Finite barrier corrections for the Kramers rate problem in the spatial diffusion regime

Abstract

Finite barrier height corrections for the rate of thermally activated escape of a particle over a barrier out of a metastable well are determined within the framework of Fokker—Planck processes in the spatial diffusion regime. Different existing concepts to obtain the rate are extended and compared to each other. The central role that is played by Kramers' stationary current carrying probability density is demonstrated. An improvement of this function by means of a perturbation theory allows one to calculate corrections to the rate by means fo the flux over population expression, the Rayleigh quotient and the mean first passage time to the stochastic separatrix. The Rayleigh quotient is in a sense superior as it allows one to obtain from the same stationary current carrying Kramers function higher order corrections than by means of other methods. Explicit results are obtained for the case of a cubic and a quartic potential and compared with existing results.