Diffusion in a tilted periodic potential. Specific properties of motion in the underdamped limit

The specific features of Brownian motion in a tilted periodic potential are analyzed in the underdamped limit. Analysis is made within two models of stochastic motion: the model of discrete activated jumps and continuum model. The analysis demonstrates the equivalence of these two models in underdamped limit, which shows itself in the similarity of the probability distribution function (PDF) of particle trapping, predicted by the models. Some peculiarities of this PDF are thoroughly considered. In particular, we analyze the anomalous strongly non-exponential behavior of the PDF of Lévy-flights type for values of external force F, close to the threshold of appearance of the running state F1. Special attention is payed to the analysis of specific features of F-dependence of the PDF in the region of the locked-to-running transition. The continuum model is shown to be able to describe time dependent properties of the process, some of which are briefly discussed.