The asymptotic behavior of diffusion and gradient flow on tilted periodic potentials

A variety of phenomena in physics and other fields can be modeled as Brownian motion in a heat bath under tilted periodic potentials. We are interested in the long time average velocity considered as a function of the external force, that is, the tilt of the potential. In many cases, the long time behavior — pinning and de-pinning phenomenon — has been observed. We use the method of stochastic differential equation to study the Langevin equation describing such diffusion. In the over-damped limit, we show the convergence of the long time average velocity to that of the Smoluchowski-Kramers approximation, and carry out asymptotic analysis based on Risken\u27s and Reimann et al.\u27s formula. In the under-damped limit, applying Freidlin et al.\u27s theory, we first show the existence of three pinning and de-pinning thresholds of the normalized tilt, corresponding to the bi-stability phenomenon; and second, as noise decays to zero, derive formulas of the mean transition times between the pinning and running states