An introduction to Monte Carlo methods

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 a 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 Worm algorithm. We conclude with an important discussion of the dynamical effects such as thermalization and correlation time.

19 pages, 6 figures, submitted to Physica A (Proceedings of the Summer School FPSPXIII, Leuven, 2013)

References in corpus (6)

• Theory of Phase Ordering Kinetics
• Worm algorithms for classical statistical models
• The Dynamic Exponent of the Two-Dimensional Ising Model and Monte Carlo Computation of the Sub-Dominant Eigenvalue of the Stochastic Matrix
• Extended Scaling in High Dimensions
• Dynamic critical exponents of Swendsen-Wang and Wolff algorithms by nonequilibrium relaxation
• A Study of Dynamic Finite Size Scaling Behavior of the Scaling Functions-Calculation of Dynamic Critical Index of Wolff Algorithm

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