Fractional-Order Phase Transition of Charged Ads Black Holes

Physics Letters B 795 (2019) 490–495

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Physics Letters B

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Fractional-order phase transition of charged AdS black holes ∗ Meng-Sen Ma a,b, a Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China b Department of Physics, Shanxi Datong University, Datong 037009, China a r t i c l e i n f o a b s t r a c t

Article history: Employing the fractional derivatives, one can construct a more elaborate classification of phase transitions Received 23 May 2019 compared to the original Ehrenfest classification. In this way, a thermodynamic system can even undergo Received in revised form 26 June 2019 a fractional-order phase transition. We use this method to restudy the charged AdS black hole and Van Accepted 28 June 2019 der Waals fluids and find that at the critical point they both have a 4/3-order phase transition, but not Available online 2 July 2019 the previously recognized second-order one. Editor: M. Cveticˇ © 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction The generalized classification can work in any thermodynamic systems that having phase structures. Black holes, being charac- In many thermodynamic systems there exist phase transitions terized by only three parameters, are the simplest thermodynamic and critical phenomena. The original classification of phase tran- system. In particular, it has been found that AdS black holes have sitions was proposed by Ehrenfest. Specifically, when the n-th fruitful phase structures [7–19]. Therefore, in this paper we choose derivative of a thermodynamic potential (usually the Gibbs free en- the charged AdS black hole to make the first attempt. ergy or Helmholtz free energy) with respect to the external fields The plan of this paper is as follows: In Sec. 2 we briefly intro- has a jump discontinuity while its lower order derivatives are all duce the history and some useful formulae on fractional deriva- continuous, the thermodynamic system undergoes a n-order phase tives. In Sec. 3 we analyze the fractional phase transition of the transition at the transition points. When n ≥ 2, it is usually called charged AdS black hole at the critical point. We also study the the continuous phase transition. Fisher generalized the scope of fractional phase transition of Van der Waals fluids for comparison. continuous phase transitions by allowing for the divergent n-th In Sec. 4 we summarize our results and discuss the possible future derivative of thermodynamic potentials near the critical point [1]. directions. (We will use the units: h¯ = c = G = kB = 1.) Mathematically, there is also non-integer order derivatives, or fractional derivatives for short. This tool has been used in many 2. Useful formulae on fractional derivatives areas of physics [2–4]. Following Ehrenfest’s idea, one can employ the fractional derivative and generalize the classification of phase In this part we only briefly introduce the history of fractional transition to include the fractional order. Therefore, the generalized derivatives and give some formulae needed in the following sec- Ehrenfest classification should have the following form [5,6]: tions. The strict definitions of fractional derivatives are left in Ap- pendix A. α α d G = + = − = d G The idea of derivatives of fractional order has a long history, lim α A A lim α , (1.1) → + → − T Tc dT T Tc dT which can date back to the era of L’Hôpital and Leibniz. After that, many mathematicians had considered the subject. The detailed his- where G = G(T , P) is the Gibbs free energy and is a positive α tory and major documents on this subject can be found in [20,21]. real number. We expect that this finer-grained classification can The first documented example on fractional derivative was given differentiate phase transitions more effectively compared to the by Lacroix (in 1819), who generalized the derivatives of power relatively coarse-grained Ehrenfest classification. function by using Gamma function to replace the factorial of an integer: Correspondence to: Institute of Theoretical Physics, Shanxi Datong University, * m n μ ν Datong 037009, China. d x n! − d x (ν + 1) − = xn m ⇒ = xν μ, (2.1) E-mail address: [email protected]. dxm (n − m)! dxμ (ν − μ + 1) https://doi.org/10.1016/j.physletb.2019.06.054 0370-2693/© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. M.-S. Ma / Physics Letters B 795 (2019) 490–495 491 where (m, n) are integers and (μ, ν) are real numbers. The ≡ ∞ μ−1 −x Gamma function is defined as (μ) 0 x e dx,(μ > 0). The 1/2 generalized formula can give us, for example, d x = √2 x1/2 and dx1/2 π 1/2 − especially d 1 = √1 x 1/2. Surprisingly, the fractional derivative dx1/2 π of a constant returns a non-zero result. This is indeed the charac- teristic of the Riemann-Liouville fractional derivative. In this paper, we prefer to choose the Caputo’s definition of fractional derivatives, which will give us more acceptable results. Here we only list some formulae that will be used next. We will α = dα write Dx dxα for simplicity. First, α = ; α = Dx c 0,(α > 0) Dx x 0,(α > 1), (2.2) where c is a constant and α is a real number. For 1 < α < 2, there are Fig. 1. The black point at (0, 0) represents the critical point. The colorful arrows − − 2x2 α 2(−x)2 α denote the various directions approaching the critical point. Dαx2 = ,(x > 0); Dαx2 =− .(x < 0). x (3 − α) x (3 − α) T 1 2Q 2 (2.3) P = − + , (3.7) v 2π v2 π v4 3. Charged AdS black holes where the specific volume v is related to the horizon radius, v = 2r+. The critical point lies at The line element of the four-dimensional RN-AdS black hole is √ given by 6 √ 1 T = , v = 2 6Q , P = . (3.8) c c c 2 − 18π Q 96π Q ds2 =−fdt2 + f 1dr2 + r2d2, (3.1) For simplicity, we define the dimensionless quantities, with the metric function P − P v − v T − T 2 = c = c = c 2M Q 2 p , ν , t . (3.9) f (r) = 1 − + + r . (3.2) Pc vc Tc r r2 3 With this new set of variables (t, p, ν), the critical point lies at The U (1) gauge potential and field strength are defined by (t = p = ν = 0). Q Now the equation of state becomes A =− dt, F = dA. (3.3) r (3p + 3)ν4 + (12p − 8t + 4)ν3 + (18p − 24t)ν2 If treating the cosmological constant as the thermodynamic pressure P =−/8π , one can construct an extended phase space + (12p − 24t)ν + 3p − 8t = 0, (3.10) for AdS black holes [22,23]. In the extended phase space, one can construct a one-to-one correspondence between the thermo- which is a quartic equation for ν and can be exactly solved in dynamic quantities of AdS black hole and those of ordinary ther- principle, although the roots are very complicated. modynamic systems [24]. After that, many other AdS black holes Using the family of dimensionless quantities (p, ν, t) to re- and other interesting critical phenomena have been extensively ex- place the r+ and P in the Gibbs free energy, we can obtain the plored [25–35]. dimensionless Gibbs free energy, In the extended phase space, there is the extended first law of 8 − ν4 − 4ν3 + 8ν − (ν + 1)4 p black hole thermodynamics: g(t, p) = √ , (3.11) 4 6(ν + 1) dM = TdS + dQ + VdP, (3.4) and the corresponding entropy can be derived from where the temperature T , the entropy S and the electric potential are ∂ g 2 2 s(t, p) =− = (1 + ν) , (3.12) ∂t 3 1 Q 2 p T = 1 − r2 − , + 2 where ν = ν(p, t) is given by Eq. (3.10). 4πr+ r+ We want to know the behaviors of the Gibbs free energy and its A 2 Q derivatives when the critical point is approached. Mathematically, S = = πr+, = , (3.5) 4 r+ for a function of two variables its continuity at certain point should be independent of the ways we choose to approach the point. As respectively. Here the r+ represents the position of the event hori- is shown in Fig. 1, there are many possible paths of approaching zon of the RN-AdS black hole. Besides, the Gibbs free energy can the critical point in the (t, p)√plane. It can be √easily found that be given by = = there are always g(0, 0) 2/ 6and s(0, 0) 2/3, no matter 2 which direction we choose to approach the critical point. There- 1 8π 3 3Q G = M − TS= r+ − Pr+ + . (3.6) fore, at the critical point the Gibbs free energy and the entropy are 4 3 r+ both continuous. So we only need to pay attention to the behavior The equation of state for the charged AdS black hole can be of higher-order derivatives of the Gibbs free energy at the critical written as point. 492 M.-S. Ma / Physics Letters B 795 (2019) 490–495

One can expand the solution ν(t, p) of Eq. (3.10)in series of (t, p). It can be shown that this series includes some terms with fractional exponents. The type of series with fractional exponents is called a Puiseux series. After substituting the ν(t, p) into the Gibbs free energy and expanding it as a series of t and p, we obtain 2 p 2 25/6 p1/3 g(t, p) = + √ + ... − − + ... t 3 2 6 3 31/6 4 × 25/6 26 × 21/6 − − + ... t2 + ... (3.13) 9 × 31/6 p2/3 9 × 35/6 p1/3 Clearly, in g the series about p also contain the fractional exponents. we will see below, this is the key element for the fractional phase transition. α One can formally express the above result in the Taylor series Fig. 2. The behaviors of Dt g near the critical point for the charged AdS black hole. of t, which is a jump discontinuity. When α > 4/3, the α-order frac- g(t, p) = A(p) + B(p)t + D(p)t2 + O [t3]. (3.14) tional derivatives of the Gibbs free energy diverge. Therefore, the 2 phase transition of the Van der Waals fluid at the critical point ∂ g The heat capacity at constant pressure C p is proportional to 2 , ∂t p is of 4/3order according to the generalized Ehrenfest classifica- so near the critical point the coefficient D(p) is proportional to C p . tion. In particular, the k dependence of the results manifests that Generally, the coefficient D(p) is divergent at the critical point, so the discontinuity is universal along any directions except for k = 0 we have the divergent C p , which is a signal of the second-order and k =∞. In fact, due to the constraint of the equation of state, phase transition according to the conventional Ehrenfest classifica- k cannot take arbitrary values. From Eq. (3.10), one can easily see tion. that However, if taking into account the fractional derivative, the 8t results can be completely different. According to Eq. (2.3), the p ≈ , (3.20) 3(ν + 1) α-order fractional derivatives of the Gibbs free energy with 1 < α ≤ 2are near the critical point. Therefore, k must be very close to 8/3for α the charged AdS black hole [24]. In Fig. 2, we show how Dt g 4 × 21/6 2 × 62/3 − 13p1/3 + ... varies with t. It can be found that the curves of Dα g are sym- Dα g(t, p) =− t t p × 5/6 − metric about the critical point (the origin). The continuity, jump 9 3 (3 α) α − discontinuity and divergent behavior of Dt g at different values of t2 α × ,(t > 0) (3.15) α are clearly presented. p2/3 For comparison, we also analyze the fractional phase transition and of Van der Waals fluids. Van der Waals equation, which was proposed to describe the behavior of real fluids, has the form × 1/6 × 2/3 − 1/3 + 4 2 2 6 13p ... a Dα g(t, p) = + − = t p 5/6 P (v b) T , (3.21) 9 × 3 (3 − α) v2 2−α (−t) where P is the thermodynamic pressure, T is the temperature, and × ,(t < 0). (3.16) p2/3 v = V /N is the specific volume of the fluid. The two parameters a and b are introduced to describe the interaction and size of the Now we should calculate the values of Dα g(t, p) in the limit t molecules in real fluids. Van der Waals equation can be used to (t → 0, p → 0). As we mentioned above, mathematically there are describe the critical phenomena of real fluids. The critical point infinitely ways of approaching the critical point. However, physi- lies at cally, due to the constraint of the equation of state, near the critical 8a a point p and t are not independent. Between them there is the re- T = , v = 3b, P = . (3.22) c c c 2 lation [36] 27b 27b Below the critical point, Van der Waals equation can describe the 3/2 p = kt + O [t ], (3.17) first-order (liquid/gas) phase transition of fluids by taking account of the Maxwell’s equal area law. with some constant k. Substituting it into Eq. (3.15) and Eq. (3.16), The Gibbs free energy of the liquid/gas system has been given and take the limit, we can obtain in [24], which is ⎧ ⎨⎪ 0forα < 4/3, (v − b)T 3/2 a × 5/6 = =− + − + α ∓ 4 2 for α = 4/3, G G(T , P ) kT 1 ln pv, (3.23) lim D g(t, p) = 2 (3.18) ± t 3×31/6k2/3 v t→0 ⎩⎪ 3 ∓∞ for α > 4/3. where is a constant charactering the gas. We also employ the dimensionless variables defined in Eq. (3.9). Obviously, in the α = 4/3case, there is Van der Waals equation turns into α α lim D g = lim D g, (3.19) 3 2 − t + t ν (3p + 3) + ν (8p − 8t) + ν(7p − 16t) + 2p − 8t = 0. (3.24) t→0 t→0 M.-S. Ma / Physics Letters B 795 (2019) 490–495 493

that case one should first expand g(t, p) as Taylor series of p, then expand the coefficients as series of t. You will find that the series about t is the Puiseux series. Following the above calculation, one can draw the same conclusions. (2). Fluid system and magnetic system are often compared to each other. In the mean field approximation, not only do they have similar thermodynamic relations, but also similar critical behaviors. Now we can differentiate their phase transitions. For magnetic system, due to the jump discontinuity of heat capacity at the critical point, it is the true second-order phase transition. While the Van der Waals fluid has 4/3-order phase transition at the critical point. (3). Hilfer also discussed the fractional phase transition and multiscaling issue [5,6]. Starting from a thermodynamic potential, such as the Gibbs free energy g(t, p) with the critical point at (t = 0, p = 0), Hilfer first introduced a curve C parameterized by Fig. 3. The behaviors of Dα g near the critical point for Van der Waals fluids. t s in the phase space, (t(s), p(s)). Thus, at the beginning the two variables (t, p) are not independent. He then calculated the frac- With the dimensionless variables, the dimensionless Gibbs free en- tional derivatives ergy and entropy can be expressed as dα g(t(s), p(s)) ± C = 3 A ( ) lim , (3.29) s→0± dsα g(t, p) =−g0(t + 1) − + (ν + 1)(p + 1) (3.25) ν + 1 with real number α. We completely follow the conventional ther- 8 2 − (t + 1) ln ν + − 4(t + 1) ln(t + 1), modynamic definitions and take (t, p) as independent variables 3 3 at the beginning. After deriving the fractional derivatives of the Gibbs free energy, Dα g, we compute their values in the limit and t (t → 0, p → 0) and judge the phase transition correspondingly. 8 2 Our conclusions are different from, even opposite to those of s(t, p) = g0 + 4 + ln ν + + 4ln(t + 1), (3.26) 3 3 Hilfer. For Van der Waals fluids, Hilfer’s results indicate that the direction labeled by k = 4is special, along which the phase tran- respectively, where g is a constant. 0 sition is of second order, while along other directions except for Similarly, we expand g(t, p) in series of t and p, = k 4the phase transition is of 4/3order. We find that the order 8 3 of the phase transition along the k = 4direction is 4/3. What is g(t, p) = −2 − g0 + ln + p + ... 3 2 more, near the critical point one can only discuss the phase transition along this direction. This issue was often overlooked in the 1/3 8 3 2 1/3 previous researches on second-order phase transition, because the + −4 − g0 + ln + 4 p + ... t 3 2 3 response functions, cp , κT , αT etc. are all divergent at the critical point, no matter from which directions the critical point is 1/3 2/3 8 2 − 16 2 − approached. − p 2/3 + p 1/3 + ... t2 3 3 9 3 4. Conclusion and discussion + O [t3]. (3.27) With the aid of fractional derivatives, one can calculate not only Clearly, Eq. (3.27) has nearly the same form as Eq. (3.13). For Van the integer-order derivatives of thermodynamic potentials, but also = der Waals fluids, near the critical point there is still (p kt) ap- any positive fractional-order derivatives of them. This allows us proximately. The only difference is that now k should take values to generalize the original Ehrenfest classification of phase transi- near k = 4. In a similar way, we find that tion. Now we can classify phase transitions according to the jump ⎧ ⎪ discontinuity of fractional-order derivatives of thermodynamic po- ⎪ 0f orα < 4/3, ⎪ 1/3 tentials. As examples, we reanalyzed the phase transition of the ⎨ 2 8 3 charged AdS black hole and the Van der Waals fluids at the criti- lim Dα g(t, p) = ∓ = (3.28) ± t for α 4/3, t→0 ⎪ k2/3 2 cal point. We find that the 4/3-order derivative of the Gibbs free ⎩⎪ 3 ∓∞ for α > 4/3. is discontinuous at the critical point. What is more, the α-order derivatives of the Gibbs free with α > 4/3all diverge at the criti- By the same token, the Van der Waals fluids should also undergo cal point. This means that the phase transition at the critical point a 4/3-order phase transition at the critical point. For comparison, is of 4/3order, but not usually recognized second order. α we also depict the curves of Dt g for different α in Fig. 3. It can be Essentially, it is the equation of state that determines the order α seen that the jump discontinuity behaviors of Dt g for the charged of the phase transition at the critical point. Firstly, for the charged AdS black holes and the Van der Waals fluids are very similar. AdS black hole and Van der Waals fluids, one can solve the equa- The only difference is that the slope of the α = 4/3-curve for the tion of state for ν = ν(t, p) and then further expand it as series of charged AdS black hole is positive near the critical point, while the t and p. You will get a Puiseux series, which leads to the fractional slope is negative for the Van der Waals fluids. phase transition if considering the generalized Ehrenfest classifica- Several comments are in order at this stage: tion. It seems that this is just a coincidence that the two systems (1). In this paper we only considered the fractional derivative both have the 4/3-order phase transition at the critical point, be- α with respect to t, namely Dt g. One can also discuss the α-order cause the cubic equation and the quartic equation happen to have fractional derivatives of the Gibbs free energy with respect to p. In the similar Puiseux series [37]. Secondly, near the critical point p 494 M.-S. Ma / Physics Letters B 795 (2019) 490–495 and t are generally not independent due to the constraint of the which is called Riemann-Liouville integral. In fact, when μ > 0, equation of state. This will affect the order of the phase transi- there are the left-handed and right-handed Riemann-Liouville in- tion. For the charged AdS black hole and Van der Waals fluids, it is tegral mathematically, which are [6,42] approximately a linear relation p ≈ kt. One can also consider rotat- x ing AdS black hole, higher-dimensional black hole and black holes μ 1 μ−1 in modified gravities [38–41], which have more complicated equa- (I + f )(x) ≡ (x − t) f (t)dt, x > a, a (μ) tion of state and more richer phase structures. We expect that they a may supply more information on the fractional phase transition. a At last, we should mention that the “order” of the fractional- μ 1 μ−1 (I − f )(x) ≡ (t − x) f (t)dt, x < a. (A.5) order derivative can not only be a fractional number, but also any a (μ) real number, even complex number. In the meaning of mathemat- x ics, they are all meaningful. Maybe this can lead to more general On the basis of the fractional integral, the (left-handed and classification of phase transition in physics. right-handed) Riemann-Liouville fractional derivative of α order can be directly defined, Acknowledgements α ≡ n · −(n−α) = n · n−α − ≤ We would like to thank Prof. Hilfer for useful correspondence. Dx f (x) RL D Dx f (x) D I f (x), (n 1 < α n), We also thank Prof. Ren Zhao for useful discussion. This work was (A.6) supported by the National Natural Science Foundation of China (Grants No. 11605107). where n is an integer. In this paper we always set the reference point a = 0. So we will drop the subscript a for clarity. When α Appendix A. Introduction to the fractional derivatives is an integer, the above fractional derivative return back to the usual integer-order derivative. For the Riemann-Liouville fractional The fractional derivative of a general function f (x) can also be derivative, there is an important property: defined, but before that we first introduce some notations. We define (γ + 1) − Dαtγ = tγ α for 0 −1 t 0 RL , α > , γ > , > . x (γ + 1 − α) d −1 D ≡ and a D f (x) ≡ (Ia f )(x) = f (t)dt, (A.1) (A.7) dx x a Another commonly used definition of fractional derivative was where a is called the reference point or fiducial point. Obviously, introduced by Caputo [43], which is defined as · −1 = −1 · = − we have D a Dx f (x) f (x), but a Dx Df(x) f (x) f (a). −(n−α) − Therefore, the orders of the operators are important in the cal- α ≡ · n = n α · n − ≤ Dx f (x) C Dx D f (x) I D f (x), (n 1 < α n). culation of derivatives. (A.8) From the first-order derivative f (x) − f (x − h) Obviously, the two definitions are different. However, there is a Df(x) ≡ lim , h→0 h relation between them − one can obtain the higher-order derivatives n 1 k−α t (k) + Dα f (t) = Dα f (t) + f (0 ). (A.9) m RL C − + 1 (k α 1) m = − k k − k=0 D f (x) lim ( 1) Cm f (x kh), h→0 hm k=0 According to Caputo’s definition, the fractional derivative of a constant is zero, namely Dα1 = 0,(α > 0). Moreover, in con- or write it in another way x C trast to the Riemann-Liouville fractional derivative, the advantage N−1 N m x − a of Caputo’s definition is that when solving differential equations it m = − k k − a Dx f (x) lim ( 1) Cm f (x k ), (A.2) is not necessary to define the fractional order initial conditions. Be- N→∞ x − a N k=0 low we will choose the Caputo’s definition of the fractional derivative and drop the subscript “C”at that point. = − k = m! = (m+1) where h (x a)/N and Cm k!(m−k)! (k+1)(m−k+1) . 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