Introduction to Relativistic Statistical Mechanics

The following sections are included:

• Contents

• Preface

• Notations and Conventions

• Introduction

The following sections are included:

• The One-Particle Relativistic Distribution Function
  • The phase space “volume element”

• The Jüttner–Synge Equilibrium Distribution
  • Thermodynamics of the Jüttner–Synge gas
  • Thermal velocity
  • Moments of the Jüttner–Synge function
  • Orthogonal polynomials
  • Zero mass particles

• From the Microcanonical Distribution to the Jüttner–Synge One

• Equilibrium Fluctuations

• One-Particle Liouville Theorem
  • Relativistic Liouville equation from the Hamiltonian equations of motion
  • Conditions for the Jüttner–Synge functions to be an equilibrium

• The Relativistic Rotating Gas

The following sections are included:

• Relativistic Hydrodynamics
  • Sound velocity
  • The Eckart approach
  • The Landau–Lifschitz approach

• The Relaxation Time Approximation

• The Relativistic Kinetic Theory Approach to Hydrodynamics

• The Static Conductivity Tensor

• Approximation Methods for the Relativistic Boltzmann Equation and Other Kinetic Equations
  • A simple Chapman–Enskog approximation

• Transport Coefficients for a System Embedded in a Magnetic Field

The following sections are included:

• Electromagnetic Quantities in Covariant Form

• The Static Conductivity Tensor

• Debye–Hückel Law

• Derivation of the Plasma Modes
  • Evaluation of the various integrals
  • Collective modes in extreme cases

• Brief Discussion of the Plasma Modes

• The Conductivity Tensor

• Plasma–Beam Instability
  • Perturbed dispersion relations for the plasma–beam system
  • Stability of the beam–plasma system

The following sections are included:

• Basic Modifications

• Thermal Equilibrium in a Gravitational Field
  • Thermal equilibrium in a static isotropic metric

• Einstein–Vlasov Equation
  • Linearization of Einstein's equation
  • The formal solution to the linearized Einstein equation
  • The self-consistent kinetic equation for the gravitating gas

• An Illustration in Cosmology
  • The two-time scale approximation
  • Derivation of the dispersion relations (a rough outline)

• Cosmology and Relativistic Kinetic Theory
  • Cosmology: a very brief overview
  • Kinetic theory and cosmology
  • Kinetic theory of the observed universe
  • Statistical mechanics in the primeval universe
  • Particle survival

The following sections are included:

• The Dynamical Problem

• Statement of the Main Statistical Problems
  • The initial value problem: observations and measures
  • Phase space and the Gibbs ensemble

• Many-Particle Distribution Functions
  • Statistics of the particles' manifolds

• The Relativistic BBGKY⁷ Hierarchy
  • Cluster decomposition of the relativistic distribution functions

• Self-interaction and Radiation
  • An alternative treatment of radiation reaction
  • Remarks on irreversibility
  • Remarks on thermal equilibrium

• Radiation Quantities

• A Few Relativistic Kinetic Equations
  • Derivation of the covariant Landau equation
  • The relativistic Vlasov equation with radiation effects
  • Radiation effects for a relativistic plasma in a magnetic field

• Statistics of Fields and Particles

The following sections are included:

• Stochastic Processes in Minkowski Space–Time
  • Basic definitions
  • Conditional currents
  • Markovian processes in space–time

• Stochastic Processes in μ Space
  • An overview
  • Markovian processes
  • An alternative approach
  • Markovian processes
  • A simple illustration

• Relativistic Brownian Motion

• Random Gravitational Fields: An Open Problem
  • A simple example
  • The case of thermal equilibrium
  • Matter-induced fluctuations
  • Random Einstein equations

The following sections are included:

• The Density Operator for Thermal Equilibrium
  • Thermodynamic properties
  • The partition function of the relativistic ideal gas
  • The average occupation number

• Relativistic Bosons in Thermal Equilibrium
  • The complex scalar field
  • Charge fluctuations
  • A few remarks on the calculation of various integrals
  • Bose–Einstein condensation
  • Interactions

• Free Fermions in Thermal Equilibrium

• Thermodynamic Properties of the Relativistic Ideal Fermi–Dirac Gas
  • Remarks on the numerical calculations of various physical quantities
  • The degenerate Fermi gas
  • Thermal corrections: Sommerfeld expansion
  • Corrections for various thermodynamic quantities
  • High temperature expansion (nondegenerate)

• White Dwarfs: The Degenerate Electron Gas
  • Cooling of white dwarfs
  • Pycnonuclear reactions

• Functional Representation of the Partition Function
  • The partition function for gauge particles (photons)
  • The photons' partition function
  • Illustration in the case of the Lorentz gauge

The following sections are included:

• The Covariant Wigner Function for Spin 1/2 Particles
  • Basic equations
  • The equilibrium Wigner function for free fermions
  • Polarized media

• Equilibrium Fluctuations of Fermions

• A Simple Example

• The BBGKY Relativistic Quantum Hierarchy

• Perturbation Expansion of the Wigner Function

• The Wigner Function for Bosons
  • The example of the λφ⁴ theory
  • Four-current fluctuations of the complex scalar field

• Gauge Properties of the Wigner Function
  • Gauge-invariant Wigner functions
  • A few remarks
  • Gauge-invariant Wigner functions for the photon field
  • Another gauge-invariant Wigner function
  • Gauge invariance and approximations

The following sections are included:

• Thermal Equilibrium

• Collective Modes

• Two-Body Correlations
  • A brief discussion
  • Exchange correlations

• Renormalization — An Illustration of the Procedure
  • Regularization of the gap equation
  • Regularization of the energy–momentum tensor
  • Determination of the constants (A_F, B_F, C_F,D_F)

• Qualitative Discussion of the Effects of Renormalization

• Thermodynamics of the System
  • The gap equation as a minimum of the free energy
  • Thermodynamics

• Renormalization of the Excitation Spectrum
  • Comparison with the semiclassical case

• A Short Digression on Bosons

The following sections are included:

• General Form of the Kinetic Equation

• An Introductory Example

• A General Relaxation Time Approximation
  • Properties of the kinetic system
  • The collision term
  • General form of F₍₁₎

The following sections are included:

• Thermodynamic Properties at Finite Temperature
  • Thermodynamics in some important cases

• Remarks on the Oscillation Spectra of Mesons

• Transport Coefficients of Nuclear Matter
  • Chapman–Enskog expansion
  • Transport coefficients: Eckart versus Landau–Lifschitz representations
  • Entropy production
  • A brief comparison: BGK versus BUU

• Discussion

• Dense Nuclear Matter: Neutron Stars
  • The static equilibrium of a neutron star
  • The composition of matter in a neutron star
  • Beyond the drip point

The following sections are included:

• Relations Obeyed by the Magnetic Field

• The Partition Function
  • Magnetization of an electron gas

• Relativistic Quantum Liouville Equation
  • Solution of the inhomogeneous equation
  • The initial value problem

• The Equilibrium Wigner Function for Noninteracting Electrons
  • Thermodynamic quantities

• The Wigner Function of the Ideal Magnetized Electron Gas
  • The nonmagnetic field limit
  • Equations of state
  • Is the pressure isotropic?
  • The completely degenerate case
  • Magnetization
  • Landau orbital ferromagnetism: LOFER states

• The Magnetized Vacuum
  • The general structure of the vacuum Wigner function
  • The wigner function of the magnetized vacuum
  • Renormalization of the vacuum Wigner function

• Fluctuations
  • Fluctuations of the four-current

• Polarization Tensors of the Magnetized Electron Gas and of the Magnetized Vacuum
  • The vacuum polarization tensor

• Remarks on the Transport Coefficients of the Magnetized Electron Gas

• Astrophysical Aspects

The following sections are included:

• Classical Fields
  • Internal symmetries and conserved currents
  • Space–time symmetries
  • A general remark

• Quantum Quasiparticles
  • Formal quantization

• Problems with the Quantization of Quasiparticles
  • A first example
  • Another example: the QED plasma
  • Migdal's approach

• The Covariant Wigner Function

• Equilibrium Properties

• A Simple Example: The λø⁴ Model

• Remarks on the Thermodynamics of Quasiparticles

• Equilibrium Fluctuations

• Remarks on the Negative Energy Modes

• Interacting Quasibosons
  • The long wavelength and low frequency limit

The following sections are included:

• Independent Quasifermions
  • Quantization and observables
  • Statistical expressions
  • Thermal equilibrium

• Interacting Quasifermions
  • The long wavelength and low frequency limit

• Kinetic Equation for Quasiparticles

• Remarks on the Relativistic Landau Theory

The following sections are included:

• Basic Equations

• Plasma Collective Modes

• The Fluctuation–Dissipation Theorem and Its Inverse

• Four-Current Fluctuations and the Polarization Tensor

• The Polarization Tensor at Order e²

• Quasiparticles in the Relativistic Plasma
  • Quasiphotons in thermal equilibrium
  • Gauge properties
  • Quasielectron modes in thermal equilibrium

The following sections are included:

• Appendix A: A Few Useful Properties of Some Special Functions

• Appendix B: γ Matrices

• Appendix C: Outline of Functional Methods

• Appendix D: Units⁺

• Appendix E: Some Useful Formulae for Wigner Functions

• Bibliography

• Index