Non-Markovian theory of activated rate processes. I. Formalism

Abstract

The escape of a particle from a potential well is treated using a generalized Langevin equation (GLE) in the low friction limit. The friction is represented by a memory kernel and the random noise is characterized by a finite correlation time. This non-Markovian stochastic equation is reduced to a Smoluchowski diffusion equation for the action variable of the particle and explicit expressions are obtained for the drift and diffusion terms in this equation in terms of the Fourier coefficients of the deterministic trajectory (associated with the motion without coupling to the heat bath) and of the Fourier transform of the friction kernel. The latter (frequency dependent friction) determines the rate of energy exchange with the heat bath. The resulting energy (or action) diffusion equation is used to determine the rate of achieving the critical (escape) energy. Explicit expressions are obtained for a Morse potential. These results for the escape rate agree with those from stochastic trajectories based on the original GLE. Non-Markovian effects are shown to have large effects on the rate of energy accumulation and relaxation within the well.