A fast and rigorous numerical tool to measure length-scale artifacts in molecular simulations

TL;DR

This work addresses finite-size effects in molecular simulations by deriving a rigorous, a priori criterion based on the two-sided Bogoliubov inequality to bound the interface free energy $\Delta F$ and define a thermodynamic quality factor $q$. For systems with two-body interactions and a known radial distribution function, the relevant six-dimensional integrals reduce to tractable computations, and the authors implement four numerical schemes (Riemann, improved Riemann, probability, and Monte Carlo) to evaluate $\Delta F$ and $q$ efficiently. They validate the approach on a Lennard-Jones binary mixture, demonstrating consistency with prior simulation results and showing that the required computation time is only minutes on a standard machine, enabling an a priori assessment of box size for bulk-property fidelity. The method provides a robust, fluctuation-aware criterion that complements traditional structure-based checks, with practical implications for designing thermodynamically reliable simulations and informing choices of system size in solvation and free-energy contexts.

Abstract

The two-sided Bogoliubov inequality for classical and quantum many-body systems is a theorem that provides rigorous bounds on the free-energy cost of partitioning a given system into two or more independent subsystems. This theorem motivates the definition of a quality factor which directly quantifies the degree of statistical-mechanical consistency achieved by a given simulation box size. A major technical merit of the theorem is that, for systems with two-body interactions and a known radial distribution function, the quality factor can be computed by evaluating just two six-dimensional integrals. In this work, we present a numerical algorithm for computing the quality factor and demonstrate its consistency with respect to results in the literature obtained from simulations performed at different box sizes.

Figures (6)

• Figure 1: Probability density functions $p_D(r)$ and $q_D(r)$ for the distance between two uniformly distributed points inside a unit cube (solid line), and between two uniformly distributed points in two halves of a unit cube (dotted line).
• Figure 2: Results for $q_\mathrm{max}$ obtained via the Riemann method, the improved Riemann method, and the probability method (the latter with direct one-dimensional integration with Riemann approach) as a function of the discretization step $n$ in one dimension for a system of $M = 50$ particles.
• Figure 3: Results for $q_\mathrm{max}$ obtained via the Monte Carlo method as a function of the number of sampled points $N$ for a system of $M = 50$ particles. The error bars have been computed using \ref{['MCvar']}.
• Figure 4: Quality factors $q_\mathrm{max}$ and $q_\mathrm{min}$ on a log-log plot as a function of the number of particles. The results are obtained using the probability method, however, all the four integration methods give similar results.
• Figure 5: The runtime and relative error for different runs and different algorithms. These calculation were performed on a desktop machine using an AMD Ryzen 7 9800X3D (2024) processor.

Theorems & Definitions (8)

• Theorem 1: Two-sided Bogoliubov inequality
• Remark 2
• Remark 3
• Remark 4
• Lemma 5
• proof
• Lemma 6
• proof