Infinite compressibility states in the Hierarchical Reference Theory of fluids. II. Numerical evidence

TL;DR

This paper provides numerical evidence supporting the idea that the Hierarchical Reference Theory of fluids exhibits stiff partial differential equations near infinite isothermal compressibility, with special focus on the hard-core Yukawa potential.

Contribution

It offers numerical validation for the stiff PDE scenario in the Hierarchical Reference Theory and explores the role of the interaction potential's Fourier transform in computational challenges.

Findings

Stiffness in the PDE occurs when the compressibility diverges.

Transient stiffness appears at intermediate cutoff at low temperatures.

Hard-core Yukawa potential presents significant computational difficulties.

Abstract

Continuing our investigation into the Hierarchical Reference Theory of fluids for thermodynamic states of infinite isothermal compressibility kappa[T] we now turn to the available numerical evidence to elucidate the character of the partial differential equation: Of the three scenarios identified previously, only the assumption of the equations turning stiff when building up the divergence of kappa[T] allows for a satisfactory interpretation of the data. In addition to the asymptotic regime where the arguments of part I (cond-mat/0308467) directly apply, a similar mechanism is identified that gives rise to transient stiffness at intermediate cutoff for low enough temperature. Heuristic arguments point to a connection between the form of the Fourier transform of the perturbational part of the interaction potential and the cutoff where finite difference approximations of the differential equation cease to be applicable, and they highlight the rather special standing of the hard-core Yukawa potential as regards the severity of the computational difficulties.