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| 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.
We have developed an algorithm [J . A. Kromer, L. Schimansky-Geier, R. Toral
Physical Review {\bf E87}, 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. Like in WE simulations the space of interest is divided into
several subregions and the probability for finding the system in them
is calculated 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 algorithm is based on WE Brownian dynamic simulations, but uses a uniform distribution of walkers within each subregion.
Our method outperforms Brownian dynamics simulation by several orders of magnitude and its efficiency is comparable to weighted-ensemble Brownian dynamic simulations in all studied systems and lead to impressive results in regions of low probability and small rates. As an example, we show in the first figure the pdf computed for the classic double well potential $U(x)=-\frac12x^2+\frac14x^4$ as well as, in the second figure, the verification of Kramers law for the probability current. Note that the vertical scale in both cases shows the high efficiency of the method in sampling low-probability events, as we are able to sample correctly events with probability of the order of $10^{-300}$ and currents of the order of $10^{-300}$. |
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