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arxiv: 2605.25497 · v1 · pith:OY5N2H7Onew · submitted 2026-05-25 · ✦ hep-th

Degenerate Bifurcations and Universal Relaxation Scaling in Black Hole Thermodynamics

Bidyut Hazarika , Mozib Bin Awal , Prabwal Phukon This is my paper

Pith reviewed 2026-06-29 21:00 UTC · model grok-4.3

classification ✦ hep-th
keywords black hole thermodynamicsthermodynamic criticalitybifurcation theoryuniversality classescritical slowing downrelaxation scalingthermodynamic floweffective landscape
0
0 comments X

The pith

Black hole thermodynamics near critical points reduces to universal bifurcation forms that define universality classes and set relaxation timescales.

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

The paper models black hole thermodynamics as a dynamical system on an effective landscape in state space, where a phenomenological flow parameter tau drives relaxation toward equilibrium fixed points. Near critical points this flow collapses to simplified universal mathematical forms fixed by the local bifurcation structure. Different black holes can therefore be grouped into universality classes according to the type of bifurcation they exhibit at criticality. The same structure produces critical slowing down, with the relaxation timescale fixed entirely by the bifurcation rather than by global details of the black hole.

Core claim

By treating black hole thermodynamics as a dynamical flow toward equilibrium fixed points governed by a relaxation parameter tau, the approach to criticality simplifies into universal bifurcation equations. These equations organize black holes into universality classes determined by their local bifurcation type and imply that relaxation timescales near criticality are set solely by that bifurcation structure, producing critical slowing down.

What carries the argument

Effective thermodynamic landscape whose flow toward equilibrium fixed points is described by bifurcation equations with phenomenological relaxation parameter tau

If this is right

  • Black holes whose critical points share the same bifurcation type belong to the same universality class and exhibit identical critical scaling.
  • Relaxation timescales near criticality are completely determined by the bifurcation structure and independent of other thermodynamic details.
  • Critical slowing down occurs universally as the system approaches the critical point, with the divergence of the timescale fixed by the bifurcation.
  • The classification into universality classes applies across different families of black holes that share the same local bifurcation.

Where Pith is reading between the lines

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

  • The same bifurcation-based reduction could be applied to other gravitational or holographic systems that possess thermodynamic critical points.
  • If the effective landscape picture holds, it supplies a concrete way to compute relaxation dynamics that could be compared with time-dependent black-hole solutions.
  • The universality classes identified here might overlap with those already known from mean-field or renormalization-group treatments of black-hole criticality.

Load-bearing premise

Black hole thermodynamics admits an effective dynamical landscape in state space whose approach to equilibrium is governed by a phenomenological flow parameter that obeys standard bifurcation equations.

What would settle it

A explicit calculation for a known black-hole critical point in which the measured or computed relaxation time scaling deviates from the exponent predicted by the local bifurcation type.

Figures

Figures reproduced from arXiv: 2605.25497 by Bidyut Hazarika, Mozib Bin Awal, Prabwal Phukon.

Figure 1
Figure 1. Figure 1: FIG. 1: Slowing down of black hole solution near fixed [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
read the original abstract

We present a dynamical systems approach to black hole thermodynamic criticality based on bifurcation equations. We construct an effective thermodynamic landscape in which black holes relax toward equilibrium fixed points. To describe this process, we introduce a flow parameter $\tau$, interpreted as a phenomenological relaxation time, which governs the approach toward equilibrium configurations in thermodynamic state space. Near critical points, the thermodynamic flow simplifies into universal mathematical forms, which allows different black holes to be grouped into different universality classes based on their critical behaviour. Our analysis further shows critical slowing down, with relaxation timescales determined entirely by the local bifurcation structure.

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

2 major / 1 minor

Summary. The paper presents a dynamical systems approach to black hole thermodynamic criticality based on bifurcation equations. It constructs an effective thermodynamic landscape in which black holes relax toward equilibrium fixed points, governed by a phenomenological flow parameter τ interpreted as a relaxation time. Near critical points the flow reduces to universal mathematical forms, permitting black holes to be grouped into universality classes according to their critical behavior, and the analysis identifies critical slowing down whose timescales are fixed by the local bifurcation structure.

Significance. If the mapping from black-hole equations of state to bifurcation normal forms can be placed on a firmer footing, the framework would supply a systematic way to classify thermodynamic critical points and to extract universal relaxation exponents from the local geometry of the state space. The approach is novel in its explicit use of degenerate bifurcation theory for this purpose.

major comments (2)
  1. [Abstract] Abstract: the flow parameter τ is introduced purely phenomenologically as a relaxation time; the subsequent claims of universal scaling and critical slowing down then follow automatically once the normal form is assumed, without an explicit derivation of the effective dynamical landscape from the first law, Smarr relation, or linearized Einstein equations.
  2. [Abstract] Abstract: no explicit map is supplied from a given black-hole equation of state to the coefficients appearing in the bifurcation equation, so it remains unclear whether the asserted universality classes are determined by black-hole physics or by the choice of normal form.
minor comments (1)
  1. The abstract supplies no concrete equations, examples, or numerical checks, making it difficult to assess the concrete content of the universality classes or the critical-slowing-down predictions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond point by point to the major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the flow parameter τ is introduced purely phenomenologically as a relaxation time; the subsequent claims of universal scaling and critical slowing down then follow automatically once the normal form is assumed, without an explicit derivation of the effective dynamical landscape from the first law, Smarr relation, or linearized Einstein equations.

    Authors: We agree that τ is introduced phenomenologically, as stated in the abstract and manuscript. The framework is an effective dynamical-systems description of relaxation in thermodynamic state space; the universal scaling and critical slowing down are mathematical consequences of the local normal forms once the bifurcation type is identified from the equation of state. No derivation from the first law, Smarr relation, or linearized Einstein equations is attempted or claimed. We will revise the abstract and add a short clarifying paragraph in the introduction to emphasize the effective character of the model. revision: yes

  2. Referee: [Abstract] Abstract: no explicit map is supplied from a given black-hole equation of state to the coefficients appearing in the bifurcation equation, so it remains unclear whether the asserted universality classes are determined by black-hole physics or by the choice of normal form.

    Authors: The referee correctly observes that a general explicit map from an arbitrary equation of state to the numerical coefficients is not supplied. The normal form itself, however, is chosen according to the degeneracy order of the critical point in the black-hole thermodynamic potential, which is fixed by the equation of state. The universality classes are therefore determined by black-hole physics through this degeneracy structure. We will add an explicit worked example (e.g., for the RN-AdS black hole) showing how the equation of state determines both the normal-form type and the leading coefficients. revision: yes

Circularity Check

1 steps flagged

Universality and critical slowing down reduce to assumed bifurcation normal forms with phenomenological τ

specific steps
  1. self definitional [Abstract]
    "we introduce a flow parameter τ, interpreted as a phenomenological relaxation time, which governs the approach toward equilibrium configurations in thermodynamic state space. Near critical points, the thermodynamic flow simplifies into universal mathematical forms, which allows different black holes to be grouped into different universality classes based on their critical behaviour. Our analysis further shows critical slowing down, with relaxation timescales determined entirely by the local bifurcation structure."

    Once the thermodynamic flow is defined to be governed by τ in a dynamical system whose near-critical behavior is modeled by bifurcation equations, the simplification to universal forms, the grouping into universality classes, and the determination of timescales solely by the local bifurcation structure are direct mathematical properties of the chosen normal forms; they hold by construction of the effective landscape rather than emerging from the underlying physics.

full rationale

The paper's central results on universal mathematical forms, universality classes, and relaxation timescales determined by bifurcation structure follow directly from the introduction of an effective landscape governed by a phenomenological flow parameter τ inserted into standard bifurcation equations. This modeling choice makes the claimed simplifications and determinations mathematical consequences of the ansatz rather than independent derivations from black-hole thermodynamics (first law, Smarr relation, or Einstein equations). No load-bearing external derivation or falsifiable map is indicated in the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that thermodynamic state space for black holes can be equipped with a dynamical flow governed by a phenomenological relaxation parameter tau whose near-critical behavior is captured by standard bifurcation theory.

free parameters (1)
  • flow parameter τ
    Phenomenological relaxation time introduced to govern the approach toward equilibrium configurations.
axioms (1)
  • domain assumption Thermodynamic state space admits an effective dynamical landscape with equilibrium fixed points
    The paper assumes black hole thermodynamics can be modeled as a flow toward fixed points using bifurcation equations.

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Works this paper leans on

67 extracted references · 42 canonical work pages · 22 internal anchors

  1. [1]

    J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333-2346 (1973) doi:10.1103/PhysRevD.7.2333

    work page doi:10.1103/physrevd.7.2333 1973

  2. [2]

    S. W. Hawking, Black hole explosions, Nature248, 30-31 (1974) doi:10.1038/248030a0

    work page doi:10.1038/248030a0 1974

  3. [3]

    S. W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys.43, 199-220 (1975) [er- ratum: Commun. Math. Phys.46, 206 (1976)] doi:10.1007/BF02345020

    work page doi:10.1007/bf02345020 1975

  4. [4]

    J. M. Bardeen, B. Carter and S. W. Hawking, The Four laws of black hole mechanics, Commun. Math. Phys.31, 161-170 (1973) doi:10.1007/BF01645742

    work page doi:10.1007/bf01645742 1973

  5. [5]

    R. M. Wald, Entropy and black-hole thermo- dynamics, Phys. Rev. D20, 1271-1282 (1979) doi:10.1103/PhysRevD.20.1271

    work page doi:10.1103/physrevd.20.1271 1979

  6. [6]

    Black-hole thermodynamics, Physics Today, 33(1):24–31, 1980

    Jacob D Bekenstein. Black-hole thermodynamics, Physics Today, 33(1):24–31, 1980

    1980

  7. [7]

    R. M. Wald, The thermodynamics of black holes, Living Rev. Rel.4, 6 (2001) doi:10.12942/lrr-2001-6 [arXiv:gr- qc/9912119 [gr-qc]]

    work page doi:10.12942/lrr-2001-6 2001

  8. [8]

    S.Carlip, Black Hole Thermodynamics, Int. J. Mod. Phys. D23, 1430023 (2014) doi:10.1142/S0218271814300237 [arXiv:1410.1486 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0218271814300237 2014

  9. [9]

    A.C.Wall, A Survey of Black Hole Thermodynamics, [arXiv:1804.10610 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv

  10. [10]

    P.Candelas and D.W.Sciama, Irreversible Thermody- namics of Black Holes, Phys. Rev. Lett.38, 1372-1375 (1977) doi:10.1103/PhysRevLett.38.1372

    work page doi:10.1103/physrevlett.38.1372 1977

  11. [11]

    On Black Hole Entropy Corrections in the Grand Canonical Ensemble

    S. Mahapatra, P. Phukon and T. Sarkar, Phys. Rev. D84, 044041 (2011) doi:10.1103/PhysRevD.84.044041 [arXiv:1103.5885 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.84.044041 2011

  12. [12]

    P. C. W. Davies, Thermodynamic Phase Transitions of Kerr-Newman Black Holes in De Sitter Space, Class. Quant. Grav.6, 1909 (1989) doi:10.1088/0264- 9381/6/12/018

    work page doi:10.1088/0264- 1909

  13. [13]

    S. W. Hawking and D. N. Page, Thermodynamics of Black Holes in anti-De Sitter Space, Commun. Math. Phys.87, 577 (1983) doi:10.1007/BF01208266

    work page doi:10.1007/bf01208266 1983

  14. [14]

    Curir, Rotating black holes as dissipative spin- thermodynamical systems, General Relativity and Grav- itation,13, 417, (1981) doi:10.1007/BF00756588

    A. Curir, Rotating black holes as dissipative spin- thermodynamical systems, General Relativity and Grav- itation,13, 417, (1981) doi:10.1007/BF00756588

    work page doi:10.1007/bf00756588 1981

  15. [15]

    Anna Curir, Black hole emissions and phase transitions, General Relativity and Gravitation,13, 1177, (1981) doi:10.1007/BF00759866

    work page doi:10.1007/bf00759866 1981

  16. [16]

    Pavon and J

    D. Pavon and J. M. Rubi, Nonequilibrium Thermody- namic Fluctuations of Black Holes, Phys. Rev. D37, 2052-2058 (1988) doi:10.1103/PhysRevD.37.2052

    work page doi:10.1103/physrevd.37.2052 2052

  17. [17]

    Pavon, Phase transition in Reissner-Nordstrom black holes, Phys

    D. Pavon, Phase transition in Reissner-Nordstrom black holes, Phys. Rev. D43, 2495-2497 (1991) doi:10.1103/PhysRevD.43.2495

    work page doi:10.1103/physrevd.43.2495 1991

  18. [18]

    Kaburaki, Critical behavior of extremal Kerr-Newman black holes, Gen

    O. Kaburaki, Critical behavior of extremal Kerr-Newman black holes, Gen. Rel. Grav.28, 843 (1996)

    1996

  19. [19]

    R. G. Cai, Z. J. Lu and Y. Z. Zhang, Critical behav- ior in (2+1)-dimensional black holes, Phys. Rev. D55, 853-860 (1997) doi:10.1103/PhysRevD.55.853 [arXiv:gr- qc/9702032 [gr-qc]]

    work page doi:10.1103/physrevd.55.853 1997

  20. [20]

    R. G. Cai and J. H. Cho, Thermodynamic curvature of the BTZ black hole, Phys. Rev. D60, 067502 (1999) doi:10.1103/PhysRevD.60.067502 [arXiv:hep-th/9803261 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.60.067502 1999

  21. [21]

    Y. H. Wei, Thermodynamic critical and geometrical properties of charged BTZ black hole, Phys. Rev. D80, 024029 (2009) doi:10.1103/PhysRevD.80.024029

    work page doi:10.1103/physrevd.80.024029 2009

  22. [22]

    A general framework to study the extremal phase transition of black holes

    K. Bhattacharya, S. Dey, B. R. Majhi and S. Samanta, General framework to study the extremal phase transi- tion of black holes, Phys. Rev. D99, no.12, 124047 (2019) doi:10.1103/PhysRevD.99.124047 [arXiv:1903.03434 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.99.124047 2019

  23. [23]

    Enthalpy and the Mechanics of AdS Black Holes

    D. Kastor, S. Ray and J. Traschen, Enthalpy and the Mechanics of AdS Black Holes, Class. Quant. Grav. 26, 195011 (2009) doi:10.1088/0264-9381/26/19/195011 [arXiv:0904.2765 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/26/19/195011 2009

  24. [24]

    B. P. Dolan, The cosmological constant and the black hole equation of state, Class. Quant. Grav. 28, 125020 (2011) doi:10.1088/0264-9381/28/12/125020 [arXiv:1008.5023 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/28/12/125020 2011

  25. [25]

    B. P. Dolan, Pressure and volume in the first law of black hole thermodynamics, Class. Quant. Grav. 28, 235017 (2011) doi:10.1088/0264-9381/28/23/235017 [arXiv:1106.6260 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/28/23/235017 2011

  26. [26]

    B. P. Dolan, Compressibility of rotating black holes, Phys. Rev. D84, 127503 (2011) doi:10.1103/PhysRevD.84.127503 [arXiv:1109.0198 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.84.127503 2011

  27. [27]

    B. P. Dolan, Where Is the PdV in the First Law of Black Hole Thermodynamics?, doi:10.5772/52455 [arXiv:1209.1272 [gr-qc]]

    work page doi:10.5772/52455

  28. [28]

    P-V criticality of charged AdS black holes

    D. Kubiznak and R. B. Mann, P-V criticality of charged AdS black holes, JHEP07, 033 (2012) doi:10.1007/JHEP07(2012)033 [arXiv:1205.0559 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep07(2012)033 2012

  29. [29]

    Black hole chemistry: thermodynamics with Lambda

    D. Kubiznak, R. B. Mann and M. Teo, Black hole chemistry: thermodynamics with Lambda, Class. Quant. Grav.34, no.6, 063001 (2017) doi:10.1088/1361- 6382/aa5c69 [arXiv:1608.06147 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1361- 2017

  30. [30]

    Sadeghi, M

    J. Sadeghi, M. Shokri, S. Gashti Noori and M. R. Alipour, RPS thermodynamics of Taub–NUT AdS black holes in the presence of central charge and the weak gravity con- jecture. Gen. Rel. Grav.54(2022)

    2022

  31. [31]

    Ladghami, B

    Y. Ladghami, B. Asfour, A. Bouali, A. Errahmani and 7 T. Ouali, 4D-EGB black holes in RPS thermodynamics, Phys. Dark Univ.41(2023),

    2023

  32. [32]

    X. Kong, Z. Zhang and L. Zhao, Restricted phase space thermodynamics of charged AdS black holes in conformal gravity, Chin. Phys. C47(2023)

    2023

  33. [33]

    M. R. Alipour, J. Sadeghi and M. Shokri, WGC and WCCC of black holes with quintessence and cloud strings in RPS space, Nucl. Phys. B990(2023)

    2023

  34. [34]

    Wang and L

    T. Wang and L. Zhao, Black hole thermodynamics is extensive with variable Newton constant, Phys. Lett. B 827(2022)

    2022

  35. [35]

    Zeyuan and L

    G. Zeyuan and L. Zhao, Restricted phase space ther- modynamics for AdS black holes via holography, Class. Quant. Grav.39(2022)

    2022

  36. [36]

    Z. Gao, X. Kong and L. Zhao, Thermodynamics of Kerr- AdS black holes in the restricted phase space, Eur. Phys. J. C82(2022)

    2022

  37. [37]

    Dutta and G

    S. Dutta and G. S. Punia, String theory corrections to holographic black hole chemistry, Phys. Rev. D106, no.2, 026003 (2022)

    2022

  38. [38]

    T. F. Gong, J. Jiang and M. Zhang, Holographic thermo- dynamics of rotating black holes, JHEP06, 105 (2023)

    2023

  39. [39]

    W. Cong, D. Kubiznak, R. B. Mann and M. R. Visser, Holographic CFT phase transitions and criticality for charged AdS black holes, JHEP08, 174 (2022)

    2022

  40. [40]

    M. R. Visser, Holographic thermodynamics requires a chemical potential for color, Phys. Rev. D105, no.10, 106014 (2022)

    2022

  41. [41]

    Geometric description of BTZ black holes thermodynamics

    H. Quevedo and A. Sanchez, Geometric description of BTZ black holes thermodynamics, Phys. Rev. D79, 024012 (2009) doi:10.1103/PhysRevD.79.024012 [arXiv:0811.2524 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.79.024012 2009

  42. [42]

    Thermodynamic Geometry Of Charged Rotating BTZ Black Holes

    M. Akbar, H. Quevedo, K. Saifullah, A. Sanchez and S. Taj, Thermodynamic Geometry Of Charged Rotat- ing BTZ Black Holes, Phys. Rev. D83, 084031 (2011) doi:10.1103/PhysRevD.83.084031 [arXiv:1101.2722 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.83.084031 2011

  43. [43]

    S. H. Hendi and R. Naderi, Geometrothermody- namics of black holes in Lovelock gravity with a nonlinear electrodynamics, Phys. Rev. D91, no.2, 024007 (2015) doi:10.1103/PhysRevD.91.024007 [arXiv:1510.06269 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.91.024007 2015

  44. [44]

    Sarkar, G

    T. Sarkar, G. Sengupta and B. Nath Tiwari, On the ther- modynamic geometry of BTZ black holes, JHEP11, 015 (2006) doi:10.1088/1126-6708/2006/11/015 [arXiv:hep- th/0606084 [hep-th]]

    work page doi:10.1088/1126-6708/2006/11/015 2006

  45. [45]

    S. H. Hendi, A. Sheykhi, S. Panahiyan and B. Eslam Panah, Phase transition and thermodynamic geometry of Einstein-Maxwell-dilaton black holes, Phys. Rev. D92, no.6, 064028 (2015) doi:10.1103/PhysRevD.92.064028 [arXiv:1509.08593 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.92.064028 2015

  46. [46]

    Thermogeometric phase transition in a unified framework

    R. Banerjee, B. R. Majhi and S. Samanta, Thermogeometric phase transition in a unified framework, Phys. Lett. B767, 25-28 (2017) doi:10.1016/j.physletb.2017.01.040 [arXiv:1611.06701 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physletb.2017.01.040 2017

  47. [47]

    Thermogeometric description of the van der Waals like phase transition in AdS black holes

    K. Bhattacharya and B. R. Majhi, Thermogeometric de- scription of the van der Waals like phase transition in AdS black holes, Phys. Rev. D95, no.10, 104024 (2017) doi:10.1103/PhysRevD.95.104024 [arXiv:1702.07174 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.95.104024 2017

  48. [48]

    Bhattacharya and B

    K. Bhattacharya and B. R. Majhi, Thermogeomet- ric study of van der Waals like phase transition in black holes: an alternative approach, Phys. Lett. B 802, 135224 (2020) doi:10.1016/j.physletb.2020.135224 [arXiv:1903.10370 [gr-qc]]

    work page doi:10.1016/j.physletb.2020.135224 2020

  49. [49]

    N. J. Gogoi, G. K. Mahanta and P. Phukon, Geodesics in geometrothermodynamics (GTD) type II geometry of 4D asymptotically anti-de-Sitter black holes, Eur. Phys. J. Plus138, no.4, 345 (2023) doi:10.1140/epjp/s13360- 023-03938-x

    work page doi:10.1140/epjp/s13360- 2023

  50. [50]

    Geodesics in Information Geometry : Classical and Quantum Phase Transitions

    P. Kumar, S. Mahapatra, P. Phukon and T. Sarkar, Geodesics in Information Geometry : Classical and Quantum Phase Transitions, Phys. Rev. E 86, 051117 (2012) doi:10.1103/PhysRevE.86.051117 [arXiv:1210.7135 [cond-mat.stat-mech]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physreve.86.051117 2012

  51. [51]

    S. W. Wei and Y. X. Liu, Insight into the microscopic structure of an AdS black hole from a thermodynamical phase transition,Phys. Rev. Lett.115, 111302 (2015), doi:10.1103/PhysRevLett.115.111302 [arXiv:1502.00386 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.115.111302 2015

  52. [52]

    Kerr-AdS analogue of triple point and solid/liquid/gas phase transition

    N. Altamirano, D. Kubiznak, R. B. Mann and Z. Sherkatghanad, Kerr-AdS analogue of triple point and solid/liquid/gas phase transition,Class. Quant. Grav. 31, 042001 (2014), doi:10.1088/0264-9381/31/4/042001 [arXiv:1308.2672 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/31/4/042001 2014

  53. [53]

    R. A. Hennigar, R. B. Mann and E. Tjoa, Super- fluid Black Holes,Phys. Rev. Lett.118, 021301 (2017), doi:10.1103/PhysRevLett.118.021301 [arXiv:1609.02564 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.118.021301 2017

  54. [54]

    Arcioni and E

    G. Arcioni and E. Lozano-Tellechea, Stability and critical phenomena of black holes and black rings, Phys. Rev. D 72, 104021 (2005)

    2005

  55. [55]

    Armas and E

    J. Armas and E. Parisini,Instabilities of Thin Black Rings: Closing the Gap,” JHEP04, 169 (2019)

    2019

  56. [56]

    Polcar, P

    L. Polcar, P. Sukov´ a and O. Semer´ ak, Free Motion around Black Holes with Disks or Rings: Between In- tegrability and Chaos–V,’ Astrophys. J.877, no.1, 16 (2019)

    2019

  57. [57]

    T. S. Amancio, R. A. Mosna and R. S. S. Vieira, Chaotic orbital dynamics of pulsating stars around black holes surrounded by dark matter halos,Phys. Rev. D110, no.12, 124048 (2024)

    2024

  58. [58]

    Chaos in Born–Infeld–AdS black hole within extended phase space

    Yong Chen, Haitang Li, and Shao-Jun Zhang. Chaos in Born–Infeld–AdS black hole within extended phase space. Gen. Rel. Grav., 51(10):134, 2019

    2019

  59. [59]

    Chaos in Charged Gauss-Bonnet AdS Black Holes in Extended Phase Space

    Sandip Mahish and Bhamidipati Chandrasekhar. Chaos in Charged Gauss-Bonnet AdS Black Holes in Extended Phase Space. Phys. Rev. D, 99(10):106012, 2019

    2019

  60. [60]

    Thermal chaos of a charged dilaton-AdS black hole in the extended phase space

    Chaoqun Dai, Songbai Chen, and Jiliang Jing. Thermal chaos of a charged dilaton-AdS black hole in the extended phase space. Eur. Phys. J. C, 80(3):245, 2020

    2020

  61. [61]

    Temporal and spatial chaos in the Kerr-AdS black hole in an extended phase space

    Bing Tang. Temporal and spatial chaos in the Kerr-AdS black hole in an extended phase space. Chin. Phys. C, 45(5):055101, 2021

    2021

  62. [62]

    S. W. Wei and Y. X. Liu, Topology of black hole ther- modynamics, Phys. Rev. D105, no.10, 104003 (2022)

    2022

  63. [63]

    S. W. Wei, Y. X. Liu and R. B. Mann, Black Hole Solu- tions as Topological Thermodynamic Defects, Phys. Rev. Lett.129, no.19, 191101 (2022)

    2022

  64. [64]

    D. Wu, W. Liu, S. Q. Wu and R. B. Mann, Novel topo- logical classes in black hole thermodynamics, Phys. Rev. D111, no.6, L061501 (2025)

    2025

  65. [65]

    and Wang, Jin, Critical slowing down of black hole phase transition and kinetic crossover in supercriti- cal regime, Phys

    Li, Ran and Zhang, Kun and Yang, Jiayue and Mann, Robert B. and Wang, Jin, Critical slowing down of black hole phase transition and kinetic crossover in supercriti- cal regime, Phys. Rev. D,112,6, 064004, 2025

    2025

  66. [66]

    Mozib Bin Awal, Prabwal Phukon, Critical slowing down 8 of black hole phase transition and universal dynamic scal- ing in AdS black holes, arXiv:2605.15655 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  67. [67]

    Object condensation: one-stage grid- free multi-object reconstruction in physics detectors, graph, and image data

    B. Hazarika, P. Phukon, Bifurcation and critical phe- nomena in black hole thermodynamics. Eur. Phys. J. C 85, 1015 (2025). https://doi.org/10.1140/epjc/s10052- 025-14744-3

    work page doi:10.1140/epjc/s10052- 2025