Stochastic kinetic Monte Carlo algorithms for long-range Hamiltonians

We present a higher order kinetic Monte Carlo methodology suitable to model the evolution of systems in which the transition rates are non-trivial to calculate or in which Monte Carlo moves are likely to be non-productive flicker events. The second order residence time algorithm first introduced by Athenes et al. [Phil. Mag. A 76 (1997) 565] is rederived from the n-fold way algorithm of Bortz et al. [J. Comput. Phys. 17 (1975) 10] as a fully stochastic algorithm. The second order algorithm can be dynamically called when necessary to eliminate unproductive flickering between a metastable state and its neighbours. An algorithm combining elements of the first order and second order methods is shown to be more efficient, in terms of the number of rate calculations, than the first order or second order methods alone while remaining statistically identical. This efficiency is of prime importance when dealing with computationally expensive rate functions such as those arising from long-range Hamiltonians. Our algorithm has been developed for use when considering simulations of vacancy diffusion under the influence of elastic stress fields. We demonstrate the improved efficiency of the method over that of the n-fold way in simulations of vacancy diffusion in alloys. Our algorithm is seen to be an order of magnitude more efficient than the n-fold way in these simulations. We show that when magnesium is added to an Al-2at.%Cu alloy, this has the effect of trapping vacancies. When trapping occurs, we see that our algorithm performs thousands of events for each rate calculation performed.

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