Cluster dynamics simulations using stochastic algorithms with logarithmic complexity in number of reactions

We show how to adapt and modify the Stochastic Simulation Algorithm (also sometimes referred to as the Gillespie algorithm or the kinetic Monte Carlo algorithm for rate equations) to efficiently simulate cluster dynamics equations for large systems, bypassing the linear complexity of the SSA by associating an internal time scale and priority queues with binary heaps for each mobile species. The internal time of a binary heap renormalises the physical time thus making it independent of the number of clusters of the mobile species concerned, while the binary heaps allow efficient sorting advances. The resulting algorithm has an algorithmic complexity that is linear with respect to the number of mobile cluster types, but logarithmic with respect to the number of immobile cluster species, thus being extremely effective when the number of mobile species is small, a situation satisfied in most models of cluster dynamics. As a physical application, we simulate the time evolution of defect clusters in a FeCu${}_{0.1\%}$ system under irradiation, by integrating the associated cluster dynamics equations using the stochastic algorithm. The speed-up of the algorithm, compared to the direct method is substantial, about several orders of magnitude, while consuming much less memory than deterministic simulations.