Brownian motion in a medium with inhomogeneous temperature

We consider the motion of a Brownian particle in a medium with inhomogeneous temperature in the presence of an external potential. We start from the Klein-Kramers equation; in this equation a thermophoretic force, proportional to the temperature gradient, should in general be included to obtain a correct description of thermodiffusion effects in the hydrodynamic stage of the evolution. With the Chapman-Enskog method we derive the correct form for the Smoluchowski equation, which reduces to van Kampen's recent result in the absence of thermophoretic forces. We also give the first correction to this equation caused by deviations from local thermal equilibrium. For the system considered, such deviations persist even in the steady state.