On relativistic kinetic gas theory (original) (raw)

Heat transport and diffusion in a canonical model of a relativistic gas

Relativistic transport phenomena are important from both theoretical and practical point of view. Accordingly, hydrodynamics of relativistic gas has been extensively studied theoretically. Here, we introduce a three-dimensional canonical model of hard-sphere relativistic gas which allows us to impose appropriate temperature gradient along a given direction maintaining the system in a non-equilibrium steady state. We use such a numerical laboratory to study the appropriateness of the so-called first order (Chapman-Enskog) relativistic hydrodynamics by calculating various transport coefficients. Our numerical results are consistent with predictions of such a theory for a wide range of temperatures. Our results are somewhat surprising since such linear theories are not consistent with the fundamental assumption of the special theory of relativity ($v\leq c$). We therefore seek to explain such results by studying the appropriateness of diffusive transport in the relativistic gas, comparing our results with that of a classical gas. We find that the relativistic correction (constraint) in the hydrodynamic limit amounts to small negligible corrections, thus indicating the validity of the linear approximation in near equilibrium transport phenomena

Transport Properties of a Modified Lorentz Gas

Journal of Statistical Physics, 2003

We present a detailed study of the first simple mechanical system that shows fully realistic transport behavior while still being exactly solvable at the level of equilibrium statistical mechanics. The system under consideration is a Lorentz gas with fixed freely-rotating circular scatterers interacting with point particles via perfectly rough collisions. Upon imposing a temperature and/or a chemical potential gradient, a stationary state is attained for which local thermal equilibrium holds for low values of the imposed gradients. Transport in this system is normal, in the sense that the transport coefficients which characterize the flow of heat and matter are finite in the thermodynamic limit. Moreover, the two flows are non-trivially coupled, satisfying Onsager's reciprocity relations to within numerical accuracy as well as the Green-Kubo relations. We further show numerically that an applied electric field causes the same currents as the corresponding chemical potential gradient in first order of the applied field. Puzzling discrepancies in higher order effects (Joule heating) are also observed. Finally, the role of entropy production in this purely Hamiltonian system is shortly discussed.

Relativistic thermodynamics of gases

Annals of Physics, 1986

Relativistic thermodynamics of degenerate gases is presented here as a field theory of the 14 lields of particle density -particle flux, and stress-energy -momentum.

A Polytropic Approach to Semi-relativistic Isothermal Gas Spheres at Arbitrary Temperature

2010

We use standard polynomial expansion technique to show the existence of a relation between polytropic model and the description of gas spheres at finite temperature. A numerical analysis is made concerning the obtained perturbative parameters. It is shown that there is a correspondence between polytropic and gas sphere thermal models for fermions and bosons. For instance, the polytropic index nnn displays evident correlation with temperature and chemical potential.

Kinetic theory of dense fluids. III. Density dependence of transport coefficients for dense gases

Annals of Physics, 1979

The viscosity coefficient obtained in a previous paper of this series is calculated as a function of density by developing the N-particle collision operator into a dynamic cluster expansion. The excess transport coefficient A7 is given in an exponential form, where TJ~ is the two-body Chapman-Enskog result for the transport coefficient, n is the density, and fir is a density-independent quantity consisting of connected cluster contributions of (I + 2) particles. Therefore, the leading term p1 consists of connected three-body cluster contributions. The excess shear viscosity coefficient is calculated for a monatomic hard-sphere fluid by computing ,8L up to the three-body contributions and the result is compared with the molecular dynamics result by Ashurst and Hoover and also with the experimental data on Ar at 75°C. In spite of the crudity of the potential model used and the approximations made the agreement is good. The result can be improved if l-body clusters (I > 4) are included in the calculation. The thermal conductivity coefficient can be obtained in a similar form by using exactly the same procedure used for the viscosity coefficient.

Statistical dynamics of conserved densities for the hard-sphere Lorentz gas

Physica A: Statistical Mechanics and its Applications, 1981

The statistical dynamics of a particle scattered by randomly positioned stationary hard spheres are investigated. We examine the somewhat unusual long-wavelength, low-frequency fluctuation spectra resulting from the existence of an infinite set of mutually coupled conserved densities of which the number density and energy density are two members. The analytically soluble infinite-mode correlation functions are compared with the corresponding functions obtained by truncating the set of slow modes at successively increasing orders. Furthermore, we evaluate the long-wavelength number density autocorrelation function for a fixed speed, v0, in terms of a frequency-dependent diffusivity D(oJ; v0) and obtain the fluctuation spectra of all conserved densities by an additional velocity average with the appropriate canonical weights. The effect of the frequency-dependent diffusivity D(~o; v0) on the density fluctuation spectra at frequencies small compared to the mean collision frequency is elucidated.

Transport properties of the Lorentz gas: Fourier's law

Journal of Statistical Physics, 1978

We investigate the stationary nonequilibrium (heat transporting) states of the Lorentz gas. This is a gas of classical point particles moving in a region A containing also fixed (hard sphere) scatterers of radius R. The stationary state considered is obtained by imposing stochastic boundary conditions at the top and bottom of A, i.e., a particle hitting one of these walls comes off with a velocity distribution corresponding to temperatures 7"1 and T2 respectively, 7"1 < T2. Letting p be the average density of the randomly distributed scatterers we show that in the Boltzmann-Grad limit, p -~-~, R-+ 0 with the mean free path fixed, the stationary distribution of the Lorentz gas converges in the Ll-norm to the stationary distribution of the corresponding linear Boltzmann equation with the same boundary conditions. In particular, the steady state heat flow in the Lorentz gas converges to that of the linear Boltzmann equation, which is known to behave as (T2 -T1)]L for large L, where L is the distance from the bottom to the top wall: i.e., Fourier's law of heat conduction is valid in the limit. The heat flow converges even in probability. Generalizations of our result for scatterers with a smooth potential as well as the related diffusion problem are discussed.

Enskog-Landau kinetic equation. Calculation of the transport coefficients for charged hard spheres

Physica A: Statistical Mechanics and its Applications, 1996

Using charged hard spheres model as an example, the dense one-component plasma is considered. For this model the Enskog-Landau kinetic equation is obtained and its normal solution is found using Chapman-Enskog method. Transport coefficients are obtained numerically and analytically and compared with the experimental data available. PACS: 05.60.+w, 05.70.Ln, 05.20.Dd, 52.25.Dg, 52.25.Fi.

Diffusion in a periodic Lorentz gas

Journal of Statistical Physics, 1987

We use a constant "driving force" F a together with a Gaussian thermostatting "constraint force" F,. to simulate a nonequilibrium steady-state current (particle velocity) in a periodic, two-dimensional, classical Lorentz gas. The ratio of the average particle velocity to the driving force (field strength) is the Lorentz-gas conductivity. A regular "Galton-board" lattice of fixed particles is arranged in a dense triangular-lattice structure. The moving scatterer particle travels through the lattice at constant kinetic energy, making elastic hard-disk collisions with the fixed particles. At low field strengths the nonequilibrium conductivity is statistically indistinguishable from the equilibrium Green-Kubo estimate of Machta and Zwanzig. The low-field conductivity varies smoothly, but in a complicated way, with field strength. For moderate fields the conductivity generally decreases nearly linearly with field, but is nearly discontinuous at certain values where interesting stable cycles of collisions occur. As the field is increased, the phase-space probability density drops in apparent fractal dimensionality from 3 to 1. We compare the nonlinear conductivity with similar zero-density results from the two-particle Boltzmann equation. We also tabulate the variation of the kinetic pressure as a function of the field strength.

Dissipation process of binary gas mixtures in thermally relativistic flow

Journal of Statistical Mechanics: Theory and Experiment, 2016

In this paper, we discuss dissipation process of the binary mixture gas in the thermally relativistic flow by focusing on the characteristics of the diffusion flux. As an analytical object, we consider the relativistic rarefied-shock layer problem around the triangle prism. Numerical results of the diffusion flux are compared with the Navier-Stokes-Fourier (NSF) order approximation of the diffusion flux, which is calculated using the diffusion and thermal-diffusion coefficients by Kox et al. [Physica A, 84, 1, pp.165-174 (1976)]. In the case of the uniform flow with the small Lorentz contraction, the diffusion flux, which is obtained by calculating the relativistic Boltzmann equation, is roughly approximated by the NSF order approximation inside the shock wave, whereas the diffusion flux in the vicinity of the wall is markedly different from the NSF order approximation. The magnitude of the diffusion flux, which is obtained by calculating the relativistic Boltzmann equation, is similar to that of the NSF order approximation inside the shock wave, unlike the pressure deviator, dynamic pressure and heat flux, even when the Lorentz contraction in the uniform flow becomes large, because the diffusion flux does not depend on the generic Knudsen number from its definition in Eckart's frame. Finally, the author concludes that the accurate diffusion flux must be calculated from the particle four flow, which is formulated using the four velocity distinguished by each species of particles.