Constructing smooth potentials of mean force, radial, distribution functions and probability densities from sampled data

TL;DR

This paper introduces a general, histogram-free method for constructing smooth, analytical estimates of probability densities, radial distribution functions, and potentials of mean force from sampled data, improving accuracy and robustness.

Contribution

It extends the Berg-Harris method to better handle radial distribution functions and potentials of mean force, addressing issues with sampling, Jacobian factors, and oscillatory fits.

Findings

Resampled, piecewise analytical fits outperform histograms.

The method effectively handles discontinuous densities and long tails.

Application to water models and diffusing particles demonstrates robustness.

Abstract

In this paper a method of obtaining smooth analytical estimates of probability densities, radial distribution functions and potentials of mean force from sampled data in a statistically controlled fashion is presented. The approach is general and can be applied to any density of a single random variable. The method outlined here avoids the use of histograms, which require the specification of a physical parameter (bin size) and tend to give noisy results. The technique is an extension of the Berg-Harris method [B.A. Berg and R.C. Harris, Comp. Phys. Comm. 179, 443 (2008)], which is typically inaccurate for radial distribution functions and potentials of mean force due to a non-uniform Jacobian factor. In addition, the standard method often requires a large number of Fourier modes to represent radial distribution functions, which tends to lead to oscillatory fits. It is shown that the issues of poor sampling due to a Jacobian factor can be resolved using a biased resampling scheme, while the requirement of a large number of Fourier modes is mitigated through an automated piecewise construction approach. The method is demonstrated by analyzing the radial distribution functions in an energy-discretized water model. In addition, the fitting procedure is illustrated on three more applications for which the original Berg-Harris method is not suitable, namely, a random variable with a discontinuous probability density, a density with long tails, and the distribution of the first arrival times of a diffusing particle to a sphere, which has both long tails and short-time structure. In all cases, the resampled, piecewise analytical fit outperforms the histogram and the original Berg-Harris method.