Rare events are ubiquitous in many biological, chemical and physical processes. Whereas the density of states is known in systems at thermal equilibrium, interesting phenomena often occur in non-equilibrium systems. Unfortunately, many such problems are inaccessible to analytic methods. Therefore computer simulations are a widely used tool to estimate the density of states or transition rates between them. Since standard Brownian dynamic simulation provides computational costs that are inversely proportional to the state's probability, specialized methods have to be used to adequately sample rare events, i.e. states with low probability or low transition rates.
In this talk I will present an algorithm [J . A. Kromer, L. Schimansky-Geier, R. Toral,
Physical Review E 87, 063311 (2013)], based on the previously developed weighted-ensemble (WE) Brownian dynamics simulations that allows one to calculate the stationary probability density function (SPDF) as well as transition rates between particular states. The idea is to divide the space of interest into several subregions and calculate the probability for finding the system in each of them by generating equally weighted walkers in each region. By moving to the underlying dynamics, the walkers transport probability between the subregions. Thus, WE methods are usually applied to systems of Brownian particles moving in a potential landscape. Our method outperforms others by several orders of magnitude in all studied systems and lead to impressive results in regions of low probability and small rates.