1. Introduction
Phase transitions in gravitational systems have attracted more interest since the discovery of Hawking–Page phase transitions in four-dimensional Anti-de Sitter (AdS) backgrounds, where the low-temperature phase is thermal AdS and the high-temperature phase is the AdS black hole [1]. Moreover, in AdS, unlike the flat spacetime, where large black holes (LBH) always have lower temperatures and the ordinary Schwarzschild has negative specific heat which is thermodynamically unstable, it was found that large AdS black holes have positive specific heat. If you make them bigger (higher energy), they get hotter. Small AdS black holes have negative specific heat and very small AdS black holes do not care about the cosmological constant at all, so are just like the asymptotically flat Schwarzschild solution. Considering that AdS black holes have the property in that the horizon is a (d − 2)-dimensional compact Einstein space of positive, zero or negative curvature, it was argued that they have different phase structures [2]. For Schwarzschild-AdS black holes with toroidal or hyperbolic horizons, it was found that they are thermally stable and the Hawking–Page phase transition does not occur. However, for Schwarzschild-AdS black holes with spherical horizons, Hawking–Page phase transition exists between large stable black holes and thermal gas in the AdS space.
By considering the cosmological constant as a thermodynamic pressure, and its conjugate variable as the thermodynamic volume, thermodynamical phase transitions have been restudied extensively; for a review please refer to [3] and references therein and [4, 5]. Witten [6] extended the four-dimensional transition to arbitrary dimensions and provided a natural explanation for a confinement/deconfinement transition in the boundary field theory via AdS/conformal field theory (CFT) correspondence. The dynamical signature of the thermodynamical phase transition was uncovered in the study of perturbations around black holes [7]. In the extended phase space, the van der Waals-like thermodynamic phase transition was further observed in quasinormal modes [8].
Black hole phase transitions have become more intriguing recently, especially in the study of the holographic superconductivity [9–20] within the context of the AdS/CFT correspondence, where it was found that phase transition happens when the scalar hair condensates onto a black hole. Further phase transition phenomena have been analyzed and classified by exploiting Ehrenfest's scheme [21–24]. Bragg–Williams' construction of a free energy function has also been applied to examine black hole phase transitions [25]. In addition to standard thermodynamic methods, geometrical ideas have been introduced to study thermodynamic phase transitions and it was argued that such a geometric method could disclose the intrinsic reason for the transition [26–28].
In this work, we want to generalize the phase transition study in black hole solutions to Einstein's theory coupled to a nonlinear electromagnetic field. Introduction of the nonlinear electrodynamics is to eliminate the problem of infinite energy of the electron [29]. Nonlinear electrodynamics naturally emerge within the modern context of the low-energy limit of heterotic string theory [30–32] and play an important role in the construction of regular black hole solutions [33–38]. Some black hole/brane solutions in nonlinear electromagnetic fields have been investigated, for instance in [39–49] and the references therein. The thermodynamics of Einstein–Born–Infeld black holes with a negative cosmological constant were studied in [50] and those of a power-law electrodynamic hole were studied in [51], where the authors showed that a set of small black holes (SBH) are locally stable by computing the heat capacity and the electrical permittivity. The thermodynamics of Gauss–Bonnet black holes for a power-law electrodynamic were studied in [52]. On the other hand, higher dimensional black hole solutions to the Einstein-dilaton theory coupled to the Maxwell field were found in [53, 54] and black hole solutions to the Einstein-dilaton theory coupled to Born–Infeld and power-law electrodynamics were found in [55, 56]. The effects of nonlinear electrodynamics on the properties of holographic superconductors have recently been investigated extensively [11–20]. In our study on black hole solutions to Einstein's theory coupled to a nonlinear power-law electromagnetic field, we take the specific value of the exponent power motivated by the existence of exact hairy black hole solutions for the p = 3/4 value [57]. Here, we want to examine phase transitions in both canonical and grand canonical ensembles. In the study of extended phase space thermodynamics and P-V criticality of d-dimensional black holes, a nonlinear source influence on the phase structure was disclosed [58]. By considering the presence of a generalized Maxwell theory, such as the power Maxwell invariant (PMI), it was shown that a first-order phase transition occurs in both canonical and grand canonical ensembles, in contrast to the situation in the Reissner–Nordström black hole with a standard Maxwell field, where phase transition only happens in the canonical ensemble. The PMI field contains richer physics than that of the Maxwell field, which reduces to a linear electromagnetic source in a special case (p = 1). The black hole solutions of the Einstein–PMI theory and their thermodynamics and geometric properties have been studied in [39–41, 52, 59–63]. Also, the effects of PMI source on strongly coupled dual gauge theory have been examined within the context of AdS/CFT correspondence [20, 42]. Here, we will concentrate on examining how the nonlinearities, especially the power-law exponent of the nonlinear electromagnetic field, influence the phase structures of black holes.
The paper is organized as follows. In section 2 we give a brief review of the background that we will study. In section 3 we give an extended thermodynamics description in canonical and grand canonical ensembles. Finally, we present our conclusions in section 4 .
2. General formalism for nonlinear electrodynamics
We consider topological black hole solutions for the power Maxwell theory coupled to gravity described by the action, see [56, 58]3 ) describes a black hole solution with an inner horizon (r−) and an outer horizon (r+). The outer horizon r+ of this black hole can be calculated numerically by finding the largest real positive root of f(r = r+) = 0, for different values of p parameter under consideration. In the following, we focus our attention on the specific value p = 3/4. Then, the metric (3 ) reads as follows7 ) we can obtain the Hawking temperature of the black hole solutions, and we can also express thermodynamic quantities as a function of the event horizon and physical charge rather than physical mass M
$\begin{eqnarray}I=\int {{\rm{d}}}^{4}x\sqrt{-g}\left(\displaystyle \frac{1}{2\kappa }(R-2{\rm{\Lambda }})+\eta \,| -{F}_{\mu \nu }{F}^{\mu \nu }{| }^{p}\right),\end{eqnarray}$
where κ = 8πG = 1, Fμν = ∂μAν − ∂νAμ and Aμ represents the gauge potential. The exponent p is a rational number and the absolute value ensures that any configuration of electric and magnetic fields can be described by these Lagrangians. One could also consider the Lagrangian without the absolute value and the exponent p restricted to being an integer or a rational number with an odd denominator [62]. The sign of the coupling constant η will be chosen such that the energy density of the electromagnetic field is positive. This condition is guaranteed in the following cases: p > 1/2 and η > 0 or p < 1/2 and η < 0 [57]. Here, we will consider the first condition and a specific value of the exponent p = 3/4. In [56–58] it was shown that the field equations have the topological nonlinearly charged black holes as solutions. The general AdS solution for a power Maxwell black hole can be written as follows $\begin{eqnarray}{{\rm{d}}{s}}^{2}=-f(r){{\rm{d}}{t}}^{2}+\displaystyle \frac{{{\rm{dr}}}^{2}}{f(r)}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}^{2},\end{eqnarray}$
where dω2 is the metric of the spatial 2-section, which can have a positive (k =1), negative (k = −1) or zero (k = 0) curvature, and $\begin{eqnarray}f(r)=k-\displaystyle \frac{m}{r}-\displaystyle \frac{{\rm{\Lambda }}}{3}{r}^{2}+\eta \displaystyle \frac{{2}^{p}{\left(2p-1\right)}^{2}}{(3-2p)}\displaystyle \frac{{q}^{2p}}{{r}^{\tfrac{2}{2p-1}}},\end{eqnarray}$
where m and q are integration constants related to the Arnowitt-Desser-Misner mass M and the electric charge Q of the black hole by $\begin{eqnarray}M=4\pi m,\end{eqnarray}$
$\begin{eqnarray}Q=\eta {2}^{2p-1}p\,{q}^{2p-1},\end{eqnarray}$
respectively, and the gauge potential is given by $\begin{eqnarray}{A}_{t}(r)=\displaystyle \frac{1-2p}{3-2p}\,q\,{r}^{\tfrac{3-2p}{1-2p}}.\end{eqnarray}$
In [57] it has been shown that equation ( $\begin{eqnarray}f(r)=k-\displaystyle \frac{m}{r}+\eta \displaystyle \frac{{q}^{3/2}}{3\sqrt[4]{2}{r}^{4}}-\displaystyle \frac{{\rm{\Lambda }}{r}^{2}}{3}.\end{eqnarray}$
In figure 1 we show the behaviour of this black hole metric function for different values of the cosmological constant and k = 1 with $p=\tfrac{3}{4}$. We can observe that, for fixed values of the cosmological constant, black hole charge and the nonlinear coupling constant η, there are different kinds of solutions; black holes with a Cauchy and an event horizon, an extremal black hole configuration and also naked singularities, when the black hole mass decreases. Now, using the surface gravity relation and equation ( $\begin{eqnarray}T=\displaystyle \frac{1}{4\pi {r}_{+}}\left(k-\eta \displaystyle \frac{{q}^{3/2}}{\sqrt[4]{2}{r}_{+}^{4}}-{\rm{\Lambda }}{r}_{+}^{2}\right).\end{eqnarray}$
The behaviour of the temperature of this black hole can be seen in figure 2. Note that the temperature vanishes for r+ = rextrem, and for r+ < rextrem we obtain a negative temperature and, therefore, it corresponds to a region with nonphysical meaning, where the thermodynamic description breaks down. Besides, the electric potential Φ, measured at infinity with respect to the horizon, is $\begin{eqnarray}{\rm{\Phi }}=-({A}_{t}({r}_{+})-{A}_{t}(\infty ))=\displaystyle \frac{q}{3{r}_{+}^{3}},\end{eqnarray}$
while the Bekenstein–Hawking entropy S is given by $\begin{eqnarray}S=8{\pi }^{2}{r}_{+}^{2}.\end{eqnarray}$
The next section is devoted to the study of the extended thermodynamics properties and its particular P-V criticality.
Figure 1. The behaviour of the lapse function as a function of r, for different values of the black hole mass, and q = 1, η = 1 and Λ = −1. |
Figure 2. The temperature for k = 1, Q = 2 and η = 1, for different values of the cosmological constant. |
3. Extended thermodynamics description
Now, in the extended thermodynamics space we consider the cosmological constant as another thermodynamics variable and, therefore, the standard extensive parameters will be the entropy, the black hole charge and the cosmological constant. To see these extended thermodynamics we consider the cosmological constant as the source of dynamical pressure using the relation $P=-\tfrac{{\rm{\Lambda }}}{8\pi }$ [64, 65], and the conjugate thermodynamics volume is given by $V=\tfrac{32{\pi }^{2}}{3}{r}_{+}^{3}$. After obtaining these quantities we verify that they satisfy the following Smarr formula
$\begin{eqnarray}M=2{TS}+\displaystyle \frac{4}{3}{\rm{\Phi }}Q-2{VP},\end{eqnarray}$
with M given by $\begin{eqnarray}M=4\pi {r}_{+}\left(k+\eta \displaystyle \frac{{2}^{3/4}{q}^{3/2}}{6{r}_{+}^{4}}-\displaystyle \frac{1}{3}{\rm{\Lambda }}{r}_{+}^{2}\right).\end{eqnarray}$
3.1. Canonical ensemble (fixed Q)
In this section we study phase transitions in the canonical ensemble, where we are considering the fixed charge as an extensive parameter. First, we compute the heat capacity for a fixed Q8 ) the heat capacity yields8 ) and (14 ) we can see that the temperature and the heat capacity are always positive with the only exception for the extreme configuration, where r+ = r− =rextreme. There is a region where CQ < 0 and, according to Davies's approach, this would indicate the presence of a type-one phase transition; also, it is important to remark that the temperature in this region is also negative, and therefore it is considered as a nonphysical region, where the thermodynamics description breaks down. Therefore, the main conclusion is that it is, according to Davies's approach, the correlation between drastic change for the stability properties of a thermodynamic black hole system and the change of the sign or divergences of the heat capacity. In reference to the canonical ensemble, it is very well known that black holes are locally stable thermodynamics systems if their heat capacity is positive or non-vanishing. Besides, at the points where the heat capacity is divergent there is a type-two phase transition. From figure 3 we can see that this black hole solution is locally unstable from a thermal point of view, because the heat capacity CQ has a divergent term, for the case where the spatial section is spherical (k = 1). For the hyperbolic (k = −1) and flat (k = 0) spatial sections we can conclude that the respective configurations of the black hole are locally stable. Then, following Davies and according to Ehrenfest's classification, second-order phase transitions occur at those points where the heat capacity diverges. These points can be obtained from the following relation8 ), (12 ) and the definition of pressure due to the cosmological constant for a fixed charge Q, we obtain the equation of state, P(V, T)16 ) can be written as
$\begin{eqnarray}{C}_{Q}=T{\left(\displaystyle \frac{\partial S}{\partial T}\right)}_{Q}.\end{eqnarray}$
So, from equation ( $\begin{eqnarray}{C}_{Q}=-\displaystyle \frac{16{\pi }^{2}{r}_{+}^{2}\left(-2{{kr}}_{+}^{4}+{2}^{3/4}\eta {q}^{3/2}+2{\rm{\Lambda }}{r}_{+}^{6}\right)}{5\times {2}^{3/4}\eta {q}^{3/2}-2{r}_{+}^{4}\left(k+{\rm{\Lambda }}{r}_{+}^{2}\right)}.\end{eqnarray}$
Now, from equations ( $\begin{eqnarray}5\times {2}^{3/4}\eta {q}^{3/2}-2{r}_{+}^{4}\left(k+{\rm{\Lambda }}{r}_{+}^{2}\right)=0.\end{eqnarray}$
Here, we can identify three regions separated by two divergent points. Then, we can recognize three different phases: two stables phases, where CQ > 0 for small and large horizon radii, and one unstable phase CQ < 0 for an intermediate radius. Therefore, there are three different phases that we call an SBH, meta-stable black hole (MBH) and LBH; some results have been reported for charged AdS black holes with Maxwell and PMIs in [66–68]. Now, to determine the second-order phase transition and its consequences we study the critical behaviour by performing a comparison of the equations of state with the van der Waals equation. First, using equations ( $\begin{eqnarray}P=\displaystyle \frac{T}{2{r}_{+}}-\displaystyle \frac{k}{8\pi {r}_{+}^{2}}+\eta \displaystyle \frac{{q}^{3/2}}{8\sqrt[4]{2}\pi {r}_{+}^{6}},\end{eqnarray}$
where r+ is the horizon radius. Following [69], we identify the geometric quantities P and T with the physical pressure and the temperature of the system using dimensional analysis ${l}_{P}^{2}={G}_{d}{\hslash }/{c}^{3}$. Then, we can identify the following relations between the geometric quantities P and T and the physical pressure and temperature: $\begin{eqnarray}\left[\mathrm{Pressure}\right]=\displaystyle \frac{{\hslash }c}{{l}_{P}^{2}}[P],\,\,[\mathrm{Temp}]=\displaystyle \frac{{\hslash }c}{k}[T].\end{eqnarray}$
Then, $\begin{eqnarray}\left[\mathrm{Pressure}\right]=\displaystyle \frac{{\hslash }c}{{l}_{P}^{2}}[P]=\displaystyle \frac{{\hslash }c}{{l}_{P}^{2}}\displaystyle \frac{T}{2{r}_{+}}+...,\end{eqnarray}$
from this expression we can identify the specific volume v of the fluid with the horizon radius of the black hole as v = 2r+; thus, equation ( $\begin{eqnarray}P=\displaystyle \frac{T}{v}-\displaystyle \frac{k}{2\pi {v}^{2}}+4\eta \displaystyle \frac{{2}^{3/4}{q}^{3/2}}{\pi {v}^{6}}.\end{eqnarray}$
Figure 3. The heat capacity for k = 1, Q = 1, η = 1 and Λ = −0.0246. This figure depicts different regions according to the sign of the heat capacity; (I) CQ < 0 and T < 0 corresponds to the nonphysical region (Non-PH) , (II) T > 0: CQ > 0 SBH region, CQ < 0 MBH region and CQ > 0 LBH region. The second-order phase transitions occur at the points where the heat capacity shows a divergence. |
In figure 4 we show the P-v diagram, and it is possible to observe similar behaviour to the van der Waals gas; this means there is a critical point corresponding to an inflection point over the critical isotherm curve.
Figure 4. The P-v diagram with k = 1, q = 1 and η = 1, for different values of the temperature. |
The critical point can be computed through
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial P}{\partial v}=0,\\ \displaystyle \frac{{\partial }^{2}P}{\partial {v}^{2}}=0,\end{array}\end{eqnarray}$
which yields $\begin{eqnarray}\begin{array}{rcl}{v}_{\mathrm{critical}}&=&\displaystyle \frac{{2}^{15/16}\sqrt[4]{15}\sqrt[4]{\eta }{q}^{3/8}}{\sqrt[4]{k}},\\ {T}_{\mathrm{critical}}&=&\displaystyle \frac{2\sqrt[16]{2}{k}^{5/4}}{5\sqrt[4]{15}\pi \sqrt[4]{\eta }{q}^{3/8}},\\ {P}_{\mathrm{critical}}&=&\displaystyle \frac{{k}^{3/2}}{6\times {2}^{7/8}\sqrt{15}\pi \sqrt{\eta }{q}^{3/4}}.\end{array}\end{eqnarray}$
This critical point only exists for a black hole with a spherical spatial section (k = 1); we can also compute the universal relation $\begin{eqnarray}{\rho }_{0}=\displaystyle \frac{{P}_{\mathrm{critical}}\times {v}_{\mathrm{critical}}}{{T}_{\mathrm{critical}}}=\displaystyle \frac{5}{12}.\end{eqnarray}$
This universal number corresponds to the critical number and it does not depend on the black hole charge or the nonlinear coupling constant η. This universal number differs from the van der Waals fluid; however, the critical point belongs to the same universality class as that of the van der Waals fluid. Now we would like to discuss the P-V criticality and the phase transition of charged black holes by considering the power Maxwell electrodynamics with $p=\tfrac{3}{4}$ in the canonical (fixed Q) ensemble. In this ensemble it is possible to compute the Gibbs free energy by evaluating the Euclidean action on-shell. To do this, we have to use the counterterm method to cancel the divergences appearing in the on-shell Euclidean action, $\begin{eqnarray}{I}_{E}={I}_{\mathrm{bulk}}+{I}_{\mathrm{ct}}+{I}_{\mathrm{GH}}+{I}_{{\rm{A}}},\end{eqnarray}$
where ${I}_{{\rm{b}}{\rm{u}}{\rm{l}}{\rm{k}}}$ corresponds to the Euclidean on-shell bulk action, ${I}_{{\rm{c}}{\rm{t}}}$ is the action coming from the counterterm method, ${I}_{{\rm{G}}{\rm{H}}}$ is the well-known Gibbons–Hawking term and ${I}_{{\rm{A}}}$ is a boundary term for the electromagnetic field needed for a well posed action principle; this term in the canonical ensemble (fixed Q) is not vanishing, and it is given by $\begin{eqnarray}{I}_{{em}}=-\displaystyle \frac{p}{4\pi }{\int }_{\partial M}{{\rm{d}}}^{n}x\sqrt{h}{\left(-F\right)}^{s-1}{n}_{\mu }{F}^{\mu \nu }{A}_{\nu }=\beta Q{\rm{\Phi }},\end{eqnarray}$
where $\beta =\tfrac{1}{T}$ and hij is the induced metric at the boundary. So, the Gibbs free energy or thermodynamics potential is given by $\begin{eqnarray}G(T,Q,P)=\frac{{I}_{E}^{\mathrm{on}-\mathrm{shell}}}{\beta }=M-{TS},\end{eqnarray}$
and the corresponding expression in our case is given by $\begin{eqnarray}G=2\pi {r}_{+}\left(k-\displaystyle \frac{8}{3}\pi {{\Pr }}_{+}^{3}+\eta \displaystyle \frac{5{q}^{3/2}}{3\sqrt[4]{2}{r}_{+}^{3}}\right).\end{eqnarray}$
In figure 5, we depict the Gibbs free energy, and clearly the critical pressure is marking the point where the system is undergoing a first-order phase transition. For P < PC, the well-known characteristic swallow tail appears, where the first-order phase transition is occurring at the intersection point between the SBH and LBH configurations. For those branches we can verify that T > 0 and CQ > 0; it does mean that the SBH and LBH are locally stable or can also be interpreted as the phases with positive specific heat in the lower and higher radius regions are stable. Now, for T > 0 and CQ < 0 we have the MBH configuration and this means the MBH branch is unstable. This meta-stable configuration can be explained because states of the lowest Gibbs free energy are preferred by the system; similar situations can occur for some states on the SBH and LBH branches. From equation (15 ) we can obtain the value of the point where the heat capacity diverges, which is exactly the critical point (21 ). The first-order phase transition takes place between the stable branches, and when P = PC the two divergent points merge to form a single divergence, removing the unstable region and where the second-order phase transition, described by CQ, occurs. For P > PC the heat capacity is positive definite and free of divergences; then the black hole configurations are stable and free of critical behaviour.
Figure 5. Gibbs free energy for nonlinear Maxwell electrodynamics for $p=\tfrac{3}{4}$, k = 1, q = 2 and η = 1 in the canonical ensemble. We can see that for P < PC (PC is represented by the dashed line) the system is undergoing a first-order phase transition described by the characteristic swallow tail. |
Note that due to the conservation of electric charge there is no Hawking–Page phase transition, with the charged black hole with a Gibbs free energy essentially non-vanishing to the thermal AdS space with a vanishing Gibbs free energy. In the next section, we consider the grand canonical ensemble, where the charge Q is not fixed and, therefore, we will be able to study the Hawking–Page phase transition.
3.2. Grand canonical ensemble (fixed Φ)
Now, in this section we discuss the P-V criticality of charged black holes by considering the power Maxwell electrodynamics with $p=\tfrac{3}{4}$ in the grand canonical (fixed Φ) ensemble. As in the previous section, an important quantity is the heat capacity CΦ at fixed Φ, which is relevant for characterization of the local stability in the grand canonical ensemble, and it is given by
$\begin{eqnarray}{C}_{{\rm{\Phi }}}=\displaystyle \frac{4S\left(2{PS}+2k\pi -3\times {2}^{\tfrac{1}{4}}\sqrt{3\pi }{S}^{\tfrac{1}{4}}\eta {{\rm{\Phi }}}^{\tfrac{1}{2}}\right)}{4{PS}-4k\pi +3\times {2}^{\tfrac{1}{4}}\sqrt{3\pi }{S}^{\tfrac{1}{4}}\eta {{\rm{\Phi }}}^{\tfrac{1}{2}}}.\end{eqnarray}$
In figure 6, we plot the behaviour of the heat capacity as a function of the entropy, and we can conclude that there is a second-order phase transition at the point where the heat capacity diverges. In fact, at the divergence point the system is undergoing a second-order phase transition between unstable SBH configurations (CΦ > 0) and stable LBH configurations (CΦ > 0), as we will see in the following.
Figure 6. Heat capacity as a function of the entropy for Φ = 0.5, k = 1, η = 0.1 and P = 5. |
In this ensemble it is possible to compute the grand canonical potential by evaluating the Euclidean action on-shell. To regularize the divergences that appear in the on-shell Euclidean action we have to use the counterterm method. However, in the grand canonical ensemble, IA is vanishing, because the electric potential Φ is kept fixed, and the grand potential becomes30 ) and the conditions ∂P = ∂2P = 0, the critical point yields
$\begin{eqnarray}{ \mathcal G }=M-{TS}-Q{\rm{\Phi }}.\end{eqnarray}$
Now, using the specific volume v = 2r+, we can obtain the temperature in this ensemble, which reads as follows $\begin{eqnarray}T=\displaystyle \frac{1}{4\pi {r}_{+}}\left(k+8\pi {{\Pr }}_{+}^{2}-\eta \displaystyle \frac{3\sqrt{3}{\left({r}_{+}{{\rm{\Phi }}}^{3}\right)}^{\tfrac{1}{2}}}{\sqrt[4]{2}}\right),\end{eqnarray}$
and the equation of state is given by $\begin{eqnarray}P=\displaystyle \frac{T}{v}-\displaystyle \frac{k}{2\pi {v}^{2}}+\eta \displaystyle \frac{3\sqrt{3}{{\rm{\Phi }}}^{3/2}}{2\times {2}^{3/4}\pi {v}^{3/2}}.\end{eqnarray}$
Then, from equation ( $\begin{eqnarray}\begin{array}{rcl}{v}_{\mathrm{critical}}&=&\displaystyle \frac{128\sqrt{2}{k}^{2}}{243{\eta }^{2}{{\rm{\Phi }}}^{3}},\\ {T}_{\mathrm{critical}}&=&-\displaystyle \frac{243{\eta }^{2}{{\rm{\Phi }}}^{3}}{128\sqrt{2}k\pi },\\ {P}_{\mathrm{critical}}&=&\displaystyle \frac{19683{\eta }^{4}{{\rm{\Phi }}}^{6}\left(8\eta {{\rm{\Phi }}}^{3/2}\sqrt{\tfrac{{k}^{2}}{{\eta }^{2}{{\rm{\Phi }}}^{3}}}-9k\right)}{65536\pi {k}^{4}}.\end{array}\end{eqnarray}$
Note that the critical point has physical meaning only for a black hole with a hyperbolic spatial section (k = −1) and loses physical meaning for k = 1 and k = 0: this point was argued in [58] for p < 1. Then, for these cases, the P-v diagram does not show a critical point because the condition ∂P = ∂2P = 0 cannot be satisfied and, therefore, there is not a critical point in the grand canonical ensemble for k = 1 and k = 0. Therefore, it is more like a solid–liquid phase transition, rather than a liquid–gas phase transition. However, for an interval of the electric potential the pressure has a maximum ${P}_{\max }$; these results are summarized in figure 7. Also, for k = −1 we can compute the universal relation ${\rho }_{0}=\tfrac{{P}_{\mathrm{critical}}\ \ast {v}_{\mathrm{critical}}}{{T}_{\mathrm{critical}}}$ for the grand canonical ensemble, giving ρ0 = 1/6.
Figure 7. The P-v diagram for k = 1, q = 1 and η = 1, for different values of the electric potential. |
On the other hand, the thermodynamic grand potential can be written as
$\begin{eqnarray}{ \mathcal G }=-\displaystyle \frac{1}{3}\pi \left(-6{{kr}}_{+}+16\pi {{\Pr }}_{+}^{3}+3\ {2}^{3/4}\sqrt{3}\eta {\rm{\Phi }}\sqrt{{r}_{+}^{3}{\rm{\Phi }}}\right).\end{eqnarray}$
An important difference between the canonical ensemble and the grand canonical ensemble corresponds to the possibility that in the latter a Hawking–Page first-order phase transition to thermal AdS can occur. Essentially, due to quantum effects, a black hole can emit energy to the external background through Hawking radiation and this allows thermal equilibrium to be reached between a stable black hole configuration and the thermal AdS space and its grand thermodynamic potential is zero. Figure 8 summarizes the main results on this subject, and we can observe the Hawking–Page phase transition between the stable black hole and thermal AdS, and there is also a second-order phase transition between SBH and LBH at the cusp temperature. Moreover, by observing the slope and concavity of the thermodynamic potential, see figure 6, we can conclude that the black holes with S < Scusp (SBH) are thermally unstable and those with S > Scusp (LBH) are thermally stable. Then, the former cannot establish the equilibrium with the thermal AdS space.
Figure 8. Gibbs free energy for nonlinear Maxwell electrodynamics for $p=\tfrac{3}{4}$, k = 1, η = 0.1 for different values of the electric potential Φ in the grand canonical ensemble. |
4. Concluding comments
In this paper we studied the thermodynamics description of black hole solutions of Einstein's theory coupled to a nonlinear power-law electromagnetic field. We considered the extended phase space, by considering the cosmological constant as a source of dynamical pressure, and we studied first-order phase transitions, including the Hawking–Page phase transition, and second-order phase transitions in both canonical and grand canonical ensembles. Mainly, we found that first- and second-order phase transitions can occur in the canonical ensemble while, for the grand canonical ensemble, we found that first-order phase transitions of the Hawking–Page type can occur. However, the first-order phase transitions depend on the parameter k, which describes the curvature of the spatial 2-section, yielding the fact that in the canonical ensemble the first-order phase transitions are allowed for k = 1 (spherical), and in the grand canonical ensemble they are allowed for k = −1 (hyperbolic).
For the canonical ensemble, we observed that the temperature and the heat capacity are always positive with the only exception for the extreme black hole configuration. From the analysis of the heat capacity at fixed charge we showed that there is a region where CQ < 0; then, according to Davies's approach, this indicates the presence of a type-one phase transition. It is also important to remark that the temperature in this region is also negative, and therefore it is considered as a nonphysical region, where the thermodynamic description breaks down. It is very well known that a black hole is a locally stable thermodynamic system if its heat capacity is positive and non-vanishing. Besides, at the points where the heat capacity is divergent there is a type-two phase transition. Also, we have shown that the black hole solution is locally unstable from a thermal point of view, because the heat capacity CQ has a divergent term, for the case where the spatial section is spherical (k = 1). For the hyperbolic (k = −1) and flat (k = 0) spatial sections the black hole configurations are stable. Then, following Davies and according to Ehrenfest's classification, second-order phase transitions occur at those points where the heat capacity diverges. Then, we obtain three different phases: two stables phases where CQ > 0 for small and large horizon radii, and one unstable phase with CQ < 0 for the intermediate radius. Therefore, there are three different phases: SMB, MBH and LBH. Also, we determined the Gibbs free energy, and we have shown that the critical pressure corresponds to the point where the system is undergoing a first-order phase transition. For P < PC the well-known characteristic swallow tail appears, where the first-order phase transition occurs at the intersection point between the SBH and LBH, with T > 0 and CQ > 0. Therefore, the SBH and LBH are locally stable; this would also be interpreted as the phases with positive specific heat in the lower and higher radius regions are stable. When T > 0 and CQ < 0 we have an unstable MBH configuration. This meta-stable configuration can be explained because states with the lowest Gibbs free energy are preferred for the system; a similar situation can occur for some states on the small and LBH branches.
From the grand canonical ensemble point of view, we found there is a critical point only for the case when k = −1, and therefore there is critical behaviour that is similar to the description in the canonical ensemble for k = 1. For k = 1 and k = 0, the P-V analysis shows that there are no critical points because the condition ∂P = ∂2P = 0 cannot be satisfied. Finally, one important difference in the grand canonical ensemble in comparison to the canonical ensemble corresponds to the possibility for a Hawking–Page first-order phase transition to thermal AdS. Essentially, due to quantum effects a black hole can emit energy to the external background through Hawking radiation and this allows thermal equilibrium to be reached between a stable black hole configuration and the thermal AdS space, and its grand thermodynamic potential is zero. Also, we have shown the existence of a Hawking–Page phase transition between a stable black hole and the thermal AdS, and there is also a second-order phase transition between SMB and LBH at the horizon radius, where CΦ is divergent: this point corresponds to the Tcusp temperature.