Deriving the Rosenfeld Functional from the Virial Expansion

TL;DR

This paper systematically derives the Rosenfeld functional for hard convex particles from Mayer cluster expansions, connecting it to intersection probabilities and extending the virial expansion by loop order.

Contribution

It provides a rigorous derivation of the Rosenfeld functional from first principles using Mayer clusters and intersection diagrams, clarifying its mathematical structure.

Findings

0-loop order reproduces Rosenfeld functional exactly

Intersection probabilities derived from generalized Blaschke, Santalo, and Chern equations

Particle geometry influences the expansion's structure

Abstract

In this article we replace the semi-heuristic derivation of the Rosenfeld functional of hard convex particles with the systematic calculation of Mayer clusters. It is shown that each cluster integral further decomposes into diagrams of intersection patterns that we classify by their loop number. This extends the virial expansion of the free-energy by an expansion in the loop order, with the Rosenfeld functional as its leading contribution. Rosenfeld's weight functions follow then from the derivation of the intersection probability by generalizing the equation of Blaschke, Santalo, and Chern. It is found that the 0-loop order can be derived exactly and reproduces the Rosenfeld functional. We further discuss the influence of particle dimensions, topologies, and geometries on the mathematical structure of the calculation.