Macroscopic Equation of Motion in Inhomogeneous media

TL;DR

This paper provides a microscopic derivation of the Langevin and Fokker-Planck equations for a Brownian particle in an inhomogeneous medium, clarifying the dynamics and stability of nonequilibrium states.

Contribution

It offers a novel microscopic derivation of the equations governing Brownian motion in inhomogeneous media, advancing understanding beyond phenomenological models.

Findings

Derived Langevin and Fokker-Planck equations microscopically

Obtained the correct overdamped (Smoluchowski) limit

Enhanced understanding of thermal ratchets and nonequilibrium stability

Abstract

The dynamical evolution of a Brownian particle in an inhomogeneous medium with spatially varying friction and temperature field is important to understand conceptually. It requires to address the basic problem of relative stability of states in nonequilibrium systems which has been a subject of debate for over several decades. The theoretical treatments adopted so far are mostly phenomenological in nature. In this work we give a microscopic treatment of this problem. We derive the Langevin equation of motion and the associated Fokker-Planck equation. The correct reduced description of the Kramers equation in the overdamped limit (Smoluchowski equation) is obtained. Our microscopic treatment may be helpful in understanding the working of thermal ratchets, a problem of much current interest.