5 Introduction to Simulation Techniques

These lectures give an introduction to Monte Carlo simulations of classical statistical physics systems and their statistical analysis. After briefly recalling a few elementary properties of phase transitions, the concept of importance sampling Monte Carlo methods is discussed and illustrated by a few standard local update algorithms (Metropolis, heat-bath, Glauber). Then emphasis is placed on thorough analyses of the generated data paying special attention to the choice of estimators, autocorrelation times and statistical error analysis. This leads to 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 discussed next. Then the general tool of reweighting techniques will be explained. This paves the way to introduce simulated and parallel tempering methods which are very useful for simulations of complex, possibly disordered systems. Finally, also the important alternative approach using multicanonical ensembles is briefly outlined.