Table of Contents Author Guidelines Submit a Manuscript
Advances in High Energy Physics
Volume 2018, Article ID 8153721, 7 pages
https://doi.org/10.1155/2018/8153721
Research Article

Criticality of a Specific Black Hole in Gravity Coupled with Yang-Mills Field

1Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla Postal 4950, Valparaíso, Chile
2Physics Department, Arts and Sciences Faculty, Eastern Mediterranean University, Famagusta, Northern Cyprus, Mersin 10, Turkey
3TH Division, Physics Department, CERN, 1211 Geneva 23, Switzerland

Correspondence should be addressed to Ali Övgün; lc.vcup@nugvo.ila

Received 17 November 2017; Revised 18 January 2018; Accepted 30 January 2018; Published 25 June 2018

Academic Editor: Douglas Singleton

Copyright © 2018 Ali Övgün. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Abstract

We study the criticality of a specific charged AdS type black hole (SBH) in gravity coupled with Yang-Mills field. In the extended phase space, we treat the cosmological constant as a thermodynamic pressure. After we study the various thermodynamical quantities, we show that the thermodynamic properties of the SBH behave as a Van der Waals liquid-gas system at the critical points and there is a first-order phase transition between small-large SBH.

1. Introduction

Important contribution on black holes’ thermodynamics in anti-de-Sitter (AdS) spacetime is made by Hawking and Page [1], where a first-order phase transition is discovered between the Schwarzschild-anti-de-Sitter (SAdS) black holes that is known as the Hawking-Page transition. Then Chamblin et al. and Cvetic et al. show that the first-order phase transition among Reissner Nordstrom (RN) AdS black holes and the similarities between charged AdS black holes and liquid-gas systems in grand canonical ensemble [24]. Moreover, in the seminal papers of Kubiznak and Mann [5], the cosmological constant is used as dynamical pressure [6]for the RN-AdS black holes in the extended phase space, instead of treating the as a fixed parameter (in standard thermodynamic), and its conjugate variable has dimension of volume Calculating the critical components and finding the phase transition of the RN-AdS black holes, it is shown that RN-AdS black holes behave similar to the Van der Waals fluid in the extended phase space where a first-order small/large black hole’s phase transition occurs at a critical temperature below [711]. The Van der Waals equation, where its pressure is , its temperature is , its specific volume is , the Boltzmann constant is , and the positive constants are and , takes into account the attractive and repulsive forces between molecules and gives an improved model for ideal gas behaviour to describe the basic properties of the liquid-gas phase transition with the ratio of at critical points [1217]. Afterwards, applications of the thermodynamical law’s to the black hole’s physics have gain attention. Different researches are done by using the variation of the first law of thermodynamics of black holes and also application of the criticality on black holes [1888]. Furthermore, AdS-CFT correspondence is the other reason for studying the AdS black hole.

In this paper we use a black hole’s solution in the Yang-Mills field which is the one of the most interesting nonabelian gauge theories. By using the string theory models they find the Yang-Mills fields equations in low energy limit and then Yasskin found the first black hole solution in the theory of Yang-Mills coupled to Einstein theory [89].

Our main aim is to check criticality of a specific charged AdS type black hole (SBH) in gravity coupled with Yang-Mills field (YMF) [90] by comparing its result with the Van der Waals system. The Yang-Mills field is acted inside the nuclei with short range and gravity which is an extension of Einstein’s General Relativity with the arbitrary function of Ricci scalar [9194]. It would be of interest to study the criticality of SBH in gravity coupled with YMF in the extended phase space treating the cosmological constant as a thermodynamic pressure. In this paper, we first study the thermodynamics in the extended phase space and then we obtain its critical exponents to show the existence of the Van der Waals like small-large black hole phase transitions.

The paper is organized as follows: in Section 2 we will briefly review the SBH in gravity coupled with YMF. In Section 3   criticality of the SBH in gravity coupled with YMF will be studied in the extended phase space by calculating its critical exponents. In Section 4 we conclude with final remarks.

2. SBH in Gravity Coupled with YMF

In this section, we briefly present a solution of SBH in gravity coupled with YMF with a cosmological constant in -dimensions [90]. Then we discuss its temperature, entropy, and other thermodynamic quantities. The action of the gravity minimally coupled with YMF () is [90] where is a function of the Ricci scalar and stands for the Lagrangian of the nonlinear YMF with , where the 2-form components of the YMF are . Here is the internal index for the degrees of freedom of the nonabelian YMF. It is noted that this nonlinear YMF can reduce to linear YM field () for and which is an integration constant. Solving Einstein field equations for the gravity coupled with YMF gives to the spherically symmetric black hole metrics (see equation in [90]) with , in which the metric function is

Note that , is the mass of the black hole, and is a constant. Furthermore for the limit of , it becomes well-known solutions in gravity. The Bekenstein-Hawking temperature [9598] of the black hole is calculated by where is the horizon of the black hole, and solving the equation , the total mass of the black hole is obtained as The entropy of the black hole can be derived as where is the area of the black hole’s event horizon. Then, in the extended phase space, we calculate the pressure in terms of cosmological constantand its thermodynamic volume is where is the volume of the unit sphere. Now the mass can be also written in terms of as follows:

The first law of the black hole thermodynamics in the extended phase space iswhere the thermodynamic variables can be obtained as , , and .

Then we write the generalized Smarr relation for the black hole, which can be derived also using the dimensional scaling, as

We introduce the cosmological constant as thermodynamic pressure in the extended phase space in (10), and it is seen that the first law of the black hole’s thermodynamics and the Smarr relations is matched well.

3. Criticality

In this section, we investigate the critical behaviour of the SBH in the extended phase space. The critical point can be defined asNow we consider the case of four dimensions , where the metric function becomesand corresponding mass of the black hole is calculated as

The temperature of the four dimensional SBH is

Then we write the temperature in terms of () as follows: Afterwards one can easily obtain the pressure in terms of the temperature :

To consider the criticality using the extended phase space, we write the black hole radius in terms of the specific volume as . Using the condition of (15), we derive the critical Bekenstein-Hawking temperature , critical pressure , and critical specific volume as follows:

One can also find this relation which is same with a Van der Waals fluid

It is noted that Figures 1 and 2 show that diagram is the same with the diagram of the Van der Waals liquid-gas system.

Figure 1: diagram of a SBH in a gravity coupled with YMF for and .
Figure 2: diagram of a SBH in a gravity coupled with YMF for and .

Let us now analyze the Gibbs free energy of the system. We first use the mass as enthalpy instead of internal energy and the Gibbs free energy in the extended phase space for the SBH in gravity coupled with Yang-Mills field is calculated as

We plot the change of the free energy with in Figure 3. There is a small-large black hole phase transition as seen in Figure 3.

Figure 3: diagram of a SBH in a gravity coupled with YMF for different values of   (, , and ) with and .

4. Conclusion

In this paper, we first treat the cosmological constant as a thermodynamical pressure and the thermodynamics and criticality of the SBH in gravity coupled with YMF is studied in the extended phase space. It is shown that there is a phase transition between small-large black holes. Furthermore, after we obtain the critical exponents, the critical behaviour of SBH in gravity coupled with YMF in the extended space behaves also similarly as Van der Waals liquid-gas systems with the ratio of at critical points. Hence it would be of great importance to obtain the criticality of SBH in gravity coupled with YMF. Hence the critical ratio is universal and independent from the modified gravities. The YMF has a parameter of but has no effect on the universal ratio of .

It is also interesting to study the holographic duality of SBH in gravity coupled with YMF. It is noted that without thermal fluctuations black hole is holographic dual with Van der Waals fluid given by , where is the Boltzmann constant [84, 85], is nonzero constant which is the size of the molecules of fluid, and the constant is a value of the interaction measurement between molecules. We leave this problem for the future projects.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the Chilean FONDECYT Grant no. 3170035 (Ali Övgün). Ali Övgün is grateful to the CERN Theory (CERN-TH) Division, Waterloo University, Department of Physics and Astronomy, and also Perimeter Institute for Theoretical Physics for hosting him as a research visitor where part of this work was done. It is a pleasure to thank Professor Robert B. Mann for valuable discussions.

References

  1. S. W. Hawking and D. N. Page, “Thermodynamics of black holes in anti-de Sitter space,” Communications in Mathematical Physics, vol. 87, no. 4, pp. 577–588, 1982/83. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, “Charged AdS black holes and catastrophic holography,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 60, no. 6, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  3. A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, “Holography, thermodynamics, and fluctuations of charged AdS black holes,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 60, no. 10, Article ID 104026, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  4. M. Cvetic, G. W. Gibbons, D. Kubiznak, and C. N. Pope, “Black hole enthalpy and an entropy inequality for the thermodynamic volume,” Physical Review D, vol. 84, 2011. View at Google Scholar
  5. D. Kubiznak and R. B. Mann, “ criticality of charged AdS black holes,” Journal of High Energy Physics, vol. 1207, article 33, 2012. View at Google Scholar · View at MathSciNet
  6. B. P. Dolan, “The cosmological constant and black-hole thermodynamic potentials,” Classical and Quantum Gravity, vol. 28, no. 12, Article ID 125020, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  7. S. Gunasekaran, D. Kubiznak, and R. B. Mann, “Extended phase space thermodynamics for charged and rotating black holes and Born-Infeld vacuum polarization,” Journal of High Energy Physics, vol. 2012, no. 11, article 110, 2012. View at Publisher · View at Google Scholar
  8. D. Kubiznak and R. B. Mann, “Black hole chemistry,” Can J. Phys, vol. 93, no. 9, p. 999, 2015. View at Google Scholar
  9. D. Kubiznak, R. B. Mann, and M. Teo, “Black hole chemistry: thermodynamics with lambda,” Classical and Quantum Gravity, vol. 34, no. 6, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  10. R. A. Hennigar, R. B. Mann, and E. Tjoa, “Superfluid black holes,” Physical Review Letters, vol. 118, no. 2, Article ID 021301, 2017. View at Publisher · View at Google Scholar · View at Scopus
  11. A. M. Frassino, R. B. Mann, and F. Simovic, https://arxiv.org/abs/1611.03525.
  12. R. A. Hennigar, E. Tjoa, and R. B. Mann, “Thermodynamics of hairy black holes in Lovelock gravity,” Journal of High Energy Physics, vol. 2017, no. 2, article no. 70, 2017. View at Publisher · View at Google Scholar · View at Scopus
  13. S. H. Hendi, R. B. Mann, S. Panahiyan, and B. Eslam Panah, “Van der Waals like behavior of topological AdS black holes in massive gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 95, no. 2, Article ID 021501, 2017. View at Publisher · View at Google Scholar · View at Scopus
  14. D. Hansen, D. Kubiznak, and R. B. Mann, “Universality of P-V Criticality in Horizon Thermodynamics,” Journal of High Energy Physics, vol. 1701, no. 47, 2017. View at Google Scholar · View at MathSciNet
  15. R. B. Mann, The Chemistry of Black Holes, vol. 170, Springer Proc.Phys., 2016. View at Publisher · View at Google Scholar · View at Scopus
  16. R. A. Hennigar, W. G. Brenna, and R. B. Mann, “P-v criticality in quasitopological gravity,” Journal of High Energy Physics, vol. 2015, no. 7, article 77, pp. 1–31, 2015. View at Publisher · View at Google Scholar
  17. T. Delsate and R. Mann, “Van Der Waals black holes in d dimensions,” Journal of High Energy Physics, vol. 2015, no. 2, article no. 70, 2015. View at Publisher · View at Google Scholar · View at Scopus
  18. S. Chen, X. Liu, C. Liu, and J. Jing, “P-V criticality of AdS black hole in f(R) gravity,” Chinese Physics Letters, vol. 30, no. 6, Article ID 060401, 2013. View at Google Scholar
  19. R.-G. Cai, L.-M. Cao, L. Li, and R.-Q. Yang, “P-V criticality in the extended phase space of Gauss-Bonnet black holes in AdS space,” Journal of High Energy Physics, vol. 2013, no. 9, article 5, 2013. View at Publisher · View at Google Scholar · View at Scopus
  20. A. Belhaj, M. Chabab, H. E. Moumni, and M. B. Sedra, “On thermodynamics of AdS black holes in arbitrary dimensions,” Chinese Physics Letters, vol. 29, no. 10, Article ID 100401, 2012. View at Publisher · View at Google Scholar · View at Scopus
  21. K. Bhattacharya, B. R. Majhi, and S. Samanta, “van der Waals criticality in AdS black holes: A phenomenological study,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 96, no. 8, 2017. View at Publisher · View at Google Scholar
  22. K. Jafarzade and J. Sadeghi, “The thermodynamic efficiency in static and dynamic black holes,” International Journal of Theoretical Physics, vol. 56, no. 11, pp. 3387–3399, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. J. Sadeghi and A. S. Kubeka, “P-V criticality of modified BTZ black hole,” International Journal of Theoretical Physics, vol. 55, no. 5, pp. 2455–2459, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  24. J. Liang, “The P-v Criticality of a Noncommutative Geometry-Inspired Schwarzschild-AdS Black Hole,” Chinese Physics Letters, vol. 34, no. 8, Article ID 080402, 2017. View at Publisher · View at Google Scholar · View at Scopus
  25. P. Pradhan, “Enthalpy, geometric volume and logarithmic correction to entropy for van der Waals black hole,” EPL (Europhysics Letters), vol. 116, no. 1, Article ID 10001, 2016. View at Publisher · View at Google Scholar · View at Scopus
  26. X.-X. Zeng and L.-F. Li, “Holographic phase transition probed by nonlocal observables,” Advances in High Energy Physics, vol. 2016, Article ID 6153435, 2016. View at Publisher · View at Google Scholar · View at Scopus
  27. J.-X. Mo, G.-Q. Li, and X.-B. Xu, “Combined effects of f(R) gravity and conformally invariant Maxwell field on the extended phase space thermodynamics of higher-dimensional black holes,” The European Physical Journal C, vol. 76, no. 10, article no. 545, 2016. View at Publisher · View at Google Scholar · View at Scopus
  28. B. R. Majhi and S. Samanta, “P-V criticality of AdS black holes in a general framework,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 773, pp. 203–207, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  29. Z.-Y. Fan, “Critical phenomena of regular black holes in anti-de Sitter space-time,” The European Physical Journal C, vol. 77, no. 4, article no. 266, 2017. View at Publisher · View at Google Scholar · View at Scopus
  30. Y. G. Miao and Y. M. Wu, “Thermodynamics of the Schwarzschild-AdS black hole with a minimal length,” Advances in High Energy Physics, vol. 2017, 14 pages, 2017. View at Publisher · View at Google Scholar
  31. S. He, L. F. Li, and X. X. Zeng, “Holographic van der waals-like phase transition in the Gauss–Bonnet gravity,” Nuclear Physics B, vol. 915, pp. 243–261, 2017. View at Publisher · View at Google Scholar
  32. A. Mandal, S. Samanta, and B. R. Majhi, “Phase transition and critical phenomena of black holes: A general approach,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 94, no. 6, Article ID 064069, 2016. View at Publisher · View at Google Scholar · View at Scopus
  33. D. Chen, G. qingyu, and J. Tao, “The modified first laws of thermodynamics of anti-de Sitter and de Sitter space–times,” Nuclear Physics B, vol. 918, pp. 115–128, 2017. View at Publisher · View at Google Scholar · View at Scopus
  34. A. Dehyadegari, A. Sheykhi, and A. Montakhab, “Critical behavior and microscopic structure of charged AdS black holes via an alternative phase space,” Physics Letters B, vol. 768, pp. 235–240, 2017. View at Publisher · View at Google Scholar · View at Scopus
  35. S. H. Hendi, S. Panahiyan, B. Eslam Panah, M. Faizal, and M. Momennia, “Critical behavior of charged black holes in Gauss-Bonnet gravity's rainbow,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 94, no. 2, Article ID 024028, 2016. View at Publisher · View at Google Scholar · View at Scopus
  36. Y.-B. Ma, R. Zhao, and S. Cao, “Q– Φ criticality in the extended phase space of (n+ 1) -dimensional RN-AdS black holes,” The European Physical Journal C, vol. 76, no. 12, article no. 669, 2016. View at Publisher · View at Google Scholar · View at Scopus
  37. H. Liu and X.-H. Meng, “P-V criticality in the extended phase-space of charged accelerating AdS black holes,” Modern Physics Letters A, vol. 31, no. 37, Article ID 1650199, 2016. View at Publisher · View at Google Scholar · View at Scopus
  38. M. B. J. Poshteh and N. Riazi, “Phase transition and thermodynamic stability in extended phase space and charged Hořava–Lifshitz black holes,” General Relativity and Gravitation, vol. 49, no. 5, article no. 64, 2017. View at Publisher · View at Google Scholar · View at Scopus
  39. X. Guo, H. Li, L. Zhang, and R. Zhao, “The critical phenomena of charged rotating de Sitter black holes,” Classical and Quantum Gravity, vol. 33, no. 13, Article ID 135004, 2016. View at Publisher · View at Google Scholar · View at Scopus
  40. H.-F. Li, M.-S. Ma, and Y.-Q. Ma, “Thermodynamic properties of black holes in de Sitter space,” Modern Physics Letters A, vol. 32, no. 2, 1750017, 12 pages, 2017. View at Google Scholar · View at MathSciNet
  41. M.-S. Ma and R.-H. Wang, “Peculiar P-V criticality of topological Hořava-Lifshitz black holes,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 96, no. 2, Article ID 024052, 2017. View at Publisher · View at Google Scholar · View at Scopus
  42. S. Upadhyay, B. Pourhassan, and H. Farahani, “P-V criticality of first-order entropy corrected AdS black holes in massive gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 95, no. 10, Article ID 106014, 2017. View at Publisher · View at Google Scholar · View at Scopus
  43. S. Fernando, “Phase transitions of black holes in massive gravity,” Modern Physics Letters A, vol. 31, no. 16, Article ID 1650096, 2016. View at Publisher · View at Google Scholar · View at Scopus
  44. S. Fernando, “P-V criticality in AdS black holes of massive gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 94, no. 12, Article ID 124049, 2016. View at Publisher · View at Google Scholar
  45. J. Sadeghi, B. Pourhassan, and M. Rostami, “P-V criticality of logarithm-corrected dyonic charged AdS black holes,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 94, no. 6, Article ID 064006, 2016. View at Publisher · View at Google Scholar · View at Scopus
  46. Y.-G. Miao and Z.-M. Xu, “Phase transition and entropy inequality of noncommutative black holes in a new extended phase space,” Journal of Cosmology and Astroparticle Physics, vol. 2017, no. 3, article no. 046, 2017. View at Publisher · View at Google Scholar · View at Scopus
  47. P. Cheng, S.-W. Wei, and Y.-X. Liu, “Critical phenomena in the extended phase space of Kerr-Newman-AdS black holes,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 94, no. 2, Article ID 024025, 2016. View at Publisher · View at Google Scholar · View at Scopus
  48. X. Guo, H. Li, L. Zhang, and R. Zhao, “The phase transition of higher dimensional charged black holes,” Advances in High Energy Physics, vol. 2016, Article ID 7831054, 10 pages, 2016. View at Publisher · View at Google Scholar · View at Scopus
  49. J. Liang, C.-B. Sun, and H.-T. Feng, “P-V criticality in the extended phase space of charged f(R) black holes in AdS space-time,” EPL (Europhysics Letters), vol. 113, no. 3, Article ID 30008, 2016. View at Publisher · View at Google Scholar · View at Scopus
  50. L. C. Zhang and R. Zhao, “The critical phenomena of Schwarzschild-de Sitter black hole,” EPL (Europhysics Letters), vol. 113, no. 1, Article ID 10008, 2016. View at Publisher · View at Google Scholar · View at Scopus
  51. J.-X. Mo, G.-Q. Li, and X.-B. Xu, “Effects of power-law Maxwell field on the critical phenomena of higher dimensional dilaton black holes,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 93, no. 8, Article ID 084041, 2016. View at Publisher · View at Google Scholar · View at Scopus
  52. M. H. Dehghani, A. Sheykhi, and Z. Dayyani, “Critical behavior of Born-Infeld dilaton black holes,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 93, no. 2, Article ID 024022, 2016. View at Publisher · View at Google Scholar · View at Scopus
  53. X. X. Zeng and L. F. Li, “Van der waals phase transition in the framework of holography,” Physics Letters B, vol. 764, pp. 100–108, 2017. View at Publisher · View at Google Scholar
  54. R. Maity, P. Roy, and T. Sarkar, “Black hole phase transitions and the chemical potential,” Physics Letters B, vol. 765, pp. 386–394, 2017. View at Publisher · View at Google Scholar · View at Scopus
  55. S. H. Hendi, R. M. Tad, Z. Armanfard, and M. S. Talezadeh, “Extended phase space thermodynamics and P–V criticality: brans–dicke–born–infeld vs. einstein–born–infeld-dilaton black holes,” The European Physical Journal C, vol. 76, no. 5, article 263, 2016. View at Publisher · View at Google Scholar · View at Scopus
  56. A. Karch and B. Robinson, “Holographic black hole chemistry,” Journal of High Energy Physics, vol. 2015, no. 12, article no. 73, pp. 1–15, 2015. View at Publisher · View at Google Scholar · View at Scopus
  57. S.-W. Wei, P. Cheng, and Y.-X. Liu, “Analytical and exact critical phenomena of d -dimensional singly spinning Kerr-AdS black holes,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 93, no. 8, Article ID 084015, 2016. View at Publisher · View at Google Scholar · View at Scopus
  58. J. Xu, L.-M. Cao, and Y.-P. Hu, “P-V criticality in the extended phase space of black holes in massive gravity,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 91, no. 12, Article ID 124033, 2015. View at Publisher · View at Google Scholar · View at Scopus
  59. J.-X. Mo and W.-B. Liu, “hase transitions, geometrothermodynamics, and critical exponents of black holes with conformal anomaly,” Advances in High Energy Physics, vol. 2014, Article ID 739454, 10 pages, 2014. View at Google Scholar
  60. S. H. Hendi and Z. Armanfard, “Extended phase space thermodynamics and P - V criticality of charged black holes in Brans–Dicke theory,” General Relativity and Gravitation, vol. 47, no. 10, article no. 125, 2015. View at Publisher · View at Google Scholar · View at Scopus
  61. Z. Sherkatghanad, B. Mirza, Z. Mirzaiyan, and S. A. H. Mansoori, “Critical behaviors and phase transitions of black holes in higher order gravities and extended phase spaces,” International Journal of Modern Physics D, vol. 26, no. 3, Article ID 1750017, 2017. View at Publisher · View at Google Scholar · View at Scopus
  62. M. Zhang, Z.-Y. Yang, D.-C. Zou, W. Xu, and R.-H. Yue, “P–V criticality of AdS black hole in the Einstein–Maxwell–power-Yang–Mills gravity,” General Relativity and Gravitation, vol. 47, no. 2, 2015. View at Publisher · View at Google Scholar · View at Scopus
  63. M. H. Dehghani, S. Kamrani, and A. Sheykhi, “P-V criticality of charged dilatonic black holes,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 90, no. 10, Article ID 104020, 2014. View at Publisher · View at Google Scholar · View at Scopus
  64. S. H. Hendi, S. Panahiyan, and B. Eslam Panah, “P - V criticality and geometrical thermodynamics of black holes with Born-Infeld type nonlinear electrodynamics,” International Journal of Modern Physics D, vol. 25, no. 1, Article ID 1650010, 2016. View at Publisher · View at Google Scholar · View at Scopus
  65. D.-C. Zou, S.-J. Zhang, and B. Wang, “Critical behavior of Born-Infeld AdS black holes in the extended phase space thermodynamics,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 89, Article ID 044002, 2014. View at Publisher · View at Google Scholar
  66. D. C. Zou, Y. Liu, and B. Wang, “Critical behavior of charged Gauss-Bonnet-AdS black holes in the grand canonical ensemble,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 90, no. 4, Article ID 044063, 2014. View at Publisher · View at Google Scholar
  67. M. Zhang, D.-C. Zou, and R.-H. Yue, “Reentrant phase transitions and triple points of topological AdS black holes in Born-Infeld-massive gravity,” Advances in High Energy Physics, Art. ID 3819246, 11 pages, 2017. View at Google Scholar · View at MathSciNet
  68. D. Zou, Y. Liu, and R. Yue, “Behavior of quasinormal modes and Van der Waals-like phase transition of charged AdS black holes in massive gravity,” The European Physical Journal C, vol. 77, no. 6, 2017. View at Publisher · View at Google Scholar
  69. D. Zou, R. Yue, and M. Zhang, “Reentrant phase transitions of higher-dimensional AdS black holes in dRGT massive gravity,” The European Physical Journal C, vol. 77, no. 4, 2017. View at Publisher · View at Google Scholar
  70. C.-Y. Zhang, S.-J. Zhang, D.-C. Zou, and B. Wang, “Charged scalar gravitational collapse in de Sitter spacetime,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 93, no. 6, Article ID 064036, 2016. View at Publisher · View at Google Scholar · View at Scopus
  71. D.-C. Zou, Y. Liu, C.-Y. Zhang, and B. Wang, “Dynamical probe of thermodynamical properties in three-dimensional hairy AdS black holes,” EPL (Europhysics Letters), vol. 116, no. 4, Article ID 40005, 2016. View at Publisher · View at Google Scholar · View at Scopus
  72. D.-C. Zou, Y. Liu, B. Wang, and W. Xu, “Thermodynamics of rotating black holes with scalar hair in three dimensions,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 90, no. 10, Article ID 104035, 2014. View at Publisher · View at Google Scholar · View at Scopus
  73. H.-H. Zhao, L.-C. Zhang, M.-S. Ma, and R. Zhao, “P-V criticality of higher dimensional charged topological dilaton de Sitter black holes,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 90, no. 6, Article ID 064018, 2014. View at Publisher · View at Google Scholar
  74. G.-Q. Li, “Effects of dark energy on P-V criticality of charged AdS black holes,” Physics Letters B, vol. 735, pp. 256–260, 2014. View at Publisher · View at Google Scholar · View at Scopus
  75. J.-X. Mo and W.-B. Liu, “P-V criticality of topological black holes in Lovelock-Born-Infeld gravity,” The European Physical Journal C, vol. 74, article 2836, 2014. View at Publisher · View at Google Scholar · View at Scopus
  76. R. Zhao, H. H. Zhao, M. S. Ma, and L. C. Zhang, “On the critical phenomena and thermodynamics of charged topological dilaton AdS black holes,” The European Physical Journal C, vol. 73, no. 12, pp. 1–10, 2013. View at Publisher · View at Google Scholar · View at Scopus
  77. X. Kuang and O. Miskovic, “Thermal phase transitions of dimensionally continued AdS black holes,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 95, no. 4, 2017. View at Publisher · View at Google Scholar
  78. X.-M. Kuang and J.-P. Wu, “Effect of quintessence on holographic fermionic spectrum,” The European Physical Journal C, vol. 77, no. 10, article no. 670, 2017. View at Publisher · View at Google Scholar · View at Scopus
  79. C.-J. Luo, X.-M. Kuang, and F.-W. Shu, “Charged Lifshitz black hole and probed Lorentz-violation fermions from holography,” Physics Letters B, vol. 769, pp. 7–13, 2017. View at Publisher · View at Google Scholar · View at Scopus
  80. L. Aránguiz, X.-M. Kuang, and O. Miskovic, “Topological black holes in pure Gauss-Bonnet gravity and phase transitions,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 93, no. 6, Article ID 064039, 2016. View at Publisher · View at Google Scholar · View at Scopus
  81. D. Kastor, S. Ray, and J. Traschen, “Enthalpy and the mechanics of AdS black holes,” Classical and Quantum Gravity, vol. 26, no. 19, Article ID 195011, 2009. View at Publisher · View at Google Scholar · View at Scopus
  82. M. Azreg-Aïnou, “Black hole thermodynamics: no inconsistency via the inclusion of the missing P-V terms,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 91, no. 6, Article ID 064049, 2015. View at Publisher · View at Google Scholar · View at Scopus
  83. M. Azreg-Aïnou, “Charged de Sitter-like black holes: quintessence-dependent enthalpy and new extreme solutions,” The European Physical Journal C, vol. 75, no. 1, pp. 1–13, 2015. View at Publisher · View at Google Scholar · View at Scopus
  84. E. Caceres, P. H. Nguyen, and J. F. Pedraza, “Holographic entanglement entropy and the extended phase structure of STU black holes,” Journal of High Energy Physics, vol. 2015, no. 9, article no. 184, 2015. View at Publisher · View at Google Scholar · View at Scopus
  85. C. V. Johnson, “Holographic heat engines,” Classical and Quantum Gravity, vol. 31, no. 20, p. 205002, 2014. View at Publisher · View at Google Scholar
  86. M. Momennia, “Reentrant phase transition of Born-Infeld-dilaton black holes,” https://arxiv.org/abs/1709.09039.
  87. S. H. Hendi and M. Momennia, “AdS charged black holes in Einstein–Yang–Mills gravity's rainbow: Thermal stability and P − V criticality,” Physics Letters B, vol. 777, pp. 222–234, 2018. View at Publisher · View at Google Scholar
  88. S. H. Hendi, B. Eslam Panah, S. Panahiyan, and M. Momennia, “Three dimensional magnetic solutions in massive gravity with (non)linear field,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 775, pp. 251–261, 2017. View at Google Scholar · View at MathSciNet
  89. P. B. Yasskin, “Solutions for gravity coupled to massless gauge fields,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 12, no. 8, pp. 2212–2217, 1975. View at Publisher · View at Google Scholar · View at MathSciNet
  90. S. H. Mazharimousavi and M. Halilsoy, “Black hole solutions in,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 84, no. 6, 2011. View at Publisher · View at Google Scholar
  91. S. Chakraborty and S. SenGupta, “Spherically symmetric brane spacetime with bulk f (R) gravity,” The European Physical Journal C, vol. 75, no. 1, article no. 11, 2015. View at Publisher · View at Google Scholar · View at Scopus
  92. S. Nojiri and S. D. Odintsov, “Anti-evaporation of Schwarzschild-de Sitter black holes in F(R) gravity,” Classical and Quantum Gravity, vol. 30, no. 12, Article ID 125003, 2013. View at Publisher · View at Google Scholar · View at Scopus
  93. S. Nojiri and S. D. Odintsov, “Instabilities and anti-evaporation of Reissner–Nordström black holes in modified F(R) gravity,” Physics Letters B, vol. 735, pp. 376–382, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  94. S. Nojiri and S. Odintsov, “Regular multihorizon black holes in modified gravity with nonlinear electrodynamics,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 96, no. 10, 2017. View at Publisher · View at Google Scholar
  95. I. Sakalli, A. Övgün, and K. Jusufi, “GUP assisted Hawking radiation of rotating acoustic black holes,” Astrophysics and Space Science, vol. 361, no. 10, article no. 330, 2016. View at Publisher · View at Google Scholar · View at Scopus
  96. I. Sakalli and A. Övgün, “Black hole radiation of massive spin-2 particles in (3+1) dimensions,” The European Physical Journal Plus, vol. 131, no. 6, article no. 184, 2016. View at Publisher · View at Google Scholar · View at Scopus
  97. I. Sakalli and A. Övgün, “Quantum tunneling of massive spin-1 particles from non-stationary metrics,” General Relativity and Gravitation, vol. 48, no. 1, article no. 1, pp. 1–10, 2016. View at Publisher · View at Google Scholar · View at Scopus
  98. I. Sakalli and A. Ovgun, “Tunnelling of vector particles from Lorentzian wormholes in 3+1 dimensions,” The European Physical Journal Plus, vol. 130, no. 6, article 110, 2015. View at Publisher · View at Google Scholar · View at Scopus