Efficiency Comparison of Single- and Multiple-Macrostate Grand Canonical Ensemble Transition-Matrix Monte Carlo Simulations

Recent interest in parallelizing flat-histogram transition-matrix Monte Carlo simulations in the grand canonical ensemble, due to its demonstrated effectiveness in studying phase behavior, self-assembly and adsorption, has led to the most extreme case of single-macrostate simulations, where each macrostate is simulated independently with ghost particle insertions and deletions. Despite their use in several studies, no efficiency comparisons of these single-macrostate simulations have been made with multiple-macrostate simulations. We show that multiple-macrostate simulations are up to 3 orders of magnitude more efficient than single-macrostate simulations, which demonstrates the remarkable efficiency of flat-histogram biased insertions and deletions, even with low acceptance probabilities. Efficiency comparisons were made for supercritical fluids and vapor-liquid equilibrium of bulk Lennard-Jones and a three-site water model, self-assembling patchy trimer particles and adsorption of a Lennard-Jones fluid confined in a purely repulsive porous network, using the open source simulation toolkit FEASST. By directly comparing with a variety of Monte Carlo trial move sets, this efficiency loss in single-macrostate simulations is attributed to three related reasons. First, ghost particle insertions and deletions in single-macrostate simulations incur the same computational expense as grand canonical ensemble trials in multiple-macrostate simulations, yet ghost trials do not reap the sampling benefit from propagating the Markov chain to a new microstate. Second, single-macrostate simulations lack macrostate change trials that are biased by the self-consistently converging relative macrostate probability, which is a major component of flat histogram simulations. Third, limiting a Markov chain to a single macrostate reduces sampling possibilities. Existing parallelization methods for multiple-macrostate flat-histogram simulations are shown to be more efficient than parallel single-macrostate simulations by approximately an order of magnitude or more in all systems investigated.