10 The Monte Carlo method
10.1 Introduction
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.^1 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). - Metropolis Monte Carlo, in which a sequence of distributions of a system is
generated in a so-called Markov chain. This method allows us to study the static
properties of classical and quantum many-particle systems. The latter will be
discussed in Chapter 12.
(^1) As explained inAppendix B, computer-generated random numbers are not truly random, hence the quotes.
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