Equilibrium of a Brownian particle in an inhomogeneous medium: An alternative approach

We look at the equilibrium of a Brownian particle in an inhomogeneous space following the alternative approach proposed in ref.[1]. We consider a coordinate dependent damping that makes the stochastic dynamics the one with multiplicative noise. Here we show that the mapping to an additive noise gives the equilibrium distribution of the generalized Langevin dynamics of a particle with mass. The procedure does not need inclusion of any ad hoc current cancelling term in the Langevin dynamics. The result shows a modified Maxwell-Boltzmann distribution with a damping dependent amplitude.