Monte Carlo Simulations in Statistical Physics -- From Basic Principles to Advanced Applications

This chapter starts with an overview of Monte Carlo computer simulation methodologies which are illustrated for the simple case of the Ising model. After reviewing importance sampling schemes based on Markov chains and standard local update rules (Metropolis, Glauber, heat-bath), nonlocal cluster-update algorithms are explained which drastically reduce the problem of critical slowing down at second-order phase transitions and thus improve the performance of simulations. How this can be quantified is explained in the section on statistical error analyses of simulation data including the effect of temporal correlations and autocorrelation times. Histogram reweighting methods are explained in the next section. Eventually, more advanced generalized ensemble methods (simulated and parallel tempering, multicanonical ensemble, Wang-Landau method) are discussed which are particularly important for simulations of first-order phase transitions and, in general, of systems with rare-event states. The setup of scaling and finite-size scaling analyses is the content of the following section. The chapter concludes with two advanced applications to complex physical systems. The first example deals with a quenched, diluted ferromagnet, and in the second application we consider the adsorption properties of macromolecules such as polymers and proteins to solid substrates. Such systems often require especially tailored algorithms for their efficient and successful simulation.