  In this paper we develop a random walk model on lattice for coordinate
dependent diffusion at constant temperature in contact with a heat bath. We
employ here a coordinate dependent waiting time of the random walker to make
the diffusivity coordinate dependent. The presence of a confining conservative
force is modeled by appropriately breaking the isotropy of the jumps of the
random walker to its nearest neighbors. We show that the equilibrium is
characterized by the position distribution which is of modified Boltzmann form.
We also show 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.
