Modelling diffusion in nonuniform solids using entropy production rate

With the help of the entropy production rate, a Smoluchowski equation is derived for a model of particle diffusion in a nonuniform solid at local thermodynamic equilibrium. It turns out that the current density in the absence of an external potential is proportional to the gradient of the product temperature x density, which can be understood as a flow driven by the partial-pressure gradient of the particles. The result can also be obtained from nonequilibrium variational principles. As an example for a consequence of this type of thermo-diffusion, we discuss the occurrence of a current maximum at a finite temperature gradient in a solid polymer with a temperature dependent mobility