Critical slowing down of black hole phase transition and kinetic crossover in supercritical regime

Jiayue Yang; Jin Wang; Kun Zhang; Ran Li; Robert B. Mann

REVIEW 3 major objections 5 minor 4 cited by

Critical slowing down of black hole phase transition and kinetic crossover in supercritical regime

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Perturbed charged AdS black holes relax ever more slowly at spinodal and critical points, where the free energy landscape flattens, and a supercritical kinetic crossover traces the thermodynamic Widom line.

desk verdict The numerical evidence for slow kinetics is plausible, but the headline analytic claim of a divergent autocorrelation time is an artifact of linearizing in a regime where the quadratic approximation is invalid. read the letter →

arxiv 2505.24148 v1 pith:6H5GQ2QZ submitted 2025-05-30 gr-qc hep-th

Ran Li , Kun Zhang , Jiayue Yang , Robert B. Mann , Jin Wang This is my paper · ORCID

classification gr-qchep-th PACS 04.70.Dy64.60.-i05.40.-a

keywords criticalslowingdownRNAdSblackholesfreeenergylandscapeLangevindynamicsWidomlineFokker-Planckequationspinodalpointssupercriticalregime

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that Reissner–Nordström–Anti-de Sitter (RNAdS) black holes in the canonical ensemble slow down dramatically as they approach their spinodal or critical points: the autocorrelation time of horizon-radius fluctuations and the variance of trajectories both grow, and the relaxation becomes power-law instead of exponential at the critical point. It attributes this to the flattening of the one-dimensional free energy landscape and confirms it numerically through the lowest nonzero eigenvalue of the Fokker–Planck equation. The paper also claims that in the supercritical regime a kinetic crossover, defined by maxima of the autocorrelation time, separates gas-like from liquid-like dynamics and traces a Widom line that closely matches the thermodynamic Widom line obtained from maxima of the isobaric heat capacity. If correct, the relaxation of a perturbed black hole becomes arbitrarily slow at these special points, and the kinetic crossover supplies a dynamical criterion for distinguishing supercritical black hole regimes.

What carries the argument

The carrying object is the generalized free energy landscape $G(r_+)$ of Eq. (1) — the gravitational-action free energy of the RNAdS black hole in the canonical ensemble with horizon radius $r_+$ as the order parameter — together with the Langevin dynamics $d^2r/dt^2=-\zeta\,dr/dt-\partial G/\partial r+\eta(t)$ and its overdamped limit, in which the relaxation rate is set by the curvature $G^{(2)}(r_e)$ of the landscape. The flattening of this landscape at spinodal and critical points makes $\tau=\zeta/G^{(2)}(r_e)$ diverge. The Fokker–Planck equation, whose smallest nonzero eigenvalue $\lambda_1$ measures the slowest relaxation rate and is computed by a pseudo-spectral method, provides the independent confirmation that slow kinetics corresponds to small $\lambda_1$.

What would settle it

Measure the relaxation of a small perturbation of a RNAdS black hole held at the critical point by an independent method, such as a fully nonlinear dynamical simulation or a microscopic horizon-fluctuation model: the paper predicts a divergent autocorrelation time and a power-law relaxation with $\beta=G^{(4)}(r_c)/(3\zeta)$, so observing exponential relaxation with a finite autocorrelation time at $T_c$ would falsify the central claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that near each spinodal branch and near the critical point of the RNAdS black hole the generalized free energy $G(r)$ flattens, so the autocorrelation time $\tau=\zeta/G^{(2)}(r_e)$ diverges; at the critical point the parabolic approximation fails and the deterministic order-parameter relaxation switches from exponential to the power law $r(t)-r_c=(r(0)-r_c)/\sqrt{(r(0)-r_c)^2\beta t+1}$ with $\beta=G^{(4)}(r_c)/(3\zeta)=12Q^2/r_c^5$ independent of $T$ and $P$. Langevin simulations show pronounced peaks of autocorrelation time and trajectory variance near the spinodal temperatures and near $(T_c,P_c)$ (with the peak slightly above $T_c$ at fixed $P_c$ and slightly below $P_c$ at fixed $T_c$), and the smallest nonzero Fokker–Planck eigenvalue is correspondingly suppressed. In the supercritical regime, the locus of autocorrelation-time maxima (and of the eigenvalue crossover) forms a kinetic Widom line that closely matches the thermodynamic Widom line, the locus of maxima of the isobaric heat capacity.

Load-bearing premise

The load-bearing premise is that the black hole's horizon radius evolves according to the one-dimensional Langevin equation with constant damping and Gaussian white noise obeying fluctuation-dissipation on the free energy landscape; this stochastic dynamics is assumed, not derived from black hole physics or quantum gravity.

Editorial extensions

If this is right

Near a spinodal branch, a perturbed small- or large-black-hole state returns to equilibrium increasingly slowly as the branch is approached, with the autocorrelation time growing without bound.
At the critical point the relaxation is a universal power law rather than an exponential, with the decay rate set by the fourth derivative of the free energy alone.
The kinetic Widom line extracted from autocorrelation-time maxima in the supercritical regime nearly coincides with the thermodynamic Widom line from isobaric-heat-capacity maxima, so dynamical measurements can serve as a surrogate for equilibrium response functions.
The slowest kinetic behavior occurs slightly above the critical temperature at fixed critical pressure and slightly below the critical pressure at fixed critical temperature, so the maximal slowing-down is displaced from the critical point itself.
The growth of autocorrelation time and the suppression of the Fokker-Planck eigenvalue both qualify as early-warning signals for the disappearance of a stable black hole state at a spinodal point.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the same free-energy-landscape Langevin dynamics applies to other asymptotically AdS black holes with van der Waals-type criticality, critical slowing down and a kinetic Widom line should appear generically, not only for the RNAdS case studied here.
The analysis assumes a one-dimensional landscape, so in the grand canonical ensemble, where both horizon radius and charge fluctuate, the predicted slowing-down may be modified or acquire additional timescales; the present results should then be read as the canonical-ensemble limit.
A holographic reading would predict that the dual field theory inherits a dynamical crossover along the same Widom line, a statement the paper does not test but that is a direct corollary of the extended phase-space dictionary.
A direct numerical-relativity test — perturbing a RNAdS black hole at the critical point and measuring the relaxation time — would either confirm the predicted divergence or falsify the Langevin assumption, since the paper's observable predictions rest entirely on that assumption.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the numerics are probably fine and the qualitative picture of slow kinetics near spinodal/critical points is credible, but the paper's central analytic claim—that the autocorrelation time diverges at those points—does not survive contact with the model's own dynamics. The divergence is an artifact of linearizing the Ornstein-Uhlenbeck equation in a regime where the quadratic approximation is invalid.

What's actually new: the authors simulate the Langevin dynamics on the free energy landscape for RNAdS, extract autocorrelation times and variances, and compute lowest Fokker-Planck eigenvalues via a pseudo-spectral method. The supercritical kinetic crossover and its rough match to the heat-capacity Widom line is a fresh application, even if the idea exists in the soft-matter literature. The appendix's validation against Dekker-van Kampen is a good sign of numerical care.

The soft spots are serious. The derivation after Eq. (7) correctly notes the parabolic expansion breaks down at the critical point, then derives the algebraic deterministic relaxation (9). But the stochastic section immediately returns to the linearized equation (10) and concludes tau = zeta/G'' diverges. That's not legitimate: for the actual quartic (or cubic) local potential, the Fokker-Planck operator has a strictly positive lowest eigenvalue, so the long-time autocorrelation decays exponentially with a finite time constant. The numerical eigenvalues in Fig. 6 agree with that—lambda_1 dips but never vanishes, and the minimum sits slightly off the critical point. So the defensible claim is 'enhanced, finite slowing down,' not divergence. This needs to be fixed before publication.

Also, the kinetic Widom line largely inherits its shape from the same curvature that controls the heat capacity, so the close agreement between blue and red lines is not an independent test. And the numerical sections give no error bars, time-step values, or run lengths; the appendix is a start but not enough for reproduction.

I'd send it to peer review but only with explicit instruction that the divergence claim be replaced by a quantitative finite-time analysis and the numerics tightened. The topic will get attention, and with those changes it could be a solid paper.

Referee Report

3 major / 5 minor

Summary. The paper studies the kinetics of Reissner-Nordström–Anti-de Sitter (RNAdS) black hole phase transitions in the canonical ensemble, modeled by a one-dimensional free energy landscape with overdamped Langevin dynamics. It claims that the autocorrelation time and the variance of stochastic trajectories increase significantly near the spinodal points and at the critical point, signaling critical slowing down, and that this is confirmed by the lowest nonzero eigenvalue of the Fokker–Planck equation. In the supercritical regime the paper identifies a kinetic crossover, defined through maxima of the autocorrelation time, and argues that this kinetic Widom line closely matches the thermodynamic Widom line obtained from maxima of the isobaric heat capacity. The paper combines an analytical relaxation-time derivation with Langevin simulations and Fokker–Planck spectral calculations.

Significance. If the claims were fully supported, the paper would add a genuinely dynamical, non-equilibrium layer to the well-established van der Waals analogy for AdS black holes: critical slowing down at spinodal and critical points, and a kinetic diagnostic of supercritical liquid-like versus gas-like regimes. The deterministic algebraic relaxation derivation in Eq. (9) is clean and internally consistent, and the numerical implementation directly addresses the intended observables, including an independent Fokker–Planck eigenvalue check. However, the advertised divergence of the stochastic autocorrelation time is not supported by the model as written, and the quantitative numerical evidence lacks uncertainty characterization. The kinetic Widom line idea remains promising, but the central analytical claim needs either correction or substantial reformulation.

major comments (3)

[Critical slowing down: analytical derivation, Eq. (10)] Equation (10) is the linearized Ornstein–Uhlenbeck equation whose validity requires G''(r_e) > 0. At the critical point Eq. (2) gives G''(r_c) = G'''(r_c) = 0, so Eq. (10) and its solution (11) are not valid at the point where the paper claims τ = ζ/G''(r_e) diverges. This is not a minor technicality: for the quartic potential implied by Eq. (9), the Fokker–Planck operator has a finite spectral gap of order sqrt(T G^{(4)}(r_c))/ζ, so the stochastic autocorrelation time is finite rather than divergent. The paper's own Fig. 6 shows λ1 remaining positive near the critical point, which is consistent with a finite relaxation time. The sound statements are the algebraic deterministic relaxation in Eq. (9) and the enhancement of fluctuations; the divergence claim should be removed or replaced by a correct finite-threshold calculation for the quartic potential.
[Critical slowing down: analytical derivation, spinodal discussion] The same linearization failure occurs at the spinodal points, where G''(r_e) = 0 makes the cubic term leading. Equation (10) is therefore inapplicable there as well, and the claimed divergence of τ at the spinodal is unsupported. The paper does not provide an analogue of Eq. (9) for the cubic case; a correct treatment should compute the relaxation of the cubic potential or the eigenvalue gap and characterize whether the timescale grows algebraically or remains finite.
[Critical slowing down: numerical results, Figs. 4 and 5] The numerical section reports autocorrelation times and variances without error bars, sample sizes, or a description of the fitting procedure used to extract τ. Since the extracted τ assumes an exponential decay form that is not valid near the critical point, the quantitative peaks in Figs. 4 and 5 need a clear fitting protocol, convergence tests in the time step h and trajectory number, and uncertainty estimates before they can support the claim that the kinetic timescale peaks at the critical point.

minor comments (5)

[Critical slowing down: numerical results] In the sentence 'The autocorrelation time t is then extracted by fitting...', the symbol t is used for the autocorrelation time; it should be τ to avoid confusion with the time variable.
[Equations (12)–(13)] There is a stray fragment 'for the autocorrelation function of the order parameter' immediately after Eq. (13); this should be integrated into the surrounding sentence.
[Appendix, Eq. (29) and Fig. 9] The comparison with the Dekker–van Kampen eigenvalues would be easier to interpret if the dimensionless units and the value of the noise strength used for Eq. (29) were stated explicitly.
[References] Reference [34] contains a typo: 'ome implications' should read 'Some implications'.
[Eq. (4) and Conclusion] The paper correctly states that the Langevin description is an assumption, but it would be helpful to state this limitation earlier, immediately after Eq. (4), together with one sentence on what would change if the damping coefficient ζ were state-dependent or if the noise were not white.

Circularity Check

2 steps flagged · score 6.0 of 10

Partial circularity: the kinetic Widom line and the analytic divergence of the autocorrelation time are by-construction consequences of the same free-energy curvature G'', not independent dynamical predictions.

self definitional [Section 'Kinetic crossover in supercritical regime', Fig. 7, and the autocorrelation-time formula following Eq. (8)]
"The autocorrelation time τ = ζ/G(2)(re) that characterizes the exponential decay is divergent at the critical point since the derivatives of generalized free energy with respect to the black hole radius vanish up to third order. ... This watershed is interpreted as the Widom line. On the right panel of Fig. 7, this kinetically determined boundary (blue line) is compared with the conventional Widom line, defined as the locus of maxima of the isobaric heat capacity (red line). We find that the two criteria are in close agreements."

In this model the autocorrelation time is τ = ζ/G^(2)(r_e), and the isobaric heat capacity is also controlled by the same curvature. From G^(1)(r_e)=0 at fixed P, differentiating with respect to T gives ∂r/∂T = 2πr/G^(2)(r_e); with S=πr^2, C_P = T(∂S/∂T)_P = 4π^2 T r^2 / G^(2)(r_e). Thus maxima of C_P and maxima of τ are both loci of minima of G^(2), up to a smooth positive prefactor. The 'kinetic crossover' is therefore not dynamically independent of the 'thermodynamic' Widom line; the close agreement in Fig. 7 is a mathematical identity of the model, not a discovered coincidence. The paper presents this as an uncovered crossover, but it is built into the construction.
self definitional [Section 'Critical slowing down: analytical derivation', Eqs. (7)-(10) and the text after Eq. (13)]
"Near the critical point, the parabolic approximation given in Eq.(7) breaks down, as the derivatives of the generalized free energy with respect to the black hole radius are zero up to third order. ... The autocorrelation time τ = ζ/G(2)(re) that characterizes the exponential decay is divergent at the critical point since the derivatives of generalized free energy with respect to the black hole radius vanish up to third order."

The paper defines the critical point by G^(1)=G^(2)=G^(3)=0 (Eq. 2), then derives an exponential relaxation rate γ=G^(2)(r_e)/ζ from the quadratic expansion (7) and defines τ=1/γ. It then concludes τ diverges at criticality because G^(2)=0. This is the critical-point condition restated in the linearized time-scale formula; it is not a property of the actual stochastic dynamics, since the paper itself says expansion (7) breaks down at criticality and supplies the quartic decay (9). The Fokker-Planck eigenvalue λ1 of the physical quartic potential remains positive (Fig. 6), so the true autocorrelation time is finite. The 'divergence' is thus an artifact of substituting the definition of criticality into an inapplicable linearization.

full rationale

The paper's numerical simulations of the assumed one-dimensional Langevin dynamics are self-contained: they show increased variance and a suppressed Fokker-Planck eigenvalue near criticality, so the qualitative slowing-down trend is genuine content conditional on the model. The circularity lies in the two headline 'predictions.' First, the analytic divergence of τ is obtained by inserting G''=0—the definition of the critical point in Eq. (2)—into the linearized Ornstein-Uhlenbeck formula τ=ζ/G'', even though the paper states the quadratic expansion (7) breaks down there and gives the quartic law (9); the stochastic relaxation time in the quartic potential is finite (their Fig. 6 shows λ1>0 at T_c), so the divergence is an artifact of linearization, not a result of the full dynamics. Second, the claimed coincidence of the kinetic Widom line with the thermodynamic Widom line is built into the model: from G'(r_e)=0 one obtains ∂r/∂T=2πr/G'', so C_P=T∂S/∂T=4π²Tr²/G'', while τ=ζ/G''; hence both lines are loci of minima of the same G''. The 'close agreement' of Fig. 7 is therefore an identity, not an independent cross-check. This reduces the central supercritical claim to a re-labeling of the thermodynamic response. No external benchmark or machine-checked derivation breaks the chain, so the circularity score is 6 rather than lower.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The model rests on a postulated Langevin dynamics on the extended-thermodynamics free energy landscape. There is one model parameter, the damping coefficient zeta, set to unity in the numerics. No new physical entities are introduced. The qualitative conclusions follow directly from the curvature of the same free energy used to define both the kinetics and the thermodynamics.

free parameters (1)

damping coefficient zeta = 1 (chosen in numerics)
The damping coefficient in the Langevin equation (4) sets the absolute relaxation timescale. The paper sets zeta = 1 in all simulations. Qualitative critical slowing down is independent of its value, but any quantitative comparison with real black hole kinetics would require a physical value.

assumptions (4)

domain assumption The order parameter r evolves according to the Langevin equation with deterministic force -dG/dr and Gaussian white noise satisfying fluctuation-dissipation (Eqs. 4-5).
The kinetic model is posited, not derived from black hole microphysics. It is load-bearing for all dynamical claims in the paper.
ad hoc to paper The overdamped limit (Eq. 6) is valid.
The inertial term in Eq. (4) is dropped without a separation-of-timescales argument. This simplifies the analytics and is used in the simulations.
domain assumption The generalized free energy G(r) in Eq. (1) is the correct canonical thermodynamic potential with r as the order parameter.
This is standard in the extended phase space approach, cited to Refs. [30-33], and is used without re-derivation.
domain assumption The heat-capacity maxima define the thermodynamic Widom line, and this line is comparable to the kinetic crossover.
The comparison in Fig. 7 assumes the standard identification of the Widom line with response-function maxima and that the same free energy governs both thermodynamics and kinetics.

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Cite this review

Pith. "Pith review of Critical slowing down of black hole phase transition and kinetic crossover in supercritical regime." pith.science (2026). https://pith.science/paper/6H5GQ2QZ

@misc{pith2026250524148,
  author       = {Pith},
  title        = {Pith review of: Critical slowing down of black hole phase transition and kinetic crossover in supercritical regime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6H5GQ2QZ}},
  note         = {Machine review of arXiv:2505.24148}
}

read the original abstract

Reissner-Nordstr\"{o}m-Anti-de Sitter (RNAdS) black holes in the extended phase space exhibit critical behavior analogous to the liquid-gas system, with critical exponents matching those of van der Waals-type phase transitions. However, the kinetics of these transitions near spinodal and critical points remain poorly understood. We demonstrate that both the autocorrelation time and the variance of trajectories increase significantly as the system approaches these special points, signaling critical slowing down. This behavior is driven by the flattening of the free energy landscape, as further confirmed by the lowest eigenvalue of the Fokker-Planck equation. Moreover, we uncover a clear dynamical crossover separating gas-like and liquid-like regimes in the supercritical region. This kinetic crossover defines the Widom line that closely matches the thermodynamic one obtained from the maxima of the isobaric heat capacity. These findings contribute to a deeper understanding of the kinetics of RNAdS black holes in the vicinity of spinodal and critical points.

Figures

Figures reproduced from arXiv: 2505.24148 by the authors.

**Figure 2.** FIG. 2: The spinodal diagram and the corresponding land [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3: Landscapes with varying the ensemble temperature [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: FIG. 5: The auto-correlation time [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 4.** Figure 4: FIG. 4: The correlation time [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 7.** Figure 7: FIG. 7: Left panel: 3D plot of the correlation time as the [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

**Figure 8.** Figure 8: FIG. 8: Left panel: 3D plot of the lowest eigenvalues for the [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]

**Figure 9.** Figure 9: FIG. 9: Eigenvalues for the Fokker-Planck equation for the [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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