Energy trapping from Hagedorn densities of states

  • Connor Behan
  • Klaus Larjo
  • Nima Lashkari
  • Brian Swingle
  • Mark Van Raamsdonk


In this note, we construct simple stochastic toy models for holographic gauge theories in which distributions of energy on a collection of sites evolve by a master equation with some specified transition rates. We build in only energy conservation, locality, and the standard thermodynamic requirement that all states with a given energy are equally likely in equilibrium. In these models, we investigate the qualitative behavior of the dynamics of the energy distributions for different choices of the density of states for the individual sites. For typical field theory densities of states (log(ρ(E)) ~ E α<1), the model gives diffusive behavior in which initially localized distributions of energy spread out relatively quickly. For large N gauge theories with gravitational duals, the density of states for a finite volume of field theory degrees of freedom typically includes a Hagedorn regime (log(ρ(E)) ~ E). We find that this gives rise to a trapping of en! ergy in subsets of degrees of freedom for parametrically long time scales before the energy leaks away. We speculate that this Hagedorn trapping may be part of a holographic explanation for long-lived gravitational bound states (black holes) in gravitational theories.


Gauge-gravity correspondence AdS-CFT Correspondence Nonperturbative Effects 


  1. [1]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].Google Scholar
  2. [2]
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].MathSciNetADSGoogle Scholar
  4. [4]
    J.M. Bardeen, B. Carter and S. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  5. [5]
    S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206-206] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    T. Jacobson, Thermodynamics of space-time: the Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260 [gr-qc/9504004] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  7. [7]
    T. Padmanabhan, Thermodynamical aspects of gravity: new insights, Rept. Prog. Phys. 73 (2010) 046901 [arXiv:0911.5004] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    E.P. Verlinde, On the origin of gravity and the laws of newton, JHEP 04 (2011) 029 [arXiv:1001.0785] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    L. Susskind, Some speculations about black hole entropy in string theory, in The black hole, C. Teitelboim and J. Zanelli eds., World Scientific, Singapore (1998), hep-th/9309145 [INSPIRE].Google Scholar
  10. [10]
    E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  11. [11]
    B. Sundborg, The Hagedorn transition, deconfinement and N = 4 SYM theory, Nucl. Phys. B 573 (2000) 349 [hep-th/9908001] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, The Hagedorn-deconfinement phase transition in weakly coupled large-N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    N.G. van Kampen, Stochastic processes in physics and chemistry, North-Holland Personal Library, The Netherlands (2007).Google Scholar
  14. [14]
    S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear fluid dynamics from gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    O. Aharony, S. Minwalla and T. Wiseman, Plasma-balls in large-N gauge theories and localized black holes, Class. Quant. Grav. 23 (2006) 2171 [hep-th/0507219] [INSPIRE].MathSciNetADSCrossRefzbMATHGoogle Scholar
  16. [16]
    J.L. Vázquez, Smoothing and decay estimates for nonlinear diffusion equations, Oxford Lecture Notes in Mathematics and its Applications volume 33, Oxford University Press, New York U.S.A. (2006).Google Scholar
  17. [17]
    J.L. Vázquez, The porous medium equation: Mathematical theory, Oxford Mathematical Monographs, Oxford University Press, New York U.S.A. (2007).Google Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  • Connor Behan
    • 1
  • Klaus Larjo
    • 2
  • Nima Lashkari
    • 1
  • Brian Swingle
    • 3
  • Mark Van Raamsdonk
    • 1
  1. 1.Department of Physics and AstronomyUniversity of British ColumbiaVancouverCanada
  2. 2.Department of PhysicsBrown UniversityProvidenceU.S.A.
  3. 3.Department of PhysicsHarvard UniversityCambridgeU.S.A.

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