KMCs

Kinetic Monte Carlo methods (KMC) are a class of stochastic simulation algorithms used to model the time evolution of systems in which the rates of discrete events are known. They are widely applied to diffusion and reaction processes on atomic or mesoscopic scales, where the system’s state is described by a discrete set of configurations and events occur with defined transition rates.

In a typical lattice KMC, the current configuration is stored together with a catalog of possible events and their rates. At each step, one event is chosen with probability proportional to its rate, the system is updated to the resulting configuration, and the simulation clock is advanced by a time increment Δt = −ln(u)/Rtot, where u is a uniformly distributed random number and Rtot is the sum of all event rates. Variants such as the Bortz-Kalos-Lebowitz (BKL) or “n-fold way” algorithm are designed to be rejection-free and efficient when many events share similar rates.

Origins and scope: KMC originated in the 1970s as an extension of Monte Carlo methods for kinetic problems. The BKL algorithm provided a practical, rejection-free approach to selecting events. Since then, continuous-time or off-lattice KMC variants have been developed for chemical kinetics, surface science, and other fields.

Applications and limitations: KMC is widely used in materials science (surface diffusion, crystal growth, defect kinetics), catalysis, battery materials, and semiconductor device modeling. Its accuracy depends on having a correct catalog of possible events and accurate rates, which can be challenging to obtain. The approach can be computationally expensive when many events are possible, and rare-event acceleration techniques are sometimes employed. KMC is most appropriate when dynamics are dominated by well-characterized, discrete transitions rather than continuous-time dynamics.

Relation to other methods: KMC is a continuous-time Markov process and is related to Gillespie’s stochastic simulation algorithm used in chemical kinetics.