A consistent kinetic Fokker-Planck model for gas mixtures

Rarefied gas dynamics is usually described by the Boltzmann equation. Unfortunately, the expense of evaluating this operator can be very prohibitive. This made it worthwhile to look for approximations that convey essentially an equivalent amount of physical information. One widely known approximative collision operator is the Bathnagar-Gross-Krook (BGK) operator. However, recently, the Foker-Planck approximation has become increasingly popular. Nevertheless, the modeling of gas mixtures in the context of the kinetic Fokker-Planck equation has so far only been addressed in a very few papers. In this paper, we propose a general multi-species Fokker-Planck model. We prove consistency of our model: conservation properties, positivity of all temperatures, H-Theorem and the shape of equilibrium as Maxwell distributions with the same mean velocity and temperature. Moreover, we derive the usual macroscopic equations.