Equilibrium of a Brownian particle with coordinate dependent diffusivity and damping: Generalized Boltzmann distribution

TL;DR

This paper derives a generalized Boltzmann distribution for Brownian particles with coordinate-dependent diffusivity and damping, providing a new framework for understanding equilibrium states in such systems.

Contribution

It introduces a generalized Boltzmann distribution incorporating an effective potential dependent on coordinate-dependent diffusivity and damping.

Findings

Derived Fick's law for coordinate-dependent diffusivity.

Obtained equilibrium solution of the Smoluchowski equation.

Proposed a generalized Boltzmann distribution for these systems.

Abstract

Fick's law for coordinate dependent diffusivity is derived. Corresponding diffusion current in the presence of coordinate dependent diffusivity is consistent with the form as given by Kramers-Moyal expansion. We have obtained the equilibrium solution of the corresponding Smoluchowski equation. The equilibrium distribution is a generalization of the Boltzmann distribution. This generalized Boltzmann distribution involves an effective potential which is a function of coordinate dependent diffusivity. We discuss various implications of the existence of this generalized Boltzmann distribution for equilibrium of systems with coordinate dependent diffusivity and damping.