Thermodynamic Geometry & Ruppeiner Metric

g_ij = −∂²S/∂X^i∂X^j — curvature R diverges at the critical point of a van der Waals gas

0.95
enabled
R(T,V) =
T/T_c = 0.95
Phase: below T_c
van der Waals:
(P + a/V²)(V−b) = RT

Critical point:
T_c = 8a/27Rb, V_c = 3b

Ruppeiner curvature:
R → ±∞ at phase transition → measures interaction correlation length
Ruppeiner (1979): thermodynamic metric connects to correlation length ξ via R ~ ξd