Diffusion of relativistic gas mixtures in gravitational fields

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 appear in Fourier's law in the presence of gravitational fields and are related to an acceleration and a gravitational potential gradient, but unlike Fourier's law these last two terms are of non-relativistic order. Furthermore, it is shown that the coefficients of diffusion depend on the gravitational potential and become smaller than those in its absence.