On Virial Expansion in Hard Sphere Model

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.