Thermodynamic Geometry, Phase Transitions, and the Widom Line
  We construct a novel approach, based on thermodynamic geometry, to characterize first-order phase transitions from a microscopic perspective, through the scalar curvature in the equilibrium thermodynamic state space. Our method resolves key theoretical issues in macroscopic thermodynamic constructs, and furthermore characterizes the Widom line through the maxima of the correlation length, which is captured by the thermodynamic scalar curvature. As an illustration of our method, we use it in conjunction with the mean field Van der Waals equation of state to predict the coexistence curve and the Widom line. Where closely applicable, it provides excellent agreement with experimental data. The universality of our method is indicated by direct calculations from the NIST database. 
George Ruppeiner, Anurag Sahay, Tapobrata Sarkar, Gautam Sengupta
cond-mat.stat-mech hep-th physics.chem-ph physics.flu-dyn