ^l[dZ^TN(Z,t)
UiZJ ' ' I
t •/AA
«=tt
x(,#>-«r/A ™ (Vi^ - vr/4, ¥#>J .- .a rc sin ,v.^/)0,£^f :t ^ & "/'."K2 « . ^ u .
'K43(3N,-2)/4"V^l ^
* E
E
n=0
EE
JFGH / + 3 + h=o
u
v i / u
' (=1
kf^-sin ^f)
=0v=0L
V
+
r ( t )
W j , arcsin v f )L(Y«>)
y
V
' °/J
x^iJFGH(G(t))4n)(r(t))^(eit')^G(e2t))E^(e^)exP[A]^X:£^ s
W
* 17
9U„ ( s ) lw ( s '|2
*• "• ry ( s ) lw ( s ) l 2
N
*
i=l
l
N
' w{a) (w ( s ) • w (s) V (11.64)
In order to get an expression for the quantity (11.64) in terms of the Kfunctional, we should calculate the quantity in the square bracket on the right hand side of the above equation. Firstly, the sum of the first two terms can be expressed in terms of the spherical harmonics in the following way. Nt
(1
^..Wiw''^
Nt
Nt
™(t)lw(t)l2
307
11.9. THE CONTRIBUTION OF G3 TO B3
V^T)
(=3
Nt
n
n\
2 w
(*)/ ( t ) x 2
(t)
E
,i,(tV,„(th2
Aft 1-1
w
y* y^ (* ~ ) u ( ik)
Nt
y/F^y
i=3 j = 2
+
w
u (wik)
y^y>
(^') 2 -iM) 2 J=2
1=3 Nt
(t)
(^-nM)2' J'=2
^ ^
_
+ l ) q - 2 ) ( r W A ( ^ t ) ) ) 3 ^3 , Q (tK v/(iVt - l)(jVt + l)(ATt + 3)1(1 - i r L ' - ^ l v /6q
A v V2F^(rWA(»W))M*>A(^))rf (t) ! fefe W-lV(ATt + l)/ - — ^^ -
(t)
^ ^
(1L65)
In the derivation of the above equation we have used the equations (A.41), (A.45) and (A.51) about spherical harmonics in the Appendix A. The last term in the square bracket on the right hand side of the equation (11.65) will be expressed in terms of the spherical harmonics as follows. 2y>
y ^ wji l w ;
1=2 j=l+i
Nt
Nt
( u ,(. t ))2 u ; (t)
w
j )
VJU^T) Nt Nt
(*) . . , ( * ) . . . ( * ) i
w)/ w,,'w ^2. E + E E E ^ Viti -1) i=2j=Tii ViU -1) ' i^j^tiw Ik
=2
Nt Nt V^ y^
(r(t)A($(t)))3 V ' -HiK*' ))
w
jk
v/27 „ 3 / __V_f^_-3
(ei 4) )
308
CHAPTER 11. A FINER K-FUNCTIONAL EQUATION
Nt-2
V2 ~3 l N / ( f c + 2)(fc + 3)
-E ±l k=j
,^
f
^
C A ( t h -4-
v W t + 3
rW/g,(«W)(rW/3fc(*W))' ^ 2
= l
fftW^
{t)
,
(t)
(11.66) In the derivation of the equation (11.66) we have used some formulas about spherical harmonics, especially, the equation (A.97) in Proposition A. 10. Through elementary, but tedious, calculations we can get the following expression of the contribution of G3 to B3: 6 R(4)
_
6
o(4) _ V ^
R (4v)
4 _ V ^ R ((4«)
v=l
l(*»)
and the expressions of B%
«^
, v = 1, • • •, 6 as follows.
D>(41) W ^ J" Tfl V"> V ^
•°3
~
K4
(11.67)
v=l
2s2s^(3Nt-
/
\
^j-J 3
x*( V W - TT/4, ^ > - arcsin ^ 5 / 3 ) £ 7/
[Y% (T™** vf) u=0 v=0
11
+y
'—• fc=i
dZ^l[dZ^TN(Z,t) (t)p
^ J ^ l Jft(Y(t)) £ r
4»>(r(*>)
n=0
-" (i'arcsin vf) ^PMEJTW at* i=l
V(iV t -l)(iV t + l)(iV t +3)/(Z-l)
(=3
m27ri
47T
Z ^ Z ^ 3 (3(3JVt 3iVt-2)/4 3 3 / 2 K 4 EE
3
2ff(Af.-l)/2
] / dZ( t )fTdZWr w (Z,i) [T((Nt - l)/2) -I ^A. s%t
,,(*)<
E ^ ( t^) r
B) W
E ^ ( Y W ) E 4 (r )exp[^] E n=0
i=l
|W
309
11.9. THE CONTRIBUTION OF G3 TO B3
arcsin 2V E E [^(i' \/f) +^ (J>™\/!) u=0 v=0 Nt
E
(t)
(«)
-1'*
1=3
V6(l + l)(l-2)(rW)3 V ( ^ t - l)(JVt + l)(JVt + 3)1(1-1)
J dZTN(Z,t) J dX^o(x)Ljk(X)£tMnY/9J^o(x))L30)
= ^
x
3
^
oo u
E S - w E E rkK*o(x)Xii(x))a i=l ^
u=0«=0
Aft
/ J7?km;JL;_,.i(X) L J=3
^
v/6(Z + l)(Z-2) V(JV t -l)(JV t + l)(JV t +3)Z(l-l).
/dZrw(Z,*) / d x ^ E
x
3
exp[A]
„
OO
K ( X
/^
( X )
S>(Yo(»))L<°>axpfl
U
E ^ W E E ^(^o(x))(i?(x))2%^JLii(x), i=\
i
(11.68)
u=0v=0
where (11.69)!
r/ f e u vJL,i(x) = Vk'uvJL , i ( X ) '
^
W
JVt
=
lim
S^---(G(t))7(^S^W^' V(l + 1)C - 2)
r„(») 'JmutJLi _ i OO ( X ) ' »
1
(n) CLjL^.iW = i CJvJOL.-.oW. •muv (n) JOOL,.,W' tmuv.
(11 69)2
'
if * — 1J if
* = 2;
if* = 3.
( n - 6 9 )3
CHAPTER 11. A FINER K-FUNCTIONAL EQUATION
310
B<42) « - m T r i ^ ^ J \ A
dZ^l[dZ^TN(Z,t)
4. i t M •"*•>>
.s_^! t.
\
m Aix
<*( i ^ - TT/4, 4 4 ) - arcsin s/2JZ J ^43(3Ar,-2)/4 ^
*E
E
3
(*) £•
^
r-(t)
' m=l
E E k ( f .arcsin Jfj
+^ih'(j,aradn»/|)]flJ(Y«)
/ + 9 +h = odd
dc
x ^ U J F G H ( G ( t ) ) 4 n ) ( r ( t ) ) 4 ( e ^ ) S ^ ( e ^ ) E ^ ( e i t > ) exp[4 £ £ ( t )
- E* E
^^^(rWA^W))
2
^^^),,
(iv t -i)v^vTi)i
( = 3 fc^i
\/2m 2
^?EE/ - - , A,
3 ^ -
A
,ft(tw
, o( t>,
-H^on^on 27r(iV.-l)/2
s
,
t
3
^U^T^W^W*
-|3
r((iVt-l)/2)J
ATt
^W^EEE^-!^ oo
oo u r
EEEE n=0
J u=0v=0
-2v
V2w
2u i ^ > a r c s i n y 3 ) +Yi2u
_ I „ i,
arcsin
9J (Y(*))
L
^»»SL.M l _ l l *N I _ 1 ,*(G ( t ) )4 B ) (r ( t ) )exp[il]
^ £ J dZTN(z,t)Y: E V ^ w ^ ) 2 t
x
m=l
S§S^(t)(^t-i)^
311
11.9. THE CONTRIBUTION OF Gz TO B3
x
oo
oo
u
n=0
J u = 0 v=0
E E E E^(*(t))fli(Y(t))»»SL.MI_1,fcNI_1.«(G(t))4w)(rW)expW 3
(t)
3
a s^/^ftoEE^B^'w^EEiw c
t
oo x
oo
m = l •y/c;^ ' j = l
i=l
kjii
u
E E E E ^ ( * ( t ) ) t o ( Y ( t ) ) » » S L . M , * N . « ( G ^ ) 4 B ) ( r ( t ) ) exp[A] n=0
J
u=0o=0
^^/-„«z,,/.|t^Sr ! <^i:E|; M X
Y,Y,Y, J
^(*o(x))5j(Yo(x))t,W
0)
ttt(MifcNii(x)4
expH,
(11.70)
u=0 u=0
where (0) % u « J M , ) t N , i ( x ) — ^fcuuJM.fcN.iW
( H . 7 1 ) I
VT^2 7
?mu«JM,fcN,t(x) ~ I i m y . I f a j M L u t N i - L i W / , ,
?
J")
X
1mmiJM1_1,ltN,_1,i( )
R(43) B
Z
_ R(43) - #31
—
_ , N /T
?mut;JM,_1N,_1o(X)'
if K = 1, » = 2 ;
'fmuoJMi-iONj.iW'
i f fc = 1 , i = 3 ;
7
i f fc = 2 , i = 1 ;
?muuJN,_1M,_1o(X)'
(11-71)2
(11-71)3
'
^muvJMi.iNi.joW'
iffc= 2, i = 3;
^muvJOMi-jNi-iW'
i f fc = 3 , i = 1 ;
. ^mmJONi-iMLiWi
i f fc = 3 , i = 2 .
K
A
t
^
,
/ t
312
CHAPTER
47T 3<3ATt-2)/4
E E E E
11. A FINER K-FUNCTIONAL
EQUATION
3 (t)c — UJm Cttkin r-(t) ' m=l
(t) .(*) sin^/273") 51 - ( ,ip\' — 7r/4, y>2 — arcsin
£
[^(J.arcsinyf) + y * * ( j , . « ^ y § ) ] W ( Y « )
n=0 J u=Ov=0 Nt
Nt
3
(n) x 1=2 ^ j=l+l E E^ tmuvJli i=l
i(G(
(i)
t
dc ))4")(r(t))exp[^]g(t)
(r(t)/3.($(t)))3
V(iv t - i ) ( M + i)(jv t + 3)(j + i)(j - 1 ) 2l/2m5/27ri
3
(t)
W ? E E /Ax ** n «
z(s)
3 /^
3
3
(t)/ ( t h 2
oo
( S L? a > . (©i 4 ) )) 2 I-i,;-i 2 7 r (7V,-l)/2
^-«) ^ 7 4 Lr((ivt -
i)/2)j
u
x £Ei £ ! E(0,^)3/2 T^^^EEE^c^^^^fexptA] J u=0 v=0
3
„
AT.i-1
7/0)
.(G(t))
TT(1)
xE^wEE —--' ti dti U t*2 V W - l)(ATt + l)(j + l)(j - 1) 21/2,
'mK7Tl 3 /2 3
3
3
- fdZTN{Z,t) E E / r f x m=lfc=l"
xE
E E ^ ( $ o ( x ) ) p j ( Y 0 ( x ) ) 4 0 ) exp[^] ^
J u=0u=0
i=l
a; m (x)(a; fc (x)) :: (Wo(x))i/2
(E(x)) 2
^ ( x ) i , m u t ) J t ( 1 ) i i ( x ) , (11.72) *
where
(n) 7'Tni»TjJL(
^muvJLtD.iW — 1m„„JL(i),i(X)'
(11.73)
"Li-l £> ,( GW ) tmuvJL;«, i),«(x) = i ™^ ES E ^ W - i ) ( ^ t + i)(i + i ) ( i - i ) '
(ll-73) 2
313
11.9. THE CONTRIBUTION OF G3 TO B3
r,(n)
(1)
r?(n)
(i)
(x), if i = l;
(n) (x) = { 77
m
(x), if s = 2;
(11.73)3
i —1,J —1
«muvJOOL £1 «) W , if* = 3. I-I,J-I _ #„(44) _ 23/2m27ri R(44) _ #3 31
- ^ E E / dz(t)ndz(s)^(z-o 3
, ,(t)i
: ^ , - J (rf> - „/4, „ < « - arcsto v ^ > ) E
^
^
EEEE kfi.^-vf)+y-•(j'"csi°\/I)
n=0 J u=0 r=0
t
V^
V ^
V ^
fe^ifc^li
t
i=i
*
"",JLi-i,*+i>»
L
i-i,fc+i
V W - l)(Nt + l)(iVt + 3)j(j - l)(fc + 2)(fc + 3)
23/2m2
^EE/^n^^^
(=1
n=0 J «=0»=0
L
V
^
'
27r(Nt-l)/2
i W -1)/2)
^
x flJ (Y( t ))(r( t )) 2 4" ) (rW)exp^] £ £ ( t )
><EE S A u = tE -i
*?(n)
(i)
(G<*>)
'""""•{I'M+I-*
V W - l)(JVt + l)(iVt + 3)j(j - l)(fc + 2)(* + 3)
'-1
314
CHAPTER
6
t
11. A FINER K-FUNCTIONAL
J
i=l m=l
V ^ O v = o *=2 ^ f e 2l/2ml/2/tl/27ri
y 4
t }
EQUATION
i=l *
#t-l)(iVt + l)j(i-l)(H2)(H3)
,
,
3
3
W;(x)(a;m(x))
2
3
^
33/2
Yk(*o(x)) f l j(YW)4 0 ) »,°>
oo u Nx-2k+u-i J
(x)
(1)
V ^ 0 u = o i S f e ^ V ( ^ x - l ) ( i V x + l)j(j-l)(/c + 2)(fc + 3)
•^
^
i=l m=l
t/wi '
i=l
x exp[A] y y y ^ ( < M x ) ) s j ( Y « ) L 3 % m u i ; J L ( 1 ) i i ( x ) , J
*
(11.74)
« = 0 ti=0
where W . J L W . i W = 'S,„j L (i) i i (x), (0)
^n)
1}
.(x) = um N^2_fc_ y y ,
„JL(D,iw - " - ^
^
(H.75)i
C \
^ *"*•«-..»+.•« ,I,„.(l) ,(X)
y fe+1
^ ( j V x - l ) ( 7 V x + l)(fc + 2)(fc + 3) ^
i
i VTCF7!)' (H.75) 2
, /(") "' 7
(1)
(x) = ^
^ljL,t+1OoW'
i f i= l !
"mi,JOLiI> lifc+1 0 (X >'
l f i= 2 ;
(n"75)3
muuJL}i ; l i f c + 1 ,i v
^L n LoOL^, t + 1 ( X ) '
i h =3
-
It is worth noting that the equations (11.75)3 and (11.73)3
are
identical, but
the quantities ^ ^ j ^ u ) .(x) and £ B j L ( 1 ) i i ( x ) , defined in (11.75)2 and (11.73)2 respectively, are different.
11.9. THE CONTRIBUTION
315
OF G3 TO B3
3
R(45) _
- ^ E E / ^>n^>iMz,t)£
R (45)
K
X
A
3(3^2)74 S (^
oo
oo
u
t
*>>
s#t
~ */*>
l^2u r 2u
*EEEE
J
arCSln
,,(t)-
fcin r-(t)
m=l
V^)
E
eX
^(*)
y -Y2( l , a r c s i „ / f ) -, arcsin /\ f -) +I +
P^
93
( Y (*))
n=0 J u=0 u=0
3
)
fe^I^m"rfN,-l,i(
A t
"'A*
]
W + W W - DM -1)
s^t
m=l
+ y 2«
(U,
I --arcsin
<« fcta' flJ(YW)exp[i4]
n=0 J u=0 v=0
•3(3JV,-2)/4
T((Nt - l)/2)
^ x
5
27r(iVt-l)/2
47T
f dZTN(z,t) f
Wt
Nt
1=2
j=l+l
(_(t)\2_(n)
(G^h
(Nt + i)v7(7^T)
g>m(x)(a;fc(x))2 y ^ dc (o; 0 (x))i/2 ^9x,'
dxj:Y:
EEE y 8 ( $ »W)»( Y »W) e x ! ) [ i ] i 3 0 ) ( f l ( x )) 2 ^N,i(x),
(11.76)
J u=01)=0
where (0)
??muuJN,i(x) — ^ m u u J N . i W '
7
(n)
/ .,
,.
v ^ o W ) )
?m^JN,i(X)=llm2^Z^
UU
''tmraJNi.LiW
i^ + VVJiF7!) '
(11.77)!
(H.77) 2
CHAPTER 11. A FINER K-FUNCTIONAL EQUATION
316
»
r
B
7mut;JNz_it( X )
if i = 1;
^tmutiJONi-iOW'
if * = 2;
. ^tmuuJOON, _ i ( X ) '
if * = 3 .
2m2
(46)=B(46)
^ E E /
'31
S
3(3Nt-2)/4
''tmraJNi.iOoW'
[n
~ */*M
dZ^Y[dZ^TN{Z,t)
~ arcsin ^ 2 / 3 1 ^
^
x
(11.77)3
r(t)'
' m=l
E E E E [*£ ( J , arcsin y f ) + r 2 ; 2 " ( J , arcsin ^f)] flJ (Y<*>) n=0 J u=0 u=0
3
Nt
Q
Nt
xLW(rW)exp[A]5:f ( t ) E E i=l
*
(iv t -i)v/jO--i)(Ar t
E ^ L a M ^ , ^ , ^ , ^ )
i=2 j = / + l fe^«
i){"M—(9fc »
+
(
^-
( 9 i
)}
2m27ri 3 3 ^ - - ,
47T ' 3 (3iVt-2)/4
OO
OO
U
r
A
A
s
3
27r(N«-l)/2
r((ivt -1)/2)
3
,
t
3
,,(t),
(t)x
y > v ^ <^m [up ) 2 m=l p=l
(t) 2
"'O
fiT\
,
EEEEkfi'Wi)
+ ^2 U 2,, l T . a r c s i n y -
93 (Y<*>)
n=0 J u=0 v=0
dc,.,v"^
v A \ - < ''tmuDjMi.^j.LltNj.Lif^
)
11.9. THE CONTRIBUTION
2-v/m/cTri
x
f dZTN(Z,t)
/rfx E
E E ^(*o)40)
317
OF G3 TO B3
ex
E
" m ( x . ) ( 7 ( f ) 2 (fl(x))2 £ g J ( Y 0 ( x ) )
9c oZ:(x) W«JM,*N,i(x),
P[^l E E
U = 0 D=0
i = l fc^i
9xj
(11.78)
where VmuvJM,kN
,i[X-)
— VmuvJM.kN.i
(11-79)!
W>
Nx j - 1 „ ( " )
&jM,ikN,i(x) =
l i m
EE
j = 3 (=2
'
-
'
(iVx-^vW !)
'
(11.79)2
(n) / ^muvJMi-i.j-iNj-iOvX
,
iffc = l , t = 2;
(n) , )mmiJM[_llj-iONj_iVx
,
if k = 1, i = 3;
(n) , ^muwJNj-iMj-i.j-iOlX,
,
if /c = 2 ,i = 1;
(n) / ''muDjOMi-i.j-iNj-i vX
,
if k = 2 , i = 3;
,
if A; = 3 , i = 1;
,
if A; = 3 , i = 2. (11-79)3
I
(n) W t i J M ,i - y . i . i i N j . i . i W
(X) 3
(n)
/
(n) , . f m u « J O N j . i M | _ i j - i ^X
Summarizing the results (11.67), (11.68), (11.70), (11.72), (11.74), (11.76) and (11.78), we have Theorem 11.14 The contribution of G3 to B3 is B. (4)
* /dZ2W(Z, t) JdxeMA] E E y
^(X»2 E ffoB%W. i=l
(11.80) where B ^ ; (X) denotes a quantity almost independent of WQ (X) , w\ (x), u>2 (x), UJ3 (x) and # ( x ) .
318
CHAPTER 11. A FINER K-FUNCTIONAL
EQUATION
The quantity B% can be formulated in terms of the ^-functional in the following way: Theorem 11.15
The contribution of G3 to B3 is
*34) = | E E E
(d^^)^)DhD-^DbmDlkK{a,h,c)^).
t=l m=l
l
fc=l
(11.81)
11.10
The Contribution of G4 to £ 3
Substituting the expression
S E a O ^ ' f ^ - " / ' . ^ - -sin V^) E W>) t
^
'
n=0
t L gt[«(f.—.^)+>=-(5.«-4) (= 1
JFGH
u
=0»=0
(t). T ,/.
xCt(itJFGH(GW)gJ(YW)U;' for d
-
^ f ' "
(t)\ 0
)5/(e< t >)H&(eW)Bfe(9i t >)
and setting Gj = 0,1 ^ 4 changing dZ into dZ in the equation (11.50)i, we
have the contribution of G4 to B3 as follows.
*i5) = - ^ E E / ^ ( t ) n«Z(S)?MZ.0exp[^] E E A
47T X-
'3(3AT t -2)/4
t
,/A
^
*
s/t
s
-Tr/4,^ - a r c s i n ^ )
£« '
E ^ ^ )
^
' n=0
E E EE Y% CJ, arcsin ^ | J + Y^
2v u
i=l
t=l
JFGH u=0t/=0 f+g + h=odd
(t)lT,,.
~
(tK
(^, arcsin y ^
11.10.
THE CONTRIBUTION
N
* 11 2U„ ( s ) lw ( s ) l 2
L
VW^)
i=3
N
'
OF GA TO B3
N
'
w ( s ) lw, ( s ) l 2
Ui^ti VJiF^)
319
N
'
w{a) (w,(s) • w ( s h
"•
VJO 3 !)
S,^i
^ E E / <«(t) II «aw^(z,o expM E E £(*) A
47T
^
t
^
"' A ^
S5*t
S
»=1
*
- * / 4 , V « - a r c s i n ^ T s ) f>(r«) n=0
3
X
E E, EE[«(i.«-.VD+^^—VD] / + 9 + h=odd
CttlJFGH(G(t))5J(Y(t))^J^E^(G(t))^G(eW)E^(e(t>
(11.82) In the above deduction we have used the Assumption C. It should be noted that we have used the notation \i instead of the A in the Assumption C for avoiding the confusion with the notation A in A\. Elementary, but tedious, calculations yield the following results: The contribution of G4 to B3 is B .
(11.83)
v=l
Now we are going to calculate the expressions B$
Bfl) = ^
, v = 1, • • •, 6.
E E / «z(t) n M{S)T»V>«) «PM E #(t)
320
:
CHAPTER 11. A FINER K-FUNCTIONAL EQUATION
Air 3(3JV t -2)/4
- v r / 4 , ^ -arcsin V 2 7 3 ) E ^ n ) ( r ( t ) ) E
j(^ ^
' n=0
Y
"" ^ r
m=l
E E E \tt (I —- vf)+ Y^2v (?•arcsin VI)] 5j(Y(t)) Nt
yV(«)
,„(tK
V6(t + l)(t-2)(rWA(* (t) )) 3
,g3
,-(*)„,
^ ( J V t - l)(ATt + l)(JVt + 3)i(l - 1 ) ^ L ' - 1 ^
Z ^muvJL,.,,^ 1=3
"
w
= H ^ W ^ E E / ^wndz TN(z,t)«xp[A]j:^(t) 27r(Wt-D/2
47T
'3(3Af,-2)/4
x E E E J
|T((JVt - l)/2)
"VV'JE^WV-IW m=l
[^(^-sinyf)
+
y-(j,arcsinyf)
JVt
=
»J
(yW)
u=0u=0
>/(' + 1)(* - 2)
^^^/ dZTjv(z ' t)exp[A] E£: (x) E ^ w ^ w ^ ' ^ w ) 2 i=l
X
*
m=l
^40) E E E^(*0(X))w(Y0(X))Cm«»JL,i(x), J
(11-84)
tt=0i;=0
where (0)
4muuJL,i(. x .) — Smui)JL,iV X /' Wx
(11.85)!
,-(»>)
C ^ j L , . 1 » ^ + i)a-2) Cmu«JL,i( X )= lim^E V(iV -l)(JV + l)J(J-l) ' x x
(11.85)2
321
11.10. THE CONTRIBUTION OF G4 TO B3
*(n) Smu v JLi _ i ,i V /
Cmui;JL, _! oo ( x )'
if * = i;
CmuuJOLi-ioM'
'f * =
2
;
if
3
-
. C m t i „ J O O L , _ i ( x )>
"W^m'i*?
-*'*>*?
- - c s i n v ^ ) E4n)(r(t))(r^)2(^)
^
' n=0
E E E E m=l
J
k(j.-sinyf)
V
^mu»JM1.i,tNM,t^
< i=3 L Lfc^i
A
Jj(Y<")
t
•/A^
r((AT t -l)/2)
x
\^
EL—
i=3 k^i
\/2mK7ri
^ J d z J
/ft(*)\\2/-l
s^t
/c,(*h\2
(t)
i=l
3 oo
E^)(r(t))('-(t))2(^))'*-*E^)
71=0
EEEk(i'--yf) .7 n=oi,=n L
2
(-M,_,l w k )) l - N , - ! ^ ))
^t>n«a(')^(z,*)exP[^5:
27r(Ar.-i)/a
47T
'3(3JV t -2)/4
),„
T^—JTA
=WT#EE/
W 2
+y2-u-(^arcsinyf)
tj=Ou=0
v , .(*) V ^ V ^ X
*-
(11.85)3
m=l
+
r
2u
V 4-,arcsm., >
SJ(Y<*>)
» "V
(Nt - l)Vl
dxTN(Z, t) exp[A) £
|^(x)(il(x)) 2 (u,o(x)r + U< 0 )
CHAPTER
322
3
oo
m=l
J
11. A FINER K-FUNCTIONAL
u
Nx
v'
2C m „„j M j _ 1 ] A . N l l i i (x) (7VX - i ) V ?
1=3 kj-i
u=Ov=0
EQUATION
(11.86) where (0)
(11.87)!
CmuvJM,feN,i(x) — C m u „jM,fcN,i( x )>
/(»)
An)
X
^muvJMi^ltkNi^i,i( )
~
V
^WJM,.1,l!N1.1,iW
(Nx - l)Vl
W l ) J M | _ ! Nj _ i O (*: ,
iffc = l, i = 2
•>muwJM|_iON|_i V*.
,
if fc = 1, i = 3
"»muj;JNi_1M(_1OVA,
,
if A: = 2, * = 1
A
,
if k = 2, i = 3
,
if A; = 3, i = 1
,
if k = 3, i = 2.
(11.87)3
\ >>mui;JMi_ 1 Ni_ 1 0\
W u t i J O M i ^ N i - i V.-*-.
u
muvJONi - i M , . , ( X
2
5(53)
x
=
3/2K
^ E E /
^n^^^oexp^s^w
a i i J ^ W f ^ - 'M ^
OO
U
r
'
J u=0u=0
L
n=0
s
, arcsin A / ^ ) + y 2 - 2 \ m=l
L<">(r<*>)(r<*>)a
- arcsin y ^ ) £
^ 3
(11.87)2
v
v <>(n) "'
(1)
T4>
arcsin
A/-
5J(Y<*))
(G<*>)
ImuvJL11, . ,,iv
° 2
' (H?.o> Of))) 2 S A i \ / W - l)(iVt + l)(JVt + 3)0P - 1) ^ I V , - .
3/2Km5/27ri
33/2«;11/2
EE/
^ t )n^)T N (Z, i )exp[^]X:^(t)
11.10.
THE CONTRIBUTION
3 oo
27r(Wt-i)/2 An 3(3AT.-2)/4 [r((JVt - l)/2)
a
oo
n=0
.
4'
arCSm
/ 2
N
V ^ «i(Y<*>)
£L tTnuvJLj^i -(G(t)) w .o> •_!)**
* *
m
^
. ^ V(ATt - l)(iVt + 1)(P - 1) 3
3/2Kml/27ri 33/2K5/2
x
7T
y^l^arcsinW^j+y^
J ti=0 ti=0
„
2
^^'(rWjlrW)2
u
*EE££ m=l
323
OF G 4 TO B3
^(x)Li0)(i?(x))2(Wo(x))"+5
I dZ I d x T N ( Z , t ) exp[A] £
EEEEye($o(x))gj(Y„(x))Um(x)(mu„JL(1)|j(x), m=l
(11.88)
J u=0i;=0
where (0)
(11.89)!
Cmu«JL( l ),i( x ) — C mu „JL(i>,t( X )' (n) mt*i)JL ( i )
JVX J - l X = liln
^m«»JL(»,i( )
C
im
(,,
WJL
C
Tt(1)
(11.89)2
^^V((^x)2-l)(j2-l)' C(n)
(n)
»
.(x)
£(n)
1
'.
(x), ift = l; . ,oov
1—1,3 — 1
(i)
"
x Q( )>
'
i f i = 2;
(11.89)3
t — i , j —i
C (n)
en
(x), ifi = 3.
1 — 1,3 — 1
B
(54) _ 2 3 / 2 Km 2 7Ti
3 ^
E E /
^
n^ ( S ) T,(Z, t) exp[A] £ |L(t)
^ ^ ^ ^ ( ^ - ^ ^ - a r c s i n ^ L r ^ r ^ ) )
2
324
CHAPTER 11. A FINER K-FUNCTIONAL EQUATION
x E E E 1 m=l
J
( J . - s i n y f ) + y 2 - ( J , arcsin y / f ) 0J (Y(*))
k
u=0 v=0
JVt-afc+1 j ' - l XU), (t)
2
C
,
yyy
(n)
tm
""JLi-i,fc+i
3/2Km5/27ri ,
S ^ E E / 13
27r(JVt-l)/2
47T •3(3Aft-2)/4
d**I[dZUTIt{Z,t)«p[A]££{t) 3
r((jv t - 1 ) / 2 )
TO=1 J
iVt-2fc+l j ' - l
t)
tmuttJLJ'j
xffJ(Y< )Ww y y y ^ m
2
3 / 2 K m l i/ /227 r i' 3 2
3 /
fe fe S r
r
^iJdZJ
oo
fc,
u
11=0 t)=0
j,i
V W - l)(iVt + l)j(j - l)(fc + 2)(fc + 3) ^
fi
d^TN(Z,t)exp[A}J2^)Lf\M^)r+HR^)) i=X
x
EEEE^^w^^w^w^to.iW, m=l
J
(ii-9o)
u=0v—0
where (11.91)!
c
<
(n) JL(1)
.(x) = i i m E F E
W„JL
W j L <J1 >i mu
" " -i't+i"
- ^
Z ^ Z . Z . V(iVx - 1)(ATX + l)j(i - 1)(* + 2)(fc + 3) (11.91)2
C(n) An)
(x\ _ ; C(n) /•(")
m (1)
(x), if i = 1; (x), if i = 2; cx)
if i = 3
(11.91)3
325
11.10. THE CONTRIBUTION OF G4 TO B3
2Km 2
B(65) :
W E E /
dZ*)n^^(Z,t)«pMi;^(t)
47T 3(37V,-2)/4
^
' n=0
x E E E E k (I -sm y|) +2uy\-—,>*( arcsin y 3 m=l
J
Aft X w
m K
) \
r
«j(Y«)
4
u=Ow=0
JVt
) I ,
2Km5/27ri S S E E 33/2
47T
2?r
' 3(3iVt-2)/4
/*(")
IN • / • = = T T ^
2 ^ /AT- , i \ / / i w
/
^n
3
(iVl-l)/2
T((Nt - 1)/2)J
( , ,
N
'-i^ « ^
(0)
^(Z,t)exp[A]j:^t)L.
3
00
u
EEEEwtfW?') m=l
J
u=0ti=0
JVt j-1 An)
(G^)\
3
00
u
)) J
J
»=1
*
m=l J
u=0i;=0
xLi 0 ) 5 J (Yo(x))a; r n (x)(wo(x))' i + 5(fi(x)) 2 C m u „jN,i(x),
(11.92)
where (11.93)!
CmtttiJN,i(X) — C m t t l ,JN,iV x )> iVx j - 1
X
CmljN,i( ) =
lim
^(0)
r
E ^E( ^ X
j = 3 (=2
'
(ic)
+ ^VJCT7!)'
(11.93)2
326
CHAPTER 11. A FINER K-FUNCTIONAL EQUATION
'
—
(11.93)3
' k
"*
K
A
if* = l;
^u„jooNI_1(G(t)),
J
t
&>>
if« = 3.
s*t
i=l
OTi
' 71=0
^
E E £ E k ( J . - s i n / I ) + y - ( J , a r c s i n / fU ) ' 0J (Y<*>)u;.(t)
m=i m =l
JT ,/=n«=n u=0 «=0 L
\^
Nt V^
Nt An) (G(V\ V ^ y ^ StmuuJMi-Lj.LfcNj-i.rV ^ ,,-,2
/=2 j = ( + l fc^i V t
= " fd f eK w E E A
t
/
^
T((Nt - l)/2) u
W
J A*
27r(JV.-l)/2
(t)oo
\^
/J
/•Q(t)\\2/=1
n ^ ' ^ ( Z , *) e*p[A] E sjit
ffiW^
2Km1/27ri
-0jdzjdxTN(Z,
i=l
£(t)
m%
T 3 oo
E^V^^^E^
n=0 Nt
m=l
Nt
xE^(Y )EE^(^)E E E ' 33/2
'
) ViO" - W W +1)
Jv
47T • 3 (3AT t -2)/4
>"/
An)
fpOOl
:-';'S:;
t) exp[A] E ^ ( x ) L i 0 ) K ( 0 ) ) " + *
x(i?(x))2 E " m ( x ) E 5 j ( Y o ( x ) ) E E ^ ( * ? ) ) E < r " « « ' J M , f c N , i ( x ) l (11.94) m=l
u=0ti=0
fc^i
11.10.
THE CONTRIBUTION
327
OF G 4 TO B3
where XO)
(11.95)!
Cmui;JM,fcN,i(x) — CmuvJM,fcN,i(X)>
~(n)
,
X
.
.. lua
Nx j-1 A°)
V > V ^
WtiJMi-i,j-i,tNj-i,:W
C„„JM,fcN,i( ) = 2^1^
An)
,
if A; = 1, i = 2;
(n) tjniivJMi-ij—iONj-i
,
if k = 1, i = 3;
(n) tmttvJ N j _ i M | _ i t j _ i O
,
iffc = 2 , i = l;
(n) tmuwJOMi_ij_iNj_i
,
if k = 2, i = 3;
(n) tmuti J N j _ i O M | _ j , j- _ i
,
if k = 3,i = 1;
,
if fe = 3,i = 2. (11.95)3
nmut/JMi-ij-iNj-iO
n An)
r'
Cxi = t
(11.95)a
TNX-I)VE1^T)
j=3l=2
U
(n) tmuuJ ONj - I M I _ I J - I
Summarizing the results (11.83), (11.84), (11.86), (11.88), (11.90), (11.92) and (11.94), we have Theorem 11.16
The contribution of G4 to B3 is B. ( 5 ) 3
p « /dZT N (Z,,t)
f dxexP[A] £
3
o; m (x)(i?(x)) 2 (o;o(x))H^ £
^-(x)B^(x), (11.96)
where B^J (x) denotes a quantity almost independent of wo (x), u>i (x), u2 (x), W3 (x) and i?(x). The quantity B% can be formulated in terms of the AT-functional in the following way: Theorem 11.17 3
B 5)
i
The contribution of G4 to B3 is
3
=2 £ £ i=l m=l
f dxB™(x)^:(x.)DhD^/2DbmK(a,h,c)(x).
(11.97)
328
CHAPTER 11. A FINER K-FUNCTIONAL EQUATION
11.11
The Contribution of G5 to B3
Substituting the expression
£ E w^m5 (^ " */4' ^ ~arcsin ^ )
*E E EEEk(5'--vl) +y2 " 2v fi' arcsin \/!) /+ 9 +h=odd JFGH
n=0
Z=l«=0t)=0
L
v
7
V
'
^(t)^)^iJFGH(G(t))pJ(Y(t))4")(r(t))4(e^)^G(e(t>)^(ef) for G5 and setting Gj = 0,1 ^ 5 changing dZ^ into dZ^) in the equation (11.50)i, we have the contribution of G5 to S3 as follows.
£ 5 5 7 4 * ( ^ - */4,
E n=n n=0
E
V
« - arcsin ^ 3 ) £
E E E k ^ . - s i n j f )
JFGH m =i„ = n„=nL JFGH m = l u = 0 « = 0 / + g + h = odd
\ *
+
J£(t)
Y - ( j , arcsin/f)
» °/
\ *
V
0/
><-(t)^)^UJFGH(G(t))^(Y(t))4")(r(t))4(e^)S^G(e^)^(ef)exp^] "
Nt
(1
9\,„W|wW|2
Wt
"«
«j(t>lw(t)l2
Nt
Wt
7»(t) ^ w ( t ) • w ( t h
(11.98) Elementary, but tedious, calculations yield the following results:
v=l
11.11. THE CONTRIBUTION OF G5 TO B3
329
We are going to calculate the explicit expressions for B^ , v — 1, • • •, 6 as follows:
* T == 3 ^ £ £ /
X
47T ^ ^ > - 7T/4, y>2 C 3(37Vt-2)/4d
E E E n=0 J
dZ^l[dZ^TN(Z,t)±^mr^exp[A}
E E
k
— arcsinv^Ts) \/2/3
( J - - s i n y | ) + Y ^ ( J , arcsin y f )
m=lu=Ov=0
Nt
m5j(Y
)L5
v^m 3 / 2 7ri 3« 5 /2
}
^
£ £ / A
t
47T
'3(3/V,-2)/4
V^K 7 / 2 7Ti 3m3/2
3 X E
fe y/(Nt - l)(iVt + l)(iVt + 3)1(1 - 1) U ' - l ( 9 i
J
"
dZ^Y[dZ^TN(Z,t)f:^(t)(r^eMA}
^
s#t
i=l
3
2 7 r (AT.-l)/2
T((Nt - l)/2)
CO
3
OO
OTi
U
EEEEEw^) n=0 J m = l u=0 u=0
3
| d Z y dxTN(Z, t) J^ ^ ( x ) ( i ? ( x ) ) 4 exp[^] £ 5 J ( Y O ( X ))
oo u E Ey£($0tVm(x)V^(x740)^mu„JL,i(x),
(11.100)
m = l u = 0 v=0
where «mui)JL,i(x) — $ m u t , J L , i ( X ) '
(11.101)!
330
CHAPTER 11. A FINER K-FUNCTIONAL EQUATION
(11.101>2
a(«) ''muvJLi.iOoW' g(") ^mtii)JL,_i,i(X)
^LOL,.I0W.
—<
if i — 1; ifi
= 2;
(11.101)3
flT - ^EE/^'na^Mj^Egm 00
3
^ J ] ^(r^)4a;W5j(Y(t))4'l)(r(t))exp^]
x^ ^ - T r / ^ ^ - a r c s i n v ^
n=0 J m = l
7T
+ V1 29„u "I -,arcsin x 1
* EE ^ ^ (Nt-l)y/(N ^ w / ^ T+^l)l^ ^ ^ - x ^ ) ) ' ^ . - ! ^ ) ) ' t
(=3 k^i
V2m3/27ri 33/2^5/2
27r(iV.-l)/2
EE/ A ^n^^^z^)^^^ i W A
l/ii
t
^
s^t
-1)/2)
* E #(*) E E (r<*>)4-^jCY<*>)4°> exp[^] l
i=l
V ^ V ^ y
2
/ ^ ( t ) - , V ^ V ^ *^
u=0 t)=0
V2K7'2
.(*)
J m=l
3
)
(ivt - i)>/i
i = 3 fe^i -
^tm"iJM|-i,iN|_i,i("
p.
00 u
7T1
33/2m3/2
/
dZ / dxTN(Z,t) £ - ^ M £ffj(Y 0 (x)) ^ ^ F E ( $ 0 ( x ) ) J
i=l
*
J
«=0«=0
11.11.
331
THE CONTRIB UTION OF G5 TO B3
3
x Y2 (-R(x)) 4 a; m (x)y/u 0 (x)L£
exp[A] ^
4 U vJM,itN,»( x ).
(11.102)
TO=1
where (11.103)!
$mm>JM,fcN,i(x) — ^ m u u jM,*;N,i( X )' <,(n)
/ x _ v ^ V y'
2)wmut)JM£_i)fcNii!i(x)
Q(«)
m»«JM,w N w o W -
if* = l , i = 2;
^muvJMj.iONi.jW'
if * = 1> * = 3;
^muvJNi-iMi.joWi
iffc= 2, i = 1; (11.103)3
x
muuJM i_i,fcN,_i,t( ) - "
(n)
0:
_iN,_,o(x)«
i f * = 2 , * = 3;
,(«)
%uvJOMMNi_i(x)'
iffc— 3, i — 1;
,_!»«,_! W»
B
(11.103)2
(iVx - 1 ) ^
1=3
if* = 3 , i = 2.
2 3 / 2 m 3 / 2 7Ti
(63) _
/
\
^
' n=0 J m=l
xL<->(rW) exp[4 f ) £
fe
(^ ™
(t)
3
00
^/wn
w f ) + Y 2 ^ ( J , arcsin j f )
u=0t;=0
ATt x
r
JVt
y
^
(n)
.(G<*))
•tini.i.JL|l> ltJ _ 1 ,f
a
^3
/Q(t)^2
S 4 >/((w - iw - ir^iw
-w^/ a /^n^(^)^
2 7 r (Ar.-i)/2
r((ivt -1)/2)
332
CHAPTER
11. A FINER K-FUNCTIONAL
(t) (t)\4 ^"1
dc l
i=l
J
2 3 / 2 K 7 / 2 7Ti 3 2
3 2
' 3 / m /
£s£s
=flJ (Y<*>)
m=l
Nt
J
EQUATION
M
Nt
1?
(0) tmuvJL''(ii -) i . j - i '
.(G ( t ) )
' f e ^ i v W ) 2 - DO'2 -1) 3
3
x
JdzJd*TN(Z,') E J|( ) E E (*(*)) Vn(x)^^^(xj 2—1
J 771=1
x 5 J ( Y o ( x ) ) 4 0 ) exp[A] E E y s ( ^ o ( x ) ) t ? m ^ J L ( 1 ) , i ( x ) ,
(11.104)
u=0 v=0
where '?mut>JL<»>,i(x) -
Wx j - 1
*<">
JL(1)
W
l? ( 0 )
(1)
(X)
(1)
mWjL 1
(x) = l i m E E
'- --"
t
.(*)
(11.105)2
SV((Jv x ) 2 -i)(j a -i)'
j=3
^
(11.105)i
^muvJLd),*^)'
. . OOv
mutJL
'
(n)
(i,
tf
'
(x), if » = 2;
(11.105)3
I — 1,J — 1
i —i,j — l '
0 (")
munJOOL.(1), . ,
(x),
if i = 3.
«r - ^^EEl-^n-^wois^x:^) 3
x*f ^
-TT/4,^
(t)
-arcsinv^J^^exp^] E - ^ (t) 0
££££k(i.™\/I)
n=0 J u=0 u=0
,arcsin ,, „ + ^2V — 4 V 3
4V*))
11.11.
THE CONTRIBUTION
333
OF G5 TO B3
Nt-2k+lj-l
$
(n) tmuvJL
(i) I —i,fc+i
(Of))2
(GW)(H3 (1)
'9J(Y<,>) S § S VW -!)(". + 1)J(J - D(* + 2)(* + 3) 27r(iVt-l)/2
- ^ ^ E E ^ ^ n ^ ^ M a s ^ r((iv 3
ac 4 1 4
(t) Wn
oo
t
-1)/2)
u
xES^ ) ^] ±E^ , .-/, ^,(t)E E E w ) 4 ( 9*, m = l A/WQ
i=l
tf(0)
Nt-2k+lj-l
(t)\l(0)
J
»=0»=0
(1,
tmu«JLJ_'j
^2 UU 23/2K7/27TJ 3
3/2m3/2
(G<*>) k+1,i
y/W ~ i ) W + i)i(j -1)(* + 2)(fc + 3) 3
3
y'dZydxTJv(Z,t)^^(x)(i?(x))4exp[J4] £ i=l
x
o^Mx/^x]
m~l
EEE^( $ »w) L l 0) ^( Y ( x ))''—(.»,i(x), J
(n.106)
u=0 v=0
where (11.107)!
£m„„JL,i(x) = ^ L j L ( D , i ( X ) ' 1?(0)
JV«-2fc+M-l
m-uuJL ( i )
CLW*) = " "* = £ £^£ V(N 2 J=3
X
- l)(iV x + l ) j ( j - 1)(* + 2)(fc + 3)' (11.107)2
tmurJLj'i'j
^LLIL^^^W ~ <
t m u t i J O L( ji )^
tmutiJOOL
(65)
B:
=-SEE/ 4
fc+100
1?(n)
0'(n)
..(*)
fc+10
(i)
(x),
if i = 1;
(x),
ifz = 2;
(11.107)3
(x), if i = 3.
i-i.it+i
<*>n^(^^££m
334
CHAPTER 11. A FINER K-FUNCTIONAL EQUATION
xsfip1? - TT/4, if™ - arcsin y/2/l J ( r ^ ) 4 exp[A)
x EE E EE k(i.-si n y|) +^-(5,™^) n=0
J
L
^
JVt
JVt
m=lu=0„=0
V
'
V
^
,o(n) ^tmtH)JNi_i,i(^
^
^"^'""'Es^Tii^19 )
»(*)))3
•|^EE/A ^ n - ^ - w o p ^ , A
t
•'"A
s
^
t
3
dc
27r(Af,-l)/2
r((jv t - i)/2)j
oo u
x(rW)*«xp[4 £ £ ( t ) ^ £ £ 5 > < * o t } ) t=l ,(t) ^m
J
m=lu=0«=0
JV, j - i .jC) m u t , J N /,- v ((tt))\^ Jr-(0) utiJN ( 0 )V V ^ VV~^v " tt m ' i.bt^
9J(Y
°
>is
J
g g w + DVJO-i) 3
oo u
(th ^
^
»=i
J
j
m=i«=o»=o
:(i?(x)) 4 u; m (x)V W o(x)3j(Yo(x))Li 0) ^ m ^j N , i (x),
(11.108)
where "mui>JN,i( x ) — " m m J N . i W i
«(«)
m
/• \ _ 1-
""JN'i(x)"
JVX J - l W
,4(0) (x\ u muiiJN|-i,i^AJ
^ f e (iVx + DV5CF1)'
(11.109)i
(11.109)2
^mu«aN|.iOo(x)' iri = l; ^mut)JNi_i,i(X)
=
S ^m«jjJON,_io(X)'
. ^muuJOON,-!^)'
" * = 2; i f
* =3 -
(11.109)3
11.11.
B
THE CONTRIBUTION
335
OF G 5 TO B3
(66) A
t
•/ii*
s
^t
/
i=l
\
3
(t)
x*( V W - * / 4 , ?<*> - arcsin v ^ ) E E ( r < ' > ) 4 - ^ f l j C Y « ) ^ ' J m=l Jii'* oo
oo
it
EEE n=0 u~0 v=0
L
!) n?(J.
*ita ( 7 > a r c s i n \l i ) + + y 2«" I T ' a r c s i n V Q
hhh 2m3/27ri 3372
( M
~ '—(fc)} (^-i( *)}
W-DV^-I)
^EE/4 ^ n ^ ^ w o j B ^ A
t
•/A>>
s^t
3
3
4 n) (r (t) )exp[A]
„
r((ivt-i)/2)
(t)
E E=i£i E |miw ^ ) ^ ( Y ( t ) ) J
4
27r(Ar.-i)/a
m;
lU),
(t)
Aft J - l T 4(°)
x E E w f W «PM E E E u=0»=0
2 f C 7 / 2 7Ti "33/2m3/2
fc#i
1=3 !=2 3
Utmu
:T:T^r-\iGit))
\
*
J-JV-JU
JJ
3
J dz Jd^TN(z,t)Y: E E ^ w w * ) ) 4 ^ ^ ^ ) J
X E E ^ ( ^ o ^ ) ^ u = 0 ti=0
m=l i=l
exp[^] E fc^i
tftm«uJM,fcN,«(x),
(11.110)
where «tmuuJM,/cN,i(XJ — "tmuuJM.fcN ,i(X)'
•T(n)
W
/ \
tmut;JM,fcN,i W
—
Nt i - 1 ,Q( n ) (CK*)"* Y ^ V ^ tmuuJMi-i^-i.fcNj-^i^" /
/ - / 2-( j = 3 (=2
(iVt - 1 ) V 5 ( H )
(li.ni)i
(11.111)2
336
CHAPTER 11. A FINER K-FUNCTIONAL
a(n) "tmtivJMi.ij.i.ltNj.i.iW
—
EQUATION
^mm>JM,_ 1 ,.,_ 1 N.;_ 1 o( x )>
if k = l,i — 2;
^mwJMi.Lj^ONj.iWi
iffc= 1 ,1 = 3;
t
iffc= 2 , 2 = 1;
'miii;JNj.iM,.1|j-io(X)'
'
(x)'
if * = 2 , i = 3;
''m«vJNj-iOM|-ilj.1 (X)'
if K = 3 , t = 1;
^mutiJON^jMi-i 3 -_i( x )'
if * = 3 , t = 2.
^muvJOMi-u-xNj-i
(11.111)3 Summarizing the results (11.99), (11.100), (11.102), (11.104), (11.106), (11.108) and (11.110), we have Theorem 11.18
The contribution of G5 to B3 is 3
B36)«
3
/*dZTw(Z,t)/rfxexp[^]^Wm(x)(i?(x))4(a;o(x))1/2^^(x)B^(x), m=l
i=l
(11.112) where BJJ (x) denotes a quantity almost independent of wo(x), u\ (x), W2(x), W3(x) and R(x). The quantity JB3 can be expressed in terms of the K-functional in the following way: Theorem 11.19 3 B
The contribution of G$ to B3 is
3
36) = 4 E E
IdxB^{K)^{*)DlDl'*DbmK{a,h,c)(-x).
i=l m=l
11.12
(11.113)
*
A Finer K-Functional
Equation
Summarizing the results (11.34)', (11.42), (11.49), (11.54), (11.58), (11.63), (11.81), (11.97) and (11.113), we have Theorem 11.20
If the distribution F(Z,t)
is of the form (10.63), then the
evolution of the .^-functional is governed by the following functional equation,
337
11.12. A FINER K-FUNCTIONAL EQUATION called the finer K-functional equation: dK an
where i?i,i?2
(a,b,c) = i ? 1 + i ? 2 + t f 3 +
tf4,
(11.114)
d $3 are defined in the equations (7.19)2, (7.19)3, (7-19)4 respec-
tively and $4 is defined as follows:
/^WEECW^^V*1-^
** = lEE fc
J = 1fc^j^
p=l g=l
+4 E E
/**$(x)^(x)D^y2i^(a,b,c)(x)
4
/dxB53fc>(x)^(x)2?^A-(o,b,C)(x)
+
E E
^ = 2EEEE/^tL(x)^(x)
+
fe=l j = l i = l m^i
xDhD^2DbjDbkDbmK(a,b,c)(x) 3 4
+ E E / dxB
1
/
2
^ K ^ b , c)(x)
3
+4
EE/ dxB l 3) ( x )^( x )^^^^( a ' b - c )( x ) <=i j / t
3
3
3
+^E E E i=l m=l 3
+
2
3 + 4
*
fdM^)^)DhD-^DbmDlkK(a,h,c)(x)
fe=l' 3
E E
[dM%(*)§l(x)DhDZ+V2DbmK(a,b,c)(x.)
3
E E i=l m=l
[dxB%(x)^(x)DlD^DbmK(a,h,c)(x.).
(11.115)
l
where the coefficients B's are the quantities stated in the preceding sections.
This page is intentionally left blank
Chapter 12
Conclusions 12.1
A View on Turbulence
Navier [63], Poisson [65], de Saint-venant [68] and Stokes [71] derived the so called Navier-Stokes equations in succession about one hundred and sixty years ago. Navier derived the equations for incompressible flows on molecular level. Poisson generalized Navier's argument to the case of compressible flows. De SaintVenant and Stokes independently based their derivations on stress analysis, i.e., on macroscopic level. As was pointed out by Anderson [1], although Navier's equations were of correct form, his reasoning was greatly flawed, and it is almost a fluke that he obtained the correct terms. The well known Reynolds' experiment for flows in straight tubes exhibited the random, or turbulent, features of fluid flows about one hundred and twenty years ago, but most contemporary experts still hold the views that Poiseuille flow is simply unstable in case of Reynolds number exceeding the critical Reynolds number and the turbulent feature of the flow is a consequence of the instability of the flow, of which each sample still conforms to the Navier-Stokes equations. Most scientists of fluid dynamics refused to consider the turbulence phenomena on molecular level. Maybe the only exception is John von Neumann [81], who tried to discuss the molecular kinetic origin of turbulence. The following passage 339
340
CHAPTER
12.
CONCLUSIONS
extracted from the famous Monin-Yaglom's monograph ([60]) on turbulence might represent the typical opinion of the contemporary experts of fluid dynamics: We must note here, however, that in the literature on the theory of turbulence, it is sometimes stated that in a turbulent flow the fluid dynamic equations, in general, are inapplicable. If one ignores completely unjustified assertions, then the only important question here is whether the molecular fluctuation can cause random splashes capable of transmitting energy to smaller-scale fluid dynamic disturbances, and, thus, for example, stimulating transition to turbulence.
At
present, it is almost widely agreed that even if such processes are possible, their role is, in every case, extremely small, so that it may be completely ignored to first
approximation.
According to my opinion, statistical mechanics concerns itself with the description of macroscopic phenomena in terms of a probability distribution of molecules or other particles. The transmission of the energy of molecular fluctuation ( I think, it might indicate the energy of thermal fluctuation in equilibrium or local equilibrium media ) to that of small-scale fluid dynamic disturbance is only a part of the by-products of the study in statistical mechanics. It is a great misunderstanding that statistical mechanics concerns itself solely with the thermal fluctuations and the statistical theory of turbulence cannot be understood on molecular level, and both theories must be studied separately and independently of each other. The present book devotes itself to finding a proper probability distribution, which, at least in principle, contains all round the information of both the thermal fluctuations and the turbulence phenomena. There is no reason to conclude that this approach is unjustified. Actually all flows, in nature or in laboratories, are turbulent, i.e., random. The so called laminar flow is also a random one, but with negligibly small variances. As was pointed out by Grad in 1983, the Gibbs, or local Gibbs, distributions and the distributions perturbed from them are
12.1. A VIEW ON TURBULENCE
341
not versatile enough to describe fluid motions with large (turbulent) fluctuations. The turbulent Gibbs distributions, defined in the equation (6.20), and the distributions perturbed from them are good candidates in describing the general fluid motions, including the fluid motion with large (turbulent) fluctuations. I think, the reason why the Chapman-Enskog techniques for the Boltzmann hierarchy are restricted by the assumption that fluctuations are small compared to the basic flow is that the 1 s t order term in the Chapman-Enskog expansion for the Boltzmann hierarchy is the local Maxwellian distribution, which corresponds to the (laminar) basic flow. In the present paper, the 1 s t order term of our asymptotic techniques is a turbulent Gibbs distribution (6.20). Every convex linear combination of turbulent Gibbs distributions is a turbulent Gibbs distribution too. This is exactly the requirement a class of molecular distributions in statistical theory of turbulence should fulfill. Hence the fluctuations of the flows specified by turbulent Gibbs distributions are not restricted to be small. This is the reason why the results of the present book have remedied the faults indicated by Grad, and therefore, are applicable to turbulence regimes.
Both the Euler iC-functional equation and finer K-functional equation are derived directly from the Liouville equation. Hence a general form of fluid motion is a turbulent flow, i.e., it is random and satisfies the Euler, or the finer, Jsf-functional equation. In classical theory on hydrodynamic stability, the basic flow is assumed to be a laminar ( or, deterministic ) flow and so is the the disturbance. Because the randomness is universal for fluid flows in nature, it is reasonable to assume that the disturbance is also random in general, even if the basic flow is deterministic. Thus we have to use the if-functional equation to study hydrodynamic stability. Of course, in certain circumstances, we should make some restrictions on the disturbance, e.g., the second order correlations, or higher order correlations, being negligibly small. Using the techniques, which have been successfully used
CHAPTER
342
12.
CONCLUSIONS
in the classical theory of hydrodynamic stability, we shall get some new results on the theory of transition from the laminar to the turbulent. Here the turbulence arises as a consequence of the sensitive dependence not only on the deterministic disturbances, but also on the disturbance of the correlations of their initial data. Hence turbulence cannot be interpreted as a deterministic chaos ( or a strange attractor ). There are a lot of papers devoting themselves to the study of hydrodynamic limits for some probability models (see, e.g., Yau and Jensen [85] and Kipnis and Landim [43]). Their logical deduction are mathematically rigorous. The hydrodynamic equations obtained in their books are equations governing deterministic flows. Although turbulence phenomena prevail everywhere, especially in nature, but the flows in laboratories can be made laminar or approximately laminar. We need to understand the mechanism of making a laminar flow, i.e., a flow with negligibly small variances. A mixture of the Liouville model and a probability model might provides us with the mechanism.
12.2
Features of the Finer /^-Functional Equation
What we have obtained in this paper are two functional equations governing the evolution of the if-functional. The first is the Euler K-functional equation, which governs the evolutions of the flows corresponding to the turbulent Gibbs distributions. The Euler K-functional equation will be reduced to the Hopf functional equation for inviscid fluids in case of incompressible flows. In classical fluid dynamics, the flows are divided into two classes: inviscid flows and viscous flows. The motions of inviscid flows are governed by Euler equations and their molecular probability distributions are local Gibbs distributions. The motions of viscous flows are governed by Navier-Stokes equations and their molecular probability
12.3. JUSTIFICATION
OF THE FINER K-FUNCTIONAL
EQUATION
343
distributions are those perturbed from local Gibbs distributions by some perturbation technique, e.g., Enskog-Chapman technique in the kinetic theory of gases. It is natural to devise a perturbation technique, similar to the Enskog-Chapman technique or something like that, to get the second X-functional equation, called the finer K-functional equation. The perturbation scheme used in the present book is presented in the First Proposal of Gross Determinism, i.e., the equation (9.4) or its special case (9.4)'. The basic idea behind the First Proposal of Gross Determinism is that the theory in the present book is applicable to those fluid flows, of which the changes of the macroscopic quantities are far much smoother than those of the microscopic ones. I think, almost all theories in non-equilibrium statistical mechanics have, explicitly or implicitly, made this restriction. To my surprise, the finer if-functional equation, obtained in the Theorem 11.20, cannot be reduced to the Hopf functional equation in case of incompressible flows. The right hand side of the finer jFC-functional equation (11.98) has nine terms, of which four have DhK and five D\K.
Hence the heat
energy (or the temperature) cannot be eliminated in the equation, even if the flow is incompressible. It means that the evolution of the flow velocity cannot be governed by a closed equation without the interference of the heat energy (or the temperature) in case of incompressible flows. The evolution of heat energy plays an important role in the description of the evolution of momentum. The interaction of the heat energy with the momentum cannot be ignored in fluid motion. This is quite different from that in the classical dynamics of incompressible flows.
12.3
Justification of the Finer if-Functional Equation
The interpretation of the fact that the results of the perturbation methods for the Liouville equations and the Boltzmann equations are so different might be as
344
CHAPTER
12.
CONCLUSIONS
follows. The solution F perturbed from the solution FQ to the Liouville equation is of the form: F ( z i , • • ;zN;t)
= F 0 (zi, • • ;zN;t)
+ F1(z1,- • -,zN;t)
(12.1)
and the solution / perturbed from the solution /o to the Boltzmann equation is of the form: / ( z ; t ) = /o(z;t) + / i ( * ; t ) .
(12-2)
In virtue of the assumption of molecular chaos, on which the whole theory of Boltzmann equations is based, the distribution function on the N particle phase space corresponding to the one particle distribution (12.2) is N
F ( z i , - • -,zN;t)
N
= Y[f{zi;t)
= JJ(/o(zi;t) + /i(z,;t)).
(=1
(12.3)
(=1
Even if N
F0(z1,---,zN;t)
= l[f0(zl;t),
(12.4)
1=1
the right hand sides of the equations (12.1) and (12.3) are of completely different structures and almost impossible to be identical. If Fo is the convex linear combination of the distribution functions Yli=i fo(zi> *) :
r
N
F0(z1,--.,zJV;i)= /d/x(i/)JI/^(z,;t), J
(12.5)
i=i
it is almost impossible to have
f
N
Fo(zi,---,zjv;t) + Fi(zi,---,zjv;<) = / dfi(u) f j J
1=1 1—t
fo(*i;t) +
K(zi;t) , (12.6)
L
because both sides of the equation (12.6) are of completely different forms. This is the very reason why we should construct a nonequilibrium statistical mechanics without the assumption of molecular chaos. Most papers on hydrodynamic limits devoted themselves to the derivations of the Euler equations or Navier-Stokes equations. Is the finer jFsT-functional equation reliable? What I can say at present is that the perturbation scheme, which is
12.4. OPEN PROBLEMS
345
used to derive the finer AT-functional equation, is reasonable, because the design of the perturbation scheme is based on the following reasonable assumption: macroscopic variables are far much smoother than the microscopic ones. Of course, the final justification of the finer /^-functional equation must be the comparison with experimental data.
12.4
Open Problems
The work of the present book is really crude. Many interesting problems are left open. Among them, the most important are the following four open problems. First Main Open Problem: The numerical determinations of the coefficients Bpojfc(x),- • •, and so on, are desirable. In order to get numerical values of the coefficients BL/. fc (x), • • • and those like that, we should solve the equation (10.47)' and those like that. I think, it is almost impossible to get exact solutions to these equations. Approximate, but explicit, solutions are also welcome. The asymptotic form of the dependence of the coefficients BJ, L(x), - o n JVX, m and K is extremely important for our theory. Second M a i n O p e n P r o b l e m : What is a condition, under which a turbulent Gibbs measure should be, in a certain sense, a Reynolds-Gibbs distribution? De Finetti-Hewitt-Savage result [40] (i.e., any symmetric measure can be expressed as a convex linear combination of product measures) holds only for infinite cartesian product of measure spaces. Counter examples can be constructed for finite cartesian product of measure spaces. Is there a version of De Finetti-Hewitt-Savage theorem for projective limit of finite cartesian product of measure spaces? Third M a i n Open Problem: What is a condition, under which the total intermolecular potential energy inside the cube can be replaced with the product of K 3 and the Gibbs mean? Is it possible to improve the P r o p o s i t i o n 11.1? What is a condition for the validity of the condition (11.17) of the P r o p o s i t i o n 11.1?
346
CHAPTER
12.
CONCLUSIONS
We have to study the asymptotic behavior of an integral, when the dimension of the domain of the integral increases infinitely. Fourth Main Open Problem: Quantization of the theory of the present book. One of the aims of Massignon's construction of his theory [58] is to make the quantization of the non-equilibrium statistical mechanics possible. He guessed that the non-equilibrium quantum statistical mechanics so constructed might be used to interpret the superconductivity and superfluidity.
Appendix A
Some Facts A b o u t Spherical Harmonics A.l
Higher Dimensional Spherical Harmonics
In the book we have tried to get an expansion of the solution to the equation (10.1) in terms of three families of spherical harmonics in ©^ , ©3
and 0 3 ,
respectively. In this Appendix A we shall collect some facts about the spherical harmonics, which have been used in the book and will be used in the future research work of determining the coefficients of the expansion. The notations for spherical harmonics used in the present paper are essentially the same as used in Vilenkin's monograph ([79]) or Vilenkin and Klimyk's monograph ([80]). A classical reference book on spherical harmonics is [26] and shorter ones [4] and [5]. In our book we have used the convention made in Vilenkin's monograph: the measure on the (Ns — 2)—dimensional unit sphere is unity, i.e., the inner product in the Hilbert space L2(SN"~2)
is defined as
An Zth order spherical harmonics in &\
is
sk(e, w ) 347
348
APPENDIX A. SOME FACTS ABOUT SPHERICAL
HARMONICS
(Al) where C^(t) denotes the Gegenbauer polynomial with the indices u and v, and K = (kl,---,kNm-4,±kN,-3),
(A.2)
a finite sequence of integers satisfying the inequalities: I = fc0 > fci > • • • > kN,-3 > 0,
(A3)
and AlK the normalization constant defined by (A — X2
iiN j 3) a ^~} T -i-'N—:K 1T 1 // \c r f c , - J': , 4-. , (cose( v^" "
n
(
0
2
(sin6» (s )) 2 ^+ 1+Ar '- J '- 3 d(9 (s) .
)
(A4) Hence we have the following formula for (A^) 2 :
{A K)2 =
'
nk1)
N„_4
2 2 f c i + > + ^ - ^ 5 ( ^ - fcj+1)!(iVs - j + 2kj - 3)r 2 (^=±^
j=0
,/*r(kj + kj+1 +
xn
N -j-a) ^
+
J
kj+1)
u
(A5)
The following two properties of spherical harmonics (see, e.g., [79] or [80]) will be frequently used in the sequel: 1. Every homogeneous polynomial / of degree / in (Na — 1) -dimensional vector x(=(xi,X2, • • • ,x^r s _i)) can be uniquely decomposed into a sum of the polynomials of the form r2k/ij_2fe(x): [1/2]
/(x) = £r 2 %_ 2fe (x), fc=0
(A6)
A.2.
A LIST OF SPHERICAL
349
HARMONICS
where -JV.-l
vl/2
j=l
/
and /ij_2fc(x) is a spherical harmonic polynomial of degree / — 2k in x. In other words, the value of any homogeneous polynomial / ( x ) of degree / at any point of the unit sphere coincides with the value of the (nonhomogeneous) harmonic polynomial /i(x) = ^ Z ^ o /i/_2fc(x) at the same point of the unit sphere. 2. Any square integrable function /(£) on the (Ns — 2) dimensional unit sphere gAr«-2 c a n b e
expancjec]
ag a series convergent in the Hilbert space
L2(SN"~2):
K
A.2
A List of Spherical Harmonics
The details about the properties of the spherical harmonics and the Gegenbauer polynomials can be found in Vilenkin's monograph cited above. In the present paper, it is enough to list the forms of the spherical harmonics, which make a part of an orthonormal system of spherical harmonic functions on the unit sphere, of orders not greater than five. Before writing down the list, we introduce the following notations for the multi-indices of the spherical harmonics used in Vilenkin's monograph ([79]): O = ( 0 1 _ ^ 0 ),
(A.9)
JV„-3 tuple
Nfc = (
1.---.1 Nm-k-l
, 0 , - - - , 0 ) , Ns-l>k>2,
(A10)
tuple fc-2 tuple
N ± = ( l . - . - . l ,±1), N.-4
tuple
(All)
350
APPENDIX
A. SOME FACTS ABOUT SPHERICAL
Mfc = ( 2,---,2 Na—k—l
Mlk = ( 2 , - - , 2
HARMONICS
,0,---,0), A T s - l > f c > 2 ,
(A.U)
t u p l e fc-2 t u p l e
, l t - - - , l , 0 , - - - , 0 ) , JV. - 1 > * : > / > 2,
(A13)
JV« —fc—1 t u p l e fe—i t u p l e 1 — 2 t u p l e
M ± = ( 2, •••,2 ,±2),
(A14)
AT.-4 tuple
M f c ± = ( 2,---,2
,1,---,1,±1), J V s - l > f c > 2 ,
(A15)
iV„ —fc —1 t u p l e k — 3 t u p l e
Lfc = ( 3,---,3
,0,---,0), J V . - l >fc>2,
(A16)
N s —fc— 1 t u p l e fc-2 tuple
L[^ = ( 3,---,3
,l,---,l,0,---,0), i V s - l > f c > / > 2 ,
(A17)
N, — k — 1 t u p l e fc—( t u p l e I —2 tuple
L((f=(
3,--,3
)2,--.,2,0,---,0),
iVs-l>A;>/>2,
(4.18)
Ns — k — l t u p l e fe—i t u p l e J —2 t u p l e
Lpife = ( 3,---,3
,2,---,2Il,---,l,0,---,0), W s - l > f c > / > p > 2 ,
(A19)
Af. —fe—lt u p l e k — l tuple i—p tuple p—2 tuple
L± = ( 3 , - - - , 3 ,±3), N,-4
L
i± = ( V - , 3
(A20)
tuple
,l,--,i,±i), iVs-l>fc>2,
(A21)
Ns — k—l tuple fe—3 tuple
4± = ( V",3
,2,-",2,±2), JVs-l>fc>2,
JV» —fc—1 t u p l e k—3 tuple
(A22)
A.2.
A LIST OF SPHERICAL HARMONICS
Lifc± = ( 3,---,3 Ns—k-l
, 2, - • -, 2 , 1 , • • •, 1, ±1), JV. - 1 > * > i > 2,
(A23)
tuple k—l tuple <—3 tuple
Kfc = ( 4,---,4 Ns—k—l
K^=(
351
4,---,4
,0,.--,0), J V s - l > f c > 2 ,
(A24)
t u p l e k—2 t u p l e
Ns-l>k>l>2,
(A25)
JV.-l>fc>I>2,
(A.26)
,3,---,3,0,---,0), J V s - l > f e > / > 2 ,
(A.27)
,l,...,l,0,---,0),
JV.-fc—1 tuple fc-i tuple (—2 tuple
K ( ( ^ = ( 4,---,4 N,—k~l
K ^ t
)2,---,2,0.---,0),
t u p l e fc—i t u p l e 1—2 t u p l e
4,---,4
iVa—/c—1 tuple A;—i tuple I — 2 tuple
K
Sfc )
=
(
4
'---'4
,2,---,2,!,-••,1,0,---,0), 7 V s - l > f c > / > p > 2 ,
Afs—*: —1 t u p l e k—l t u p l e I—p t u p l e p—2 t u p l e
(A28)
K
S'=(
V"-,4
,3,---,3,l,---,l,0,---,0), A r s - l > f c > / > p > 2 ,
iV8—fc—1 t u p l e &—/ t u p l e I—p t u p l e p—2 t u p l e
(A29)
K
S*)=(
4,-.-,4
,3,-.-,3>2,-..,2,0>---,0), J V . - l > f c > I > p > 2 ,
iV, —fc— l t u p l e k—l t u p l e (—p t u p l e p—2 t u p l e
(A30)
Kq„*fc = ( 4, - • -, 4
,3,---,3,2,---,2,l,---,l,0,---,0),
iVs —A;—1 tuple /c—/tuple l—p tuple p—g tuple q — 2 tuple
352
APPENDIX A. SOME FACTS ABOUT SPHERICAL
HARMONICS
Ns-l>k>l>p>q>2,
(A31)
K± = ( 4 , - - - , 4 , ± 4 ) ,
(A32)
NB—4 tuple
**!
4
=(
'---'4
,!,•••,1,±1), i V s - l > f c > 2 ,
(A33)
N,—k—l tuple k—3 tuple
K
fc± = ( V " > 4
,2,---,2,±2)I A T s - l > f c > 2 ,
(A34)
jV.—fc—l tuple k—3 tuple
K
fc± = ( ^•••A
,3,--,3,±3), JV8-l>fc>2,
(A35)
NB—k—1 tuple k—3 tuple
K|fc±=(
4,---,4
, 2, • • •, 2 , 1 , • • -, 1, ±1), JV. - 1 > fc > i > 2,
(A36)
N, — k — l tuple fc—( tuple Z—3 tuple
K ^
= ( 4,---,4
, 3, • • •, 3 , 1 , • • •, 1, ±1), JV. - 1 > fc > i > 2,
(A37)
Na — k — 1 tuple fe —( tuple I — 3 tuple
K
{k±=(
4
----'4
)3l-.-,3)2,---,2,±2),
iV,-l>fc>/>2,
(A38)
AT. — k — 1 tuple fc —( tuple J—3 tuple
Kp^± = ( 4 , - - , 4 N,-k—l
, 3 , - . - 1 3 , 2 , - - - , 2 , l , - - . , l , ± l ) , iVs - 1 > fc > / > p > 2,
tuple fc-i tuple i—3 tuple p—3 tuple
(A39) It is easy to get the following list of corresponding spherical harmonics S ^ (©) ) • H° 0 (ei s ) ) = 1.
(4.40)
A.2.
A LIST OF SPHERICAL
353
HARMONICS
AT.-2
Hk<e<") = ,/Si=T««f2 M n n - • » - ^ S ))\ -k+l,i'W
(^•41)
w
a
j=k
where sin
if i = 1;
A(*(8))={ sin^s)cos^s),
sin
if i = 2;
cos if2 \
V 2
(A42)
if i = 3.
j=2
^/iVs~1
rw(s)+iW(s)i
(A43)
It is convenient to introduce the following spherical harmonics with fictitious multi-indices N2 and N i :
^MS)) ^(e<s))
1
V2
zhM^ + ^AQ?) ^ + (e| s) )-^_(ef))
\/2i .
(») w.
(4.43),
(s)
(A43) 2
r(s)ft($(s)) m2,i-
N —2 l t (0<
s)
iV —2
) = ^(Ns + 1)(N. - l ) c o s ^ M c o S 0 ( ( i > M f [ s i n a # > ft sin 6 $ j=k
j=l
V(JV S +i)(jv. - i)^r+i,i^+i
(A44)
(r(s)A(*(s)))2 ^2
, n (s)s _ VHNs ~ 1)(JV. + 1) V2(fc - 1)
Vfc(W. - 1)(NS + 1) V2(fc - l)(r«ft(*W))2
W„-2
«*"'&.<-r I l s i n 2 « j=fe
(^/-iD^)2 J=2
(A45)
354
APPENDIX
=2
S>( s ) .
A. SOME FACTS ABOUT SPHERICAL
_ y/(Ns + l)(Na - 1)
^MM )
HARMONICS
n si*2*s ( c o s ^ ± i s i n C )
N.-2
2V2
3(8)
(<»)\2
i=2
V(iV s + 1)(JV. - 1) 2 v / 2(r( s >A($( s ))) 2
(^) 2 ±2h^£-(^) 2
(A46)
It is convenient to introduce the following spherical harmonics with fictitious multi-indices M2 and M12:
^Ms)) = TIL's2d+(e, 1
w
V(iV g + 1)(JV. - 1) 2(r( s )A($( s ))) 2
:
) + E&_(e«)
(-£)2 - (^) 2 V(ivs + i)(jvs-i)
^ + (e{ s) )-sg,_(eW)
Mu(®i )
(i4.46)j
>/2iL
(rW/3,(*W))
(s) (s)
J
^s.i^.i(A46) 2
=•2 / p w - i
>/(". + W
-1)
c Q s
^ ^ ^2s.n2 ^
V2
j=fc
fc-1 n
sin
^(cog,(s)
± isin,(s)}
J=2
y(^i)(^i)„^i]i(^±iwg)_
(A47)
V^(rW/8i(*W)) 2
It is convenient to introduce the following spherical harmonics with fictitious multi-indices M2* and M n : 1 B) / (8) M 2 f c ( feti ) - ~7=y^ + (ei )
= S2
+ ^_(e«)
^(JV. + 1)(ATS - 1) (rW/3j(*<">))2
u/
(s) (s) *+i.*u'3.*-
(A47) x =•?. / fi (»h _ _ 1 V^i
3U©n =
•=•2 "2
/ of (O s )^
(s)> A = 2 CftW-v
V(NS + 1)(NS-1) (r( s )ft($( s ))) 2
(>) u,
(s)
*+i.«tu2.<(A47) 2
A.2.
A LIST OF SPHERICAL
355
HARMONICS
^ ( e , ) = y/{N. + 3)(iVs + 1)(JV. - 1) cosC M cos9\ m J l t i cos0™ lti N —2
x
k—1
1—1
3
2
IIsin (^)nsin (^)nsin(^) i=k
3=1
(s) . »
,» (A48)
( r (s) / 3 i ($(s)))3
-3
,QCh,
V(iVs + 3)(iVs + l ) ( i V s - l ) I
JV.-2
*:-!
x cos0fc_M ( cos2 0;_ M - y ) 1 3 sin 3 8jti J J sin 2 6jti l
^
>
V(iY. + 3)(iY. + l ) ( i V . - i y
w
j=k
j=i
(*.*)2-yi;MS)a
(A49)
i=2
-3
, Q ( . h _ \/(k + 2)(JV. - l)(Na + 1)(NS + 3)
'
J=fc
y/(k + 2)(Na - 1)(JV. + 1)(7VS + 3)
V2(fc + l)(r( s )A($^)) 3
j=l
H (s)
W J + l,»
fc+1
a
2 <™&.«) -rro2>i?) j=2 (A50)
sL(©is)) y^iV. - l)(Na + 1)(JV. + 3)(fcT2) v/6(fc - 1)
, 3 z,(s)
fc —1,»
3cos^M ifc + 2
AT.-2
n
356
APPENDIX A. SOME FACTS ABOUT SPHERICAL HARMONICS
y/(Na - 1)(N. + 1)(JV. + 3)(fc + 2) y/6(k
- l)(r( s )A(*( s ))) 3
^,» ± (©*)
^
W.-2 j=k
+
3w
(s) fc+1
( ^ i ^ - T Tfc^+E2 M J ) 3
(A51)
J=2
3)(^+1)(^-1)COS^MCOS^^
fe-1
1-1
j=l
j=2
s
4- ; „ > h V(iVs + 3)(ATS + l)(iVs - 1) < (A) , i„< » i , iA...0O «i ± ™w) ^2 (r(s)ft($(s)))3
(A52)
It is convenient to introduce the following spherical harmonics with fictitious multi-indices Li2ik and Ijiik'-
HL (ei)
-
~7i
Ku+M+
= V(NS + 3)(JV, + 1)(JVS - 1)
^(©0
*$«>-&)
W
(s) (s) (s) t+l,itl,l+l,iW3,i (r(s)&($(s)))3 '
= ^ [ sL : +(e«)-sL_(e«) (s)
= V(NS + 3)(NS +
-3
2 x cos
,ft(.h
(i4.52),
1)(NS-1)
(s)
(s)
( r W/3i(*(-))) 3
(A52) 2
:
V(iVs + 3)(iVs + l)(JV s -l)(fc + 2)
2 3
isin
*&,, - rxo) Tft ^ $ n ™*jj(««*i3 ± <') fc + 2
j=fc
j=2
A.2.
A LIST OF SPHERICAL
357
HARMONICS
y/(N3 + 3)(N. + 1)(JV. - l)(fc+~2)
2X/F+T(r(s)ft($(8)))3
(^i^-fc^D^J)2
(«,g±i«,g>)
(A53)
It is convenient to introduce the following spherical harmonics with fictitious multi-indices L ^ and L ^ : =3
^C»(S)\_L773
(»)>
V2 ^(Ns
+ 3)(iV. + 1)(JV. - l)(fcT2)
fc+i
a
a (*,«) -ri«E(™S5) fc + 2 J=2
(i4.53)i
"3
=3 /'ftW'i T;3 f£»( s h - T ( i H y i ) - ^ r (i) l ^ i )
r<=i(sh — •*•
y/(Ns + 3)(JV. + 1)(JV. - l)(fc+~2) (s) \2 s s «!,«) v /2(FTT)(r( )/3 i ($( )))3
fe+1
1 fc + 2
(s)\2 E(«15)' i=2
(s)
(A53) 2 £*<2>(©i)
V(iV s + 3)(Ars + l ) ( i V s - l ) 2 ^
L
fc± JV»-2
fc-1
xcos^x, I I s m 3 ^ n s i n 2 ^ ( c o s ^ ± i s i n ^ ) 2 J=fc
N /(fl.
+ 3)(JV. + l ) ( J V . - l ) 2 v / 2>( s )/? i ($( s ))) 3
3=2
»I3I,« ( ( ^ )
2
±*«#«,« - («®)2)] . (A54)
It is convenient to introduce the following spherical harmonics with fictitious multi-indices Li,? and L•U2A:: 2fc •=3 CP)( S )\ _ " ¥ (2) V U i )
1
V2
^ , ( 0 0 + 5^,(90
358
APPENDIX A. SOME FACTS ABOUT SPHERICAL
_V(Na
+ 3)(N8 + l)(Na-l)
(s)
HARMONICS
((w(s))2_(w(3))2\
(A
.
s ^ , > = ^fc«<*>-^<*> v /(iV s
+ 3)(iVs + l ) ( 7 V s - l ) ^ ( s ) mit(M)mjB) W k+l,iW3,iW2,i(r( s )A($ ( s ) )) 3
(.), _ >/(W. + 3)(iV. + l ) ( J V . - l )
s£±(ej"0
4^3
W.-2
n sin *g 3
(A54) 2
(cos^s]±isin^s])3
3=2
- ^ ^ f t l w y " " ' ^ ' ' ± ^ ' ' ^ " 3,"» <"4')2 T
i(aSn (A55)
It is convenient to introduce the following spherical harmonics with fictitious multi-indices L2 and li\2 :
sL(e<w) = VH 4 s£+(e«) + sL(ej">)" V(iVs + 3)(JVs + l ) ( i V s - l ) 2V6(rWft(*W))3
•=•3
fA(sh _
L
l(
(s) 3i }
_
( 1 ) ( | ) 3i ( 2i } J
,
"
(i4.55)!
^+(e(s))-HL(eis))
\/2i .' A /(jy.
+ 3)(jy. + i ) ( j y , - i ) 2^(r«A(*W))3
S
K„lt(6i
[ (
w 2 (S) 3i ] 2i
_ (
(S) 3, 2i } J
) = V(Na + 5)(Ns + 3)(Ns +
x cose
* - i , i c o s e i - i , i c o s f l ? - i , i c o s e J-i,i
-
l)(Ns-l)
(A55) 2
A.2.
359
A LIST OF SPHERICAL HARMONICS
-li^^u^^ii^^ii^s j=k
3=1
1=1
3=P
(s)
= V(Ns + 5)(ATS + 3)(WS + 1)(JV, - 1)
„4
(s)
M
(s)
(.4.56)
(rWft(*<»>))4
, f i (.h _ v W + 5)(JV3 + 3)(NS + 1)(JV. - lfr
AT.-2 s X C 0 8( >^
(s)
s 0™<, 0 8fl(^ )
^^M-J'
M
y/(N. + S)(N. + 3)(N. + 1)(JVS"^T)P (s) («) «'fc+l,<«'l+l,i V2(p-l)(rWft($W))4
k-1
Z-l
7=1
3=P
(^ 2 -;X>£) 2 i=2
(A57) -4
, ft (.K _ V(iVs + 5)(JV. + 3)(JV. + 1)(JVS - l)(t + 2)
k-1
T Ns-2
x
(S)
^cfl(S>
«»**-w«»«r-i,«
«» a #>
l-l
*
Z+ 2
j=k
j=l
3=P
^(Ns + 5)(iVs + 3)(JV. + l)(iVs - l)(t + 2) + l)(r( s )ft($W))4 v /2(/ , (s)
-4 , 0 <-h ^ ( 2 1 ) (©i )
cos20<s>
(A58)
_ V W + 5)(ATS + 3)(JVS + l)(iVs - l)(fc + 4) V2(fc + 3) N.-2
*
H-l
(s)
fc + 4
4 k-1 s i n 2
cos^ooBtfW^ n sm ^n
^n s i ^s l-l
J=P
360
APPENDIX A. SOME FACTS ABOUT SPHERICAL HARMONICS
_ V(ATS + 5)(iV. + 3)(JV. + l)(jy. - l)(fc + 4) -/2(/fc + 3)(r«/3i(*«)) 4 fc+1 w
,.«
w
\2
(A59)
fc + 4 W^-iiiM)
* l+l,i p+l,i
J=2
„4
, Q (.h _ VU^+ 5)(*s + 3)(ATS + l)(Nm - l)(l + 2)
„ ,.v '-1>*
xcosflW
3 cos 0, ,,. J+2
^(Ns + 5)(ATS + 3)(ATS + l)(Na - l)(f + 2) V6^1)(rW/a,(*W))*
8
3iu (s)
l+l
(«a,) -^p^)
*«#!.*
(A60)
J=2
-4 / e (.)x = V W + 5)(7VS + 3)(N. + !)(#. - l)l(k + 4) "K" * ' 2y/(l-l)(k + 3) AT.-2
. 2 /)(•)
fc-l,t
fc
+
4
fc-1
cos^^M-y
y^iV. + 5)(JV. + 3)(ATS + 1)(7VS - l)l(k + 4)
2(r(s)A(*(s)))V(i-1)(fc + 3) ,.(s)
^2a
1
fc+i
2
a ( * , ) - 1J=2 B^) «i.*) -rriEMS) fc + 4J'=2
2
y/(JVs + 5)(iVs + 3)(iVs + l)(JV. - 1)/ •"•JI.
2(rWft($W))V(( - l)(fc + 3)(fc + 4)'
(A61)
A.2. A LIST OF SPHERICAL HARMONICS
fc+1
1
361
u
2
1
3
l+1
(«&,<)' - \ j=2£(-2) rfi.*) - 73=1E(«©2 2V(ATS + 5)(iVs + 3)(JV. + 1)(JV.^1)I
+ (rW/8(($C))) V C
T J+l
(.)
2
- 1)(* + 3 )( fc + 4 )
i=2
(A61)' -4
,ftCK
XCOS^
y/W + 5)(AT3 + 3)(ATS + 1)(N8 - l)(k + 4)
COS
3c08g
3fl(.) #fc-l
fc-l,t
fc + 4
(8).
~ j=k
j=l
y/(Na + 5){Na + 3)(JY. + 1)(ATS - l)(fc + 4) (rWA(*W))V6(fc+l) Q
(»)
Ww)
*wl+l,iwk+l,i
^2 :
fc+l
-E(-?) 2
(A62)
fc + 4j = 2
, 4 , Q (-h _ \/(iVs + 5)(ATS + 3)(AT8 + 1)(JVS - l)(fc + 4)(fc + 2) 2 x /6(fc-l)(fc+l)
sue")
iV.-2
cos4*!"*
k ;— -^-
(fc + 4)(fc + 2)
n
•
4 /i(s)
sin4 6) J
i=k
y/(Ns + 5)(NS + 3)(iV. + 1)(JV. - l)(fc + 4)(fc + 2) 2V6(fc-l)(A; + l)(r(s>A($<s)))4 (s)
\2 fc+1
2
• k+l
; (-i^-^^E^?) ^ (fc + 4)(fc + 2 ) VE(-S) fc + 4 t J'=2
^4 ( e (.K " Kfc '
=
j 2
v P T l - 5)(JV3 + 3)(JV. + 1)(JV. - l)(fc + 4), 2x/6(fc - l)(Jfc + l)(fc + 2)(rWft($(s)))4
. (A63)
362
APPENDIX A. SOME FACTS ABOUT SPHERICAL HARMONICS
-.
3fc
fe+l
-i r
A
fc + 4j = 2
k+1
-2(fc-l)(^M)4}.
J"=2
(A63)' s
s J U . (eW) = ^
+ 5)(iVs + 3)(iVs + l
p
) ^ -•*^( sC - / ^» C « I-*<•> M ^CW
x " j f sin4 *}J J ] sin3 *$ J J sin2 ^ j=fe
i=J
J ] sin $
j=p
(cos « « ± sin fljj)
j=2
V(iV s + 5)(iVs + 3)(iVs + i ) ( A r 3 - i )
(s)
(s)
(s)
, ( . ) ( . ) .
M 6 4 )
It is also convenient to introduce the following spherical harmonics with fictitious multi-indices K 2 p ^ and Kipjfc: "4 Cft( s )\ , = 4 00 > ) - —7= - K p l f c + ( e i ) + S K p i f c _ ( 0 j )
C4
K2plfc(0i
V(JVS + 5)(7VS + 3)(iVs + 1)(JV7^1) (s) (s) (s) (s) «'fc+l,i«' I Vl,4 U 'lH-l,i u '3,i. (r«ft(*M))4
34
rft (s)s
_ J_
-Kplt+(0i
V(Ars + 5)(iVs + 3)(iVs + l ) ( i V s - l ) (r(s)/3.($(s)))4
^ 1
) -SKpifc_(®i
(s)
(s)
(s)
k+l,» l+l,i">H,i '2,»-
2^/^+T
x c o s ^(«) _1
)
" u/ i . , ! : W u / / , i „-"A~i_1 »-U/o u
,a>h) — _ ^N- + 5)(^» + 3 ) ^ + ^
K -(31)(®i
(s)
cos 2^)
JL
(A64)j
~ lW + V
(^.64) 2
363
A.2. A LIST OF SPHERICAL HARMONICS
N.-2 x
l-l
fc-1 4
sin ^>nsin ^nsin^i(cos^±isin^)
II j=k
3
j=l
j=2
V(JV. + 5)(NS + 3)(JVS + l)(N^l)(l 2v / iTT(r( s >A($ (s) )) 4
(s)
x w fc+l,i
rfi^-rbB^S)2
+ 2)
K»/ +_Li «; „0»-
(A65)
It is convenient to introduce the following spherical harmonics with fictitious multi-indices K ^
and K.\lk : (pi(s>\
=4 2lk
/ftW\
=4 „ y2
* * •
1 1 .
(S)N
,=4
I-
* * •
1
1
—
^/(JV. + 5)(ATS + 3)(iV. + 1)(JV. - ! ) ( / + 2) 2 v / r+T(r( s )A($ ( s ) )) 4 '+1
1
(s) (s) 3,iwk+l,i
Xw
(^.65)!
i + 2 3=2
I
^ ( e i 8 ' ) - ^V2i:
=4
tC\(s)\
=4
(s)>
y/(Ns + 5)(JV. + 3)(NS + 1)(JV. - l)(i + 2) 2VT+T(r(s>ft($(s)))4
(s)
(s)
l+l
1
(-m,)2-r^E(-S)2 1 +
l
j=2
(A65)2
364
APPENDIX A. SOME FACTS ABOUT SPHERICAL HARMONICS
x cos Aww cos e\% n si°4 *s n sin3 ^-2,i n sin2 *s? ^ <$± [ ^ «s )2 7V.-2
j=k
fe-1
1-1
3=1
J'=2
V(7V8 + 5)(iVs4-3)(iVs + l ) ( i V s - l )
(3)
(s)
A w ^ ^ W ^ . ) . -
(.M (A66)
It is convenient to introduce the following spherical harmonics with fictitious multi-indices Ki,. 21k and K>-121kS
i(*h)
K(32)(0t
^21*:
=•4
-
-^= 2 V2 L
V(Na + 5)(Na + 3)(Ns + l)(Ns-l)
WN (0W) =
S*
v^i
/o(s)\
(32)(ei '*+
w
(32)(©»(*h ) i ^-Ik-
,~4
)+H
(s)
/ , ( s ) ) 2 _ , (s) )2 \
00 <, c4 /C*( s )\ =4 - K ( 3 2 ) ( @ j ) — S (32)(©j )
V(iv. + 5)(jy. + 3)(jv. + i ) ( j v . - i ) (rWft($W)) 4
U;
w
(s)
(s)
(s)
fc+l,t U 7+l,i U ; 3,i 1 / ; 2,i"
(.)* _ V(NS + 5)(JV. + 3)(iVs + 1)(JV. - l)(fc + 4) ) 2VF+3
-K(21)(0i
(s)
„2 /)(s)
c o s ' C , , , - ^ cosd, iV„-2
4fc-1 s i n 2 1-1 s i n
>< n sin ^n j=k
j=l
,
^n ^( c o ^s ± i s i ^S) j=2
V(iVs + 5)(iVs + 3)(N. + 1)(JV. - l)(fc + 4) 2v/fcT3(r(s)/3i($(s)))4
(A66)2
365
A.2. A LIST OF SPHERICAL HARMONICS
2 (»Si,«)afc-rzii;W) +4 3=2
,.(•)
/...(•)-1-«...(•)
(A67)
< M « ± i < ) -
It is convenient to introduce the following spherical harmonics with fictitious L L multi-indices K.LJ 2(fe and K,,.': iifc
^(iV. + 5)(#. + 3)(7V. + 1)(JV. - l)(fc + 4) V2(fc + 3)(rWft(*W)) 4 fc+i,t
a
<«#i,,) - Am; + 4 £ (»g)
(s)
(s)
(A67)i
i=2
. cat8)1!
^
/ft( 8 )>
(s)\ _ - 4
"•lit
V^
1
L
lk
+
Ik-
^(Ns + 5)(JV. + 3)(iV5 + 1)(JV. - l)(fc + 4) >/2(ife
+ 3)(rW / 9 i (* ( ' ) )) 4 k+l,i
(^M)
~4 Ki
2
- FXI E (^5>: fc + 4 J"=2
(s)
, f l (.h _ V(NS + 5)(NS + 3)(NS + 4v/3
(s)
(A67)2
1)(NS-1)
>< cos ^w "if sin4 e$ nsin3e% (c°s *a ± *sin eu) (s)\3
J=fe
J=2
V(ATS + 5)(iVs + 3)(JV. + ^ ( A ^ I ) 4v / 3(r( s ) / 5 i (* (s) )) 4
366
APPENDIX A. SOME FACTS ABOUT SPHERICAL HARMONICS
< sah 2 „ » _ o„ 1 1 W/.„W\2 ;f.„W\3l 3 >> r r „ » \ 3 ± 3i«i) *
(AM)
It is convenient to introduce the following spherical harmonics with fictitious multi-indices K2feJ and K j 2 f c . •=4
s
(P>(sh —
>< h 4. ^ H^(3,(er) + ^ ( 3,(er)
^ ( i V s + 5)(ATS + 3)(AT3 + l)(iV 3 - 1 ) _ J s ) 2 v / 6(r( s >/? i ($( s ))) 4
(«)>
(s)
. <.),
(
.h2l
^ i K i r - K W (A68)>
-K(3) (Vi
J — /JJ.
« \ _ 4 3,(er) ca(s)> s^3,(er)-^ ( =
v^i .'
V(iVs + 5)(iVs + 3)(JVS + 1)(JVS - 1) ( s ) w<.).2(.) W K K 2 v / 6(r( s )A($( s ))) 4 -44
2
cos
f(.).3.
- W (A68)a
, fi (-h _ V(iVs + 5)(JV. + 3)(JVS + 1)(N. - l)(k + 4)
K(2,(en
2 fl (s) fc-i,i
4yfcT3
fc
+
fc-1
N.-2
1 4
] ^ sin 4fl}5J ] sin 2 *SJ (cos flJJ ± i sin * « ) 2 J=
fe
j=2
y/(N. + 5)(JV. + 3)(JV. + 1)(JV. - l)(fc + 4) 4\/fcT3(r( s )A(* (s) )) 4
Si.*)2 -fcT45 > S ) 2 ] ( (W S )2 ± 2 i w S w S " ( * ) ) 2 ) •
( A .69)
It is convenient to introduce the following spherical harmonics with fictitious multi-indices K2fc and Kj 2fc : s (s), * =4,,m((C\^)\ h = i ^,4, ( e, rq )M )++ ss^)(er) V2 =4 K
A.2.
A LIST OF SPHERICAL
367
HARMONICS
y/(Na + 5)(7VS + 3)(JV. + 1)(N. - l)(fc + 4) 2v/2(fc + 3)(r( s )ft($( s ))) 4 T fc + 1
(fS w )
2 2 — YV s) ) 2 (^g) - (™g)
a
(A69)!
,
it + 4 A; *' j=2
?4
l
ccv(s)\ _
H^,(e, ( s ) )-s^(0! s »)
y/(ATs + 5)(JV. + 3)(N. + l)(iV. - l)(fc + 4) y/2{k + 3)(r( s )A($ ( s ) )) 4
n
fc+1 (s)
~4
,ft(.h
(s)
(A69) 2
V(iV s + 5)(iVs + 3)(JV. + l)(iV. - l)(fc + 4)
2^/3{kTT)
- s
3
^
M
- ^ c o s ^
M
f] sin4^ Hsin^cosflg
± i s i n
^)
V(JV. + 5)(JV. + 3)(iVs + 1)(JV. - l)(fc + 4) 2v^3(A; + l)(r(s)/?l(<E>(s)))4 fc+1
xw
(•)
u%r - ^
EK?> 2 ] (»a ± i»S)-
(^-70)
It is convenient to introduce the following spherical harmonics with fictitious multi-indices Kb-2fc L ' and K12A: Lt: L
•=;* ...CA^h -
<sh -L.^4 . . / f l W l
368
APPENDIX A. SOME FACTS ABOUT SPHERICAL HARMONICS
V(iVs + 5)(JV. + 3)(ATS + 1)(JVS - l)(fc + 4) V6(fc + l)(rWft(*(»>))4
*«45«&.«
fc+i
2
(^M) -^E(-S)2
(A70)i
j=2
"•it
v2i L
fc
fc
+
-
V(iV. + 5)(iVs + 3)(JV. + l)(jy. - l)(fc + 4) V6(fc + l)(r(8)ft($(«)))4 fc+i
2 ""gfKw ("Si^-jbTiP^) J=2
«4
(A70)2
/•Cl( s )^
y/(Ars + 5)(7V5 + 3)(JV. + 1)(JY. =1) ^ .4 w . (s) 4 (>) i —^ | | sm 0W(cos0j ' ±isin0i^) )t 8^6 J=2
V(iVs + 5)(JV. + 3)(ATS + 1)(JV.^1) 8V^(rW/3i(*W))4
( U ;g) 4 ±4i(^) 3 ( U ;g)-6( W g) 2 ( U ;g) 2 T 4i( W g)( W g) 3 + (^g) 4
(-4-71)
It is convenient to introduce the following spherical harmonics with fictitious multi-indices K2 and K^2 : ?4 /•ft(s)\ _ J _ -K 2 (ei ) - ^
s{C(i)(ei')) + s{c(1)(ei"))
A.3.
PRODUCTS
OF SOME SPHERICAL
V(ivs + 5)(JV. + 3)(jy. + i)(JvT^i) s
8v/3(r( )A($
(s)
))
4
HARMONICS
369
(<]) 4 -6(^g) 2 (< ) ) 2 + (^) 4 J, (A71)!
H
K-(^
)=
^[HK-(^"))-Sl«(e* )
y/(Ns + 5)(JVS + 3)(JV. + 1)(ATS - 1) 2v / 3(r( s >A($ ( s ) )) 4
A.3
("SMS-fg^g)8
(A71) 2
Products of Some Spherical Harmonics
In order to solve the equation (10.21), we should get a list of the products of S
N f c ( e i S ) ) S p ( e i S ) ) ' P = 0,1,2,3,4, which will be used in Ritz-Galerkin method
for solving the equation (10.21). Firstly it is plain to prove the following two propositions. Proposition A . l
sU©!s))-°o(©!s)) = s U e h .
(A.72)
Proposition A.2
ShMS))^MS))
= ][^^^llkM%ifk
* l.
(A73)
In order to get more expressions for the products of two spherical harmonics, we need the following elementary lemmas from linear algebra: Lemma A . l (
-ai —a2 -a3
Let n x n matrix 1 - ai -a2 -a3
0 1 — a2 -a3
0 0 1 - a3
0 0 0
0
\
0 0
A =
, -a„_i 1
—an_i — a„_i 1 1 1
— an-\
—a„_i 1
1 — an-i 1 /
(A74)
370
APPENDIX A. SOME FACTS ABOUT SPHERICAL
HARMONICS
then bln\ &21
&22
fan
(A75)
"nn ' where 0,
if i < * — 1 ;
1,
iij =
i-l;
if0
hj = {
ai-iU^ZH1 ~ ak),
ifi>l,j
- 1 + E C i ai llCj+iC 1 - ak),
if i = 1, j < n;
11 - Td=i ai UkZi+iO- ~ ak),
iii = l,j = n.
= n;
The above lemma can be proved through routine computations. Here and henceforth we have made the following convention:
i€0
Setting a,j = l/(j + 1) into the Lemma A . l yields the following Lemma A.2
The inverse of the (Ns — 1) x (Ns — 1) matrix
't
0 0
0
\
0
B =
(A77) -1
-1
-1
-1
Ns-3
Ns-3
N„-3
N.-3
N.-2
iV 8 -2
iV„-2
N„-2
N.-l
N.-l
N.-l
-1
-1
1
-1
-1
-1
1
-1
-1
1
-1
N.-l
1
N.-3 -1 N.-2 -1 N.-l
1
U
JV 8 -3 N.-2 -1 N.-l
1
0 0 N.-2 N.-l
1
/
A.3.
PRODUCTS
OF SOME SPHERICAL
/- 111 =1 ¥ -l
A
Tf2 1
0
-1
-1
3
N.-3
N.-2
3
JV.-3
AT.-2
AT„-3
N.-2
^i
=i
371
HARMONICS
-1
M 1
-1
-1
-1
AT.-l 1
Nm-1
B~l =
(A78)
0 0 0 x N.-2 N.-l \ 0 0 0 As a consequence of the Lemma A.2, combining with the equations (A.40), (A.41), (A43)i, 2 , (A.44), (A.45), (A46)i, 2 and (A.47)i, 2 , we have the following two propositions. Proposition A.3 (•=1 fft(s)\\2 _
V2(Ns ~ 1)
Proposition A.4
V
^ s ^ e n + s^ej"). (A79)
If k > l,we have
(sk(©!s)))2 y/2(Nn - 1) r
/ 7k — q 1 "
~
-=2
,ft(s)s
N
2
'~ ST
1
-=2
/ft(sK
+ =0o(©is))(A80)
It is worth mentioning that the Proposition A.3 is a special case of the Proposition A.4 under the convention that E ^ (&\ ) is a function of arbitrary form. The following lemma, which is a consequence of the Lemma 10.5, is useful in deriving the expressions of the product E ^ (9j Hjj(0- 8 ') in terms of a linear combination of spherical harmonics. Lemma A.3
For a fixed k, let 1/(1 + 1), at = < 3/(fc + 2), l/(/ + 3),
ifl
fc-l;
iffc
(A81)
372
APPENDIX
A. SOME FACTS ABOUT SPHERICAL
HARMONICS
then the quantities bij defined in equation (A.76) are of the forms: 0,
ifj
1,
if j =
-1/j,
if 0 < i — 1 <j < fc-1;
b^ = < - l / ( j + 2),
if 0 < i - 1 < j < Ns - l,i=t k < j ; .
-3/C7 + 2),
if* =
1/(N. + 1),
iil
U / ( ^ . + l),
i-l;
(A.82)
fc<j
iii = k,j =
Ns-l;
Ns-l;
In other words, let (JVS — 1) x (7VS — 1) matrix Bk
JVs-2
fc-1 fc 1
-l 2 -1 3
2
fc-2 fc-1 fc
iVs-2 N.-l
-1 fe-1
-1 fe-1
0
0
0
0
0
0
0
0
0
0
fc-1 fc+2
0
0
-1 fe+3
0
0
-1 iV„ + l
-1 JV8 + 1
AT. N.+l
1
1
fc-2
-1 fe-1
Na
fe-1
-3
-3
-3
-3
fe+2
fe+2
fe+2
fc+2
-1 fe+3
-1
fe+3
-1 fe+3
fe+3
-1 iVs + l
-1 iVs + l
-1 iV„ + l
1
1
1
-1
-1 N.+l
1
J (A83)
where the (fc - l ) t h row is of the form: -3
-3
fc-1
fc + 2 fc + 2'fc + 2' y v fc—1 tuple
'
0---0
Ns-k—1 tuple
A.3.
PRODUCTS OF SOME SPHERICAL
373
HARMONICS
the j t h row above the (k - l ) t h row, i.e., j < k - 1, is of the form: -1 j+1
-1 j j + 1' j + 1
,
0 - 0 N.-J-2
tuple
j tuple
the j t h row below the (k — l ) t h row but above the last row, i.e., & — 1 < j < Ns — 1, is of the form:
i + 2,
f^l...^l \j + l
J + 3'
J+ 3
0---0 AT,-j-2 tuple
j tuple
and the last row is of the form: 1,
,1 ,
then its inverse is of the form B-l
fc - 2 fc - 1
1
_1
2
1
3
0
1
0
0
k
x
1
k
0
=1
=1
2
3
^
0
^
0 0
-l fc-2 -1 fc-2
-1 fc-1
-1 fc-2
-1 fc-1
1
-1 fc-1
-1 fc-1
0
fc
fc
+ 1
-1 fc+2 -1 fc+2 -1 fc+2
-1 fc+3 -1 fc+3 -1 fc+3
-1 fc+2 -3 fc+2
-1 fc+3 -3 fc+3
•••
iVs-2
AT. + l
-1 N.
N.+l
-1 Ns
N. + l
-1 NB -3 -1
-1
0
0
0
0
Ns-
0
0
0
0
0
0
0
iVs - 1 V 0
0
0
0
0
0
0
1
-1 N.
-1 fc+3
jfc + l
ATS
1 1
JV„ + 1 3 Af. + l 1 N. + l
N. + l 1 NB + 1 '
(A84) It is plain to prove the following
\
374
APPENDIX
A. SOME FACTS ABOUT SPHERICAL
HARMONICS
L e m m a A.4 BB71
'(fc-l)x(k-l)
^(k-l)x(JV.-k)
=
(A85) 0(NB-k)x(k-i)
G(jv e _ f c ) X (Ar._ f c )
where B, I and O denote the matrix (A.77), the unit and zero matrices respectively, AT. - fc - 1
1
AT, - fc
/ F,(fc-l)x(JV.-fc) k-
\
—
0
fc-2 -2(fc-l) -2(fc-l) fc(fc+2) fc(fc+3)
fc-1
-2(fc-l) fcJV.
0 2(fc-l) fc(JV. + l) /
(i4.85)! and *{N.-k)x(Nm-k)
Nu-k-l (fc+l)(fc+3)
1
(fc+l)(fc+4)
2 (k+l)N.
-2 (fc+l)(iV. + l)
(k+2)(k+4)
2 (fc+2)iVs
-2 (fc+2)(JV. + l)
(NB-2)N,
(JV.-2)(N. + 1)
Ns-k-2
0
0
0
Na-k-l
0
0
0
1
U
-2 (NB-1)(N, + 1)
0
0
0
1 (A85) 2
Ns-k
. /ftW^ - ? .Ij (©J rft( s;h) and Furthermore, it follows from the definitions of Ejj,.(0j '), E^ S
L P ( @ i S ) ) > (^e, (A.41), (A43)i,(A43) 2 , (A.44), (A,46) 2 , (A47) 1 ; (A47) 2 , (A.48),
(A52)i,(A52) 2 ,(A54) 2 ) that
A.3.
PRODUCTS OF SOME SPHERICAL
HARMONICS
375
Proposition A. 5
sk(ei B) )sk,(eW)
= <
J^Zlje?), JW^IJQ?), (N.-l)VN^i(
ill
(s) s2 (s)
j f
. (r< s )ft(*< s ))) 3 V^p+1^ ^ i + 1'
(AM)
, _,
lt«
The quantities (^(+i) 2 ^p+i and (Wp+i) 2 ^;^! on the right hand side of the equation (A.85) can be expressed as linear combinations of spherical harmonics of order three in the following way. The formulas (A.50),(A.51),(A.49) and (A54)i with an appropriate change of indices can be rewritten as follows: V^3!) y/(N, + 3)(N. + l)(N.-l)l
/C\(s)\
•=•3
L
U-i
(•)
2 < Z
(rWft($W)) 3 ww
(A49)'
j=2
\/6(fc - 2) -3 V(W. - 1)(JV. + \){NS + 3)(fc + 1)
(<0 w ( r W f t ^ W ) ) 3 fe,i
(Gi(s)\
(™S) a -fcfiD^J) a ].
V2(/ + 1)
(eis))
- T (1) V " i
V(i + 2)(ATS - 1)(NS + 1)(JV. + 3) L i - i M-i
(s)
( r ( S ) / 3 i ($(s)))3 w*,<
3 ("ft^-n^EK?) Z +2 J =2
, 2 < A; - 1 < I,
(A50)'
376
APPENDIX A. SOME FACTS ABOUT SPHERICAL
HARMONICS
The equations (A51)', (A49)' and (A50)' together with the trivial identity
J—2
can be written in a matrix form as follows. Setting /
1 =3 (QW) y/(N.+3)(N.+l)(N.-l) "L™ V * '
>/i
y/{Nm+3)(N.+l)(N.-l)3~h<£^ , > ( © !
S)
y/2(k-2) „3 V(N.+3)(JV. + l ) ( J V . - l ) ( f c - l ) " L ^ Vfc =
-y/6(fc-l)
,Q(sk « '
V
L
>/2(fc+2) „3 ^/(fc+aXAf.-iXJV.+iXJV.+a) s?m " L ^ +
/2ivr
1
."3
(W. + 1 ) V ' W - 1 ) ( J V . + 3 ) " L < ^ 1 . 1 V 1
»
)
sa4''(ei">)
v ^(iV s -l)(Ar.+l)(JV.+3)(fc+2)"
\
(A.88)
(©i s) ) (»h'
*
(P)(a)\
•=•!
and
Ufe
(r(">&(*(»)))
«Bi>8) a
3
(A.89)
v^'i,,^;,,)2/ we have Vfc =
BkVLk,
(A90)
1
vk.
(A90)'
BB^Vk,
(AM)
or, equivalently, Lemma A.5 Ufe = B. Therefore, we have Lemma A.6 Buk where the matrix BB^1
=
is of the form (A.85).
A.3.
PRODUCTS
OF SOME SPHERICAL
377
HARMONICS
By virtue of the equations (A.41) and (A.45), we have
TJI re>(sh"2
(Ns-i)^k{Ns+l)
(S)
I"
v 2 _ i y i / w^a Ki)
(s)
-N,(ei J-Mje, ) - (r(s)A($(s)))3y2(fc3i)u;i+14K+1'i)
fc^
(4.92) The following three propositions are the consequences of the Lemma A.6 and the equation (A.92). Proposition A.6
If 2 < k < I — 1, we have (A93)
Proposition A.7
If /c > 2, we have
-N.(Q« )-Mt(e, ) - ^
^
x / ( f c - l ) ( i V s - l ) ^3 k
j=k+2 V 3\3 + l)(^s + 3)
L
( J v ; +
3)(fc +
2)-L^
e
« J
,n(.K , V 2 ( f c - l ) ( J V . - l ) g l
*,;-i
(s)
V/c(Ars + l) (A94)
Proposition A.8 If 2 < I — 1 < k, we have
AT.-1
+ V
2ViV v s s - 1
~3 (e (s) )
~
A i VKk - 1)0" + W + 2)(JV. + 3)~Lu
*
V2(JV _1) ' =^(©,(S))Vfc(fc - l)(iVs + 1)
(A95)
The form (A.84) of the matrix BjT1, combining with the equations (A49)', (A50)', (A51)' and (A87), yields the following three propositions and one lemma. Proposition A.9 =1 fC(( s h" 2
If 1 < j < k — 1, we have (ft( s h - (Ns-l)y/Ns
+l
(8)
(s)
2
378
APPENDIX
A. SOME FACTS ABOUT SPHERICAL
k
_V2(j-l)(Ns-T)^
^i
(s)
y/2(N.-l)
V(fc-l)(fc + 2)(ATs + 3 ) " L t V ^y _ 2
V £
HARMONICS
^
(s)
* '
*
V 2(N» ~ 1) g3 /Q(S)X V ^ s - l g i ,ft(.)> } V( P +2)( P+ 3)(Ar s + 3 ) " ^ + 1 ( 0 ^ ) + vmr=r^{9i -
P r o p o s i t i o n A. 10
Mqfix (A96)
If j > k + 1, we have
_ y/2(j + l ) ( i V s - l ) ^ 3
^
(s) l
V ( j + 2)(iVs + 3 ) ^ i r
+
^
V2(AT S -1) L
V a + 2)G + 3)(iVs + 3 ) ^ M + /
^TTsS»(e!"')-
(s) i
^
(A97)
As a by-product, we have L e m m a A.7
(N. - 1)VN7TT
(.)
(rC)ft($C)))3
^ f c + 1 .* ;
^A
>/2(JV. - 1)
3
^3
^2>/i0-+i)(^+3)^a-,
V6(fc-l)(JV.-l)o3 V(fc + 2)(7Vs + 3 ) ~ L j
( .)
/ oV
(o< ) + 3
iVs-l„1
(s)
' '
w
^rr^(0i}-
(A98)
It is easy to see that the P r o p o s i t i o n A.5 with the supplement of the P r o p o sitions A.9-10 shows us all the forms of the expressions of of the quantities
x Hjy^ (©i ) as linear combinations of spherical harmonics of orders three and one.
A.3.
PRODUCTS
OF SOME SPHERICAL
379
HARMONICS
Now we are going to find out the forms of the products Ejj.(0^ JS^©* ) as linear combinations of the spherical harmonics of orders four, two and zero. Firstly it is easy to show that Proposition A. 11
f), ' ^^ ^f we jC^B (© <*>), * P
. ^ ^ ( © f ) ,
(A.99)
ifj
In order to get expressions of the other quantities Sjj. (©j
)SL(6J
) we need
the following lemma, which is a consequence of the Lemma A . l : ( Let the matrix Bik =
Lemma A.8
bl
\
t>2
(A.100)
\bN.-J with the row vectors hj defined as -1 *
i
-1 1
'
'
•
j + 1
i
3 i
'
•
i
i
'
0, • ••, 0
j + 1 j + 1 j tuple
-3
1 tuple
-3
j + 2'
(i — 1) tuple
-1 j + 3'
1 tuple
- 3 fc + 1 'fc+ 4 ' fc + 4 '
(A:-l) tuple
-1 j + 5'
1 tuple
-1 j +2 ' j + 3' j + 3'
j tuple
-3 /c + 4 '
I-I
'1 + 2' i + 2 '
1 tuple
-1 j +4 ' j + 5' j + 5'
j tuple
1 tuple
( A f . - j - 2 ) tuple
( N . - 1 ) tuple
(A100)i
380
APPENDIX
A. SOME FACTS ABOUT SPHERICAL
HARMONICS
Then the inverse of the matrix Bik is of the form (
dn
di,Ne-i d2,N.-l
dii
\
(A.IQ1)
B
ik \djV„-l,l
dff.-l,2
•••
dN.-l,NB-l'
where
fo, 1,
iij
-},
*%]
ifl<»
- jL,
if i < i < j , I < j < k _ i, i± I-
-j|a,
if J < j < f c - 1 , t = J;
-jfi,
if 1 < * < J, k<j
-jfi,
iffc < j < i V s - 2 , i = / o r A;;
^ g ,
ifj=iV.-l,
(4.102) ij^k;
k^i^l;
K~N7+3> if j = Na-l,
i = l oi k.
In other words, / B
lk
(-Dll)(<-l)x(J-l)
(-Ol2)((-l)x(fc-()
(D2l)(k-l)x(l-l)
(D22) (k-l)x(k-l)
\(-C>3l)(JV.-fc)x(i-l)
(•Dl3)(i-l)x(Ar s -fc)
\
(D23)(k-l)x(N.-k) (•^ > 33)(iV 8 -fc)x(Ns-fc)'
(-D32)(Af.-fe)x(fc-0
(.4.102)' where the blocks Dy are of the following forms: x
/
2
1 •
(Dn)(i-i)x(i-i)
_I •
_I
2
3
0 0
1 0
-| 1
V0
0
0
/ (•Dl2)(Z-l)x(fe-i)
•
._!_
3
=
1+2 1 Z+2
L_\
i-2 1 '1-2 1 "(-2
Z-l \ 1_ l-l 1_ i-1
1-2
l-l
1
L.
(4.102)!
-TV 1+3 1_ (+3
3_\
k+1 \ 1_ fe+1
\
L.
L_
l-l
\
1+2
1+3
k+1 /
(4.102)2
A.3.
PRODUCTS OF SOME SPHERICAL
/ (Dis\i-
1)X(NB-1)
~
_1_ fc+5 fe+5 1 fc+5
' N.+2 1 Ws+2
Na+3 1 AT8+3
fc+4
1 "fe+5
1 N.+2
1 N„+3 •
—
/
( 0 0
0 0
o 0
0
Vo o
0
0 /
1+2
1 0
(-D22) (fc-i)x(fe-f)
381
1 +4 " t fc+4 1 'fe+4
\
{D2l)(k-l)x(l-\)
HARMONICS
3
3
"i+3 1 (+3
";+4
ix (A102) 4
"fc+l\ 1 'fc+1 1 "fe+1
1 '1+4 1 "i+4
1
(^23)(fe- •J)x(JV.-J)
3 fc+4 "fc+4 1 'fc+4
3 fe+5 1 'fc+5
1 "fc+4
1 'fc+5
1 ' Ns+2
1 W s +3 •
0 0
0 0
0 0
1 0
0
0
0
0
'fc+5 1 'fe+5
1
0
(A102) 7
(A102) 8
3 fc+6 1 "fc+6 1 "fc+6
3
fc+4 1
Ns+2 1 ' JVs+2 1 ' N.+2
V 0' L e m m a A.9
(A102) 6
W a +3
1 iVs+2
(•D32)(Ar.-fc)x(fc-J)
(-C*33)(iVs-fc)x(Ns-fc) —
N.+3 1
"ATB+2
= 0,(AT.-fc)x(l-l)i
(£>3l) (JV.-k)x(Z-l)
/
(A102) 5
1 "fc+1
V '0' (
(A102) 3
1
Ns+3\ 1 iV s +3 1 N.+3 1 N.+3
(A102) 9
Let
/ »&WS(s)M(«43)a \ ( s )
u(fe = :
™iu)
>
$
)
2
4
(rWft(*W))
\
(s) 'l+l,iwk+l,i\wNB,iJ
-^Mrf,)V
(A103)
382
APPENDIX A. SOME FACTS ABOUT SPHERICAL
HARMONICS
and 1
C0(S))
-=4
\
V(N.+5)(iV 8 +3)(iV. + l ) ( A r 8 _ l ) ^ K < ^ , V » ' V4 (32)1 <e."') i/(iV.+5)(Ar 8 +3)(JV s + l ) ( A r s - l ) 3 " K !3Ifc y/2(l-2) =4 V / (^s+5)(iV.+3)(Ar B + l)(Af 8 -l)(J-l)^K i ( i 2 : l ) i f c
V6(t-1)
V
» '
:Ht(.,(e«) •4
+ l)(N.+3)(N.+5)(l+2) ~K< 3 ) y^2(t+2) •=•4 ^(3.) > /(i+3)(iV.-l)(Ar. + l)(N.+3)(JV.+5) " K ^l + l J, f c y/(Nm-l)(N.
(©i S) ) (A104)
Vifc =
V2fc % /(fc+l)(iV.-l)(JV 8
=4 + l)(iV 8 +3)(Ar 8 +5)"'K -K-Oi)
s
/G»( h (e'O
n*Mm))
l,fc-l,k 7:4
yfyfc+l) r
V'(A .-l)(iVs + l)(iV.+3)(Ar8+5)(fc+4) K} y/2(k+4) =4 (QK001 ">) i ,/(N,+5)(NB+3)(Ns + l)(N.-l)(k+5) ~K<**> + 1 V * ' - • = *
V2ATs+4 774 C0^Sh ( V (AT.+3) v /(iV.-l)(iV. + l ) ( J V . + 5 ) " K 1 2 ^ . _ 1 * ' 1 n2 («00\
\
V(w.-i)(JV.+i)" M,fcV *
/
have u.fc = Bj^vifc,
(A105)
where B ^ 1 denotes the matrix defined in (A.101) and (A.102). Proof
Since v;fe =
BikUik,
which is a paraphrase of the equations (10.121), (10.124), (10.122), (10.126), (10.123) together with the trivial identity ^(iV, - 1)(JV. + 1) 3 U © I
S)
)
(•h2
(r<»>/3.(*<'>))4 £E
The following three propositions are consequences of the Lemma A.9 Proposition A.12 4<s)^3
W\_ sMenswen-
( A r s - i ) v / ( A r s + i)(iv s + 3) (r(s)/3.($(s)))4
u,
(s) (s) (s) j+i,tu,fc+i,i(wp+i,i)
A.3.
PRODUCTS
OF SOME SPHERICAL
y/2(p-l)(NB-l)^
£4
(B)
y/6(Ns - 1 ) ?4 ( x / ( / - l ) G + 2)(iVs + 5 ) ^ r
( .) 4
383
HARMONICS
y/2{N.-l)
^
(.)
^ >/2(JV.-l) ^4 ( .) K V ^ ^ V ( j + 2)(j + 3)(iV. + 5 ) " i ^ i > *
v W - 1) ^4 fe(sh v/(fc + l)(fc + 4)(iVs + 5 ) " < A ' ' iV a -2
V
V2(N* -1)
.
4
~Tk V U + 4 0 + 5 P . + 5
v^T7!
(S)
a
Ml0fix
v
"-ifc.i+i
V^ s + 3
P r o p o s i t i o n A.13 =i , e ( . ) ^ 3
f e (s) v _..
V2(i + l ) ( i V . - l ) g 4 x/(r+2)(]v7T5) ~ K ^
(JV s -l)V(iVs + l)(iVs + 3)
V
(s)
(s)
2
_ ^4 \/2(iV s ~ 1) ^4 rQ (.)v 3 1 ' ^ ^ V ' ( j + 2)(i + 3)(iVs + 5 ) ^ 3 k / * '
(s)
v W - 1) ^4 , V(fc + l)(fc + 4)(iVs + 5) K U ; N„-2
(>)
y2(;vs-1)
4
(s)
~Tk VU + 4 ) U + 5)(iVs + 5 ) "-pfc.i+i
( .)
yTV.-i
(.)
2
.
V-Ws + 3
P r o p o s i t i o n A. 14 =i
re(-)^3
reCK
V2(fc + 3 ) ( J V . - l ) g 4
W - l ) x / ( W s + l)(iVs + 3)
( .)
ATs-2
" ^
(s)
>/2(JV.-l)
(s)
(s)
2
(s)
384
APPENDIX A. SOME FACTS ABOUT SPHERICAL HARMONICS
As by-products of the Lemma A.9 we have the following two lemmas. Lemma A.10 If / < k, we have
V^T^
, 0 (.K
v/(ATs + 5)(JV. + 3)(JVS + l)(iVs - !)(/ + 2)~K«*
*
fc-2
^
V(^« + 5 )(^s + 3)(JV. + l)(iVs - l)(j + 2)(j + 3)"Kif/ii,»
3
^
^4
(Q(s)x
V(7VS + 5)(Na + 3)(JVS + 1)(JV, - l)(fc + 1)(* + 4 ) " < U _
V Z^
^
V
'
l
774
/0(.)x >
W.- + i ^4)(j /.' + , ^5)" " KKi< (2 . ^' . . . ^ V(JV. + 5)(7VS + 3)(iVs + 1)(NS _- 1 l)(j +1 . /^r
l
W
+
W
^IMAT
2
^IVAT
1
. = 5 * (ej'h. (JV. + 3 V W + 1)(JV. - 1) "
(A109)
Lemma A. 11 If i < k, we have 1
,>)
/,>)
\3
>/6(fc + l) ._(.), =4 sL K (x,(er) V(iVs + 5)(WS + 3)(WS + 1)(JVS - l)(fc + 4) ,Y
V
3 V
^
^4
/Q(8)X
£ £ >/(#, + 5)(NS + 3)(ATS + 1)(JV. - l)(j + 4)(j + 5)" K ffi + . V *
A.3.
PRODUCTS OF SOME SPHERICAL
+
+ l)(Na-l)~M^
(Ns+3)^(Ns
385
HARMONICS
In order to calculate the matrix BkBf^,
i
(A110)
h
where the matrices B^ and Bf,}
denote the matrices in (A.83) and (A.101) respectively, we need to write the matrix B^ as a block matrix in the following form: / (Cll)(I-l)x(l-l) Bk =
{Cl2)(l-l)x(k-l)
(C2l)(fc-()x(;-l)
(Cl3)(l-l)x(N.-k)
{C22)(k-l)x (k-l)
\(C3l)(;v„-fe)x(i-l)
\
, (Alll)
(C23)(k-l)x(N.-k)
(C32)(N,-k)x(k-l)
(C33)(N.-fc)x(N,-fe)/
where the blocks Cy- are of the following forms:
( d i ) (( f - l ) x ( i - l )
=
3
1 3
w
1 I
(Alll)!
1 I
/ 0 0 (Cl2)(l-l)x(k-l)
0 < 0
2 3
0 0
1 I
"
• •• • ••
0
0
(Alll)2
=
V £ "o" ••'• 0 (Alll)3
(Cl3)(J-l)x(W.-fc) = 0((-l)x(AT 9 -k)>
(C2l)(fc-i)x(i-l) =
/ (C"22)( fe-i)x(fe-i)
J+l 1_ (+2 1_ k-1
v
k+2
M
1 1+ 1
1 'M-l
1+
1 fe-1 3 k+2
1 'fe-1 3
1_ fc-1 3_ /
"fc+2
fe+2 /
I l+l 1 1+2 1_ fe-1 3 k+2
(Alll)4
0
1+1 (+2 1_ k-1 3
fc+2
0 0
(Alll)5 fe-2 fe-1
1 'fe-1 3
3_
fc+2 0
fc+2
\ (Alll)6
(C23)(fe-J)x(AT.-fc) fe-1
\
0
0
fe+2
0/
386
APPENDIX
A. SOME FACTS ABOUT SPHERICAL
'
=
(C3l)(N.-k)x(l-l)
1
^r 1
/
L_
/
k+3
=
(C32)(N,-k)x(k-l)
V
(C33)(N,-k)x(N.-k)
1 JV. + l
1 JV. + l I
1
1 fe+3
_
1
fc+2 fe+3
1 fe+3
1 N, + l
1
1
)
(.4-111)8
N. Ns+1
(Alll)9
/ 0
0
1 Ns+1
(Alll)7
/
i N. + l
1
i
J
f e + 3 "^
1 Ns + 1
1 N.+l
V
fc+3 \
1
1 N,+l
/-
1 fe+3
k+3
I
HARMONICS
1
1
By virtue of the equations (A101)', (A102)i_ 9 ) (^4.111), and ( A l l l ) i - g , we have the following Lemma A.12 /
BfeBffe1 =
(EU)(l-l)x(l-l)
(El2)(l-l)x(k-l)
(El3)(l-l)x(N,-k)
(^2l)(fc-()x((-l)
(^22)(fe-i)x(fe-()
(£ , 23)(fc-;)x(N 8 -fe)
\{E3l)(Ns-k)x(l-l)
(E32) (Na-k)x(k-l)
(E33)
\
(N.-k)x(N.-k)'
(A112) where ( £ 1 1 ) (l-l)x(l-l)
/ (•Bi2)(/-i)x(fc-;)
/ (-El3)((-l)x(iV s -fc)
0
0 0
\ \
-
2(1-1) 1(1+2)
0
0
2(1-1) ' 1(1+3)
2(1-1) ' 1(1+4)
2(i-l) l(k+4)
(E2i)(k-i)x(i-i)
0 2(i-l) l(k+5)
0 2(f—1) ((fe+6)
= 0(fc-;)x((-i)!
\
0
(A112) 2
2(1-1) . 'l(k+l) I
0
0 0
. \
(A112)!
— -<(i-l)x((-l)l
0
2(1-1) ~l(N*+2)
0
\
0
2(1-1) l(N,+3) I
(A112) 3 (A112) 4
A.3.
PRODUCTS OF SOME SPHERICAL
(J+1KJ+3)
1
0
(i + l)fc 2 ((+2)fc 2 (J+3)fc
1
0
0 0
Vo
(E23)(k_
(i+l)(i+4) 2 (I+2)(I+4)
0
(E22)(k-l)x(k-l)
387
HARMONICS
(/+l)(fe+l) \ 2 (/+2)(fc+l) 2 ((+3)(fc+l) (fc-l)(fc+l)
1
.
/ (A112) 5
2 / (i + l)(fc+4) 2 (i+2)(fc+4)
2 (J+l)(fc+5) 2 (i+2)(fc+5)
(l + l)(Ns+2) 2 (i+2)(JV„+2)
(J+l)(iV.+3) \ 2 (l+2)(N,+3)
(fc-l)(fe+4) 6 (fc+2)(fc+4)
(fc-l)(fc+5) 6 (fc+2)(fe+5)
2 (fc-l)(JV.+2) 6 (fc+2)(ATs+2)
2 '(fc-l)(JV.+3) 6 (fc+2)(Afs+3) '
l)x(N.-k)
(A112) 6 (A112) 7
(•^3l)(iV s -fc)x(i-l) = 0 ( J V . - f c ) x ( l - l ) ,
=
(E32)(Ns-k)x(k-l)
(i4.112)8
0(N,-k)x(k-l),
(E33)(Ns-k)x(Ns-k)
(
1± 0
(fc+3)(fc+5)
(fc+3)(fc+6)
I
2
*•
(fc+4)(fc+6)
0
0
1
U
(fc+3)(JVs+2) 2 (k+4)(N3+2) 2 (k+5)(N,+2)
0
"(fe+3)(iV»+3) 2 (k+4)(N„+3) 2 ' (k+5)(Ns+3)
\
(Na + l)(N,+3)
Vo
1
(A112) 9 L e m m a A.13
li I < k, we have
s U e j - V * = ViVs - lBfcBii^jfc, where v/t and v;t denote the vectors (A.88) and (A.104) respectively. Proof
Since v f c = BfcUfc,
(A113)
388
APPENDIX A. SOME FACTS ABOUT SPHERICAL
HARMONICS
which is a paraphrase of the equations (A49)', (A50)', (A51)', we have
^,(ei s) ) Vfc = BfcSj^e^H = VNS - iBkulk = ^NS - iBkB[k\lk. As the consequences of the Lemma A.12 and Lemma A.13, we have the following five propositions. Proposition A.15
If 2 < j < I - 1 < k — 2, we have
—Ni V W i
'~L(2) ^ *
-K-(32)
' ~~
v^wTTs K ^
(A114)
(©H-
Proposition A.16 If / < fc, we have =1 CA(S)\=3
fc 2
-££-?
/ v
(M*)\ -
V3l(N* - 1 )
=4
/ f i (sh
2V(I - l)(Na - 1 ) ,QW. =4 —lf(31) (©» ) K ' ( j + 2 ) ( j + 3)(iVs + 5 ) " L I ,l' ±11, ,IXV
2V3(t-l)(AT.-l) ^ (s) U V/(fc + l)(fc + 4)(7Vs + 5 ) " < *
"'
£
j?k
2
2V(i - 1)(ATS - 1)
=4
/Q(.h
_ V2(i-l)(JV.-l)g.
%<«> (©D +
sjl{j + 4)(j + 5)(ATS + 5 ) " K ! * J + . ^ * ' '
y/l(NB + 3)
- M l l t ( © i )•
(4.115) Proposition A.17 If Z < j < fc — 1, we have "N,(.Wi
j ^ x (2) ( U j 3*
j
-
0»h
v^T
V ( j + 2)(7Vs + 5) K}«-
fc-1 +
„ ^ t 1 V j ( ^ + l ) ( " + 2)(iVs + 5)
2^ v /j(fc + l)(fc + 4)(JVs + 5)
K
" T K^ U I © * )
-
i)(©iB)) u
A.3.
PRODUCTS OF SOME SPHERICAL
,
JV.-l V^
f
u=fe+1
v/j(u + 3)(u + 4 ) ( i V s + 5 )
389
HARMONICS
^4
/ft(s)x
,
V^
-=2
/ft(s)\
6"(#. + 3) (4.116)
Proposition A. 18 If Z < fc, we have
y ( f c + 4)(7Vs + 5) AT.-2
+ V j?k
2
V^ TT4 /Q(SK V(fc + 2)(j + 4)(j + 5)(JVS + 5 ) ^ + / '
V^
~2
/Q(-)X
M tl
V^fc + 2 ) ( i V s T 3 ) " "
'
(A.117)
,
Proposition A. 19 If I < k < j , we have V ( j + 2)(j + 3)
g4
V(j + 4)(ivs + 5) N.-2
+ ~£-
7?4
/ v
0 + 2 ) ( w + 4)(u + 5)(iVs + 5) v^ ^ ( j + 2)(iVs + 3) ^
In order to calculate the matrix BiBff},
- «K -(21)
^,:+1
K
(s)
(©| S) )
(<>)>
(A118)
OD
I < k, we need to write the matrix
Bi as a block matrix in the following form: / (Fll)(l-l)x(l-l) Bl —
(Fl2)(l-l)x(k-l)
(Fl3)(l-l)x(N.-k)
\
(F2l)(k-l)x(l-l)
{F22)(k-l)x(k-l)
(F23)(Lk-l)x(LNt-k)
\(F3l)(N.-k)x(l-l)
(F32)(N.-k)x(k-l)
(^33)(AT.-fc)x(iV.-fc)'
(A119)
where the blocks Fij are of the following forms / (•Fn)(i-i)x(i-i) =
1
1 3 1 (-1 3 1+2
1
h
0
3
2 3
1 (-1 3 1+2
1 (-1 3 '1+2
0 (A119), 1 ";-i 3 "7+2
f-2 i-i 3 ~ 1+2 •
390
APPENDIX
A. SOME FACTS ABOUT SPHERICAL
(° (Fl2)(l-l)x(k-l)
0 0
0
=
0
HARMONICS
\ (A119) 2
0
i-i 1+2
0/ (-f113)(/-l)x(ATs-fe) =
1 "(+3
1 '1+3
1 'fc+1 1 'k+2
1 'fc+1 1 "fc+2
lL . (+3 (+3 1 1+4
1+2 «±2 1+3 f+3 1 1+4
1+3 1+4
1 fc+1 1 'fc+2
1 fc+1 1 "fc+2
(•P2l)(fe-i)x((-l)
/ /
(F 22 ) (fc-i)x(fc-l)
(A119) 3
0(i_i)x(AT.-fc),
M
1+3
(A119) 4
1 'k+1 1 'k+2
0
\
0 , 1 k+1 . ^ _ fc+2
1_ fc+1 L_ fc+2
/° (•f23)(fc-J)x(W.-fc)
=
0 \ k+1 X
fc+2
k fc+1 L fc+2
0 0
(A119) 5
/ '
(A119) 6
o U
1 'fc+3
1 'fc+3
1 'fc+3
Ws + 1
Ns + 1
ATS + 1
(A119) 7
( * 3 l ) (JV.-fc)x(i-l)
1
1 /
1
fc+3
1 'fc+3
• - 1 -
Ns + 1
iVs + 1
Ns+1
.
1
/
\
fc+3
\
(A119) 8
(F32) (N.-k)x(k-l)
1
1
1 "fc+3
fc+2 fc+3
Ws + 1
Ns+1
0
(F33)(Ns-k)x(N,-k)
1
1
Ns+1
Na Ns+1
1
In parallel with the L e m m a A. 12, we have the following
1
(A119) 9
A.3.
PRODUCTS OF SOME SPHERICAL
391
HARMONICS
L e m m a A.14 B B
l 7k
=
I
(Gll)(J-l)x(i-l)
(G'l2)((-l)x(fe-i)
(G2l)(fc-i)x(i-l) r (G 3l)(Af.-fe)x(J-l) l
(G22)(k-l)x(k-l) (C r 32)(N.-*:)x(fc-I)
(Gl3)(l-\)x{N.-k) (G f 23)(fe-i)x(Ar.-fe) I , (G33)(iv.-fc)x(N.-/t),
(A120) where (Gll)(i_l) x (I-l) = -fy-l)x(f-l).
(A120)i
( G r i 2 ) ( i - l ) x ( f c - 0 = 0(Z-l)x(fc-Z)>
(A120) 2
(G f 13)(i-l)x(iV s -fe) = 0(J-l)x(JV s -*;)>
(A120) 3
(G2l)(fe-J)x(J-l) = 0(k-«)x(l-l)>
(A120) 4
(G22)(fc-J)x(fc-l)
=
hk-l)*
(A120) 5
(*-»)'
(G r 23)(*:-i)x(W B -fc)
/
0 0
0 0
••• •••
0 0
0 0
(A120) 6
v-
0 2(fc+l) (fc+2)(A:+4)
0 2(fc+l) (fc+2)(fc+5)
•••
0 2(fc+l) (fc+2)(Ws+2)
0 2(fc + l) (fc+2)(JVs+3)
(G3l)(JV.-fc)x(J-l) = 0 ( A r . _ f c ) x ( i _ i ) ,
(A120) 7
(
(A120) 8
392
APPENDIX A. SOME FACTS ABOUT SPHERICAL
HARMONICS
(G33)(N,-k)x(N,-k)
n
1
(fe+3)(JV.+2) 2 (fc+4)(N s +2) 2 (fc+5)(Af.+2)
(fc+3)(JV.+3) 2 (fe+4)(iV.+3) 2 (fe+5)(iV.+3)
0 0
1 0
2 '(Af. + l)(JV.+3)
(fe+3)(fc+5)
(fc+3)(fe+6)
0 0
1 0
(fc+4)(fc+6)
0
0 0
Vo
1
\
I (A120) 9
Lemma A.15
If I < k, we have (A121)
where vj and v;fe denote the vectors (A.88) and (A.104) respectively. Proof It can be proved in a way similar to that of the L e m m a A.13. As consequences of the Lemma A.15, we have the following propositions. Proposition A.20 If 2 < j ; < / < k, we have
^Ms)n^M8))=#=k(32)(eh.
(Am)
Proposition A.21 If 2 < / < k, we have = 1 fft(»)\n3 f fl (s)\ _ V-^s ~ 1 ^4
fft(»)\
(A123)
-Nt(©< W © * ) - - 7 = = = ^ K j ( s ) ( o i ). Proposition A.22 If I + 1 < j < k — 1, we have
(A124)
-Nt(e, j ^ O , )- 7 ===^ K <. ) (e i ). Proposition A.23 li I < k, we have =i
r o (-K=3
ffi(s)v_
JV.-l
- AEI
y/3(fc + 2 ) ( A r s - l ) = 4
>('h V(* + 4)(JV. + 5) "Kfi^ * '
2 V ( * + 1)(JV.-1) u
3
." 4
V( + )(" + 4)(fc + 2)(iVs + 5) Ki*
(»h
A.3.
PRODUCTS OF SOME SPHERICAL
y/2(k +
+ y/(k
HARMONICS
l){N,-l)a.
+ 2)(N.
393
(S)N
(4.125)
+ 3) s& Ifc (en-
Proposition A.24 If / < k < j , we have =1
ro
W^3
, e C K _ V(J + 2)(j + 3 ) ( J V . - l ) g 4
-N.(e, K u . O , ) -
(*)\
K(21)(©j
V ( j + 1 ) ( i + 4)(iVs + 5)-K;:;.
)
AT.-l
+
2y/N7=T (»)\ s^(.i)(©"0 K J ^ VO' + 1)0' + 2)(« + 3)(u + 4)(iVs + 5)" u y/2(NB - 1) H^(eis>). / ( j + l ) ( j + 2)(iV + 3) x s
E^(e\a))E3,^(O^H^e^)-*,^)
In order to get the expressions of and - ^ ( © f ^ H ^ e f 0 )
(A.126)
for 2 < / < fc < A^s - 1 as linear combinations of
S k ( e £ ° ) , S i L I ( e ^ ) ) and S2j(ej s ) ), we introduce the following matrix: /
R-s
Rii
R12
R-l.JV.-l
R"21
R.22
R-2,iVB-l
\RAr._l,l
RjV,-l,2
•••
\
(A127)
RiV._l,ATs-l/
where TLuv r»(l)
,
***uv V
R
(AT.—u-2) tuple
R£>= fl, 2 ~ ^ 2 , - 2 u - 4 ,
(u—w) tuple
0~^To
Yw
(A127) 2
(AT B -«—2) tuple
R4^- - = (Y 2 ~ ^ 2 , l - u , («—v) tuple
(A127)!
(AA.-u-l)
(u—v) tuple
U 1)
for 1 < u < JVS - 1
1~~~1
Yv
(k — 1) tuple
R ^ = Tl, 2 7 ^ 2 , 1 - u , TT^Tl ,-(fc + ti + 4),
(A127) 3
( W s - f e - u - 2 ) tuple
6~"^0
Y
394
APPENDIX
A. SOME FACTS ABOUT SPHERICAL
HARMONICS
for 1 < k < Na - u - 2, v < u < N3 - 2, (N.-U-2)
Ri% = ( ^ ,
(A127) 4
tuple
o,..,o
(A.127)5
) , " < Na-2,
NB—u—l tuple
HK+"
_1)
= ( -u,-",-« ),«<JV.-2, (Af.-fe-u-2) tuple
k tuple (k)
I —u, • • •, — u, (k + w +
R 'u,u+l
(A127) 6
4)M,
)•
(A127) 7
for 1 < k < Ns - u - 2, (N.-v) }
R^ =(
tuple
O^To
J,
for 0 < f c < W . - u - l , u + 2 < u < iVs - 1, (A127) 8
(N,-v-l) R
K t
= f 1.
tuple
'2. • • • .2"
(A127) 9
J, for 1 < u < JV. - 1,
Since the equations (A.61), (A69)i, (A.45), (A46)i, (A.63), (A71)i can be reformulated as follows: 2 ( r ( s ) f t ( ^ W ) ) V « ( n + 1)( U + k + 4)(u + fc + 5 ) ^ 4
(s)
(2)
K i+l,u + k+l
V^(7V. + 5)(WS + 3)(iV. + 1)(JV, - 1)
(eD
(u + fc + 5 ) ( u + l ) u+fc+2 ..(•) ( U; «+*:+2,j)
U+k+5
(•)\2 a a £K?) («&.«> -^D«#)
*
3=2 u
u+1
u+1
) 2
1
2
j=2
2
u+lu+/c+l
2E E (^: ) (^ ) +( -")E(-?) (^2,) +E E (-?)2(^))2 , j = 2 p=j"+l
) 2
j=2 p=u+3
A.3.
PRODUCTS
OF SOME SPHERICAL
395
HARMONICS
u+l
u+k+l
j=2
j=u+3
u+l
J=2
(.4.128)
for 1 < k < Ns - u - 2, 1 < u < Na - 2,
V2u( M + l)(r(*>A((s)))4=2 (B) V(iVs - l)(iVs + 1) - ^ u + 1 ( e D JV.
_ ^ _^^u(+s )2 , i )- ) 2 _ ^ _ 2V- ^W( Si) ^1 2 + 1 U + 1
-(ti + i ) £ W(s)\2 ) J=2
ti
u+l
U+l
*E E rfW-^-iiE^A) i=2
j = 2 J=j + 1
i+l
N.
2
E E
2
N.
u+l
j=u+3
j=2
2
E (»SV + B^?>4-«(w&.«)4.
rf) ^) -^,)
i=2 j = u + 3
(A129)
for 1 < u < Na - 2, 2^2u(u + 2)(u + 3)(u + 5)(r( s )A($ ( s ) )) 4 j 4 ,0(.K K +A y/3(Ns + 5)(NS + 3)(JVS + 1)(NS - 1) ~ " * fi/,„(s)
(u + 5)(u + 3) (W
2
"+ ^
u+ 5
s 2 u+2
^
/U
^
} +
u
u+l
+2
(u + 5)(u + 3)1, y
J—2
2
=
. Ji }
J—2
u+l
u+l
W\4
E E ( ^ J V f f J ' - ^ + ^ ^ + ^ E ^ ^ + E^?) j=2p=j + l
j=2
j=2
;
396
APPENDIX A. SOME FACTS ABOUT SPHERICAL
+
(r
«(tH^(f|JWaii)4f
(s)A($(s)))4H0)(e(s))
f o r l
N.
-,2
3=1
••
N.-l
HARMONICS
2 )
(A130)
N.
N.
=
j=2 u=j+l
j=2
(A131) we have the following L e m m a A. 16 V = RsU,
(A132)
U = R^V,
(A132)'
or
where R s denotes the matrix (A.127), and Ul
/
\
:
U=
(A133)
UN.-2
(^'+l.i)4
/
1
\
(«# W )>&,.)
3
, for 1 < j < 7 V S - 1 ,
(r(s)^($(s)))4
(Al33)i
v M+V;) 2 ^) 2 / (A134) \ViV.-l/
2^JU^T)
^4
V'3(iV.+5)(iV.+3)(JV. + l)(Ar.-l)(j+5)C7+3)
K
(e(s)}\
' +l V
* '
2VjQ + l)Q+5)(j+6) s/(.N.+5)(N.+3)(N. + l)(N.-l)
"K<2+>1|i+2 V
i
'
for 1 < j < Ns - 2, 2y/j(3 + l)(JVs+2)
= 4
_
,Q(S)N
y(Afs+5)(Afs + l)(JV 8 -l)"K< 2 + ) 1|Na _ 1 V2j(j + 1)
„2
, f t (s)x
(A134)!
A.3.
PRODUCTS
OF SOME SPHERICAL
397
HARMONICS
vN._1 = (E0o(eis))).
(A134) 2
In order to compute the inverse matrix ( R ) _ 1 of the matrix R in (A. 127), we introduce several matrices as follows:
(Au)(jv._u)x(iV.-u) =
0
2(u+2) (u+5)(u+6) 1 (u+5)(u+6) 1 u+6 0
2(u+2) ( u+6)(u+7) 1 ( u+6)(u+7) 1 ( u+6)(u+7) 1 u+7
2(u+2) (JV„ + l)(iV.+2) 1 (iV„ + l)(iV.+2) 1 (JV. + l)(ATs+2) 1 (N.+l)(N.+2)
2(u+2) (N.+2)(N,+3) 1 (iV s +2)(iV.+3) 1 (iV.+2)(iV s +3) 1 (iV.+2)(iV.+3)
0
0
0
V 0
0
0
1 N.+2 0
1 (JV s+2)(AT.+3) 1 N,+3
/ 1—u ' u+5 1 u+5 0
2(u+2) , JV.+3 \ 1 N.+3 1 N.+3 1 N,+3 1 iVB+3 JV.+3 ' (A135)
/
(B„)(JV. - u ) x ( N . - u )
0
u (u+l)(u+3) 2 u +2u+3 (u+l)(u+3) 0
0
0
o
0
2u+3 (u+l)(u+3) 2u(u+2) (u+l)(u+3)
=
V
0
0
0
0
1 u+1 0
0 1 u+1
0
0
n \ U 1
•• •
0
•• •
0
•
0
••
J
u+1 / ( A l 36)
/u(u+2) f 0
0 u u
0 0
0
0
u
0\ 0 0 0
V o
0
0
u )
(C„) (N.-u)x(N.-u-l)
Q
Q
(A137)
(DU)(JV.-U)X(JV.-U-1) = (BUCU)(WB_„)X(JV8-U-1) / '
2u(u2+3u+3) 3(u+l)(u+3) u(2u 3 +7u 2 +6u-3) 3(u+l)(u+3) 0 0
0
0
0
0 u u+1 0
0
0
0 u u+1
0
0
0
\
0
-**- I u+1 /
(A138)
APPENDIX
A. SOME FACTS ABOUT
SPHERICAL
HARMONICS
u>v, —u)x(N,—v)
_ 1 = (n \°(N.-u-l)xl
n (J)lX("-") U(N.-u-l)x(u-v)
(J)ix(«-«) = ( - 2
( W s ) iV.(iV.-l) 2 *
Af.(JV.-l) 2
-2
0
( P g ) JV.(JV.-l)
( A 1 3 9 )
••• - 2 ) l x ( „ _ 1 , ) ,
/Ai 0 0 A2 V 0
E21
\(N.-u-l)\ -i-N.-u-l )
0
(A139)!
\
0
(.4.140)
• • • AAr._i/
JV.(Af.-l)
IiV.-2 E32
lAfa-3
Ew._2,Ar„-3
I2 EAT,-l,iV.-2
1/ (A141)
( Q s ) iV.(Af.-l) A . , JV«(W.-1) 2 2
Qn
Q12
QI,JV.-I^
Q21
Q22
Q2.Ar.-1
'QAT,-1,1
QjV.-l,2
• • • Ql,iVs-l/
(A142)
A.3.
PRODUCTS OF SOME SPHERICAL
399
HARMONICS
where
(Quv)(Nt-u)x(N.-v)
0(jv.-u)x(Ar.-«)> Bu, = { Du D„_iB„, Du---D„_i, Ii,
Hu> v; ifu = v
(A143)
Having introduced the matrices (A.135)-(A.143), we can formulate the following L e m m a A. 17 Rs" 1 = P s Q s W s M12 M22
Mn M2i
MI.JV._I
M
M N.-l,2
1,1
\
M 2 ,iv.-i N.-l
(A144)
N.-IS
where
(M u „)(jv a -u)x(7V.—v) =
0(jv.-«)x(Af.-«). F„Au_i, G„B„A„, < GUDU D^-iB,^, G u D„---DAr s -2Aiv s -i, LGjv.-iAjv.-i,
i f _ > u + l; ifu = i; + l; if u = v < Na - 2 (A145) if _ < v < i V s - 2 iiu
and (Fu)(jv.-«)x(JVB-u+i) = E„ t U _iB„_i -2u2+u-5 u(u+2)
4u-*-2tt-5 u(u+2) ,°(N,-u-l)xl
0,
0(JV.-u-l)xl
3
2
Proof
( A 146)
-u^N.-u-l
(G u )(jv.-«)x(iv s -u) = E U ) U _iD u _i 4M4-4U ti 3-3u - 3 H-2 - 1 2 M + 2 4
^lx(Na-u-l)
+IN,-U
Q
3u(u+2)
'lx(JV.-u-l)
0( A r . - u - i ) x i
^IAT.-U-I
(A147)
The form of the inverse of the matrix R s can be obtained through a
series of elementary row operations. Of course, as soon as the form of the inverse matrix is available, it can be verified simply by matrix multiplication.
400
APPENDIX A. SOME FACTS ABOUT SPHERICAL
HARMONICS
It is easy to see that =1
(Ns-l)y/(Na
to(s)\c3
+ 3)(Na + l),
y/2(l - l ) f ( r « f t ( * M ) ) 4
L
fc»(s)\
(8)
«i.<)
,2 a
i+i
a
'(«O -B«#>a j=2
(Ns-l)y/(N^+3)(Ns
+ l)
y/2(l - 1)Z
HU,
(A148)
where (4.148),
H = (HI,---,HAT.-I) ,
— (<Si,k-t+i, • • • ><^iv.-i,fe-i+i),
if i < / — 1;
Hj = { (I - l)(^i,fc—i+i,- • •,<5w.-z,k-i+i),
iii = I;
(4.148)2
if i > I + 1. Hence we have the following Proposition A.25 S
i
fft<*h=3
m (*K
. ( ^ s - l ) V ( i V s + 3)(iVs + l)
HR;^.
(4.149)
Since
zkms)KvMs)) (Ns - l)y/(fcT 2)(JV3 + 1)(NS + 3 ) , (., V2(* + l)(rWft(*W)) <
,2
Wi.,)
8
2
2 rfM) -TT^E(-S) A; + 2 i=2
(iV s -l)V(7V s + l)(iV s + 3) ' >/2(Jt + l)(* + 2)(r(->&(*«))
fc+i
h2 <*
i=2
+(t»ffM)4 + (-ffu)2 [ E (-S)2 - ( * + i ) d O a l I j=f+2
A.3.
PRODUCTS OF SOME SPHERICAL
(Nm-l)y/(N.
401
HARMONICS
+ l){Nm + 3)
y/2(k + l)(k + 2)
(A. 150)
KU,
where K = (Ki,---,KAr._i), ' —{Si,i-i+i,- • • fSffm-i,i-i+i), K i = < (1,---,1 ,-(fc+ l), (fc-i)tuple
<),•••,0
(i4.150)! if i < I - 1;
),
if * = 2;
(A150) 2
(iV.-fc-l)tuple
.0lx(Jv.-i),
i f i > / + l.
Hence we have the following Proposition A.26
Ef,,(e<->)E;(„(e<-»). W - W W + q w + UKB.-y,
(A151)
V2(A; + l)(fc + 2)
Finally, since
Hk(e!s))sL(eis)) (W. - 1)V(W. + 1)(JV. + 3) 4 > /6(fc-l)(fc + 2)(rW/3 i ($W))
(fc-i)rfM)4-3rfM)2E(^S)
(S)N2
j=2
(JV.-l)>/(JV. + l)(iV. + 3) LU, / v 6(fc-l)(fc + 2)
(A152)
where L = ( L i , - - - ,LAT.-I ) ,
' -3(#i,fc-i+i, • • •, <5^s_i,fe_i+i), Li = < ( f c - 1 ,
0,---,0
),
if i < k - 1; if i = fc;
(iV B -/s-l)tuple
L°lx(W.-i).
(A152)i
if i > k + 1.
(A152) 2
402
APPENDIX
A. SOME FACTS ABOUT SPHERICAL
HARMONICS
Hence we have the following Proposition A.27
si,,(e«)E£fc(e<->) = * ^y/6(k-l)(k f f i 2 f f i +± 2)5 L R . - i v . A.4
(A153)
Derivatives of Some Spherical Harmonics
It follows from the definition of the spherical harmonics H(0^ SJ ) that Proposition A.28 If j ^ i, we have (.4.154) h
It follows from the equation (A.40) that Proposition A.29
^( f )H° 0 (ei s ) ) = o.
(A155)
It follows from the equation (A.41) that Proposition A.30
S)
dw^hM ) =
V W . - l r l ' > f t ( * l , » ) 5 N ^ e ' ) S N , _ 1 ( 0 i )'
l f / ^ f c + 1;
s°o(eis))-i^ri(^(e|8)))2
if Z = fc + 1. (A156)
According to the Proposition A.2 and Proposition A.4, the right hand side of the equation (A.156) can be written as a linear combination of the basic spherical harmonics. Hence we have the following two equations. In case of / ^ k + 1 we have 1
»(*h a ^, = i o n - - A + l r W f t ( W ) ^.,(fln. /Q(»)
(^156)
A.4.
DERIVATIVES
OF SOME SPHERICAL
HARMONICS
403
and
d c
v^
,
=1
(&(*)\
_
1
f ^8 ~ 2
=Q
.
(S)
^rHk(e'>,-g^iE-«<e?,)]}-
v/^Ti
(A156)
*
Taking account of theProposition A.2 and Proposition A.4, the equations (Al56)i and (A. 156)2 are reformulations of the equation (A.156). All the equations in the following propositions can be reformulated in a similar way, but we will not write down the reformulations explicitly. Proposition A.32
y/N.- lr<*)ft (*(•>) " N j - i V ^ i
^-Mn^i
^
+7^ife) S N,(©i s) ), (sh
».c„E^(ej"') = <
2
ifi = * + l; s
— 1 /•ft( )\'='2
s
fo( )\
if j = 1 + 1; I -VTC=Ir(-» A (*C)) B i» J -i( e « ) ^ (
8
«
>'
°therWiSe(A157)
Of course, using the Propositions A.5, A.9-10, the term
y/Na - lrW/8i($W)
^(ehs^eh
on the right hand side of the equation (A. 157) can be expressed as a linear combination of the basic spherical harmonics of order four.
404
APPENDIX A. SOME FACTS ABOUT SPHERICAL
HARMONICS
Proposition A.33 ^/jV^ : T^(•)/3 i (*(»))"Nj_ 1 V c '^ y/2(N. + l) 2
»„vn*M ) = \
=
»<"h)
J-Mk^i
(.)
p l
V(fc-l)ferC)/9i(*(»))"NiV
m)
( s h=2
(•ft( s )\'=2
1
if 2 < j < k;
« '' /«(s)\
l Va(fc-i)(Af.+i) g l . _ ( . ) . Vfcr(')A(*<-)) "N^Wi ).
, ,. t , it J - fc + 1,
i f
+
2 TTl CA^h"2 CCi( s )\ c >/A?.-lr<»)/3 i (*<">)~N i _i^ 'i ^ M t V 0 ; />
if j > k + 1. (4.158) Of course, using the Propositions A.6-8, the term 2
„,
-^(s)-,^
>/3?TTrWiJj(*W)"Ni-11
* ;~M^Wi
(s)s ;
on the right hand side of the equation (A. 158) can be expressed as a linear combination of the basic spherical harmonics of order four. Proposition A.34
'^k^Ek-^)ElJ^) 2 ^
>
+ ^ f U r y ^ O , ) , ifi = I + l;
b r ^ - » (e<)sU (e<) + ?riSpjy^, (e*)> ** = * + !; 3
TTl
otherwise. (A159)
2 % /wr=i' : i N i-i( 0 i ) H L p i f c ( e O:
Of course, using the Propositions A . l l - 1 4 , the term 3
\-=3
"27^^t H ^- i ( 0 i ) H ^' t ( e i ) on the right hand side of the equation (A. 159) can be expressed as a linear combination of the basic spherical harmonics of order four. Proposition A.35 ^<.>^)(©!s))
405
A.4. DERIVATIVES OF SOME SPHERICAL HARMONICS
~ 2VJV.-1 S N j - i ( e i ) H L i ( ^ ) ( ^ 3
=1
+
e n i i V 2 ('- 1 )( j V .+ 3 )^2
cft\33
if j = fc + l;
r(')/3i"(*<->)SM|(e»)i rft.\
if
„•_/,•,.
= <
- ^ ^ ( e , ) ^ ) -
>/i^gttM)^.1,l(e<),
if 2
Of course, using t h e P r o p o s i t i o n s A . 1 5 - 1 7 , A . 2 0 , A . 2 5 , t h e t e r m
V^T^-.< e ') H C< e '> on t h e right hand side of t h e equation (A.160) can be expressed as a linear combination of t h e basic spherical harmonics of order four. Proposition A.36
^
(
3
~i
CA1"3
[V(fc-l)(fc+2)„ 2 I
2
v^fc-1
, f t (g). 1
(Pi \ I
(e<°>)
VW.+3
_^TT„
p2
2
, (s)s
(ft( s h
2vTvr=T-N i _A t '»;- L (i)(.« :: '»; + Vk+2r^pi{^-))^Mlk\L>i
if j = l + l; )> « j - « + 1 ,
= < - 2 V J V . - l S N i _ i ( e 0 S L ( i ) (@i) / v
V2(iV.+3) ^2 , (,) (fe+l)(fc+2)r(»)ft(* (B) )^ M ^- 1 '^ * ; '
H ^ J ^ t ,
~2VN.-lSN3-1(0i)SL(1> (0i) V2(Af.+3) 0»h .M-M,i:,-_i(0i ) V'(fc+l)(fc+2)r(»)ft(*("))
[-27fcHN3_1(©i)S3j(1)(0,),
if / + 2 < j < k; otherwise. (A161)
406
APPENDIX A. SOME FACTS ABOUT SPHERICAL
HARMONICS
Of course, using the Propositions A.22-24, A.19, A.26, the term 3
=1
/ft.\s3.
v^ 3 f *- ( e , ) E « ( e J on the right hand side of the equation (A. 161) can be expressed as a linear combination of the basic spherical harmonics of order four. Proposition A.37 3^
_
2 V J V . - l S N i _ 1 ( e » ) 5 L , t ( e i ) + Vfc+2rC),li(*<•>) H M t (©i )'
^ 3 = k + V,
^ V J V . - l 5 ^ . , (ei)SLfc ( 9 i )
.-a73i=TSNi-i(e*)^(e').
otherwise. (A162)
Of course, using the Propositions A.18, A . 2 1 , A.27, the term 3
Wl
^Q.\=3
- 5 7 R j- I * l . 1 (8,) B t.(ft) on the right hand side of the equation (A. 162) can be expressed as a linear combination of the basic spherical harmonics of order four.
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Verlag, New York. [71] G.G.Stokes (1845), Trans. Camb. Phil. Soc. 8, 271. [72] M.E.Taylor (1996), Partial Differential Equations, vol.11. Springer-Verlag, New York. [73] R.L.Taylor, P.Z.Daffer and R.F.Patterson (1985), Limit Theorems for Sums of Exchangeable Random
Variables. Rowman & Allanheld Publishers, To-
towa, New Jersey. [74] C.Truesdell and R.G.Muncaster (1980), Fundamentals Theory of a Simple Monatomic
of Maxwell
Gas. Academic Press, New York.
Kinetic
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[75] S.Tsuge (1970), Phys. Lett, 3A, 145-148. [76] S.Tsuge and K.Sagara (1975), J. Stat. Phys., 12, 403-430. [77] N.N.Vahanya, V.I.Tarieladze and S.A.Chobanyan (1985), Probability Distributions in Banach Spaces,(in Russian). Nauka, Moscow. [78] N.G.van Kampen (1985), Phys.
Reports , 124, 69-160.
[79] N.Ja.Vilenkin (1968), Special Functions
and the Theory of Group Rep-
resentations. American Mathematical Society, Providence. [80] N.Ja.Vilenkin and A.U.Klimyk (1993), Representation Special Functions, vol.2: Class I Representations,
of Lie Groups and
Special Functions, and
Integral Transforms. Kluwer Academic Publishers, Dordrecht. [81] J.von Neumann (1962), Recent Theories of Turbulence, in Collected Works of John von Neumann, vol.6, 437-472. [82] E.T.Whittaker (1928), Treatise on Analytical Dynamics
of Particles and
Rigid Bodies. Cambridge University Press, London. [83] E.T.Whittaker and G.N.Watson (1927), A
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New York. [85] H.T.Yau and L. Jensen (1996), Hydrodynamical
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[87] J.Yvon (1966), Les Correlatiions et I'Entropie en Mecanique
Statistique
Classique. Dunod, Paris. [88] V.N.Zhigulev (1966), Sov. Phys.-Doklady , 10, 1003-1006. [89] V.N.Zhigulev and A.M.Tumin (1987), Vozniknovenie
Turbulentnosti (Onset
of Turbulence) (in Russian).Izdatelstvo Nauka, Novosibirsk.
Index Anderson, J.D.jr. 339,407
BBGKY hierarchy 3,4,12,18,90,91,180
Andrews, G.E. 407
Bogoliubov, N.N. 3,12,407,408
antiderivative of H-functional 32,126,135
Bogoliubov,N.N.Jr. 408
approximate Liouville equation 176
Boltzmann, L. 3,6,7,408
Arnous, E. 3,407
Boltzmann equation 5,9,13,24,54,70,79,
Askey, R. 407
80,92,98,178,179,180,184,
assumption of molecular chaos 57
253,271,272,343
assumption A 107,109
Boltzmann-Grad limit 7.9,10,92,179
assumption B 164,165
Boltzmann-Gronwall theorem on
assumption C 281,287,319 asymptotic analysis of Liouville equation 81
summational invariants 104 Boltzmann hierarchy 5,6,7,341 Born, M. 3,12,408
Avery, J. 407
Bruns 71,108
Bach,A. 407
Carleman, T. 408
balance equations 13,54,55
Cercignani, C. 408
for mean mass 57
Chapman, S. 4,7,408
for mean momentum 60
characteristic functional of random
for mean kinetic energy 62,63
fields 11
for mean potential energy 62,63
Chen, T.Q. 5,6,8,408,409
for mean total energy 63
Chobanyan, S.A. 413
Bass, J. 3,407
Chow, Y.S. 409
416
INDEX
Conway, J.H. 409
finer if-functional equation 24,136,252,
correct variables of Euler 4,57,180
271,272,336,341,342,343,344,
Cowling, T.G. 408
345 first proposal of gross determinism 177,
Daffer, P.Z. 412
178,343
Daletskii, Yu.L. 409
Fomin, S.V. 409
de Boer, J. 407
formal derivative of i^-functional with
de Finetti 106,345 definition of Euler fluid flows 165
respect to heat energy 290 fractional derivative of AT-functional 289
energy equation 155 Enskog, D. 4,7,387
Gallavotti, G. 409
Enskog-Chapman expansion
Galerkin-Ritz technique 249,369
(technique) 4,6,8,24,92,
Gegenbauer polynomial 348
178,180,183,341,343
Gelfand, I.M. 409
equation of continuity 149 equipartition of heat energy 127,263, 269 Erdelyi 409 Esposito, R. 409 Euler equations 152,153,179,180,271, 342,344 Euler if-functional equation 23,24,
generalized Gibbs canonical distribution 100 Gibbs, J.W. 2,6,7,98,409 Gibbs canonical distribution 14,22,98, 100,102,340 Gibbs mean 109,115,116,151,153,281 Gibbs micro-canonical distribution 98 Gibbs power 115
119,136,140,144,147,148,149,
Goldstein, H. 28,409
153,155,156,271,341,342
Gora,E.K. 407
exchangeable random variables 57
Grad, H. 5,10,19,23,91,92,184,340, 341,410
fast variables 88
Grad's thirteen moments 6,31,64,71
417
INDEX Gradshteyn, I S . 410
integral Gibbs power 115,116
Green, H.S. 3,12,408 gross determinism 175,177,184 Grothendieck, A. 410
Jensen, L. 342,413 Johnson, G.W. 410
Hadamard, J. 410 Hagihara, Y. 410 Hamilton's system 28 Hamiltonian 28 Hewitt, E. 106,410 /f-functional 12,13,14,27,29,30,32,51, 54,55,56,64,65,66,70,73,75, 79 if-functional equation 13,33,51,54,55, 62,64,66,67,73,78,79,136, 155 Hopf, E. 5,8,12,147,169,410
^-functional 14,24,25,54,69,70,71, 73,75,78,79,108,119,127,131, 143,144,147,149,157,162, 163,164,167,169,173,180, 182,279,318,327,336,342 Kipnis, C. 342,410 Kirkwood, J.G. 3,12,410,411 Klimyk, A.U. 346,413 Kogan, M.N. 411 Kolmogorov, A.N. 5,169 Kothe, G. 411
Hopf functional 8 Hopf functional equation for turbu-
Landim, C. 342,410
lent inviscid incompressible
Lanford, O.E.III. 91,411
flows 9,80,147,180,342
Laplace asymptotic theory of inte-
hydrodynamic functional 30 hydrodynamic random fields 30
grals 263,282 Lapidus, M.L. 410 law of large numbers 57
Ikenberry, E. 6
Lewis, M. 5,6,411
Illner, R. 408
linearized Boltzmann integral opera-
infinite dimensional pseudo-differential operators 143,279,280,281
tor 70,80,253 Liouville 29
418
INDEX
Liouville equation 5,7,8,9,10,12,13,15, 20,21,24,25,29,51,57,60,66,
Muncaster, R.G. 54,69,92,177,178, 184,412
70,73,80,81,82,85,99,166,176, 179,212,253,272,341,343 Liouville operator 70,80,95,96,244, 252,253
Navier 339,412 Navier-Stokes equations 1,2,105,178, 179,180,339,344
local Gibbs distribution 22,100,103, 105.106,108,340,342 local Maxwellian distribution 79,182, 271,341 local stationary Liouville equation 24,
Painleve 71,108 partie finie de Hadamard 40 Patterson, R.F. 412 Poincare, H. 71,108
93,177,212,225,232,233,249,
Poiseuille flow 339
265
Poisson, S.D. 339,412
Loomis, L.H. 411
poly-spherical coordinates 161,234 projective limit of K-functionals 165,
macroscopic delta function 56,126
166
Marra, R. 409
pull-back 148
Massignon,D. 3,4,5,7,9,18,24,25,57,
Pulvirenti, M. 408
156,178,179,180,182,407,411 Maxwell, J.C. 3,6,7,411
random fields 11
Maxwellian distribution 6
reformulation
Maxwellian iteration of Ikenberry and Truesdell 6 method of stretched fields 184 Minlos, R.A. 409 Monin, A.S. 147,340 Morrey, C.B.,Jr. 7,9,10,18,90,91,92 Morrey's fluid limit 7,10,180
of H-functional equation 64 of Euler K-functional equation 140 Reynolds, O. 1,2,3,5,105,156,169,233, 412 Reynolds-Gibbs distribution 107,227, 228,229,231,233,234
419
INDEX Roy, R. 407
Stokes 339,412
Ryzhik, I.M. 410
Sagara, K. 5,413
Tarieladze, V.I. 413
Saint-Venant, B.de 339,412
Taylor, G.I. 5,169
Savage, L.J. 106,345,410
Taylor, M.E. 412
Schlichting. 6
Taylor, R.L. 412
Schwartz, L. 84,176,412
Teicher, H. 409
second order approximate solution to
temporal part of material derivative
the Liouville equation 227, 232,233 second proposal of gross determinism 184
of TN 184 third proposal of gross determinism 96,210,211,220,222,223,242, 246,255
Shapiro, Z.Ya. 409
Tollmien 6
signed K-measure 167,168
Tollmien-Schlichting waves 6
Sloane, N.J.A. 409
topological tensor product 65,66
slow variables 88
Truesdell, C. 6,54,69,177,184,412
spatial part of material derivative of
Tsuge, S. 5,6,413
TN 219,220 spherical harmonics 234,247,252,262,
Tumin, A.M. 5,414 turbulent Gibbs distribution 14,21,22,
277,307,308,347,348,349,352,
23,24,79.81,84,85,90,93,94,
353,365,366,367,368,369,371,
95,96,97,98,99,103,105,106,
375,378,379,402,403,404,405,
107,108,109,113,119,127,131,
406
136,155,157,158,162,163,164
Spohn, H. 412
165,166,169,173,175,176,180
Sternberg, S. 411
181,182,184,211,219,220,221
Stirling formula for Gamma function
228,231,233,244,249,269,271
170,284
272,296,341
420
INDEX
turbulent Gibbs measure 108,164,165, 166,167,168,169
Uhlenbeck, G.E. 407 Vahanya, N.N. 413 van Kampen, N.G. 88,109,413 Vilenkin, N.Ja. 347,349,409,413 Volterra's functional calculus 147 von Neumann, J. 339,413
Watson, G.N. 413 Whittaker, E.T. 413 Wong, R. 413 Yaglom, A.M. 147,340 Yau, H.T. 342,413 Yuan.M.E. 5,409 Yvon,J. 3,5,12,413,414
Zhigulev, V.N. 5,6,414
A NON-EQUILIBRIUM STATISTICAL MECHANICS Without the Assumption of Molecular Chaos
*m m This book presents the construction of an asymptotic technique for solvirg the Liouville equation, which is to some degree an analogue of the Enskog-Chapman technique for solving the Boltzmann equation. Because the assumption of molecular chaos has been given up at the outset, the macroscopic variables at a point, defined as arithmetic means of the corresponding microscopic variables inside a small neighborhood of the point, are random in general. They are the best candidates for the macroscopic variables for turbulent flows. The outcome of the asymptotic technique for the Liouville equation reveals some new terms showing the intricate interactions between the velocities and the internal energies of the turbulent fluid flows, which have been lost in the classical theory of BBGKY hierarciy.
ISBN 981-238-378-6
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