Monte Carlo methods in statistical physics: Mathematical foundations and strategies

Monte Carlo is a versatile and frequently used tool in statistical physics and beyond. Correspondingly, the number of algorithms and variants reported in the literature is vast, and an overview is not easy to achieve. In this pedagogical review, we start by presenting the probabilistic concepts which are at the basis of the Monte Carlo method. From these concepts the relevant free parameters—which still may be adjusted—are identified. Having identified these parameters, most of the tangled mass of methods and algorithms in statistical physics Monte Carlo can be regarded as realizations of merely a handful of basic strategies which are employed in order to improve convergence of a Monte Carlo computation. Once the notations introduced are available, many of the most widely used Monte Carlo methods and algorithms can be formulated in a few lines. In such a formulation, the core ideas are exposed and possible generalizations of the methods are less obscured by the details of a particular algorithm.