Abstract
Thermodynamics of phase transition for a black hole in 5D Einstein-Gauss-Bonnet gravity with a negative cosmological constant is studied. As the Bekenstein-Hawking entropy is adopted, we find the heat capacity, volume expansion coefficient and isothermal compressibility are divergent at the critical points, which implies the existence of phase transitions. The fact that the phase transitions are indeed second order is revealed by studying the Ehrenfest’s equations and the Prigogine-Defay ratio. Furthermore near the critical points, we also explicitly calculate the critical exponents of the relevant thermodynamic quantities at fixed charge or fixed temperature. It is shown that the corresponding critical exponents satisfy the thermodynamic scaling law. The Generalized Homogeneous Function hypothesis is also checked by studying the Helmholtz free energy, which is shown to be consistent with the thermodynamic scaling law.
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References
Hawking S W. Black hole explosions. Nature, 1974, 248: 30–31; Hawking S W. Particle creation by black holes. Commun Math Phys, 1975, 43: 199–220
Bardeen J M, Carter B, Hawking S W. The four laws of black hole mechanics. Commun Math Phys, 1973, 31: 161–170
Davies P C W. The thermodynamic theory of black holes. Proc R Soc London A, 1977, 353: 499–521; Davies P CW. Thermodynamic phase transitions of Kerr-Newman black holes in de Sitter space. Class Quantum Grav, 1989, 6: 1909–1914
Curir A. Rotating black holes as dissipative spin-thermodynamical systems. Gen Rel Grav, 1981, 13: 417–423
Pavón D. Phase transition in Reissner-Nordstrom black holes. Phys Rev D, 1991, 43: 2495–2497
Weinhold F. Metric geometry of equilibrium thermodynamics. J Chem Phys, 1975, 63: 2479–2483; Weinhold F. Metric geometry of equilibrium thermodynamics. II. Scaling, homogeneity, and generalized Gibbs-Duhem relations. J Chem Phys, 1975, 63: 2484–2487; Weinhold F. Metric geometry of equilibrium thermodynamics. III. Elementary formal structure of a vector-algebraic representation of equilibrium thermodynamics. J Chem Phys, 1975, 63: 2488–2495; Weinhold F. Metric geometry of equilibrium thermodynamics. IV. Vector-algebraic evaluation of thermodynamic derivatives. J Chem Phys, 1975, 63: 2496–2501; Weinhold F. Metric geometry of equilibrium thermodynamics. V. Aspects of heterogeneous equilibrium. J Chem Phys, 1976, 65: 559–564
Ruppeiner G. Thermodynamics: A Riemannian geometric model. Phys Rev A, 1979, 20: 1608–1613
Quevedo H. Geometrothermodynamics of black holes. Gen Rel Grav, 2008, 40: 971–984
Quevedo H. Geometrothermodynamics. J Math Phys, 2007, 48: 013506
Salamon P, Ihrig E, Berry R S. A group of coordinate transformations which preserve the metric of Weinhold. J Math Phys, 1983, 24: 2515–2520
Mrugala R, Nulton J D, Schön J C, et al. Statistical approach to the geometric structure of thermodynamics. Phys Rev A, 1990, 41: 3156–3160
Maldacena J M. The large-N limit of superconformal field theories and supergravity. Adv Theor Math Phys, 1998, 2: 231–252
Witten E. Anti de Sitter space and holography. Adv Theor Math Phys, 1998, 2: 253–291
Gubser S S. Breaking an Abelian gauge symmetry near a black hole horizon. Phys Rev D, 2008, 78: 065034
Hartnoll S A, Herzog C P, Horowitz G T. Building a holographic superconductor. Phys Rev Lett, 2008, 101: 031601
Policastro G, Son D T, Starinets A O. Shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma. Phys Rev Lett, 2001, 87: 081601; Kovtun P, Son D T, Starinets A O. Holography and hydrodynamics: diffusion on stretched horizons. J High Energy Phys, 2003, 0310: 064; Buchel A, Liu J T. Universality of the shear viscosity from supergravity duals. Phys Rev Lett, 2004, 93: 090602; Kovtun P, Son D T, Starinets A O. Viscosity in strongly interacting quantum field theories from black hole physics. Phys Rev Lett, 2005, 94: 111601
Witten E. Anti-de Sitter space, thermal phase transition, and confinement in gauge theories. Adv Theor Math Phys, 1998, 2: 505–532
Liu H S, Lu H, Luo M X, et al. Thermodynamical metrics and black hole phase transitions. J High Energy Phys, 2010, 1012: 054
Hawking S W, Page D N. Thermodynamics of black holes in anti-de Sitter Space. Commun Math Phys, 1983, 87: 577–588
Banerjee R, Modak S K, Samanta S. Glassy phase transition and stability in black holes. Eur Phys J C, 2010, 70: 317–328
Nieuwenhuizen Th M. Ehrenfest relations at the glass transition: solution to an old paradox. Phys Rev Lett, 1997, 79: 1317–1320
Abramowitz M, Stegun I A. Handbook of Mathematical Functions. New York: Dover Publications, 1972
Zemansky M W, Dittman R H. Heat and Thermodynamics: An Intermediate Textbook. California: McGraw-Hill Publications, 1997
Banerjee R, Roychowdhury D. Thermodynamics of phase transition in higher dimensional AdS black holes. J High Energy Phys, 2011, 1111: 004
Banerjee R, Ghosh S, Roychowdhury D. New type of phase transition in Reissner Nordstrom-AdS black hole and its thermodynamic geometry. Phys Lett B, 2011, 696: 156–162
Banerjee R, Modak S K, Samanta S. Second order phase transition and thermodynamic geometry in Kerr-AdS black hole. Phys Rev D, 2011, 84: 064024
Banerjee R, Modak S K, Roychowdhury D. Thermodynamics of phase transitions in AdS black holes. J High Energy Phys, 2012, 1210: 125
Banerjee R, Roychowdhury D. Critical phenomena in Born-Infeld AdS black holes. Phys Rev D, 2012, 85: 044040
Lousto C O. The fourth law of black-hole thermodynamics. Nucl Phys B, 1993, 410: 155–172
Lau Y K. On the second order phase transition of a Reissner-Nordstrom black hole. Phys Lett A, 1994, 186: 41–46
Jain S, Mukherji S, Mukhopadhyay S. Notes on R-charged black holes near criticality and gauge theory. J High Energy Phys, 2009, 0911: 051
Sahay A, Sarkar T, Sengupta G. On the thermodynamic geometry and critical phenomena of AdS black holes. J High Energy Phys, 2010, 1007: 082
Niu C, Tian Y, Wu X N. Critical Pphenomena and thermodynamic geometry of RN-AdS Black Holes. Phys Rev D, 2012, 85: 024017
Mann R B, Pourhasan R. Gauss-Bonnet black holes and heavy fermion metals. J High Energy Phys, 2011, 1109: 062
Wu J P. Holographic fermions in charged Gauss-Bonnet black hole. J High Energy Phys, 2011, 1107: 106
Yoshino H. Black hole initial data in Gauss-Bonnet gravity: momentarily static case. Phys Rev D, 2011, 83: 104010
Taj S, Quevedo H. Geometrothermodynamics of five dimensional black holes in Einstein-Gauss-Bonnet-theory. arXiv: 1104.3195 [hep-th]
Kim H C, Cai R G. Slowly rotating charged Gauss-Bonnet black holes in AdS spaces. Phys Rev D, 2008, 77: 024045
Cai R G. Gauss-Bonnet black holes in AdS spaces. Phys Rev D, 2002, 65: 084014
Myers R C, Simon J Z. Black-hole thermodynamics in Lovelock gravity. Phys Rev D, 1988, 38: 2434–2444; Cai R G. A note on thermodynamics of black holes in Lovelock gravity. Phys Lett B, 2004, 582: 237–242
Cvetic M, Nojiri S, Odintsov S D. Black hole thermodynamics and negative entropy in de Sitter and anti-de Sitter Einstein-Gauss-Bonnet gravity. Nucl Phys B, 2002, 628: 295–330
Brihaye Y, Radu E. Black objects in the Einstein-Gauss-Bonnet theory with negative cosmological constant and the boundary counterterm method. J High Energy Phys, 2008, 0809: 006
Wiltshire D L. Black holes in string-generated gravity models. Phys Rev D, 1988, 38: 2445–2456
Quevedo H, Sanchez A. Geometrothermodynamics of asymptotically anti-de Sitter black holes. J High Energy Phys, 2008, 0809: 034
Jäckle J. Models of the glass transition. Rep Prog Phys, 1986, 49: 171–231
Hankey A, Stanley H E. Systematic application of generalized homogeneous functions to static scaling, dynamic scaling, and universality. Phys Rev B, 1972, 6: 3515–3542
Stanley H E. Introduction to Phase Transitions and Critical Phenomena. New York: Oxford University Press, 1987
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Hu, C., Zeng, X. & Liu, X. Phase transition and critical phenomenon of AdS black holes in Einstein-Gauss-Bonnet gravity. Sci. China Phys. Mech. Astron. 56, 1652–1663 (2013). https://doi.org/10.1007/s11433-013-5107-4
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DOI: https://doi.org/10.1007/s11433-013-5107-4