Drift and diffusion in periodic potentials: upstream and downstream step times are distributed identically.

This note deals with particles diffusing in one dimension in the presence of a periodic potential and a uniform driving force. We show that (i) the probabilities for the particle to make a step of the length L, where L is the period, in the upstream and downstream directions are independent of the periodic potential, and (ii) the distributions of the step time are independent of the step direction. These two characteristics are used to derive expressions for the effective drift velocity and diffusion coefficient.