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

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. In simulations of condensed matter systems, one can barely overstate the importance of the radial distribution function (RDF) g(r). To name only a few applications, g(r) is used to (i) link thermodynamic properties to microscopic details [1-3]; (ii) compute structure factors for comparison with X-ray diffraction [4,5]; and more recently, (iii) calibrate interparticle forces for coarse-grained (CG) molecular dynamics (MD) [6][7][8][9][10][11]. Indeed, the RDF is such a key property that in the past few years, much work has been devoted to estimating g(r) via parallel processing on GPUs [12]. Given these observations, it is thus surprising that state-of-the-art techniques still construct g(r) by binning simulated pair-separations into histograms, with little thought given to developing more efficient methods [3,13].In this letter, we address this issue by proposing a spectral Monte Carlo (SMC) method for computing simulated RDFs. The key idea behind our approach is to express g(r) in an appropriate basis set and determine the mode coefficients via Monte Carlo estimates. Relative to binning, we show that this approach decreases subjectivity of the analysis, thereby reducing both the noise in g(r) and the number of pair separations needed to generate useful RDFs. To support these claims, we also discuss how traditional L 2 (or sum-of-squares) metrics are insufficient for assessing convergence of g(r) and propose a Sobolev norm [14] as an appropriate alternative.The motivation for this work stems from the fact that g(r) is increasingly being used in settings in which the details of its functional form play a critical role. For example, scientists now routinely simulate untested materials in an effort to tailor their structural properties without the need for expensive experiments [15,16]; in such applications, objectively computing RDFs is a key task. Along * paul.patrone@nist.gov related lines, structural properties are increasingly being used to calibrate coarse-grained force-fields [6][7][8][9][10][...

Classification Under Uncertainty: Data Analysis for Diagnostic Antibody Testing

“…In particular, we require that the probability densities P (r) and N (r) be known. In previous works [13], we have developed methods for objectively reconstructing PDFs with high-fidelity given many measurements, as may be available when testing large portions of a population. However, in general (and especially at the beginning of an emerging outbreak), there may not be sufficient data reconstruct P (r) and N (r) without empirical assumptions (e.g.…”

“…In particular, limited training data leads to situations in which we must empirically determine parameterized distributions of measurement outcomes associated with positive or negative samples. However, this problem diminishes with increasing amounts of data and, in the case of testing at a nation-wide scale, likely becomes negligible [13]. 4 Moreover, all classification schemes contain subjective elements, so that our analysis is not unique in this regard.…”

“…In particular, large amounts of empirical data can be used to accurately reproduce probability density functions using spectral methods; see, e.g. Ref [13]…”

The role of data analysis in uncertainty quantification: case studies for materials modeling

“…[The spectral Monte Carlo (SMC) estimate of the probability density is discussed in Ref. 33.] Note that the range of plausible T g values associated with this method spans roughly 150 K, which is too large for practical purposes.…”

“…Spread in T g value computed according to the method in figure4when iterated 50,000 times. [The spectral Monte Carlo (SMC) estimate of the probability density is discussed in Ref 33…”