  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.
