Stationary points approach to thermodynamic phase transitions

TL;DR

This paper introduces a stationary points approach to analyze thermodynamic phase transitions, linking nonanalyticities in entropy to stationary points of the potential energy, and provides criteria for phase transition existence.

Contribution

It develops an analytic framework connecting stationary points to phase transitions and derives criteria for their occurrence in the thermodynamic limit.

Findings

Nonanalyticities in entropy are caused by stationary points.

Only 'asymptotically flat' stationary points influence phase transitions.

The approach can be used to compute transition energies in classical spin models.

Abstract

Nonanalyticities of thermodynamic functions are studied by adopting an approach based on stationary points of the potential energy. For finite systems, each stationary point is found to cause a nonanalyticity in the microcanonical entropy, and the functional form of this nonanalytic term is derived explicitly. With increasing system size, the order of the nonanalytic term grows, leading to an increasing differentiability of the entropy. It is found that only "asymptotically flat" stationary points may cause a nonanalyticity that survives in the thermodynamic limit, and this property is used to derive an analytic criterion establishing the existence or absence of phase transitions. We sketch how this result can be employed to analytically compute transition energies of classical spin models.