A microscopic characterization, based on the thermodynamic curvature R, is proposed for first-order liquid-gas phase transitions. Near the critical point, where R is proportional to the correlation volume ξ(3), we propose that R takes the same value in the coexisting phases. This proposal allows a determination of the liquid-gas coexistence curve with no use of the problematic Maxwell equal area construction. Furthermore, |R| ~ ξ(3) allows a direct determination of the Widom line in the supercritical regime. We illustrate with input from the van der Waals model and the National Institute of Standards and Technology Chemistry WebBook.

In this paper, we study various aspects of the equilibrium thermodynamic state space geometry of AdS black holes. We first examine the ReissnerNordstrom-AdS (RN-AdS) and the Kerr-AdS black holes. In this context, the state space scalar curvature of these black holes is analysed in various regions of their thermodynamic parameter space. This provides important new insights into the structure and significance of the scalar curvature. We further investigate critical phenomena, and the behaviour of the scalar curvature near criticality, for KN-AdS black holes in two mixed ensembles, introduced and elucidated in our earlier work arXiv:1002.2538 [hep-th]. The critical exponents are identical to those in the RN-AdS and KerrAdS cases in the canonical ensemble. This suggests an universality in the scaling behaviour near critical points of AdS black holes. Our results further highlight qualitative differences in the thermodynamic state space geometry for electric charge and angular momentum fluctuations of these. *

We investigate phase transitions and critical phenomena in Kerr-Newman-Anti de Sitter black holes in the framework of the geometry of their equilibrium thermodynamic state space. The scalar curvature of these state space Riemannian geometries is computed in various ensembles. The scalar curvature diverges at the critical point of second order phase transitions for these systems. Remarkably, however, we show that the state space scalar curvature also carries information about the liquid-gas like first order phase transitions and the consequent instabilities and phase coexistence for these black holes. This is encoded in the turning point behavior and the multivalued branched structure of the scalar curvature in the neighborhood of these first order phase transitions. We re-examine this first for the conventional Van der Waals system, as a preliminary exercise. Subsequently, we study the Kerr-Newman-AdS black holes for a grand canonical and two "mixed" ensembles and establish novel phase structures. The state space scalar curvature bears out our assertion for the first order phase transitions for both the known and the new phase structures, and closely resembles the Van der Waals system. *

Thermodynamic fluctuation metrics in Ruppeiner's formalism are worked out for Kerr-AdS black holes in the extended state space. The implications of constraints upon the state space geometry and their correspondence with thermodynamical ensembles are explicitly worked out in the most general setting. The state space scalar curvature for a given ensemble is found to be sensitive to the instabilities or phase transitions therein. In particular, it is found that the appropriate Ruppeiner scalar curvature does encode critical phenomena in the Kerr-AdS black holes. A detailed study is undertaken of the curvature contour of the state space of the 4D Kerr-AdS black hole, and suitable inferences are drawn. In particular, thermodynamic geometry suggests an instability in the Schwarzschild-AdS limit for all the ensembles except the pressure ensemble, which is equivalent to the unextended state space of the Kerr-AdS black holes. The extrinsic geometry of the ensemble hypersurfaces is introduced, and its relevance to constrained thermodynamic fluctuations is discussed. A new interpretation for the thermodynamic curvature of black hole systems is suggested.