Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions

TL;DR

This paper computes high-order virial coefficients for hard spheres across multiple dimensions, revealing sign changes that suggest the series' singularity lies off the positive real axis in higher dimensions.

Contribution

It provides the first extensive calculation of virial coefficients up to order 10 for hard spheres in dimensions 2 to 8, highlighting sign patterns and their implications.

Findings

Virial coefficients with even k are negative for D>=5.

Evidence suggests the virial series' singularity is off the positive real axis in higher dimensions.

Negative virial coefficients are likely for some k>10 in D=4 and possibly in D=3.

Abstract

We evaluate the virial coefficients B_k for k<=10 for hard spheres in dimensions D=2,...,8. Virial coefficients with k even are found to be negative when D>=5. This provides strong evidence that the leading singularity for the virial series lies away from the positive real axis when D>=5. Further analysis provides evidence that negative virial coefficients will be seen for some k>10 for D=4, and there is a distinct possibility that negative virial coefficients will also eventually occur for D=3.