Phase Structures and Transitions of Born-Infeld Black Holes in a Grand Canonical Ensemble

To make a Born-Infeld (BI) black hole thermally stable, we consider two types of boundary conditions, i.e., the asymptotically anti-de Sitter (AdS) space and a Dirichlet wall placed in the asymptotically flat space. The phase structures and transitions of these two types of BI black holes, namely BI-AdS black holes and BI black holes in a cavity, are investigated in a grand canonical ensemble, where the temperature and the potential are fixed. For BI-AdS black holes, the globally stable phases can be the thermal AdS space. For small values of the potential, there is a Hawking-Page-like first order phase transition between the BI-AdS black holes and the thermal-AdS space. However, the phase transition becomes zeroth order when the values of the potential are large enough. For BI black holes in a cavity, the globally stable phases can be a naked singularity or an extremal black hole with the horizon merging with the wall, which both are on the boundaries of the physical parameter region. The thermal flat space is never globally preferred. Besides a first order phase transition, there is a second order phase transition between the globally stable phases. Thus, it shows that the phase structures and transitions of BI black holes with these two different boundary conditions have several dissimilarities.


I. INTRODUCTION
The study of black hole thermodynamics has continued to fascinate researchers since the pioneering work [1][2][3], where Hawking and Bekenstein found that black holes possess the temperature and the entropy. However, it is well known that a Schwarzschild black hole in asymptotically flat space is thermally unstable because of its negative specific heat. To study black hole thermodynamics in a thermally stable system, one can impose appropriate boundary conditions. For example, putting black holes in the anti-de Sitter (AdS) space can make them thermally stable since the AdS boundary acts as a reflecting wall for the Hawking radiation. The investigations of the thermodynamic properties of AdS black holes have come a long way since the discovery of the Hawking-Page phase transition [4], i.e., a phase transition between the thermal AdS space and the Schwarzschild-AdS black hole.
Later, with the advent of the AdS/CFT correspondence [5][6][7], there has been much interest in studying the phase transitions of AdS black holes [8][9][10][11][12][13]. From the holographic perspective, we are eager to find out whether the duality is independent of the details of the boundary conditions of the bulk spacetime. It is therefore interesting to study the thermodynamics and phase structures of black holes under different boundary conditions and look for similarities or dissimilarities to the AdS case.
On the other hand, placing a Schwarzschild black hole in a cavity in the asymptotically flat space, York showed that the black hole can be thermally stable and has similar phase structure and transition to these of a Schwarzschild-AdS black hole [14]. Specifically, the Schwarzschild black hole in a cavity undergoes a Hawking-Page-like transition to the thermal flat space as the temperature decreases. The thermodynamics and phase structure of a Reissner-Nordstrom (RN) black hole in a cavity have been studied in a grand canonical ensemble [15] and a canonical ensemble [16,17], which showed that the phase structures of the RN black hole in a cavity and the RN-AdS black hole have extensive similarities. In a series of paper [18][19][20][21][22][23], the phase structures of various black brane systems in a cavity were investigated in a grand canonical ensemble and a canonical ensemble, and it was found that Hawking-Page-like or van der Waals-like phase transitions always occur except for some special cases. In [24][25][26][27], boson stars and hairy black holes in a cavity were considered, and it showed that the phase structure of the gravity system in a cavity is strikingly similar to that of holographic superconductors in the AdS gravity. The stabilities of solitons, stars and black holes in a cavity were also studied in [28][29][30][31][32][33][34][35], which showed that the nonlinear dynamical evolution of a charged black hole in a cavity could end in a quasi-local hairy black hole. The thermodynamic behavior of de Sitter black holes in a cavity has been discussed in the extended phase space [36]. Recently, McGough, Mezei and Verlinde [37] proposed that the holographic dual of TT deformed CFT 2 is a finite region of AdS 3 with the wall at finite radial distance, which further motivates us to explore the properties of a black hole in a cavity.
The Born-Infeld (BI) electrodynamics is a particular example of a nonlinear electrodynamics, which is an effective model incorporating quantum corrections to Maxwell electromagnetic theory. BI electrodynamics was first proposed to smooth divergences of the electrostatic self-energy of point charges by introducing a cutoff on electric fields [38]. Later, it is realized that BI electrodynamics can emerge from the low energy limit of string theory, which encodes the low-energy dynamics of D-branes. Coupling the BI electrodynamics field to gravity, the BI black hole solution was first obtained in [39,40]. For the BI black holes in asymptotically AdS space, the thermodynamic behavior and phase transitions have been investigated in . Specifically, the phase structures and transitions of 4D BI-AdS black holes in a canonical ensemble were studied in [47,55,58], which showed that a reentrant phase transition was always observed in a certain region of the parameter space. Meanwhile, the thermodynamics and phase transitions in a grand canonical ensemble have been analyzed in [42], which showed that the system undergoes the first and zeroth order phase transitions between the black hole solutions and the thermal AdS space. On the other hand, by placing a BI black hole in a spherical thermal cavity, we recently discussed the phase structures and transitions of the canonical ensemble of this system [63], which were found to have dissimilarities from these of the BI-AdS black holes.
In this paper, we study the phase structures and transitions of the grand canonical ensemble of BI black holes using both asymptotically AdS and the Dirichlet wall boundary conditions. So the gauge potential is fixed rather than the charge on the boundaries in this paper. In the framework of the AdS/CFT duality, the grand canonical ensemble is more relevant than the canonical ensemble. Although the phase structures and transitions of BI-AdS black holes in the grand ensemble have already been investigated in [42], we carry out the analysis in a more through way with a broader survey of the parameter space. The phase diagrams in the parameter space are obtained, which can be used to make a comparison with these of BI black holes in a cavity. In the second part of this paper, we analyze the phase structures and transitions of BI black hole in a cavity in the grand canonical ensemble. We find that the thermal flat space, which is the counterpart of the thermal AdS space in the BI-AdS case, can never be the globally stable phase. Moreover, the system has no zeroth order transition, but instead a second order transition occurs. It turns out that the results of the BI black holes in a cavity and BI-AdS black holes have several dissimilarities.
The rest of this paper is organized as follows. In section II, we study the phase structures and transitions of BI-AdS black holes and give the phase diagrams, e.g., FIGs. 2 and 4.
In section III, we discuss the phase structures and transitions of BI black holes in a cavity.
The related phase diagrams are given in FIGs. 7 and 10, from which one can read the phase structures and transitions. Section IV is devoted to our discussion and conclusion.

II. BORN-INFELD ADS BLACK HOLES
In this section, we consider the phase structures and transitions of BI-AdS black holes in a grand canonical ensemble. The action of a (3 + 1) dimensional model of gravity coupled to a Born-Infeld electromagnetic field A µ is where the cosmological constant Λ = −3/l 2 , and we take 16πG = 1 for simplicity. The

Born-Infeld electrodynamics Lagrangian density is
where F µν = ∂ µ A ν − ∂ ν A µ , and the Born-Infeld parameter a is related to the string tension α ′ as a = (2πα ′ ) 2 > 0. When a → 0, L BI (F ) reduces to the Lagrangian of the Maxwell field. The Born-Infeld AdS black hole solution was obtained in [39,40]: where Here M and Q are the mass and the charge of the back hole, respectively, and 2 F 1 (a, b, c; x) is the hypergeometric function.
At the horizon r = r + , one has that f (r + ) = 0, and the Hawking temperature is given by Requiring A t (r) at the horizon to be zero, it can show that the gauge potential measured with respect to the horizon is In the limit of r + → +∞, BI-AdS black holes would reduce to RN-AdS black holes, and we find that As r + → 0, eqns. (4) and (5) gives where Φ c ≡ 4 √ 2πΓ 1 4 Γ 5 4 ∼ 32.95. So when Φ > Φ c , T → −∞ as r + → 0, which means that r + has a nonzero minimum value. On the other hand, T → +∞ as r + → 0 for Φ < Φ c , and hence r + can go to zero in this case.
To study the phase structures and transitions, we need to consider the free energy of the black hole. The free energy of a BI-AdS black hole in a canonical ensemble was obtained by computing the Euclidean action in [58], where an extra boundary term S surf was introduced to keep the charge fixed instead of the potential. However for the grand canonical ensemble, S surf is not needed any more. Excluding the contribution of S surf , the computation of the Euclidean action in [58] then gives the free energy of the BI-AdS black hole in the grand canonical ensemble: where S = 16π 2 r 2 + is the entropy of the black hole. For the later convenience, we can express quantities in units of l: Note that the potential Φ is dimensionless.
To find the phase structures of the black hole, one needs to use eqns. (4) and (5) to express the horizon radiusr + in terms of the temperatureT and the potential Φ:r + =r + (T , Φ).
When Φ < Φ c , T → +∞ in the limits of r + → 0 and r + → +∞, which implies that The thermal stable black holes have C Φ ≥ 0, which means ∂r + /∂T > 0. AtT min , the black hole appears, and its free energy is larger than that of the thermal AdS space. AsT increases fromT min , the free energy of Large BH decreases while that of the thermal AdS space is constant. They cross each other at some point, where a first-order transition occurs, and Large BH then becomes globally stable.

III. BORN-INFELD BLACK HOLES IN A CAVITY
In this section, we consider a thermodynamic system with Born-Infeld electrodynamics charged black holes inside a cavity, on the boundary of which the temperature and the potential are fixed. On a (3 + 1) dimensional spacetime manifold M with a time-like boundary ∂M, the action is given by where K is the extrinsic curvature, γ is the metric on the boundary, and K 0 is a subtraction term to make the boundary term vanish in flat spacetime. The BI black hole solution of the action (11) is [63] ds 2 = −f (r) dt 2 + dr 2 f (r) + r 2 dθ 2 + sin 2 θdφ 2 , where Here M and Q are the mass and the charge of the back hole, respectively. Note that M plays no role in our paper since we always use the horizon radius r + to eliminate M.
Suppose that the wall of the cavity enclosing the BI black holes is at r = r B , and the wall is maintained at a temperature of T and a gauge potential of Φ, where we assume that Φ > 0 without loss of generality. For this system, the Euclidean continuation of the action S was calculated in [63]: where S = 16π 2 r 2 + is the entropy of the black hole. In the semiclassical approximation, the free energy F is related to S E by Expressing the mass M in terms of the horizon radius r + , one finds that the free energy F is a function of the temperature T , the potential Φ, the charge Q, the cavity radius r B and the horizon radius r + : where T , Φ and r B are parameters of the grand canonical ensemble. The locally stationary points of the free energy F can be determined by extremizing F (r + , Q; T, Φ, r B ) with respect to r + and Q: where is the Hawking temperature of the black hole. Usually, it is convenient to express quantities in units of r B : The potential Φ is dimensionless. In terms of x and tilde quantities, f (r B ) andF can be expressed as respectively.
For the BI black holes residing in a cavity, there appears to be some constraints imposed on x andQ. As shown in [63], whenQ 2 < 4ã, BI black holes are Schwarzschild-like type, which exist for 0 < r + < r B , or 0 < x < 1 in tilde variables. WhenQ 2 ≥ 4ã, BI black holes are RN type, which can have the extremal BI black hole solution with the nonzero horizon radius r e = Q 2 − 4a/2. Requiring that r e < r + < r B leads to Q2 − 4ã/2 < x < 1 and x = x B1min is a Schwarzschild black hole in a cavity, which is dubbed as Schwarzschild State. However, one finds that which means that neither the thermal flat space nor Schwarzschild State can be the global minimum of the free energy over the whole physical region of x andQ.
• B2: x = 0 and 0 ≤Q ≤ 2 √ã . For the state on B2, its metric and Ricci scalar are respectively. Although the metric is regular, the spacetime has a physical singularity at r = 0. So the state on B2 is a naked singularity since it has no horizon. The global minimum of the free energy on B2 is atQ =Q B2min with 0 <Q B2min < 2 √ã when Φ ≤ Φ B2c and atQ = 2 √ã otherwise. For simplicity, we denote the boundary state atQ =Q B2min and x = 0 as NS State. Note that the thermal flat space, which is at Q = 0 and x = 0 on B2, is never the global minimum of the free energy on B2.  x(T , Φ) is multivalued, there are more than one branch of different sizes. As x → 0, we find that there is a critical potential Φ c1 such thatT → +∞ (−∞) when Φ < Φ c1 (Φ > Φ c1 ). As x → 1, one has thatT where Φ c2 = 8 √ 1+aπ √ 1+2a < Φ c1 . Therefore at x = 1,T > 0 (< 0) when Φ < Φ c2 (Φ > Φ c2 ). We also find that, for Φ > Φ c1 ,T is always negative, and henceF has no locally stationary   in the grand ensemble were already studied in [42], where the branches of BI-AdS black holes and the BH/Thermal AdS first and zeroth order phase transitions were found. In this paper, we investigated phase structures of BI-AdS black holes in the grand ensemble in a more thorough way. To our knowledge, the phase diagrams 2 and 4 has yet to be reported. Moreover, Region III of FIG. 2 was not observed in [42]. One can also study the phase structures and transitions of BI-AdS black holes in the context of the extended phase space thermodynamics, where the cosmological constant is interpreted as thermodynamic pressure, i.e., P = 6/l 2 [64,65]. Our results can simply be generalized to the extended phase space case by making replacements T = T 6/P ,ã = aP/6 andF ≡ F P/6.
To determine the phase structure of BI black holes in a cavity, we computed the locally stationary points of the free energy BH and NS State and a second-order phase transition between BH and Extremal State. In this paper, we only focus on spherical topology, and hence it is possible that there are some other states of lower free energy in a different topological sector with the same potential and temperature. If this happens, the globally stable phases discussed above could be only metastable.
For BI black holes in a cavity, the flat thermal space is on the boundary of the physical region of the system. However, NS state or Extremal state is always preferred over the flat thermal space. So the flat thermal space is never the globally stable phase of the system.
As shown in FIGs. 8(a) and 8(b), there are some regions of the parameter space, in which NS state is globally stable while there is an unstable branch of the BI black hole solution.
Perturbing the unstable black hole, we find that the black hole radiates away more energy than it absorbs, and the system would eventually settle down to a naked singularity. Finding a time-dependent solution, which describes a BI black hole evolving to a naked singularity, is very tempting, since such solution can provide a counterexample to the weak cosmic censorship conjecture [66].
Finally, we found that, in the grand canonical ensemble, there are some dissimilarities between the phase structures and transitions of BI-AdS black hole and those of BI black holes in a cavity: 1) For BI-AdS black holes, the thermal AdS space is sometimes preferred over the black hole solutions. Inspired by the phase structure of RN black holes in a cavity [17], one would expect that, for BI black holes in a cavity, the thermal flat space could sometimes be globally preferred. However, our results showed that the thermal flat space is never globally preferred. Instead, NS state or Extremal State can be the globally minium  I  II  III  IV