L. Y. Chen
Department of Physics, University of Texas at San Antonio
San Antonio, TX 78249
S. C. Ying
Department of Physics, Brown University
Providence, RI 02912
The numerical computation of rate of rare events such as activated processes at low temperatures poses a great challenge, since most of the computation time are 'wasted' in the time interval between jumps. Various schemes have been proposed to speed up the processes and overcome this problem. We present here a novel approach for this rare event problem by solving the Langevin equation for the activated dynamics through a path integral formalism. As a demonstration of the method, we study the Brownian motion of a particle in a periodic potential subject to stochastic forces. This is a prototype problem with many applications such as surface adatom diffusion or the mobility of vortices in type-II superconductors. At temperatures much lower than the activation barrier, we find that the stationary path approximation to the path integral gives excellent accuracy and greatly simplifies numerical efforts in the solution of the Langevin equation. As the temperatures is lowered, our stationary path approximation to the path integral becomes increasingly more accurate without requiring any greater numerical effort. This is in direct contrast with standard numerical simulation methods where the computation become increaingly more costly (even impossible in 2D) at low temperatures. For one-dimension, the results that we obtain with this new approach for the surface diffusion constant are in full agreement with existing results in the literature through either analytic means or standard numerical simulations. Furthermore, we have derived analytical formulae for the probabilities of jumps of different lengths. The method is particularly powerful and provides additional insights in the low friction regime where long jumps are of frequent occurence. For the problems with colored noises (memory effects) and for two-dimension problems, the numerical implementation of the stationary path approximation to the path integrals is still feasible and provides a method for calulating the rates at low temperatures that is beyond the means of conventional simulation approaches.