Monte Carlo Simulation

So what does a computer simulation technique have in common with the world famous Monte Carlo casino, in the second smallest country in the world, the principality of Monaco? - The element of chance is used in both, so that the desired result occurs in the long run. The random nature of a game of chance is designed so that the owners of the casino can be assured, that in the long run, the casino will make a profit while the individual gambler has a reasonable chance of winning. The random nature of a Monte Carlo simulation is designed so that the programmer can be assured, that in the long run, the simulation will approach equilibrium values, while an individual move has a realistic chance of taking the simulation away from equilibrium.

Monte Carlo simulation methods typically use (pseudo) random number generators to generate the element of chance. Similar techniques are also used in electronic games of chance such as draw poker machines and one armed bandits.

The type of Monte Carlo simulation we will be discussing here is the type used to obtain material properties given an inter-atomic potential (such as the ubiquitous Lennard- Jones 6-12 potential). The properties that are calculable are those which can be expressed statistical mechanically as a weighted average over the configuration space of an ensemble of systems. The specific Monte Carlo method can be generalized to study any material for which a set of potential functions are available.

The following discussion is aimed at students with an elementary knowledge of statistical mechanics and some knowledge of at least one programming language. No specific knowledge of any specific programming language is expected.