Mobility in a periodic potential: A simple derivation

A simple derivation is given for the mobility of Brownian particles in a periodic potential in the overdamped regime. The method makes use of the fact that the steady state current density, in response to a uniform external force, is uniform in space and can be expressed as a product of the particle density and mean velocity field. To lowest order in the external force, the particle density is given by the equilibrium density in the absence of the external force.