On Virial Expansion in Hard Sphere Model

TL;DR

This paper analyzes the virial expansion of the hard-sphere model, fitting known coefficients to a polylogarithm function, and finds no evidence of phase transition within the physical parameter space, highlighting limitations of the virial series.

Contribution

It provides a detailed analysis of the virial coefficients up to 12th order and explores their asymptotic behavior, offering insights into the convergence and physical implications of the virial expansion.

Findings

Virial coefficients fit well to a polylogarithm function with exponent 1.90.

No singular behavior or phase transition indicated within eta ≤ 0.74.

Asymptotic behavior of cluster coefficients resembles large dimension results.

Abstract

Virial expansion is a traditional approach in statistical mechanics that expresses thermodynamic quantities, such as pressure $p$, as power series of density or chemical potential. Its radius of convergence can serve as a potential indicator of phase transition. In this study, we investigate the virial expansion of the hard-sphere model, using the known dimensionless virial coefficients $\tilde{B}_k^{}~(k=1,2,\cdots)$ up to the $12$th order. We find that it is well fitted by $\tilde{B}_k^{}=1.28\times k^{1.90}$, corresponding to the analytic continuation of the virial expansion of the pressure as $\sim \mathrm{Li}_{-1.90}^{}(\eta)$, where $\eta$ is the packing fraction and $\mathrm{Li}_s^{}(x)$ is the polylogarithm function. This implies the absence of singular behavior in the physical parameter space $\eta\leq \eta_{\mathrm{max}}^{}\approx 0.74$ and no indication of phase transition in the virial expansion approach. In addition, we calculate the cluster-integral coefficients $\{b_l^{}\}_{l=1}^\infty$ and observe that their asymptotic behavior resembles the results obtained in the large dimension limit ($D\rightarrow \infty$), suggesting that $D=3$ might be already regarded as large dimension. However, the existence of phase transition in the hard-sphere model has been confirmed by numerous simulations, which clearly indicates that a naive extrapolation of the virial series can lead to unphysical results.