Random walk model for coordinate-dependent diffusion in a force field

Abstract

In this paper we develop a random walk model on a lattice for coordinate-dependent diffusion at constant temperature. We employ here a coordinate-dependent waiting time of the random walker to get coordinate dependence of diffusion. Such a modeling of the coordinate dependence of diffusion keeps the local isotropy of the process of diffusion intact which is consistent with the nature of thermal noise. The presence of a confining conservative force is modeled by appropriately breaking the isotropy of the jumps of the random walker to its nearest lattice points. We show that the equilibrium is characterized by the position distribution which is of a modified Boltzmann form as is obtained for an Itô process. We also argue that, in such systems with coordinate-dependent diffusivity, the modified Boltzmann distribution correctly captures the transition over a potential barrier as opposed to the Boltzmann distribution.