The Exit Problem: A New Approach to Diffusion Across Potential Barriers

Abstract

We consider the problem of a Brownian particle confined in a potential well of forces, which escapes the potential barrrier as the result of white noise forces acting on it. The problem is characterized by a diffusion process in a force field and is described by Langevin’s stochastic differential equation. We consider potential wells with many transition states and compute the expected exit time of the particle from the well as well as the probability distribution of the exit points. Our method relates these quantities to the solutions of certain singularly perturbed elliptic boundary value problems which are solved asymptotically. Our results are then applied to the calculation of chemical reaction rates by considering the breaking of chemical bonds caused by random molecular collisions, and to the calculation of the diffusion matrix in crystals by considering random atomic migration in the periodic force field of the crystal lattice, caused by thermal vibrations of the lattice.