Monte Carlo Methods

Systems with a large number of degrees of freedom are often studied in physics. Many-body systems ranging from many electron atoms to a chunk of metal are examples of such systems. The solution of those systems often requires integrals over multi-dimensional spaces, typically 3A dimensions. An example is the partition function for a system at temperature T ($k$ is Boltzman constant and $E$ the energy),

$$Z = \int d^3r_1 ... d^3r_A \exp( - \frac{1}{k T} E( r_1, ..., r_A ) )$$

We will use this function later to solve the Ising Model.

The difficulty with calculating numerically integrals in multi-dimensional spaces is the necessity to compute the integrand a very large number of times. For instance, calculating $Z$ using Trapezoidal or Simpson Rules with only 10 points per degree of freedom implies $10^{ 3 A}$ grid points, and therefore function evaluations. Even for a tiny system with $A = 20$ this corresponds to $10^{ 60}$ grid points!

This section deals with a method of integration which is based on a random of exploration of phase space. It is dubbed a Monte Carlo Integration Method. The name, of course, stems from the random or chance character of the method in connection to the casino in Monte Carlo. The method is based on a representative random sampling of phase space. As such it is analogous to the prediction of the results of an election based on a poll of a small number of voters.