We will investigate various limits concerning the long time average
velocity V of a Brownian particle diffusing on a periodic potential. The
prototype model is Langevin dynamics which incorporates inertia (mass) and
friction. The key feature of the current work is the consideration of an
additional macroscopic tilt, F. The goal is to understand how the average
velocity V depends on F. Interesting thresholds for the value of F can be
obtained, in particular under the limit of vanishing friction and noise.
Using the averaging theory of Freidlin-Wentzel, the current work provides
rigorous mathematical justification of some formulas obtained by Risken.
An earlier, fundamental result by Tanaka for Brownian particle diffusing
on a random (Brownian) potential will also be discussed.|