Phase Transitions from Saddles of the Potential Energy Landscape

TL;DR

This paper investigates how saddle points in the potential energy landscape influence the thermodynamic properties and phase transitions of many-particle systems, revealing conditions under which they cause nonanalyticities in entropy.

Contribution

It provides a detailed analysis of the role of saddle points in the potential energy landscape and their impact on phase transitions in the thermodynamic limit.

Findings

Saddle points cause nonanalyticities in finite systems' entropy.

The nonanalyticity order increases with system size, affecting entropy differentiability.

Saddle points may or may not induce phase transitions depending on circumstances.

Abstract

The relation between saddle points of the potential of a classical many-particle system and the analyticity properties of its thermodynamic functions is studied. For finite systems, each saddle point is found to cause a nonanalyticity in the Boltzmann entropy, and the functional form of this nonanalytic term is derived. For large systems, the order of the nonanalytic term increases unboundedly, leading to an increasing differentiability of the entropy. Analyzing the contribution of the saddle points to the density of states in the thermodynamic limit, our results provide an explanation of how, and under which circumstances, saddle points of the potential energy landscape may (or may not) be at the origin of a phase transition in the thermodynamic limit. As an application, the puzzling observations by Risau-Gusman et al. on topological signatures of the spherical model are elucidated.