Finite-size estimates of Kirkwood-Buff and similar integrals

TL;DR

This paper extends a method for approximating improper integrals of oscillatory functions to arbitrary dimensions, demonstrating that choosing a non-physical embedding dimension can significantly improve the accuracy of finite-size integral estimates for Kirkwood-Buff and related integrals.

Contribution

The paper generalizes a recent integral approximation method to any embedding dimension, showing that non-physical dimensions can enhance accuracy in estimating integrals in statistical mechanics.

Findings

Choosing an embedding dimension different from the physical dimension reduces approximation error.

The method effectively estimates Kirkwood-Buff integrals and static structure factors.

Higher or non-physical dimensions can improve integral approximation accuracy.

Abstract

Recently, Kr\"uger and Vlugt [Phys. Rev. E 97, 051301(R) (2018)] have proposed a method to approximate an improper integral $\int_0^\infty \text{d}r\, F(r)$, where $F(r)$ is a given oscillatory function, by a finite-range integral $\int_0^L \text{d}r\, F(r) W(r/L)$ with an appropriate weight function $W(x)$. The method is extended here to an arbitrary (embedding) dimensionality $d$. A study of three-dimensional Kirkwood-Buff integrals, where $F(r)=4\pi r^2h(r)$, and static structure factors, where $F(r)=(4\pi/q) r\sin(qr) h(r)$, $h(r)$ being the pair correlation function, shows that, in general, a choice $d\neq 3$ (e.g., $d=7$) for the embedding dimensionality may significantly reduce the error of the approximation $\int_0^\infty \text{d}r\, F(r)\simeq \int_0^L \text{d}r\, F(r) W(r/L)$.