Two Hard Spheres in a Spherical Pore: Exact Analytic Results in Two and Three Dimensions

Abstract

The partition function and the one- and two-body distribution function 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. We 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.