Simulation of order transitions on a parallel computer

The traditional Monte Carlo simulation of order transitions in a ferromagnet (or in a binary crystalline alloy) is carried out by generating a sequence of local swaps (spin reversals or exchanges of atoms) on a finite lattice, each of which in turn is accepted or rejected according to probabilities prescribed by Metropolis et al., and thus creating a sample from the limit distribution of the resulting Markov chain. The limit can, however, take a long time to reach, especially near critical temperatures, if the lattice used is large enough to mask boundary effects. Since the probability is fairly small that successive swaps in the sequence involve directly interacting lattice sites, it makes sense to save time by performing them simultaneously where possible, by running the simulation on a parallel computer. This report discusses, at an elementary level, some of the theoretical and practical questions that can arise in doing so.