Supercritical Black Hole Thermodynamics

• Supercritical black hole thermodynamics is defined as the region beyond the critical point where first-order phase transitions vanish, yet continuous fluid-like crossovers occur.
• The regime is characterized by quantifiable crossover boundaries marked by Widom lines, which separate gas-like and liquid-like dynamical behaviors through response function extrema.
• Universal scaling laws and critical slowing down within this regime provide key insights, linking gravitational phenomena to conventional liquid–gas criticality.

The supercritical regime of black hole thermodynamics refers to the domain above the critical point in the phase diagram of gravitational systems, such as Reissner–Nordström–AdS (RNAdS) and related spacetimes, when formulated within the extended phase space where the cosmological constant is interpreted as a thermodynamic pressure. This regime is characterized by the absence of any true first-order phase transition or coexistence line, yet it retains rich structure reminiscent of fluid supercriticality, including quantitative crossover boundaries, dynamical slowing-down phenomena, and universal scaling laws. The landscape is further structured by the Widom line(s), which sharply demarcate gas-like and liquid-like regimes via response-function extrema and dynamical indicators.

1. Extended Phase Space and Criticality

In the extended phase space framework, the negative cosmological constant $\Lambda$ serves as a thermodynamic pressure, $P = -\Lambda/(8\pi)$, and its conjugate as the thermodynamic volume. The black hole equation of state, typically for the RNAdS solution in four dimensions, takes the form

$$P=\frac{T}{2r_h} - \frac{1}{8\pi r_h^2} + \frac{Q^2}{8\pi r_h^4}, \qquad S = \pi r_h^2,$$

where $T$ is the Hawking temperature, $Q$ is the charge, and $r_h$ is the horizon radius. The system exhibits a van der Waals–type first-order phase transition (small/large black hole) terminating at a critical point $(T_c, P_c)$ found from the inflection point conditions on the equation of state. In the supercritical regime—defined by $T > T_c$ and $P > P_c$—the canonical first-order transition line vanishes, yielding a single, globally stable “fluid” phase.

2. Definition and Structure of the Supercritical Regime

Although no genuine phase separation persists above $(T_c,P_c)$, the supercritical domain exhibits nontrivial structure. Two principal observations hold:

• All isotherms for $T>T_c$ are single-valued and monotonic. Response functions such as isothermal compressibility, $\kappa_T$, and heat capacity, $C_P$, remain finite and continuous throughout the regime.
• The system displays a smooth, yet tangible, crossover between gas-like (“small black hole”) and liquid-like (“large black hole”) dynamical behavior, distinguished now not by latent heat but rather by the properties of equilibrium and dynamical response functions.

This crossover is sharply demarcated by the Widom line or, in more intricate models, multiple Widom lines, which generalize the notion of a phase boundary into the supercritical region.

3. The Widom Line: Thermodynamic and Kinetic Characterization

The Widom line is defined as the locus in the $T$–$P$ plane along which an appropriate response function achieves an extremum. Two main classes of Widom line definitions dominate:

• Thermodynamic Widom Line: Determined by the loci of peaks in the isobaric heat capacity $C_P = T(\partial S/\partial T)_P$ or higher-order derivatives of the Gibbs free energy, e.g., the scaled variance $$\Omega = (\partial^2 G/\partial T^2) / (\partial G/\partial T)$$. Along fixed $P>P_c$, the condition $(\partial C_P/\partial T)_P=0$ (or analogously $(\partial \Omega/\partial T)_P=0$) identifies the thermodynamic Widom line (Li et al., 30 May 2025, Zhao et al., 7 Apr 2025).
• Dynamical (Kinetic) Widom Line: Operationalized via kinetic measures (e.g., autocorrelation time $\tau$ from Langevin simulations on the effective free-energy landscape), where $\tau$ reaches a maximum as a function of $T$ for given $P>P_c$. This “ridge” in $(T, P, \tau)$ space separates distinct dynamical regimes and closely tracks the thermodynamic Widom line (Li et al., 30 May 2025). In the formalism of the Fokker–Planck equation, the lowest nonzero eigenvalue $\lambda_1$ provides an equivalent marker, with the ridge of minimal $\lambda_1$ tracing the dynamical Widom line.

These two lines—the equilibrium (thermodynamic) and out-of-equilibrium (kinetic)—are found to nearly coincide, demonstrating that both the equilibrium response and non-equilibrium relaxation properties are governed by the same underlying crossover.

4. Dynamics, Free-Energy Landscape, and Critical Slowing Down

The approach to the critical point and Widom line is accompanied by pronounced dynamical phenomena:

• The free-energy landscape $G(r)$, with $r=r_+$ as the order parameter, undergoes progressive flattening near spinodal points (where $G''=0$) and the critical point (where $G'=G''=G'''=0$). At the spinodal, $G(r) \approx G(r_s) + (1/3!) G'''(r_s)\,(r-r_s)^3$, leading to diverging autocorrelation time $\tau \sim \zeta/ G''(r_{e})$ and variance $V \sim T/ G''(r_{e})$. At the critical point, the quartic expansion ($G(r) \sim G(r_c) + (1/4!)G''''(r_c)(r - r_c)^4$) yields even stronger slowing down.
• Power-law scaling of dynamical quantities is observed: $\tau \sim |T-T_c|^{-\Delta}$ and $V \sim |T-T_c|^{-\gamma}$, with $\Delta \approx 1$, $\gamma \approx 1$—consistent with mean-field exponents for the liquid–gas universality class (Li et al., 30 May 2025).
• In the supercritical region, both $\tau(T,P)$ and $V(T,P)$ display ridges of maximal slowing-down. Numerical studies of the Fokker–Planck spectrum confirm that the relaxation time $1/\lambda_1$ peaks along the Widom line.

These features establish the persistence of critical kinetic phenomena (critical slowing down) in the supercritical regime, even when no phase separation survives.

5. Generalizations: Scaling Laws and Multicriticality

Universal scaling theory governs the crossover structure near the critical point. For the RNAdS black hole, the Widom line(s) obey

$$P^{\pm}_{\rm crossover}(T) = P_c \left[ 1 + \alpha_\pm\, \left( \frac{T}{T_c} - 1 \right)^{\beta+\gamma} \right],$$

with mean-field exponents $\beta=1/2$, $\gamma=1$ ($\beta+\gamma=3/2$), and coefficients $\alpha_\pm$ determined numerically (Wang et al., 12 Jun 2025). The deviation of the order parameter ($\rho=1/v$) along the Widom lines scales as $(T-T_c)^{\beta}$.

In higher-curvature and multidimensional black hole theories (e.g., Gauss–Bonnet AdS), the number of supercritical Widom lines equals the number of coexisting phases at the highest multicritical point minus one. For instance, six-dimensional charged Gauss–Bonnet AdS black holes with a triple point exhibit two Widom lines, carving the supercritical domain into three regimes (small/intermediate/large BH–like), in direct correspondence with triple-phase coexistence (Li et al., 13 Nov 2025). This correspondence encodes the analytic continuation of Lee–Yang singularities, with each phase branch above the multicritical point spawning a distinct supercritical sector.

6. Model Variants and Robustness Across Theories

The qualitative structure of the supercritical regime is robust across gravitational settings:

• Higher-dimensional and de Sitter Black Holes: The supercritical regime ($T>T_c$, $P>P_c$) exhibits finite, analytic response functions ($C_P$, $\kappa_T$) and single-valued $P$–$V$ isotherms, with the critical exponents and crossover structure mirroring the van der Waals and liquid–gas paradigm (Zhang et al., 2014).
• Nonsingular and Regular Black Holes: Deformations such as spacetime regularization simply rescale the critical parameters and expand the supercritical domain, but do not alter universality (Kumar et al., 2021). The response functions, scaling behavior, and absence of coexistence persist.
• Quantum Black Holes (e.g., Quantum BTZ): In quantum-corrected scenarios, features such as multibranched heat capacities and “super-entropic” behavior emerge. The supercritical region can retain multiple unstable branches or unusual stability domains, indicating that quantum degrees of freedom can enrich the phase structure (Johnson et al., 2023).
• Rotating and Fractal Black Holes: In Kerr–AdS and systems with fractal horizons or Yang–Mills couplings, the disappearance of first-order transitions and the emergence of a globally stable, monotonic supercritical phase is generic, though controllable by system-specific parameters (e.g., central charge, Barrow exponent) (Du et al., 29 Oct 2024, Gao et al., 2021).
• Lee–Yang and Complex Phase Diagram Perspective: The supercritical Widom line is precisely the real-plane projection of the Lee–Yang zeros that, for $P>P_c$, detach from the real axis and persist as loci of analytic but rapid crossover (response-function extrema) (Xu et al., 8 Apr 2025, Li et al., 13 Nov 2025).

7. Physical Interpretation and Universality

The existence and universality of the supercritical regime in black hole thermodynamics vindicate the analogy between gravitational and fluid systems. Above the critical point, the sharp first-order small/large black hole distinction is effaced, but the system retains a memory of phase structure through rapid, though continuous, crossovers in both equilibrium and dynamical characteristics.

Crucially, the position and scaling of such crossovers, as well as the number of supercritical sectors, are dictated by underlying universality classes and the analytic structure of the Euclidean partition function’s Lee–Yang zeros. This grants gravitational systems a direct parallel to the supercritical phenomena in laboratory fluids, including water’s liquid–liquid transition, and provides a powerful route to translate statistical mechanical universality into quantum gravity and AdS/CFT contexts (Wang et al., 12 Jun 2025).

In summary, the supercritical regime of black hole thermodynamics is an analytically well-defined, physically rich domain, characterized by robust crossover lines, persistence of dynamical criticality, and universal scaling. It stands as a direct probe of both the macroscopic thermodynamics and the microscopic statistical structure of gravitational systems above their critical points.