The Boltzmann Equation and Its Applications

466 Pages·1988·37.438 MB·English

by  Carlo Cercignani (auth.)

Applied Mathematlca' Sciences
Volume 67
Editors
F. John 1. E. Marsden L. Sirovich
Advisors
M. Ghil J. K. Hale 1. Keller
K. Kirchgassner B. Matkowsky
J. T. Stuart A. Weinstein
Applied Mathematical Sciences
I. John: Partial Differential Equations, 4th ed.
2. Sirovich: Techniques of Asymptotic Analysis.
3. Hale: Theory of Functional Differential Equations, 2nd ed.
4. Percus: Combinatorial Methods.
5. von Mises/Friedrichs: Fluid Dynamics.
6. Freiberger/Grenander: A Short Course in Computational Probability and
Statistics.
7. Pipkin: Lectures on Viscoelasticity Theory.
9. Friedrichs: Spectral Theory of Operators in Hilbert Space.
II. Wolovich: Linear Multivariable Systems.
12. Berkovitz: Optimal Control Theory.
13. BlumaniCole: Similarity Methods for Differential Equations.
14. Yoshizawa: Stability Theory and the Existence of Periodic Solution and Almost
Periodic Solutions.
15 .. Braun: Differential Equations and Their Applications, 3rd ed.
16. Lefschetz: Applications of Algebraic Topology.
17. CollatzlWetterIing: Optimization Problems.
18. Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol I.
20. Driver: Ordinary and Delay Differential Equations.
21. Courant/Friedrichs: Supersonic Flow and Shock Waves.
22. Rouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method.
23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory.
24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. II.
25. Davies: Integral Transforms and Their Applications, 2nd ed.
26. Kushner/Clark: Stochastic Approximation Methods for Constrained and
Unconstrained Systems
27. de Boor: A Practical Guide to Splines.
28. Keilson: Markov Chain Models-Rarity and Exponentiality.
29. de Veubeke: A Course in Elasticity.
30. Sniatycki: Geometric Quantization and Quantum Mechanics.
31. Reid: Sturmian Theory for Ordinary Differential Equations.
32. Meis/Markowitz: Numerical Solution of Partial DilTerential Equations.
33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III.
34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics.
35. Carr: Applications of Centre Manifold Theory.
36. Bengtsson/GhiJ/Kallen: Dynamic Meterology: Data Assimilation Methods.
37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces.
38. Lichtenberg/Lieberman: Regular and Stochastic Motion.
39. Piccini/StampacchiaiVidossich: Ordinary Differential Equations in RO.
40. NayJor/Sell: Linear Operator Theory in Engineering and Science:
41. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors.
42. Guckenheimer/Holmes: Nonlinear Oscillations, Dynamical Systems and
Bifurcations of Vector Fields.
43. Ockendon/Tayler: Inviscid Fluid Flows.
(continued following index)
Carlo Cercignani
The Boltzmann Equation
and Its Applications
With 42 Illustrations
Springer Science+Business Media, LLC
Carlo Cercignani
Department of Mathematics
Politecnico di Milano
20133 Milano (1)
Italy
Editors
F. John J. E. Marsden L. Sirovich
Courant Institute of Department of Division of Applied
Mathematical Sciences Mathematics Mathematics
New York University University of Brown University
New York, NY 10012 California Providence, RI 02912
U.S.A. Berkeley, CA 94720 U.S.A.
U.S.A.
AMS Classifieations: 76P05, 82A40
Library of Congress Cataloging-in-Publieation Data
Cereignani, Carlo.
The Boltzmann equation and its applieations / Carlo Cercignani.
p. em. - (Applied mathematieal sciences; v.67)
Inc1udes index.
1. Transport theory. 2. Rarefied gas dynamics. 1. Title.
II. Series: Applied mathematieal sciences (Springer-Verlag New York
Inc.)
QAl.A647 voI. 67
[QC175.2]
510 s-dc19
[530.1'38] 87-26654
Portions of this work previously appeared as Theory and Application of the Boltzmann Equation.
© Scottish Academic Press Ltd., 1975.
<.el 1988 by Springer Seience+Business Media New York
Originally published by Springer-Verlag New York me. in 1988
Softcover reprint ofthe hardcover lst edition 1988
AU rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher Springer Seience+Business Media, LLC,
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9 8 7 6 543 2 1
ISBN 978-1-4612-6995-3 ISBN 978-1-4612-1039-9 (eBook)
DOI 10.1007/978-1-4612-1039-9
To Silvana
PREFACE
Statistical mechanics may be naturally divided into two branches, one dealing
with equilibrium systems, the other with nonequilibrium systems. The
equilibrium properties of macroscopic systems are defined in principle by
suitable averages in well-defined Gibbs's ensembles. This provides a frame
work for both qualitative understanding and quantitative approximations to
equilibrium behaviour. Nonequilibrium phenomena are much less understood
at the present time. A notable exception is offered by the case of dilute gases.
Here a basic equation was established by Ludwig Boltzmann in 1872.
The Boltzmann equation still forms the basis for the kinetic theory of gases
and has proved fruitful not only for a study of the classical gases Boltzmann
had in mind but also, properly generalized, for studying electron transport in
solids and plasmas, neutron transport in nuclear reactors, phonon transport in
superfluids, and radiative transfer in planetary and stellar atmospheres.
Research in both the new fields and the old one has undergone a considerable
advance in the last thirty years.
In the last ten years, a new wave of interest has surrounded the Boltzmann
equation, stemming from its unique role in the theory of time-dependent
phenomena in large systems. In fact, the Boltzmann equation appears as a
prototype of a reduced description taking into account only partial informa
tion about the underlying microscopic state (fully described by the coordinate
and momenta of all the molecules), but nevertheless undergoing an auton
omous time evolution. Thus the problem of the rigorous derivation of the
Boltzmann equation from the microscopic description has attracted a certain
amount of interest among physicists and mathematicians. This, in turn, has
revived the interest in the theory of existence and uniqueness of the solutions of
the Boltzmann equation, since this problem has proved to he intimately tied
with the previous one.
This justifies the appearance of the present book (which tries to present a
unified approach to the problems arising in the different fields mentioned
above) by exploiting the similarities whenever they exist and underlining the
differences when necessary. The main line of exposition, however, is tied to the
classical equation established by Boltzmann, and hence the detailed descrip
tions of some applications almost exclusively refer to monatomic neutral gases.
Appropriate references are given, however, to papers dealing with similar
problems arising in other fields, with particular concern for neutron transport,
gas mixtures, and polyatomic gases.
viii PREFACE
The material dealing with the basic properties and applications known
before 1975 was covered in a previous book by the author"'; thus, it was natural
to incorporate in the present book the material of the first seven chapters of
that book. This material is updated in an extensive Appendix, covering the
developments from 1975 to 1987.
But the main feature of the book is Chapter VIII, completely rewritten and
covering the important studies resulting from new mathematical approaches
to the old problem of existence and uniqueness, and the new ones tied up with
the question of validity. There is still no satisfactory proof for existence of
solutions under general reasonable initial conditions. The material presented
here indicates, however, that a great deal of progress has been achieved in
recent times. There is no doubt that the better understanding resulting from
this progress will be essential to the further penetration to be expected in the
next few years.
It is hoped that the book will be useful as a textbook for an advanced course
in kinetic theory and as a reference for mathematicians, physicists, and
engineers interested in the kinetic theory of gases and its applications.
Milano, Italy CARLO CERCIGNANI
September 1987
*Theory and Application of the Boltzmann Equation, Scottish Academic Press, Edinburgh, 1975.
CONTENTS
vii
PREFACE
I. BASIC PRINCIPLES OF THE KINETIC THEORY OF GASES
1. Introduction
2. Probability 3
3. Phase space and Liouville's theorem 9
4. Hard spheres and rigid walls. Mean free path 13
5. Scattering of a volume element in phase space 20
6. Time averages, ergodic hypothesis and equilibrium states 25
Appendix 36
References 39
II. THE BOLTZMANN EQUATION
1. The problem of nonequilibrium states 40
2. Equations for the many particle distribution functions for a gas of
rigid spheres 44
3. The Boltzmann equation for rigid spheres 52
4. Generalizations 57
5. Details of the collision term 67
6. Elementary properties of the collision operator. Collision
invariants 72
7. Solution of the equation Qif, f) = 0 78
8. Connection between the microscopic description and the
macroscopic description of gas dynamics 79
9. Non-cutoff potentials and grazing collisions. Fokker-Planck
equation 86
10. Model equations 95
Appendix 98
References 102
III. GAS-SURFACE INTERACTION AND THE H-THEOREM
1. Boundary conditions and the gas-surface interaction 104
2. Computation of scattering kernels 108
X CONTENTS
3. Reciprocity 111
4. A remarkable inequality 11S
S. Maxwell's boundary conditions. Accommodation coefficients 118
6. Mathematical models for gas-surface interaction 122
7. Physical models for gas-surface interaction 130
8. Scattering of molecular beams 134
9. The H-theorem. Irreversibility 137
10. Equilibrium states and Maxwellian distributions 142
Appendix 149
References 156
IV. LINEAR TRANSPORT
1. The linearized collision operator 158
2. The linearized Boltzmann equation 161
3. The linear Boltzmann equation. Neutron transport and radiative
transfer 165
4. Uniqueness of the solution for initial and boundary value problems 172
5. Further investigation of the linearized collision term 174
6. The decay to equilibrium and the spectrum of the collision operator 180
7. Steady one-dimensional problems. Transport coefficients 189
8. The general case 200
9. Linearized kinetic models 205
10. The variational principle 212
11. Green's function 21S
12. The integral equation approach 222
References 229
V. SMALL AND LARGE MEAN FREE PATHS
1. The Knudsen number 232
2. The Hilbert expansion 234
3. The Chapman-Enskog expansion 239
4. Criticism of the Chapman-Enskog method 24S
S. Initial, boundary and shock layers 248
6. Further remarks on the Chapman-Enskog method and the
computation of transport coefficients 260
7. Free molecule flow past a convex body 262
CONTENTS xi
8. Free molecule flow in presence of nonconvex boundaries 271
9. Nearly free-molecule flows 278
References 283
VI. ANALYTICAL SOLUTIONS OF MODELS
1. The method of elementary solutions 286
2. Splitting of a one-dimensional model equation 286
3. Elementary solutions of the simplest transport equation 288
4. Application of the general method to the Kramers and Milne
problems 294
S. Application to the flow between parallel plates and the critical
problem of a slab 299
6. Unsteady solutions of kinetic models with constant collision
frequency 306
7. Analytical solutions of specific problems 310
8. More general models 315
9. Some special cases 319
10. Unsteady solutions of kinetic models with velocity dependent
collision frequency 322
11. Analytic continuation 330
12. Sound propagation in monatomic gases 334
13. Two-dimensional and three-dimensional problems. Flow past
solid bodies 338
14. Fluctuations and light scattering 344
Appendix 345
References 348
VII. THE TRANSITION REGIME
1. Introduction 351
2. Moment and discrete ordinate methods 351
3. The variational method 355
4. Monte Carlo methods 359
S. Problems of flow and heat transfer in regions bounded by planes
or cylinders 361
6. Shock-wave structure 369
7. External ftows 377
8. Expansion of a gas into a vacuum 380
References 385