Properties of the Virial Expansion and Equation of State of Ideal Quantum Gases in Arbitrary Dimensions

TL;DR

This paper investigates the properties of the virial expansion for ideal quantum gases across various dimensions, highlighting unique convergence behaviors at specific dimensions and analyzing the underlying singularities affecting the expansion's asymptotic behavior.

Contribution

It provides a detailed analysis of the convergence radius and singularity structure of the virial expansion in arbitrary dimensions, revealing special properties at certain dimensions like d=3.

Findings

Convergence radius peaks sharply at d=3

Singularities govern the asymptotic behavior of the virial expansion

Special properties occur at specific non-integer dimensions

Abstract

The virial expansion of ideal quantum gases reveals some interesting and amusing properties when considered as a function of dimensionality $d$. In particular, the convergence radius $\rho_c(d)$ of the expansion is particulary large at {\em exactly\/} $d=3$ dimensions, $\rho_c(3) = 7.1068\ldots \times \lim_{d\to3} \rho_c(d)$. The same phenomenon occurs in a few other special (non-integer) dimensions. We explain the origin of these facts, and discuss more generally the structure of singularities governing the asymptotic behavior of the ideal gas virial expansion.