Monte Carlo methods in classical statistical physics

The purpose of this chapter is to give a brief introduction to Monte Carlo simulations of classical statistical physics systems and their statistical analysis. To set the general theoretical frame, first some properties of phase transitions and simple models describing them are briefly recalled, before the concept of importance sampling Monte Carlo methods is introduced. The basic idea is illustrated by a few standard local update algorithms (Metropolis, heat-bath, Glauber). Then methods for the statistical analysis of the thus generated data are discussed. Special attention is payed to the choice of estimators, autocorrelation times and statistical error analysis. This is necessary for a quantitative description of the phenomenon of critical slowing down at continuous phase transitions. For illustration purposes, only the two-dimensional Ising model will be needed. To overcome the slowing-down problem, non-local cluster algorithms have been developed which will be described next. Then the general tool of reweighting techniques will be explained which is extremely important for finite-size scaling studies. This will be demonstrated in some detail by the sample study presented in the next section, where also methods for estimating spatial correlation functions will be discussed. The reweighting idea is also important for a deeper understanding of so-called generalized ensemble methods which may be viewed as dynamical reweighting algorithms. After first discussing simulated and parallel tempering methods, finally also the alternative approach using multicanonical ensembles and the Wang-Landau recursion are briefly outlined.