Scalar curvature density as a new invariant in thermodynamic geometry: metric dependence and critical exponents
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: $R \sim t^{-dν}$ is governed by the correlation length exponent, whereas $\mathcal{R} \sim t^{-(1+β)}$ scales solely with the order parameter exponent $β$, 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.