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An introduction to Monte Carlo methods

Author

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  • Walter, J.-C.
  • Barkema, G.T.

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.

Suggested Citation

  • Walter, J.-C. & Barkema, G.T., 2015. "An introduction to Monte Carlo methods," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 418(C), pages 78-87.
    • Handle: RePEc:eee:phsmap:v:418:y:2015:i:c:p:78-87
    DOI: 10.1016/j.physa.2014.06.014
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    Cited by:

    1. Jiang, Zhong-ming & Feng, De-Cheng & Zhou, Hao & Tao, Wei-Feng, 2021. "A recursive dimension-reduction method for high-dimensional reliability analysis with rare failure event," Reliability Engineering and System Safety, Elsevier, vol. 213(C).
    2. David Yevick, 2022. "Variational autoencoder analysis of Ising model statistical distributions and phase transitions," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 95(3), pages 1-15, March.
    3. Shafi, Khuram & Latif, Natasha & Shad, Shafqat Ali & Idrees, Zahra & Gulzar, Saqib, 2018. "Estimating option greeks under the stochastic volatility using simulation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 1288-1296.

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