Simple relationship between the virial-route hypernetted-chain and the compressibility-route Percus--Yevick values of the fourth virial coefficient

TL;DR

This paper reveals a simple, exact relationship between the fourth virial coefficient values obtained from the virial and compressibility routes in hypernetted chain and Percus--Yevick theories, confirmed across various models.

Contribution

It establishes a universal, exact relationship between the virial-route HNC and compressibility-route PY values of B4, independent of interaction potential, components, or dimensionality.

Findings

The virial-route B4 in HNC is 1.5 times the compressibility-route B4 in PY.

The relationship holds for isotropic and non-isotropic interactions.

Confirmed through analytical and numerical models across different systems.

Abstract

As is well known, approximate integral equations for liquids, such as the hypernetted chain (HNC) and Percus--Yevick (PY) theories, are in general thermodynamically inconsistent in the sense that the macroscopic properties obtained from the spatial correlation functions depend on the route followed. In particular, the values of the fourth virial coefficient $B_4$ predicted by the HNC and PY approximations via the virial route differ from those obtained via the compressibility route. Despite this, it is shown in this paper that the value of $B_4$ obtained from the virial route in the HNC theory is exactly three halves the value obtained from the compressibility route in the PY theory, irrespective of the interaction potential (whether isotropic or not), the number of components, and the dimensionality of the system. This simple relationship is confirmed in one-component systems by analytical results for the one-dimensional penetrable-square-well model and the three-dimensional penetrable-sphere model, as well as by numerical results for the one-dimensional Lennard--Jones model, the one-dimensional Gaussian core model, and the three-dimensional square-well model.