Solution of the Percus-Yevick equation for hard hyperspheres in even dimensions

TL;DR

This paper solves the Percus-Yevick equation for hard hyperspheres in even dimensions, providing numerical results for pair correlation functions and virial coefficients, and analyzing the series' convergence properties.

Contribution

It generalizes a previous approach to even dimensions, computes key fluid properties, and reveals the nature of virial series convergence in higher dimensions.

Findings

Confirmed the alternating character of the virial series for d ≥ 6.

Provided first evidence of alternating series for d=4.

Identified the branch point on the negative real axis as the cause of sign alternation.

Abstract

We solve the Percus-Yevick equation in even dimensions by reducing it to a set of simple integro-differential equations. This work generalizes an approach we developed previously for hard discs. We numerically obtain both the pair correlation function and the virial coefficients for a fluid of hyper-spheres in dimensions $d=4,6$ and 8, and find good agreement with available exact results and Monte-Carlo simulations. This paper confirms the alternating character of the virial series for $d \ge 6$, and provides the first evidence for an alternating character for $d=4$. Moreover, we show that this sign alternation is due to the existence of a branch point on the negative real axis. It is this branch point that determines the radius of convergence of the virial series, whose value we determine explicitly for $d=4,6,8$. Our results complement, and are consistent with, a recent study in odd dimensions [R.D. Rohrmann et al., J. Chem. Phys. 129, 014510 (2008)].