Multilevel simulation of hard-sphere mixtures

TL;DR

This paper introduces a multilevel Monte Carlo simulation approach for multi-scale physical systems, specifically applied to size-asymmetric hard-sphere mixtures, demonstrating improved estimator performance with an added hierarchical level.

Contribution

The paper develops a three-level multilevel Monte Carlo method that enhances simulation efficiency for complex mixtures by interpolating between detailed and coarse-grained models.

Findings

The three-level method outperforms the two-level version in estimator efficiency.

The approach achieves numerically-exact results at the most detailed level.

Analysis of asymptotic variance explains the improved performance.

Abstract

We present a multilevel Monte Carlo simulation method for analysing multi-scale physical systems via a hierarchy of coarse-grained representations, to obtain numerically-exact results, at the most detailed level. We apply the method to a mixture of size-asymmetric hard spheres, in the grand canonical ensemble. A three-level version of the method is compared with a previously-studied two-level version. The extra level interpolates between the full mixture and a coarse-grained description where only the large particles are present -- this is achieved by restricting the small particles to regions close to the large ones. The three-level method improves the performance of the estimator, at fixed computational cost. We analyse the asymptotic variance of the estimator, and discuss the mechanisms for the improved performance.