Stability of the iterative solutions of integral equations as one phase freezing criterion

TL;DR

This paper critically examines the proposed link between the stability of iterative solutions to integral equations and phase transitions in fluids, revealing that the connection does not hold in one-dimensional systems and questioning its general applicability.

Contribution

The study demonstrates that the stability criterion based on iterative solutions does not reliably predict phase transitions, especially in 1D systems where no transition occurs.

Findings

Lyapunov exponent analysis shows similar behavior in 1D and 3D systems.

The proposed stability criterion fails in 1D hard rods fluid.

Numerical stability is influenced by fluid structure and iteration scheme.

Abstract

A recently proposed connection between the threshold for the stability of the iterative solution of integral equations for the pair correlation functions of a classical fluid and the structural instability of the corresponding real fluid is carefully analyzed. Direct calculation of the Lyapunov exponent of the standard iterative solution of HNC and PY integral equations for the 1D hard rods fluid shows the same behavior observed in 3D systems. Since no phase transition is allowed in such 1D system, our analysis shows that the proposed one phase criterion, at least in this case, fails. We argue that the observed proximity between the numerical and the structural instability in 3D originates from the enhanced structure present in the fluid but, in view of the arbitrary dependence on the iteration scheme, it seems uneasy to relate the numerical stability analysis to a robust one-phase criterion for predicting a thermodynamic phase transition.