Extension of Kirkwood-Buff Theory to the Canonical Ensemble

TL;DR

This paper introduces a new theory to accurately compute Kirkwood-Buff integrals and direct correlation functions in finite canonical systems, overcoming previous convergence issues and enabling precise calculations in simulations.

Contribution

The paper develops a comprehensive method to determine direct correlation functions and Kirkwood-Buff integrals directly in the canonical ensemble, improving convergence and accuracy over prior approaches.

Findings

The new method accurately computes KB integrals in finite systems.

Finite-size corrections for chemical potential and KB coefficients are derived.

Simulation results on 1D Ising and Lennard-Jones models validate the theory.

Abstract

Kirkwood-Buff (KB) integrals are notoriously difficult to converge from a canonical simulation because they require estimating the grand-canonical radial distribution. The same essential difficulty is encountered when attempting to estimate the direct correlation function of Ornstein-Zernike theory by inverting the pair correlation functions. We present a new theory that applies to the entire, finite, simulation volume, so that no cutoff issues arise at all. The theory gives the direct correlation function for closed systems, while smoothness of the direct correlation function in reciprocal space allows calculating canonical KB integrals via a well-posed extrapolation to the origin. The present analysis method represents an improvement over previous work because it makes use of the entire simulation volume and its convergence can be accelerated using known properties of the direct correlation function. Using known interaction energy functions can make this extrapolation near perfect accuracy in the low-density case. Because finite size effects are stronger in the canonical than the grand-canonical ensemble, we state ensemble correction formulas for the chemical potential and the KB coefficients. The new theory is illustrated with both analytical and simulation results on the 1D Ising model and a supercritical Lennard-Jones fluid. For the latter, the finite-size corrections are shown to be small.