On Heat Properties of AdS Black Holes in Higher Dimensions

We investigate the heat properties of AdS Black Holes in higher dimensions. We consider the study of the corresponding thermodynamical properties including the heat capacity explored in the determination of the black hole stability. In particular, we compute the heat latent. To overcome the instability problem, the Maxwell construction, in the (T,S)-plane, is elaborated. This method is used to modify the the Hawking-Page phase structure by removing the negative heat capacity regions. Then, we discuss the thermodynamic cycle and the heat engines using the way based on the extraction of the work from a black hole solution.


Introduction
Recently, many efforts have been devoted to study the thermodynamical properties of black holes, in connection with higher dimensional supergravity models [1,2]. These properties have been extensively studied via different methods including numerical computation using various codes [3]. In fact, several models based on mathematical methods have been explored to study critical behaviors of black holes having different geometrical configurations in arbitrary dimensions. A particular emphasis has been put on AdS black holes [4,5,6,7,8,9,10].
More precisely, a nice interplay between the behaviors of the RN-AdS black hole systems and the Van der Waals fluids has been shown [11,12,13,14,15]. In this context, several landmarks of statistical liquid-gas systems, such as the P-V criticality, the Gibbs free energy, the first order phase transition and the behavior near the critical points have been derived. Also, in arbitrary dimensions of the spacetime, the authors [11,13] studied the critical behaviors of charged RN-AdS black holes. Extension to other solutions considered as a subject of interest in gravity theory, has also been performed and their corresponding phase transitions and statistical properties have been investigated using different approaches [19,20,21,22]. More recently, some authors have worked out the heat properties of AdS charged black holes and their solutions in four dimensions [23,24].
The aim in this paper is to reconsider the heat properties of AdS black holes in higher spacetime dimensions. More precisely, we will study the corresponding thermodynamical properties including the sign of the heat capacity explored when discussing the stability problem. In particular, we will derive the expression of the latent heat considered as a trivial consequence of the Hawking-Page phase transition. To overcome the instability problem, the Maxwell construction in (T, S)-plane is then elaborated to modify the Hawking-Page phase structure. Finally, we will discuss the thermodynamic cycle and the holographic heat engines.
The paper is organized as follows. In section 2, we reconsider the study of thermodynamics of AdS black holes along with the latent heat. Section 3 is devoted to Maxwell's construction of higher dimensional AdS black holes. In section 4, we discuss the thermodynamic cycle and the holographic heat engines. Finally, section 5 contains conclusions.

Thermodynamics and latent heat
This section concerns the study of the latent heat properties of Ads black holes in higher dimensions. This investigation could be supported by the existence of higher dimensional supergravity theories including superstring models, M-theory, and related topics. Here, we consider a non-rotating, neutral, asymptotically anti-de Sitter black holes. The corresponding metric solution reads as where dΩ 2 n−2 represents the volume of (n − 2)-dimensional sphere. This solution is characterized by the function f (r) taking the following general form It is worth recalling that the parameter M indicates the ADM mass of such a black hole solution while its horizon radius r + can be identified to the largest real root of f (r) = 0.
These two parameters are linked via the relation In the non-rotating AdS black holes, M should be interpreted as an enthalpy [4] which can be written as follows where the Bekenstein-Hawking entropy is given in terms of the horizon In this equation, the quantity ω reads as In fact, many other thermodynamical quantities can be also computed using similar technics.
Indeed, the pressure can be associated with the cosmological constant Λ and they are related , wheras the temperature is given in terms of the horizon radius r + via the following form By combining thermodynamical relations, the temperature can be re-expressed as One can see that the temperature presents a minimum associated with the following entropy value This minimum is given by A similar computation shows that the heat capacity can be expressed as Note that this quantity is negative for S < S T min . It becomes positive for S > S T min , but diverges at S = S T min .
Besides, recalling that the Gibbs free energy G is the Legendre transform of the enthalpy, one finds where the entropy function given in terms of T and P reads as It is noted that variation of G It is observed that for PS the Gibbs free energy of the black hole is negative. Lower than the solution associated with G Ads = 0, the corresponding situation can be considered as a more stable thermodynamical configuration. For PS however, the pure AdS space-time is the more stable. It is realized that the black hole with will evaporate. This is associated with the Hawking-Page transition explained by the following equation It is worth nothing that P| coexistance can be computed for any dimension. Indeed, we list the following expressions   The Gibbs free energy as a function of temperature at fixed P = 1 for 4 ≤ n ≤ 6. Right: Higher dimensional cases associated with 7 ≤ n ≤ 11 Schwarzschild-AdS black hole.
One can notice a minimum temperature T min for which no black holes can survive. However, above this temperature, two branches of the black holes have appeared. In fact, the upper branch describes an unstable small (Schwarzschild-like) black hole involving a negative specific heat. For (T > T min ), the black holes, at lower branch, become stable with positive specific heat. It is observed a Hawking-Page temperature T HP associated with vanishing values of the Gibbs free energy. Moreover, the black hole Gibbs free energy becomes negative for T > T HP . In fact, at T = T HP , a first order Hawking-Page phase transition shows up between the thermal radiations and large black holes as reported in [16,17,23].
Moreover, from Fig.1 we also note a jump in entropy which becomes more relevant in terms of the dimension of the space as shown in the following equation It is worth noting that the latent heat can be computed using the following thermodynamical expression L = T∆S.
Indeed, using Eq. (8), the calculation results in which is equal to the mass on the coexistence curve in the black hole phase. Here, note that for n = 4, we reproduce the result reported in [23]. It is shown that the latent heat is nonzero for any finite T and vanishes for very large values of T. In fact, in the case of asymptotically flat space-time with P = 0 the latent heat becomes infinite. We then deduce that the black hole cannot decay.
It is clear now that the sign of the heat capacity plays an important role in the determination of the stability of the black hole. More precisely, it has been shown that its negative values render the black hole unstable. In the next section we will show how to overcome this problem, by using the Maxwell construction to modify the Hawking-Page phase structure.

On Maxwell's construction of high dimensional AdS black holes
In this section, we investigate the corresponding Maxwell's construction. To do so, we first recall that the equal area law was introduced by Maxwell in order to explain the experimental behaviors of real fluids. Usually, this construction is elaborated in the (P, V) plane while keeping constant temperature [19,21,22]. However, fixing the pressure in (14), such a construction can also be done in the (T, S) plane. For Schwarzschild-AdS black hole, the choice of this plane has been explained in many papers including [16,19,20]. The starting point is the temperature as a un function of the entropy given by eq.(8). Then, we plot this function in Fig.3. It is observed that this function involves minimums at S T min and T min given by eq.(9) and eq.(10) respectively. For any value of the cosmological constant, these quantities produce the following reduced forms It is recalled that the Maxwell area law can be obtained using the fact the Gibbs free energy is the same for coexisting black holes. Exploring Gibbs free energy (12) where T ⋆ is the temperature of the equal area isotherm. The equal area law, in the reduced variables, gives the entropy of the liquid and gaseous phases. This solves the following Introducing new variable x ≡ s 1 n−2 , we get the following equation  Table 1: roots of the polynomial form and the corresponding entropy and temperature.
We should eliminate the states corresponding to either complex or negatives values since It is observed that a pure radiation phase can survive beyond T min up to the higher temperature given by the isotherm T ⋆ = t ⋆ T min . For T = T ⋆ , the black holes with different entropy values have the same free energy. They are more stable than the pure radiation. For T > T min , there exists a single and a stable black hole with a positive heat capacity. When we go back to four dimensions, we recover the results reported in [20] and [19] describing neutral case.

Thermodynamic cycle and the heat engines
Having discussed some thermodynamical proprieties of the Schwarzschild-AdS black hole including thermodynamical quantities associated with stability and phases transitions, we move to extract the corresponding work from the heat energy according to a cycle between two sources (cold/hot) with temperature T C and T H respectively. Then, we make contact with the Carnot cycle defined as a simple cycle described by two isobars and two isochores as taken in [24] and presented in Fig. 5. Exploring the equation (7) and using the link between the cosmological constant and the pressure, we can get the equation of state of the studied black holes. The calculation gives the following equation where the thermodynamical volume V is related to the horizon radius r + via the relation It is possible to extract the work of the cycle. Expressing the volume in terms of the entropy, which will be used to reduce the number of variables in the final expression for the efficiency, the work takes the following form where the subscripts refer to the quantities evaluated at the corners labeled (1,2,3,4). To derive the efficiency, one has to compute the heat quantity. The upper isobar transformation will produce the net inflow of the heat which will be identified with Q H . The latter is expressed as follows This integration do not look nice due to the non-trivial entropy dependence of C p giving non-trivial T dependence. It is recalled that the efficiency is To determine such a quantity, we will use the large value limit of T and P. In this way, the equation (13) n−2 (n − 2) 3 T 6 + · · · In this limit, the heat capacity (11) reduces to which infers the following expression Then, we get At leading order, associated with P ∼ (n 2 −3n+2)TV 1 1−n 4ω + · · · , we can calculate the efficiency. Identifying T C = T 4 and T H = T 2 , corresponding to the lowest and highest temperatures of the engine, we obtain In this way, the efficiency can be compared with the one of the Carnot cycle η Carnot = 1 − T C T H . Again, it is worth noting that for n = 4, we recover the result reported in [24].

Conclusion
In this work, we have investigated the heat properties of AdS black holes in higher dimensions. We have considered the study of corresponding thermodynamical properties along with the sign of the heat capacity explored in the determination of the stability of such black hole. More precisely, we have computed the latent heat as a trivial consequence of the Hawking-Page phase transition. To overcome the instability problem, the Maxwell' construction have been elaborated to modify the Hawking-Page phase structure in the (T, S) − plane.
We have derived the equal area isotherm for any dimension in the range 4 ≤ n ≤ 11. Then, we have analyzed the thermodynamic cycle and the holographic heat engines using the expression of the extracted work and the efficiency.
By following [24], it is possible to make contact with Maldacena conjecture known by AdS/CFT holographic correspondence [18,25]. This could be useful to bring new features in the gauge theories embedded in string theory and related topics.