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

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 to this problem based on already established methods. 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/modified Maxwell-Boltzmann distribution. We investigate two distinct possibilities of the Stokes-Einstein relation not holding locally and holding locally everywhere. In the former case we get a local proportional relation between the coordinate dependent diffusivity and damping which is consistent with other requirements of equilibrium. The latter case requires restrictions on the upper limit of the local velocity of the Brownian particle to make the modified Maxwell-Boltzmann relation obtain the correct over-damped limit.