Monte Carlo methods in statistical physics: Mathematical foundations and strategies

TL;DR

This pedagogical review explains the probabilistic foundations of Monte Carlo methods in statistical physics, categorizing various algorithms into basic strategies to improve convergence and providing clear formulations of key methods.

Contribution

It offers a unified framework for understanding Monte Carlo algorithms in statistical physics by identifying core strategies and simplifying their formulation.

Findings

Identifies fundamental strategies underlying Monte Carlo algorithms.

Provides a unified formulation for widely used Monte Carlo methods.

Highlights adjustable parameters influencing convergence.

Abstract

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.