Statistical Mechanics of Two Hard Spheres in a Spherical Pore, Exact Analytic Results in D Dimension

TL;DR

This paper provides exact analytical solutions for the statistical mechanics of two hard spheres confined in a spherical pore across arbitrary dimensions, exploring confinement effects, thermodynamic properties, and relations to many-body systems.

Contribution

It introduces exact analytical expressions for the partition function and distribution functions of two hard spheres in a spherical pore in any dimension, advancing understanding of confined inhomogeneous systems.

Findings

Analytical evaluation of partition functions and distribution functions.

Detailed analysis of high confinement and low density limits.

Derived relations between different confined two-sphere systems.

Abstract

This work is devoted to the exact statistical mechanics treatment of simple inhomogeneous few-body systems. The system of two Hard Spheres (HS) confined in a hard spherical pore is systematically analyzed in terms of its dimensionality >. The canonical partition function, and the one- and two-body distribution functions are analytically evaluated and a scheme of iterative construction of the system properties is presented. We analyse in detail both the effect of high confinement, when particles become caged, and the low density limit. Other confinement situations are also studied analytically and several relations between, the two HS in a spherical pore, two sticked HS in a spherical pore, and two HS on a spherical surface partition functions are traced. These relations make meaningful the limiting caging and low density behavior. Turning to the system of two HS in a spherical pore, we also analytically evaluate the pressure tensor. The thermodynamic properties of the system are discussed. To accomplish this statement we purposely focus in the overall characteristics of the inhomogeneous fluid system, instead of concentrate in the peculiarities of a few body system. Hence, we analyse the equation of state, the pressure at the wall, and the fluid-substrate surface tension. The consequences of new results about the spherically confined system of two HS in D dimension on the confined many HS system are investigated. New constant coefficients involved in the low density limit properties of the open and closed system of many HS in a spherical pore are obtained for arbitrary. The complementary system of many HS which surrounds a hard sphere (a cavity inside of a bulk HS system) is also discussed.