KRAMERS PROBLEM IN PERIODIC POTENTIALS - JUMP RATE AND JUMP LENGTHS

The Kramers problem in periodic potentials is solved separating the in trawell and interwell dynamics. Both the jump rate and the probability distribution of the jump lengths are obtained by a Fourier analysis o f the decay function f(q); at high and intermediate potential barriers , in the first Brillouin zone, f (q) essentially coincides with the en ergy half-width of the quasielastic peak of the dynamic struture facto r. The method is applied to the Klein-Kramers dynamics; numerical resu lts are obtained in a wide damping range by solving the Klein-Kramers equation with cosine potential and homogeneous friction, at high (16k( B)T) and intermediate (6k(B)T) potential barriers. The jump rate exhib its the expected turnover behavior; an increasing deviation from the e xponential decay of the jump-length distribution is found as the dampi ng decreases. The low-friction, multiple-jump regime is quantitatively characterized. The comparison with asymptotic analytical approximatio ns of the Mel'nikov and Meshkov kind suggests that finite-barrier corr ections are significant even at high potential barriers, especially in the underdamped regime.