Theory of activated rate processes for arbitrary frequency dependent friction: Solution of the turnover problem

Abstract

An analytical theory is formulated for the thermal (classical mechanical) rate of escape from a metastable state coupled to a dissipative thermal environment. The working expressions are given solely in terms of the quantities entering the generalized Langevin equation for the particle dynamics. The theory covers the whole range of damping strength and is applicable to an arbitrary memory friction. This solves what is commonly known as the Kramers turnover problem. The basic idea underlying the approach is the observation that the escape dynamics is governed by the unstable normal mode coordinate—and not the particle system coordinate. An application to the case of a particle moving in a piecewise harmonic potential with an exponentially decaying memory‐friction is presented. The comparison with the numerical simulation data of Straub, Borkovec, and Berne [J. Chem. Phys. 8 4, 1788 (1986)] exhibits good agreement between theory and simulation.