Nonanalyticities of the entropy induced by saddle points of the potential energy landscape

TL;DR

This paper investigates how saddle points in the potential energy landscape of many-particle systems cause nonanalyticities in entropy and explores their role in phase transitions, especially in the thermodynamic limit.

Contribution

It provides a detailed analysis of the relationship between saddle points and entropy nonanalyticities, deriving conditions for phase transitions based on saddle point distributions.

Findings

Finite systems exhibit entropy nonanalyticities at saddle points.

As system size increases, entropy becomes more differentiable, but phase transitions can still occur.

Saddle points with vanishing curvature induce phase transitions in mean-field models.

Abstract

The relation between saddle points of the potential of a classical many-particle system and the analyticity properties of its Boltzmann entropy 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 the generic case of potentials having the Morse property. With increasing system size the order of the nonanalytic term grows unboundedly, leading to an increasing differentiability of the entropy. Nonetheless, a distribution of an unboundedly growing number of saddle points may cause a phase transition in the thermodynamic limit. Analyzing the contribution of the saddle points to the density of states in the thermodynamic limit, conditions on the distribution of saddle points and their curvatures are derived which are necessary for a phase transition to occur. With these results, the puzzling absence of topological signatures in the spherical model is elucidated. As further applications, the phase transitions of the mean-field XY model and the mean-field k-trigonometric model are shown to be induced by saddle points of vanishing curvature.