A Singular Perturbation Approach to Kramers’ Diffusion Problem

Abstract

We consider Kramers’ diffusion problem, which seeks to calculate the rate of escape of a particle from one potential well over a barrier, to another presumably deeper and therefore more stable well. Though Kramers introduced the problem as a model for chemical reactions, it applies to numerous rate processes, including atomic migration and ionic conductivity in crystals, and transitions due to noise, between stable states of dynamical systems with multi-stable states, to name but a few. We propose a new approach, not based on a Fokker–Planck equation, but rather on the solution of a singularly perturbed boundary value problem. Specifically, we relate the rate of escape to the first passage time from the domain of attraction of the stable point corresponding to the first well. The first passage time is then characterized via the Ito calculus, as a solution of an elliptic partial differential equation of singular perturbation type. Finally this equation is solved asymptotically by methods previously developed by the authors. We obtain some new results on the rate of escape, which reduce to those of Kramers for the cases he considered, and in addition discuss the validity of the various results derived by Kramers. Finally, in contrast to other approaches, our methods readily extend to higher dimensions.