2026/07/31 by José Torres-Arenas, Jaime Jaramillo-Gutiérrez, Juan Becerra-Zamudio
Physics and Astronomy · #cond-mat.stat-mech
arxiv created 2026/07/31 · arxiv updated 2026/08/03
We compare two Ruppeiner metrics constructed under fixed volume and fixed particle number conditions (gV and gV) by analyzing the scalar curvature R and introducing the scalar curvature density R = √(|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 R exhibit distinct critical scaling: R ∼ t-dν is governed by the correlation length exponent, whereas R ∼ 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 RV, RN, RV, and RN 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, RV = RN) and the Curvature-Density Equality Curve (CDEC, RV = RN), with the CDEC enclosing a substantially larger region of the phase diagram in all cases. These results establish 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.