Solution of the Langevin equation for rare event rates using a path-integral formalism

We present an approach to the problem of evaluating the rates of rare activated events by solving the Langevin equation through a path-integral formalism. At temperatures much lower than the activation barrier, we find that the minimal path approximation to the path integral yields excellent accuracy, and greatly simplifies numerical efforts in the solution of the Langevin equation. In addition, the extremal paths allow one to locate the saddle points without presuming any particular physical mechanisms for getting from one configuration to another. As a demonstration of this approach, we study the Brownian motion of a particle in a periodic potential subject to stochastic forces. This model has many applications in varied fields besides physics, such as chemistry and communication theory. We focus specifically in this paper on the application to the problem of surface adatom diffusion. For one dimension, the results we obtain with this approach are in full agreement with standard analytical and numerical methods. Furthermore, we have derived analytical formulas for the probability distribution of jump lengths. [S0163-1829(99)02248-1].