Ruppeiner Geometry
Thermodynamic curvature R diverges at phase transitions
Thermodynamic state space (T,V) with curvature R
Scalar curvature R vs reduced temperature T/Tc
The Ruppeiner metric gij = −∂²S/∂Xi∂Xj defines a geometry on thermodynamic state space.
Scalar curvature R → ∞ at phase transitions — divergence marks thermodynamic instability.
R < 0: attractive interactions (bosons, van der Waals). R > 0: repulsive (fermions).