Monte Carlo Method in Statistical Physics - Overview

Overview

The general motivation to use the Monte Carlo method in statistical physics is to evaluate a multivariable integral. The typical problem begins with a system of which the Hamiltonian is known, it is at a given temperature and it follows the Boltzmann statistics. To obtain the mean value of some macroscopic variable, say A, the general approach is to compute, over all the phase space, PS for simplicity, the mean value of A using the Boltzmann distribution:

.

where is the energy of the system for a given state defined by - a vector with all the degrees of freedom (for instance, for a mechanical system, ), and

is the partition function.

One possible approach to solve this multivariable integral is to exactly enumerate all possible configurations of the system, and calculate averages at will. This is actually done in exactly solvable systems, and in simulations of simple systems with few particles. In realistic systems, on the other hand, even an exact enumeration can be difficult to implement.

For those systems, the Monte Carlo integration (and not to be confused with Monte Carlo method, which is used to simulate molecular chains) is generally employed. The main motivation for its use is the fact that, with the Monte Carlo integration, the error goes as, independently of the dimension of the integral. Another important concept related to the Monte Carlo integration is the importance sampling, a technique that improves the computational time of the simulation.

On the following sections, the general implementation of the Monte Carlo integration for solving this kind of problems is discussed.