Thermodynamic curvature measures interactions

Thermodynamic fluctuation theory originated with Einstein who inverted the relation $S=k_B\ln \mathrm{\Omega }$ to express the number of states in terms of entropy: $\mathrm{\Omega }=\exp (S\mathrm{/}{k}_B)$. The theory’s Gaussian approximation is discussed in most statistical mechanics texts. I review work showing how to go beyond the Gaussian approximation by adding covariance, conservation, and consistency. This generalization leads to a fundamentally new object: the thermodynamic Riemannian curvature scalar $R$, a thermodynamic invariant. I argue that $\mathrm{\mid }R\mathrm{\mid }$ is related to the correlation length and suggest that the sign of $R$ corresponds to whether the interparticle interactions are effectively attractive or repulsive.