Two-phase equilibrium properties in charged topological dilaton AdS black holes

In this paper we discuss phase transition of the charged topological dilaton AdS black holes by Maxwell equal area law. The two phases involved in the phase transition could be coexist and we depict the coexistence region in $P-v$ diagrams. The two-phase equilibrium curves in $P-T$ diagrams are plotted, the Clapeyron equation for the black hole is derived, and the latent heat of isothermal phase transition is investigated. We also analyze of the parameters of the black hole that have an effect on the two phases coexistence. The results show that the black hole may go through a small-large phase transition similar to those of usual non-gravity thermodynamic systems.


I. INTRODUCTION
In recent years, the cosmological constant in n-dimensional AdS and dS spacetime has been regarded as state variable, pressure, of black hole thermodynamic system.
Although some encouraging results about black hole thermal properties in AdS and dS spacetimes have been achieved, the statistical mechanics background for black hole thermodynamic system is still not clear. It is significant to further explore phase equilibrium and phase structure in black holes, which can offer more statistical information about black hole thermodynamic system, and help to further the understanding of black hole thermodynamic properties. Recently phase transition below critical temperature and phase structure of some black holes have received attention [45][46][47].
A scalar field called the dilaton appears in the low energy limit of string theory. The presence of the dilaton field has important consequences on the causal structure and the thermodynamic properties of black holes. Thus much interest has been focused on the study of the dilaton black holes in recent years [48][49][50][51][52][53][54][55][56][57][58]. The isotherms in P ∼ v diagrams of charged topological dilaton AdS black hole in Ref. [13] show there exists thermodynamic unstable region with ∂P/∂v > 0 when temperature is below critical temperature and the negative pressure emerges when temperature reduce to a certain value. This situation also exists in van der Waals-Maxwell gas-liquid system, which has been resolved by Maxwell equal area law. In this paper the Maxwell equal area law is extended to charged topological dilaton AdS black holes and a rational simulated phase transition process is generated, where the issues about unstable states and negative pressure are removed. By the simulated phase transition the information about two phase equilibrium is acquired. The results show the simulated phase transition is the first order phase transition, but phase transition at critical point belongs to the second-order one. We expect to provide some more relevant information for exploring quantum gravity properties and behaviors by studying phase transition and phase structure of charged topological dilaton AdS black holes.
The paper is arranged as follow. The charged topological dilaton AdS black hole as a thermodynamic system is briefly introduced in section 2. In section 3, by Maxwell equal area law the simulated phase transition process at some certain temperatures and the boundary of two phase equilibrium region are depicted in P − v diagram for charged topological dilaton AdS black holes. And then some parameters of the black hole are analyzed to find the relevance with two phase equilibrium. In section 4, P − T phase diagrams and phase change latent heat are investigated. We make some discussions and conclusions in section 5. (we use the units G d = = k B = c = 1 in this paper)

II. CHARGED DILATON BLACK HOLES IN ANTI-DE SITTER SPACE
The Einstein-Maxwell-Dilaton action in (n + 1)-dimensional (n ≥ 3)spacetime is [57,58] where the dilaton potential is expressed in terms of the dilaton field and its coupling to the cosmological constant: where R is the Ricci scalar curvature, Φ is the dilaton field and V (Φ) is a potential for Φ, α is a constant determining the strength of coupling of the scalar and electromagnetic field, F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field tensor and A µ is the electromagnetic potential. The topological black hole solutions take the form [57,58] with γ = α 2 /(α 2 + 1) and b is an arbitrary constant. The cosmological constant is related to spacetime dimension n by where l denotes the AdS length scale. In (2.5), m appears as an integration constant and is related to the ADM (Arnowitt-Deser-Misnsr) mass of the black hole. According to the definition of mass due to Abbott and Deser, the mass of the solution (2.5) is The electric charge is where ω n−1 represents the volume of constant curvature hypersurface described by dΩ 2 The thermodynamic quantities satisfy the first law of thermodynamics The Hawking temperature and entropy of the topological black hole where r + represents the position of black hole horizon and meets f (r + ) = 0. The electric 13) and the pressure and volume are respectively Using the Eqs. (2.7), (2.11) and (2.14) for a fixed charge Q, one may obtain the equation where specific volume [12] v = 4(α 2 + 1)b 2γ (n − 1) . (2.17) In Fig.1 we plot the isotherms in P −v diagrams in terms of Eq. (2.15) at different dimension n, charge Q, and parameters b and α. One can see from Fig.1 that there are thermodynamic unstable segments with ∂P/∂v > 0 on the isotherms as temperature T < T c , where T c is critical temperature, and the negative pressure emerges when temperature is below certain valueT .T and the corresponding specific volumeṽ can be derived. The state equation of the topological black hole is exhibit by isotherms in Fig.1, where both the thermodynamic unstable states with ∂P/∂v > 0 that will lead to the system expansion or contraction automatically and the negative pressure situation have no physical meaning. The case occurs also in van der Waals equation but it has been resolve by considering liquid-gas phase transition process and by Maxwell equal area law.
We generate the Maxwell equal area law to n + 1-dimensional charged topological dilaton AdS black hole to study phase transition of the black hole thermodynamic system. On the isotherm with temperature T 0 in P − v diagram, the two points (P 0 , v 1 ) and (P 0 , v 2 ) meet the Maxwell equal area law, where the two points (P 0 , v 1 ) and (P 0 , v 2 ) are seen as endpoints of isothermal phase transition. Considering and setting x = v 1 /v 2 , we can get Substituting (3.6) into (3.4) and setting T 0 = χT c (0 < χ < 1), we obtain When x → 1, the corresponding state is critical point state.
From Table 1, we can see that x is unrelated to b but it is incremental with χ and α. v 2 increases with increasing b, but decreases with increasing χ and α as the other parameters are fixed respectively. P 0 is incremental with χ and α, but decreases with increasing b while the other parameters are determined respectively. So phase transition process become shorter with increasing α, but it lengthens as b increases.   Here we investigate the two phase equilibrium coexistence P − T curves and the slope of them for the topological dilaton AdS black hole Rewrite EqS. (3.4) and (3.5) as where We plot the P − T curves as 0 < x ≤ 1 in Fig.3 when the parameters b, α, Q take different values respectively. The curves represent two phase equilibrium coexistence condition for the topological dilaton AdS black hole and the terminal points of the curves represent corresponding critical points. Fig.3 shows that for fixed α and Q, both the critical temperature and critical pressure decrease as b increases. Both critical pressure and temperature are incremental with α, but two phase equilibrium pressure decreases with increasing α at certain temperature. The change of two phase equilibrium coexistence curve with parameter Q is similar to that with parameter b. As Q becomes larger the critical pressure and critical temperature become smaller, but at certain temperature the corresponding pressure on P − T curves is larger for where y ′ (x) = dy dx . The Eq. (4.4) represents the slope of two phase coexistence P − T curve as function of x.
From Eqs.(4.1) and (4.4) we can get the phase change latent heat as function of x for n + 1-dimensional charged topological dilaton AdS black hole, The rate of change of phase change latent heat with temperature for usual thermodynamic where C β P and C α P are molar heat capacity of phase β and phase α. For n + 1-dimensional charged topological dilaton AdS black hole, the rate of change of phase change latent heat with temperature can be obtained from Eqs. (4.5) and (4.2), . (4.7) Using Eqs. (4.5) and (4.2) we plot L − T curves in Fig.4 as the parameters b, α and Q

V. DISCUSSIONS AND CONCLUSIONS
The charged topological dilaton AdS black hole is regarded as a thermodynamic system, and its state equation has been derived. But the state equation contains some region which has no physical meaning. When temperature is below critical temperature thermodynamic unstable situation appears on isotherms, and when temperature reduces to a certain value the negative pressure emerges, which can be seen from Fig.1 and Fig.2. However, considering the phase transition process the problems can be resolved. The phase transition process at a defined temperature is deemed happening at constant pressure, where the system specific volume changes with the ratio of the two coexistent phases, so according to Ehrenfest scheme the phase transition belongs to the first order one. Using Maxwell equal area law we draw the isothermal phase transition process and depict the boundary of two phase coexistence region in Fig.2.
Taking black hole as thermodynamic system, many investigations show the phase transition of some black holes in AdS spacetime and dS spacetime is similar to that of van der Waals-Maxwell liquid-gas system [3, 5, 13-20, 36-38, 40-44], and the phase transition of some other AdS black hole is alike to that of multicomponent superfluid or superconducting system [6,[8][9][10]. It would make sense to see if we can seek some observable system, such as van der Waals gas, to back analyze physical nature of black holes by their similar thermodynamic properties. That would help to further understand the thermodynamic quantities, such as entropy, temperature, capacity and so on, of black hole and that is significant for improving self-consistent thermodynamics geometric theory for black holes.
The Clapeyron equation of usual thermodynamic system agrees well with experiment result. In this paper we have plotted the two phase equilibrium coexistence curves in P − T diagrams, derived the slope of the curves, and acquired information on phase change latent heat by Clapeyron equation, which could create condition for finding some usual thermodynamic systems similar to black holes in thermodynamic properties and provide theoretical basis for experimental research of analogous black holes.