Brownian motion in a two-dimensional potential: A new numerical solution method

Studying the Brownian motion of a particle in a two-dimensional potential with two saddle-point passages connecting two wells, we compute the activation rate of the particle from one well into the other and illustrate a new technique for obtaining numerical solution to the Langevin equation for transition probability. By virtue of a Langevin equation with negative friction, this new method directly traces the active part of an activation event, without having to simulate the long period of small fluctuations in a well between two successful events, and computes the statistical weight for each successful activation. It makes feasible for us to numerically integrate the Langevin equation for transition probability even when the activation energy barrier (i.e. the potential difference between the saddle point and the well) is much greater than thermal energy k(B)T where other methods fail to be tractable.