First-Passage Kinetic Monte Carlo. I: Method and Basic Theory

Tomas Oppelstrup, 2 Vasily V. Bulatov, Aleksandar Donev, Malvin H. Kalos, George H. Gilmer, and Babak Sadigh Lawrence Livermore National Laboratory, Livermore, California 94551, USA Royal Institute of Technology (KTH), Stockholm S-10044, Sweden Abstract In this first part of a series of two papers we present a new efficient method for Monte Carlo simulations of diffusion-reaction processes. First introduced by us in [Phys. Rev. Lett., 97:230602, 2006], the new algorithm skips the traditional small diffusion hops and propagates the diffusing particles over long distances through a sequence of super-hops, one particle at a time. By partitioning the simulation space into non-overlapping protecting domains each containing only one or two particles, the algorithm factorizes the N -body problem of collisions among multiple Brownian particles into a set of much simpler single-body and two-body problems. Efficient propagation of particles inside their protective domains is enabled through the use of time-dependent Green’s functions (propagators) obtained as solutions for the first-passage statistics of random walks. The resulting Monte Carlo algorithm is event-driven and asynchronous; each Brownian particle propagates inside its own protective domain and on its own time clock. The algorithm reproduces the statistics of the underlying Monte-Carlo model exactly. Extensive numerical examples demonstrate that for an important class of diffusion-reaction models the new algorithm is efficient at low particle densities, where other existing algorithms slow down severely.