  The dynamical evolution of a Brownian particle in an inhomogeneous medium
with spatially varying friction and temperature field is important to
understand conceptually. It requires to address the basic problem of relative
stability of states in nonequilibrium systems which has been a subject of
debate for over several decades. The theoretical treatments adopted so far are
mostly phenomenological in nature. In this work we give a microscopic treatment
of this problem. We derive the Langevin equation of motion and the associated
Fokker-Planck equation. The correct reduced description of the Kramers equation
in the overdamped limit (Smoluchowski equation) is obtained. Our microscopic
treatment may be helpful in understanding the working of thermal ratchets, a
problem of much current interest.
