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5 - More on importance sampling Monte Carlo methods for lattice systems

Published online by Cambridge University Press:  24 November 2021

David Landau  and
Kurt Binder
David Landau
Affiliation:
University of Georgia
Kurt Binder
Affiliation:
Johannes Gutenberg Universität Mainz, Germany
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Summary

Advances in simulational methods sometimes have their origin in unusual places; such is the case with an entire class of methods which attempt to beat critical slowing down in spin models on lattices by flipping correlated clusters of spins in an intelligent way instead of simply attempting single spin-flips. The first steps were taken by Fortuin and Kasteleyn (Kasteleyn and Fortuin, 1969; Fortuin and Kasteleyn, 1972), who showed that it was possible to map a ferromagnetic Potts model onto a corresponding percolation model. The reason that this observation is so important is that in the percolation problem states are produced by throwing down particles, or bonds, in an uncorrelated fashion; hence there is no critical slowing down. In contrast, as we have already mentioned, the q-state Potts model when treated using standard Monte Carlo methods suffers from slowing down. (Even for large q where the transition is first order, the time scales can become quite long.) The Fortuin–Kasteleyn transformation thus allows us to map a problem with slow critical relaxation into one where such effects are largely absent. (As we shall see, not all slowing down is eliminated, but the problem is reduced quite dramatically.)

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A Guide to Monte Carlo Simulations in Statistical Physics , pp. 161 - 242
Publisher: Cambridge University Press
Print publication year: 2021

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