Kramers problem in periodic potentials: Jump rate and jump lengths

Abstract

The Kramers problem in periodic potentials is solved separating the intrawell and interwell dynamics. Both the jump rate and the probability distribution of the jump lengths are obtained by a Fourier analysis of the decay function f(q); at high and intermediate potential barriers, in the first Brillouin zone, f(q) essentially coincides with the energy half-width of the quasielastic peak of the dynamic struture factor. The method is applied to the Klein-Kramers dynamics; numerical results are obtained in a wide damping range by solving the Klein-Kramers equation with cosine potential and homogeneous friction, at high (16${\mathit{k}}_{\mathit{B}}$T) and intermediate (6${\mathit{k}}_{\mathit{B}}$T) potential barriers. The jump rate exhibits the expected turnover behavior; an increasing deviation from the exponential decay of the jump-length distribution is found as the damping decreases. The low-friction, multiple-jump regime is quantitatively characterized. The comparison with asymptotic analytical approximations of the Mel’nikov and Meshkov kind suggests that finite-barrier corrections are significant even at high potential barriers, especially in the underdamped regime.