Extended Phase Space Thermodynamics for Dyonic Black Holes with a Power Maxwell Field
CCTP-SCU/2020027
Extended Phase Space Thermodynamics for Dyonic Black Holeswith a Power Maxwell Field
Feiyu Yao a ∗ and Jun Tao a † a Center for Theoretical Physics, College of Physics,Sichuan University, Chengdu, 610064, China
Abstract
In this paper, we investigate the thermodynamics of dyonic black holes with the presence ofpower Maxwell electromagnetic field in the extended phase space, which includes the cosmologicalconstant Λ as a thermodynamic variable. For a generic power Maxwell black hole with the electriccharge and magnetic charge, the equation of state is given as the function of rescaled temperature˜ T in terms of other rescaled variables ˜ r + , ˜ q and ˜ h , where r + is the horizon radius, q is the electriccharge and h is some magnetic parameter. For some values of ˜ q and ˜ h , the phase structure ofthe black hole is uniquely determined. Moreover the peculiarity of multiple temperature withsome fixed parameter configurations results in more rich phase structures. Focusing on the powerMaxwell Lagrangian with L ( s ) = s , we obtain the corresponding phase diagrams in the ˜ q -˜ h plane,then analyse the black holes phase structure and critical behaviour. For this case, the critical lineis semi-infinite and extends to ˜ h = ∞ . We also examine thermal stabilities of these black holes. ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ g r- q c ] A ug ontents I. Introduction II. Dyonic PM AdS Black Hole III. 8-d Dyonic PM AdS Black Hole with p=2 IV. Discussion and Conclusion V. Acknowledgement References I. INTRODUCTION
Black holes are intriguing concepts from the two cornerstones of modern theoreticalphysics: General Relativity and Quantum Field Theory. Classically black holes absorbed allmatter and emitted nothing. Superficially they had neither temperature nor entropy, andwere characterized by only a few basic parameters: mass, angular momentum, and charge(if any) [1]. However all of this were changed by the advent of quantum field theory incurved spacetime. The first indication linking black holes and thermodynamics came fromHawking’s area theorem [2], which states that the area of the event horizon of a black holecan never decrease. Bekenstein subsequently noticed the resemblance between this area lawand the second law of thermodynamics [3], proposing that each black hole should be as-signed an entropy proportional to the area of its event horizon [4]. Analogous to the laws ofthermodynamics, Bardeen, Carter and Hawking soon established the four laws of black holemechanics [5], where the surface gravity corresponds to the temperature. Since Hawkingdiscovered that black holes do emit radiation with a blackbody spectrum [6], the idea ofblack hole thermodynamics has convinced most physicists. And over the past four decades,a preponderance of evidence suggested that it is a meaningful subject.According the sign of cosmological constant Λ, black holes can be classified into asymp-totically de Sitter (dS) black holes (Λ > <
0) and asymptotically flat black holes (Λ=0). Sufficiently large asymptotically AdS2as compared to the AdS radius l ) black holes, unlike asymptotically flat black holes, havepositive specific heat and can be in stable equilibrium at a fixed temperature [7]. Moreoverasymptotically AdS black holes, unlike asymptotically dS BH [8], only have one horizon andone can define a good notion of the asymptotic mass. So the asymptotically AdS black holesare always popular subjects. Studying the phase transitions of asymptotically AdS blackholes is primarily motivated by AdS/CFT correspondence [9]. Hawking and Page showedthat a first-order phase transition occurs between Schwarzschild AdS black holes and thermalAdS space [7], which was later understood as a confinement/deconfinement phase transitionin the context of the AdS/CFT correspondence [10]. For Reissner-Nordstrom (RN) AdSblack holes, authors of [11, 12] showed that their critical behavior is similar to that of a Vander Waals liquid gas phase transition.Later, the asymptotically AdS black holes were studied in the context of extended phasespace thermodynamics, where the cosmological constant is interpreted as thermodynamicpressure [13, 14]. In this case, the black hole mass should be understood as enthalpy insteadof the internal energy; the first law was modified [15]. The P - V criticality study has beenexplored for various AdS black holes [16–21]. It showed that the P - V critical behaviors ofAdS black holes are similar to that of a Van der Waals liquid gas system.Nonlinear electrodynamics (NLED) is an effective model incorporating quantum correc-tions to Maxwell electromagnetic theory. Coupling NLED to gravity, various NLED chargedblack holes were derived and discussed in a number of papers [22–31]. The thermodynamicsof generic NLED black holes in the extended phase space have been considered in [32–36].And various particular NLED black holes were also considered, e.g., Born-Infeld AdS blackholes [37, 38], power Maxwell invariant black holes [39–41], nonlinear magnetic-charged dSblack holes [42].Among the various NLED, straightforward generalization of Maxwell’s theory leads to theso called power Maxwell (PM) theory described by a Lagrangian density of the form L ( s ) = s p , where s is the Maxwell invariant, and p is an arbitrary rational number. Clearly thespecial value p = 1 corresponds to linear electrodynamics. The solutions of PM charged blackholes and their interesting thermodynamics and geometric properties have been examinedin [26, 43–50].The PM theory is a toy model to generalize Maxwell theory which reduces to it for p = 1. One of the most important properties of the PM model in d -dimensions occurs3hen p = d/ , where the PM theory becomes conformally invariant and the trace of energy-momentum tensor vanishes, such as Maxwell theory in four-dimensions. On the other hand,taking into account the applications of the AdS/CFT correspondence to superconductivity,it has been shown that the PM theory makes crucial effects on the condensation as well asthe critical temperature of the superconductor and its energy gap [51].A substantial gap in these studies is the absence of dyonic solution. Authors of [52]derived a scheme of finding dyonic solution in NLED coupled to GR by quadratures foran arbitrary Lagrangian function L ( s ) and a dyonic solution for the truncated Born-Infeldtheory. However there are still not many papers devoted to studies of specific cases withdyonic solution.In this paper, We first investigate the thermodynamic behavior of generic d -dimensionaldyonic PM black holes in the extended phase space. Then, we turn to study the phase struc-ture and critical behavior of 8-dimensional dyonic PM black holes with an power exponentof 2 by studying the phase diagrams in the q/l . - h/l . plane. After this Introduction, wederive d -dimensional dyonic PM black hole solutions and discuss thermodynamic proper-ties of the black hole in section II. In section III, we study the phase structure and criticalbehavior of 8-dimensional dyonic PM AdS black holes with an power exponent of 2. Thephase diagram for the black hole in the q/l . - h/l . plane is given in FIG. 5, from whichone can read the black hole’s phase structure and critical behavior. We also explore thermalstabilities of these black holes. We summarize our results in section IV. We will use theunits (cid:126) = c = 16 πG = 1 for simplicity. II. DYONIC PM ADS BLACK HOLE
In this section, we derive the d -dimensional dyonic PM asymptotically AdS black holesolution in the Einstein gravity and verify the thermodynamic properties of the black hole.We first consider a d -dimensional model of gravity coupled to a PM nonlinear electromagneticfield with action given by S Bulk = (cid:90) d d x √− g [ R −
2Λ + L ( s )] , (1)where Λ = − ( d −
1) ( d − l (2)4s cosmological constant, s = 14 F µν F µν (3)is the maxwell invarient, F = dA = ∂ µ A ν − ∂ ν A µ and A µ is the gauge potential. In our casethe Lagrangian density has the following form L ( s ) = s p . (4)Taking the variation of the action (1) with respect to g µν and A µ , one can get the equationsof motion, they are R µν − Rg µν − ( d −
2) ( d − l g µν = T µν ∂ µ (cid:0) √− gG µν (cid:1) = 0, (6)where T µν = g µν L ( s ) + ∂ L ( s ) ∂s F ρµ F νρ (7)is energy-momentum tensor and we defined the auxiliary field G µν = ∂ L ( s ) ∂s F µν . (8)To construct a dyonic black hole solution in asymptotically AdS spacetime, we take thefollowing ansatz for the metric and the gauge pential ds = − f ( r ) dt + 1 f ( r ) dr + r d Ω d − , (9) A = A t ( r ) dt − h (cid:32) d − (cid:89) i =1 sin θ i (cid:33) cos θ d − dθ d − , (10)where d Ω d − is the metric of ( d − − sphere (only consider the case of positive constantcurvature, i.e. k = 1), d Ω = dθ , (11) d Ω n +1 = d Ω n + (cid:32) n (cid:89) i =1 sin θ i (cid:33) dθ n +1 . (12)Then the equations of motion read( d − rf ( r ) (cid:48) + ( d −
2) ( d − f ( r ) − − ( d −
2) ( d − r l = r (cid:2) L ( s ) + G rt A (cid:48) t ( r ) (cid:3) , (13) ∂ r (cid:0) r d − G rt (cid:1) = 0, (14) ∂ θ d − (cid:0) sin θ d − G θ d − θ d − (cid:1) = 0, (15)5nd plugging eqn. (10) into eqn. (3) and eqn. (8) results in s = A (cid:48) t ( r )2 − h r and G rt = − ∂ L ( s ) ∂s A (cid:48) t ( r ) . (16)Eqn. (15) can result in ∂ θ d − h = 0 and the rest equations of motion can be derived fromeqn. (13) and eqn. (14) . Now A (cid:48) t ( r ) can be determined by eqn. (14), which leads to ∂ L ( s ) ∂s A (cid:48) t ( r ) = qr d − , (17)where q is a constant. Moreover integrating eqn. (13), we have f ( r ) = 1 + r l − mr d − − d − r d − (cid:90) ∞ r drr d − (cid:34) L (cid:32) A (cid:48) t ( r )2 − h r (cid:33) − qr d − A (cid:48) t ( r ) (cid:35) , (18)where m is a constant. At the horizon r = r + , the Hawking temperature of the black holeis given by T = f (cid:48) ( r + )4 π , (19)so one can have T = 14 πr + (cid:40) d − d − r l + 1 d − r (cid:34) L (cid:32) A (cid:48) t ( r )2 − h r (cid:33) − qr d − A (cid:48) t ( r + ) (cid:35)(cid:41) , (20)which results from plugging f ( r + ) = 0 into eqn. (13) . Then the electric charge is [53] Q = (cid:90) S ¯ F = (cid:90) (cid:32) d − (cid:89) dθ i (cid:33) ¯ F = (cid:90) (cid:32) d − (cid:89) i =1 dθ i (cid:33) √− g qr d − = ω d − q, (21)where ¯ F = ∂ L ∂s ( ∗ F ) , (22)with ω d − being the volume of the unit ( d − ω d − = 2 π d − Γ (cid:0) d − (cid:1) . (23)Moreover, the mass can be extracted by comparison to a reference background, e.g.,vacuum AdS. So the mass can be determined by the Komar integral M = d − π ( d − (cid:90) ∂ Σ dx d − (cid:112) γ (cid:48) ( σ µ n ν ∇ µ K ν ) − M AdS , (24)6here K µ is the Killing vector associated with t , and M AdS is Komar integral associatedwith K µ for vacuum AdS space M AdS = d − π ( d − (cid:90) ∂ Σ dx d − (cid:112) γ (cid:48) (cid:16) rl (cid:17) , (25)and γ (cid:48) is the induced metric of ∂ Σ, which is the boundary of Σ. σ µ is the unit normal vectorof Σ and n µ is the unit outward-pointing normal vector. Setting Σ and ∂ Σ are a constant- t hypersurface and a ( d − r = ∞ .Using σ µ = ( − f , , , ...... ) , (26) n µ = (0 , f − , , ...... ), (27)one can have σ µ n ν ∇ µ K ν = 12 f (cid:48) ( r ) . (28)It is shown that in some case, which is d − < p < d −
2, (29)one can have 12 f (cid:48) ( r ) = rl + ( d − m r d − + O ( r − d ) (30)at spatial infinity, whether it hold or not is determined by the relationship of power exponent p and dimension d . When it hold, we have M = d − π ω d − m. (31)In the following, we study the thermodynamics of the dyonic PM AdS black hole solutionin the extended phase space, where the cosmological constant is interpreted as thermody-namic pressure and treated as a thermodynamic variable in its own right. The mass of theblack hole is no longer regarded as internal energy, it is identified with the chemical enthalpy.In terms of the horizon radius r + , the mass can be rewritten as M = d − π ω d − (cid:26) r d − + r d − l − d − (cid:90) ∞ r + drr d − (cid:20) L (cid:18) A (cid:48) t ( r )2 − h r (cid:19) − A (cid:48) t ( r ) qr d − (cid:21)(cid:27) (32)7here we have used eqn. (31).Adding that the Gibbs free energy F can be expressed by the Euclidean action S E [32]: F = M − T S, (33)where the entropy of the black hole is one-quarter of the horizon area S = r d − ω d − . (34)We have expressed thermodynamics quantities ( F, M and S ) as the functions of the horizonradius r + , q (proportional to the electric charge Q ), h (associated with magnetic charge)and the AdS radius l (the pressure P = ( d −
1) ( d − /l ). Now we need to express thethermodynamics quantities in terms of T , q , h and P by solving the equation of state for r + = r + ( T, q, l, h ) . (35)So we first rescale the T , which becomes˜ T = 14 π ˜ r + (cid:40) d − d −
1) ˜ r + ˜ r d − (cid:34)(cid:32) ˜ A (cid:48) t ( r + )2 − ˜ h r (cid:33) p − ˜ A (cid:48) t ( r + ) ˜ q ˜ r d − (cid:35)(cid:41) , (36)where ˜ r + = r + l − , ˜ q = ql − p − d +4 , ˜ A (cid:48) t ( r + ) = l p A (cid:48) t ( r + ) , ˜ h = hl p − , ˜ T = T l, (37)and p is the power of L ( s ) = s p . (38)Then ˜ A (cid:48) t ( r + ) is determined by (cid:32) p ˜ A (cid:48) t ( r + )2 − ˜ h r (cid:33) p − ˜ A (cid:48) t ( r + ) = ˜ q ˜ r d − , (39)which usually cannot be solved analytically and has multiple solutions when p is large. Afterthat, solving eqn. (36), ˜ r + can be expressed as a function of ˜ T , ˜ q and ˜ h : ˜ r + = ˜ r + ( ˜ T , ˜ q, ˜ h ).With ˜ r + = ˜ r + ( ˜ T , ˜ q, ˜ h i ), one can express the thermodynamic quantities in terms of ˜ T , ˜ q and˜ h , e.g., the Gibbs free energy is given by˜ F ≡ F/l d − = ˜ F ( ˜ T , ˜ q, ˜ h ) . (40)The rich phase structure of the black hole comes from solving eqn. (36), i.e., ˜ T =˜ T (˜ r + , ˜ q, ˜ h ), for ˜ r + . If ˜ T (˜ r + , ˜ q, ˜ h ) is a monotonic function with respect to ˜ r + for some values8 mall BHLarge BH r / lTl Small BHLarge BH
Tlr / l Small BHLarge BH
TlF / l d - (a) Branches around a local minimum of ˜ T = ˜ T min . Small BHLarge BHSmaller BHLarger BH r / lTl Small BHLarge BHSmaller BHLarger BH
Tlr / l Small BHLarge BHSmaller BHLarger BH
TlF / l d - (b) Branches around a local maximum of ˜ T = ˜ T max when ˜ T = ˜ T (˜ r + ; ˜ q, ˜ h ) is multivalued. FIG. 1:
Branches of black holes around local extremums of ˜ T = ˜ T min . Right panels: Gibbs free energy vstemperature. The blue branches are thermodynamically preferred and thermally stable. The red ones arethermally unstable. Below column: the case of more than one ˜ T (˜ r + , ˜ q, ˜ h ) with some fixed ˜ q and ˜ h. of ˜ q and ˜ h , there would be only one branch for the black hole. More often, with fixed ˜ q and˜ h , there exists a local minimum/maximum for ˜ T (˜ r + , ˜ q, ˜ h ) at ˜ r + = ˜ r + , min / ˜ r + = ˜ r + , max . In thiscase, there are more than one branch for the black hole. In FIG. 1(a), we plot two branches,namely small BH and large BH, around a local minimum of ˜ T = ˜ T min . The Gibbs free energyof these two branches is displayed in the right panel of FIG. 1(a). Since ∂ ˜ F ( ˜ T , ˜ q, ˜ h ) /∂ ˜ T = − π ˜ r , the upper branch is small BH while the lower one is large BH, which means thatthe large BH branch is thermodynamically preferred. Similarly, there are also two branchesaround a local maximum of ˜ T . In this case the upper/lower branch is large/small BH sinceit has more/less negative slope and the small BH branch is thermodynamically preferred inthis case. In general, one might need to figure out how the existence of local extremumsdepends on values of ˜ Q and ˜ h to study the phase structure of the black hole.Moreover, since there are more than one solution by solving the eqn. (39) for some valuesof ˜ q and ˜ h , there are more than one ˜ T (˜ r + , ˜ q, ˜ h ) with some fixed ˜ q and ˜ h , which means moreone set of ˜ r + i = ˜ r + i ( ˜ T , ˜ q, ˜ h ). And every set of ˜ r + i ( ˜ T , ˜ q, ˜ h ) maybe still have many branches.In FIG. 1(b), we show a possible simple case, every panel is corresponding to those of FIG.9(a). And of couse it could be more complicated.After the black hole’s branches are obtained, it is easy to check their thermodynamicstabilities against thermal fluctuations. The thermal stability of the branch follows from thespecific heat C >
0. The specific heat we need is C Q,h,P = T (cid:18) ∂S∂T (cid:19) Q,h,P = l d − ( d −
2) ˜ T ˜ r d − ω d − (cid:18) ∂ ˜ r + ∂ ˜ T (cid:19) , (41)since ω d − is also positive, the sign of ˜ T (cid:48) (˜ r + ) determines the thermodynamic stabilities. III. 8-D DYONIC PM ADS BLACK HOLE WITH P=2
In this section, we focus on a specific example. For simplicity, we consider the case thatthe power exponent is 2, and we consider 8 − d spacetime in order to satisfying the conditionof eqn. (29).When d = 8, p = 2 the mass becomes M = 25 π (cid:26) r + r l − (cid:90) ∞ r + drr (cid:20) L (cid:18) A (cid:48) t ( r )2 − h r (cid:19) − A (cid:48) t ( r ) qr (cid:21)(cid:27) (42)and the electric charge Q = 16 π q, (43)where we have used eqn. (32) and eqn. (21) . And the eqn. (36) and eqn. (34) become˜ T = 14 π ˜ r + (cid:40) r + ˜ r (cid:34) L (cid:32) ˜ A (cid:48) t ( r + )2 − ˜ h r (cid:33) − ˜ A (cid:48) t ( r + ) ˜ q ˜ r (cid:35)(cid:41) , (44) S = 415 π r , (45)where ˜ r + = r + l − , ˜ q = ql − . , ˜ A (cid:48) t ( r + ) = l . A (cid:48) t ( r + ) , ˜ h = hl − . , ˜ T = T l. (46)Moreover, the Gibbs free energy is given by F = M − T S (47)and ˜ F ≡ F/l = ˜ F ( ˜ T , ˜ q, ˜ h ) . (48)10he case of p = 2 is described by the Lagrangian density L ( s ) = s , (49)and solving eqn. (39) for ˜ A (cid:48) t ( r ) gives˜ A (cid:48) ti ( r + ) = 2˜ h √ r cos (cid:34)
13 arccos (cid:32) √ q h (cid:33) − π i − (cid:35) , i = 1 , ,
3, (50)where ˜ A (cid:48) t ( r + ) and ˜ A (cid:48) t ( r + ) exist only if 3 √ q/ h ≤ h ˜ q >
0; ˜ A (cid:48) t ( r + ) and ˜ A (cid:48) t ( r + )exist only if 3 √ q/ h ≥ − h ˜ q < . The equations of state (36) become˜ T i = 14 π ˜ r + r + ˜ r (cid:32) ˜ A (cid:48) ti ( r + )2 − ˜ h r (cid:33) − A (cid:48) ti ( r + ) ˜ q ˜ r (51)= 14 π ˜ r + r + ˜ r (cid:32) h r C i (cid:16) ˜ h, ˜ q (cid:17) − ˜ h r (cid:33) − h ˜ q √ r C i (cid:16) ˜ h, ˜ q (cid:17) (52)= 14 π ˜ r + (cid:34) r + ˜ h r D i (cid:16) ˜ h, ˜ q (cid:17)(cid:35) , (53)and we have defined two functions that are independent of ˜ r + for later use C i (cid:16) ˜ h, ˜ q (cid:17) = cos (cid:34)
13 arccos (cid:32) √ q h (cid:33) − π i − (cid:35) ( i = 1 , , D i (cid:16) ˜ h, ˜ q (cid:17) = C i (cid:16) ˜ h, ˜ q (cid:17) − − C i (cid:16) ˜ h, ˜ q (cid:17) q √ h . (55)Setting 3 √ q/ h ≡ x , we have C i ( x ) ≡ cos (cid:20)
13 arccos x − π i − (cid:21) ( i = 1 , , , (56) D i ( x ) ≡ (cid:40)(cid:20)(cid:18) C i ( x )3 − (cid:19) (cid:21) − xC i ( x ) 49 (cid:41) , (57)and we can see that ˜ h D i ( x ) completely determines the dependence of ˜ T i (˜ r + ) on ˜ r ,˜ T i (cid:16) ˜ r + ; ˜ h, x (cid:17) = 14 π ˜ r + (cid:34) r + ˜ h r D i ( x ) (cid:35) , (58)11 ( x ) C2 ( x ) C3 ( x )- - - - (a) C ( x ) D1 ( x ) D2 ( x ) D3 ( x )- - - - - - (b) D ( x ) FIG. 2: C ( x ) and D ( x ), where x ≡ √ q h . which also tells us that three curves of ˜ T i (cid:16) ˜ r + ; ˜ h, x (cid:17) are never intersect with fixed ˜ h, x .Moreover, we can write the rescaled Gibbs free energy as˜ F = 12 π ˜ r π ˜ r − π ˜ r ˜ T , (59)and it is also easy to see that three curves of F (cid:16) ˜ T i ; ˜ h, x (cid:17) are never intersect with fixed ˜ h, x .To study the behavior of local extremums of ˜ T i (˜ r + ), we consider the equation ˜ T (cid:48)(cid:48) i (˜ r + ) = 0,which becomes d ˜ T i d ˜ r = 104 π ˜ r + 7˜ h π ˜ r D i (cid:16) ˜ h, ˜ q (cid:17) = 0 . (60)For ˜ h ˜ q >
0, since D (cid:16) ˜ h, ˜ q (cid:17) > D (cid:16) ˜ h, ˜ q (cid:17) > , which is shown in FIG. 2(b), only˜ T (cid:48)(cid:48) (˜ r + ) = 0 has a root and only has one root. And when ˜ h ˜ q <
0, only ˜ T (cid:48)(cid:48) (˜ r + ) = 0 has andonly has a root. From FIG. 2(b), we can find D ( x ) is symmetry with D ( x ), which means˜ T (˜ r + ) / ˜ T (˜ r + ) for ˜ h ˜ q > T (˜ r + ) / ˜ T (˜ r + ) for ˜ h ˜ q <
0. So we only discuss thecase of ˜ q >
0, ˜ h > T (cid:48)(cid:48) i (˜ r + ) = 0, it is easy to analyzethe existence of the local extremums of ˜ T (cid:48) (˜ r + ), results of which are summarized in Table I.When solving eqn. (53) for ˜ r + in terms of ˜ T , the solution ˜ r + ( ˜ T ) is often a multivaluedfunction like what is shown in FIG. 1(b). The parameters ˜ h and ˜ q determine the number ofthe branches of ˜ r + ( ˜ T ) and the phase structure of the black hole. In what follows, we findfour regions in the ˜ h -˜ q plane if we only consider the number of branches of ˜ r + ( ˜ T ), takinginto account for phase transition, Region I has five subregions. Each region has the distinctbehavior of the branches and the phase structure:12 mallest BHIntermediate BHLargest BHSmall BHLarge BHSmaller BHLarger BH r + / l (a) Region I: h/l . = 1 and q/l . = 0 . / l (b) Region I5: h/l . = 0 . q/l . = 0 .
02. There is a firstorder phase transition between Smallest BH and Large BH. - - - - - / l (c) Region I4: h/l . = 1 and q/l . = 0 . - - - - - / l (d) Region I3: h/l . = 1 and q/l . = 0 .
02. There is a firstorder phase transition between Smallest BH and Larger BHand a zeroth order phase transition between Larger BH andLarge BH. - - - - - / l (e) Region I2: h/l . = 1 and q/l . = 0 .
05. There is azeroth order phase transition between Smallest BH andLarger BH and a zeroth order phase transition betweenLarger BH and Large BH. - - - - - / l (f) Region I1: h/l . = 1 and q/l . = 0 .
1. There is a firstorder phase transition between Smallest BH and Largest BH,a zeroth order phase transition between Largest BH andLarger BH and a zeroth order phase transition betweenLarger BH and Large BH.
FIG. 3:
Plot of ˜ r + , ˜ F against ˜ T for PL-AdS black holes in Region I. the first of this is ˜ r + − ˜ T of RegionsI. The shape of ˜ r + − ˜ T is same in these five subregions. h > , ˜ q > T (cid:48) (0) ˜ T (cid:48) (+ ∞ ) Solution of ˜ T (cid:48)(cid:48) (˜ r + ) = 0 Extremums of ˜ T (cid:48) (˜ r + ) T , ˜ h D (cid:16) ˜ h, ˜ q (cid:17) > − (cid:0) (cid:1) ∞ π ˜ r > T (cid:48) min (˜ r ) < T , ˜ h D (cid:16) ˜ h, ˜ q (cid:17) < − (cid:0) (cid:1) ∞ π ˜ r > T (cid:48) min (˜ r ) > T , x < −∞ π None, ˜ T (cid:48)(cid:48) (˜ r + ) > T , x < −∞ π None, ˜ T (cid:48)(cid:48) (˜ r + ) > T (cid:48)(cid:48) i (˜ r + ) = 0 and the local extremums of ˜ T (cid:48) i (˜ r + ) in various cases. • Region I: x < T (cid:48) (˜ r ) <
0, where ˜ r is the solution of ˜ T (cid:48)(cid:48) (˜ r + ) = 0. In this region,˜ T (cid:48) (˜ r + ) = 0 has two solutions ˜ r + = ˜ r max and ˜ r + = ˜ r min with ˜ r max < ˜ r < ˜ r min . Since˜ T (+ ∞ ) = + ∞ , ˜ T (˜ r + ) has a local maximum of ˜ T = ˜ T (˜ r max ) at ˜ r + = ˜ r max and alocal minimum of ˜ T = ˜ T (˜ r min ) at ˜ r + = ˜ r min . There are three branches for ˜ r +1 ( ˜ T ):smallest BH for 0 ≤ ˜ T ≤ ˜ T , intermediate BH for ˜ T ≤ ˜ T ≤ ˜ T and largestBH for ˜ T ≥ ˜ T , there are two branches for ˜ r +2 ( ˜ T ): small BH for 0 ≤ ˜ T ≤ ˜ T and large BH for ˜ T ≥ ˜ T , and there are also two branches for ˜ r +3 ( ˜ T ): smaller BHfor 0 ≤ ˜ T ≤ ˜ T and larger BH for ˜ T ≥ ˜ T . The dependence of ˜ r + to T isdisplayed in the left panel of FIG. 3(a). The Gibbs free energy of the three branchesis plotted in the follow five panels for different subregions. The smallest BH, largeBH, larger BH and largest BH branches are thermally stable. (When x = 1 , there arethree subregions, and they are similar with the case I5, I2, I1 respectivly, except ˜ T and ˜ T merge into one.) • Region II: x < T (cid:48) (˜ r ) ≥
0. In this region, ˜ T (cid:48) (˜ r + ) ≥ ˜ T (cid:48) (˜ r ) ≥ T (˜ r + )is an increasing function. So there is only one branch for ˜ r +1 ( ˜ T ): largest BH, whichis thermally stable, there are two branches for ˜ r +2 ( ˜ T ): small BH for 0 ≤ ˜ T ≤ ˜ T and large BH for ˜ T ≥ ˜ T , and there are two branches for ˜ r +3 ( ˜ T ): smaller BH for0 ≤ ˜ T ≤ ˜ T and larger BH for ˜ T ≥ ˜ T , and these are displayed in the left panelof FIG. 4(a). The Gibbs free energy of the three branches is plotted in the right panel.The large BH, larger BH and largest BH branches are thermally stable. (If x = 1 , itis similar except ˜ T and ˜ T merge into one.) • Region III: x > T (cid:48) (˜ r ) <
0. In this region, ˜ T (cid:48) (˜ r + ) = 0 has two solutions˜ r + = ˜ r max and ˜ r min with ˜ r max < ˜ r < ˜ r min . Since ˜ T (+ ∞ ) = + ∞ , ˜ T (˜ r + ) has a local14 argest BHSmall BHLarge BHSmaller BHLarger BH r + / l 0.0425 0.0430 0.0435 0.0440 0.0445 0.0450Tl - - - - - / l (a) Region II: h/l . = 1 and q/l . = 0 .
2. There is a zeroth order phase transition between Largest BH and Larger BH and azeroth order phase transition between Larger BH and Large BH.
Smallest BHIntermediate BHLargest BH r + / l 0.041 0.042 0.043 0.044 0.045 0.046Tl - / l (b) Region III: h/l . = 0 . q/l . = 0 .
1. There is a first order phase transition between Smallest BH and Largest BH.
Largest BH r + / l 0.042 0.043 0.044 0.045 0.046 Tl - - - - - / l (c) Region IV: h/l . = 0 . q/l . = 0 .
3. There is no phase transition.
FIG. 4:
Plot of ˜ r + , ˜ F against ˜ T for PL-AdS black holes in RegionsII, III and IV. The number of branchsis different in these regions. The intermediate BH, small BH and smaller are always thermally unstable,others are always thermally stable. T = ˜ T (˜ r max ) at ˜ r + = ˜ r max and a local minimum of ˜ T = ˜ T (˜ r min )at ˜ r + = ˜ r min . There are three branches for ˜ r +1 ( ˜ T ): smallest BH for 0 ≤ ˜ T ≤ ˜ T ,intermediate BH for ˜ T ≤ ˜ T ≤ ˜ T and largest BH for ˜ T ≥ ˜ T . ˜ r +2 ( ˜ T ) and˜ r +3 ( ˜ T ) don’t exist, so there is a first order phase transition. These are displayed inthe left panel of FIG. 4(b). The Gibbs free energy of the three branches is plotted inthe right panel. The smallest BH and largest BH branches are thermally stable. • Region IV: x > T (cid:48) (˜ r ) ≥
0. In this region, ˜ T (cid:48) (˜ r + ) ≥ ˜ T (cid:48) (˜ r ) ≥ T (˜ r + ) is an increasing function. So there is only one branch for ˜ r +1 ( ˜ T ): largestBH, which is thermally stable, ˜ r +2 ( ˜ T ) and ˜ r +3 ( ˜ T ) don’t exist, so there is no phasetransition in this region. These are displayed in the left panel of FIG. 4(c). The Gibbsfree energy is plotted in the right panel.We have marked first order phase transition with black point and zeroth order phasetransition with arrow. In FIG. 5, we plot these eight regions in the ˜ h -˜ q plane.We can now discuss the critical behavior and phase structure of black holes in two cases.The critical line is the boundary between the region in which ˜ T (˜ r + ) has n extremums andthat in which ˜ T (˜ r + ) has n + 2 extremums, determined by ∂ ˜ T (˜ r + , ˜ Q, ˜ a ) ∂ ˜ r + = 0 and ∂ ˜ T (˜ r + , ˜ Q, ˜ a ) ∂ ˜ r = 0 . (61)In the first case, q and h are fixed parameters, and the AdS radius l (the pressure P )varies. With fixed values of q and h , varying l would generate a curve in the ˜ h -˜ q plane,which is determined by ˜ q l (cid:16) ˜ h (cid:17) = qh ˜ h . (62)In FIG. 6, we plot ˜ q l (cid:16) ˜ h (cid:17) for various values of q/h . It shows that, there is always criticalpoint for black holes. For q/h > √ /
9, as one starts from P = 0, ˜ q l (cid:16) ˜ h (cid:17) always crossesthe critical line and enters Region IV, in which there is no phase transition, from RegionIII, in which there is a first phase transition between smallest BH and largest BH. For q/h < √ /
9, as one starts from P = 0, ˜ q l (cid:16) ˜ h (cid:17) always passes through five subregions, thencrosses the critical line and enters Region II, in which there are two zeroth phase transitions,from largest BH to larger BH and from larger BH to large BH.In the second case, h and P ( l ) are fixed parameters, and one varies ˜ q . As one increases ˜ q from ˜ q = 0, the black hole would experience different regions. And there is always a critical16 IIIIIIV0.0 0.5 1.0 1.5 2.00.00.10.20.30.4 h / l q / l . (a) The four main regions in the ˜ h -˜ q plane, each of whichpossesses the distinct behavior of the branches and the phasestructure. I1I2I3I4I50.0 0.5 1.0 1.5 2.00.000.050.100.150.20 h / l q / l . (b) The five subregions of Region I in the ˜ h -˜ q plane, each ofwhich possesses the distinct phase structure. FIG. 5:
The eight regions in the ˜ h -˜ q plane for dyonic PMI AdS black holes. The color represents for thebranch which it enters at first phase transition with increasing ˜ T from 0 (except Region IV), the darker colorrepresents for zeroth order phase transition, the lighter color represents for first order phase transition. point for any ˜ h since the critical line only has one end point at ˜ h = 0. In FIG 7, I plot thatin the case of ˜ h = 0 .
5, ˜ h = 0 .
6, ˜ h = 0 . h = 1.For ˜ h ≤ ˜ h (cid:39) .
51, as one increases ˜ q from ˜ q = 0, the black hole would experience threeregions, in which there occur the smallest BH/large BH first order phase transition → thesmallest BH/largest BH first order phase transition → no phase transition, showed in FIG.7(a).For ˜ h ≥ ˜ h , there is another region to across between the two regions mentioned above,in which there occurs the smallest BH/large BH zeroth order phase transition → a small-est BH/larger BH first order phase transition and a larger BH/large BH zeroth phasetransition → a smallest BH/larger BH zeroth order phase transition and a larger BH/largeBH zeroth phase transition. These are showed in FIG. 7(b).When ˜ h ≥ ˜ h (cid:39) .
63, the range of region that has a smallest BH/largest BH first orderphase transition will expand to the left, which is showed in FIG. 7(c).17 / h = / h = / h = / h = / h = IIIIIIIV0.0 0.5 1.0 1.5 2.00.00.10.20.30.4 h / l q / l . I1I2I3I4I5 q / h = / h = / h = / h = / h = / l q / l . FIG. 6:
In the case of varying P with fixed q and h , the system moves along ˜ q l (cid:16) ˜ h (cid:17) , which is displayed forvarious values of q/h . There is always a critical point and the corresponding largest BH/smallest BH firstorder phase transition. When ˜ h ≥ ˜ h (cid:39) .
76, the smallest BH/large BH first order phase transition at leftdisappears, showed in FIG. 7(a) and the smallest BH/large BH zeroth order phase transitiondisappears when ˜ h ≥ ˜ h (cid:39) .
14, it is reminiscent of FIG. 7(d), just cutting the Red linebefore Blue line at left.
IV. DISCUSSION AND CONCLUSION
We have investigated the thermodynamic behavior of d -dimensional dyonic PM AdS blackholes in an extended phase space, which includes the conjugate pressure/volume quantities.It showed that the black hole’s temperature T , charge q , horizon radius r + (thermodynamicvolume V ), the AdS radius l (pressure P ) and the magnetic parameter h could be connectedby T l = ˜ T (cid:0) r + /l, q/l a , h/l b (cid:1) , (63)where a and b depend on the dimension d and the power exponent p . In the canonicalensemble with fixed T and q , we found that the critical behavior and phase structure of theblack hole are determined by ˜ q ≡ q/l a and ˜ h ≡ h/l b .18 mallest BH Largest BHLarge BH (a) h/l . = 0 . LargerBH Smallest BHLargest BHSmallest BHLarge BH / l (b) h/l . = 0 . Large BHLarger BHSmallest BH Largest BH / l (c) h/l . = 0 . Largest BHSmallest BHLarger BHLarge BH / l (d) h/l . = 1 FIG. 7:
The phase diagram in the ˜ q - ˜ T plane for PMI-AdS black holes with h/l . = 0 . h/l . = 0 . h/l . =0 . h/l . =1 . The Blue line (point) represent for first order phase transition and the Red line (point) represent for zeroth order transition. The Black point represent for the critical point.
For 8-dimensional PM dyonic AdS black holes with an power exponent of 2, we examinedtheir critical behavior and phase structure, whose dependence on ˜ q and ˜ h was plotted inFIG. 5. There are 8 regions in FIG. 5, and each region has a different phase behavior. Unlikeother black holes, the temperature ˜ T ( r + /l, q/l . , h/l . ) of PM dyonic AdS black holes couldbe more than one when some parameter configurations of q/l . and h/l . vary. If we discussthem separately, one of them ( ˜ T in this case) likes the case of RN-AdS black holes and theother two (if exist) like the case of Schwarzschid-AdS black holes. The combination of themresults in these rich phase structures and phase behaviors.The thermodynamically preferred phases, along with the zeroth and first order phasetransitions and critical points, were displayed in FIG. 7 for the black holes. We examinedthermal stabilities of the black holes and found that all the thermodynamically preferred19hases are thermally stable. V. ACKNOWLEDGEMENT
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