Numerical study of the tight-binding approach to overdamped Brownian motion on a tilted periodic potential

We present a numerical study of the tight-binding approach to overdamped Brownian motion on a tilted periodic potential. In the tight-binding method the probability density is expanded on a basis of Wannier states to transform the Smoluchowski equation to a discrete master equation that can be interpreted in terms of thermal hopping between potential minima. We calculate the Wannier states and hopping rates for a variety of potentials, including tilted cosine and ratchet potentials. For deep potential minima the Wannier states are well localized and the hopping rates between nearest-neighbor states are qualitatively well described by Kramers' escape rate. The next-nearest-neighbor hopping rates are negative and must be negligible compared to the nearest-neighbor rates for the discrete master equation treatment to be valid. We find that the validity of the master equation extends beyond the quantitative applicability of Kramers' escape rate.