An introduction to Monte Carlo methods

J. -C. Walter*, G. T. Barkema

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

Monte Carlo simulations are methods for simulating statistical systems. The aim is to generate a representative ensemble of configurations to access thermodynamical quantities without the need to solve the system analytically or to perform an exact enumeration. The main principles of Monte Carlo simulations are ergodicity and detailed balance. The Ising model is a lattice spin system with nearest neighbor interactions that is appropriate to illustrate different examples of Monte Carlo simulations. It displays a second order phase transition between disordered (high temperature) and ordered (low temperature) phases, leading to different strategies of simulations. The Metropolis algorithm and the Glauber dynamics are efficient at high temperature. Close to the critical temperature, where the spins display long range correlations, cluster algorithms are more efficient. We introduce the rejection free (or continuous time) algorithm and describe in details an interesting alternative representation of the Ising model using graphs instead of spins with the so-called Worm algorithm. We conclude with an important discussion of the dynamical effects such as thermalization and correlation time. (C) 2014 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)78-87
Number of pages10
JournalPhysica. A, theoretical and statistical physics
Volume418
DOIs
Publication statusPublished - 15 Jan 2015

Funding

J-CW thanks Christophe Chatelain for his careful reading of the manuscript and the various collaborations that have largely inspired these notes. We also thank Raoul Schram for stimulating discussions and the reading of the manuscript. J-CW is supported by the Laboratory of Excellence Initiative (Labex) NUMEV, OD by the Scientific Council of the University of Montpellier 2. This work is part of the D-ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).

Keywords

  • Monte Carlo simulations
  • Ising model
  • Algorithms
  • COUPLED CHEMICAL-REACTIONS
  • SPIN SYSTEMS
  • ISING-MODEL
  • SIMULATION
  • ALGORITHM

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