In Chapter 8 we saw how a classical many-particle system can be simulated by the MD method, in which the equations of motion are solved for all the particles involved. This enables us to calculate statistical averages of static and dynamic physical quantities. There exists another method, called the Monte Carlo (MC) method, for simulating classical many-particle systems by introducing artificial dynamics based on ‘random’ numbers. The artificial dynamics used in the MC method prevent us from using it for determining dynamical physical properties in most cases, but for static properties it is very popular.
In fact, every numerical technique in which random numbers play an essential role can be called a ‘Monte Carlo’ method after the famous Mediterranean casino town, and we shall discuss the method not only as a tool for studying classical many-particle systems, but also as a way of dealing with the more general problem of calculating high-dimensional integrals. In fact, three main types of Monte Carlo simulations can be distinguished:
Direct Monte Carlo, in which random numbers are used to model the effect of complicated processes, the details of which are not crucial. An example is the modelling of traffic where the behaviour of cars is determined in part by random numbers.
Monte Carlo integration, which is a method for calculating integrals using random numbers. This method is efficient when the integration is over high-dimensional volumes (see below). […]