Energy landscapes and their relation to thermodynamic phase transitions

TL;DR

This paper explores how stationary points in energy landscapes relate to phase transitions, showing that in the thermodynamic limit, only flat stationary points can induce phase changes, with implications for understanding complex systems.

Contribution

It establishes a direct link between energy landscape stationary points and thermodynamic phase transitions, especially highlighting the role of flat stationary points in the thermodynamic limit.

Findings

Finite systems show nonanalyticities in entropy caused by stationary points.

As system size increases, nonanalyticity order grows, leading to smoother entropy.

In the thermodynamic limit, only flat stationary points can cause phase transitions.

Abstract

In order to better understand the occurrence of phase transitions, we adopt an approach based on the study of energy landscapes: The relation between stationary points of the potential energy landscape of a classical many-particle system and the analyticity properties of its thermodynamic functions is studied for finite as well as infinite systems. For finite systems, each stationary point is found to cause a nonanalyticity in the microcanonical entropy, and the functional form of this nonanalytic term can be derived explicitly. With increasing system size, the order of the nonanalytic term grows unboundedly, leading to an increasing differentiability of the entropy. Therefore, in the thermodynamic limit, only asymptotically flat stationary points may cause a phase transition to take place. For several spin models, these results are illustrated by predicting the absence or presence of a phase transition from stationary points and their local curvatures in microscopic configuration space. These results establish a relationship between properties of energy landscapes and the occurrence of phase transitions. Such an approach appears particularly promising for the simultaneous study of dynamical and thermodynamical properties, as is of interest for example for protein folding or the glass transition.