Introduction to Computational Materials Science Fundamentals to Applications

In Chapter 7 we introduced the Monte Carlo method, with the focus being on calculating equilibrium properties based on sampling of the degrees of freedom in a Hamiltonian (energy) function. In this chapter, the focus is on the Monte Carlo method applied to rates. We shall see that for certain classes of problems, we can find an association between the Monte Carlo “time” and actual time, opening the door to a new class of simulation methods that can model time-dependent processes at time scales far beyond what is possible with standard molecular dynamics.

The fundamental input to the kinetic Monte Carlo method is a list of possible events such as a jump from one site to another in a diffusion problem, a chemical reaction, etc. Associated with each event will be a rate, which will be related to a probability that the event will occur. An understanding of rates is thus very important, so in Appendix G.8 we give a brief review of kinetic rate theory.

A number of researchers independently developed what has come to be known as the kinetic Monte Carlo method. The N-fold way as a methodology for accelerating the simulations of the Ising model is probably the first example [38], which will be discussed in Chapter 10. Voter introduced a similar approach as a way to study the dynamics of cluster diffusion on surfaces [327]. We will discuss his calculation later in this chapter as one of two examples of the complexities, and limitations, of the kinetic Monte Carlo approach.

THE KINETIC MONTE CARLO METHOD

Consider a system whose properties are dominated by thermally activated processes, such as diffusion.