Phase transitions induced by saddle points of vanishing curvature

TL;DR

This paper establishes a new geometric criterion based on saddle point curvatures in potential energy landscapes that predicts the occurrence of phase transitions in classical many-particle systems, linking microscopic properties to macroscopic phenomena.

Contribution

It introduces a necessary condition involving saddle point curvatures for phase transitions, extending the topological approach with a geometric perspective.

Findings

The criterion accurately predicts phase transition energies in exactly solvable models.

It excludes phase transitions at all energies except the transition energy.

The criterion enhances understanding of the topology-geometry relationship in phase transitions.

Abstract

Based on the study of saddle points of the potential energy landscapes of generic classical many-particle systems, we present a necessary criterion for the occurrence of a thermodynamic phase transition. Remarkably, this criterion imposes conditions on microscopic properties, namely curvatures at the saddle points of the potential, and links them to the macroscopic phenomenon of a phase transition. We apply our result to two exactly solvable models, corroborating that the criterion derived is not only valid, but also sharp and useful: For both models studied, the criterion excludes the occurrence of a phase transition for all values of the potential energy but the transition energy. This result adds a geometrical ingredient to an established topological condition for the occurrence of a phase transition, thereby providing an answer to the long standing question of which topology changes in configuration space can induce a phase transition.