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PHYSICAL REVIEW D 102, 064014 (2020)

Thermodynamics and Van der Waals of charged black holes in flat spacetime via R´enyi statistics

† ‡ Chatchai Promsiri ,1,2,* Ekapong Hirunsirisawat ,2,3, and Watchara Liewrian 1,2,4, 1Department of Physics, King Mongkut’s University of Technology Thonburi, Pracha Uthit Road, Bangkok 10140, Thailand 2Theoretical and Computational Physics (TCP), Theoretical and Computational Science Center (TaCS), King Mongkut’s University of Technology Thonburi, Pracha Uthit Road, Bangkok 10140, Thailand 3Learning Institute, King Mongkut’s University of Technology Thonburi, Pracha Uthit Road, Bangkok 10140, Thailand 4Thailand Center of Excellence in Physics, Ministry of Higher Education, Science, Research and Innovation, 328 Si Ayutthaya Road, Bangkok 10400, Thailand

(Received 13 June 2020; accepted 5 August 2020; published 9 September 2020)

The phase structure and of the 3 þ 1 dimensional charged black holes in asymptotically flat spacetime are investigated within the R´enyi statistics. As the nonextensive parameter λ above zero, a charged black hole can be in thermodynamic equilibrium with surrounding thermal radiation and have a Hawking–Page phase transition. This gives more evidence supporting the proposed conjectured equivalence between the black hole thermodynamics in asymptotically flat spacetime via R´enyi statistics and that in asymptotically anti–de Sitter (AdS) spacetime via Gibbs–Boltzmann statistics. The present work also provides another aspect of supporting evidence through exploring the extended phase space within the R´enyi statistics. Working on a modified version of the Smarr formula, the thermodynamic pressure P and volume v of a charged black hole are found to be related to λ. The thermodynamics of asymptotically flat charged black holes via R´enyi statistics has the Van der Waals phase structure, P − v criticality and universal constant, in a similar way as that of asymptotically AdS charged black hole via Gibbs–Boltzmann statistics. This raises an interesting question of how λ in the former system relates to jΛj in the latter one.

DOI: 10.1103/PhysRevD.102.064014

I. INTRODUCTION AND MOTIVATIONS analogy between the laws of black hole mechanics and of thermodynamics has led us to the notion that a black hole In general relativity, a black hole was initially expected could behave as a thermal object. to be a dark object as a result of the existence of the event Usually, a zero-charge black hole in an asymptotically flat horizon inside, which nothing can escape. However, spacetime background can only be in the phasewith negative according to black hole thermodynamics initiated by heat capacity at an arbitrary temperature, such that it cannot Bekenstein [1] and Hawking [2], a black hole can have be in a thermal equilibrium with a heat bath of radiation. and nonzero temperature. Later on there have been However, a rich phase structure can be found in some further works, namely the laws of black hole mechanics, spacetime backgrounds through thermodynamic stability suggesting that the area and the surface gravity of a black analysis. For instance, in an asymptotically anti–de Sitter ’ hole s event horizon correspond to entropy and temper- (AdS) space there are possibly two branches of uncharged ature, respectively [3]. Furthermore, the electrostatic poten- black holes, namely the small and large black holes. The tial and angular velocity at the event horizon can be treated small black hole phase has negative heat capacity implying as the chemical potentials. Surprisingly, the mathematical that it is thermodynamically unstable. On the other hand, the large black hole phase has positive heat capacity, hence it is *[email protected] thermodynamically stable. However, these two black hole † [email protected] phases can exist only above a certain temperature, let us call ‡ T T [email protected] it min here. Below min, the pure thermal radiation bath occupies the AdS space. Intriguingly, there is the Hawking– Published by the American Physical Society under the terms of T Page phase transition at the temperature HP, slightly above the Creative Commons Attribution 4.0 International license. T T T Further distribution of this work must maintain attribution to min. At the temperature in the range between min and HP, the author(s) and the published article’s title, journal citation, the thermal radiation is mostly thermodynamically preferred and DOI. Funded by SCOAP3. over the small and large black hole phases. The large black

2470-0010=2020=102(6)=064014(15) 064014-1 Published by the American Physical Society PROMSIRI, HIRUNSIRISAWAT, and LIEWRIAN PHYS. REV. D 102, 064014 (2020)

S hole phase turns out to be the most thermodynamically words, the black hole entropy BH should encode the black T>T preferred at HP [4]. hole information with nonlocal and nonextensive nature. There have been some controversial issues for a long Consequently, we need a non-Boltzmannian approach to time, such as whether it is appropriate to apply the standard deal with this. A new type of entropic function can be Gibbs–Boltzmann (GB) approach to a self-gravitating introduced by relaxing the Shannon–Khinchin axiomatic system. Gravitation is a long-range attractive force, such definition of the entropic function, i.e., additivity, to the that the average potential energy hVi between particles in a weaker nonadditive composition rule. With composability, self-gravitating system is negative. Given that hKi is a the entropic function can remain to be physically mean- positive value of average kinetic energy, we can use the ingful. According to the derivation of Abe, the most general virial theorem to show that the total average energy of nonadditive entropy composition rule is in the form [18] the system hUi has negative value, namely, hUi¼hKiþ hVi¼−hKi. It is well known that the kinetic energy hKi is HλðS12Þ¼HλðS1ÞþHλðS2ÞþλHλðS1ÞHλðS2Þ; ð1Þ linear in the temperature T up to some constants due to the equipartition theorem. As a consequence, we obtain the where Hλ is a differentiable function of S, and λ ∈ R is a C ¼ dhUi=dT < 0 heat capacity as [5,6]. Importantly, a constant parameter. One of the simplest versions of non- negative heat capacity indicates that the self-gravitating extensive entropy, obeying Tsallis entropy, can be written system is thermodynamically unstable. This implies that in the form [10] the system cannot become thermodynamically equilibrium when it is in thermal contact with a heat bath. While this is a 1 XW weird behavior of the self-gravitating system, it might be S ¼ pq − 1 ; ð Þ T 1 − q i 2 possible that it is an incorrect conclusion. A complete i¼1 understanding of this situation has still been a challenging issue. Generically, the standard GB statistical approach where pi are the probabilities of microstates of the system, should be violated in the case of long-range interactions W is the total number of microstates, and q ∈ R is the due to some clues such as the existence of a divergent dimensionless parameter of nonextensivity. Clearly, the partition function. This has been pointed out by Gibbs [7] standard GB entropy is recovered when q → 1.Thecom- and later on by others [8–10] (see also [11] and references position rule of nonadditive Tsallis entropy can be written as therein). In other words, applying the standard GB 12 1 2 1 2 approach in the case of a self-gravitating system may lead ST ¼ ST þ ST þð1 − qÞSTST; ð3Þ us to obtain an incomplete result. In conventional thermodynamic systems of ordinary which satisfies the Abe’s nonadditive entropy composition matter, the entropy of a whole system can be written as rule,asshownin(1), when HλðSÞ¼ST and λ ¼ 1 − q. the sum of the entropy of subsystems. In this way, the However, there is a long-standing problem about the thermal entropy of the system typically scales with its volume. As a equilibrium for nonextensive systems relating to the com- “ ” result, it is said to be an extensive variable. However, patibility with the zeroth law of thermodynamics. Namely, if Bekenstein argued that a black hole system carries entropy two systems are in the thermal equilibrium, then the total proportional to the surface area of its event horizon rather entropy has the maximum value dSAB ¼ dðSA þ SBÞ¼0, “ ” than the volume. Such behavior is called an area law. which implies the existence of an empirical temperature Therefore, the black hole’s entropy is nonextensive. 1 ∂S ∂S ¼ A ¼ B . The problem with nonextensive entropy is Intriguingly, this black hole area law guides us to the T ∂EA ∂EB holographic principle, which states that the information in a that its composition rule is in the nonadditive form, hence it higher dimensional bulk spacetime can be encoded into its is not clear whether we can define an empirical temperature boundary [12,13]. Later on, this principle becomes more [19]. Recently, Biró and Ván proposed a way to solve this established by the developments of the AdS/CFT corre- problem by transforming the nonadditive entropy into spondence [14–16]. Moreover, the area law can be found in another one that has an additive composition rule, which the entanglement entropy of the reduced state of a sub- satisfies the zeroth law of thermodynamics [20].Their “ ” region in strongly correlated quantum systems [17]. This method is called the formal logarithmic approach. They similarity raises the question about whether the quantum showed that, for a homogeneous system, the Tsallis entropy origin of black hole entropy may be somehow related to the can be transformed into the well-defined entropy function as entanglement entropy. Currently, the nonextensive nature 1 of entropy has received wide attention in several fields, and LðSTÞ¼ ½lnð1 þð1 − qÞSTÞ ≡ SR: ð4Þ it might possibly improve our insights about the micro- 1 − q scopic nature of black holes. As discussed above, the GB statistics might not be Interestingly, this result is a well-known R´enyi entropy SR appropriate to use in black hole thermodynamics. In other defined up to an arbitrary real parameter q as [21]

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XW 3 1 where the cosmological constant Λ ¼ − 2, and L is the S ¼ pq; ð Þ L R 1 − q ln i 5 AdS radius. The consideration in this way is called the i¼1 “extended phase space” [28–30]. With this approach, there where 0

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dr2 where κ, Ω, and Φ are surface gravity, angular velocity, and ds2 ¼ −fðrÞdt2 þ þ r2dΩ2; ð Þ fðrÞ 2 9 electric potential at the event horizon, respectively. This can be seen as the first law of thermodynamics when one 2 2 2 2 A κ where dΩ2 ¼ dθ þ sin θdϕ is the square of line element identifies the event horizon area , surface gravity with on 2-sphere and the function fðrÞ is given by entropy, and temperature of the black hole, respectively. For a nonrotating charged black hole, Q is the number of 2M Q2 particles in thermodynamic description since the charge fðrÞ¼1 − þ : ð Þ Φ r r2 10 simply counts the number of particles, and its conjugate plays a role of chemical potential. The black hole horizon can be determined from the condition fðrÞ¼0, where its roots consist of A. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi One can consider thermodynamics of black holes in r ¼ M M2 − Q2; ð Þ 11 either grand canonical or canonical ensemble. When the black hole exchanges charge Q with the surrounding heat where rþ and r− are the radii of the outer and inner horizon, bath, the chemical potential Φ can be held to be fixed. In respectively. The black hole horizon is at rþ, from which this way, the system is being considered in the grand the Hawking radiation is generated. The black hole mass M canonical ensemble. Consequently, the charged black hole relates to the event horizon radius and the charge Q as thermodynamic quantities can be written as follows: 2 rþð1 þ Φ Þ r Q2 E ¼ M ¼ ; ð16Þ M ¼ þ 1 þ : ð Þ 2 2 12 2 rþ f0ðr Þ 1 − Φ2 T ¼ þ ¼ ; ð Þ The charge Q can generate the gauge field of the form H 17 4π 4πrþ Q A A ¼ Atdt ¼ − − Φ dt: ð13Þ S ¼ ¼ πr2 ; ð Þ r BH 4 þ 18 By setting At ¼ 0 at the horizon, we can relate the electric ∂S 2πr2 ð1 þ Φ2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi BH þ 2 2 CΦ ¼ TH ¼ − ; ð Þ potential Φ with M and Q using rþ ¼ M þ M − Q . 2 19 ∂TH Φ 1 − Φ Thus, we have 2 rþð1 − Φ Þ Q Q G ¼ E − THS − ΦQ ¼ ; ð20Þ Φ ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð14Þ BH 4 rþ M þ M2 − Q2

where E is the , TH is the Hawking Obviously, the outer and inner horizons are degenerate temperature, S is the Bekenstein–Hawking entropy, M ¼ Q BH when . Namely, this extremal black hole has only CΦ is the heat capacity at a fixed electric potential Φ, one horizon, i.e., re ¼ rþ ¼ r−. On the other hand, the and G is the , respectively. Because the M

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FIG. 1. Left: R´enyi temperature of a charged black hole TR versus the event horizon radius rþ for a fixed value Φ ¼ 0.5 is plotted with λ ¼ 0.3 (solid blue), 0.6 (solid green), and 1.0 (solid red), compared with the case of the GB statistics, λ ¼ 0 (dashed black). Right: heat capacity CR of a charged black hole versus rþ with Φ ¼ 0.5 is plotted in the case of λ ¼ 0.3 (solid blue), its value is negative at rþ rc. However, the heat capacity cannot be positive at all values of rþ in the GB statistics, λ ¼ 0 (dashed black).

2 ðrþ − QÞ where all thermodynamic variables are defined as in the E ¼ M − Me ¼ ; ð21Þ C 2rþ grand canonical ensemble, except that Q here is the heat capacity at a constant number of particles, and F is the f0ðr Þ r2 − Q2 T ¼ þ ¼ þ ; ð Þ . From (24), we have two branches of H 3 22 4π 4πrþ charged black holes in which each black hole has either positive or negative heat capacity. The heat capacity is A S ¼ ¼ πr2 ; ð Þ positive when BH 4 þ 23 pffiffiffi 2 2 2 ∂S 2πr ðr − Q Þ Q 3Q. At the critical ðrþ − QÞðrþ − 3QÞ F ¼ E − T S ¼ ; ð Þ point rc ¼ 3Q, the heat capacity diverges as the Hawking H BH 25 4rþ temperature reaches its maximum value

FIG. 2. Left: Gibbs free energy vs temperature for fixed Φ ¼ 0.5 in the R´enyi entropy. Right: Gibbs free energy vs temperature by varying R´enyi parameters as λ ¼ 0.3, 0.6 and 1 corresponding to blue, green, and red curves, respectively. The plot from the GB statistics is shown in this graph with black dashed curve.

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1 r rc, where rc ¼ 1=λπ. Accordingly, there are two possible black hole configurations: one with In contrast with typical systems, the phase transition of RN- negative value of heat capacity and another with positive flat black holes depends on the ensemble in consideration. value. We will refer to these as small and large black holes, This might result from the long-range nature of gravita- respectively. Substituting rc into (12), we obtain the critical tional and electromagnetic interactions of the system [38]. mass

III. THERMODYNAMICS AND THERMAL PHASE 1 þ Φ2 Mc ¼ pffiffiffiffiffi : ð32Þ TRANSITION VIA RÉNYI STATISTICS 2 λπ Now, we turn to an alternative approach to study thermodynamic properties of charged black holes. Here, In R´enyi statistics, a RN-flat black hole is in the small black hole branch when MM namic quantities. Then, we will investigate the thermal hole branch when c. The heat capacity CR is inversely proportional to the phase structure of RN-flat black holes in both the fixed T r potential and fixed charge ensemble. slope of the graph R versus þ, as shown in Fig. 1 (left). As suggested in previous works [23,24], black holes can Namely, we can write be treated in the way that follows the nonadditive Tsallis 1 ð1 þ Φ2Þ 1 statistics, whose composition rule is in the simplest form as C ¼ : ð Þ R 2 2 33 shown in (3). Even though the Tsallis entropy function 2 ð1 − Φ Þ ∂TR=∂rþ tends to be one of the proper choices representing black hole entropy, it turns out to have difficulty defining the Therefore, the critical radius rc is just a turning point at empirical temperature through the zeroth law of thermo- which the slope of TRðrþÞ changes the sign, i.e., T0 ðr Þ¼0 dynamics. To avoid this, the R´enyi entropy in the form of R þ . With the presence of this turning point, the the formal logarithm of the Tsallis entropy is proposed [20]. RN-flat black holes with λ ¼ 0.3 (solid blue), 0.6 (solid Using this transformation rule, we obtain the R´enyi entropy green), and 1.0 (solid red) can be stable with positive heat function of a black hole as capacity when rþ >rc and negative heat capacity at rþ < rc while there is no turning point in the case of GB statistics 1 S ¼ ð1 þ λS Þ; ð Þ (dashed black). The discussion of these results is actually R λ ln BH 28 equivalent to the plots in Fig. 1 (right). Interestingly, the turning point in Fig. 1 (left) also shows which has an additive property. The R´enyi temperature can that the lower bound of a Hawking temperature becomes be expressed as larger with a higher level of nonextensivity. This can be seen through solving (30) for rþ, such that we have 1 T ¼ ¼ T ð1 þ λS Þ: ð Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ∂S =∂M H BH 29 ! R 2T λð1 − Φ2Þ2 r ¼ R 1 1 − ; ð Þ L;S 2 2 34 Based on this formula of the R´enyi entropy and its λð1 − Φ Þ 4πTR corresponding temperature, we consider in this section the thermal phase transitions of RN-flat black holes in the where rL and rS denote the event horizon of the large and grand canonical and the canonical ensemble. small black hole configurations at fixed temperature, respectively. The solution is real when the discriminant A. Grand canonical ensemble in the above equation is greater than zero. This implies that there is a minimum temperature at r ¼ rc, which is For the grand canonical ensemble with a fixed value of Φ, the corresponding R´enyi temperature as a function of the rffiffiffi ð1 − Φ2Þ λ event horizon rþ and the electrical potential Φ is given by T ¼ : ð Þ R;min 2 π 35 2 2 ð1 − Φ Þð1 þ λπrþÞ TR ¼ : ð30Þ Note that for the uncharged case, the minimum temperature 4πrþ qffiffi T ¼ 1 λ reduces to the form R;min 2 π as found in [23]. The heat capacity can be obtained in a usual way As shown in the work of Gross et al. [39], the 3 þ 1 ∂S 2πr2 ð1 þ Φ2Þ dimensional hot flat space can be in the unstable state due C ¼ T R ¼ − þ : ð Þ to the nucleation of black holes without the lower bound of R R ∂T ð1 − Φ2Þð1 − λπr2 Þ 31 R Φ þ temperature. In other words, in the GB statistics, a black

064014-6 THERMODYNAMICS AND VAN DER WAALS PHASE TRANSITION … PHYS. REV. D 102, 064014 (2020) hole can be formed at an arbitrarily low temperature of thermal radiation. Considering this through R´enyi statistics, however, gives a different conclusion. A black hole is not T T – hole only when R R;min. This shows that a Hawking Page phase transition of black holes in the Minkowski background is, in a similar way, what occurs in the black holes in the AdS background via GB statistics. Therefore, FIG. 3. Hawking–Page phase transition line in the T − Φ phase our result is in contrast with that derived from the diagram. The line dividing the space into two regions, the thermal consideration of Gross et al. [39]. radiation phase, and large black hole phase. The thermodynamic stability can be investigated through considering the free energy as a function of the temper- temperature at the cusp, as seen clearly through the ature. In the grand canonical ensemble, we use the Gibbs 2 ∂ GR divergence of heat capacity CR ¼ −TRð 2 Þ at the cusp, free energy as thermodynamical potential. With the R´enyi ∂TR Φ statistics, the Gibbs free energy function should be modi- as shown in Fig. 2. fied by R´enyi entropy and its corresponding temperature as 1 ð1 − Φ2Þ SRc ¼ ln 2;GRc ¼ pffiffiffiffiffi ð1 − ln 2Þ: ð38Þ GR ¼ M − TRSR − ΦQ; λ 2 λπ   r ð1 − Φ2Þ 1 ¼ þ 2 − 1 þ ð1 þ λπr2 Þ : ð Þ S λ 4 λπr2 ln þ 36 Notice that Rc is a universal constant at a fixed .Itis þ obviously independent of the black hole’s mass M and charge Q. At the cusp, the Gibbs free energy is at the The plots of GR versus TR are shown in Fig. 2 (left). While G the GB statistics (λ ¼ 0) tell us that the charged black hole highest value Rc, as shown in the equation above. Φ is unstable as its free energy is positive at any nonzero In the AdS/CFT corespondence, the electric potential temperature [dashed, Fig. 2 (right)], the R´enyi parameter in the bulk has the holographic dual to the chemical μ (λ > 0) allows that the black hole can be stable above a potential of the gauge theory at the boundary. The T G Hawking–Page phase transition in the bulk of AdS space certain temperature HP, where the R has negative value as indicated in Fig. 2 (left). From (36), we can find this is dual to the phase transition between a cold confined temperature at which the first order Hawking–Page phase phase and a hot deconfined phase at finite chemical G ¼ 0 potential. Intriguingly, the phase diagram from our results, transition occurs by setting R . Numerically, we can T − μ find that as shown in Fig. 3, looks similar to the phase diagram of confining/deconfining phase transition at finite chemical rffiffiffi λ potential. This provides one more supporting evidence that T ≈ 0 64ð1 − Φ2Þ ; ð Þ the physics of asymptotically flat black holes using the HP . π 37 R´enyi approach somehow corresponds to that of asymp- totically AdS black holes using GB statistics, as suggested which is about 1.28 times TR; . Figure 3 shows the min in previous works [23,24]. Hawking–Page temperature as a function of the electro- statics potential. This phase transition behavior from our results are very similar to the RN-AdS via standard GB B. Canonical ensemble statistics [25–27]. Obviously, we can see that for Φ ¼ 1 and Here, we investigate the thermal properties and the phase λ ¼ 0 cases the Hawking–Page temperature vanishes, structure of black holes in canonical ensemble (fixed Q) via therefore, the first-order phase transition cannot exist either R´enyi statistics. Using (22) and (29) the Hawking temper- in the extremal black hole or in the consideration of black ature with fixed charge can be written as holes via the GB statistics. G T ðr2 − Q2Þð1 þ λπr2 Þ Let us focus on the presence of the cusp of R at R;min, T ¼ þ þ : ð Þ T R 3 39 which is less than HP. The cusp corresponds to the point of 4πrþ small/large black hole phase transition. This is of the second-order type due to the discontinuity of the second The relation between TR and rþ with fixed Q ¼ 1 derivative of the Gibbs free energy with respect to the at different λ are plotted as shown in Fig. 4 (left).

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FIG. 4. Left: Plots of the R´enyi temperature of charge black hole TR versus the event horizon radius rþ with fixed charge Q ¼ 1 at different value of parameter λ. Right: Heat capacity CR versus rþ with charge Q ¼ 1 at different value of λ.

These isocharge curves of the black hole temperature In contrast with GB statistics, the R´enyi approach allows versus the horizon radius look similar to the case of three branches of charged black hole configurations in the nonzero charge black holes in AdS background [25–27]. canonical ensemble, one negative and two positive heat For different values of λ parameter, the isocharge curves capacities as shown in Fig. 4 (right). We denote them as behave differently where a critical phase transition can branch 1, branch 2, and branch 3, respectively. The addi- occur with the condition tional branch 3 now appears and shows an interesting thermal phase of the black hole. From (45), the heat ∂T ∂2T 0 ¼ R ¼ R : ð Þ capacity grows without an upper bound when the horizon 2 40 r r r ∂rþ Q ∂rþ Q radius þ equals 1 and 2. The corresponding temperatures at radii r1 and r2 are T1 ¼ 0.0299 and T2 ¼ 0.0296, By solving this equation, we obtain the critical Renyi respectively, for the chosen parameters in the Fig. 4. parameter λc, critical horizon rc, and critical temperature Tc In the canonical ensemble, the thermodynamic potential at the critical point as is the Helmholtz free energy, which can be generalized to pffiffiffi satisfy the R´enyi statistics as 7 − 4 3 λ ¼ ; ð Þ c 2 41 πQ FR ¼ E − TRSR;  pffiffiffi Q − rþ r ¼ð3 þ 2 3Þ1=2Q; ð Þ ¼ 2πr2 ðQ − r Þ c 42 4πr3 þ þ þ  2 1 2 2 Tc ¼ pffiffiffi : ð43Þ þ ðQ þ rþÞð1 þ λπrþÞ lnð1 þ λπrþÞ ; ð46Þ ð3 þ 2 3Þ3=2πQ λ

For small R´enyi parameter λ < λc, there are two local where E ¼ M − Me is the energy of the system relative to extrema of the isocharge curve in Fig. 4 (left) at that of the extremal black hole. Figure 5 (left) shows the relation of the free energy versus the R´enyi temperature of 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 2 2 2 2 4 2 RN-flat black holes at a small λ parameter. Interestingly, the r1;2 ¼ ð1 − λπQ Þ ∓ λ π Q − 14λπQ þ 1 ; 2πλ swallowtail behavior occurs when λ < λc, which is very ð44Þ similar to a Van der Waals type of liquid/gas phase transition. In Fig. 5 (left), there is only branch 1 that exists T where the horizon radii r1 and r2 correspond to a local at low temperature. At a certain temperature 2, branch 2 maximum and local minimum of the isocharge curve, and 3 emerge simultaneously with larger free energy than respectively. The discriminant in (44) is zero as λ ¼ λc, branch 1. However, when the temperature increases to T1,it hence two extremal radii r1 and r2 are degenerate into rc. can be seen that branch 1 and 2 combine and disappear. Just The corresponding heat capacity is a little bit below T1, the free energy of branch 1 and 3 are T equal at temperature HP, and the latter has more negative ∂S 2πr2 ðr2 − Q2Þ free energy than the first at T>T . This result implies the C ¼ T R ¼ − þ þ : HP R R 2 2 2 2 2 – ∂TR Q rþ − 3Q þ λπrþðrþ þ Q Þ Hawking Page phase transition from branch 1 to branch 3 at this point. Furthermore, for λ > λc there is only one large ð Þ 45 black hole configuration with positive heat capacity,

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FIG. 5. Left: Branch 1, 2, and 3 of the RN-flat are shown in the swallow-tail-shape graph of the Helmholtz free energy FR versus the R´enyi temperature TR at a fixed charge Q ¼ 1 and 0 < λ < λc. Right: Graphs of FR versus TR at different values of λ are shown with the graph of that of the RN-flat via GB statistics (λ ¼ 0, dashed). The discussion about these graphs can be seen in the text. therefore, it is thermodynamically stable. The graphs of the Although the parameter λ can be a large negative real free energy versus R´enyi temperature at different values of number, we consider the black hole here with the λ are shown in Fig. 5 (right). assumption that there should be a small deviation from the conventional GB entropy due to the effect of non- IV. THE EMERGENCE OF VAN DER WAALS extensivity corresponding to 0 < λ ≪ 1. From the Smarr PHASE TRANSITION formula above, we can use the relations In the extended phase space approach with the GB T eλSR − 1 T ¼ R S ¼ ; ð Þ statistics, a complete analogy between VdW liquid and RN- H and BH 48 eλSR λ AdS black holes in canonical ensemble was established by identifying the thermodynamic pressure with cosmological Λ where they are derived from (28) and (29), to rewrite [41] in constant P ¼ − . However, in the asymptotically flat 8π the R´enyi description. In addition to this, we can expand the spacetime, the thermodynamic pressure P ¼ 0 and the first term of [41] in the power series of the λ parameter to black hole phase transition in the F–T plane is not in obtain the VdW type. As discussed in the previous section, the λ parameter can give an effect to the thermodynamic descrip- M ¼ 2T S − λT S2 þ ΦQ þ Oðλ2Þ; tion in the same way as the existence of the negative R R R R cosmological constant Λ. In this section, we will derive a λð1 − Φ2Þ ¼ 2T S − πr3 þ ΦQ þ Oðλ2Þ: ð Þ consistent Smarr formula for R´enyi statistics and show the R R 4 þ 49 results that the phase structure of a RN-flat black hole through “R´enyi extended phase space” can have the critical To obtain the second term of the last line, we have 2 behavior like the VdW liquid-gas system in the canonical substituted TR from (30) and expanded SR ¼ πrþ − 1 2 4 2 ensemble. 2 λπ rþ þ Oðλ Þ. Then, from the second term of (49), we can identify A. Generalized Smarr formula To achieve this, we will extend the usual Smarr formula 3λð1 − Φ2Þ 4 P ¼ ;V¼ πr3 : ð50Þ in terms of the R´enyi entropy and its corresponding 32 3 þ temperature instead of Bekenstein–Hawking entropy and Hawking temperature. In four-dimensional RN-flat black Therefore, the lowest order approximation of λ in (49) can holes, the Smarr formula is [41] be written in the form

M ¼ 2T S þ ΦQ: ð Þ M ¼ 2T S − 2PV þ ΦQ: ð Þ H BH 47 R R 51

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We call this relation here a modified Smarr formula via well. To describe the black hole microscopic structure, each R´enyi statistics. Obviously, we have suggested that the new individual microstate of the black hole has been proposed term in the Smarr formula is the product of thermodynamic to contain γ Planck area pixels of event horizon surface pressure P and the thermodynamic volume V. The thermo- [35]. Thus, the total number of degrees of freedom is dynamic volume is a conjugate quantity to the pressure, A which can be obtained through the standard relation N ¼ ; ð Þ ∂M 2 55 V ¼ð Þ at the leading order of λ. Hence, it is not a γlp ∂P Φ;SR geometric spherical volume with horizon radius rþ. qffiffiffiffiffi l ¼ ℏG l2 Typically, the black hole mass corresponds to the internal where p c3 is the Planck length, and p is the area of energy in conventional black hole thermodynamics. one Planck area pixel. We can say that each individual However, this thermodynamical interpretation could be constituent of a black hole has “specific volume.” Using changed when we consider the nonextensive thermody- (50) and (55), the black hole has the specific volume of namics from the R´enyi statistics. Recall that the first law of the form black hole thermodynamics from the standard GB statistics can be written in the form 2 V γlp v ¼ ¼ rþ: ð56Þ δM ¼ T δS þ ΩδJ þ ΦδQ: ð Þ N 3 H BH 52 Note that the specific volume v scales linearly in the It is interesting to ask a question about what the first law of horizon radius rþ of the black holes. In the remaining of black hole thermodynamics is in the framework of R´enyi this section, we will explore about the and statistics. To address this, we substitute (48) into (52) in the thermal phase diagram from this approach in both grand case of zero-charge and nonrotating, for simplicity, then we canonical and canonical ensembles. arrive at the relation 1. Grand canonical ensemble δM ¼ THδS ; BH λSR With the macroscopic perspective, the equation of state TR e − 1 ¼ δ ; of the RN-flat black hole in the fixed Φ ensemble can be eλSR λ obtained from the formula of P, v, TR. Summarizing from 1 ¼ T δS þ πr3 δλ þ Oðλ2Þ; (30), (50), and (56), we now have R R 8 þ ð1 − Φ2Þð1 þ λπr2 Þ ≈ TRδSR þ VδP; ð53Þ þ TR ¼ ; 4πrþ P ¼ 3 λ V ¼ 4 πr3 where 32 and 3 þ. Our result shows that the 3λð1 − Φ2Þ M S P P ¼ ; mass of black hole is a function of R and , therefore, it 32 should be interpreted as the HðSR;PÞ rather than EðS ;VÞ 8 the internal energy R . This is, in a way, similar to the v ¼ rþ: ð57Þ results from the extended phase space approach [28].Asit 3 is well known, these two quantities can be related through Combining these equation with the elimination of λ,we the Legendre transformation E ¼ H − PV, and we can then will arrive at the equation of state obtain the first law of thermodynamics, in the R´enyi thermodynamics, as T 2ð1 − Φ2Þ P ¼ R − ; ð Þ v 3πv2 58 δE ¼ TRδSR − PδV; ð54Þ

where we have used lp ¼ 1 and γ ¼ 8 in (58). As will be E where the internal energy is now defined as the black hole clear later, choosing the value of γ like this allows us to mass M subtracted by PV. obtain the universal constant Zc consistent with that in VdW fluid. Remark that the equation of state is written in B. The equation of state the form of the specific volume v, which is proportional r V Typically, in a thermodynamic system, the matter to þ rather than the thermodynamic volume . In the P v changes its temperature when its microscopic components grand canonical ensemble, the relations between and in T emit or absorb photons. In the same way, a black hole can isothermal process with different values of R are plotted Φ ¼ 0 5 change its temperature through gaining or losing its masses. at . , 0.7 in Fig. 6. By solving the condition ð∂PÞ ¼ 0, the pressure has a maximum value P By this analogy, we may assume that the black hole could ∂v TR;Φ max have its own microstate carrying the degrees of freedom, as at vc as

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FIG. 6. P − v diagram of charged black holes for fixed potential Φ ¼ 0.5 (left) and Φ ¼ 0.7 (right). In both graphs, the temperature of isotherms decreases from top to bottom. For a given temperature, there are two branches of black holes, one is the small black hole (P increases as v increases) and the other is the large black hole (P decreases as v increases).

3πT2 4ð1 − Φ2Þ T 2 128Q2 P ¼ R ;v¼ : ð Þ P ¼ R − þ : ð Þ max 2 c 59 2 4 61 8ð1 − Φ Þ 3πTR v 3πv 27πv Q P − v For a given temperature, there are two black hole configu- At a fixed value of , this formula indicates that the rations below a maximum pressure P corresponding diagram shows a critical behavior like the VdW liquid-gas max system, as shown in Fig. 7. The phase structure can be with small and large black holes when vvc, respectively. The small black hole phase is unstable critically changed when the temperature crosses the critical because P increases as v increases, therefore, the com- point, which is the point of inflection with these two pression coefficient is negative. In the large black hole conditions P phase, the compression coefficient is positive because ∂P ∂2P decreases as v increases, hence it is thermodynamically ¼ 0; ¼ 0: ð Þ ∂v ∂v2 62 stable. We can find an “ideal gas” behavior for large black TR;Q TR;Q holes at high temperature. However, as seen from (59), the ð∂2PÞ v Solving these conditions leads to the critical thermody- second derivative ∂v2 cannot vanish at c. This means TR;Φ namic quantities of the charged black holes as follows: that there is no critical behavior for the RN-flat black hole in the grand canonical ensemble. 1 P ¼ ; ð Þ c 128πQ2 63 2. Canonical ensemble rffiffiffi 2 For the canonical ensemble, it has been known that the v ¼ 8 Q; ð Þ c 3 64 transition of small/large RN-AdS black holes in extended phase space is analogous to the liquid/gas phase transition 1 of the VdW type. Therefore, it is interesting to see whether Tc ¼ pffiffiffi : ð65Þ 3 6πQ the P − v criticality appears or not for the R´enyi extended phase space approach in this ensemble. As in the previous These formulas of critical values lead us to the critical section, we identify the macroscopic quantities as compressibility factor ðr2 − Q2Þð1 þ λπr2 Þ þ þ Pcvc 3 TR ¼ ; Z ≡ ¼ ; ð Þ 4πr3 c T 8 66 þ c 3λ Q2 P ¼ 1 − ; 32 r2 which is a universal constant in the sense that it is þ independent of a black hole’s mass and charge. 8 v ¼ r ; ð Þ Interestingly, this universal constant from our results has 3 þ 60 3 the same value as in the VdW fluid. Remark that Zc ¼ 8 γ ¼ 8 Q exactly when the factor . This in turn indicates that for where we have used Φ ¼ , lp ¼ 1 and γ ¼ 8. The rþ consistent values of Zc, we need 1 degree of freedom of the equations of state in terms of P, v, and TR can be written black hole to consist of 8 Planck area pixels on the event in the form horizon in the R´enyi extended phase space model. In other

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FIG. 7. P − v diagram of charged black holes for fixed charge Q ¼ 0.5 (left) and Q ¼ 1 (right). In both graphs, the temperature of isotherms decreases from top to bottom. The dashed green line is denoted as the critical isotherm T ¼ Tc. The lower solid lines correspond to small temperatures TTc, analogous to the ideal gas phase of VdW system. words, the value of the critical compressibility factor statistics, our results indicate that it is possible, with depends on the number of Planck area pixels, which 0 < λ < 1, to have the small and large black hole branches represents 1 degree of freedom of the black hole. For in the grand canonical ensemble while this cannot occur in the RN-AdS black hole extended phase space, it is, the GB statistics approach. Moreover, we have shown that however, necessary to choose γ ¼ 6 to obtain the result the Hawking–Page phase transition between thermal radi- 3 Zc ¼ 8 [29]. ation and large black holes in RN-flat can happen and The P − v diagram in Fig. 7 is the same as in the VdW crucially depend on the λ parameter. liquid-gas system. The left and right graphs of this figure For the canonical ensemble, in contrast with the grand correspond to the case of fixed charge Q ¼ 0.5,1, canonical case, there is a critical behavior such that a small/ respectively. The critical isothermal curve is depicted by large black hole first-order phase transition can occur when λ < λ λ the green dashed line. For TR Tc, the black holes behave like an ideal gas and λ no phase transition occurs. mula for R´enyi statistics is derived. We suggest that the parameter contributes to the thermodynamic pressure P for black holes, as shown in (50). Then, the quantity conjugate V. CONCLUSION to the pressure may be quantified as a thermodynamic In this paper, we investigate the thermodynamics of volume of black holes. This result shows that the black hole charged black holes in asymptotically flat spacetime or RN- mass represents the enthalpy rather than the internal energy. flat via an alternative R´enyi entropy both in the grand Introducing a specific volume, which is thermodynamic canonical and canonical ensembles. Applying the R´enyi volume of black holes for each degree of freedom, we write

064014-12 THERMODYNAMICS AND VAN DER WAALS PHASE TRANSITION … PHYS. REV. D 102, 064014 (2020) the equation of state and study the thermal behavior of and Tanapat Deesuwan for helpful discussions. This work charged black holes in flat space with nonzero λ parameter. has been supported by the Petchra Pra Jom Klao Ph.D. In the P − v diagram, we find that its thermal phase Research Scholarship from King Mongkut’s University of structure in the canonical ensemble has the VdW-like Technology Thonburi. Moreover, W. Liewrian would like phase transition. Intriguingly, taking γ ¼ 8, we obtain to thank the Ministry of Higher Education, Science, 3 the critical compressibility factor Zc ¼ 8 as in the VdW Research and Innovation, Thailand, for financial support- equation of state. The consideration of our setup in the ing this research by grant fund under the Thailand Center of extended phase space approach gives the result correspond- Excellence in Physics. ing to the VdW-like phase structure found in the case of AdS charged black holes via the GB statistics. As a consequence, this gives more supporting evidence of the APPENDIX: NONEXTENSIVE EFFECT ON existence of the equivalence, but it is in a different way BLACK HOLE STABILITY from the aforementioned works of Czinner et al. Taking into account a nonextensive effect, one can show The microstates of a self-gravitating system could be the emergence of an additional energy density from the correlated, in some ways, among themselves within the presence of nonzero R´enyi parameter λ. In a microcanon- system and also with those in the environment. We suggest ical ensemble of Schwarzschild black holes together with that, in nonextensive description, some energy density thermal radiation in asymptotically flat, we have a fixed emerged from the nontrivial correlation between the black E ¼ E þ E ¼ E total energy density bh r const, where bh and hole and heat bath may be encoded in the nonextensive Er stand for the energy density of black holes and thermal parameter λ. We discuss this in more detail in the Appendix. radiation, respectively. Basically, the thermal equilibrium This emergent energy density from nonextensive nature exists when the system is in the most probable state. jΛj induces a pressure to the black holes in the same way as This results in the maximum value of the total entropy plays a role in the case of AdS black holes through S ¼ S þ S bh r. Hence, the conditions of thermal equilibrium the extended phase space approach. In other words, the consist of nonextensive parameter λ then behaves effectively as the cosmological constant jΛj, which emerges from the non- ∂S ∂2S ¼ 0; < 0: ð Þ trivial thermodynamical behavior. and 2 A1 ∂Er ∂E If the nonextensivity allows us to have the emergent r energy from the correlations, as suggested above, the The first condition gives vacuum around the black hole can gain some energy density from the system-environment correlations resulting 0 ¼ dS ¼ dS þ dS bh r from long-range gravitation. Considering in the R´enyi dE dE extended phase space, using (53) and (54), we can associate ¼ bh þ r M ¼ E þ PV T Tr the black hole mass to the enthalpy, i.e., , bh rather than the internal energy E. This has a physical 1 1 ¼ − dE ; ð Þ implication that the creation of a black hole mass costs not T T bh A2 bh r only the internal energy, but also the additional energy responsible to the correlations. where we have used the conservation law of energy As it seems that the thermodynamic behaviors of these dE ¼ −dE r bh. The above equation gives us the condition two physical systems with different statistics are possible to T ¼ T bh r, as it should be. On the other hand, the second be equivalent, our results also indicate that there should be condition reads some connections between the R´enyi parameter λ and the absolute value of AdS space cosmological constant jΛj. ∂2S ∂2S ∂2S ¼ bh þ r As shown in (8) and (50), both λ and jΛj play the roles as 2 2 2 ∂Er ∂E ∂Er thermodynamic pressure up to some constants. This con- bh 1 ∂T 1 ∂T nection could guide us to understand more about the ¼ − bh − r λ T2 ∂E T2 ∂E physical meaning of the R´enyi parameter , and the running bh bh r r of this parameter might work, in some ways, similar to the 1 1 1 ¼ − þ < 0: ð Þ running of jΛj in AdS black holes with the GB statistics. 2 A3 T C Cr However, this is beyond the scope of this paper, thus it may bh be studied in future works. Obviously, we obtain the inequality condition on the heat capacity in the form ACKNOWLEDGMENTS 1 1 We are grateful to Pitayuth Wongjun, Supakchai þ > 0: ð Þ C C A4 Ponglertsakul, Sirachak Panpanich, Krai Cheamsawat, bh r

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˜ 4 −2 Using Er ¼ σVTr , we have the heat capacity of a radiation in the geometric unit λ has the dimension of ðlengthÞ . filled in volume V˜ as As defined in (50), the pressure in the form P ∼ λ thus has the dimension of energy density. In the R´enyi version of ˜ 3 VδP Cr ¼ 4σVTr : ðA5Þ extended phase space, the appearance of an extra term in (53) can be thought of as the contribution from the Using (39), (45), (A5), and setting the electric charge nonextensivity in the form of an additional energy. With Q ¼ 0 for simplicity, the second condition (A4) becomes this point of view, we can also think that this additional λ 1 − λπr2 1 energy contributes as the term with in (A6). − þ þ > 0: ð Þ Interestingly, (A7) can be rewritten in the form 2 ˜ 3 A6 2πrþ 4σVT ˜ This condition of thermal equilibrium relates the volume V rþ >Lλ; ðA8Þ of the space filled with thermal radiation, the black hole’s pffiffiffiffiffi size rþ, and nonextensive parameter λ. Interestingly for the where Lλ ≡ 1= πλ. This indicates that the black hole in black hole in the heat bath of infinitely large volume asymptotically flat spacetime can be stable only if the event V˜ → ∞,wehave horizon radius is larger than the characteristic length scale 1 Lλ. The presence of this nonextensivity length implies that λ > ; ð Þ 4πM2 A7 there is a particular value of gravitational energy beyond bh which the conventional is no longer M ¼ rþ where we have used bh 2 . The black holes can be in applicable. This is because the nonextensive effect turns out thermal equilibrium with an infinite heat reservoir when the to take an important role in describing black hole thermo- black hole mass satisfies the above inequality. Notice that dynamics when rþ is larger than Lλ.

[1] J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, [12] G. Hooft, Dimensional reduction in quantum gravity, Conf. 2333 (1973). Proc. C 930308, 284 (1993). [2] S. W. Hawking, Particle creation by black holes, Commun. [13] L. Susskind, The world as a hologram, J. Math. Phys. (N.Y.) Math. Phys. 43, 199 (1975). 36, 6377 (1995). [3] J. M. Bardeen, B. Carter, and S. W. Hawking, The four laws [14] J. Maldacena, The large N limit of superconformal field of black hole mechanics, Commun. Math. Phys. 31, 161 theories and supergravity, Int. J. Theor. Phys. 38, 1113 (1973). (1999). [4] S. W. Hawking and D. Page, Thermodynamics of black [15] E. Witten, Anti-de Sitter space and holography, Adv. Theor. holes in anti-de Sitter space, Commun. Math. Phys. 87, 577 Math. Phys. 2, 253 (1998). (1983). [16] E. Witten, Anti-de Sitter space, thermal phase transition, and [5] D. Lynden-Bell, Negative specific heat in astronomy, confinement in gauge theories, Adv. Theor. Math. Phys. 2, physics and chemistry, Physica (Amsterdam) 263A, 293 505 (1998). (1999). [17] M. Srednicki, Entropy and Area, Phys. Rev. Lett. 71, 666 [6] W. Thirring, System with negative specific heat, Z. Phys. (1993). 235, 339 (1970). [18] S. Abe, General pseudoadditivity of composable entropy [7] J. W. Gibbs, Elementary Principles in Statistical Mechanics prescribed by the existence of equilibrium, Phys. Rev. E 63, Developed with Special Reference to the Rational Founda- 061105 (2001). tion of Thermodynamics/by J. Willard Gibbs (C. Scribner, [19] M. Nauenberg, The critique of q entropy for thermal New York, 1902). statistics, Phys. Rev. E 67, 036114 (2003). [8] P. T. Landsberg, Thermodynamics and Statistical Mechan- [20] T. S. Biró and P. Ván, Zeroth law compatibility of ics (Oxford University Press, New York, 1978). nonadditive thermodynamics, Phys. Rev. E 83, 061147 [9] P. Landsberg, Is equilibrium always an entropy maximum?, (2011). J. Stat. Phys. 35, 159 (1984). [21] A. R´enyi, On the dimension and entropy of probability [10] C. Tsallis, Possible generalization of Boltzmann-Gibbs distribution, Acta Math. Acad. Sci. Hung. 10, 193 (1959). statistics, J. Stat. Phys. 52, 479 (1988). [22] C. Tsallis and L. J. L. Cirto, Black hole thermodynamical [11] C. Tsallis, Introduction to Nonextensive Statistical entropy, Eur. Phys. J. C 73, 2487 (2013). Mechanics: Approaching a Complex World (Springer [23] V. G. Czinner and H. Iguchi, R´enyi entropy and the Science and Business Media, New York, NY, USA, thermodynamic stability of black holes, Phys. Lett. B 2009). 752, 306 (2016).

064014-14 THERMODYNAMICS AND VAN DER WAALS PHASE TRANSITION … PHYS. REV. D 102, 064014 (2020)

[24] V. G. Czinner and H. Iguchi, Thermodynamics, stability and [33] S. Hendi, S. Panahiyan, and B. Eslam, Panah Extended Hawking-Page transition of Kerr black holes from R´enyi phase space of black holes in Lovelock gravity with non- statistics, Eur. Phys. J. C 77, 892 (2017). linear electrodynamicss, Prog. Theor. Exp. Phys. 2015, [25] A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, 103E01 (2015). Charged AdS black holes and catastrophic holography, [34] C. Niu, Y. Tian, and X. N. Wu, Critical phenomena and Phys. Rev. D 60, 064018 (1999). thermodynamic geometry of RN-AdS black holes, Phys. [26] A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, Rev. D 85, 024017 (2012). Holography, thermodynamics and fluctuations of charged [35] S.W. Wei and Y.X. Liu, Insight into the Microscopic AdS black holes, Phys. Rev. D 60, 104026 (1999). Structure of an AdS Black Hole from a Thermodynam- [27] P. Burikham and C. Promsiri, The mixed phase of charged ical Phase Transition, Phys. Rev. Lett. 115, 111302 AdS black holes, Adv. High Energy Phys. 2016, 5864672 (2015). (2016). [36] S. W. Wei, Y. X. Liu, and R. B. Mann, Ruppeiner geometry, [28] D. Kastor, S. Ray, and J. Traschen, Enthalpy and the phase transitions, and the microstructure of charged AdS mechanics of AdS black holes, Classical Quantum Gravity black holes, Phys. Rev. D 100, 124033 (2019). 26, 195011 (2009). [37] M. Cadoni, E. Franzin, and M. Tuveri, Van der Walls-like [29] D. Kubiznak and R. Mann, P-v criticality of charged AdS behaviour of charged black holes and hysteresis in the dual black holes, J. High Energy Phys. 07 (2012) 033. QFTs, Phys. Lett. B 768, 393 (2017). [30] D. Kubiznak and R. Mann, Black hole chemistry: Thermo- [38] F. Bouchet and J. Barr´e, Classification of phase transitions dynamics with Lambda, Classical Quantum Gravity 34, and ensemble inequivalence, in systems with long range 063001 (2017). interactions, J. Stat. Phys. 118, 1073 (2005). [31] S. Gunasekaran, R. Mann, and D. Kubiznak, Extended [39] D. J. Gross, M. J. Perry, and L. G. Yaffe, Instability of phase space thermodynamics for charged and rotating black flat space at finite temperature, Phys. Rev. D 25, 330 holes and Born-Infeld vacuum polarization, J. High Energy (1982). Phys. 11 (2012) 110. [40] R. Penrose, Gravitational Collapse and Spacetime Singu- [32] R. G. Cai, L. M. Cao, L. Li, and R. Q. Yang, P-V criticality larity, Phys. Rev. Lett 14, 57 (1965). in the extended phase space of Gauss-Bonnet black holes in [41] L. Smarr, Mass Formula for Kerr Black Holes, Phys. Rev. AdS space, J. High Energy Phys. 09 (2013) 005. Lett. 30, 71 (1973).

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