Phase Transition and Clapeyron Equation of Black Holes in Higher Dimensional AdS Spacetime

Phase Transition and Clapeyron Equation of Black Holes in Higher Dimensional AdS Spacetime

Hui-Hua Zhao, Li-Chun Zhang, Meng-Sen Ma, Ren Zhao111corresponding author: Email:zhao2969@sina.com(Ren Zhao) Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China Department of Physics, Shanxi Datong University, Datong 037009, China
Abstract

By Maxwell equal area law we study the phase transition in higher dimensional Anti-de Sitter (AdS) Reissner-Nordström (RN) black holes and Kerr black holes in this paper. The coexisting region of the two phases involved in the phase transition is found and some coexisting curves are shown in figures. We also analytically investigate the parameters which affect the phase transition. To better compare with a general thermodynamic system, the Clapeyron equation is derived for the higher dimensional AdS black holes.

Keywords: Reissner-Nordström AdS black hole, Kerr AdS black hole, phase transition, two phase coexistence, Clapeyron equation

pacs:
04.70.-s, 05.70.Ce

I Introduction

Black hole is an ideal platform for studying many interesting behaviors in classical gravity theory. It also can be regarded as a macroscopic quantum system in that its peculiar thermodynamic properties and thermodynamic quantities, as entropy, temperature, and its gravity holographic nature are quantum mechanical intrinsically. Hence research of black hole thermodynamics is an important channel for studying quantum gravity JDB1 (); JDB2 (); Hawking1 (); Lu (); ZhaoZhang (). Although thermodynamics of black hole has been researched for many years, the exact statistical interpretation for thermodynamics state of black hole is not clear. So the thermodynamics of black hole remains an important subject.

It has been found that a black hole possesses not only standard thermodynamic quantities but abundant phase structures and critical phenomena, such as those involved in Hawking-Page phase transitionHawking2 (), similar to the ones of a general thermodynamic system. Even more interesting is that the studies on the charged black holes show they may have an analogous phase transition with that of van der Waals-Maxwell liquid-gasCham1 (); Cham2 (). Recently critical behaviors and phase transition of black holes have been extensively investigated by considering the cosmological constant as thermodynamic pressureKubi1 (); Dolan (); Guna (); Cvet (); Kastor:2009wy (),. People have been trying to construct a complete liquid-gas analogue system for black holes.

In ref.Kubi1 (); Guna (); Banerjee1 (); Banerjee2 (); Banerjee3 (); Banerjee4 (); Banerjee5 (); Banerjee6 (), the phase transition and critical behaviors of some AdS black holes have been studied, which exhibits several different phase transition behaviors. On the basis of the thermodynamic volume of dS spacetime given in ref.Dolan (); Kastor:2009wy (); Alta1 (), considering the connection between black hole horizon and cosmological horizon, we derived the effective temperature and effective pressure in some dS spacetimes in our previous worksZhaohh (); Zhao1 (); Mams1 (); Zhanglc (). The relation among the effective thermodynamic quantities was investigated, which shows the critical phenomena like that in van der Waals liquid-gas system. Using Ehrenfest scheme, the second order phase transition was proved to exist in AdS spacetime black hole at the critical statesBanerjee1 (); Banerjee4 (); Mo1 (); Mo2 (); Mo3 (); Mo4 (); Mo5 (); Mo6 (); Mams2 (); Lala (). Alike conclusion has been reached by investigating thermodynamics and state space geometry of black holes in ref.Banerjee1 (); Lala (); Wei1 (); Suresh (); Mans (); Thar (); Niu (). The phase transition behaviors of black holes is found to be related to not only the spacetime metric but also the theory of gravity or other factorsMams2 (); Mams3 (); Mams4 (); Gim (); Posh (); Arci ().

Although some encouraging results have been achieved about the thermodynamics of black holes in AdS and dS spacetimes, many thermodynamic properties need to be investigated more specifically. It is significant to study the critical behaviors and the process of noncritical phase transition in detail. The analyses on the relation of some black holes in AdS spacetimes show a negative pressure appear as the temperature is below a certain value and a thermodynamic unstable region exists with Kubi1 (); Guna (); Zhao2 (); Zhao3 (); Alta2 (); Cai (); Zou1 (); Zou2 (); Fras (); Li (); Wei2 (); Kubi2 (); Xu (); Liu (); Hendi (). Negative pressure, that is, positive , doesn’t correspond to AdS spacetime. The situation is found in var der Waals-Maxwell liquid-gas system, where it is solved by the remarkable Maxwell’s equal area lawSpal1 (); Zhaojx (); Spal2 (); ZhangAHEP ().

In this paper Maxwell’s equal area law is extended to analyze the phase transition in the RN-AdS black holes and the Kerr-AdS black holes. It can help to resolve the puzzles about the negative pressure and the unstable region. By Maxwell’s equal area law we discover the possible two phase coexistence curves in the process of phase transition and the boundary curves of the coexistence region in plots. We expect to provide some particular information about phase transition and properties of black hole thermodynamic system in AdS spacetimes to contribute to search for more stable black holes and to explore properties of quantum gravity.

The paper is arranged as follow: the thermodynamic quantities of the -dimensional RN AdS black hole and the -dimensional Kerr AdS black hole are introduced firstly in section 2 and in section 3 respectively. Then the phase transition of the black holes are studied by Maxwell equal area law, and the effect of the dimension and the thermodynamic quantities, as electric charge in section 2 and angular momentum in section 3, on the phase transition is analyzed . In section 4, we make some discussion. (we use the units

Ii The charged black holes

ii.1 Thermodynamics

Reissner-Nordstrom black holes are characterized by the spherically symmetry and electrical nature. The solution for -dimensional RN-AdS spacetime with a negative cosmological constant, , is defined by the line elementCham1 ()

(1)

where

(2)

The ADM mass and the electric charge have been defined as (,

(3)

in which the volume of the unit -sphere can be expressed as

(4)

The corresponding Hawking temperature, entropy, and electric potential of the system are defined as

(5)

In our consideration we interpret the cosmological constant as a thermodynamic pressure Kastor:2009wy (),

(6)

The corresponding conjugate quantity, the thermodynamic volume, is given byKastor:2009wy ()

(7)

with being the position of black hole horizon determined from . Combining Eqs. (5) and (6), we can obtain

(8)

Comparing (8) with the Van der Waals equation, we conclude that the specific volume should be identified with the horizon radius of the black hole asKubi1 (); Guna ()

(9)

In geometric units we have

(10)

and the equation of state reads

(11)

In Fig.1, the isotherms are plotted in diagrams for different and . It shows there are thermodynamic unstable regions with in the extended phase space, and the situation of negative pressure appears when . The isotherm corresponding to is tangent to horizontal axis with , in which can be obtained by

(a)  ; , , 
(b)  ; , , 
(c)  ; , , 
Figure 1: Isotherms in diagrams. In each diagram the three groups of curves correspond to the three given values of , the lower ones (red curves) correspond to the smallest , the middle ones (blue curves) meet with the medium , and the upper ones (green curves) are with the biggest . The five curves in every group represent five different temperatures, the lower curve the lower temperature.
(12)
(13)

ii.2 Maxwell equal area law and two phase equilibria

The state equation of the -dimensional RN-AdS black hole is exhibited in the diagrams in Fig.1, which show the existence of negative pressure and thermodynamic unstable region with , which makes a thermodynamic system contract and expand automatically in classical thermodynamics. The case occurs in van der Waals equation, where the problems have been resolved by reference to some practical process of phase transition in real fluid and by Maxwell equal area law.

Using Maxwell equal area law to deal with the state equation of the -dimensional RN-AdS black hole, the two-phase equilibrium lines, where the two phases coexist, are acquired.

Take as critical temperature of the -dimensional RN-AdS black hole, and suppose at temperature ( the boundary state parameters for two-phase coexistence are and . According to Maxwell equal area law,

(14)

we can get

(15)
(16)

Set , from Eqs. (15) and (16), we can obtain

(17)

where

(18)

While , from (17) we get the critical specific volume

(19)

According to (16),

(20)

and the critical temperature and critical pressure are obtained as ,

(21)
(22)

These critical thermodynamic quantities are consistent with those in Ref.Guna ().

Combining (17), (20) and (21), as with , we can get

(23)

For a fixed , i.e. a fixed , we can evaluate with (23), and then according to (17) and (16), the , and are specified while is determined. Join the points and on the isotherm with in the diagram, which generate an isobar representing the process of phase transition like that of van der Waals system. Fig.2 shows the isobars with solid (red) straight lines and the boundary of the region of two phase coexistence by the dot-dashed (green) curve. In Fig.2 each combined solid line simulates the process of thermodynamic state change and phase transition of -dimensional RN-AdS black hole at a certain temperature. The phase transition process becomes shorter as temperature goes up until it turns into a single point at a certain temperature, which is the critical temperature, and the point corresponds to critical state of -dimensional RN-AdS black hole.


Figure 2: The simulated phase transition(red solid lines) and the boundary of two phase coexistence (green dot-dashed curve) on the base of isotherms in diagram for RN-AdS black hole with , .

When taking , we have calculated the quantities , , as =0.5, 1, 1.5 in the spacetimes with the dimension respectively. The results are shown in Table 1.



0.1 1.79E-6 5.74E5 1.51E-8 1.86E-4 3.17E3 1.72E-5 3.26-4 1.40E3 5.80E-5

0.3 0.00712 155. 1.61E-4 0.0378 16.0 0.00919 0.0470 9.99 0.0217

0.5 0.0496 24.3 0.00151 0.149 4.24 0.0497 0.173 2.80 0.110

0.7 0.149 9.15 0.00475 0.324 2.06 0.123 0.361 1.40 0.265

0.9 0.387 4.36 0.00995 0.595 1.23 0.225 0.635 0.862 0.477

0.1 1.79E-6 1.15E6 3.77E-9 1.86E-4 3.64E3 1.31E-5 3.26E-4 1.55E3 4.76E-5

0.3 0.00712 310. 4.03E-5 0.0378 18.4 0.00696 0.0470 11.0 0.0178

0.5 0.0496 48.7 3.78E-4 0.149 4.87 0.0377 0.173 3.09 0.0903

0.7 0.149 18.3 0.00119 0.324 2.37 0.0933 0.361 1.55 0.218

0.9 0.387 8.71 0.00249 0.595 1.42 0.170 0.635 0.951 0.391

0.1 1.79E-6 1.72E6 1.68E-9 1.86E-4 3.95E3 1.11E-5 3.26E-4 1.64E3 4.24E-5

0.3 0.00712 465. 1.79E-5 0.0378 20.0 0.00592 0.0470 11.7 0.0159

0.5 0.0496 73.0 1.68E-4 0.149 5.28 0.0320 0.173 3.28 0.0804

0.7 0.149 27.5 5.27E-4 0.324 2.57 0.0793 0.361 1.642 0.194

0.9 0.387 13.1 0.00111 0.595 1.54 0.145 0.635 1.01 0.348
Table 1: The parameters of two phase coexistence for RN-AdS black hole.

From Table 1, it can be seen that is unrelated to , but incremental with the increase of / at certain /. decreases with the increasing / and increases with the incremental when the other parameters are fixed respectively . increases with the incremental / and decreases with the increasing with others determined parameters respectively.

ii.3 curves of two phase coexistence

From isothermal curves in Fig.1, we know that when temperature , the negative pressure section on the isotherm emerges and becomes larger with lower temperature. From Fig.2, we can see that the lower temperature, the smaller the value of , which also can be seen in Table 1. The doubt is whether is negative when temperature is low enough.

Considering (17), (16) can be rewritten as

(24)

From (23), (24) and , one can get the relation between and , which is shown in Fig.3. Fig.3 exhibits the phase diagrams at fixed and , in which the curves represent the states of two phase coexistence and the terminal points are the critical points. From Fig.3 it can be seen that the influence of the electric charge and spacetime dimension on the phase diagrams, however pressure tends toward zero with decreasing temperature for all of the fixed and cases. That the pressure is always positive means Maxwell equal area law is appropriate to resolve the doubts about negative pressure and unstable states in phase transition of the -dimensional RN-AdS black hole. In Fig.4, the connection is depicted at different spacetime dimension , which explicitly exhibits the effect of the dimension on the distribution of the black hole phases.

(a)  ; , , 
(b)  ; , , 
(c)  ; , , 
Figure 3: phase diagrams at fixed and for -dimensional RN-AdS black hole.

Figure 4: phase diagram for -dimensional RN-AdS black hole with ,,

From (15)-(17), one can get

(25)

in which

(26)

From (25)

(27)

with . (27) stands for the Clapeyron equation of the thermodynamic system of the -dimensional RN-AdS black holes.

Iii Rotating Black Holes

iii.1 Thermodynamics

The AdS rotating black hole solution is given by the Kerr-AdS metricHawking3 (); Gibbons (); Alta1 (),

(28)

where

(29)

and denotes the metric element on a -dimensional sphere. The associated thermodynamic quantities are

(30)

in which meets , that is . Then

(31)

for in terms of and . While expressing , we expandedAlta1 ()

(32)

to some given order and solve Eq. (31) order by order. The first term yields the relation

(33)

Using Eqs.(6) and (30), the equation of can be obtained

(34)

where

(35)

While the isotherms are plotted in diagrams at respectively, which are shown in Fig.5. The situations of negative pressure and the thermodynamics unstable region with are also existent in the state equation the of Kerr-AdS black hole. When , where is the critical temperature of the Kerr-AdS black hole, unstable thermodynamic state appears, and while , negative pressure emerges. The expression of can be derived from (34) as in (12), and the corresponding is obtained by the way.

(a)  ;
(b)  ;
(c)  ;
Figure 5: the isotherms in diagrams . The four curves in each diagram correspond to , , , respectively.
(36)

iii.2 Maxwell equal area law

We use Maxwell equal area law to simulate a possible phase transition process of the -dimensional Kerr-AdS black hole as a thermodynamic system. We also take and as the specific volumes of the two associated phases in the phase transition respectively, and as the constant pressure throughout the simulated process of the phase transition at constant temperature (), which can be derived by Maxwell equal area law.

(37)

which can be rewritten as

(38)

Combining with

(39)

and setting , one can get

(40)

where

(41)

while , critical thermodynamics quantities can be got (40),

(42)

The results are consistent with that in Ref.Alta1 (). As and , we can derive from (40) and (42),

(43)

So we can solve at certain , then from (40) we can obtain the , and . Take , respectively. In Fig.6, we plot the isotherms (blue curves) for different in diagrams and the isobars (red solid lines) which represent the simulated phase transition processes derived from Maxwell equal area law. Like that in RN-AdS black hole the processes become shorter with increasing temperatures until it turns into a point at a certain temperature, which is the critical temperature, and the point is the critical point of the black hole thermodynamic system. Every connected solid curve represents a processes of state change and phase transition, where no negative pressure and thermodynamic unstable state appear. The boundary of the two-phase coexistence region is delineated by the dot-dashed line.


Figure 6: The simulated phase transition(red solid lines) and the boundary of two phase coexistence (green dot-dashed curve) on the base of isotherms in diagram for Kerr-AdS black hole with , .

In Table 2, we calculate , , as , , and respectively to find the influence of these parameters on the simulated phase transition process and the two-phase coexistence region.




0.1 1.98E-5 7.21E4 1.28E-7 2.56E-4 1.71E3 4.56E-5 3.95E-4 913. 1.17E-4

0.3 0.0166 90.9 2.86E-4 0.0428 10.5 0.0199 0.0506 7.29 0.0389

0.5 0.0859 18.8 0.00201 0.162 2.87 0.104 0.182 2.09 0.193

0.7 0.218 8.12 0.00556 0.345 1.42 0.253 0.375 1.05 0.460

0.9 0.475 4.36 0.0108 0.617 0.862 0.457 0.650 0.654 0.824

0.1 1.98E-5 1.14E5 5.12E-8 2.56E-4 1.99E3 3.36E-5 3.95E-4 1.02E3 9.31E-5

0.3 0.0166 144. 1.14E-4 0.0428 12.2 0.0146 0.0506 8.17 0.0309

0.5 0.0859 29.8 8.03E-4 0.162 3.34 0.0763 0.182 2.34 0.153

0.7 0.218 12.8 0.00222 0.345 1.65 0.186 0.375 1.18 0.367

0.9 0.475 6.89 0.00432 0.617 1.00 0.337 0.650 0.732 0.655

0.1 1.98E-5 1.61E5 2.56E-8 2.56E-4 2.23E3 2.67E-5 3.95E-4 1.12E3 7.83E-5

0.3 0.0166 203. 5.72E-5 0.0428 13.7 0.0116 0.0506 8.91 0.0260

0.5 0.0859 42.1 4.01E-4 0.162 3.75 0.0606 0.182 2.55 0.129

0.7 0.218 18.2 0.00111 0.345 1.86 0.148 0.375 1.29 0.308

0.9 0.475 9.75 0.00216 0.617 1.13 0.267 0.650 0.799 0.551
Table 2: The parameters of two phase coexistence for Kerr-AdS black hole

From Table 2, one can see that is unrelated to , but incremental with the increase of / at certain / . decreases with increasing / and increases with the incremental when the other parameters are fixed respectively. Similarly increases with the incremental / and decreases with increasing while the other parameters are given respectively.

iii.3 the relation of phase transition pressure to temperature

From (39) and , we get

(44)

Combining (43), (44) and , we plot the diagram and diagrams in Fig.7 and Fig.8.


Figure 7: phase diagram for -dimensional Kerr-AdS black hole with ,,.
(a)  ;
(b)  ;
(c)  ;
Figure 8: phase diagrams at fixed and for -dimensional Kerr-AdS black hole.

From (40), we get

(45)

where

Then

(46)

which is the Clapeyron equation of the Kerr-AdS black hole.

We can see from Fig.7, , and as . So there is no the situation of negative pressure and the thermodynamic unstable states in the simulated phase transition processes of -dimensional Kerr-AdS black hole, as that of the -dimensional RN-AdS black hole.

Iv Discussions

In this paper we take AdS black holes as thermodynamic systems and find the state equations in some region yield negative pressure and thermodynamic unstable states with . Comparing with phase transition in a usual liquid-gas system and using Maxwell equal area law, we simulate the phase transition process in AdS black holes. From Fig.2 and Fig.6, one can see that, for both the -dimensional RN-AdS black hole and the -dimensional Kerr-AdS black hole, as a part of each isotherm could be replaced with an isobar, which means phase transition occurs and two phases coexist in the section. The simulated phase transition belongs to the first order phase transition, and as the phase transition process shortens gradually until it become a point at , where the phase transition is the second order phase transition. The dimension and the electric charge (the angular momentum ) also have effect on the simulated phase transition process. The larger the dimension , the greater the change of specific volume before and after the phase transition. The larger the parameter , the lower the pressure , in the simulated phase transition.

Due to lack of the knowledge of chemical potential, the curve for two phase coexistence state of an usual thermodynamic system are obtained by experiment. However the slope of the curves can be calculated from the Clapeyron equation,

(47)

in which , and are the molar volumes of phase and phase respectively, and the Clapeyron equation accords with experiment result, which directly verifies its thermodynamic correctness.

In a general thermodynamic system, the phase transition means the change of organization structure of substance (including the changes of position and the orientation of atom, ion, and electron) and a different phase appear. Phase transition is a physical process, and it does not involve the chemical reaction, so the chemical compositions do not change in the process.

Appropriate theoretical interpretation to the phase structure of AdS black hole thermodynamic system can help to know more about black hole thermodynamic properties, such as entropy, temperature and heat capacity, and it is significant for improving a self-consistent black hole thermodynamic theory. This paper displays the diagrams for the two phase coexistence state and the Clapeyron equations of the black holes, which provide theoretical basis for investigation of phase transition and phase structure of black holes and help to explore the theory of gravity.

Acknowledgements.
This work is supported by NSFC under Grant Nos.(11475108,11175109;11075098), by the Shanxi Datong University doctoral Sustentation Fund Nos. 2011-B-03, China, Program for the Innovative Talents of Higher Learning Institutions of Shanxi, and the Natural Science Foundation for Young Scientists of Shanxi Province,China (Grant No.2012021003-4).

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