Multiple time-scale approach for a system of Brownian particles in a non-uniform temperature field

The Smoluchowski equation for a system of interacting
Brownian particles in a temperature gradient is derived from
the Kramers equation by means of a multiple time-scale
method. The interparticle interactions are assumed to be represented
by a mean-field description.
We present numerical results that compare well with
the theoretical prediction together with an extensive discussion on the
prescription of the Langevin equation in overdamped systems.