# Generalization of Stokes-Einstein relation to coordinate dependent damping and diffusivity: An apparent conflict

## TL;DR

This paper investigates the equilibrium behavior of Brownian particles with coordinate-dependent damping and diffusivity, clarifying when the Stokes-Einstein relation applies and proposing conditions for its local validity.

## Contribution

It introduces an alternative approach using Kramers-Moyal expansion to analyze equilibrium distributions with coordinate-dependent parameters, and discusses conditions for the Stokes-Einstein relation's applicability.

## Key findings

- The Stokes-Einstein relation holds with constant diffusivity and damping.
- No homogeneous limit exists for coordinate-dependent diffusivity and damping when the relation fails locally.
- Imposing a maximum local velocity ensures the modified Maxwell-Boltzmann distribution aligns with the classical one.

## Abstract

Brownian motion with coordinate dependent damping and diffusivity is ubiquitous. Understanding equilibrium of a Brownian particle with coordinate dependent diffusion and damping is a contentious area. In this paper, we present an alternative approach based on already established methods to this problem. We solve for the equilibrium distribution of the over-damped dynamics using Kramers-Moyal expansion. We compare this with the over-damped limit of the generalized Maxwell-Boltzmann distribution. We show that the equipartition of energy helps recover the Stokes-Einstein relation at constant diffusivity and damping of the homogeneous space. However, we also show that, there exists no homogeneous limit of coordinate dependent diffusivity and damping with respect to the applicability of Stokes-Einstein relation when it does not hold locally. In the other scenario where the Stokes-Einstein relation holds locally, one needs to impose a restriction on the local maximum velocity of the Brownian particle to make the modified Maxwell-Boltzmann distribution coincide with the modified Boltzmann distribution in the over-damped limit.