Thermodynamic curvature: pure fluids to black holes
  Thermodynamics unavoidably contains fluctuation theory, expressible in terms of a unique thermodynamic information metric. This metric produces an invariant thermodynamic Riemannian curvature scalar $R$ which, in fluid and spin systems, measures interatomic interactions. Specifically, $|R|$ measures the size of organized fluctuating microscopic structures, and the sign of $R$ indicates whether the interactions are effectively attractive or repulsive. $R$ has also been calculated for black hole thermodynamics for which there is no consensus about any underlying microscopic structures. It is hoped that the physical interpretation of $R$ in fluid and spin systems might offer insight into black hole microstructures. I give a brief review of results for $R$ in black holes, including stability, the sign of $R$, R=0, diverging |R|, and various claims of "inconsistencies" in thermodynamic metric geometry. 
George Ruppeiner
gr-qc