Three-dimensional diffusion in inhomogeneous media

Abstract

The problem of diffusion in a medium with non-uniform temperature was first considered by Landauer, and subsequently discussed by a number of authors. The conclusion is that there is no universal diffusion equation; rather its precise form depends on the details of the underlying mechanism. Some examples were worked out for one-dimensional systems, but here we are concerned with more dimensions. In that case it may happen that the stationary distribution of the particle density involves a non-vanishing diffusion flow. The equations are established and solved for a simple toy system. Consideration is given to the subsequently more interesting case of dust particles diffusing in a quiescent atmosphere whose temperature varies irregularly in space. The resulting correction to the barometric formula is computed by means of perturbation theory.