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arxiv: 2605.15655 · v1 · pith:SGAB7A7Fnew · submitted 2026-05-15 · ✦ hep-th

Critical slowing down of black hole phase transition and universal dynamic scaling in AdS black holes

Mozib Bin Awal , Prabwal Phukon This is my paper

Pith reviewed 2026-05-20 18:10 UTC · model grok-4.3

classification ✦ hep-th
keywords black hole phase transitionscritical slowing downuniversal scalingAdS black holesLangevin dynamicsfree energy landscaperelaxation timeFokker-Planck equation
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The pith

Black hole phase transitions exhibit critical slowing down with relaxation time scaling as |ε|^{-2/3} across multiple systems.

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

The paper extends a stochastic model of free energy landscape dynamics to Kerr-AdS black holes by evolving the entropy variable with a Langevin equation. Near the critical point of the phase transition, the autocorrelation time grows sharply, confirming critical slowing down through both direct simulation and the lowest eigenvalue of the Fokker-Planck operator. The relaxation time follows the power-law relation τ = |ε|^{-2/3} when the system is driven toward or away from criticality along different thermodynamic paths. The identical exponent appears for RN-AdS, Kerr-AdS, and Bardeen black holes, indicating a common dynamical scaling that links the geometry of the free energy surface to nonequilibrium critical phenomena in black hole thermodynamics.

Core claim

Modeling the nonequilibrium dynamics of Kerr-AdS black holes via a stochastic Langevin equation for the entropy on the free energy landscape reveals critical slowing down near the phase transition. The autocorrelation time increases markedly as the critical point is approached, and the lowest eigenvalue of the Fokker-Planck equation tracks this divergence. Along paths varying temperature, pressure, or angular momentum, the relaxation time obeys the scaling τ = |ε|^{-2/3}. The same exponent is recovered for RN-AdS and Bardeen black holes, establishing an underlying universal dynamical behavior across distinct black hole families.

What carries the argument

the stochastic Langevin evolution of entropy on the free energy landscape, whose autocorrelation time diverges near criticality and yields the scaling exponent -2/3.

If this is right

  • The scaling exponent -2/3 appears identically in RN-AdS, Kerr-AdS, and Bardeen black holes.
  • Autocorrelation time grows as the critical point is approached from any thermodynamic direction.
  • The scaling persists for both approach to and departure from criticality.
  • Geometry of the free energy landscape is directly tied to the observed universal critical slowing down.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The shared exponent suggests the scaling may appear in other AdS black holes with similar thermodynamic phase structures.
  • Holographic duals of these black holes could be examined for matching dynamical scaling in boundary field theories.
  • The result offers a concrete route to test whether black hole thermodynamics obeys the same dynamic universality classes as condensed-matter systems.

Load-bearing premise

The stochastic Langevin framework previously used for RN-AdS black holes applies unchanged when the entropy variable is evolved for Kerr-AdS black holes and accurately describes the dynamics near the phase transition.

What would settle it

Numerical integration of the Langevin equation for entropy at a small reduced parameter ε = 0.001 that yields an autocorrelation time inconsistent with |ε|^{-2/3} would falsify the reported scaling.

read the original abstract

We investigate the dynamical critical behaviour of black hole phase transitions in anti de Sitter spacetime by extending the stochastic framework of free energy landscape dynamics to Kerr AdS black holes. By analyzing the Langevin evolution of the entropy (in contrast to the horizon radius in the RN-AdS case) near criticality, we demonstrate that the system exhibits pronounced critical slowing down, characterized by a significant increase of the autocorrelation time as the critical point is approached. This behaviour is further confirmed by the lowest eigenvalue of the Fokker-Planck equation. By analysing the dynamics along different thermodynamic paths, including variations in temperature, pressure, and angular momentum, and considering both directions - towards and away from criticality, we find that the relaxation time obeys a robust scaling relation, $\tau=|\epsilon|^{-2/3}$ near criticality. The same scaling exponent is obtained for RN-AdS, Kerr-AdS, and Bardeen black holes, suggesting the existence of an underlying universal dynamical behaviour across distinct black hole systems. Our results establish a connection between the geometry of the free energy landscape, stochastic nonequilibrium dynamics, and universal critical phenomena in black hole thermodynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript extends the stochastic free energy landscape framework to Kerr-AdS black holes by applying Langevin dynamics to the entropy variable (contrasted with horizon radius for RN-AdS). It reports critical slowing down via increased autocorrelation time and the lowest eigenvalue of the Fokker-Planck operator, with the relaxation time obeying the scaling τ = |ε|^{-2/3} near criticality. The same exponent is obtained for RN-AdS, Kerr-AdS, and Bardeen black holes, which is presented as evidence for universal dynamical behavior across distinct black hole systems.

Significance. If the scaling holds after addressing the variable-choice issue, the result would indicate a universal dynamic exponent for critical slowing down in AdS black hole thermodynamics. This connects the geometry of the free-energy landscape to nonequilibrium stochastic dynamics and critical phenomena, with the consistency across three black hole families serving as a strength of the universality argument.

major comments (1)
  1. [Kerr-AdS Langevin/Fokker-Planck analysis] The abstract (and corresponding analysis) switches the stochastic variable from horizon radius r+ (RN-AdS) to entropy S (Kerr-AdS) while retaining the same Langevin/Fokker-Planck framework. No explicit derivation is supplied showing that the drift and diffusion coefficients transform such that the lowest eigenvalue of the Fokker-Planck operator (or autocorrelation time) retains the identical |ε|^{-2/3} scaling near the critical point. Because the free-energy landscape is expressed in different coordinates, changes in effective potential curvature and measure could alter the relaxation dynamics; this step is load-bearing for the universality claim across systems.
minor comments (1)
  1. The abstract states that dynamics are analyzed along paths varying temperature, pressure, and angular momentum in both directions toward and away from criticality; explicit definitions of the reduced parameter ε along each path and the fitting procedure used to extract the exponent would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment regarding the choice of stochastic variable and the need for an explicit derivation of the scaling invariance below.

read point-by-point responses

  1. Referee: [Kerr-AdS Langevin/Fokker-Planck analysis] The abstract (and corresponding analysis) switches the stochastic variable from horizon radius r+ (RN-AdS) to entropy S (Kerr-AdS) while retaining the same Langevin/Fokker-Planck framework. No explicit derivation is supplied showing that the drift and diffusion coefficients transform such that the lowest eigenvalue of the Fokker-Planck operator (or autocorrelation time) retains the identical |ε|^{-2/3} scaling near the critical point. Because the free-energy landscape is expressed in different coordinates, changes in effective potential curvature and measure could alter the relaxation dynamics; this step is load-bearing for the universality claim across systems.

    Authors: We thank the referee for highlighting this important technical point. The choice of entropy S for the Kerr-AdS analysis (versus r+ for RN-AdS) follows from the natural thermodynamic coordinate in which the phase structure and free-energy landscape are expressed for rotating black holes. To strengthen the universality argument, we will add a new appendix in the revised manuscript that explicitly derives the transformation of the Langevin equation and Fokker-Planck operator under the change of stochastic variable. Using the Itô calculus chain rule, the drift term acquires a Jacobian correction while the diffusion coefficient transforms with the square of the derivative; near criticality the effective potential retains its characteristic cubic inflection point, so the vanishing curvature that controls the slowest relaxation mode yields the same |ε|^{-2/3} scaling for the lowest eigenvalue and autocorrelation time. This coordinate-independent behavior near the mean-field critical point supports the reported universality across the three black-hole families. The revised manuscript will contain this derivation to make the argument fully self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity: scaling emerges from Langevin/Fokker-Planck analysis on distinct free-energy landscapes

full rationale

The paper applies the stochastic Langevin framework to the entropy variable for Kerr-AdS (contrasted explicitly with horizon radius for RN-AdS) and extracts the relaxation-time scaling τ = |ε|^{-2/3} by direct analysis of autocorrelation times and the lowest eigenvalue of the Fokker-Planck operator near criticality. This is presented as a computed result along multiple thermodynamic paths rather than a fitted parameter or self-defined quantity. The same exponent appearing for RN-AdS, Kerr-AdS and Bardeen cases is reported as an observed outcome of the dynamics, not inserted by construction or justified solely via self-citation. No load-bearing step reduces the central claim to a prior input by definition; the derivation remains self-contained against the free-energy landscape and stochastic evolution.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of stochastic dynamics to black hole entropy evolution and on standard thermodynamic relations in AdS spacetime; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The stochastic Langevin evolution of entropy on the free energy landscape accurately models the nonequilibrium dynamics of black hole phase transitions.
    The paper extends this framework from RN-AdS to Kerr-AdS without additional justification in the abstract.

pith-pipeline@v0.9.0 · 5733 in / 1221 out tokens · 68323 ms · 2026-05-20T18:10:17.361502+00:00 · methodology

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