Kinetic Monte Carlo

The kinetic Monte Carlo (KMC) method is a Monte Carlo method computer simulation intended to simulate the time evolution of some processes occurring in nature. Typically these are processes that occur with known transition rates among states. It is important to understand that these rates are inputs to the KMC algorithm, the method itself cannot predict them. The KMC method is essentially the same as the dynamic Monte Carlo method and the Gillespie algorithm.

One possible classification of KMC algorithms is as rejection KMC (rKMC) and rejection free KMC (rfKMC).

A rfKMC algorithm, often only called KMC, for simulating the time evolution of a system, where some processes can occur with known rates r, can be written for instance as follows: and numerical comparisons between the algorithms were provided.

The key property of the KMC algorithm (and of the FRM one) is that if the rates are correct, if the processes associated with the rates are of the Poisson process type, and if different processes are independent (i. e. not correlated) then the KMC algorithm gives the correct time scale for the evolution of the simulated system. There was some debate about the correctness of the time scale for rKMC algorithms, but this was also rigorously shown to be correct. in which case detailed balance need not be obeyed. The rfKMC algorithm is efficient in the sense that every iteration is guaranteed to produce a transition. However, in the form presented above it requires operations for each transition, which is not too efficient. In many cases this can be much improved on by binning the same kinds of transitions into bins, and/or forming a tree data structure of the events. A constant time scaling algorithm of this type has recently been developed and tested. The major disadvantage with rfKMC is that all possible rates and reactions have to be known in advance. The method itself can do nothing about predicting them. The rates and reactions must be obtained from other methods, such as diffusion (or other) experiments, molecular dynamics or density functional theory simulations.

The KMC method can be subdivided by how the objects are moving or reactions occurring. At least the following subdivisions are used:

Lattice KMC (LKMC) signifies KMC carried out on an atomic lattice. Often this variety is also called atomistic KMC, (AKMC). A typical example is simulation of vacancy diffusion in alloys, where a vacancy is allowed to jump around the lattice with rates that depend on the local elemental composition. Object KMC (OKMC) means KMC carried out for defects or impurities, which are jumping either in random or lattice specific directions. Only the positions of the jumping objects are included in the simulation, not those of the 'background' lattice atoms. The basic KMC step is one object jump. Event KMC (EKMC) or First passage KMC (FPKMC) signifies an OKMC variety where the following reaction between objects (e. g. clustering of two impurities or vacancy interstitial annihilation) is chosen with the KMC algorithm, taking the object positions into account, and this event is then immediately carried out.