Two novel approaches to the Kramers rate problem in the spatial diffusion regime

Abstract

At present, there are two general theoretical approaches to calculating the rate of thermally activated escape of a Brownian particle over a barrier out of a metastable well in the spatial diffusion regime. A direct approach involves techniques entirely based on the underlying Fokker–Planck equation, such as the Kramers flux over population method, the mean first passage time formalism, and the eigenmode expansion. An alternative consists of replacing the original one-dimensional stochastic dynamics by an infinite dimensional Hamiltonian system. The rate is then calculated using reactive flux methods. Both approaches are rather efficient when treating bistable potentials with high parabolic barriers. However, complications arise if the barrier is not parabolic. In such a case, large deviations of theoretical predictions from exact numerical rates are observed in the intermediate friction region. The latter holds true even though the barrier is infinitely high, to say nothing of low barriers for which the problem of finite barrier height corrections remains effectively unresolved. Based on the expansion of the Fokker–Planck equation in reciprocal powers of the friction coefficient, two novel methods for calculating analytically the rate of escape over an arbitrarily shaped barrier are presented. These are a continued fraction expansion method and a self-similar renormalization technique developed recently for summation of divergent field-theoretical series, respectively. In this way, two different rate expressions are constructed that agree in the limiting case of high friction with the rate following from the corresponding Smoluchowski equation and reduce to the transition state theory rate at zero damping. Comparison with a known rate expression for a purely parabolic barrier and from numerical simulations for bistable potentials with cusped and smooth barriers of different heights show excellent agreement between the present theories and exact numerical results. As long as the escape dynamics is dominated by spatial diffusion across the barrier top, the maximal relative errors attained with the continued fraction method and the self-similar renormalization technique are less than 3% and 7%, respectively. This is in drastic contrast to known rate formulas derived by other means, whose relative errors are larger by factors and even by orders of magnitude.