Towards a unification of HRT and SCOZA

TL;DR

This paper explores unifying two liquid state theories, HRT and SCOZA, by combining their key features through analytical and numerical methods, aiming for a consistent and comprehensive description of phase behavior.

Contribution

It proposes a framework to unify HRT and SCOZA by integrating their core principles and analyzing the conditions for consistent solutions in simplified models.

Findings

Mean Spherical Approximation as a solution

Necessary conditions for closure consistency

Potential for practical unification scheme

Abstract

The Hierarchical Reference Theory (HRT) and the Self-Consistent Ornstein-Zernike Approximation (SCOZA) are two liquid state theories that both furnish a largely satisfactory description of the critical region as well as phase separation and the equation of state in general. Furthermore, there are a number of similarities that suggest the possibility of a unification of both theories. As a first step towards this goal we consider the problem of combining the lowest order gamma expansion result for the incorporation of a Fourier component of the interaction with the requirement of consistency between internal and free energies, leaving aside the compressibility relation. For simplicity we restrict ourselves to a simplified lattice gas that is expected to display the same qualitative behavior as more elaborate models. It turns out that the analytically tractable Mean Spherical Approximation is a solution to this problem, as are several of its generalizations. Analysis of the characteristic equations shows the potential for a practical scheme and yields necessary conditions any closure to the Ornstein Zernike relation must fulfill for the consistency problem to be well posed and to have a unique differentiable solution. These criteria are expected to remain valid for more general discrete and continuous systems, even if consistency with the compressibility route is also enforced where possible explicit solutions will require numerical evaluations.