Two hard spheres in a spherical pore: Exact analytic results in two and three dimensions

TL;DR

This paper provides exact analytical solutions for the behavior of two hard spheres in a spherical pore across two and three dimensions, including partition functions and distribution functions, and explores their implications for virial coefficients and density limits.

Contribution

It establishes a general relation between partition functions and virial coefficients for inhomogeneous systems in spherical pores, applicable to both spheres and disks, and evaluates specific cluster integrals.

Findings

Derived exact partition functions and distribution functions for two hard spheres in a spherical pore.

Established a relation connecting inhomogeneous virial coefficients across dimensions.

Analyzed behavior of the system near maximum density.

Abstract

The partition function and the one- and two-body distribution functions are evaluated for two hard spheres with different sizes constrained into a spherical pore. The equivalent problem for hard disks is addressed too. We establish a relation valid for any dimension between these partition functions, second virial coefficient for inhomogeneous systems in a spherical pore, and third virial coefficients for polydisperse hard spheres mixtures. Using the established relation we were able to evaluate the cluster integral $b_{2}(V)$ related with the second virial coefficient for the Hard Disc system into a circular pore. Finally, we analyse the behaviour of the obtained expressions near the maximum density.