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 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.