Brownian motion in a periodic potential under an applied bias : the transition from hopping to free conduction

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Abstract
We consider particles in an arbitrary large bias, subject to a periodic potential ; the friction is such that the oscillation in the potential wells is underdamped. We develop a stochastic formulation based on the assumption that the friction is Markovian from one well to the next. The relevant parameter is the mean energy δ lost in crossing one well, compared to either T or to the drop in bias energy y. We show that a rapid change of regime occurs at y = δ. When δ >> y, T, we recover the old results of Frenkel. When δ << T, the mobility is increased : we give a detailed solution in this limit for arbitrary bias. We thus extend the results obtained by Ambegaokar and Halperin [2] in the overdamped case

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