Virial series for fluids of hard hyperspheres in odd dimensions

TL;DR

This paper investigates the convergence of the virial series for fluids of hard hyperspheres in odd dimensions using an exact solution of the Percus-Yevick equation, confirming its alternating nature and determining the radius of convergence.

Contribution

It extends the analysis of virial series convergence for hard hypersphere fluids to higher odd dimensions and explicitly calculates the radius of convergence for each dimension.

Findings

Confirmed the alternating character of the virial series for d≥5.

Explicitly determined the radius of convergence for each odd dimension up to 13.

Validated the limiting behavior of scaled density as d approaches infinity.

Abstract

A recently derived method [R. D. Rohrmann and A. Santos, Phys. Rev. E. {\bf 76}, 051202 (2007)] to obtain the exact solution of the Percus-Yevick equation for a fluid of hard spheres in (odd) $d$ dimensions is used to investigate the convergence properties of the resulting virial series. This is done both for the virial and compressibility routes, in which the virial coefficients $B_j$ are expressed in terms of the solution of a set of $(d-1)/2$ coupled algebraic equations which become nonlinear for $d \geq 5$. Results have been derived up to $d=13$. A confirmation of the alternating character of the series for $d\geq 5$, due to the existence of a branch point on the negative real axis, is found and the radius of convergence is explicitly determined for each dimension. The resulting scaled density per dimension $2 \eta^{1/d}$, where $\eta$ is the packing fraction, is wholly consistent with the limiting value of 1 for $d \to \infty$. Finally, the values for $B_j$ predicted by the virial and compressibility routes in the Percus-Yevick approximation are compared with the known exact values [N. Clisby and B. M. McCoy, J. Stat. Phys. {\bf 122}, 15 (2006)]