Diffusion of relativistic gas mixtures in gravitational fields

TL;DR

This paper derives Fick's law for relativistic gas mixtures in gravitational fields using a BGK model, showing how gravity influences diffusion coefficients and introduces additional non-relativistic terms.

Contribution

It provides a novel derivation of Fick's law in a relativistic gravitational context, highlighting the gravitational effects on diffusion coefficients and non-equilibrium distributions.

Findings

Diffusion coefficients increase with gravitational potential.

Fick's law includes acceleration and gravitational potential gradient terms.

Gravity modifies the non-equilibrium distribution functions in relativistic gases.

Abstract

A mixture of relativistic gases of non-disparate rest masses in a Schwarzschild metric is studied on the basis of a relativistic Boltzmann equation in the presence of gravitational fields. A BGK-type model equation of the collision operator of the Boltzmann equation is used in order to compute the non-equilibrium distribution functions by the Chapman-Enskog method. The main focus of this work is to obtain Fick's law without the thermal-diffusion cross-effect. Fick's law has four contributions, two of them are the usual terms proportional to the gradients of concentration and pressure. The other two are of the same nature as those which appears in Fourier's law in the presence of gravitational fields and are related with an acceleration and gravitational potential gradient, but unlike Fourier's law these two last terms are of non-relativistic order. Furthermore, it is shown that the coefficients of diffusion depend on the gravitational potential and they become larger than those in the absence of it.