The Relativistic Boltzmann Equation: Theory and Applications

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.

Contenu

1 Special Relativity.- 1.1 Introduction.- 1.2 Lorentz transformations.- 1.3 Tensors in Minkowski spaces.- 1.4 Relativistic mechanics.- 1.4.1 Four-velocity.- 1.4.2 Minkowski force.- 1.4.3 Elastic collisions.- 1.4.4 Relative velocity.- 1.5 Electrodynamics in free space.- 1.5.1 Maxwell equations.- 1.5.2 Energy-momentum tensor.- 1.5.3 Retarded potentials.- 2 Relativistic Boltzmann Equation.- 2.1 Single non-degenerate gas.- 2.2 Single degenerate gas.- 2.3 General equation of transfer.- 2.4 Summational invariants.- 2.5 Macroscopic description.- 2.6 Local Lorentz rest frame.- 2.7 Equilibrium distribution function.- 2.8 Trend to equilibrium. H-theorem.- 2.9 The projector ???.- 2.10 Equilibrium states.- 3 Fields in Equilibrium.- 3.1 The general case.- 3.2 Non-degenerate gas.- 3.2.1 Modified Bessel function of second kind.- 3.2.2 Expressions for n, e and p.- 3.2.3 Non-relativistic limit.- 3.2.4 Ultra-relativistic limit.- 3.3 Degenerate relativistic Fermi gas.- 3.3.1 Completely degenerate relativistic Fermi gas.- 3.3.2 White dwarf stars.- 3.3.3 Strongly degenerate relativistic Fermi gas.- 3.4 Degenerate relativistic Bose gas.- 3.4.1 Some useful integrals.- 3.4.2 Relativistic Bose-Einstein condensation.- 4 Thermomechanics of Relativistic Fluids.- 4.1 Introduction.- 4.2 Thermodynamics of perfect fluids.- 4.3 Eckart decomposition.- 4.4 Landau and Lifshitz decomposition.- 4.5 Thermodynamics of a single fluid.- 5 Chapman-Enskog Method.- 5.1 Introduction.- 5.2 Simplified version.- 5.3 The integrals Il, I2 and I3.- 5.4 Transport coefficients.- 5.4.1 Hard-sphere particles.- 5.4.2 Israel particles.- 5.5 Formal version.- 5.5.1 Integral equations.- 5.5.2 Second approximation.- 5.5.3 Orthogonal polynomials.- 5.5.4 Expansion in orthogonal polynomials.- 5.6 Appendix.- 6 Method of Moments.- 6.1 Introduction.- 6.2 Grad distribution function.- 6.3 Constitutive equations for Taßry and Paß.- 6.4 Linearized field equations.- 6.5 Five-field theory.- 6.5.1 Laws of Navier-Stokes and Fourier.- 6.5.2 Linearized Burnett equations.- 6.6 Maxwellian particles.- 6.7 Combined method of Chapman-Enskog and Grad.- 7 Chemically Reacting Gas Mixtures.- 7.1 Introduction.- 7.2 Boltzmann and transfer equations.- 7.3 Maxwell-Jüttner distribution function.- 7.4 Thermodynamics of mixtures.- 7.5 Transport coefficients.- 7.6 Onsager reciprocity relations.- 8 Model Equations.- 8.1 Introduction.- 8.2 The characteristic time.- 8.3 Single non-degenerate gas.- 8.3.1 The model of Marle.- 8.3.2 The model of Anderson and Witting.- 8.3.3 Comparison of the models.- 8.4 Single degenerate gas.- 8.4.1 Non-zero rest mass.- 8.4.2 Zero rest mass.- 8.5 Relativistic ionized gases.- 8.5.1 Boltzmann and balance equations.- 8.5.2 Decomposition with respect to the four-velocity.- 8.5.3 Ohm's law.- 8.6 Appendix.- 9 Wave Phenomena in a Relativistic Gas.- 9.1 Introduction.- 9.2 Propagation of discontinuities.- 9.3 Small oscillations.- 9.3.1 Boltzmann equation.- 9.3.2 Continuum-like theories.- 9.4 Shock waves.- 9.4.1 Continuum theory.- 9.4.2 Boltzmann equation.- 10 Tensor Calculus in General Coordinates.- 10.1 Introduction.- 10.2 Tensor components in general coordinates.- 10.3 Affine connection.- 10.4 Covariant differentiation.- 10.5 Spatial metric tensor.- 10.6 Special relativity in general coordinates.- 11 Riemann Spaces and General Relativity.- 11.1 Introduction.- 11.2 Tensors in Riemannian spaces.- 11.3 Curvature tensor.- 11.4 Physical principles of general relativity.- 11.5 Mechanics in gravitational fields.- 11.5.1 Four-velocity.- 11.5.2 Equations of motion.- 11.6 Electrodynamics in gravitational fields.- 11.7 Perfect fluids.- 11.8 Einstein's field equations.- 11.9 Solution for weak fields.- 11.10 Exact solutions of Einstein's field equations.- 11.11 Robertson-Walker metric.- 11.11.1 Geometrical meaning.- 11.11.2 Determination of the energy density.- 11.11.3 Determination of K(t).- 12 Boltzmann Equation in Gravitational Fields.- 12.1 Introduction.- 12.2 Transformation of volume elements.- 12.3 Boltzmann equation.- 12.4 Transfer equation.- 12.5 Equilibrium states.- 12.6 Boltzmann equation in a spherically symmetric gravitational field.- 12.7 Dynamic pressure in a homogeneous and isotropic universe.- 13 The Vlasov Equation and Related Systems.- 13.1 Introduction.- 13.2 The Vlasov-Maxwell system.- 13.3 The Vlasov-Einstein system.- 13.4 Steady Vlasov-Einstein system in case of spherical symmetry.- 13.5 The threshold of black hole formation.- 13.6 Cosmology with the Vlasov-Einstein system.- Physical Constants.- Modified Bessel Function.