Information geometry, phase transitions, and the Widom line: 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 a conjectured equality of the correlation lengths ξ in co-existing phases, where ξ 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 toy 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 location of the Widom line. Our analysis points towards the possibility of multiple Widom lines in liquid systems.