The long time behavior of Brownian motion in tilted periodic potentials
نویسندگان
چکیده
• We study the Langevin equation describing diffusion in tilted periodic potentials. • In the over-damped limit the long time average velocity converges in probability. • In the over-damped and vanishing noise limit the convergence rate varies as the tilt crosses threshold. • In the under-damped limit we recover Risken's results about the bi-stability. • In the under-damped limit we derive asymptotics of the transition times. a b s t r a c t 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's and Reimann et al.'s formula. In the under-damped limit, applying Freidlin et al.'s 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 approaches zero, derive formulas of the mean transition times between the pinning and running states.
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