Thermally driven escape over a barrier of arbitrary shape

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 Fokke r-Planck equation in the spatial diffusion regime. With the use of the Mel' nikov-Meshkov result for the underdamped Brownian motion an overall rate ex pression is constructed, which interpolates the correct limiting behavior f or both weak and strong friction. It generalizes in a natural way various d ifferent rate expressions that are already available in the literature for parabolic, cusped, and quartic barriers. Applications to symmetric paraboli c and cusped double-well potentials show good agreement between the theory and estimates of the rates from numerical calculations. (C) 1999 American I nstitute of Physics. [S0021-9606(99)01404-X].