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Fractional-Order Phase Transition of Charged Ads Black Holes Physics Letters B 795 (2019) 490–495 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 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 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 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 α is a positive 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 d1/2x 2 1/2 generalized formula can give us, for example, = √ x and dx1/2 π d1/21 1 −1/2 especially = √ x . 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.
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