Extended phase space thermodynamics for AdS black holes

Transcription

1 Extended phase space thermodynamics for AdS black holes Liu Zhao School of Physics, Nankai University Nov based on works with Wei Xu and Hao Xu arxiv: [EPJC (2014) 74:2970] arxiv: [EPJC (2014) 74:3074] arxiv: [PLB 736 (2014) ] Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 1 (2014) / 85

2 1 Introduction 2 P V criticality in AdS and van der Waals analogy 3 Gauss-Bonnet black holes 4 P V criticality for Lovelock black holes 5 P V criticality in conformal gravity 6 Summary of open problems Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 2 (2014) / 85

3 Introduction Some ancient feelings and key features of gravity: Gravity acts universally on all objects (matter and energy); Gravity manifests itself only in macroscopic configurations (gravity caused by an isolated single particle with only a few degrees of freedom has not been observed yet); The behavior of gravity resembles that of a macroscopic system very much (either in the bulk through black hole thermodynamics, or on the boundary via holographic duality). Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 3 (2014) / 85

4 Introduction Some reasons that lie behind the above observations: Gravity couples to matter only kinematically through actions of the form d n x gg µν µ Φ ν Φ +, which makes gravity distinguished from all other interactions which couples through potential energy. This seems to be the reason why gravity is universal whilst other interactions are not; The Einstein equation, G µν = κt µν, and most of its generalizations, imply that gravity ignores the microscopic details and only remembers the macroscopic behaviors of the source, because all gravitational field equations of the above kind contains implicitly the relativistic fluid equation µ T µν = 0; Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 4 (2014) / 85

5 Introduction Thus studying the distinguished features of concrete sources does not provide us with more understandings about gravity. Rather, studying macroscopic properties such as thermodynamics and/or fluid mechanics might do the job; In the light of such reasonings, it is understandable why current researches on gravity focuses on subjects like holographic phase transitions, gravity/fluid duality and black hole thermodynamics; I also feel some sympathy about the idea of emergent paradigm for gravity (...) Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 5 (2014) / 85

6 Introduction One big difference between thermodynamics and fluid mechanics is that the former does not contain time (thermodynamic equilibrium = static?), while the latter involves nontrivial time evolution (and especially breaks time reversal); Why the same gravitational theory captures the features of these two drastically different aspects of macroscopic physics? Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 6 (2014) / 85

7 Introduction Essentially Einstein gravity is timeless (look at the ADM construction for the Hamiltonian formalism for Einstein gravity: S = (NH + N i P i ), the action is a sum of Hamiltonian and momentum constraints, so there is no generator for time evolution); On the other hand, Einstein gravity is diffeomorphism invariant (which induces boosts in the tangent space). Boosts in the tangent space introduces time dependence and breaks time reversal. This information is remembered in the dual picture which gives time evolution in the dual fluid. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 7 (2014) / 85

8 Introduction The rest of this talk concentrate on the thermodynamic aspect of gravity. For black holes in asymptotically flat backgrounds, four classic laws of thermodynamics founds their analogues: Thermodynamics Black holes Zero-th law: T = const. κ = const. First law: de = T ds pdv + µdn δm = κ 8π δa + ΩδJ + V δq Second law: δs 0 δa 0 Third law: T 0 impossible κ 0 impossible Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 8 (2014) / 85

9 Introduction For static (J = 0) neutral (Q = 0) black holes, the first law is quite uninteresting: δm = κ 8π δa, only a single extensive variable A is present and there can be no analogues of nontrivial thermodynamical behavior such as phase transitions. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 9 (2014) / 85

10 Introduction Things become more interesting when black hole in asymptotically non-flat, especially in AdS background is under consideration: The AdS background provide a natural confining box which makes it possible to form a thermal equilibrium between a stable large black hole and a thermal gas (Hawking-Page transition, CMP 87 (1983)); On the other hand, the cosmological constant Λ can be regarded as a thermodynamic pressure (P Λ 8π ), and together with its conjugate volume V, the thermodynamic phase space for AdS black holes can be extended and richer thermodynamic behavior is naturally expected (P V criticality). [Kastor, Ray, Traschen ( ); B. Dolan ( )] Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 10 (2014) / 85

11 Introduction It is surprising and feels unnatural to think of Λ as a free thermodynamic variable, because Λ is a constant which appear explicitly in the action, and changing Λ feels like changing the theory. However, one can think of the classical theory of gravity as an effective theory that follows from a fundamental theory in which all the presently physical constants are actually moduli parameters and can run from place to place in the moduli space of the fundamental theory. In this view point, one can include as many as possible new thermodynamic variables in the first law of black hole dynamics and study the corresponding critical behavior. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 11 (2014) / 85

12 Introduction In ordinary thermodynamics, the thermodynamic potential (e.g. the enthalpy H) must be a homogeneous function of all extensive variables E a in the system, H = H(E a ), so that when the size of the system is scaled by a factor λ, we have λ d H H = H(λ d Ea Ea ), where d H and d Ea are the scaling dimensions for H and E a, respectively. Taking the first derivative with respect to λ and setting λ = 1 in the end, we get the Gibbs relation d H H = ( ) H d Ia E a I a, I a = E a a Here I a are thermodynamic conjugate of E a.. E b (b a) Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 12 (2014) / 85

13 Introduction Applying the same scaling argument to black hole thermodynamics, it would be inevitable that any dimensionful parameter with nonzero scaling dimension must appear in the Smarr relation for the black hole thermodynamics; Consequently, in differential form, the first law of black hole thermodynamics must contain the differential of each extended variable with nonvanishing scaling dimension, yielding terms like P dv in de (or equally well V dp if one considers the enthalpy differential). Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 13 (2014) / 85

14 P V criticality in AdS and van der Waals analogy [Kubiznak & Mann, arxiv: ] Consider the RN-AdS black hole solution Here, This metric is the solution to the field equations of the Einstein-Maxwell action Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 14 (2014) / 85

15 P V criticality in AdS and van der Waals analogy Thermodynamic variables: The mass M is taken as the enthalpy H; The temperature T is The entropy is S = A 4, where A = 4πr2 +, r + is the black hole radius; We have also the electric charge Q and its conjugate, the Coulomb potential Φ = Q r + ; Finally, we have the thermodynamic pressure P = Λ 8π conjugate volume V = 4πr and its Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 15 (2014) / 85

16 P V criticality in AdS and van der Waals analogy The first law for RN-AdS black hole is written in the extended phase space as The corresponding Smarr relation is The equation of state (EOS) at constant Q reads in which r + is taken as an EOS parameter instead of V. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 16 (2014) / 85

17 P V criticality in AdS and van der Waals analogy Introducing v = 2l 2 P r + (where l P = 1 in our unit), the EOS can be rewritten as This resembles very much to the EOS for van der Waals liquid, Therefore, it is naturally expected that the thermodynamic behavior for RN-AdS black holes should be quite similar to that of the van der Waals system. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 17 (2014) / 85

18 P V criticality in AdS and van der Waals analogy (Taken from Kubiznak & Mann, arxiv: ) Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 18 (2014) / 85

19 P V criticality in AdS and van der Waals analogy Criticalities appear at the inflection points of the isotherms, i.e. it obeys This leads to the critical parameters These parameters obey which is exactly the same relation obeyed by the critical parameters for van der Waals system. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 19 (2014) / 85

20 P V criticality in AdS and van der Waals analogy (Taken from Kubiznak & Mann, arxiv: ) Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 20 (2014) / 85

21 P V criticality in AdS and van der Waals analogy The scaling properties can be obtained by expanding the dimensionless EOS near the critical point, where The resulting expansion reads with Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 21 (2014) / 85

22 P V criticality in AdS and van der Waals analogy After some procedure, the following scaling relations and critical exponents results, ( ) T Tc α C r+, T c r +l r +s (T T c ) β, ( ) T Tc γ κ T, T c P P c (r + r c ) δ, α = 0, β = 1/2, γ = 1, δ = 3. All these exponents are identical to those of the van der Waals system. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 22 (2014) / 85

23 Gauss-Bonnet black holes It is natural to consider the P V criticality for other black holes, or extend such studies to other pairs of thermodynamic variables. The P V criticality has been extensively studied for most familiar black hole solutions in AdS background. The Q Φ criticality has also been studied for many charged black holes. For black holes in extended theories of gravity (such as higher curvature gravities, f(r) gravities etc.), extra coupling constants might be present, and these can all be thought of as remnants of moduli parameters in some fundamental theory. So, it is reasonable to consider also these extra couplings as free thermodynamic variables. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 23 (2014) / 85

24 Gauss-Bonnet black holes We will be interested in black hole solutions to Gauss-Bonnet (GB) gravity. The action reads I = 1 d d x [ ] g R 2Λ + α(r µνγδ R µνγδ 4R µν R µν + R 2 ) 16π 1 d d x gf µν F µν. 4 Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 24 (2014) / 85

25 Gauss-Bonnet black holes This theory has a black hole solution with line element ds 2 = f(r)dt f(r) dr2 + dω 2 d 2,k, ( ) f(r) = k + r α 2 α l π αm (d 2)r d 1 2 αq 2 (d 2)(d 3)r 2d 4, where α = (d 3)(d 4)α and k = 0, ±1. The corresponding Maxwell field reads A µ dx ν = Φ dt, Φ = Q 8π(d 3)r+ d 3. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 25 (2014) / 85

26 Gauss-Bonnet black holes The GB-AdS black hole does not asymptotes to a pure AdS background with cosmological constant Λ. Rather, it asymptotes to an AdS background with the effective cosmological constant ( ) (d 1) (d 2) α Λ Λ eff = 1 δ 1 + 8, 4 α (d 1) (d 2) where δ = +1 corresponds to GR branch and δ = 1 corresponds to a GB branch with ghost degrees of freedom. We shall take only δ = +1. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 26 (2014) / 85

27 Gauss-Bonnet black holes By a simple analogy with the previously mentioned P V criticality for AdS black holes, one may tends to consider P eff Λ eff as a thermodynamic pressure. However, for two reasons, we will not take this point of view: 1 Λ eff is a complicated combination of two parameters Λ and α and we wish to understand the role of each parameter independently; 2 In the enthalpy description of the first law, P eff appear in the term V eff dp eff, and it is reasonable to decompose dp eff as a combination of dλ and dα. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 27 (2014) / 85

28 Gauss-Bonnet black holes Moreover, we have a good reason to consider P GB 1 8πα instead of α as a component of thermodynamic pressure, because it is 1 α which scales like a pressure. This is where our considerations depart from the previous work [R. -G. Cai, L. -M. Cao, L. Li and R. -Q. Yang, JHEP 1309, 005 (2013)], which took α and its conjugate A as a pair of thermodynamic quantities. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 28 (2014) / 85

29 Gauss-Bonnet black holes The mass M is now identified with the enthalpy H, [ (d 2)r d 3 ( + H = k + r2 + 16π l 2 + k 2 ) ] Q 2 8πr+ 2 P + GB 32π(d 3)r+ d 3. The Hawking temperature of the black hole is given by T = 1 4π f (r + ) = (d 1)r 4 + l 2 + (d 3)kr (d 5)k2 8πP GB 4πr + (r k 4πP GB ) Q 2 2(d 2)r 2d 8 +, and the associated entropy for the black hole reads S = rd ( 1 + (d 2)k 4π(d 4)r 2 + P GB ). Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 29 (2014) / 85

30 Gauss-Bonnet black holes The thermodynamic volumes associated with P = Λ 8π and P GB are given respectively as ( ) H V = = r d 1 + P S,Q,P GB d 1, ( ) H (d 2)k2 V GB = P GB S,Q,P 128π 2 PGB 2 r+ d 5. The Smarr relation that follows from scaling arguments and the first law: (d 3)H = (d 2)T S + (d 3)QΦ 2P V 2P GB V GB, dh = T ds + ΦdQ + V dp + V GB dp GB. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 30 (2014) / 85

31 We shall consider the criticality associated with the new variable P GB, taking P, V and Q, Φ as fixed parameters. Gibbs free energy: G = G(T, Q, P, P GB ) = H T S ( ) = rd 3 + (d 2) k + r2 + 16π l 2 T rd k2 (d 2)r+ d 5 128π 2 + Q2 r+ 3 d P GB 32π (d 3). ( 1 + The EOS arises from the horizon condition f(r + ) = 0: (d 2) k 4πP GB (d 4) r 2 + ) P GB = ((5 d)k + 8 T π r + ) (d 2)k [ ( r 2 ) ]. 8π + (d 1) + (d 3)k 4 T π r l 2 + (d 2)r+ 2 Q2 2r 2d 8 + Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 31 (2014) / 85

32 Gauss-Bonnet black holes P V criticality at fixed GB coupling α: [Cai R.-G., Cao L.-M., Li L. and Yang R.-Q. (CCLY), arxiv: ] (from CCLY [arxiv: ]) Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 32 (2014) / 85

33 Gauss-Bonnet black holes (from CCLY [arxiv: ]) Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 33 (2014) / 85

34 Gauss-Bonnet black holes (b is the dimensionless charge ) (from CCLY [arxiv: ]) Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 34 (2014) / 85

35 Gauss-Bonnet black holes Criticality associated with P GB neutral case: Enthalpy: H = Gibbs free energy: G = EOS: (k = 0 is trivial) (d 2)rd π (d 2)rd π T rd ( k + r2 + l 2 + k 2 ) 8πP GB r+ 2, ( 1 + ( k + r2 + l 2 + k 2 8πP GB r 2 + (d 2) k 4π (d 4) r 2 + P GB ) ). ((5 d)k + 8 T π r + ) k P GB = ( r 2 ). 8π + (d 1) + (d 3)k 4 T π r l 2 + r+ 2 Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 35 (2014) / 85

36 The critical point is determined as the inflection point on the P GB r + diagram, i.e., P GB r + = 2 P GB = 0. r+ =r c,t =T c r + =r c,t =T c r 2 + For k = 1, the above pair of equations can never have a solution with real and positive r c. Therefore, we are left with only the choice k = +1. For k = +1, eliminating T c from the above pair of equations, we get a single simplified equation determining the critical radius r c, 36(d 1) 2 R 2 c 12(d 1)(2d 9)R c + (d 3)(7d 39) = 0, R c = r2 c l 2. The solutions to this equation read R c = 2 d 9 ± 3(d 2)(d 6). 6(d 1) Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 36 (2014) / 85

37 Only when d = 5 or d = 6 R c can be real positive; For d = 5, only the + branch of the solution is allowed, and this is indeed a critical point; For d = 6, the two branches of solutions degenerate, and 2 P GB does r+ 2 not change its signature around the corresponding r c, so, there is no critical point in six dimensions. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 37 (2014) / 85

38 Gauss-Bonnet black holes In five dimensions, we have R c = 1 6, r c = 6 6 l, P c GB = 9 2πl 2, T c = 6 2πl, from which we can easily find P c GB r c T c = 3 2. This relation is universal in the sense that it is independent of all parameters. This result is very similar to the one in the Van der Waals system, which has Pcvc T c = 3 8. Omitting the details, we conclude that all the critical exponents at this case are identical to those of Von der Waals gas-liquid system. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 38 (2014) / 85

39 Gauss-Bonnet black holes Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 39 (2014) / 85

40 Gauss-Bonnet black holes In six dimensions, each isotherm contains one and only one extremum (which is a maximum). This means that at each temperature, the value of P GB has an upper bound, black holes with P GB bigger than the upper bound simply could not exist. For smaller values of P GB, there are two different black holes at each temperature: a small unstable black hole and a large stable black hole. The small black hole phase cannot physically persist because of its thermal instability, and there is no phase equilibrium in this case. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 40 (2014) / 85

41 Gauss-Bonnet black holes Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 41 (2014) / 85

42 Gauss-Bonnet black holes Charged cases: Recall the EOS: P GB = ((5 d)k + 8 T π r + ) (d 2)k [ ( r 2 ) ]. 8π + (d 1) + (d 3)k 4 T π r l 2 + (d 2)r+ 2 Q2 2r 2d 8 + When d = 5, the critical point conditions reduces into 24R 3 c 4kR 2 c 5Q2 l 4 = 0, R c r2 c l 2, (1) T c = k + 9 R c 5πl R c. (2) Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 42 (2014) / 85

43 Gauss-Bonnet black holes Correspondingly, PGB c = P GB r+ =r c,t =T c = 3 20π PGB c r c 3k = T c 4(k 3 R c ). Positivity of T c and P c GB requires k + 9 R c > 0, k (k 3 R c ) > 0. k + 9 R c l 2 kr c (k 3 R c ), There are two sub cases: when k = 1, the critical horizon radius is in the region 0 < R c < 1 3 ; when k = 1, we have R c > 1 9. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 43 (2014) / 85

44 Gauss-Bonnet black holes On the other hand, the squared charge Q 2 can be taken as a function of R c at the critical point due to (1), Q 2( ) 4R 2 R c = cl 4 (6R c k). 5 For k = +1, positivity of the squared charge gives a tighter bound for R c, i.e. 1 6 < R c < 1 3, with 0 Q < l2. For k = 1, the bound on R c does not get tighter, however Q 2 is still bounded: Q > l2. In both cases we have 1215Q 2 + 4k 3 l 4 > 0. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 44 (2014) / 85

45 Gauss-Bonnet black holes Eq.(1) has 3 analytical roots, ( R c1 = 1 1 l 2 36 x1/3 c + 1 k 2 l 4 9 x 1/3 c ( R c2 = 1 l 2 1 [ x1/3 c + 1 k 2 l 4 9 x 1/3 c ( R c3 = 1 l 2 1 [ x1/3 c + 1 k 2 l kl2 x 1/3 c ) ] + i ] i, [ x1/3 c 1 9 [ 1 36 x1/3 c 1 9 x c = 4860 Q 2 l 2 + 8k 3 l Q 2 + 4k 3 l 4 Ql 2. k 2 l 4 ] ) kl2 x 1/3 c k 2 l 4 ] ) kl2 x 1/3 c When 1215Q 2 + 4k 3 l 4 > 0, only the first root is real positive. Therefore, there is only a single critical point in five dimensions. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 45 (2014) / 85

46 Gauss-Bonnet black holes Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 46 (2014) / 85

47 Gauss-Bonnet black holes All the critical exponents are kept unchanged and are identical to those for Van der Waals system. The difference from neutral case lies in that the critical radius and electric charge are both bounded for the critical point to exist: 1 for k = 1 (spherical horizon), the critical horizon radius and charge need be in this region: 1 6 R c < 1 3, 0 Q < l2 ; 2 for k = 1 (hyperbolic horizon), the bounds are given as follows: R c > 1 9, Q > l2. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 47 (2014) / 85

48 Gauss-Bonnet black holes In d > 5 dimensions, the situation becomes much more complicated. The critical point conditions become a set of higher order algebraic equations which is difficult to solve. Significant simplifications occur if we choose the following specific value for the electric charge Q: Q = ϱ R (d 3)/2 c l d 3, where ϱ is a dimensionless parameter because Q has the dimension [length] d 3. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 48 (2014) / 85

49 Gauss-Bonnet black holes In this case, the equation determining R c can be reduced into The corresponding temperature reads Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 49 (2014) / 85

50 Gauss-Bonnet black holes For d > 5, there can be up to two different critical points of critical radius r c = l R c for a given ϱ with appropriate value. However, please bear in mind that these two critical points do not correspond to the same electric charge. The explicit solution for R c reads R c± = 1 6(d 1) [ (2d 9)k (2d 7)(5d 14)ϱ ± 3 = 3 (2 d 7) 2 ϱ k (2 d 1) (d 4) ϱ 16 k 2 (d 2) (d 6). ], Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 50 (2014) / 85

51 Gauss-Bonnet black holes Numerical result for d = 6 at fixed charge Q = ql 3 : The critical point condition reduces into 175 q 4 48σ 6 c q q 2 σ 8 c + 144σ 12 c 2880σ 14 c σ 16 c = 0, where r c = σ c l, with σ c and q both being dimensionless parameters. Setting q = 0.01, the critical radius parameter can be worked out numerically, giving rise to two different critical radii: 1 critical radius σ s 0.208, critical temperature T s l, critical pressure PGB s l ; 2 2 critical radius σ l 0.377, critical temperature T l l, critical pressure PGB l l. 2 Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 51 (2014) / 85

52 Gauss-Bonnet black holes Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 52 (2014) / 85

53 P V criticality for Lovelock black holes Third order Lovelock gravity: A.M. Frassino, D. Kubiznak, R. B. Mann, and F. Simovic [ ] studied cases when α 2 and α 3 are not related. Black hole solution Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 53 (2014) / 85

54 P V criticality for Lovelock black holes Thermodynamic quantities: First law: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 54 (2014) / 85

55 P V criticality for Lovelock black holes Gibbs free energy: EOS: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 55 (2014) / 85

56 P V criticality for Lovelock black holes Critical points: k = 0: ideal gas, no criticality; k = 1: A single critical point exist in all dimensions d 7 (7 is the lowest dimension for the third order Lovelock gravity to be meaningful). Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 56 (2014) / 85

57 P V criticality for Lovelock black holes k = 1: two sets of solutions for critical point conditions: iu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 57 (2014) / 85

58 P V criticality for Lovelock black holes d > 12: A is complex, no critical points; d = 12: A = 0, degenerate case, not a critical point; When d = 7, both v c1 and T c1 vanish, leaving only a single critical point; When d = 8, 9, 10, 11, two critical points; in A. Belhaj, M. Chabab, H.E. Moumni, K. Masmar, M.B. Sedra, arxiv: , only a single critical point was found, and the identification of d = 12 as a dimension allowing for criticality is wrong. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 58 (2014) / 85

59 P V criticality for Lovelock black holes Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 59 (2014) / 85

60 P V criticality for Lovelock black holes k = 1 and d = 7: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 60 (2014) / 85

61 P V criticality for Lovelock black holes k = 1 and d = 7: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 61 (2014) / 85

62 P V criticality for Lovelock black holes k = 1 and d = 7: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 62 (2014) / 85

63 P V criticality for Lovelock black holes k = 1 and d = 7: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 63 (2014) / 85

64 P V criticality for Lovelock black holes k = 1 and d = 7: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 64 (2014) / 85

65 P V criticality for Lovelock black holes k = 1 and d = 7: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 65 (2014) / 85

66 P V criticality for Lovelock black holes k = 1 and d = 7: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 66 (2014) / 85

67 P V criticality for Lovelock black holes k = 1 and d > 7: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 67 (2014) / 85

68 P V criticality for Lovelock black holes k = 1 and d > 7: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 68 (2014) / 85

69 P V criticality for Lovelock black holes k = 1 and d > 7: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 69 (2014) / 85

70 P V criticality for Lovelock black holes k = 1 and d > 7: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 70 (2014) / 85

71 P V criticality for Lovelock black holes k = 1 and d = 7: (this case is very similar to the case of RN-AdS black hole) [J.-X. Mo, W.-B. Liu, ] Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 71 (2014) / 85

72 P V criticality for Lovelock black holes k = 1 and d = 8, 9: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 72 (2014) / 85

73 P V criticality for Lovelock black holes k = 1 and d = 8, 9: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 73 (2014) / 85

74 P V criticality for Lovelock black holes k = 1 and d = 8, 9: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 74 (2014) / 85

75 P V criticality for Lovelock black holes k = 1 and d = 10, 11: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 75 (2014) / 85

76 P V criticality for Lovelock black holes k = 1 and d = 10, 11: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 76 (2014) / 85

77 P V criticality for Lovelock black holes k = 1 and d = 10, 11: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 77 (2014) / 85

78 P V criticality in conformal gravity Conformal gravity and its black hole solution: [J. Li, H.-S. Liu, H. Lu, Z.-L. Wang, JHEP 1302(2013) 109] Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 78 (2014) / 85

79 P V criticality in conformal gravity Technically more involved (EOS is 4-th order in P ), but the result seems to be neat: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 79 (2014) / 85

80 P V criticality in conformal gravity All phase transitions are of zeroth order: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 80 (2014) / 85

81 P V criticality in conformal gravity All phase transitions are of zeroth order: Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 81 (2014) / 85

82 Summary of open problems Extended phase space thermodynamics for AdS black holes possesses unexpected richness and complexity. Though for some AdS black holes the Von der Waals analogy seems to be perfect, there are also instances where the analogy is completely gone; Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 82 (2014) / 85

83 Summary of open problems We have not yet fully understood what is implied by such complicated thermodynamic behaviors, e.g. 1 why the phase transition is of the first order in some theories and some are of the zeroth or the second order? 2 why in some models phase transitions occur at temperatures below the critical value whilst in some other models they occur at temperatures above the critical value? 3 when multiple critical points exist, why in some models criticalities appear in between two critical points and in some other models they appear either above the higher critical point or below the lower critical point but not in between? Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 83 (2014) / 85

84 Summary of open problems We have not a clear idea on what is the coexistence phase for a small and a large black hole: it can either be small or be large, but can it be small and large simultaneously? Perhaps the proper understanding of such thermodynamics relies on the holographic interpretation of the gravitational theories, in which case different phases are interpreted as actual phases of the dual condensed matter system. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 84 (2014) / 85

85 Thank you. Liu Zhao School of Physics, Nankai University Extended Nov. phase 2014 space basedthermodynamics on works with Wei forxu AdSand black Haoholes Xu arxiv: [EPJC 85 (2014) / 85