The Relativistic Boltzmann Equation

by Cercignani, Carlo; Kremer, Gilberto Medeiros

• ISBN13: 9783764366933
• ISBN10: 3764366931
• Format: Hardcover
• Copyright: 2002-05-01
• Publisher: Birkhauser

Summary

The aim of this book is to present the theory and applications of the relativistic Boltzmann equation in a self-contained manner, even for those readers who have no familiarity with special and general relativity. Though an attempt is made to present the basic concepts in a complete fashion, the style of presentation is chosen to be appealing to readers who want to understand how kinetic theory is used for explicit calculations. The book will be helpful not only as a textbook for an advanced course on relativistic kinetic theory but also as a reference for physicists, astrophysicists and applied mathematicians who are interested in the theory and applications of the relativistic Boltzmann equation.

Table of Contents

Preface	ix
Special Relativity
Introduction	1	(1)
Lorentz transformations	1	(7)
Tensors in Minkowski spaces	8	(6)
Relativistic mechanics	14	(7)
Four-velocity	14	(1)
Minkowski force	15	(2)
Elastic collisions	17	(2)
Relative velocity	19	(2)
Electrodynamics in free space	21	(10)
Maxwell equations	21	(3)
Energy-momentum tensor	24	(2)
Retarded potentials	26	(5)
Relativistic Boltzmann Equation
Single non-degenerate gas	31	(8)
Single degenerate gas	39	(2)
General equation of transfer	41	(1)
Summational invariants	42	(3)
Macroscopic description	45	(2)
Local Lorentz rest frame	47	(2)
Equilibrium distribution function	49	(4)
Trend to equilibrium. H-theorem	53	(5)
The projector Δαβ	58	(1)
Equilibrium states	59	(6)
Fields in Equilibrium
The general case	65	(2)
Non-degenerate gas	67	(7)
Modified Bessel function of second kind	67	(2)
Expressions for n, e and p	69	(2)
Non-relativistic limit	71	(1)
Ultra-relativistic limit	72	(2)
Degenerate relativistic Fermi gas	74	(16)
Completely degenerate relativistic Fermi gas	74	(3)
White dwarf stars	77	(7)
Strongly degenerate relativistic Fermi gas	84	(6)
Degenerate relativistic Bose gas	90	(9)
Some useful integrals	90	(2)
Relativistic Bose--Einstein condensation	92	(7)
Thermomechanics of Relativistic Fluids
Introduction	99	(1)
Thermodynamics of perfect fluids	99	(3)
Eckart decomposition	102	(3)
Landau and Lifshitz decomposition	105	(2)
Thermodynamics of a single fluid	107	(4)
Chapman--Enskog Method
Introduction	111	(7)
Simplified version	118	(1)
The integrals I1, I2 and I3	118	(5)
Transport coefficients	123	(5)
Hard-sphere particles	124	(2)
Israel particles	126	(2)
Formal version	128	(16)
Integral equations	128	(4)
Second approximation	132	(2)
Orthogonal polynomials	134	(5)
Expansion in orthogonal polynomials	139	(5)
Appendix	144	(5)
Method of Moments
Introduction	149	(1)
Grad distribution function	149	(5)
Constitutive equations for Tαβγ and Pαβ	154	(2)
Linearized field equations	156	(1)
Five-field theory	157	(6)
Laws of Navier-Stokes and Fourier	158	(1)
Linearized Burnett equations	159	(4)
Maxwellian particles	163	(3)
Combined method of Chapman--Enskog and Grad	166	(5)
Chemically Reacting Gas Mixtures
Introduction	171	(1)
Boltzmann and transfer equations	171	(4)
Maxwell-Juttner distribution function	175	(2)
Thermodynamics of mixtures	177	(3)
Transport coefficients	180	(7)
Onsager reciprocity relations	187	(4)
Model Equations
Introduction	191	(3)
The characteristic time	194	(3)
Single non-degenerate gas	197	(11)
The model of Marle	197	(4)
The model of Anderson and Witting	201	(4)
Comparison of the models	205	(3)
Single degenerate gas	208	(10)
Non-zero rest mass	215	(3)
Zero rest mass	218	(1)
Relativistic ionized gases	218	(11)
Boltzmann and balance equations	218	(2)
Decomposition with respect to the four-velocity	220	(2)
Ohm's law	222	(7)
Appendix	229	(4)
Wave Phenomena in a Relativistic Gas
Introduction	233	(1)
Propagation of discontinuities	234	(4)
Small oscillations	238	(21)
Boltzmann equation	238	(7)
Continuum-like theories	245	(14)
Shock waves	259	(12)
Continuum theory	259	(4)
Boltzmann equation	263	(8)
Tensor Calculus in General Coordinates
Introduction	271	(1)
Tensor components in general coordinates	271	(6)
Affine connection	277	(3)
Covariant differentiation	280	(5)
Spatial metric tensor	285	(2)
Special relativity in general coordinates	287	(4)
Riemann Spaces and General Relativity
Introduction	291	(1)
Tensors in Riemannian spaces	291	(1)
Curvature tensor	292	(6)
Physical principles of general relativity	298	(1)
Mechanics in gravitational fields	299	(3)
Four-velocity	299	(1)
Equations of motion	300	(2)
Electrodynamics in gravitational fields	302	(2)
Perfect fluids	304	(3)
Einstein's field equations	307	(3)
Solution for weak fields	310	(2)
Exact solutions of Einstein's field equations	312	(5)
Robertson--Walker metric	317	(10)
Geometrical meaning	318	(2)
Determination of the energy density	320	(2)
Determination of k(t)	322	(5)
Boltzmann Equation in Gravitational Fields
Introduction	327	(1)
Transformation of volume elements	327	(3)
Boltzmann equation	330	(2)
Transfer equation	332	(2)
Equilibrium states	334	(5)
Boltzmann equation in a spherically symmetric gravitational field	339	(3)
Dynamic pressure in a homogeneous and isotropic universe	342	(5)
The Vlasov Equation and Related Systems
Introduction	347	(2)
The Vlasov-Maxwell system	349	(1)
The Vlasov-Einstein system	350	(1)
Steady Vlasov-Einstein system in case of spherical symmetry	351	(6)
The threshold of black hole formation	357	(6)
Cosmology with the Vlasov--Einstein system	363	(8)
Physical Constants	371	(2)
Modified Bessel Function	373	(2)
Index	375