# Virial inversion and density functionals

## TL;DR

This paper introduces a new mathematical inversion theorem for functionals in infinite-dimensional spaces, with applications to density function theory and improved convergence estimates for the virial expansion in statistical mechanics.

## Contribution

It develops a novel inversion method using fixed point equations and combinatorial identities, enhancing convergence analysis for density functionals and the virial expansion.

## Key findings

- Proves a new inversion theorem for power series functionals.
- Provides rigorous convergence framework for inhomogeneous systems.
- Achieves improved radius of convergence for the virial expansion of the hard sphere gas.

## Abstract

We prove a novel inversion theorem for functionals given as power series in infinite-dimensional spaces and apply it to the inversion of the density-activity relation for inhomogeneous systems. This provides a rigorous framework to prove convergence for density functionals for inhomogeneous systems with applications in classical density function theory, liquid crystals, molecules with various shapes or other internal degrees of freedom. The key technical tool is the representation of the inverse via a fixed point equation and a combinatorial identity for trees, which allows us to obtain convergence estimates in situations where Banach inversion fails. Moreover, the new method for the inversion gives for the (homogeneous) hard sphere gas a significantly improved radius of convergence for the virial expansion improving the first and up to now best result by Lebowitz and Penrose (1964).