Non-linear determination of the distribution function of degenerate gases with an application to semiconductors

In this paper we consider an extended thermodynamic model for degenerate gases in which the first twenty moments of the gas distribution function are used as state variables. Exploiting the maximum entropy principle, we determine an analytic expression for the non-equilibrium distribution function of Bose and Fermi gases by means of a non-linear expansion with respect to the local thermodynamical equilibrium. Once the distribution function is given, we find the constitutive functions which appear in the fluxes of the balance equations, Explicit results can be obtained for classical ideal gases and strongly degenerate Bose and Fermi gases. We also consider an application to silicon semiconductors for which we test the accuracy of the closure relations by a comparison with Monte Carlo results.