Problem with the Boltzmann distribution in small systems with boundary induced inhomogeneity.

The Boltzmann distribution characterizes equilibrium of a system at thermodynamic limit where inhomogeneity in the bulk is due to some conservative force. In mesoscopic systems, boundary induced inhomogeneities are often present which a conservative force cannot capture. Moreover, there is controversy (It\^o vs Stratonovich) arising from handling of multiplicative noise in the Langevin dynamics of such mesoscopic systems. To make things worse, modification of Fick's law for coordinate dependent diffusion is also considered controversial in existing literature. It is shown here that, the demand of the Boltzmann distribution is untenable for thermodynamic equilibrium in the presence of coordinate (conformation) dependent damping and diffusion. To alleviate confusion about the form of Fick's law, its exact modification for coordinate dependent diffusivity is derived from first principle and the same is used in the relevant Fokker-Planck dynamics. Equilibrium distribution of mesoscopic systems with space dependent damping and diffusion, in general, is shown to be a modified Boltzmann distribution.