Waste-recycling Monte Carlo with optimal estimates: application to free energy calculations in alloys

TL;DR

This paper evaluates the Delmas-Jourdain waste-recycling Monte Carlo estimator for free energy calculations in alloys, demonstrating near-optimal variance reduction and improved ergodicity in complex alloy systems.

Contribution

It provides a numerical assessment of the Delmas-Jourdain estimator's performance in alloy free energy calculations, highlighting its variance reduction and applicability to complex systems.

Findings

Variance of the estimator is significantly reduced.

Reduction approaches the theoretical maximum despite control variate inaccuracies.

Gradual transmutations improve ergodicity and enable phase coexistence determination.

Abstract

The estimator proposed recently by Delmas and Jourdain for waste-recycling Monte Carlo achieves variance reduction optimally with respect to a control variate that is evaluated directly using the simulation data. Here, the performance of this estimator is assessed numerically for free energy calculations in generic binary alloys and compared to those of other estimators taken from the literature. A systematic investigation with varying simulation parameters of a simplified system, the anti-ferromagnetic Ising model, is first carried out in the transmutation ensemble using path-sampling. We observe numerically that (i) the variance of the Delmas-Jourdain estimator is indeed reduced compared to that of other estimators; and that (ii) the resulting reduction is close to the maximal possible one, despite the inaccuracy in the estimated control variate. More extensive path-sampling simulations involving a FeCr alloy system described by a many-body potential additionally show that (iii) gradual transmutations accommodate the atomic frustrations, thus alleviating the numerical ergodicity issue present in numerous alloy systems and eventually enabling the determination of phase coexistence conditions.