Nonequilibrium Statistical Mechanics

Fundamental Theories of Physics

An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application

Editor: Alwyn van der Merwe, University of Denver, U.S A.

Editoral Advisory Board: Lawrence P. Horwitz, Tel-Aviv University, Israel Brian D. Josephson, University of Cambridge, U.K. Clive Kilmister, University of London, U.K. Pekka J. Lahti, University ofTurku, Finland Giinter Ludwig, Philipps-Universitiit, Marburg, Germany Asher Peres, Israel Institute of Technology, Israel Nathan Rosen, Israel Institute of Technology, Israel Eduard Prugovecki, University of Toronto, Canada Mendel Sachs, State University of New York at Buffalo, U.SA. Abdus Salam, International Centre for Theoretical Physics, Trieste, Italy Hans-Jiirgen Treder, Zentralinstitut fiir Astrophysik der Akademie der Wissenschaften,

Germany

Volume93

Statistical Mechanics Nonequilibrium

Ensemble Method

by

Byung Chan Eu Department of Chemistry, McGill University, Montreal, Quebec, Canada

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-5007-6 ISBN 978-94-017-2438-8 (eBook) DOI 10.1007/978-94-017-2438-8

Printed on acid-free paper

All Rights Reserved © 1998 Springer Science+ Business Media Dordrecht

Originally published by Kluwer Academic Publishers in 1998 Softcover reprint of the hardcover 1st edition 1998

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

This is for Hui Young and

my children

Contents

Preface xi

1 Introduction 1

References 10

2 Thermodynamics of Irreversible Processes 12 2.1 The Zeroth Law of Thermodynamics . 13 2.2 The First Law of Thermodynamics . . 14 2.3 Local Forms of the Conservation Laws 16 2.4 The Second Law of Thermodynamics . 19 2.5 Local Intensive Quantities:

Temperature, Pressure, etc. . . . . . . . . . . . . . . . . 30 2.5.1 Local Temperature . . . . . . . . . . . . . . . . . 30 2.5.2 Local Pressure and Local Chemical Potentials 32 2.5.3 Local Generalized Potentials . . . . . . . 33

2.6 Local Differential Form for Calortropy . . . . . 34 2. 7 Irreversible Thermodynamics in the space 1-lJ . 40

2.7.1 The Inverse Temperature as an Integrating Factor . . . . . . . . 40

2.7.2 Integrability Conditions and Generalized Maxwell Relations . 41

2.8 Irreversible Thermodynamics of Steady States . 44 2.9 Thermodynamics of Linear Irreversible Processes . 46 2.10 Inclusion of Chemical Reactions . . . . . . . . . . . . 48

References 52

3 Boltzmann Equation 55 3.1 Boltzmann Equation . . . . . 56 3.2 Mean Macroscopic Variables 58 3.3 Collisional Invariants . . . . . 59 3.4 Broken Time Reversal Symmetry 60 3.5 Generic Boltzmann Equation . 61

References 64

viii Contents

4 Equilibrium Solution and Local Variables 65 4.1 The H Theorem and Boltzmann Entropy 65 4.2 Equilibrium Solution . . . . . . . . . . . . . 67 4.3 Local Variables and Matching Conditions . 72 4.4 Stability of Equilibrium . . . . . . . . . . . . 77 4.5 Conservation Laws and Balance Equations 78

4.5.1 Conservation Laws . . . . . . . . . . . 78 4.5.2 Boltzmann Entropy Balance Equation 79

References 85

5 Mathematical Preparation 87 5.1 Reduced Kinetic Equation 87 5.2 Tensor Hermite Polynomials 89

References 95

6 The Chapman-Enskog and Moment Methods 96 6.1 The Chapman-Enskog Method

and Irreversible Processes . . . . . . . . . . . . . . . . . 96 6.1.1 First-Order Solutions . . . . . . . . . . . . . . . . 97 6.1.2 Formal Expansions for Transport Coefficients 106 6.1.3 The Chapman-Enskog Method

and Irreversible Thermodynamics . . . . . . 113 6.2 The Moment Method and Irreversible Processes 117

6.2.1 The Maxwell-Grad Moment Method . 118 6.2.2 The Moment Method

and Irreversible Thermodynamics. 120

References 122

7 Classical Nonequilibrium Ensemble Method 123 7.1 Thermodynamic Branch of fa . . . . . . . 127 7.2 Thermodynamics of Irreversible Processes 131

7.2.1 Calortropy Differential . . . . . . . . . 131 7.2.2 Thermodynamic Correspondence . . 142

7.3 Nonequilibrium Statistical Thermodynamics 143 7.4 Relative Boltzmann Entropy . . . . . . . . . . 146

7.4.1 Relative Boltzmann Entropy and Fluctuations 146 7.4.2 Evolution of Fluctuations . . . . 154

7.5 Statistical Treatment of Fluctuations . . . 159 7.5.1 Probability of Fluctuations . . . . . 160 7.5.2 Thermodynamic Complementarity 161

7.6 Resummation of the Moment Expansion 163 7.6.1 Algebraic Method . . . . . . . . . . . 164

7.6.2 Linear Integral Equations for Generalized Potentials .

Contents

7.7 Constitutive Equations for Fluxes .......... . 7.7.1 Relation between Two Representations of Ia 7.7.2 Kinematic Terms in the h-Representation. 7. 7.3 Cumulant Approximations for Aqa

7.8 Generalized Hydrodynamics Equations 7.9 Collision Bracket Integrals . . . . . 7.10 Nonequilibrium Partition Function

7.10.1 Expansion Method ..... . 7.10.2 Cumulant Method ...... . 7.10.3 Method of Differential Equations for OCa . 7.10.4 The Case of Diffusion, Heat Flow,

and Shear Only . . . . . . . . . . . . . . 7.10.5 The Case of Shear Only . . . . . . . . .

7.11 Nonequilibrium Grand Canonical Ensemble 7.12 Synopsis ...................... .

References

8 Transport Processes in Fluids 8.1 Approximate Constitutive Equations ....... . 8.2 Reduced Generalized Hydrodynamics Equations 8.3 Chapman-Cowling First-Order Theory 8.4 Steady Nonlinear Transport Processes . 8.5 Applications . . . . . . . . . .

8.5.1 Plane Couette Flow . . . . . . . . . 8.5.2 Tube Flow . . . . . . . . . . . . . . . 8.5.3 Boundary Layer and the Question of Slip . 8.5.4 Thermoviscous Coupling .

8.6 Shock Wave . . . . . . . . . . . . . . . . . . . . . . . .

References

9 Quantum Nonequilibrium Ensemble Method 9.1 Kinetic Equations for Density Matrices 9.2 Wigner Representations for Operators . 9.3 Kinetic Equation for Ia . . . . . . . . . 9.4 Evolution Equations for Observables . 9.5 Equilibrium Distribution Function . . · 9.6 Nonequilibrium Distribution Function

9.6.1 Matching Conditions ...... . 9.6.2 Nonequilibrium Canonical Form .

9.7 Quantum Nonequilibrium Ensemble Method.

ix

166 171 171 173 174 179 182 186 188 189 190

195 197 201 204

207

210 210 215 219 222 227 227 231 234 240 246

262

264 265 268 272 280 284 286 287 289 292

x Contents

9.7.1 Calortropy Differential and the Second Law of Thermodynamics 292

9. 7.2 Nonequilibrium Statistical Thermodynamics for Quantum Gases . . . . . . . . . . . . . . 296

9.8 Relative Boltzmann Entropy and Fluctuations . 298 9.9 Cumulant Approximations for Aka . . . . 301 9.10 Transport Processes in Quantum Fluids . 305 9.11 Collision Bracket Integrals

for Heat and Mass Transports . . . . . . . 311 9.12 Applicability to Radiation Kinetic Theory 313

References 315

10 Nonequilibrium Ensemble Method for Dense Fluids 317 10.1 Irreversible Kinetic Equation . . . . . . . . 318 10.2 Boltzmann Entropy and the H Theorem . 329 10.3 Evolution Equations . . . . . . . . . . 332

10.3.1 Equation of Continuity. . . . . . . . 332 10.3.2 Momentum Conservation Law . . . 333 10.3.3 Internal Energy Conservation Law 335 10.3.4 Evolution Equations

for Nonconserved Variables . . . . . 337 10.4 Nonequilibrium Ensemble Method . . . . 339

10.4.1 Nonequilibrium Grand Canonical Form 339 10.4.2 Statistical Calortropy

and its Balance Equation . . . . . . . . . 340 10.4.3 Pfaffian Form for Calortropy . . . . . . . 342 10.4.4 Nonequilibrium Statistical Thermodynamics

for Dense Fluids . . . . . . . . . . . . . . . . 346 10.5 Relative Boltzmann Entropy and Fluctuations . 350 10.6 Summing up . . . . . . . . . . . . . . . . . . . . . . . 354

References 358

A Addition Theorem of Tensor Hermite Polynomials 359

B Density Matrix and Evolution Equations 361 B.1 Quantum Operator Calculus . . . . . . . . . . . . 361 B.2 The H theorem . . . . . . . . . . . . . . . . . . . . . 363 B.3 Evolution Equations for Macroscopic Variables 365

B.3.1 Boltzmann Entropy Balance Equation 372 B.3.2 The H theorem for the BNUU model 37 4

References 376

Index 377

Preface The statistical mechanics of J. C. Maxwell and L. Boltzmann is for dy­

namical processes in dilute monatomic gases and is based on the evolution equation-kinetic equation-for the singlet distribution function. Starting from the basic ideas of Maxwell and Boltzmann on the statistical treat­ment of systems of many particles, J. W. Gibbs synthesized a general for­mal theory of statistical mechanics, at least, for equilibrium phenomena which covers all states of aggregation of matter. This equilibrium ensem­ble theory of Gibbs-statistical thermodynamics-is a deductive approach based on a set of postulates which aims to give the thermodynamics of reversible processes-equilibrium thermodynamics-a molecular represen­tation. It does succeed in the attempt because the mathematical structure of the phenomenological theory of reversible thermodynamic processes is well founded on the thermodynamic laws through the concept of entropy enunciated by R. Clausius. The generalization achieved by the Gibbs the­ory, however, came with a cost since the theory, with no kinetic equation, is limited to equilibrium phenomena and the underlying thermodynamics is necessarily that of reversible processes.

The Chapman-Enskog method and the Maxwell-Grad moment method for the Boltzmann kinetic equation are, in one aspect, theories attempting to generalize the Gibbsian ensemble theory to nonequilibrium theory, al­beit for dilute gases. On the other hand, the theory of linear irreversible processes represents the phenomenological line of efforts to generalize equi­librium thermodynamics and receives statistical foundations through the first-order Chapman-Enskog solution of the Boltzmann equation. It makes the local equilibrium assumption to begin the development, but the as­sumption made for the basic structure of thermodynamics of irreversible processes has been troubling the subject field as it had J. Meixner, who may be regarded as one of founders of the linear theory of irreversible pro­cesses, simply because the Clausius entropy is a concept which is valid for reversible processes only. Therefore, the conclusion can be drawn that ef­forts toward a rational formulation of extended Gibbsian ensemble theory must first of all establish a rigorous thermodynamical theory of irreversible processes, albeit formal by necessity, firmly grounded in the laws of ther­modynamics.

The realization of this essential point has come late in my winding course of study on kinetic theory and irreversible thermodynamics since 1975. This had a lot to do with the unfortunate fact that we often do not know exactly what shape our object of scientific inquiry will eventually take and even what we are looking for. The prevailing thoughts and modes of thinking in the community in which we found ourselves did not help in the mat­ter, and if they turned out to be misleading they only detained progress. In the last few years it has been possible to acquire a mathematical rep­resentation of the second law of thermodynamics, as originally stated by

xi

xii

Clausius and Kelvin, which appears to be capable of serving the role for a nonequilibrium theory that the equilibrium thermodynamics does for the Gibbsian equilibrium ensemble theory. Another important piece of the mo­saic of the nonequilibrium ensemble theory presented in this work is the demonstration that it is not necessary to solve the Boltzmann equation before constructing a rigorous formal theory of nonequilibrium statistical thermodynamics and a similarly rigorous formal theory of irreversible pro­cesses. It has been found sufficient to exploit the irreversibility and collision invariants of the Boltzmann kinetic equation and treat the equation for­mally, but exactly, for the purpose. In this manner the Boltzmann equation at long last is given a well deserved exact treatment so as to serve as the statistical mechanical foundations of irreversible thermodynamics. It turns out that what we have learned with the Boltzmann equation serves as a paradigm for other kinds of fluid, including dense fluids, since, after all, thermodynamics is a universal theory holding for all states of aggregation of matter, and different substances and states of aggregation are distin­guished only by the constitutive equations which will have to be described by appropriate kinetic equations, if the processes are dynamic. The work presented in this monograph is in essence a set of examples showing that the aforementioned paradigm is capable of delivering nonequilibrium en­semble methods parallel with the equilibrium ensemble method of Gibbs. As is the Gibbsian theory based on postulates on the distribution function, the kinetic equations used are postulates that should satisfy a set of condi­tions on them. Therefore, practical applications of the theory should start with modelling kinetic equations meeting the conditions stated, and the justification of the theory is a posteriori, as is the case for the equilibrium ensemble theory. The modelling should be implemented on the collision term in the kinetic equation and its goodness will be determined by its ability to give transport properties experimentally studied. Our journey in the realm of science is personal in its essence, and one mode of journey, delightful to one, may be an intense discomfort to another. The modes are indeed diverse, and let them be. The solitary journey in the course of this work has been often lightened by the distant echoes which seem to rever­berate from the depth of the Old Master Boltzmann's soul: "Alas, without me for eternity, the sun will rise in the east and will set in the west. Why suffer the pains but for just even a glimpse of Nirvana?" One still hopes for the day of the ultimate privilege.

This monograph has been written and prepared without a typesetter's assistance, thanks to the modern word processing softwares. The sophis­ticated word processing softwares, as powerful and versatile they are, can be heartbreaking to a novice. Much valuable technical assistance has been unstintingly given to me by my graduate student Mazen Al-Ghoul, who has also drawn many of the figures, and by Dr. Juan Gallego. I am grateful for their assistance. I would also like to thank Ms Margaret Deignan of Kluwer Academic Publishers for the interest in this work.

xiii

Finally, I would like to express my gratitudes to my wife Hui Young who has patiently carried with me the mental burden of writing a book throughout the course of this work. Without her constant and generous encouragements and understanding and her seeing to it that there is suf­ficient time for me to devote to this work, this monograph could not have been realized, incomplete as it seems, now that it is in a form to present.

Montreal February 1997

B.C.E.