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

TL;DR

This paper derives a modified Smoluchowski equation for interacting Brownian particles in a temperature gradient using a multiple time-scale approach, highlighting the effects of non-uniform temperature fields on particle dynamics.

Contribution

It introduces a novel derivation of the Smoluchowski equation for non-uniform temperature fields from the Kramers equation, incorporating mean-field interactions.

Findings

Numerical results agree with theoretical predictions.

The approach clarifies the Langevin equation formulation in overdamped systems.

Provides insights into particle behavior in temperature gradients.

Abstract

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.