First-passage kinetic Monte Carlo method

Abstract

We present an efficient method for Monte Carlo simulations of diffusion-reaction processes. Introduced by us in a previous paper [Phys. Rev. Lett. 97, 230602 (2006)], our algorithm skips the traditional small diffusion hops and propagates the diffusing particles over long distances through a sequence of superhops, one particle at a time. By partitioning the simulation space into nonoverlapping 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 algorithm is efficient at low particle densities, where other existing algorithms slow down severely.

An exact and efficient first passage time algorithm for reaction–diffusion processes on a 2D-lattice

“…Thus particles nucleate when they are in adjacent lattice sites. The assumption of no interaction for particles that are more than one lattice spacing apart also justifies the fact that only particles who see their zones violated need to be updated [22]. It is therefore computationally favorable to maintain a chronological event queue.…”

“…Moreover, accurate simulation of nucleation events is critically important since long-term growth patterns are influenced by atomic-scale events. We report here a variation of the first-passage-time-KMC approach, tailored to 2D on-lattice surface processes [23,22].…”

Coupled Mesoscopic and Microscopic Simulation of Stochastic Reaction-Diffusion Processes in Mixed Dimensions

“…The GFRD idea is to decompose the problem into one-body and two-body problems by choosing a time step ∆t such that molecules are unlikely to react with more than one other molecule or the boundary of the domain ∂Ω. Details how to choose ∆t can be found in [29,40,48,52]. If the molecule is located at z n at time t = t n , the new position z n+1 at t = t n+1 = t n + ∆t is sampled from the cumulative distribution function (CDF) derived from the corresponding PDF.…”

“…In the microscopic model, individual molecules move by Brownian motion and when they are close they can react with each other. The molecules diffuse in the simulation either by taking small time steps in a solution of a Langevin equation [5,32] or by sampling a probability distribution for the new position [29,40,52]. Such microscopic models are much more computationally demanding than a corresponding mesoscopic simulation, at least for reasonable mesh resolutions.…”

Average first-passage time of a quasi-Hamiltonian Mathieu oscillator with parametric and forcing excitations

“…In a stochastic context, this time is a random variable. It is characterized by its probability density function which can be obtained by Monte Carlo simulations [9,29] or by resolution of the Chapman-Kolmogorov equation, via use of numerical [22] or semi-analytical methods such as a numerical path integration [48,21], approximation of the solution by a Galerkin scheme [37,36], a Poisson distribution based assumption [3]. These numerical approaches are important because there are very few problems in which the distribution of the first passage time can be established in closed-form [44,40].…”