arXiv:1705.04923v2 [gr-qc] 16 May 2017
Thermodynamic stability, Q-Φ criticality,
microstructure of charged AdS black holes in
f (R) gravity and solution categorization
Gao-Ming Deng1,2∗, Yong-Chang Huang1†
1
Institute of theoretical physics,
Beijing University of Technology, Beijing 100124, China
2
Institute of theoretical physics,
China West Normal University, Nanchong 637002, China
Abstract
The electric charge Q and its potential Φ are employed as state parameters to further investigate properties of charged AdS black holes in f (R)
gravity, involving the thermodynamic stability, criticality and microstructure. Attempt in this new picture is launched from both macroscopic and
microscopic perspectives. In the macroscopic view, the charged AdS black
holes in f (R) gravity is thermally unstable and qualitatively behaves like
Van der Waals fluid. However, of especial interest, it is when the temperature T >Tc that the striking Q-Φ “oscillation” arises. Moreover, unlike the
ordinary case in general relativity, phase structures of the black holes in
f (R) theory exhibit an interesting dependence on gravity modification parameters. Microscopic analyses are implemented by using the state space
geometry. The Ruppeiner curvature shares the same divergence with the
heat capacity. Although the charged f (R) AdS black hole possesses the
property as ideal Bose gas, it manifests a completely opposite microscopic
behaviour to Van der Waals system. In addition, we distinguish black hole
solutions by performing a novel classification scenario.
∗
†
[email protected]
[email protected]
1
1
Introduction
There is now a general consensus that our Universe is experiencing an accelerating expansion [1]. To give a persuasive theoretical interpretation of such puzzling
acceleration, f (R) theory has been advanced as a promising candidate. This
theory has been widely applied in gravitation and cosmology [2, 3]. Especially,
it has succeeded in mimicking the historical evolution of the cosmology. For
more comprehensive reviews, see, e.g. [4, 5]. Since f (R) theory is considered as
an intriguing modification of general relativity, some distinctive properties from
Einstein’s gravity are highly expected. Further studies of this theory, for example,
black hole solutions and the thermodynamics etc., are of particular interest [6–8].
In general, it is a challenge to construct an exact f (R) black hole solution due to
high nonlinearity of field equations. Sheykhi achieved a higher dimensional black
hole solution [9] in f (R) gravity with a conformally invariant Maxwell field, abbreviated here as f (R)-CIM black hole solution. Motivated by these facts, this
present work will specialize in further investigating the thermodynamic properties of the charged f (R)-CIM AdS black holes [9], such as the thermodynamic
stability(instability), criticality and microstructure.
Black hole thermodynamics has been a long-standing and fascinating subject [10–14], which provides interesting clues to the underlying structure of quantum gravity. In particular, promoted by the development of the AdS/CFT correspondence [15, 16], the investigation of black hole phase transitions in AdS
spacetime has received popular attentions [17–22]. The pioneering study of these
transitions can be traced back to the well-known Hawking-Page phase transition [23], which describes a first-order phase transition between Schwarzschild
AdS black holes and the thermal AdS space. Inspired by the AdS/CFT, Witten
has explained such an appealing phenomenon as a confinement/deconfinement
transition in the dual quark gluon plasma [16]. For Reissner-Nordström AdS
black holes, Van der Waals-like phase transition was first observed by Chamblin
et al. [24, 25]. Soon afterwards, Cvetic et al. extended to elaborate phase transitions in gauged supergravity theory [26] and triggered popular discussions in
various theories like massive gravity [27, 28]. Caldarelli, Dolan, Banerjee et al.
later disclosed similar behaviours in the rotating case [17, 29, 30]. In Ref. [31],
Kubizňák, Mann completed the analogy of charged AdS black hole and obtained
the same critical exponents as that in Van der Waals fluid. Likewise, Hendi
et al. [32] generalized this analogy to higher dimensional Reissner-Nordström AdS
black holes and depicted the coexistence curves of large-small charged black holes.
Up to now, it is found that the Van der Waals-like phase transition widely exists
such as in BTZ black holes [33], Gauss-Bonnet black holes [34, 35], Born-Infeld
black holes [36, 37].
It is noteworthy that a great many previous phase structure analyses in AdS
spacetime were based on a popular proposal [38], which identifies the cosmology
constant Λ and its conjugate quantity as thermodynamic variables [31, 39–41],
1
i.e., pressure P and volume V , respectively. As a result, the temperature T
and the newly-introduced variables P , V were chosen as sate parameters to delineate phase structures and critical behaviours in various AdS spacetime backgrounds [42–46]. Nevertheless, aside from the variables P and V , if there exits
another pair of parameters available to describe black hole thermodynamics, such
as electric charge Q and its potential Φ. If so, some underlying thermodynamical
properties are expected. The answer is positive. For charged black holes, electric
charge Q is a natural intrinsic character. That is to say, a black hole is absolutely
endowed with electric charge(s) as long as it’s charged. In fact, Davies [47] has
implied the feasibility of state parameters (T, Q, Φ) in describing black hole thermodynamics. Consequently, our one attempt is focused on choosing (T, Q, Φ)
as state parameters to describe the physical picture of black hole phase transitions in AdS background. In particular, we will put this idea into practice with
the charged f (R)-CIM AdS black hole [9]. A well-defined statistical description still unclear, analyzing from diverse perspectives has an important physical
significance. After all, the more directions we view a problem from, the more
comprehensive and transparent picture we capture. In this sense, our approach
provides one more alternative candidate other than (P, V ) description.
The thermodynamic geometry was initiated by Weinhold [48] and has been
developed by Ruppeiner, Aman, Quevedo et al. [49–53]. Ruppeiner has argued
that the Riemannian geometry could give insight into the underlying statistical
mechanical system [50]. In this picture, divergence of the scalar curvature R is
closely related to phase transition. By calculating R, microscopic interactions
and the resulting macroscopic phase transition can be linked. More interestingly,
in Ref. [54], Ruppeiner pointed out that the sign of the curvature may be an
indicator of the interactions among the microstructure of thermodynamic system.
For instance, the curvature R<0 manifests the interaction as repulsive [55, 56],
corresponding to ideal Fermi gases, and conversely, R>0 manifests the interaction
as attractive, corresponding to ideal Bose gases. We will also further detect
the stability and possible properties of the microstructure for f (R)-CIM black
holes [9] from the geometrical perspective of thermodynamics.
The remainders of this Letter are outlined as follows. In the next section,
we briefly introduce the charged f (R)-CIM AdS black hole [9] and discuss its
thermodynamics. In sec. 3, both the local and global stabilities are analyzed, we
concentrate on employing (T, Q, Φ) as state parameters to investigate the critical
behaviours in detail. Sec. 4 is devoted to adopting the thermodynamic geometry
to further study the phase transition and underlying properties of microstructure.
In sec. 5, we perform a novel classification [57, 58] of the black hole configurations depending on the number of phase transitions. Finally, the summary and
discussions are given in sec. 6.
2
Charged f (R)-CIM AdS black holes and thermodynamics
2
To begin with, we briefly review the charged f (R)-CIM AdS black hole solution
[9] and its thermodynamics. Behaviours of the charged f (R)-CIM AdS black
hole are governed by the line element
ds2 = −N(r)dt2 +
with
N(r) = 1 −
dr 2
+ r 2 dθ2 + sin2 θdφ2 ,
N(r)
Q2 R0 2
2M
+ 2 −
r ,
ξr
r
12
(1)
ξ = 1 + f ′ (R0 ) ,
where the constant curvature scalar R0 is defined as R0 = 4Λ, Q and M are the
mass and electric charge of the black hole, respectively. Note that, the cosmology
constant Λ usually defines the spacetime background [59], for the AdS sapcetime,
Λ<0. If r+ denotes the radius of the outer event horizon, by setting N(r) = 0,
the mass of the black hole are given by
Q2 R0 2
ξr+
1+ 2 −
.
(2)
M=
r
2
r+
12 +
The electric potential, entropy and temperature can be calculated as
Φ=
T =
3
ξQ
,
r+
2
S = πξr+
,
4
2
−R0 r+
+ 4r+
− 4Q2
N ′ (r+ )
=
.
3
4π
16πr+
(3)
(4)
Instability and critical behaviours in Q-Φ plane
This section concentrates on exploring the instability and criticality. Interestingly, analyses will be done by employing (T, Q, Φ) as state parameters. We first
derive the equation of state (EoS). Combining Eqs.(3) and (4), the EoS has the
following form
Q=
i
p
2Φ h
2 ξ 2 T 2 − R (Φ2 − ξ 2 ) .
−4πξT
+
16π
0
R0 ξ 2
(5)
We express the electric charge Q as a function of the potential Φ and temperature
T . Moreover, Q relates to the parameters ξ and R0 as well.
Behaviours of the electric charge Q can be witnessed intuitively by plotting
Q-Φ diagram under different temperature, see Figure 1. As is depicted by gray
isotherms in the left diagram, there exists an interval where electric charge Q
3
Q
Q
T1
0.24
T5 > T4 > T3 > T2 > T1
T2
T > Tc
0.22
T3
D
Q2
0.20
A
QA
T4
C
E
0.18
T5
Q1
B
0.16
0.4
0.6
0.8
F
1.0
0.4
0.5
0.6
0.7
0.8
0.9
1.0
F
Figure 1: (color online) . Q-Φ diagram (isotherm) for different temperature. We
have set ξ=1.5 and R0 =−8. The right diagram exhibits a gray isotherm of the
left diagram.
increases as potential Φ increases. More details have been presented in the right
diagram. Obviously, one Q corresponds three possible values of Φ when Q1 < Q
) > 0, which
< Q2 . Along the curve segment between the points B and D, ( ∂Q
∂Φ T
does not satisfy the thermodynamical equilibrium condition and implies that the
black holes are in a thermally unstable phase. Fortunately, as is presented in
the left diagram, the isotherm will undergo a noticeable phase transition with a
further decrease of the temperature. Next, we will proceed to investigate phase
transitions and critical behaviours of the charged f (R)-CIM AdS black hole [9].
Taking into account the derivative of equation (5),
2 ∂ Q
∂Q
= 0,
= 0,
(6)
∂Φ T
∂Φ2 T
one obtains the critical point as
√
−R0
Tc = √
,
3 2π
Qc = √
1
,
−3R0
ξ
Φc = √ ,
6
(7)
where critical entities have been labelled with subscript “c”, R0 =4Λ takes negative values for the AdS backgrounds. These critical values are in agreement
with the outcomes in P -V description [60]. This fact implies validity of such a
description in Q-Φ phase plane. The existence of the solution confirms that the
charged f (R)-CIM AdS black hole is indeed thermally unstable. With the temperature decreasing, the black hole undergoes a phase transition at the critical
point (Tc , Qc , Φc ). The physical picture is shown in Figure 2. It is apparent that
this Q-Φ diagram qualitatively resembles the P -V behaviours of a Van der Waals
liquid-gas system. An (gray solid) isotherm of a f (R)-CIM AdS black hole has
a local minimum and maximum located at Φ1 and Φ2 respectively (seen from the
top-left diagram of Figure 2). As is mentioned above, along the curve segment
) > 0, which means that the black holes are in a
between the two points, ( ∂Q
∂Φ T
4
Q
Q
T11
0.24
0.24
T14 > T13 > Tc1 > T12 > T11
T12
0.22
T24 > T23 > Tc2 > T22 > T21
T21
0.22
Tc1
T22
HTc1 , Qc1 , Fc1 L
Qc1
0.20
0.20
T13
Tc2
HTc2 , Qc2 , Fc2 L
Qc2
0.18
0.18
T23
T14
0.16
T24
0.16
F1
0.2
0.4
Fc1 F2
0.6
0.8
1.0
1.2
Fc2
F
0.2
(a) ξ = 1.5, R0 = −8
0.4
0.6
0.8
1.0
1.2
F
(b) ξ = 1.5, R0 = −10
Q
Q
T31
0.24
0.24
T44 > T43 > Tc4 > T42 > T41
T34 > T33 > Tc3 > T32 > T31
T32
0.22
T41
0.22
Tc3
T42
HTc3 , Qc3 , Fc3 L
Qc3
0.20
0.20
T33
Tc4
Qc4
0.18
HTc4 , Qc4 , Fc4 L
0.18
T43
T34
0.16
0.16
0.14
0.04
Fc3
0.06
0.08
0.10
0.12
0.14
0.16
0.14
0.04
F
(c) ξ = 0.2, R0 = −8
T44
Fc4
0.06
0.08
0.10
0.12
0.14
0.16
F
(d) ξ = 0.2, R0 = −10
Figure 2: (color online) . Q-Φ diagram (isotherm) for different temperature. The
temperature of isotherms decreases from bottom to top. The critical isotherm
T = Tc is depicted by the red dashed line, the black dot denotes the critical point
(Tc , Qc , Φc ). Each diagram corresponds to different ξ and R0 .
thermally unstable phase, while the ( ∂Q
) < 0 parts denote that the black holes
∂Φ T
stably exist in the form of small black holes or the big case. In fact, the unphysical “oscillating” part of the isotherm (blue dashed curve in the right diagram of
Figure 1) can be replaced with an isocharge (red solid) line segment AE based
on the Maxwell’s equal area law (the shaded area below the isocharge line equals
to the above shaded part). Interestingly, differing from P -V description, the
striking “oscillation” presents when the temperature is higher than the critical
value. When the temperature comes down and reaches the critical temperature
Tc , the local minimum and maximum located at Φ1 and Φ2 approach and eventually merge into the critical potential Φc1 . Proceeding with lower temperature,
T < Tc , along the isotherm (blue solid curves in Figure 2), the electric charge
Q monotonously decreases with the potential Φ, the black hole exists stably. It
is worth emphasizing that, unlike the ordinary case in general relativity, Figure
2 exhibits an interesting dependence of phase structures on the parameters ξ
and R0 . The remarkable influence of R0 on isotherms is shown in the horizonal
direction, while the effect of ξ is listed in the vertical direction.
As a characteristic thermodynamical quantity, the heat capacity CQ at constant charge plays a crucial role in measuring the local stability against a small
5
perturbation. With Eqs.(3) and (4) in hand, CQ can be written as
2
4
2
2ξπr+
R0 r+
− 4r+
+ 4Q2
∂S
=
.
CQ = T
4
2
∂T Q
R0 r+
+ 4r+
− 12Q2
(8)
The heat capacity CQ varies with Q and the radius of the even horizon r+ . We
illustrate the relation between them in Figure 3. In general, the local stability
Figure 3: (color online) . Heat capacity CQ with r+ and Q for ξ=1.5, R0 =−8.
can be reflected by the heat capacity. In other words, the positive heat capacity
ensures stable existence of a black hole, while the negative sign means that a black
hole cannot establish a thermal equilibrium with the environment. In addition,
divergences signal the onset of a phase transition where the physics of the system
undergoes an abrupt change.
Observed from Figure 3, when Q takes a small value, there exists two sharp
divergences at the points r1 and r2 . The heat capacity CQ takes positive values for
r+ <r1 or r+ >r2 , while it is negative for r1 <r+ <r2 . Actually, r1 and r2 represent
two phase transition points. It is at the point r+ =r1 that a small stable black hole
with CQ > 0 completes an evolution and grows up to an intermediate unstable
black hole with CQ < 0. Proceeding with evolution, at the point r+ =r2 , the
intermediate unstable black hole develops into a large stable one.
When electric charge Q increases, the phase transition points located at r1
and r2 degenerate into one, eventually, disappear. To better understand this
interesting evolution process, we plot CQ versus r+ diagram in Figure 4. It is
evident that divergent behaviour vanishes and a f (R)-CIM AdS black hole ends
with a large stable state when Q>Qc .
The heat capacity CQ reflects the local stability of the black holes, while the
global thermodynamical properties of physical systems are characterized by the
Gibbs free energy. In what follows, we will step forward to discuss the Gibbs free
6
CQ
Q<Qc
60
Q=Qc
Q>Qc
30
0.4
r1
rc
r2
0.8
r+
1.2
-30
Figure 4: (color online) . CQ versus r+ for different electric charge. The red
dashed isocharge corresponds to the case of critical charge Qc . We have set
ξ=1.5 and R0 =−8.
energy. In an extended phase space which the cosmology constant Λ is included
in, the Gibbs free energy can be defined as
3
3Q2
ξ R0 r+
,
(9)
+ r+ +
G = M − TS =
4
12
r+
where M has the meaning of the enthalpy H [38–40], r+ = r+ (T, Q) acts as a
function of electric charge Q and temperature T through Eq.(4). The behaviour
of the Gibbs free energy G is depicted in the left diagram of Figure 5. Of particular
G
G
0.6
2.0
0.5
Q=1.5Qc1
1.5
Q=Qc1
1.0
0.4
0.3
Q=0.8Qc1
0.2
L=0.4Lc
Q=0.6Qc1
0.5
L=0.8Lc
L=0.6Lc
L=1.5Lc
L=Lc
0.1
0.0
0.10
0.15
0.20
0.25
0.0
T
0.02
0.04
0.06
0.08
Figure 5: (color online) . The Gibbs free energy G versus T for (left): different
charge Q, fixed R0 =4Λ=−8, (right): different Λ, fixed Q=1. We have set ξ=1.5.
The red solid curve corresponds to the critical case in both diagrams.
interest, in terms of the thermodynamical variables (T, Q, Φ), G-T diagram also
displays a characteristic “swallow tail” analogous to Van der Waals fluid. This
interesting behaviour reveals that there exists a first order phase transition for the
charged f (R)-CIM AdS black hole. Paralleling with the (T, Q, Φ) description,
we also plot G-T curves for different Λ when Q is fixed in the right diagram of
Figure 5.
7
T
By now, we have surveyed the local and global stabilities of the charged f (R)CIM AdS black hole, and investigated its critical behaviours from the macroscopic perspective. The results strongly imply that this black hole thermodynamics system qualitatively behaves like the Van der Waals fluid when adopting the
state parameters (T, Q, Φ). It is noteworthy that, unlike P -V description [60],
it is in the region of T >Tc in Q-Φ plane that the thermodynamic instability
arises. Next, we will turn to disclose the microscopic properties by using the
thermodynamic state space geometry.
4
Thermodynamic geometry and microstructure
A systematic microscopic statistical description of black holes is still a lack. According to Ruppeiner’s proposal, the Riemannian geometry could give insight into
the underlying statistical mechanical system. Inspired by close connection with
phase transitions, thermodynamic geometry has been widely used for black hole
research [42, 61–65]. The metric as defined by Ruppeiner [50] has the following
form
∂ 2 S(x)
ds2R = − i j dxi dxj ,
(10)
∂x ∂x
where xi denotes the extensive variables of the system. Practically, it’s convenient
to adopt Weinhold’s definition [48]
ds2W = −
∂ 2 U(x)
dxi dxj ,
∂xi ∂xj
(11)
where U is the internal energy of the system. It is found that these two definitions
are conformally related with each other
ds2R =
1 2
ds .
T W
(12)
As for the charged f (R)-CIM AdS black hole, U = M − QΦ, and we define
the coordinates x1 = S, x2 = Φ. Combining Eqs.(2-4), the Ruppeiner metric can
be calculated as
4
2
R0 r+
+ 4r+
− 4Q2
,
2
4
2
2πξr+
(−R0 r+
+ 4r+
− 4Q2 )
8Qr+
=
= gΦS ,
4
2
ξ (−R0 r+ + 4r+
− 4Q2 )
4
16πr+
.
=
4
2
ξ (−R0 r+
+ 4r+
− 4Q2 )
gSS =
(13)
gSΦ
(14)
gΦΦ
(15)
Programming with Mathematica 10.2, one can obtain the scalar curvature as
R=
4
ξπ (−R0 r+
+
2
4r+
8∆
,
4
2
− 4Q2 ) (R0 r+
+ 4r+
− 12Q2 ) 2
8
(16)
with
2
2
2
4
4
2
4
∆ = 4Q4 2 − 5R0 r+
− Q2 r+
16 − 18R0 r+
− 3R02 r+
+ r+
8 − 6R0 r+
− R02 r+
.
It is clear that, considering the nonextremal condition (4), the scalar curvature
(16) diverges when
4
2
R0 r+
+ 4r+
− 12Q2 = 0 .
(17)
Obviously, it shares the same denominator with the heat capacity CQ in Eq.(8).
Hence the curvature diverges exactly at the point where the heat capacity diverges. One can appreciate this consistency in Figure 6. Perhaps more interRCQ
800
CQ -r+
R-r+
600
400
200
r+
0.4
rc
0.6
0.8
Figure 6: (color online) . CQ versus r+ and R versus r+ for ξ=1.5, R0 =−8. The
scalar curvature shares the same divergence point with the heat capacity CQ .
estingly, the charged f (R)-CIM AdS black holes possess the property as ideal
Bose gases. As is alluded in sec. 1, Ruppeiner claimed that different sign of the
curvature manifests different interaction among the microstructure of the thermodynamic system [54]. For ideal Bose gas, the scalar curvature R>0 [55, 56]
and goes to positive infinity at critical point, and it corresponds to Bose-Einstein
condensation. While for ideal Fermi gas, R<0 [55, 56] and diverges to negative
infinity at critical point, which appeared as the Pauli’s exclusion principle forbidding two particles to stay in the same state with unlimited repulsive force.
For the Van der Waals liquid-gas system, R is always nonpositive, and this fact
means that attractive force cannot exist in such system.
In contrast with Van der Waals liquid-gas system, the charged f (R)-CIM
AdS black hole shows a remarkable difference in microscopic property. Figure
6 intuitively displays that the scalar curvature is always nonnegative and positively diverges. Just as ideal Bose gas, R>0 signals that attractive interaction is
permitted in the microscopic system of the charged f (R)-CIM AdS black hole.
9
5
Classification of black hole solutions in terms
of the number of phase transitions
Recently, Cembranos et al. [8, 57, 58] adopted a novel classification based on the
number of phase transitions to distinguish black hole solutions. For the charged
f (R)-CIM AdS black holes, as is aforementioned, phase transition was triggered
at the point where the heat capacity diverges, namely
p
2 −1 ± 1 + 3R0 Q2
4
2
2
R0 r+
+ 4r+
− 12Q2 = 0
⇒
r+
=
,
(18)
R0
where R0 =4Λ takes negative values for the AdS backgrounds. According to the
distinctive classification scenario [8,57,58], Eq.(18) allow us to classify the charged
f (R)-CIM AdS black hole solutions into two classes, in detail,
1
, Eq.(18) has two different solutions
• Slow black holes. When 0 < Q < √−3R
0
corresponding to signs “ ± ” respectively. It means that there exist two
different phase transition points for these black hole configurations. We
refer to these kinds of solutions as slow black holes.
1
2
, the equation about r+
in Eq.(18)
• Fast black holes. When Q > √−3R
0
have no solution, which means there is no any phase transition occurrence
for these black hole configurations. Consequently, we call these kinds of
solutions as fast black holes.
In Figure 7, we schematically display the regions in which black hole behaves
as the “slow” or “f ast” black hole respectively. It should be pointed out that,
Q
0.5
0.4
II
Fast BH
0.3
0.2
I
Slow BH
L-6
-5
-4
-3
-2
0.1
-1
Figure 7: (color online) . The distribution region of the slow black holes and f ast
black holes.
according to the analysis above, this classification scenario has nothing to do
with the gravitation modification parameter ξ for the charged f (R)-CIM AdS
black holes, while strongly depends on the electric charge Q and the cosmology
constant Λ.
10
6
Discussions and conclusions
In this Letter, we choose the intrinsic quantities (Q, Φ) as state parameters to
further investigated the thermodynamical stability, critical behaviours and phase
structure of the charged AdS black hole in f (R) theory. This work, on the one
hand, provided one more alternative candidate other than (P, V ) description, on
the other hand, disclosed some underlying properties which contribute to better
understanding f (R) gravity theory. A point worth stressing is that not only
does the f (R)-CIM AdS black hole possess critical phenomena akin to Van der
Waals liquid-gas system, but also some fascinating differences between them were
dug out in this Letter, no matter from macroscopic or microscopic perspective.
The detailed calculations and graphical illustrations confirmed that the charged
f (R)-CIM AdS black hole is thermally unstable and it undergoes a first order
phase transition. In the macroscopic view, the charged f (R)-CIM AdS black
hole qualitatively shares some critical behaviours with Van der Waals liquid-gas
system, including the characteristic Q-Φ “oscillation” curves and G-T “swallow
tail” phenomenon. However, unlike Van der Waals fluid, the Q-Φ “oscillation”
arises in the range of T >Tc . As a modification theory, f (R) gravity is highly
expected to find some distinctive properties from general relativity. Excitingly,
differing from black holes in Einstein’s gravity, phase structures of the black holes
in f (R) theory exhibit an interesting dependence on the modification parameters
ξ and R0 .
From the microscopic perspective, we adopted the thermodynamic geometry
and discussed the Ruppeiner scalar curvature R. It turns out that both the
curvature R and heat capacity CQ share the same divergence with each other.
This fact implies that the information of phase transition is contained in the
curvature. Furthermore, the charged f (R)-CIM AdS black hole behaves as ideal
Bose gas. In detail, the result that scalar curvature R>0 and diverges to positive
infinity corresponds to attractive interaction among the microscopic system. But
for Van der Waals liquid-gas system, it presents a completely opposite outcome.
Finally, we classified the black hole solutions into two types based on the number
of phase transitions, namely, the slow and f ast black holes.
As an attractive candidate for unraveling the puzzling accelerated expansion
of the universe, the f (R) theory deserves in-depth explorations. Research on
f (R) gravity in the holographic view may lead to another exciting highlights.
We leave this for future plan.
Acknowledgements
The authors would like to thank Y.-B. Deng for useful discussions. This work is
supported by the National Natural Science Foundation of China (Nos. 11275017
and 11173028).
11
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