# Many-body correlations from integral geometry

## TL;DR

This paper develops a morphometric approach based on integral geometry to predict many-body correlations in hard sphere liquids, extending scaled particle theory and validating with simulations.

## Contribution

It introduces a new morphometric theory for many-body correlations in hard sphere liquids, generalizing scaled particle theory and demonstrating its effectiveness through numerical tests.

## Key findings

- New morphometric theory for many-body correlations
- Effective in predicting local structures in liquids
- Validated against simulation data

## Abstract

In a recent letter we presented a framework for predicting the concentrations of many-particle local structures inside the bulk liquid as a route to assessing changes in the liquid approaching dynamical arrest. Central to this framework was the morphometric approach, a synthesis of integral geometry and liquid state theory, which has traditionally been derived from fundamental measure theory. We present the morphometric approach in a new context as a generalisation of scaled particle theory, and derive several morphometric theories for hard spheres of fundamental and practical interest. Our central result is a new theory which is particularly suited to the treatment of many-body correlation functions in the hard sphere liquid, which we demonstrate by numerical tests against simulation.