Gilberto M. Kremer

The relativistic Boltzmann equation : theory and applications

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-Juttner 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 Ohms 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 Einsteins field equations.- 11.9 Solution for weak fields.- 11.10 Exact solutions of Einsteins field equations.- 11.11 Robertson-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-Juttner 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 Ohms 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 Einsteins field equations.- 11.9 Solution for weak fields.- 11.10 Exact solutions of Einsteins 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.

Fermions as sources of accelerated regimes in cosmology

In this work it is investigated if fermionic sources could be responsible for accelerated periods during the evolution of a universe where a matter field would answer for the decelerated period. The self-interaction potential of the fermionic field is considered as a function of the scalar and pseudoscalar invariants. Irreversible processes of energy transfer between the matter and gravitational fields are also considered. It is shown that the fermionic field could behave like an inflaton field in the early universe and as dark energy for an old universe.

Cosmological models described by a mixture of van der Waals fluid and dark energy

The Universe is modeled as a binary mixture whose constituents are described by a van der Waals fluid and by a dark energy density. The dark energy density is considered either as quintessence or as the Chaplygin gas. The irreversible processes concerning the energy transfer between the van der Waals fluid and the gravitational field are taken into account. This model can simulate (a) an inflationary period where the acceleration grows exponentially and the van der Waals fluid behaves like an inflaton, (b) an accelerated period where the acceleration is positive but it decreases and tends to zero whereas the energy density of the van der Waals fluid decays, (c) a decelerated period which corresponds to a matter dominated period with a non-negative pressure, and (d) a present accelerated period where the dark energy density outweighs the energy density of the van der Waals fluid.

Noether symmetry for non-minimally coupled fermion fields

A cosmological model where a fermion field is non-minimally coupled with the gravitational field is studied. By applying Noether symmetry the possible functions for the potential density of the fermion field and for the coupling are determined. Cosmological solutions are found showing that the non-minimally coupled fermion field behaves as an inflaton describing an inflationary scenario, whereas the minimally coupled fermion field describes a decelerated period, behaving as a standard matter field.

Viscous cosmological models and accelerated universes

It is shown that a present acceleration with a past deceleration is a possible solution to the Friedmann equation by considering the Universe as a mixture of a scalar with a matter field and by including a nonequilibrium pressure term in the energy-momentum tensor. The dark energy density decays more slowly with respect to the time than the matter energy density does. The inclusion of the nonequilibrium pressure leads to a less pronounced decay of the matter field with a shorter period of past deceleration.

Cosmological model with interactions in the dark sector

A cosmological model for the present Universe is analyzed whose constituents are a non-interacting baryonic matter field and interacting dark matter and dark energy fields. The dark energy and dark matter are coupled through their effective barotropic indexes, which are considered as functions of the ratio of their energy densities. Two asymptotically stable cases are investigated for the ratio of the dark energy densities which have their parameters adjusted by considering best fits to Hubble function data. It is shown that the deceleration parameter, the density parameters, and the luminosity distance have the correct behavior which is expected for a viable present scenario of the Universe.

Cosmological model with non-minimally coupled fermionic field

A model for the Universe is proposed whose constituents are: a) a dark energy field modeled by a fermionic field non-minimally coupled with the gravitational field, b) a matter field which consists of pressureless baryonic and dark matter fields and c) a field which represents the radiation and the neutrinos. The coupled system of Diracs equations and Einstein field equations is solved numerically by considering a spatially flat homogeneous and isotropic Universe. It is shown that the proposed model can reproduce the expected red-shift behaviors of the deceleration parameter, of the density parameters of each constituent and of the luminosity distance. Furthermore, for small values of the red-shift the constant which couples the fermionic and gravitational fields has a remarkable influence on the density and deceleration parameters.

Constraining non-minimally coupled tachyon fields by the Noether symmetry

A model for a homogeneous and isotropic Universe whose gravitational sources are a pressureless matter field and a tachyon field non-minimally coupled to the gravitational field is analyzed. The Noether symmetry is used to find expressions for the potential density and for the coupling function, and it is shown that both must be exponential functions of the tachyon field. Two cosmological solutions are investigated: (i) for the early Universe whose only source of gravitational field is a non-minimally coupled tachyon field which behaves as an inflaton and leads to an exponential accelerated expansion and (ii) for the late Universe whose gravitational sources are a pressureless matter field and a non-minimally coupled tachyon field which plays the role of dark energy and is responsible for the decelerated–accelerated transition period.

Letter: Irreversible Processes in a Universe Modelled as a Mixture of a Chaplygin Gas and Radiation

The evolution of a Universe modelled as a mixture of a Chaplygin gas and radiation is determined by taking into account irreversible processes. This mixture could interpolate periods of a radiation dominated, a matter dominated and a cosmological constant dominated Universe. The results of a Universe modelled by this mixture are compared with the results of a mixture whose constituents are radiation and quintessence. Among other results it is shown that: (a) for both models there exists a period of a past deceleration with a present acceleration; (b) the slope of the acceleration of the Universe modelled as a mixture of a Chaplygin gas with radiation is more pronounced than that modelled as a mixture of quintessence and radiation; (c) the energy density of the Chaplygin gas tends to a constant value at earlier times than the energy density of quintessence does; (d) the energy density of radiation for both mixtures coincide and decay more rapidly than the energy densities of the Chaplygin gas and of quintessence.

Free molecular sound propagation

The sound propagation through a gas in the free-molecular regime is studied on the basis of the linearized collisionless Boltzmann equation. The two principal quantities that characterize the sound propagation, namely the phase and amplitude of the perturbation, are determined by taking into account the influence of the receptor. It is shown that at a small distance between the source and the receptor the presence of the last changes qualitatively the sound characteristics. Two phase velocities are introduced: a differential and an integral, which are different in the free molecular regime.