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Fick's law for coordinate dependent diffusivity is derived. Correspondingdiffusion current in the presence of coordinate dependent diffusivity isconsistent with the form as given by Kramers-Moyal expansion. We have obtainedthe equilibrium solution of the corresponding Smoluchowski equation. Theequilibrium distribution is a generalization of the Boltzmann distribution.This generalized Boltzmann distribution involves an effective potential whichis a function of coordinate dependent diffusivity. We discuss variousimplications of the existence of this generalized Boltzmann distribution forequilibrium of systems with coordinate dependent diffusivity and damping.

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