Retrapping and velocity inversion in jump diffusion

A method for the solution of the Kramers problem in periodic potentials is proposed in the general case of a tilted periodic potential. The method is then applied to the Fokker-Planck equation with a cosine potential without tilt, and the results for the jump-length probability distribution are compared to simulation data concerning the lengths crossed by a hopping particle before its first velocity inversion. It is shown that, at high dissipation, the diffusing particle thermalizes in most cases in the cell where it inverts the velocity for the first time. On the contrary, at low dissipation the actual length of the jump and the length crossed before the first velocity inversion display significant differences, showing that the first velocity inversion is not a good criterion for the length of the jump, just in the case where long jumps are more important.