# Random Close Packing and the Hard Sphere Percus-Yevick Theory

## TL;DR

This paper investigates the Percus-Yevick theory for hard spheres across different dimensions, revealing a critical density where the pair distribution function becomes negative, aligning with the onset of random close packing.

## Contribution

It identifies a critical volume fraction where the pair distribution function turns negative, linking it to the random close packing transition across dimensions.

## Key findings

- Pair distribution function becomes negative at a critical density.
- Critical densities align with random close packing thresholds.
- Supports implications for systems modeled by Percus-Yevick theory.

## Abstract

The Percus-Yevick theory for monodisperse hard spheres gives very good results for the pressure and structure factor of the system in a whole range of densities that lie within the liquid phase. However, the equation seems to lead to a very unacceptable result beyond that region. Namely, the Percus-Yevick theory predicts a smooth behavior of the pressure that diverges only when the volume fraction $\eta$ approaches unity. Thus, within the theory there seems to be no indication for the termination of the liquid phase and the transition to a solid or to a glass. In the present article we study the Percus-Yevick hard sphere pair distribution function, $g_2(r)$, for various spatial dimensions. We find that beyond a certain critical volume fraction $\eta_c$, the pair distribution function, $g_2(r)$, which should be positive definite, becomes negative at some distances. We also present an intriguing observation that the critical $\eta_c$ values we find are consistent with volume fractions where onsets of random close packing (or maximally random jammed states) are reported in the literature for various dimensions. That observation is supported by an intuitive argument. This work may have important implications for other systems for which a Percus-Yevick theory exists.