Beyond histograms: efficiently estimating radial distribution functions via spectral Monte Carlo

TL;DR

This paper introduces a spectral Monte Carlo method for estimating radial distribution functions that overcomes limitations of traditional histogram-based approaches, providing more accurate, objective, and efficient results with analytical formulas.

Contribution

The authors propose a novel spectral Monte Carlo method for RDF estimation and a Sobolev norm for quality assessment, improving accuracy and convergence over traditional methods.

Findings

SMC reduces noise in RDF estimates by orders of magnitude.

SMC requires fewer pair separations for convergence.

Provides differentiable formulas useful for force-field calibration.

Abstract

Despite more than 40 years of research in condensed-matter physics, state-of-the-art approaches for simulating the radial distribution function (RDF) g(r) still rely on binning pair-separations into a histogram. Such methods suffer from undesirable properties, including subjectivity, high uncertainty, and slow rates of convergence. Moreover, such problems go undetected by the metrics often used to assess RDFs. To address these issues, we propose (I) a spectral Monte Carlo (SMC) method that yields g(r) as an analytical series expansion; and (II) a Sobolev norm that assesses the quality of RDFs by quantifying their fluctuations. Using the latter, we show that, relative to histogram-based approaches, SMC reduces by orders of magnitude both the noise in g(r) and the number of pair separations needed for acceptable convergence. Moreover, SMC reduces subjectivity and yields simple, differentiable formulas for the RDF, which are useful for tasks such as coarse-grained force-field calibration via iterative Boltzmann inversion.