Scalar curvature density as a new invariant in thermodynamic geometry: metric dependence and critical exponents

Abstract

We compare two Ruppeiner metrics constructed under fixed volume and fixed particle number conditions ($g_{{}_V}$ and $g_{{}_V}$) by analyzing the scalar curvature $R$ and introducing the scalar curvature density $\mathcal{R}=\sqrt{|g|}R$ as a complementary geometric invariant. Three fluid models of increasing physical realism are considered: van der Waals, Lennard-Jones, and argon described by a multiparameter equation of state that correctly reproduces non-mean-field critical behavior consistent with the Ising universality class. We find that $R$ and $\mathcal{R}$ exhibit distinct critical scaling: $R\sim t^{-d\hat{I}½}$ is governed by the correlation length exponent, whereas $\mathcal{R}\sim t^{-(1+{\hat{I}}^2)}$ scales solely with the order parameter exponent ${\hat{I}}^2$, a result that follows analytically from hyperscaling and the Rushbrooke relation independently of the universality class. The $N$-metric consistently outperforms the $V$-metric in reconstructing the vapor-liquid coexistence curve away from criticality, and the four Widom lines defined by the minima of $R_{{}_V}$, $R_{{}_N}$, ${\mathcal{R}}_{{}_V}$, and ${\mathcal{R}}_{{}_N}$ display a characteristic fourfold structure reproduced across all three models. The loci where both representations yield identical geometric descriptions define two new objects: the Curvature Equality Curve (CEC, $R_{{}_V}=R_{{}_N}$) and the Curvature-Density Equality Curve (CDEC, ${\mathcal{R}}_{{}_V}={\mathcal{R}}_{{}_N}$), with the CDEC enclosing a substantially larger region of the phase diagram in all cases. These results establish $\mathcal{R}$ as a meaningful complement to $R$ in thermodynamic geometry and highlight the nontrivial role of metric choice in the description of phase transitions and supercritical behavior.