Asymptotic expression of the virial coefficients for hard sphere systems

TL;DR

This paper analyzes the asymptotic behavior of high-order virial coefficients in hard sphere systems, revealing their geometric series nature and deriving explicit equations of state at high densities, with implications for phase transition points.

Contribution

It provides an explicit asymptotic expression for virial coefficients and the equation of state at high densities, linking theoretical predictions with numerical simulation data.

Findings

Virial coefficients follow a geometric series at high order

Explicit equation of state with a simple pole at close packing density

Estimates of freezing and melting densities consistent with simulations

Abstract

We evidence via a computation in the reciprocal space the asymptotic behaviour of the high order virial coefficients for a hard sphere system. These coefficients, if their order is high enough, are those of a geometric series. We thus are able to give an explicit expression of the equation of states of the hard sphere system at high density when the fluid phase is no longer the stable one; in the disordered phase this equation of states exhibits a simple pole at the random close packing density. We can then estimate the packing densities of the freezing point of the disordered phase and also of the melting point of the fcc ordered phase. The results are compared with those of the numerical simulations.