Supercritical-subcritical correspondence, asymmetric effects and antisymmetric corrections near a critical point

TL;DR

The paper addresses how asymmetry influences scaling near a critical point, including the supercritical regime defined by $L^{\pm}$ lines, by developing a complete scaling framework with linear mixing of three fields. It predicts universal antisymmetric corrections that drive nonrectilinear behavior of the supercritical diameter, with pressure and density corrections scaling as $\delta P^{\Delta}$ and $\delta \rho^{\beta}$ (where $\Delta=\beta+\gamma$), plus higher-order antisymmetric terms. These predictions are tested against NIST liquid-gas data for several fluids and a mean-field two-state model of a liquid–liquid transition; higher-order cumulants $\kappa_3$ and $\kappa_4$ are shown to exhibit the same asymmetric scaling near the critical point. The results support a subcritical–supercritical correspondence and reveal a universal, antisymmetric scaling structure across criticality, with potential implications for interpreting supercritical phenomena in diverse systems.

Abstract

The second-order phase transitions in the Ising model and liquid-gas systems share a universality class and critical exponents, despite the absence of $Z_2$ symmetry in the liquid-gas Hamiltonian. This discrepancy highlights a central puzzle in critical phenomena: what is the influence of asymmetry on scaling laws? For over a century, this question has been explored through examining violations of the empirical ``rectilinear diameter law'' for the subcritical coexistence curve, where asymmetry could generate singular corrections. Here, we extend this investigation to the supercritical regime. We propose a supercritical-subcritical correspondence, drawing a formal analogy between the subcritical coexistence curve and recently defined supercritical boundary lines ($L^\pm$ lines). Our theory predicts that the linear mixing of physical fields - a hallmark of asymmetric systems - produces universal scaling corrections, with antisymmetric coefficients, in these supercritical loci. We verify these predictions using liquid-gas data from the NIST database and a model liquid-liquid transition. Furthermore, we demonstrate that the same asymmetric scaling framework governs the behavior of higher-order cumulants in the order parameter distribution.

Figures (12)

• Figure 1: Phase diagrams and EOSs for symmetric and asymmetric systems. (A) $H-T$ phase diagram of the 2D mean-filed Ising model. (B) $P-T$ phase diagram of $\rm{CO_2}$, obtained from the NIST database NIST. The thick black line is the first-order transition (coexistence) line, and the red circle the critical point. The thin contour lines are constant order parameter EOSs, with one typical EOS highlighted by a thicker line. In (B), the dashed black line is the critical isochore ($\rho=\rho_{\rm c}$), and the red lines are $L^{\pm}$ lines li2023thermodynamic.
• Figure 2: Supercritical-subcritical correspondence. The following correspondence is visualized in the $\hat{T}$-$\hat{\rho}$ phase diagram: $L^\pm$ lines vs coexistence curve; supercritical diameter $\rho_{\rm d}^>=\frac{1}{2}(\rho^+ + \rho^-)$ vs subcritical diameter $\rho_{\rm d}=\frac{1}{2}(\rho_{\rm L} + \rho_{\rm G})$. The lines are drawn based on the NIST data.
• Figure 3: Asymmetric effects in supercritical fluids. The black solid lines are obtained form fitting: (A) $y=0.27 x^{0.36}$; (B) $y=0.191 x^{0.21} + 0.07 x^{0.36}$. The red dashed lines in (B) represent the two terms in Eq. (\ref{['eq:ratio']}).
• Figure 4: Asymmetric effects for the "symmetry line" with $\kappa_3 = 0$.
• Figure 5: Asymmetric effects in supercritical hydrogen. The solid fitting line in (A) represents $y=0.18 x^{0.36}$, and the one in (B) represents $y= 0.28 x^{0.36}$.