Nonequilibrium Statistical Mechanics

Transcription

1 Nonequilibrium Statistical Mechanics

2 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

3 Nonequilibrium Statistical Mechanics Ensemble Method by Byung Chan Eu Department of Chemistry, McGill University, Montreal, Quebec, Canada SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

4 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / 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

5 This is for Hui Young and my children

6 Contents Preface xi 1 Introduction 1 References 10 2 Thermodynamics of Irreversible Processes The Zeroth Law of Thermodynamics The First Law of Thermodynamics Local Forms of the Conservation Laws The Second Law of Thermodynamics Local Intensive Quantities: Temperature, Pressure, etc Local Temperature Local Pressure and Local Chemical Potentials Local Generalized Potentials Local Differential Form for Calortropy Irreversible Thermodynamics in the space 1-lJ The Inverse Temperature as an Integrating Factor Integrability Conditions and Generalized Maxwell Relations Irreversible Thermodynamics of Steady States Thermodynamics of Linear Irreversible Processes Inclusion of Chemical Reactions References 52 3 Boltzmann Equation Boltzmann Equation Mean Macroscopic Variables Collisional Invariants Broken Time Reversal Symmetry Generic Boltzmann Equation. 61 References 64

7 viii Contents 4 Equilibrium Solution and Local Variables The H Theorem and Boltzmann Entropy Equilibrium Solution Local Variables and Matching Conditions Stability of Equilibrium Conservation Laws and Balance Equations Conservation Laws Boltzmann Entropy Balance Equation 79 References 85 5 Mathematical Preparation Reduced Kinetic Equation Tensor Hermite Polynomials 89 References 95 6 The Chapman-Enskog and Moment Methods The Chapman-Enskog Method and Irreversible Processes First-Order Solutions Formal Expansions for Transport Coefficients The Chapman-Enskog Method and Irreversible Thermodynamics The Moment Method and Irreversible Processes The Maxwell-Grad Moment Method The Moment Method and Irreversible Thermodynamics. 120 References Classical Nonequilibrium Ensemble Method Thermodynamic Branch of fa Thermodynamics of Irreversible Processes Calortropy Differential Thermodynamic Correspondence Nonequilibrium Statistical Thermodynamics Relative Boltzmann Entropy Relative Boltzmann Entropy and Fluctuations Evolution of Fluctuations Statistical Treatment of Fluctuations Probability of Fluctuations Thermodynamic Complementarity Resummation of the Moment Expansion Algebraic Method

8 7.6.2 Linear Integral Equations for Generalized Potentials. Contents 7.7 Constitutive Equations for Fluxes Relation between Two Representations of Ia Kinematic Terms in the h-representation Cumulant Approximations for Aqa 7.8 Generalized Hydrodynamics Equations 7.9 Collision Bracket Integrals Nonequilibrium Partition Function Expansion Method Cumulant Method Method of Differential Equations for OCa The Case of Diffusion, Heat Flow, and Shear Only The Case of Shear Only Nonequilibrium Grand Canonical Ensemble 7.12 Synopsis.... References 8 Transport Processes in Fluids 8.1 Approximate Constitutive Equations Reduced Generalized Hydrodynamics Equations 8.3 Chapman-Cowling First-Order Theory 8.4 Steady Nonlinear Transport Processes. 8.5 Applications Plane Couette Flow Tube Flow Boundary Layer and the Question of Slip 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 Evolution Equations for Observables. 9.5 Equilibrium Distribution Function Nonequilibrium Distribution Function Matching Conditions Nonequilibrium Canonical Form. 9.7 Quantum Nonequilibrium Ensemble Method. ix

9 x Contents Calortropy Differential and the Second Law of Thermodynamics Nonequilibrium Statistical Thermodynamics for Quantum Gases Relative Boltzmann Entropy and Fluctuations Cumulant Approximations for Aka Transport Processes in Quantum Fluids Collision Bracket Integrals for Heat and Mass Transports Applicability to Radiation Kinetic Theory 313 References Nonequilibrium Ensemble Method for Dense Fluids Irreversible Kinetic Equation Boltzmann Entropy and the H Theorem Evolution Equations Equation of Continuity Momentum Conservation Law Internal Energy Conservation Law Evolution Equations for Nonconserved Variables Nonequilibrium Ensemble Method Nonequilibrium Grand Canonical Form Statistical Calortropy and its Balance Equation Pfaffian Form for Calortropy Nonequilibrium Statistical Thermodynamics for Dense Fluids Relative Boltzmann Entropy and Fluctuations Summing up References 358 A Addition Theorem of Tensor Hermite Polynomials 359 B Density Matrix and Evolution Equations 361 B.1 Quantum Operator Calculus B.2 The H theorem 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 374 References 376 Index 377

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

11 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.

12 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.