Distribution Functionals for Hard Particles in N Dimensions

TL;DR

This paper develops a comprehensive framework for distribution functionals of hard particles in N dimensions, deriving correlation functionals and analyzing their properties, including exact and approximate solutions for particle correlations.

Contribution

It introduces a novel representation of correlation functionals using rooted and unrooted Mayer diagrams, extending previous mathematical methods to these new forms.

Findings

Derived the generic functional for all r-particle distributions.

Reproduced known solutions for contact probabilities of spheres.

Demonstrated the failure of the Kirkwood superposition approximation.

Abstract

The current article completes our investigation of the hard-particle interaction by determining their distribution functionals. Beginning with a short review of the perturbation expansion of the free-energy functional, we derive two representations of the correlation functionals in rooted and unrooted Mayer diagrams, which are related by a functional derivative. This map allows to transfer the mathematical methods, developed previously for unrooted diagrams, to the current representation in rooted graphs. Translating then the Mayer to Ree-Hoover diagrams and determining their automorphism groups, yields the generic functional for all r-particle distributions. From this we derive the examples of 2- and 3-particle correlations up to four intersection centers and show that already the leading order reproduces the Wertheim, Thiele, Baxter solution for the contact probability of spheres. Another calculation shows the failure of the Kirkwood superposition approximation for any r-particle correlation.