An Introduction to Theory of the BO (Dover Books on Physics)

237 Pages·2004·17.46 MB·English

by  Stewart Harris

An Introduction to
the Theory of the
BOLTZMANN
EQUATION
Stewart Harris
Stewart Harris (1937-2004) was a faculty member in
the Department of Mechanical Engineering at Stony
Brook University from 1966 until his death. He served as
chairman of the department from 1978 to 1981 and was
the dean of the College of Engineering from 1981 to
1992. Professor Harris held many visiting appointments
in the U.S. and in Europe; these included positions at
Princeton University, Imperial College (London), the
Materials Directorate at Wright Patterson Air Force Base,
and Columbia University.
Dr. Harris received a B.S.E. from Case Institute of
Technology (1959) and an M.S. (I960) and Ph.D. (1964)
from Northwestern University, followed by postdoctoral
work at the Courant Institute from 1964 to 1966. His
research focused on the theory of transport processes at
the microscopic level and on molecular beam epitaxy—
the growth of atomically precise metal and semiconductor
crystal structures. Dr. Harris,s research interests encom­
passed a number of distinct areas related to the study of
reaction-diffusion systems; most recently these included
the dispersal of genes and organisms, including early
human populations.
Copyright
Copyright © 1971 by Holt, Rinehart and Winston, Inc.
Copyright © renewed 1999 by Stewart Harris
All rights reserved.
Bibliographical Note
This Dover edition, first published in 2004, is an unabridged republication of
the work originally published by Holt, Rinehart and Winston, Inc., New York,
in 1971.
Library of Congress Cataloging-in-Publication Data
Harris, Stewart.
An introduction to the theory of the Boltzmann equation / Stewart Harris,
p. cm.
Originally published: New York : Holt, Rinehart, and Winston, 1971.
Includes bibliographical references and index.
ISBN 0-486-43831-7 (pbk.)
1. Transport theory. I. Title.
QC175.2.H33 2004
530.13'8—dc22
2004050198
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
43831702
To My Parents, Seymour and Goldie
And My Wife, Helen
Faith is a fine invention
For gentlemen who see;
But microscopes are prudent
In an emergency I
Emily Dickinson
Preface
The past twenty years have seen the steady maturation of the
theory of the Boltzmann equation. In this period the Boltzmann equation
has gone from a rather formidable mathematical curiosity with a limited
but distinguished following, to a much used methodology having wide­
spread application. With the advent of modern high-speed flight through
the rarefied regions of the outer atmosphere Boltzmann's equation has
taken on a particular practical significance. However, it is no longer the
exclusive property of the fluid physicist. One finds Boltzmann's equation,
or Boltzmann-like equations, extensively used in such disparate fields as
laser scattering, solid-state physics, nuclear transport, and even outside
the conventional boundaries of physics and engineering in the fields of
cellular proliferation and automobile traffic flow. It is especially significant
that at a time when the method of time correlation functions seemed to be
usurping much of the Boltzmann equation's earlier domain of transport
theory such a veritable cornucopia of new uses for this equation should
arise.
Many of the advances which have been made in the theory of Boltz-
mann's equation since the publishing, some thirty years ago, of Chapman
and Cowling's now classic monograph are contained in the Handbuch
article of Grad, who is himself responsible for much of this work. There
exist two other outstanding expositions on the theory of the Boltzmann
equation, the book by Carleman, and the Handbuch article by Wald­
mann一neither, unfortunately, are at present available in English. The
above-mentioned sources are intended for a relatively sophisticated
audience and for that reason are not basically suitable for use as a text in
vii
viii
PREFACE
an introductory graduate level course on the theory of Boltzmann's equa­
tion; in fact, strangely enough, there seems to be no book at the present
time which satisfies this need： (Shorter, less complete accounts of the
theory of Boltzmann's equation appear as an isolated chapter or two in
books mainly devoted to equilibrium statistical mechanics.) This situation
has been evident to me for some time, both as a graduate student and as a
teacher of graduate students, and I have finally become emboldened
enough to attempt to remedy what I consider to be a serious void in text­
book literature.
In this book I have attempted to present in some detail the basic
modern theory of Boltzmann's equation, and to include representative
applications using both Boltzmann's equation and the model Boltzmann
equations which are developed in the text. It is hoped that adherence to
detail will enable this book to also serve as a reference source for research
workers using the Boltzmann equation. The book is primarily intended
for use in a graduate level course on the theory of the Boltzmann equation
for physicists and engineers, and to this end I have tended to emphasize
the physical aspects of the theory. The problems following each chapter
are intended as learning examples, and these have quite often been used
to extend and generalize the text material. This book has been the direct
product of a course on the "Theory of the Boltzmann EquationM which
I have taught at Stony Brook for the past few years. The point of view
represents a distillation and, hopefully, a coherent synthesis of ideas
presented to me from several sources. In particular, courses by Professors
I. M. Krieger, M. B. Lewis, I. Prigogine, and G. E. Uhlenbeck, and the
books by Chapman and Cowling, and by Carleman have been especially
stimulating. A special debt must be acknowledged to Professor H. Grad
who, through both the spoken and written word, has taught me much of
what I know about the Boltzmann equation. Any successes which this
book may enjoy are due in part to the above men; its failures must be
my own.
Stony Brook, New York Stewart Harris
April 1971
Historical Introduction
In the Foreword to Part 2 of his Lectures on Gas Theory, written in
1898, Boltzmann comments on the increasing attack which was being
mounted by the so-called "school of energetics^ on all theories based on
the atomic model of matter. It is unfortunate that it was only after
Boltzmann^ untimely death, eight years later, that the atomic theory of
matter became universally accepted by the scientific community, and the
twentieth century began for physics and chemistry.
The idea that matter is composed of atoms arose to conflict with the
earth, water, air, and fire theory sometime in the fifth century b.c., and
is usually credited to the Greek philosophers Leucippus and Democritus.
Although discredited by Aristotle, the idea survived somehow until the
first century a.d., when it was championed by the Roman poet-philoso­
pher Lucretius. In the Middle Ages the ideas of Aristotle became firmly
established as Church dogma, and atomism was relegated to the list of
heresies. For this, among other heresies, the ex-monk and philosopher
Giordano Bruno fell victim to the flames of the Inquisition in 1600. Ideas
cannot be killed as easily as men, however, and in. the seventeenth century,
in a somewhat more tolerant atmosphere, the atomic theory was again
taken up by Gassendi, and then by Boyle and Newton. Finally, in 1738,
Daniel Bernoulli used the atomic model as the basis for postulating the
first kinetic theory of gases. After another century of inactivity, Herapath,
Waterson, Clausius, and finally the masters, Alaxwell and Boltzmann,
formulated and developed a rigorous kinetic theory of gases. Herapath
and Waterson found their ideas rejected by the scientific establishment
of their country (they were English), although Waterson^ paper on the
ix
X HISTORICAL INTRODUCTION
subject was finally published in 1891 by Lord Rayleigh, eight years after
Waterson's death and forty-five years after he first submitted it to the
Royal Society. Clausius, Maxwell, and Boltzmann, however, were
important scientists in their countries, and although they were swimming
against the scientific current of their day, they were able to publish their
results. The work of Maxwell and Boltzmann, especially, can be con­
sidered as forming the basis of the modern kinetic theory of rarefied gases
which we will consider in this book.
Contents
Preface vii
Historical Introduction ix
Chapter 1 Statistical Mechanical Preliminaries 1
11- The Microscopic Description 3
12- Relationship Between Microscopic and Macroscopic
Descriptions 4
13- The Gibbs Ensemble 5
14- The Liouville Equation 6
1- 5 Proof of Lionville's Theorem 8
References 9
Problems 10
Chapter 2 Contraction of the Statistical
Mechanical Description 13
2- 1 Reduced Distribution Functions 15
22- Reduction of the First BBGKY Equation to the
Boltzmann Equation 19
23- The Moments of f 23
24- The Hydro dynamical Equations 26
References 27
Problems 28
xi
xii CONTENTS
Chapter 3 Description of Binary Collisions 31
31- General Formulation 33
32- An Alternate Representation of the Collision Integral 38
33- Hard Sphere Molecules Ifi
34- General Intermolecular Potentials 41
35- Power Law Potentials 站
3- 6 The Collision Cross Section 46
References 50
Problems 50
Chapter 4 Properties of the Collision Term 53
4- 1 Symmetry Properties of J(/) 55
42- The Summational Invariants 56
43- The Linearized Collision Term, L(g) 59
44- An Alternate Representation of L(g) 61
4- 5 The Eigentheory of L(g) 62
References 63
Problems 64
Chapter 5 The H Theorem and Irreversibility 67
5- 1 Spatially Homogeneous Systems 69
52- The H Theorem for a Nonuniform System 71
5- 3 Irreversibility and Boltzmann's Equation 74
References 77
Problems 77
Chapter 6 Normal Solutions of Boltzmann^
Equation 79
6- 1 The Fredholm Theorems 81
62- Hilbert's Uniqueness Theorem 82
63- The Chapman-Enskog Procedure 86
64- The First-Order Chapman-Enskog Solution 90
65- The Second-Order Chapman-Enskog Solution 90
66- Expressions for the Thermal Conductivity and
Viscosity 93
67- Calculation of the Transport Coefficients 96
68- Validity of the Chapman-Enskog Results 99
References 100
Problems 101