Multiple time-scale approach for a system of Brownian particles in a   non-uniform temperature field
  The Smoluchowsky 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. 
Cristobal Lopez and Umberto Marini Bettolo Marconi