On nonlinear heat equations and diffusion in porous media

The problem of diffusion of a Brownian particle in a liquid under the influence of an exterior field, taking into account thermal effects, as given by Streater [1–3], is reformulated for the case in which the thermal conductivity coefficient is nonhomogeneous. This modified formulation is applied to the description of diffusion in a liquid filling a porous medium. Macroscopic equations of Brownian diffusion for that case are obtained using the two-scale asymptotic method. As a result, in the macroscopic diffusion equation the friction term is absent, and the external force appears explicitly as a heat production term only; implicitly the drift force is found in the first correction f(1) in the asymptotic expansion of distribution function f.