Infinite compressibility states in the Hierarchical Reference Theory of fluids. I. Analytical considerations

TL;DR

This paper analytically explores the behavior of the Hierarchical Reference Theory PDE for fluids near critical points, identifying classes of solutions with infinite compressibility and discussing their numerical implications.

Contribution

It systematically classifies asymptotic solutions of the PDE for diverging compressibility in the Hierarchical Reference Theory, clarifying their properties and resolving previous contradictions.

Findings

Identified three classes of asymptotic solutions for infinite compressibility.

Analyzed the numerical properties of these solution classes.

Resolved a contradiction with earlier asymptotic solutions.

Abstract

In its customary formulation for one-component fluids, the Hierarchical Reference Theory yields a quasilinear partial differential equation for an auxiliary quantity f that can be solved even arbitrarily close to the critical point, reproduces non-trivial scaling laws at the critical singularity, and directly locates the binodal without the need for a Maxwell construction. In the present contribution we present a systematic exploration of the possible types of behavior of the PDE for thermodynamic states of diverging isothermal compressibility kappa[T] as the renormalization group theoretical momentum cutoff approaches zero. By purely analytical means we identify three classes of asymptotic solutions compatible with infinite kappa[T], characterized by uniform or slowly varying bounds on the curvature of f, by monotonicity of the build-up of diverging kappa[T], and by stiffness of the PDE in part of its domain, respectively. These scenarios are analzyed and discussed with respect to their numerical properties. A seeming contradiction between two of these alternatives and an asymptotic solution derived earlier [Parola et al., Phys. Rev. E 48, 3321 (1993)] is easily resolved.