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

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.