Rigid Disks and Spheres at High Densities: Bounds on the Partition Function

Abstract

Upper and lower bounds on the partition functions for finite systems of N rigid disks and N rigid spheres are obtained for the high‐density limit. The lower bounds are also valid in the thermodynamic limit of N→∞ and at all densities. Consider the following asymptotic form for the specific Helmholtz free energy for a classical mechanical system of N ν‐dimensional (ν=2 or 3) hard spheres confined in a volume V in the limit V→V₀ (the close‐packed volume), $$A_N{\mathrm{/Nk}}_B\mathrm{T\cong \nu }\ln \mathrm{(\Lambda /\sigma )-\nu }\ln {\text{[1−(V}}_0{\mathrm{/V)]+C}}_{\nu }\mathrm{(N),}$$ where Λ is the de Broglie wavelength and σ the sphere diameter. The lower bounds obtained for C_ν(N) are C₂(N) > −0.7576··· and C₃(N) > −0.05074···. Two upper bounds found for C₂ are C₂<1.530··· and C₂<1.289···. An upper bound for C₃ is 3.6423···.