Master equation reduction using state projection and singular perturbation arguments

Abstract

The kinetic Monte Carlo (KMC) method is becoming increasingly popular in the research of heterogeneous catalysts, yielding fundamental understanding of, e.g., their activity, selectivity, and rate of poisoning. KMC simulations can provide a link between ab initio electronic structure calculations and experiments by simulating at larger scales than is feasible for other methods, such as molecular dynamics (MD). A common problem for many systems studied using KMC, however, is the timescale separation of events considered, often leading to considerable computational effort and time spent on modeling fast events that are quasi-equilibrated and, thus, “uninteresting” from a kinetics perspective. In an attempt to make these simulations more efficient, two broad classes of schemes have been developed: those separating the state-space into “superbasins” and treating the fast events in approximate ways, and those employing absorbing Markov chain theory. In this paper, we consider the master equation underpinning KMC simulations in cases where the slow and fast events can be clearly defined, allowing us to “lump” the states into superbasins and use a perturbation expansion to derive approximations of the full solution. We thus develop a framework that unifies these two classes of schemes and enables us to derive a hierarchy of approximations that transcend the quasi-equilibration approximation. The validity of these approximations is then examined by testing them on four model systems, two of them relevant to on-lattice reaction kinetics.