Thermally driven escape over a barrier of arbitrary shape

Abstract

The Kramers theory for the thermally activated rate of escape of a Brownian particle from a potential well is extended to a barrier of arbitrary shape. The extension is based on an approximate solution of the underlying Fokker–Planck equation in the spatial diffusion regime. With the use of the Mel’nikov–Meshkov result for the underdamped Brownian motion an overall rate expression is constructed, which interpolates the correct limiting behavior for both weak and strong friction. It generalizes in a natural way various different rate expressions that are already available in the literature for parabolic, cusped, and quartic barriers. Applications to symmetric parabolic and cusped double-well potentials show good agreement between the theory and estimates of the rates from numerical calculations.