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

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, $τ=|ε|^{-2/3}$ 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.

• The paper establishes that black hole phase transitions display critical slowing down with a universal exponent of 2/3 across various settings.
• The paper employs a stochastic free energy landscape framework using Langevin and Fokker-Planck dynamics to model entropy fluctuations near criticality.
• The paper highlights that the universality of the dynamic scaling law connects holographic principles with nonequilibrium gravitational physics.

Critical Slowing Down and Universal Dynamic Scaling in AdS Black Hole Phase Transitions

Introduction

The study examines the dynamics of phase transitions in AdS black holes, focusing on the phenomenon of critical slowing down—a hallmark of continuous phase transitions characterized by diverging relaxation times near criticality. By extending the stochastic free energy landscape framework to Kerr-AdS black holes, and benchmarking results against RN-AdS and Bardeen black holes, the work establishes the existence of a universal dynamical critical exponent governing the relaxation near critical points.

Stochastic Free Energy Landscape and Black Hole Phase Transitions

Departing from classical equilibrium thermodynamics, the research leverages a stochastic dynamical approach to black hole phase transitions. The free energy landscape, with black hole entropy as the order parameter, forms the basis for analyzing dynamics under thermal fluctuations. The Langevin equation, incorporating dissipation and noise, models the evolution of entropy, while the associated Fokker-Planck equation characterizes the probabilistic relaxation toward equilibrium states.

This approach contextualizes black hole phase transitions within the broader paradigm of nonequilibrium statistical mechanics, aligning with frameworks used in condensed matter and molecular systems where phase kinetics are controlled by the underlying landscape geometry and fluctuation-dissipation mechanisms.

Critical Slowing Down in Kerr-AdS Black Holes

Analytical and numerical investigations both reveal pronounced critical slowing down in Kerr-AdS black holes. As the critical point (in temperature, pressure, or angular momentum) is approached, the curvature of the free energy landscape at equilibrium flattens, dramatically reducing the effective restoring force. This flattening leads to a divergence of the autocorrelation time for entropy fluctuations, accompanied by an increase in fluctuation variance, and is corroborated by a vanishing smallest nonzero eigenvalue of the Fokker-Planck operator.

Distinct thermodynamic paths (e.g., varying temperature at fixed angular momentum or vice versa) consistently display these signatures, with the dynamical behavior remaining robust under the choice of order parameter (entropy or horizon radius).

Universal Dynamic Scaling and the Critical Exponent

A central result is the extraction of a universal scaling law for the relaxation time:

$\tau \sim |E|^{-\lambda}$

where $E$ is the reduced distance from the critical point ($E = (T-T_c)/T_c$, or analogous expressions for pressure and angular momentum), and the dynamical exponent $\lambda \approx 2/3$. This scaling relation is numerically validated not just for Kerr-AdS black holes, but also for RN-AdS and regular Bardeen black holes, and for all relevant thermodynamic routes to criticality.

The exponent’s universality is further endorsed by analysis of the Fokker-Planck spectrum: the smallest non-zero eigenvalue $\lambda_1$ scales as $\lambda_1 \sim |E|^{2/3}$, confirming that relaxation times diverge with the same critical exponent regardless of the specific black hole type. Analytical derivation from the critical expansion of the free energy landscape supports the robustness and generality of this critical exponent.

Theoretical and Practical Implications

The findings indicate that the critical slowing down in AdS black hole phase transitions is governed by the structure and flattening of the free energy landscape near criticality, rather than the microphysical peculiarities of different black hole spacetimes. The emergence of identical scaling exponents in charged, rotating, and regular black hole backgrounds points to a universality class for black hole dynamical critical phenomena, analogous to those in statistical physics models of continuous phase transitions.

This universality has several implications:

• Holography and AdS/CFT: Since black hole thermodynamics in AdS spacetime underlies quantum field theory dynamics via AdS/CFT, the results suggest that critical slowing down and associated dynamic scaling may have field-theoretic analogs in large-$N$ gauge theories.
• Dynamical Probes of Black Hole Microstructure: The dynamical critical exponent provides a new invariant for classifying black hole phase transitions, complementary to standard thermodynamic quantities.
• Connections to Nonequilibrium Gravitational Physics: The framework paves the way for further research intersecting black hole dynamics, nonequilibrium statistical mechanics, and dynamical systems theory, including potential links to early-warning signals for critical gravitational transitions.

Future Directions

The study identifies several promising avenues for further research:

• Extending analysis to higher-dimensional, multi-parameter, or modified gravity black holes to test the limits of the observed universality.
• Investigating connections between the dynamical critical exponent and alternative diagnostic tools, such as quasinormal mode spectra, Lyapunov exponents, or thermodynamic geometry.
• Exploring the interplay between stochastic critical dynamics in gravity and holographic nonequilibrium phenomena, potentially yielding new insights into quantum criticality in dual gauge theories.

Conclusion

The work rigorously establishes that critical slowing down in the dynamical phase transitions of AdS black holes is characterized by a universal scaling law with the dynamical exponent $\lambda = 2/3$, independent of black hole charge, rotation, or regularity. This universality emerges from the generic features of the free energy landscape near criticality and extends the toolkit of black hole thermodynamics into the nonequilibrium regime, embedding black hole critical phenomena firmly within the broader landscape of statistical physics and dynamical systems. The results prompt further theoretical investigations into black hole dynamics and their holographic implications, potentially enriching our understanding of gravitational and quantum criticality.

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Reference: "Critical slowing down of black hole phase transition and universal dynamic scaling in AdS black holes" (2605.15655)