The idea that black holes possess genuine thermodynamic properties dates back to the foundational work of Jacob Bekenstein, Stephen Hawking and their collaborators, who established that black holes carry a well-defined temperature and entropy proportional to the area of their event horizons. That analogy, however, raised a deeper question: do black holes exhibit the full machinery of thermodynamics, including phase structure and critical phenomena? Research beginning in the 1970s by P.C.W. Davies and P. Hut suggested they might, and the discovery gained fresh momentum after Juan Maldacena’s 1997 AdS/CFT correspondence made asymptotically AdS black holes central to modern theoretical physics.

A pivotal conceptual advance came with the reinterpretation of the cosmological constant as a thermodynamic pressure. In this extended framework, the phase behaviour of charged and rotating AdS black holes closely mirrors the van der Waals liquid-gas transition of ordinary fluids: a small black hole phase corresponds roughly to the gas, a large black hole phase to the liquid, and a first-order transition connects them, complete with a critical point where the distinction between the phases dissolves. The new study asks what happens to the dynamics of such transitions as the critical point is approached, treating the black hole not as a static equilibrium object but as a stochastic system buffeted by thermal fluctuations.

The researchers build on a framework known as free energy landscape dynamics, which has proven powerful in physics, chemistry and biology, from protein folding to chemical reactions. In this picture, the thermodynamic states of a system are valleys on a landscape defined by its free energy, and thermal noise jiggles the system between them. Previously, Rong-Gen Li and Jin Wang and their collaborators applied this framework to the Hawking-Page transition and to the small-large black hole transition of Reissner-Nordström AdS (RN-AdS) black holes, showing that stochastic switching between phases can occur in both directions. More recently, it was shown that near the critical and spinodal points of the RN-AdS system, the relaxation dynamics slows dramatically.

The Dibrugarh team extends this analysis to rotating Kerr-AdS black holes, and in doing so makes a deliberate technical choice: rather than using the horizon radius as the fluctuating order parameter, they treat the black hole’s entropy as the dynamical variable evolving on the free energy landscape. For Kerr-AdS black holes, the generalized Gibbs free energy in the canonical ensemble is most naturally written as a function of the entropy, and the entropy uniquely labels each equilibrium macrostate. The authors also show that this choice does not affect the universal long-time behaviour, because entropy and horizon radius are related by a smooth transformation that merely relabels coordinates on the same thermodynamic manifold.

The mathematical core of the analysis is a Langevin equation: a stochastic differential equation in which the entropy evolves under a deterministic driving force generated by the slope of the free energy, a friction term describing dissipation into the thermal environment, and a Gaussian white noise term constrained by the fluctuation-dissipation relation. Far from criticality, the free energy well surrounding a stable state is approximately parabolic, and perturbations decay exponentially with a characteristic time given by the damping coefficient divided by the curvature of the free energy at equilibrium. At the critical point, however, the first three derivatives of the free energy with respect to entropy vanish, and the quadratic approximation collapses entirely.

That collapse is the origin of critical slowing down. As the landscape flattens, the restoring force that pulls fluctuations back toward equilibrium weakens and eventually disappears, so fluctuations persist for ever longer times. The researchers demonstrate this both analytically and numerically. Simulating the Langevin equation with a Heun predictor-corrector scheme, they extract the autocorrelation time and the variance of the entropy trajectories, finding that both rise sharply near the critical point and near the spinodal lines where one of the black hole phases ceases to exist. Independently, they solve the associated Fokker-Planck equation, which describes the probability distribution of the entropy, and find that its smallest nonzero eigenvalue, which sets the slowest relaxation rate, is strongly suppressed near criticality, confirming the same physics from the spectral side.

The study’s most striking result concerns universality. Fitting the numerically obtained relaxation time to a power law of the form tau proportional to the reduced distance from criticality raised to a negative exponent, the team recovers a dynamical critical exponent of approximately two-thirds along every path they examined, whether varying the temperature at fixed pressure or angular momentum, varying the pressure at fixed temperature, or varying the angular momentum at fixed temperature, and regardless of whether the critical point is approached from above or below. Analytically, this exponent follows from a mean-field argument: near a critical inflection point the order parameter scales as the cube root of the distance from criticality, so the free energy curvature scales as the two-thirds power, and the relaxation time, its inverse, diverges as the minus two-thirds power.

Remarkably, the same exponent emerges for three physically distinct black hole families: charged RN-AdS black holes, rotating Kerr-AdS black holes, and Bardeen black holes, an early example of regular black holes whose cores are nonsingular and which satisfy the weak energy condition. Despite radically different spacetime geometries and thermodynamic variables, all three systems realize identical dynamical scaling, placing them in the same mean-field dynamical universality class. The conclusion is that critical slowing down is governed not by the microscopic details of the black hole solution but by the generic structure of the free energy landscape, specifically its quartic form at a critical inflection point, echoing the logic of universality that underpins conventional critical phenomena.

The work connects black hole thermodynamics to a web of ideas spanning the Kibble-Zurek mechanism of defect formation in cosmological phase transitions, early-warning signals of critical transitions in complex systems, and the time-dependent Ginzburg-Landau theory of near-critical dynamics. The authors suggest several future directions, including extensions to higher-dimensional black holes, modified gravity theories and multicritical systems, and possible links to quasinormal modes, thermodynamic geometry, Lyapunov exponents and holographic nonequilibrium phenomena. For now, the message is conceptually simple and profound: a black hole poised at a thermodynamic critical point forgets its past ever more slowly, and the way it forgets obeys a law that charged, rotating and even regular black holes all share. In the slow drift toward a phase transition, gravity and statistical mechanics appear to speak the same universal language.

Subject of Research: Universal dynamical scaling and critical slowing down in black hole phase transitions in anti-de Sitter spacetime

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

Article References: Awal, M. B., & Phukon, P. (2026). Critical slowing down of black hole phase transition and universal dynamic scaling in AdS black holes. The European Physical Journal C, 86(9), Article 1058. https://doi.org/10.1140/epjc/s10052-026-16329-0

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16329-0

Keywords: black hole thermodynamics, phase transitions, critical slowing down, AdS black holes, free energy landscape, Langevin equation, Fokker-Planck equation, Kerr-AdS black holes, RN-AdS black holes, Bardeen black holes, universality, dynamical critical exponent