Phase Transition and Clapeyron Equation of Black Holes in Higher Dimensional AdS Spacetime
Abstract
By Maxwell equal area law we study the phase transition in higher dimensional Antide Sitter (AdS) ReissnerNordström (RN) black holes and Kerr black holes in this paper. The coexisting region of the two phases involved in the phase transition is found and some coexisting curves are shown in figures. We also analytically investigate the parameters which affect the phase transition. To better compare with a general thermodynamic system, the Clapeyron equation is derived for the higher dimensional AdS black holes.
Keywords: ReissnerNordström AdS black hole, Kerr AdS black hole, phase transition, two phase coexistence, Clapeyron equation
pacs:
04.70.s, 05.70.CeI Introduction
Black hole is an ideal platform for studying many interesting behaviors in classical gravity theory. It also can be regarded as a macroscopic quantum system in that its peculiar thermodynamic properties and thermodynamic quantities, as entropy, temperature, and its gravity holographic nature are quantum mechanical intrinsically. Hence research of black hole thermodynamics is an important channel for studying quantum gravity JDB1 (); JDB2 (); Hawking1 (); Lu (); ZhaoZhang (). Although thermodynamics of black hole has been researched for many years, the exact statistical interpretation for thermodynamics state of black hole is not clear. So the thermodynamics of black hole remains an important subject.
It has been found that a black hole possesses not only standard thermodynamic quantities but abundant phase structures and critical phenomena, such as those involved in HawkingPage phase transitionHawking2 (), similar to the ones of a general thermodynamic system. Even more interesting is that the studies on the charged black holes show they may have an analogous phase transition with that of van der WaalsMaxwell liquidgasCham1 (); Cham2 (). Recently critical behaviors and phase transition of black holes have been extensively investigated by considering the cosmological constant as thermodynamic pressureKubi1 (); Dolan (); Guna (); Cvet (); Kastor:2009wy (),. People have been trying to construct a complete liquidgas analogue system for black holes.
In ref.Kubi1 (); Guna (); Banerjee1 (); Banerjee2 (); Banerjee3 (); Banerjee4 (); Banerjee5 (); Banerjee6 (), the phase transition and critical behaviors of some AdS black holes have been studied, which exhibits several different phase transition behaviors. On the basis of the thermodynamic volume of dS spacetime given in ref.Dolan (); Kastor:2009wy (); Alta1 (), considering the connection between black hole horizon and cosmological horizon, we derived the effective temperature and effective pressure in some dS spacetimes in our previous worksZhaohh (); Zhao1 (); Mams1 (); Zhanglc (). The relation among the effective thermodynamic quantities was investigated, which shows the critical phenomena like that in van der Waals liquidgas system. Using Ehrenfest scheme, the second order phase transition was proved to exist in AdS spacetime black hole at the critical statesBanerjee1 (); Banerjee4 (); Mo1 (); Mo2 (); Mo3 (); Mo4 (); Mo5 (); Mo6 (); Mams2 (); Lala (). Alike conclusion has been reached by investigating thermodynamics and state space geometry of black holes in ref.Banerjee1 (); Lala (); Wei1 (); Suresh (); Mans (); Thar (); Niu (). The phase transition behaviors of black holes is found to be related to not only the spacetime metric but also the theory of gravity or other factorsMams2 (); Mams3 (); Mams4 (); Gim (); Posh (); Arci ().
Although some encouraging results have been achieved about the thermodynamics of black holes in AdS and dS spacetimes, many thermodynamic properties need to be investigated more specifically. It is significant to study the critical behaviors and the process of noncritical phase transition in detail. The analyses on the relation of some black holes in AdS spacetimes show a negative pressure appear as the temperature is below a certain value and a thermodynamic unstable region exists with Kubi1 (); Guna (); Zhao2 (); Zhao3 (); Alta2 (); Cai (); Zou1 (); Zou2 (); Fras (); Li (); Wei2 (); Kubi2 (); Xu (); Liu (); Hendi (). Negative pressure, that is, positive , doesn’t correspond to AdS spacetime. The situation is found in var der WaalsMaxwell liquidgas system, where it is solved by the remarkable Maxwell’s equal area lawSpal1 (); Zhaojx (); Spal2 (); ZhangAHEP ().
In this paper Maxwell’s equal area law is extended to analyze the phase transition in the RNAdS black holes and the KerrAdS black holes. It can help to resolve the puzzles about the negative pressure and the unstable region. By Maxwell’s equal area law we discover the possible two phase coexistence curves in the process of phase transition and the boundary curves of the coexistence region in plots. We expect to provide some particular information about phase transition and properties of black hole thermodynamic system in AdS spacetimes to contribute to search for more stable black holes and to explore properties of quantum gravity.
The paper is arranged as follow: the thermodynamic quantities of the dimensional RN AdS black hole and the dimensional Kerr AdS black hole are introduced firstly in section 2 and in section 3 respectively. Then the phase transition of the black holes are studied by Maxwell equal area law, and the effect of the dimension and the thermodynamic quantities, as electric charge in section 2 and angular momentum in section 3, on the phase transition is analyzed . In section 4, we make some discussion. (we use the units
Ii The charged black holes
ii.1 Thermodynamics
ReissnerNordstrom black holes are characterized by the spherically symmetry and electrical nature. The solution for dimensional RNAdS spacetime with a negative cosmological constant, , is defined by the line elementCham1 ()
(1) 
where
(2) 
The ADM mass and the electric charge have been defined as (,
(3) 
in which the volume of the unit sphere can be expressed as
(4) 
The corresponding Hawking temperature, entropy, and electric potential of the system are defined as
(5) 
In our consideration we interpret the cosmological constant as a thermodynamic pressure Kastor:2009wy (),
(6) 
The corresponding conjugate quantity, the thermodynamic volume, is given byKastor:2009wy ()
(7) 
with being the position of black hole horizon determined from . Combining Eqs. (5) and (6), we can obtain
(8) 
Comparing (8) with the Van der Waals equation, we conclude that the specific volume should be identified with the horizon radius of the black hole asKubi1 (); Guna ()
(9) 
In geometric units we have
(10) 
and the equation of state reads
(11) 
In Fig.1, the isotherms are plotted in diagrams for different and . It shows there are thermodynamic unstable regions with in the extended phase space, and the situation of negative pressure appears when . The isotherm corresponding to is tangent to horizontal axis with , in which can be obtained by
(12) 
(13) 
ii.2 Maxwell equal area law and two phase equilibria
The state equation of the dimensional RNAdS black hole is exhibited in the diagrams in Fig.1, which show the existence of negative pressure and thermodynamic unstable region with , which makes a thermodynamic system contract and expand automatically in classical thermodynamics. The case occurs in van der Waals equation, where the problems have been resolved by reference to some practical process of phase transition in real fluid and by Maxwell equal area law.
Using Maxwell equal area law to deal with the state equation of the dimensional RNAdS black hole, the twophase equilibrium lines, where the two phases coexist, are acquired.
Take as critical temperature of the dimensional RNAdS black hole, and suppose at temperature ( the boundary state parameters for twophase coexistence are and . According to Maxwell equal area law,
(14) 
we can get
(15) 
(16) 
Set , from Eqs. (15) and (16), we can obtain
(17) 
where
(18) 
While , from (17) we get the critical specific volume
(19) 
According to (16),
(20) 
and the critical temperature and critical pressure are obtained as ,
(21) 
(22) 
These critical thermodynamic quantities are consistent with those in Ref.Guna ().
Combining (17), (20) and (21), as with , we can get
(23) 
For a fixed , i.e. a fixed , we can evaluate with (23), and then according to (17) and (16), the , and are specified while is determined. Join the points and on the isotherm with in the diagram, which generate an isobar representing the process of phase transition like that of van der Waals system. Fig.2 shows the isobars with solid (red) straight lines and the boundary of the region of two phase coexistence by the dotdashed (green) curve. In Fig.2 each combined solid line simulates the process of thermodynamic state change and phase transition of dimensional RNAdS black hole at a certain temperature. The phase transition process becomes shorter as temperature goes up until it turns into a single point at a certain temperature, which is the critical temperature, and the point corresponds to critical state of dimensional RNAdS black hole.
When taking , we have calculated the quantities , , as =0.5, 1, 1.5 in the spacetimes with the dimension respectively. The results are shown in Table 1.



0.1  1.79E6  5.74E5  1.51E8  1.86E4  3.17E3  1.72E5  3.264  1.40E3  5.80E5 

0.3  0.00712  155.  1.61E4  0.0378  16.0  0.00919  0.0470  9.99  0.0217 

0.5  0.0496  24.3  0.00151  0.149  4.24  0.0497  0.173  2.80  0.110 

0.7  0.149  9.15  0.00475  0.324  2.06  0.123  0.361  1.40  0.265 

0.9  0.387  4.36  0.00995  0.595  1.23  0.225  0.635  0.862  0.477 

0.1  1.79E6  1.15E6  3.77E9  1.86E4  3.64E3  1.31E5  3.26E4  1.55E3  4.76E5 

0.3  0.00712  310.  4.03E5  0.0378  18.4  0.00696  0.0470  11.0  0.0178 

0.5  0.0496  48.7  3.78E4  0.149  4.87  0.0377  0.173  3.09  0.0903 

0.7  0.149  18.3  0.00119  0.324  2.37  0.0933  0.361  1.55  0.218 

0.9  0.387  8.71  0.00249  0.595  1.42  0.170  0.635  0.951  0.391 

0.1  1.79E6  1.72E6  1.68E9  1.86E4  3.95E3  1.11E5  3.26E4  1.64E3  4.24E5 

0.3  0.00712  465.  1.79E5  0.0378  20.0  0.00592  0.0470  11.7  0.0159 

0.5  0.0496  73.0  1.68E4  0.149  5.28  0.0320  0.173  3.28  0.0804 

0.7  0.149  27.5  5.27E4  0.324  2.57  0.0793  0.361  1.642  0.194 

0.9  0.387  13.1  0.00111  0.595  1.54  0.145  0.635  1.01  0.348 
From Table 1, it can be seen that is unrelated to , but incremental with the increase of / at certain /. decreases with the increasing / and increases with the incremental when the other parameters are fixed respectively . increases with the incremental / and decreases with the increasing with others determined parameters respectively.
ii.3 curves of two phase coexistence
From isothermal curves in Fig.1, we know that when temperature , the negative pressure section on the isotherm emerges and becomes larger with lower temperature. From Fig.2, we can see that the lower temperature, the smaller the value of , which also can be seen in Table 1. The doubt is whether is negative when temperature is low enough.
Considering (17), (16) can be rewritten as
(24) 
From (23), (24) and , one can get the relation between and , which is shown in Fig.3. Fig.3 exhibits the phase diagrams at fixed and , in which the curves represent the states of two phase coexistence and the terminal points are the critical points. From Fig.3 it can be seen that the influence of the electric charge and spacetime dimension on the phase diagrams, however pressure tends toward zero with decreasing temperature for all of the fixed and cases. That the pressure is always positive means Maxwell equal area law is appropriate to resolve the doubts about negative pressure and unstable states in phase transition of the dimensional RNAdS black hole. In Fig.4, the connection is depicted at different spacetime dimension , which explicitly exhibits the effect of the dimension on the distribution of the black hole phases.
Iii Rotating Black Holes
iii.1 Thermodynamics
The AdS rotating black hole solution is given by the KerrAdS metricHawking3 (); Gibbons (); Alta1 (),
(28) 
where
(29) 
and denotes the metric element on a dimensional sphere. The associated thermodynamic quantities are
(30) 
in which meets , that is . Then
(31) 
for in terms of and . While expressing , we expandedAlta1 ()
(32) 
to some given order and solve Eq. (31) order by order. The first term yields the relation
(33) 
Using Eqs.(6) and (30), the equation of can be obtained
(34) 
where
(35) 
While the isotherms are plotted in diagrams at respectively, which are shown in Fig.5. The situations of negative pressure and the thermodynamics unstable region with are also existent in the state equation the of KerrAdS black hole. When , where is the critical temperature of the KerrAdS black hole, unstable thermodynamic state appears, and while , negative pressure emerges. The expression of can be derived from (34) as in (12), and the corresponding is obtained by the way.
(36) 
iii.2 Maxwell equal area law
We use Maxwell equal area law to simulate a possible phase transition process of the dimensional KerrAdS black hole as a thermodynamic system. We also take and as the specific volumes of the two associated phases in the phase transition respectively, and as the constant pressure throughout the simulated process of the phase transition at constant temperature (), which can be derived by Maxwell equal area law.
(37) 
which can be rewritten as
(38) 
Combining with
(39) 
and setting , one can get
(40) 
where
(41) 
while , critical thermodynamics quantities can be got (40),
(42) 
The results are consistent with that in Ref.Alta1 (). As and , we can derive from (40) and (42),
(43) 
So we can solve at certain , then from (40) we can obtain the , and . Take , respectively. In Fig.6, we plot the isotherms (blue curves) for different in diagrams and the isobars (red solid lines) which represent the simulated phase transition processes derived from Maxwell equal area law. Like that in RNAdS black hole the processes become shorter with increasing temperatures until it turns into a point at a certain temperature, which is the critical temperature, and the point is the critical point of the black hole thermodynamic system. Every connected solid curve represents a processes of state change and phase transition, where no negative pressure and thermodynamic unstable state appear. The boundary of the twophase coexistence region is delineated by the dotdashed line.
In Table 2, we calculate , , as , , and respectively to find the influence of these parameters on the simulated phase transition process and the twophase coexistence region.





0.1  1.98E5  7.21E4  1.28E7  2.56E4  1.71E3  4.56E5  3.95E4  913.  1.17E4 

0.3  0.0166  90.9  2.86E4  0.0428  10.5  0.0199  0.0506  7.29  0.0389 

0.5  0.0859  18.8  0.00201  0.162  2.87  0.104  0.182  2.09  0.193 

0.7  0.218  8.12  0.00556  0.345  1.42  0.253  0.375  1.05  0.460 

0.9  0.475  4.36  0.0108  0.617  0.862  0.457  0.650  0.654  0.824 

0.1  1.98E5  1.14E5  5.12E8  2.56E4  1.99E3  3.36E5  3.95E4  1.02E3  9.31E5 

0.3  0.0166  144.  1.14E4  0.0428  12.2  0.0146  0.0506  8.17  0.0309 

0.5  0.0859  29.8  8.03E4  0.162  3.34  0.0763  0.182  2.34  0.153 

0.7  0.218  12.8  0.00222  0.345  1.65  0.186  0.375  1.18  0.367 

0.9  0.475  6.89  0.00432  0.617  1.00  0.337  0.650  0.732  0.655 

0.1  1.98E5  1.61E5  2.56E8  2.56E4  2.23E3  2.67E5  3.95E4  1.12E3  7.83E5 

0.3  0.0166  203.  5.72E5  0.0428  13.7  0.0116  0.0506  8.91  0.0260 

0.5  0.0859  42.1  4.01E4  0.162  3.75  0.0606  0.182  2.55  0.129 

0.7  0.218  18.2  0.00111  0.345  1.86  0.148  0.375  1.29  0.308 

0.9  0.475  9.75  0.00216  0.617  1.13  0.267  0.650  0.799  0.551 
From Table 2, one can see that is unrelated to , but incremental with the increase of / at certain / . decreases with increasing / and increases with the incremental when the other parameters are fixed respectively. Similarly increases with the incremental / and decreases with increasing while the other parameters are given respectively.
iii.3 the relation of phase transition pressure to temperature
From (39) and , we get
(44) 
Combining (43), (44) and , we plot the diagram and diagrams in Fig.7 and Fig.8.
We can see from Fig.7, , and as . So there is no the situation of negative pressure and the thermodynamic unstable states in the simulated phase transition processes of dimensional KerrAdS black hole, as that of the dimensional RNAdS black hole.
Iv Discussions
In this paper we take AdS black holes as thermodynamic systems and find the state equations in some region yield negative pressure and thermodynamic unstable states with . Comparing with phase transition in a usual liquidgas system and using Maxwell equal area law, we simulate the phase transition process in AdS black holes. From Fig.2 and Fig.6, one can see that, for both the dimensional RNAdS black hole and the dimensional KerrAdS black hole, as a part of each isotherm could be replaced with an isobar, which means phase transition occurs and two phases coexist in the section. The simulated phase transition belongs to the first order phase transition, and as the phase transition process shortens gradually until it become a point at , where the phase transition is the second order phase transition. The dimension and the electric charge (the angular momentum ) also have effect on the simulated phase transition process. The larger the dimension , the greater the change of specific volume before and after the phase transition. The larger the parameter , the lower the pressure , in the simulated phase transition.
Due to lack of the knowledge of chemical potential, the curve for two phase coexistence state of an usual thermodynamic system are obtained by experiment. However the slope of the curves can be calculated from the Clapeyron equation,
(47) 
in which , and are the molar volumes of phase and phase respectively, and the Clapeyron equation accords with experiment result, which directly verifies its thermodynamic correctness.
In a general thermodynamic system, the phase transition means the change of organization structure of substance (including the changes of position and the orientation of atom, ion, and electron) and a different phase appear. Phase transition is a physical process, and it does not involve the chemical reaction, so the chemical compositions do not change in the process.
Appropriate theoretical interpretation to the phase structure of AdS black hole thermodynamic system can help to know more about black hole thermodynamic properties, such as entropy, temperature and heat capacity, and it is significant for improving a selfconsistent black hole thermodynamic theory. This paper displays the diagrams for the two phase coexistence state and the Clapeyron equations of the black holes, which provide theoretical basis for investigation of phase transition and phase structure of black holes and help to explore the theory of gravity.
Acknowledgements.
This work is supported by NSFC under Grant Nos.(11475108,11175109;11075098), by the Shanxi Datong University doctoral Sustentation Fund Nos. 2011B03, China, Program for the Innovative Talents of Higher Learning Institutions of Shanxi, and the Natural Science Foundation for Young Scientists of Shanxi Province,China (Grant No.20120210034).References
 (1) J. D. Bekenstein, “Extraction of Energy and Charge from a Black Hole”, Phys. Rev. D 7, 949(1973).
 (2) J. M. Bardeen, B. Carter, S. W. Hawking, “The Four laws of black hole mechanics”, Commun. Math. Phys. 31, 161(1973).
 (3) S. W. Hawking, “Black Hole Explosions”, Nature 248, 30(1974).
 (4) J. X. Lu, “The Thermodynamical Phase Structure of black branes in String/M Theory” (in Chinese), Sci SinPhys Mech Astron 42, 1099(2012).
 (5) Ren Zhao, Li Chun Zhang, “A new explanation for statistical entropy of charged black hole”, Sci ChinaPhys Mech Astron 56, 1632(2013).
 (6) S. Hawking, D. N. Page, “Thermodynamics of black holes in antide Sitter space”, Commun. Math. Phys. 87, 577(1983).
 (7) A. Chamblin, R. Emparan, C. Johnson, R. Myers, “Charged AdS black holes and catastrophic holography”, Phys. Rev. D 60, 064018(1999), [hepth/9902170].
 (8) A. Chamblin, R. Emparan, C. Johnson, R. Myers, “Holography,thermodynamics and uctuations of charged AdS black holes”, Phys. Rev. D 60, 104026(1999), [hepth/9904197].
 (9) D. Kubizňák, R. B. Mann, “PV criticality of charged AdS black holes”, JHEP 1207, 033(2012), arXiv:1205.0559.
 (10) S. Gunasekaran, D. Kubiz, R. B. Mann, “Extended phase space thermodynamics for charged and rotating black holes and BornInfeld vacuum polarization”, JHEP 1211,10(2012), arXiv:1208.6251.
 (11) B. P. Dolan, D. Kastor, D. Kubizňák, R. B. Mann, Jennie Traschen, “Thermodynamic Volumes and Isoperimetric Inequalities for de Sitter Black Holes”, Phys. Rev. D 87, 104017(2013), arXiv:1301.5926.
 (12) M. Cvetic, G. W. Gibbons, D. Kubizňák, C. N. Pope, “Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume”, Phys. Rev. D 84, 024037(2011), arXiv:1012.2888.
 (13) D. Kastor, S. Ray, J. Traschen, “Enthalpy and the Mechanics of AdS Black Holes,” Class. Quant. Grav. 195011 (2009), arXiv:0904.2765 [hepth].
 (14) R. Banerjee, S. K. Modak, S. Samanta, “Second Order Phase Transition and Thermodynamic Geometry in KerrAdS Black Hole”, Phys. Rev. D 84, 064024(2011), arXiv:1005.4832.
 (15) R. Banerjee, D. Roychowdhury, “Critical behavior of Born Infeld AdS black holes in higher dimensions”, Phys. Rev. D 85, 104043(2012), arXiv:1203.0118.
 (16) R. Banerjee, D. Roychowdhury, “Critical phenomena in BornInfeld AdS black holes”, Phys. Rev. D 85, 044040(2012), arXiv:1111.0147.
 (17) R. Banerjee, D. Roychowdhury, “Thermodynamics of phase transition in higher dimensional AdS black holes”, JHEP 1211, 004(2011), arXiv:1109.2433.
 (18) R. Banerjee, S.t Ghosh, D. Roychowdhury, “New type of phase transition in Reissner Nordström  AdS black hole and its thermodynamic geometry“, Phys. Lett. B. 696, 156(2011), arXiv:1008.2644.
 (19) R. Banerjee, S. K. Modak, S. Samanta, “Glassy Phase Transition and Stability in Black Holes”, Eur. Phys. J. C 70, 317(2010), arXiv:1002.0466.
 (20) Natach Altamirano, David Kubiznak, Robert B. Mann, and Zeinab Sherkatghanad, “Thermodynamics of Rotating Black Holes and Black Rings: Phase Transitions and Thermodynamic Volume”, Galaxies 2, 89(2014), arXiv:1401.2586.
 (21) H. H. Zhao, L. C. Zhang, M. S. Ma, R. Zhao, “PV criticality of higher dimensional charged topological dilaton de Sitter black holes”, Phys. Rev. D 90, 064018(2014).
 (22) Ren Zhao, Mengsen Ma, Huihua Zhao, Lichun Zhang. “On the critical phenomena and Thermodynamics of the ReissnerNordstromde Sitter black hole”, Advances in High Energy Physics 2014, 124854(2014).
 (23) MengSen Ma, HuiHua Zhao, LiChun Zhang, Ren Zhao, “Existence condition and phase transition of ReissnerNordströmde Sitter black hole”, Int. J, Mod. Phys. A 29, 1450050(2014), arXiv:1312.0731.
 (24) LiChun Zhang, MengSen Ma, HuiHua Zhao, Ren Zhao, “Thermodynamics of phase transition in higher dimensional ReissnerNordströmde Sitter black hole”, Eur. Phys. J. C 73, 3052(2014), arXiv:1403.2151.
 (25) JieXiong Mo, WenBiao Liu, “Ehrenfest scheme for PV criticality of higher dimensional charged black holes, rotating black holes, and GaussBonnet AdS black holes”, Phys. Rev. D 89, 084057(2014).
 (26) J. X. Mo, W. B. Liu, “Ehrenfest scheme for PV criticality in the extended phase space of black holes”, Phys. Lett. B 727, 336(2013).
 (27) JieXiong Mo, WenBiao Liu, “PV Criticality of Topological Black Holes in LovelockBornInfeld Gravity”, Eur. Phys. J. C 74, 2836(2014), arXiv:1401.0785.
 (28) JieXiong Mo, GuQi Lia, WenBiao Liu, “Another novel Ehrenfest scheme for PV criticality Of RNAdS black holes”, Phys. Lett. B 730, 111(2014).
 (29) JieXiong Mo, XiaoXiong Zeng, GuQiang Li, Xin Jiang, WenBiao Liu, “A unified phase transition picture of the charged topological black hole in HoavaLifshitz gravity”, JHEP 1310, 056(2013), arXiv:1404.2497.
 (30) J. X. Mo, “Ehrenfest scheme for the extended phase space of f(R) black holes”, Eur. phys. Lett. 105, 20003(2014).
 (31) MengSen Ma, Fang Liu, Ren Zhao, “Continuous phase transition and critical behaviors of 3D black hole with torsion”, Class. Quantum Grav. 31 095001(2014), arXiv:1403.0449.
 (32) Arindam Lala, Dibakar Roychowdhury, “Ehrenfest’s scheme and thermodynamic geometry in BornInfeld AdS black holes”, Phys. Rev. D 86, 084027(2012).
 (33) ShaoWen Wei, YuXiao Liu, “Critical phenomena and thermodynamic geometry of charged GaussBonnet AdS black holes”, Phys. Rev. D 87, 044014(2013). arXiv:1209.1707
 (34) Jishnu Suresh, R. Tharanath, Nijo Varghese, V. C. Kuriakose, “The thermodnamics and thermodynamic geometry of the Park black hole”, Eur. Phys. J. C. 74, 2819(2014).
 (35) Seyed Ali Hosseini Mansoori, Behrouz Mirza, “Correspondence of phase transition points and singularities of thermodynamic geometry of black holes”, Eur. Phys. J. C. 74, 2681(2014), arXiv:1308. 1543.
 (36) R. Tharanath, Jishnu Suresh, V. C. Kuriakose, “Phase transitions and Geometrothermodynamics of Regular black holes”, arXiv:1406.3916.
 (37) Chao Niu, Yu Tian, Xiaoning Wu, “Critical Phenomena and Thermodynamic Geometry of RNAdS Black Holes”, Phys. Rev. D 85, 024017(2012), arXiv:1104.3066.
 (38) MengSen Ma, Ren Zhao, “Phase transition and entropy spectrum of the BTZ black hole with torsion”, Phys. Rev. D 89, 044005(2014).
 (39) MengSen Ma, “Thermodynamics and phase transition of black hole in anasymptotically safe gravity”, Phys. Lett. B 735, 45(2014).
 (40) Yongwan Gim, Wontae Kim, “Thermodynamic phase transition in the rainbow Schwarzschild black hole”, JCAP 2014, 003(2014), arXiv:1406.6475.
 (41) Mohammad Bagher Jahani Poshteh, Behrouz Mirza, “Phase transition, critical behavior, and critical exponents of MyersPerry black holes”, Phys. Rev. D 88, 024005 (2013), arXiv:1306.4516.
 (42) Gustavo Arciniega, Alberto S¨¢nchez, “Geometric description of the thermodynamics of a black hole with power Maxwell invariant source”, arXiv:1404.6319 [mathph].
 (43) Ren Zhao, MengSen Ma, HuaiFan Li, LiChun Zhang, “On Thermodynamics of charged and rotating asymptotically AdS black strings”, Advances in High Energy Physics 2013, 371084(2013).
 (44) Ren Zhao, HuiHua Zhao, MengSen Ma, LiChun Zhang, “On the critical phenomena and thermodynamics of charged topological dilaton AdS black holes”, Eur. Phys. J. C. 73, 2645(2013), arXiv:1305.3725.
 (45) Natacha Altamirano, David Kubiznak, Robert B. Mann, “Reentrant phase transitions in rotating antide Sitter black holes”, Phys. Rev. D 88, 101502(2013).
 (46) RongGen Cai, LiMin Cao, Li Li, RunQiu Yang, “PV criticality in the extended phase space of GaussBonnet black holes in AdS space”, JHEP 1309, 005(2013), arXiv:1306.6233[grqc].
 (47) DeCheng Zou, ShaoJun Zhang, Bin Wang, “Critical behavior of BornInfeld AdS black holes in the extended phase space thermodynamics”, Phys. Rev. D 89, 044002(2014), arXiv:1311.7299.
 (48) DeCheng Zou, Yunqi Liu, Bin Wang, “Critical behavior of charged GaussBonnet AdS black holes in the grand canonical ensemble”, Phys. Rev. D 90, 044063(2014), arXiv:1404.5194.
 (49) Antonia M. Frassino, David Kubiznak, Robert B. Mann, Fil Simovic, “Multiple Reentrant Phase Transitions and Triple Points in Lovelock Thermodynamics”, arXiv:1406.7015.
 (50) GuQiang Li, “Effects of dark energy on PVcriticality of charged AdS black holes”, Phys. Lett. B, Volume 735, 256(2014).
 (51) ShaoWen Wei, YuXiao Liu, “Triple points and phase diagrams in the extended phase space of charged GaussBonnet black holes in AdS space”,Phys. Rev D 90, 044057(2014), arXiv:1402.2837.
 (52) D. Kubiznak, and R. B. Mann, “Black Hole Chemistry”, arXiv:1404.2126.
 (53) Wei Xu, Hao Xu, Liu Zhao, “GaussBonnet coupling constant as a free thermodynamical variable and the associated criticality”, Eur. Phys. J. C 74, 2970(2014), arXiv:1311.3053.
 (54) Yunqi Liu, DeCheng Zou, Bin Wang, “Signature of the Van der Waals like smalllarge charged AdS black hole phase transition in quasinormal modes”, arXiv:1405.2644.
 (55) S. H. Hendi, M. H. Vahidinia, “Extended phase space thermodynamics and PV criticality of black holes with nonlinear source”, Phys. Rev. D 88, 084045 (2013), arXiv:1212.6128 [hepth].
 (56) Euro Spallucci, Anais Smailagic, “Maxwell’s equalarea law for charged Antide Sitter black holes”, Phys. Lett. B. 723, 436(2013).
 (57) JunXin Zhao, MengSen Ma, LiChun Zhang, HaiHun Zhao, Ren Zhao, “The equal area law of asymptotically AdS black holes in extended phase space”, Astrophysics and Space Science 352, 763(2014).
 (58) Euro Spallucci, Anais Smailagic, “Maxwell’s equal area law and the HawkingPage phase transition”, J. Grav. 2013, 525696 (2013), arxiv:1310.2186.
 (59) Lichun Zhang, Hui Hua Zhao, Ren Zhao, Mengsen Ma, “Equal area laws and latent heat for ddimensional RNAdS black hole”, Advances in High Energy Physics, to be published.
 (60) S. W. Hawking, C. J. Hunter, M. M. TaylorRobinson, “Rotation and the AdSCFT correspondence”, Phys. Rev. D 59 (1999) 064005.
 (61) G. W. Gibbons, H. L¨¹, Don N. Page, C.N. Pope, “The general Kerrde Sitter metrics in all dimensions”, J. Geom. Phys. 53 (2005) 49.