Information Geometry, Phase Transitions, and Widom Lines : Magnetic and Liquid Systems

TL;DR

This paper explores the use of information geometry to analyze phase transitions in magnetic and liquid systems, establishing a universal microscopic characterization and identifying Widom lines through scalar curvature of thermodynamic state space.

Contribution

It introduces a geometric framework linking correlation lengths to phase transitions and Widom lines, applicable to various models including Ising, Curie-Weiss, and liquid-liquid coexistence.

Findings

Information geometry correctly describes phase behavior in mean-field models.

Correlation lengths are related to scalar curvature in thermodynamic space.

Multiple Widom lines are identified in liquid systems.

Abstract

We study information geometry of the thermodynamics of first and second order phase transitions, and beyond criticality, in magnetic and liquid systems. We establish a universal microscopic characterization of such phase transitions via the equality of correlation lengths $\xi$ in coexisting phases, where $\xi$ is related to the scalar curvature of the equilibrium thermodynamic state space. The 1-D Ising model, and the mean-field Curie-Weiss model are discussed, and we show that information geometry correctly describes the phase behavior for the latter. The Widom lines for these systems are also established. We further study a simple model for the thermodynamics of liquid-liquid phase co-existence, and show that our method provides a simple and direct way to obtain its phase behavior and the locations of the Widom lines. Our analysis points towards multiple Widom lines in liquid systems.