Thermodynamics and phase transition of Bardeen black hole via Rényi statistics in grand canonical ensemble and canonical ensemble

The thermodynamics of the Bardeen black hole in asymptotically flat space is investigated with the corrected first law of thermodynamics via Rényi statistics. The nonextensive parameter λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} gives the possibility to the thermal stability of Bardeen black hole, and there is a Hawking–Page phase transition in the grand canonical ensemble (fixed the potential), which is similar to the cases of Bardeen black hole and corrected Bardeen black hole in asymptotically anti-de Sitter (AdS) space via standard Gibbs–Boltzmann (GB) statistics. By introducing the general Smarr formula via Rényi statistics, the thermodynamic pressure P is defined with the parameter λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} and its conjugate quantity V is the thermodynamic volume (not a geometric spherical volume with horizon radius rh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{h}$$\end{document}). The thermodynamics of the asymptotically flat Bardeen black hole via Rényi statistics in the canonical ensemble (fixed the charge q) behaves like the van der Waals system, which is also same as the asymptotically Bardeen-AdS black hole via GB statistics. The analogy between the thermodynamics of the asymptotically flat Bardeen black hole from Rényi statistics and the Bardeen-AdS black hole from GB statistics makes us to consider what is the relation between the nonextensive parameter λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} and the cosmological constant Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}.


Introduction
Black hole is excepted by general relativity inside whose event horizon nothing can escape. At first, the black hole is considered to only have mass, angular momentum and charge (if exists) and no temperature or entropy [1][2][3], but the quantum field theory in curved space changes this view. Then the a e-mail: jhchen@hunnu.edu.cn (corresponding author) black hole thermodynamics was founded by Bekenstein and Hawking et al. [4][5][6][7][8], suggested that a black hole can have entropy and nonzero temperature. According to the laws of black hole thermodynamics, the entropy of the black hole is proportional to its area and the surface gravity of black hole plays the role of its temperature. The black hole thermodynamics pushes us to consider the black hole as a thermal system [6,9].
A zero-charged black hole in asymptotically flat space can only have negative heat capacity with standard Gibbs-Boltzmann (GB) statistics, which means the black hole cannot be in thermal equilibrium [10]. But Hawking and Page proved that there are two branches of uncharged black hole in asymptotically anti-de Sitter space, i.e. the small black hole branch is unstable with negative heat capacity and the large black hole branch can be stable with positive heat capacity. Moreover, there is a minimum temperature T min below which the two branches of black hole cannot occur, and there is another certain temperature T H P which is the point of Hawking-Page phase transition [9,10]. With the development of AdS/CFT correspondence, other researches on the black hole thermodynamics were done for the charged AdS black holes [11][12][13].
Some researches [14,15] suggested that the standard GB statistics will be violated for the long-range interactions (black hole system and etc.). Recently, some authors [9,[16][17][18] also pointed out that the standard stability analysis may not be applicable for the black hole system because it depends strongly on the additivity of entropy function, which fails to hold on the Bekenstein-Hawking entropy. The mentioned problems all suggested that applying GB statistics to a selfgravitation system may bring an uncomplete result. It is well known that the black hole entropy is proportional to its area rather than its volume, which indicates that the black hole entropy is nonextensive. The difficulties force us to consider the black hole thermodynamics with nonextensive statistics and nonextensive entropy. Fortunately, A weaker composition rule [19] which is the famous Abe's composition rule can be taken to apply for the nonextensive approach, where H is a differentiable function of entropy S, A, B are two independent systems, and λ ∈ R is a constant parameter. One of nonextensive entropy, Tsallis entropy [20][21][22][23] takes the form where W ∈ N is the total number of possible (microscopic) configurations and p i is the associated possibilities. When q → 1, the standard GB entropy S BG = − W i=1 p i lnp i will be recovered. The composition rule of nonextensive Tsallis entropy can be written as which satisfies the Abe's pseudoadditive rule by setting H λ (S) = S q and λ = 1 − q.
The empirical temperature cannot be well defined from the pseudoadditivity of Tsallis entropy, so Biró and Ván [15] considered the logarithmic formal of Tsallis entropy Interestingly, it is just the well-known Rényi entropy [24,25], which is additive under Abe's composition rule. By defining a nonextensive parameter λ = 1 − q, we have The Rényi entropy is compatible with the zeroth law of thermodynamics and the empirical temperature can be defined as where E is the energy of system. In Ref. [9] the nonextensive parameter λ plays the role of pressure just like the cosmological constant in anti-de Sitter background. The Rényi statistics has also been applied into several different black holes in different background [14,[16][17][18]26,27], but still not been used to consider the regular black holes.
When we consider the thermodynamics of regular black holes, the different temperatures will be obtained by the surface gravity approach and the thermodynamic approach, which may lead to giving up one of the first law of thermodynamics or Bekenstein area law. To avoid the different temperatures obtaining by different approaches, Ma and Zhao [51] corrected the first law of black hole thermodynamics.
The thermodynamics of regular black holes has been investigated with several different approaches, but it still has some difficulties which are nonadditivity of entropy function with GB statistics and different temperatures obtaining with different approaches. To solve these problems, we apply the corrected first law of thermodynamics of black hole and Rényi statistics to the thermodynamics of Bardeen black hole.
This paper is organized as follows. In Sect. 2, we briefly review the Bardeen black hole spacetime and its corrected thermal properties. In Sect. 3, we introduce the general Smarr formula of Bardeen black hole via Rényi statistics and derive the corrected first law of regular black hole of Rényi statistics. In Sect. 4, we investigate the thermodynamics quantities of Bardeen black hole with GB statistics and corrected first law of thermodynamics in the grand canonical ensemble and the canonical ensemble. In Sect. 5, the thermodynamic quantities in the Rényi model will be studied. The brief conclusions and discussions are given in the last section. In this paper, the geometric units G =h = c = k B = 1 will be chosen.

The first law of thermodynamics for regular black holes
In this section, we briefly review the first law of thermodynamics of regular black holes. The line element of spherically symmetric Bardeen black hole is where the function f (r ) is the effective mass m(r ) in function f (r ) is where the parameters M and q are the black hole mass and the magnetic charge, respectively. When q > 4 √ 3M 9 , there are inner horizon (Killing horizon) and outer horizon (event horizon). Two horizons meet when q = 4 √ 3M 9 and there is no horizon when q > 4 √ 3M 9 . Setting r h as the radius of event horizon and solving f (r h ) = 0 for M, we can get The temperature is defined by surface gravity: On the other hand, the temperature can be calculated from the first law of thermodynamics. For Bardeen black hole the first law of black hole thermodynamics takes the form where S is the entropy of black hole which equals to quarter of the area of black hole event horizon and h is the magnetic potential of black hole. The temperature calculated with the first law is where the parameter C(M, q, A is the area of black hole horizon, H is the corrected magnetic potential and E is the internal energy. Thus we have The magnetic potential of Bardeen black hole can be obtained by the corrected first law

Thermodynamics and general Smarr formula from Rényi statistics
The thermodynamics and general Smarr formula in the Rényi model will be reviewed in this section. As a self-gravitation system, the black hole thermodynamics can be considered under the nonadditive Tsallis statistics which has the general composition rule. Though the Tsallis statistics can be applied for the non-extensity nature of black hole, the empirical temperature through the zeroth law of thermodynamics cannot be well defined. To avoid this difficulty, the formal logarithm of the Tsallis entropy, which is the well known Rényi entropy, will be taken. The Bekenstein-Hawking entropy of black hole can be considered as the Tsallis entropy from which the Rényi entropy function of black hole takes the form which is additive with the Abe's composition rule. The Rényi temperature can be written as The Smarr formula i.e. the mass formula of Bardeen black hole in asymptotically flat space can be written as in which the surface gravity term is actually the temperature term 2T S. From the discussion above, the relation of temperature and entropy between the Gibbs-Boltzmann statistics and the Rényi statistics is From the Ref. [9], the temperature term can be expanded as where the S R can be expanded as The pressure P and the volume V will be defined by expanding the temperature term of M, which will be shown latter, and the P − v criticality and the van der Waals like phase transition of the canonical ensemble will occur naturally. Moreover, we can derive the corrected first law of regular black hole of Rényi statistics. Following Ref. [9], substituting Eq. (20) into the corrected first law of regular black hole Eq. (14), we obtain   14), the E cannot be interpreted as the internal energy, but reinterpreted as the enthalpy H in the Rényi extended phase space. From the Legendre transformation E in = H − PV , the corrected first law of regular black hole in Rényi statistics is where the E in is the internal energy in Rényi statistics. The internal energy now be defined as E in = E − PV .

Thermodynamics of Bardeen black hole from standard Gibbs-Boltzmann statistics with corrected first law of thermodynamics
The physical quantities of a black hole can be treated as thermodynamic quantities because of the analogy between black hole mechanics and the laws of thermodynamics. The thermodynamics of Bardeen black hole can be considered in the grand canonical ensemble and the canonical ensemble.

Grand canonical ensemble
In the grand canonical ensemble, the magnetic charge q of black hole changes with the surrounding heat bath, and the magnetic potential H can be set to be fixed. As mentioned in Sect. 2, the magnetic potential of Bardeen black hole can be written as Consequently, the internal energy of black hole can be considered as and the Bekenstein-Hawking entropy is Other quantities can be calculated as where T h is the Hawking temperature, C H is the heat capacity at fixed magnetic potential H , and G H is the Gibbs free energy. The black hole cannot be in stable because the heat capacity C H is negative at arbitrary r h . This can be seen from the dashed black line in Fig. 1.

Canonical ensemble
In the canonical ensemble we fixed the magnetic charge q of black hole. Considering two horizons meet, the mass and the magnetic charge of extremal black hole can be written as M e = 3 √ 3q/4, and the radius of the extremal black hole is r e = √ 2q. Consequently, the internal energy of black hole is the Bekenstein-Hawking entropy is still S B H = πr 2 h and other quantities can be calculated as where the C q is the heat capacity of black hole with fixed the magnetic charge q and F H is the Helmoholtz free energy. From the heat capacity C q , two branches of black hole can be founded, i.e., the one with positive heat capacity and the other with negative heat capacity. From C q > 0, r h can be solved as The heat capacity will be negative when r h > which means that at r h = √ √ 57+7q √ 2 the capacity diverges and temperature gets maximum It is found that the phase transition of Bardeen black hole depends on the choice of ensemble, which may be resulted from the long-range gravitation and electromagnetic interactions of the system.

Thermodynamics of Bardeen black hole from Rényi statistics with corrected first law of thermodynamics
In this section, the thermodynamics of Bardeen black hole will be studied with the Rényi approach in the grand canonical ensemble (fixed potential) and the canonical ensemble (fixed charge).

Grand canonical ensemble
The thermodynamic quantities of the grand canonical ensemble will be calculated in the Rényi model firstly, then the state equation P − v will be discussed. As mentioned in Sect. 3, the Rényi temperature in the grand canonical ensemble can be obtained with the above temperature, the heat capacity can be calculated The heat capacity is negative when r h < r c and is positive when r h > r c . For the critical point r c = √ 1/λπ , C R is divergent. So there are two possible black hole branches, which are the small black hole branch with negative heat capacity and the large black hole branch with positive heat capacity. Substituting r c into Eq. (10), the critical mass can be obtained which means that the Bardeen black hole in Rényi statistics is in the small black hole branch when M < M c and in the large black hole branch when M > M c . The heat capacity C R is proportional to the inverse of the slope of the temperature T R , we have The critical radius r c is the extremum point of T R i.e. T R (r c ) = 0. With this relation, the Bardeen black hole can be stable, which has positive heat capacity when r h > r c and negative heat capacity when r h < r c . While there is no extremum point in the case of GB statistics. We can also solve Eq. (37) for r h (T R ) function of temperature T R and the minimum temperature can be obtained by solving r h (T R ) = r c = 1/ √ λπ and the minimum temperature T R,min = 1 2 λ π corresponds to the Schwarzschild case when H = 0. It is also interesting The Rényi statistics, in which black hole can be locally stable in asymptotically flat space with certain level of nonextensivity, gives different conclusions from GB statistics. There is no black hole phase when T R < T R,min , because the event horizon cannot exist due to the imaginary value r h (T R ). The hot thermal radiation in asymptotically flat space will collapse to a black hole with higher temperature than a minimum temperature i.e. T R > T R,min , which also occurs in the case of black hole in the AdS background from GB statistics.
In the grand canonical ensemble, the Gibbs free energy will be considered in discussing the global stability of the black hole. So in the Rényi model, the corresponding Gibbs free energy function should be written with Rényi entropy and temperature as The Fig. 2 shows that the Bardeen black hole is globally unstable in the GB case as its free energy is always positive. From Fig. 2, we can see the free energy of black hole will be negative when the temperature is high enough, so the black hole is global stable with the Rényi parameter λ. The Hawking-Page temperature T H P can be obtained from which is about 1.24 times T R,min . The Hawking-Page temperature T H P versus magnetic potential H is plotted in Fig. 3. The range of magnetic potential can be obtained from the horizon structure of Bardeen black hole mentioned in Sect. 2 and the extreme value of magnetic potential obtained when q = 4 √ 3M/9. From Fig. 3, the Hawking-Page transition line is similar to the T − μ phase diagram of confining/deconfining phase transition at finite chemical potential. This shows that the properties of asymptotically flat black hole using Rényi statistics is similar to the asymptotically AdS black hole using GB statistics [9,[11][12][13].
Now we turn to the cusp of G R versus T R , which occurs at T R,min . The cusp corresponds to the small/large black hole phase transition. This cusp corresponds to a second-order phase transition because the heat capacity, which is the second order derivative of Gibbs free energy respect to temper- Gibbs free energy has the maximum value G Rc and the Rényi entropy is a constant with a fixed λ at the cusp point. Then we focus on the P −v phase. As mentioned in Sect. 3, the general Smarr formula with Rényi statistics can be written as For the grand canonical ensemble, substituting the T R and S R into Eq. (45), we obtain with which the pressure P and the volume V can be defined as and the general Smarr formula can be written as It needs to be emphasized that the V = 4πr 3 h /3 is not the geometric spherical volume with the horizon radius r h , but the thermodynamic volume which is the conjugate quantity of the pressure. The specific volume is v = 8r h /3, not v = 2r h in AdS case, which allows us to get a better ratio Pv/T in Rényi statistics approaching to the standard van der Waals system [9]. With the specific volume, the equation of state can be written as The pressure P versus the specific volume v is plotted in Fig. 4 which shows that there is a maximum value of P. The maximum value P max can be obtained by solving the condition ( ∂ P ∂v ) T R , H = 0. The corresponding critical volume for P max is v c . For a given temperature, there are two branches of black hole when P < P max which are small black hole branch with v < v c and the large black hole branch with v > v c . For the large black hole branch, the pressure P decreases with the specific volume v, which means the thermodynamic stability for the large black hole branch. In contrary to the large black hole branch, the small black hole branch is unstable. The second derivative ( ∂ 2 P ∂v 2 ) T R , H at v c is nonzero, which means that there is no critical behavior for Bardeen black hole in the grand canonical ensemble.

Canonical ensemble
The thermal properties of Bardeen black hole in the canonical ensemble will be investigated in this section. From Eqs. (18) as the function of event horizon r h is plotted with fixed q = 0.3 and different values of λ in Fig. 5. These temperature curves of Bardeen black hole in the Rényi model are similar to the uncharged black holes in AdS background. Interestingly, the critical phase transition will occur at different values of λ, which can be obtained with the condition For this condition, the critical horizon r c , critical nonextensitive parameter λ c and the critical Rényi temperature T c can be numerically calculated When λ < λ c , there are local maximum and local minimum in the temperature curve. When λ = λ c , these two extremum points meet at the critical event horizon r c . The Rényi heat capacity is Unlike the GB case, three branches of black hole are allowed in the canonical ensemble via Rényi statistics, two positive branches and one negative branch as shown in Fig. 5. It shows that when horizon r h equals to the extremum points mentioned above, the heat capacity is divergent, which means there is a phase transition. The temperature at these two extremum points are T 1 = 0.0695703 and T 2 = 0.0730847 for the blue curve in Fig. 5.
In the canonical ensemble, it is better to consider the global stability of Bardeen black hole with Helmholtz free energy. The Helmholtz free energy in the Rényi statistics is where E = C(M, q, r h )(M − M e ). Figure 6 shows the Helmholtz free energy versus the Rényi temperature of Bardeen black hole. The swallowtail behavior occurs only at λ < λ c , which is similar to the van der Waals liquid/gas system.
The general Smarr formula in the Rényi model is Substituting Eqs. (17) and (50) into Eq. (55) and holding the first order of λ, we have Fig. 6 The Rényi Helmholtz free energy F R versus the Rényi temperature T R with different magnetic charge q and different λ. The swallowtail-shape occurs only at λ < λ c . As the charge q increases, the swallow-tail-shape deviates from the standard shape, which may be resulted from the nonlinear magnetic charge of Bardeen black hole Defining the pressure P and the volume V as the general Smarr formula can be written as Using the relation v = 8r h /3, the equation of state can be obtained which the P −v diagram will behavior like the van der Waals system as shown in Fig. 7. The critical point can be obtained with the conditions The critical pressure P c , specific volume v c and temperature T c can be numerically calculated So, the critical compressibility factor is which is independent on the black hole's mass and magnetic charge. Interestingly, this result is similar to the ratio of standard van der Waals fluid 3 8 . The relation P versus v in Fig. 7 is same as van der Waals system. The critical isothermal curve at T R = T c is plotted with purple dashed line. When T R < T c , there is a small black hole phase corresponding to the liquid phase and a large black hole phase corresponding to the gas phase. The small black hole and the large black hole phase are stable with positive compression coefficient (P decreases with v). There is a unstable part between the small black hole and large black hole phase, which has the negative compression coefficient (P increases with v) and stands for the mixture of liquid and gas phase. Solving the following conditions the minimum temperature T 0 can be obtained at v 0 = 4 3 2 √ 57 + 7 q, When T 0 < T < T c , there exists the phase transition similar to van der Waals system. The black hole will be like the ideal gas and there is no phase transition when the temperature is high enough T > T c .

Conclusion and discussion
In this paper, to avoid the incompatible temperature of regular black hole obtained with different approaches, we introduced the corrected first law of thermodynamics and studied the thermodynamics of the Bardeen black hole in both the grand canonical ensemble and the canonical ensemble via Rényi statistics. In the grand canonical ensemble (fixed charge potential), the Bardeen black hole can be locally stable and there are small black hole and large black hole branches with the nonextensive parameter 0 < λ < 1, while these two branches will not occur in GB statistics. Moreover, there is a Hawking-Page phase transition between the thermal radiation phase and the large black hole phase depending on the nonextensive parameter λ. In the canonical ensemble (fixed charge q): (1) when λ < λ c (critical nonextensive parameter), there is a small/large black hole phase transition (first order); (2) when λ > λ c , the phase transition above will disappear and the large black hole phase will just be a possibility. These behaviors are similar to the cases of Bardeen black hole and corrected Bardeen black hole in AdS background via GB statistics (see [48][49][50]). Furthermore, by introducing the general Smarr formula via Rényi statistics, which the parameter λ plays the thermodynamic pressure P, we found that the thermal phase structure in the canonical ensemble just behaves as the van der Waals system. The same phase transition also occurs in the case of Bardeen-AdS black hole via GB statistics. Both the nonextensive parameter λ in Rényi statistics and the cosmological constant in AdS spacetime play as the thermodynamic pressure P. The analogy between the case of asymptotically flat Bardeen black hole via Rényi statistics and Bardeen-AdS black hole via GB statistics suggests that the nonextensive parameter λ should have some connections with the cosmological constant . This connection may lead us to understand the nonextensive parameter λ more deeply.