# Emergent Phase, Thermodynamic Geometry and Criticality of Charged Black
Holes from Rényi Statistics

Ekapong Hirunsirisawat<EMAIL_ADDRESS>Theoretical and Computational
Physics (TCP); Theoretical and Computational Science Center (TaCS), Faculty of
Science, King Mongkut’s University of Technology Thonburi (KMUTT), Pracha
Uthit Road, Bangkok, 10140, Thailand  Learning Institute, King Mongkut’s
University of Technology Thonburi (KMUTT), Pracha Uthit Road, Bangkok, 10140,
Thailand  Ratchaphat Nakarachinda<EMAIL_ADDRESS>The Institute
for Fundamental Study, Naresuan University, Phitsanulok, 65000, Thailand
Chatchai Promsiri<EMAIL_ADDRESS>Department of Physics, Faculty
of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Pracha
Uthit Road, Bangkok, 10140, Thailand  Theoretical and Computational Physics
(TCP); Theoretical and Computational Science Center (TaCS), Faculty of
Science, King Mongkut’s University of Technology Thonburi (KMUTT), Pracha
Uthit Road, Bangkok, 10140, Thailand

###### Abstract

Recently, a novel emergent phase can occur from thermodynamic consideration of
the asymptotically flat Reissner-Nordström black hole (RN-AF) using Rényi
statistics. We present an analysis of the thermodynamical and mechanical
stabilities of the RN-AF in both the Gibbs-Boltzmann (GB) and the alternative
Rényi statistics when charge $q$ and electrostatic potential $\phi$ are
treated as pressure and volume, respectively. Interestingly, the emergent
phase of the RN-AF can be both thermodynamically and mechanically stable in
some range of parameters in the framework of Rényi thermodynamics. With the
construction of the Maxwell equal area law in $q-\phi$ plane, the coexistence
line between the near-extremal black hole phase and the emergent phase can be
found in some values of charge which can be associated as the vapor pressure
at which the liquid and gas phases coexist. In the aspect of thermodynamic
geometry, the microscopic interaction between the black hole microstructures
can be repulsive in the Rényi description. This implies that a novel
correlation between the microstates of a self-gravitating system could be
emerged via the nonextensive nature of long-range interaction systems.
Finally, we also investigate the critical phenomena of the RN-AF in Rényi
statistics compared to that of the van der Waals (vdW) fluid and find that the
critical exponents of the relevant physical quantities of both systems are
identical. This implies that both systems are in the same universality class
of the phase transition.

##  I Introduction and Motivations

According to the Gibbs-Boltzmann (GB) statistical approach, Bekenstein argued
that the entropy of black hole is proportional to its horizon area Bekenstein
(1973). Later on, by taking into account the quantum fluctuation around the
black hole’s horizon, Hawking had shown that black holes emit thermal
radiation at the Hawking temperature Hawking (1975). Moreover, both
quantities, namely, the Bekenstein-Hawking entropy $S_{\text{BH}}$ and Hawking
temperature $T_{\text{H}}$ are related through the first law of black hole
thermodynamics Bardeen  _et al._ (1973). However, black holes are fascinating
compact objects where the spacetime surrounded them is severely curved in such
a way that an observer, being outside their event horizons, cannot observe
anything inside the black hole event horizons. Therefore, the statistical
origin of the black hole entropy is hidden from the rest of the universe,
which leads to some important open questions on counting black hole
microstates and calculating its corresponding entropy from the first principle
Strominger and Vafa (1996); Callan and Maldacena (1996); Horowitz and
Strominger (1996); Emparan and Horowitz (2006); Dabholkar (2002); Giveon and
Itzhaki (2012); Brustein and Zigdon (2021); Giveon (2021).

One of the most peculiarities about black holes is that their entropy obeys
the area law and thus it is seemingly nonextensive Tsallis and Cirto (2013);
Tsallis (2020). Consequently, when two black holes of entropies $S_{\text{A}}$
and $S_{\text{B}}$ merge adiabatically, the resulting entropy of the black
hole system $S_{\text{AB}}$ can be expressed in a nonadditive form Davies
(1977). To investigate the effects from nonextensitivity, it has been proposed
that $S_{\text{AB}}$ can be written in the form of the Abe’s generalized
nonadditve composition rule Abe (2001)

$H\left(S_{\text{AB}}\right)=H\left(S_{\text{A}}\right)+H\left(S_{\text{B}}\right)+\lambda
H\left(S_{\text{A}}\right)H\left(S_{\text{B}}\right),$ (1)

where $H(S)$ is a differentiable function of $S$ and $\lambda$ is a
nonextensivity parameter. From statistical mechanics, one can derive the
thermodynamic quantities of macroscopic system from the microscopic
description via the standard statistical methods. In the GB approach, it is
usually assumed that the interaction between the microstructures of the system
is much smaller than the size of the system. In this way, the GB entropy
formula might not be appropriate for strong-gravitating systems because the
long-range interaction cannot be ignored. This is a microscopic description to
the origin of nonextensive nature for black hole thermodynamical systems.

However, there is an obscure notion of thermal equilibrium and empirical
temperature in thermodynamics of nonextensive entropy $H(S)$ with nonadditive
composition rule (1). Fortunately, this problem can be solved by transforming
$H(S)$ into another entropic functional form Biró and Ván (2011)

$\displaystyle L(S)=\frac{1}{\lambda}\ln\left[1+\lambda H(S)\right],$ (2)

which makes the composition rule (1) turning out to be additive

$\displaystyle L(S_{\text{AB}})=L(S_{\text{A}})+L(S_{\text{B}}).$ (3)

In the simplest case, we identify the differentiable function $H(S)$ to be the
Tsallis entropy $H(S)=S_{\text{T}}$ Tsallis (1988). In this case, the
logarithmic functional form of entropy in Eq. (2) is indeed the well-known
Rényi entropy Rényi (1959)

$\displaystyle L(S_{\text{T}})=\frac{1}{\lambda}\ln(1+\lambda
S_{\text{T}})\equiv S_{\text{R}}.$ (4)

Since the Rényi entropy is additive despite the presence of nonextensive
nature, this tends to a well-defined thermal equilibrium state for
nonextensive systems and then the empirical temperature function can be
uniquely determined from the standard relation

$\displaystyle\frac{1}{T_{\text{R}}}=\frac{\partial S_{\text{R}}}{\partial
E},$ (5)

where $T_{\text{R}}$ is the corresponding so-called Rényi temperature and $E$
is the energy of the system.

Interestingly, by considering $S_{\text{BH}}$ as $S_{\text{T}}$ Tsallis and
Cirto (2013), one can obtain the Rényi entropy associated to a black hole. The
results show that the black holes can be thermodynamically stable with its
surrounding environment in some range of parameter space in the cases of
asymptotically flat spacetime Biró and Czinner (2013); Czinner and Iguchi
(2016, 2017); Promsiri  _et al._ (2020, 2021); Alonso-Serrano  _et al._ (2021)
and asymptotically dS spacetime Tannukij  _et al._ (2020); Nakarachinda  _et
al._ (2021); Samart and Channuie (2020). In addition, the stability of a black
string is also investigated using the Rényi statistics. Sriling  _et al._
(2021). Recently, as suggested in Refs. Promsiri  _et al._ (2020, 2021), the
nonextensivity parameter $\lambda$ can be identified as a thermodynamic
pressure $P=\frac{3\lambda}{32}$ and its conjugate variable as the
thermodynamic volume $V=\frac{4}{3}\pi r_{h}^{3}$, where $r_{h}$ is the
horizon radius. This framework is called the Rényi extended phase space
approach. In this formalism, it was first indicated that, in a canonical
ensemble, the Reissner-Nordström black hole in asymptotically flat spacetime
(RN-AF) allows a first-order phase transition which is reminiscent of the vdW
liquid-gas phase transition and a Hawking-Page phase transition of charged AdS
black holes Chamblin  _et al._ (1999a, b); Peca and Lemos (1999); Burikham and
Promsiri (2016); Kubiznak and Mann (2012); Gunasekaran  _et al._ (2012); Majhi
and Samanta (2017) (for review, see Kubiznak  _et al._ (2017); Altamirano  _et
al._ (2014); Mann (2015) and references therein).

In a conventional thermodynamic system, the well-known Maxwell equal area law
or Maxwell construction is a qualitative explanation for the first-order phase
transition between two different locally stable phases without passing the
unstable phase. The vdW fluid undergone in the isothermal compression or
expansion is an example of the Maxwell equal area law. On the $P-V$ plane, it
has been found that there possibly exist three phases along each isothermal
curve. Two of them are the liquid and gas phases with positive
compressibility. Another one is the phase with negative compressibility.
Hence, the phase transition of the fluid via the isothermal compression or
expansion should pass the mechanically unstable phase. The argument states
that the fluid will evolve under the first-order phase transition with a
certain pressure instead of passing through the unstable phase. From the fact
that the Gibbs free energy does not change David (2016), the certain pressure
is evaluated in the way that the upper and lower areas between the isothermal
curve and the line of constant pressure are the same. Interestingly, this
equal area law is held in the charged black holes (see e.g. Spallucci and
Smailagic (2013); Ma  _et al._ (2016); Zhou and Wei (2019)). In our case, the
aforementioned equal area law should be obtained by considering the isothermal
curve on the $q-\phi$ plane. It will be seen that the charged black hole with
Rényi description undergoes the transition between two stable black hole
phases in a similar way to what occurs in the vdW fluid.

One may argue that the correction due to nonextensive effect should be also
seen in the microscopic structure of black hole systems. In the phase
transition, the thermodynamic systems are undergoing some reorganization
processes of its microscopic structure. As mentioned above, the charged black
hole cannot be only a thermal object, but it also interestingly undergoes a
phase transition as the vdW liquid-gas system does. Therefore, it is believed
that a black hole should have microscopic degrees of freedom even if its
constituents are still unknown. There is a proposal on possible ways to study
the microstructure or the interaction between black hole molecules by using
the geometrical approach of the thermodynamic phase space. The thermodynamic
geometry method has been developed by Weinhold Weinhold (1975) and Ruppeiner
Ruppeiner (1995). It is an interesting approach to investigate some
microscopic aspects of thermodynamic system from their macroscopic quantities.
Ruppeiner Ruppeiner (1995) proposed that an entropy plays a role of the
thermodynamic potential rather than the internal energy Weinhold (1975), and
then derived a Riemannian thermodynamic line element, which has physical
meaning as the distance between two neighbouring thermal fluctuating states.
The scalar curvature $R$ corresponding to the Ruppeiner metric can be computed
in the same way as the Ricci scalar in the general relativity. By applying
this formalism to various thermodynamic systems Ruppeiner (1979); Brody and
Rivier (1995); Brody and Hook (2009); Ruppeiner (1981); Janyszek and Mrugal/a
(1989); Oshima  _et al._ (1999), it has been argued that the sign of $R$ can
be used to identify the type of interaction between the microstructure of a
system. The positiveness and negativeness in $R$ are, respectively, related to
the repulsive and attractive interactions, whereas $R=0$ corresponds to the
systems with no interaction between its constituents, for a review see
Ruppeiner (2010) and the references therein.

There is a connection between the microscopic fluctuation of matters and
macroscopic response function, such as the heat capacity and compressibility.
At the second-order phase transition, these response functions go to infinity
which means the correlated fluctuation being over large distance. Typically,
such correlation occurs over a characteristic distance $\xi$, called the
correlation length. Ruppeiner has shown that $R$ is proportional to the
correlation volume of the system as Ruppeiner (1995)

$\displaystyle R\sim\xi^{d},$ (6)

where $d$ is the physical dimension of the system. This is a further evidence
about the connection between $R$ and microscopic correlation of the system.
According to the connections between $R$ and some microscopic aspects, the
thermodynamic geometry approach might be suitable for probing some information
about microstructure of a black hole. The interesting studies on thermodynamic
geometry of various black holes in different theories of gravity can be seen
in Refs. Aman  _et al._ (2003); Aman and Pidokrajt (2006); Shen  _et al._
(2007); Niu  _et al._ (2012); Wang  _et al._ (2020); Wei and Liu (2013, 2015);
Wei  _et al._ (2019a); Wei and Liu (2021); Yerra and Bhamidipati (2020); Wu
_et al._ (2021); Ghosh and Bhamidipati (2020); Yerra and Bhamidipati (2021).

In the present work, we study further in a critical phenomena and microscopic
origin of the Rényi entropy associated to the black hole. The rest of this
paper is organized as follows. In section II, we introduce an analogy
$(q,\phi)\leftrightarrow(P,V)$ between RN-AF and vdW liquid-gas system from
the first law of thermodynamics. In this way, we shall revisit thermal
stability criteria based on the signs of the associated response functions
such as the heat capacity and compressibility. In section III, the thermal
phase structure of RN-AF and its stability are studied within the Rényi
statistics. In section IV, we present two different expressions of the Maxwell
equal area law in the $q-\phi$ plane when the systems are near and far away
from the critical point. At this point, the coexistence region and phase
diagram are also studied. In section V, we first briefly review the
thermodynamic geometry of RN-AF with the GB statistics. The phase structure
and types of interaction between microstructure of black hole within Rényi
statistics are then studied in the thermodynamic geometry framework. In
section VI, we investigate the thermal behavior of the RN-AF via the Rényi
statistics near the critical point. All critical exponents have been
calculated. We summarize and discuss our results in section VII .

##  II Thermodynamics of Reissner-Nordström black holes from the Gibbs-
Boltzmann statistics

In the asymptotically flat spacetime, the Reisser-Nordström solution to the
Einstein field equation with the U(1) electric charge can be written in the
form

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ $\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1-\frac{2M}{r}+\frac{q^{2}}{r^{2}}.$ (7)

By solving the horizon equation, $f(r)=0$, there exist the inner and outer
horizon with the radii $r_{-}$ and $r_{+}$, following the relation

$\displaystyle r_{\pm}=M\pm\sqrt{M^{2}-q^{2}}.$ (8)

The black hole is extremal when the inner and outer horizons are identical at
$M=q$. For $M<q$, the black hole cannot exist because the event horizons in
Eq. (8) are complex numbers, and, hence, the cosmic censorship hypothesis will
be violated Penrose (1969). The mass and electric charge can be, respectively,
expressed in terms of the horizon radii as follows:

$\displaystyle M$ $\displaystyle=$ $\displaystyle\frac{r_{+}+r_{-}}{2},$ (9)
$\displaystyle q$ $\displaystyle=$ $\displaystyle\sqrt{r_{+}r_{-}}.$ (10)

In the standard approach, the Bekenstein-Hawking entropy of the black hole is
obtained from the area law Bekenstein (1973)

$\displaystyle S_{\text{BH}}=\frac{\mathcal{A}}{4}=\pi r_{+}^{2}.$ (11)

where $\mathcal{A}$ is the surface area of the outer horizon radius. The first
law of black hole thermodynamics reads Bardeen  _et al._ (1973)

$\displaystyle dM=T_{\text{H}}dS_{\text{BH}}+\phi dq,$ (12)

which implies that the Hawking temperature $T_{\text{H}}$ and electrostatic
potential $\phi$ on the outer event horizon are, respectively, given by

$\displaystyle T_{\text{H}}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial M}{\partial
S_{\text{BH}}}\right)_{q}=\frac{r_{+}-r_{-}}{4\pi r_{+}^{2}},$ (13)
$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
M}{\partial q}\right)_{S_{\text{BH}}}=\frac{q}{r_{+}}.$ (14)

Note that Eq. (13) manifests the fact that that the Hawking temperature is
vanishing at the extremal limit, where the outer and inner horizon radii are
degenerate.

By considering the first and second derivatives of the temperature
$T_{\text{H}}=T_{\text{H}}(r_{+},q)$ with respect to $r_{+}$, this temperature
has a maximum value given by $T_{\text{max}}=(6\sqrt{3}\pi q)^{-1}$ at the
outer horizon radius being equal to

$\displaystyle r_{0}=\sqrt{3}q.$ (15)

---
Figure 1: The dimensionless Hawking temperature $T_{\text{H}}q$ (left) and
heat capacity $C_{q}/q^{2}$ (right) of the RN-AF via the GB statistics with
respect to the dimensionless outer horizon radius $r_{+}/q$.

Therefore, the profile of this temperature can be divided into two different
behaviours which are increasing and decreasing function in $r_{+}$ for
$r_{+}<r_{0}$ and $r_{+}>r_{0}$, respectively. This is shown in the left panel
in Fig. 1. As discussed previously, a black hole horizon exists when
$r_{+}\geq q$. Using Eq. (14), this implies that the electric potential is in
the range $0<\phi\leq 1$, where its upper limit corresponds to the extremal
black hole.

The heat capacity for fixed charge is computed as

$\displaystyle C_{q}=T_{\text{H}}\left(\frac{\partial S_{\text{BH}}}{\partial
T_{\text{H}}}\right)_{q}=\frac{2\pi r^{2}_{+}(r_{+}-r_{-})}{3r_{-}-r_{+}}.$
(16)

As a result, there are two possible black hole configurations, namely, the
small black hole branch with positive heat capacity, and the large black hole
branch with negative heat capacity in which we will henceforth denote as the
pBH, and nBH, respectively. One also calls that the pBH (nBH) is locally
stable (unstable). The nBH-pBH phase transition occurs when the outer event
horizon equals $r_{+}=r_{0}$. At the radius $r_{0}$, the temperature reaches
its maximum and the heat capacity diverges. The profiles of the temperature
and heat capacity for the non-zero charged black hole with respect to the
outer horizon are illustrated in Fig. 1. In this figure, the red (green)
curves represent the quantities for pBH (nBH) branch. Since the entropy is
indeed a monotonic function of $r_{+}$ as obviously seen in Eq. (11), the sign
of the heat capacity can be obtained from that of the slope of temperature
profile graph (the left panel of Fig. 1). Note that eliminating $r_{-}$ in
Eqs. (13) and (16) by using Eq. (10) allows us to express the outer horizon in
the scale of $q$, i.e. $r_{+}/q$, as seen in Fig. 1.This can be done due to
the fact that the phase structure of this system is not changed for different
values of non-zero charge $q$. However, it is very important to emphasize that
the scaling by $q$ is not appropriate for black hole thermodynamics from the
Rényi statistics because the phase structure depends on $q$ in that
circumstance, as will be discussed in the next section.

It has been observed that the charge $q$ and electrostatic potential $\phi$ of
the charged black holes can be identified as the pressure $P$ and volume $V$
of the vdW fluid, respectively Chamblin  _et al._ (1999a, b); Peca and Lemos
(1999). From the first law of black hole thermodynamics, we have

$\displaystyle dM$ $\displaystyle=$ $\displaystyle
T_{\text{H}}dS_{\text{BH}}+\phi dq$ $\displaystyle=$ $\displaystyle
T_{\text{H}}dS_{\text{BH}}-qd\phi+d(q\phi),$

then we can write

$\displaystyle d(M-q\phi)$ $\displaystyle=$ $\displaystyle
T_{\text{H}}dS_{\text{BH}}-qd\phi.$ (18)

By comparing this relation to the conventional first law, $dU=TdS-PdV$, with
identifying $qd\phi$ as $PdV$, one can define the appropriate internal energy
of the RN-AF as

$\displaystyle U=M-q\phi,$ (19)

where the contribution of the electric potential energy to the mass of black
hole is excluded Shen  _et al._ (2007). In this analogy, the black hole mass
$M$ is indeed a function of $S_{\text{BH}}$ and $q$, $M=M(S_{\text{BH}},q)$.
Therefore, the mass should be interpreted as the enthalpy rather than the
internal energy.

In order to investigate the phase structure of the thermodynamic system, one
needs to consider the Gibbs free energy which is defined as
$\mathcal{G}=U-TS+PV$. As mentioned previously, the internal energy of the
black hole should be taken in the form as expressed in Eq. (19). The charge
and its conjugate variable, i.e., the electrostatic potential are then
interpreted as a pressure and volume, respectively. We, therefore, define an
appropriate Gibbs free energy for RN-AF in the following form

$\displaystyle\mathcal{G}=U-T_{\text{H}}S_{\text{BH}}+q\phi=M-T_{\text{H}}S_{\text{BH}}=\frac{1}{4}(r_{+}+3r_{-}).$
(20)

---
Figure 2: The profile of the dimensionless Gibbs free energy $\mathcal{G}/q$
of the RN-AF via GB statistics with respect to the dimensionless outer horizon
radius $r_{+}/q$ (left) and the Hawking temperature $T_{\text{H}}/q$ (right).

The infinitesimal change of $\mathcal{G}=\mathcal{G}(T_{\text{H}},q)$ can be
expressed as follows:

$\displaystyle d\mathcal{G}$ $\displaystyle=$ $\displaystyle dM-
T_{\text{H}}dS_{\text{BH}}-S_{\text{BH}}dT_{\text{H}}$ (21) $\displaystyle=$
$\displaystyle(T_{\text{H}}dS_{\text{BH}}+\phi
dq)-T_{\text{H}}dS_{\text{BH}}-S_{\text{BH}}dT_{\text{H}}$ $\displaystyle=$
$\displaystyle-S_{\text{BH}}dT_{\text{H}}+\phi dq,$

where we have used (II) for $dM$, this yields

$\displaystyle\left(\frac{\partial\mathcal{G}}{\partial
T_{\text{H}}}\right)_{q}=-S_{\text{BH}},\hskip
28.45274pt\left(\frac{\partial\mathcal{G}}{\partial
q}\right)_{T_{\text{H}}}=\phi.$ (22)

It is seen that the Gibbs free energy is always positive for both pBH and nBH
branches, and reaches its minimum value $\mathcal{G}_{\text{min}}=\sqrt{3}q/2$
at the same point where the nBH-pBH phase transition occurs ($r_{+}=r_{0}$).
The profile of the free energy is illustrated in Fig. 2. Note that, for the
black hole with a certain value of free energy $\mathcal{G}$, there are two
possible black hole phases which are small and large ones. The small (large)
branch is locally stable (unstable). As seen in a right panel in Fig. 2, the
nBH-pBH phase transition occurs at the cusp where the second derivative of
$\mathcal{G}$ proportional to the heat capacity as
$\Big{(}\frac{\partial^{2}\mathcal{G}}{\partial
T_{\text{H}}^{2}}\Big{)}_{q}=-\frac{C_{q}}{T_{\text{H}}}$ is discontinuous.
Hence, the nBH-pBH transition is actually the second-order type of phase
transition.

Moreover, it is possible to construct a constraint surface
$F(q,\phi,T_{\text{H}})=0$ similar to the equation of state $F(P,V,T)=0$ in
the conventional thermodynamics. Using Eqs. (10) and (14), the Hawking
temperature in Eq. (13) can be expressed in terms of $\phi$ and $q$ in the
form

$\displaystyle T_{\text{H}}=\frac{(1-\phi^{2})\phi}{4\pi q}.$ (23)

From the above expression, we can write

$\displaystyle q=\frac{(1-\phi^{2})\phi}{4\pi T_{\text{H}}},$ (24)

which is indeed the equation of state of this system. The isothermal curves on
the $q-\phi$ plane are plotted in the left panel in Fig. 3. The heat capacity
can be expressed as

$\displaystyle C_{q}=-\frac{2\pi
q^{2}}{\phi^{2}}\left[\frac{(1-\phi^{2})}{(1-3\phi^{2})}\right].$ (25)

Note that the sign change of $C_{q}$ as well as its divergence occur at the
value

$\displaystyle\phi_{0}=\frac{1}{\sqrt{3}},$ (26)

which actually corresponds to $r_{0}$. Each isothermal curve is, therefore,
divided into the branches of pBH ($\phi>1/\sqrt{3}$) and nBH
($\phi<1/\sqrt{3}$).

Furthermore, the thermodynamical properties of RN-AF as discussed above will
make sense if a black hole can be in locally thermal equilibrium against
microscopic fluctuations. The conditions for local stability are studied in
terms of both the heat capacity $C_{q}$ and compressibility
$\kappa_{T_{\text{H}}}$ as

$\displaystyle C_{q}>0,\hskip 28.45274pt\kappa_{T_{\text{H}}}>0.$ (27)

According to the Le Chatelier principle which states that if a extensive
variable fluctuate from its equilibrium value, its conjugate intensive
variable will change in such a way that extensive variable is restored to its
equilibrium Landau and Lifshitz (1980); Callen (1985); Sekerka (2015). In this
way, a thermal fluctuation will increase (decrease) the entropy so that the
increasing (decreasing) temperature resists the heat flowing between the
system and its environment, resulting to the reach into equilibrium.
Therefore, the entropy monotonically increases with temperature. This means
that the heat capacity is positive, $C_{q}>0$.

---
Figure 3: Left: The isothermal curves on the $q-\phi$ plane with different
temperatures. Right: the profiles of the heat capacity $C_{q}/q^{2}$ (solid
lines) and the compressibility $\kappa_{T_{\text{H}}}q$ (dashed lines) with
respect to the electrostatic potential $\phi$, where the green (red) lines are
the quantities for nBH (pBH) branch.

Let us consider another local stability. In the conventional fluid system, the
work done on (by) a system decreases (increases) its volume. After that the
pressure should increase (decrease) in order to enforce the system to restore
its equilibrium state. This behavior also leads us to define the the
isothermal compressibility, $\kappa_{T}=-\frac{1}{V}\big{(}\frac{\partial
V}{\partial P}\big{)}_{T}$. The minus sign is introduced in order to identify
the mechanically stable (unstable) system as a system with positive (negative)
compressibility.

We have mentioned that $q$ and $\phi$ of the charged black holes play the
similar roles to pressure $P$ and volume $V$ of the vdW fluid, respectively,
in the aspect of phase structure. This might be interpreted that the
mechanical stability in conventional system is actually the mechanical
stability for charged black holes. Hence, the isothermal compressibility for
RN-AF can be defined as

$\displaystyle\kappa_{T_{\text{H}}}=-\frac{1}{\phi}\left(\frac{\partial\phi}{\partial
q}\right)_{T_{\text{H}}}=-q\left[\frac{(1-\phi^{2})}{(1-3\phi^{2})}\right].$
(28)

Obviously, the vanishing and divergent points of $\kappa_{T_{\text{H}}}$ and
$C_{q}$ are the same (see the terms in the square brackets in Eqs. (25) and
(28)). The signs of $C_{q}$ and $\kappa_{T_{\text{H}}}$ are also identical for
any value of $\phi$. Therefore, the black hole with positive (negative) heat
capacity has positive (negative) compressibility. In other words, the pBH
(nBH) is both thermally and mechanically stable (unstable). The profiles of
$C_{q}$ and $\kappa_{T_{\text{H}}}$ are illustrated in the right panel in Fig.
3. In addition, the discontinuous point of the compressibility
$\kappa_{T_{\text{H}}}$ [i.e., $\phi=1/\sqrt{3}$] is indeed the place where
the second-order phase transition occurs in the isothermal process, since the
second derivative of the free energy $\mathcal{G}$ is proportional to
$\kappa_{T_{\text{H}}}$ as follows:
$\Big{(}\frac{\partial^{2}\mathcal{G}}{\partial
q^{2}}\Big{)}_{T_{\text{H}}}=-\kappa_{T_{\text{H}}}\phi$.

It has been found that there does not exist a critical phenomenon in the RN-AF
using the GB statistics. It is because there is no solution for $q$, $\phi$
and $T_{\text{H}}$ satisfying the criticality condition,
$\big{(}\frac{\partial
q}{\partial\phi}\big{)}_{T_{\text{H}}}=\big{(}\frac{\partial^{2}q}{\partial\phi^{2}}\big{)}_{T_{\text{H}}}=0$.
In the next section, the nonextensivity in the black hole entropy is
introduced via the Rényi statistics. The critical phenomena and other
interesting behaviors as well as the difference between the black holes with
the GB and Rényi statistics will be studied.

##  III The emergent phase and swallowtail behavior of RN-AF in Rényi
statistics

For an alternative Rényi statistics, the Rényi entropy $S_{\text{R}}$ of the
black holes can be expressed in terms of the Bekenstein-Hawking entropy
$S_{\text{BH}}$ described by the Tsallis entropy as discussed in Refs Czinner
and Iguchi (2016, 2017); Promsiri  _et al._ (2020, 2021); Alonso-Serrano  _et
al._ (2021); Tannukij  _et al._ (2020); Nakarachinda  _et al._ (2021); Samart
and Channuie (2020),

$\displaystyle S_{\text{R}}=\frac{1}{\lambda}\ln(1+\lambda S_{\text{BH}}),$
(29)

where $\lambda$ is the nonextensivity parameter. It is also found that the
above entropic function obeys the additive composition rule so that the zeroth
law is compatible Biró and Ván (2011). As mentioned in Refs. Promsiri  _et
al._ (2020, 2021), this approach has an energy (length) scale defined via the
nonextensivity parameter $\lambda$ by

$\displaystyle L_{\lambda}=\frac{1}{\sqrt{\pi\lambda}}.$ (30)

The first law of black hole Rényi thermodynamics is given by

$\displaystyle dM=T_{\text{R}}dS_{\text{R}}+\phi dq.$ (31)

Note that the mass parameter still plays the same role of the enthalpy. Only
the heat term for Rényi statistics $T_{\text{R}}dS_{\text{R}}$ is modified
from that for GB statistics. The Rényi temperature $T_{\text{R}}$ and
electrostatic potential $\phi$ corresponding to the Rényi entropy,
respectively, are

$\displaystyle T_{\text{R}}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial M}{\partial
S_{\text{R}}}\right)_{q}=\left(\frac{r_{+}-r_{-}}{4\pi
r_{+}^{2}}\right)\left(1+\frac{r_{+}^{2}}{L^{2}_{\lambda}}\right),$ (32)
$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
M}{\partial q}\right)_{S_{\text{R}}}=\frac{q}{r_{+}}.$ (33)

The heat capacity and the Gibbs free energy are, respectively, computed as

$\displaystyle C_{\text{R}}$ $\displaystyle=$ $\displaystyle
T_{\text{R}}\left(\frac{\partial S_{\text{R}}}{\partial
T_{\text{R}}}\right)_{q}=\frac{2\pi
r^{2}_{+}(r_{+}-r_{-})}{3r_{-}-r_{+}+\frac{r^{2}_{+}}{L^{2}_{\lambda}}(r_{+}+r_{-})},$
(34) $\displaystyle\mathcal{G}_{\text{R}}$ $\displaystyle=$ $\displaystyle
M-T_{\text{R}}S_{\text{R}}=\frac{r_{+}+r_{-}}{2}-\frac{(r_{+}-r_{-})(r_{+}^{2}+L_{\lambda}^{2})}{4r_{+}^{2}}\ln\left(1+\frac{r_{+}^{2}}{L_{\lambda}^{2}}\right).$
(35)

One can check that, in the limit $L_{\lambda}\to\infty$ (or $\lambda\to 0$),
$T_{\text{R}},C_{\text{R}}$ and $\mathcal{G}_{\text{R}}$ are reduced to
$T_{\text{H}},C_{q}$ and $\mathcal{G}$ for the GB statistics case,
respectively.

As mentioned previously, the RN-AF via Rényi statistics has a critical
behavior. In other words, there exist the nontrivial solutions for $q$ and
$\phi$ of the conditions determining the critical point,

$\displaystyle\left(\frac{\partial
q}{\partial\phi}\right)_{T_{\text{R}}}=\bigg{(}\frac{\partial^{2}q}{\partial\phi^{2}}\bigg{)}_{T_{\text{R}}}=0.$
(36)

Using Eqs. (10) (32) and (33), the Rényi temperature is written as

$\displaystyle T_{\text{R}}=\frac{(1-\phi^{2})\Big{(}\phi+\frac{q^{2}}{\phi
L_{\lambda}^{2}}\Big{)}}{4\pi q}.$ (37)

From the above equation of state, the conditions (36) can be solved for the
charge, potential and temperature at critical point which are, respectively,
obtained in the following:

$\displaystyle q_{c}^{2}$ $\displaystyle=$ $\displaystyle
L_{\lambda}^{2}\big{(}7-4\sqrt{3}\,\big{)},$ (38) $\displaystyle\phi_{c}^{2}$
$\displaystyle=$ $\displaystyle\frac{2-\sqrt{3}}{\sqrt{3}},$ (39)
$\displaystyle T_{c}$ $\displaystyle=$ $\displaystyle
T_{\text{R}}\big{|}_{q=q_{c},\phi=\phi_{c}}=\frac{2}{3\pi
L_{\lambda}}\sqrt{2\sqrt{3}-3}.$ (40)

To investigate the phase structure of RN-AF via Rényi statistics, our analysis
can be simplified by rescaling the parameters in the system as follows:

$\displaystyle q\rightarrow\bar{q}=\frac{q}{L_{\lambda}},\hskip
28.45274pt\phi\rightarrow\bar{\phi}=\phi,\hskip
28.45274ptr_{\pm}\rightarrow\bar{r}_{\pm}=\frac{r_{\pm}}{L_{\lambda}}.$ (41)

---
Figure 4: The rescaled Rényi temperature $\bar{T}_{\text{R}}$ (left) and heat
capacity $\bar{C}_{\text{R}}$ (right) of the RN-AF via Rényi statistics with
respect to the rescaled outer horizon radius $\bar{r}_{+}$ for various values
of the rescaled charge $\bar{q}$. The radii $\bar{r}_{1}$ and $\bar{r}_{2}$ in
these plots are the values only for the case $\bar{q}=0.2$.

The Rényi temperature, heat capacity, compressibility and Gibbs free energy
can be rescaled as

$\displaystyle T_{\text{R}}$ $\displaystyle\rightarrow$
$\displaystyle\bar{T}_{\text{R}}=2\pi L_{\lambda}T_{\text{R}},\hskip
28.45274ptC_{\text{R}}\rightarrow\bar{C}_{\text{R}}=\frac{1}{L_{\lambda}^{2}}C_{\text{R}},$
$\displaystyle\kappa_{T_{\text{R}}}$ $\displaystyle\rightarrow$
$\displaystyle\bar{\kappa}_{T_{\text{R}}}=L_{\lambda}\kappa_{T_{\text{R}}},\hskip
34.14322pt\mathcal{G}_{\text{R}}\rightarrow\bar{\mathcal{G}}_{\text{R}}=\frac{1}{L_{\lambda}}\mathcal{G}_{\text{R}},$
(42)

respectively. Note that, for the Rényi case, we do not consider the
dimensionless quantities by treating the charge $q$ as a scaling parameter, as
done in the GB case, since the system now has a fundamental length scale
$L_{\lambda}$ due to the existence of the nonextensivity. Therefore it is more
appropriate to measure the thermodynamic quantities with respect to this
length scale $L_{\lambda}$ instead of $q$. The quantities at the critical
point now become

$\displaystyle\bar{q}_{c}$ $\displaystyle=$
$\displaystyle\sqrt{7-4\sqrt{3}}\approx 0.268,$ (43)
$\displaystyle\bar{\phi}_{c}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2-\sqrt{3}}{\sqrt{3}}}\approx 0.393,$ (44)
$\displaystyle\bar{T}_{c}$ $\displaystyle=$
$\displaystyle\frac{4}{3}\sqrt{2\sqrt{3}-3}\approx 0.908.$ (45)

Let us consider the local stability in the aspect of the heat capacity first.
The behaviors of the rescaled temperature and heat capacity are illustrated in
Fig. 4. In these plots, we present the quantities with three different values
of $\bar{q}$ which are $\bar{q}=0.2<\bar{q}_{c}$ (the magenta curves),
$\bar{q}=\bar{q}_{c}$ (the gray curves), and $\bar{q}=0.4>\bar{q}_{c}$ (the
orange curves). By considering its first and second derivatives with respect
to $\bar{r}_{+}$, it is seen that, for $\bar{q}<\bar{q}_{c}$, there exist two
local extrema in $\bar{T}_{\text{R}}$, where there is no finite maximum in
this case, corresponding to the divergence of $\bar{C}_{\text{R}}$ occuring at
$\bar{r}_{+}=\frac{1}{\sqrt{2}}\Big{(}1-\bar{q}^{2}\mp\sqrt{1-14\bar{q}^{2}+\bar{q}^{4}}\Big{)}^{1/2}\equiv\bar{r}_{1,2}$.
The temperature reaches the local maximum and local minimum at $\bar{r}_{1}$
and $\bar{r}_{2}$, respectively. The three possible configurations exist due
to the appearance of the nonextensivity. Two of them are the phases with
positive heat capacity denoted as pBH1 (for the small black hole branch,
$\bar{r}_{e}<\bar{r}_{+}<\bar{r}_{1}$) and pBH2 (for the large black hole
branch, $\bar{r}_{+}>\bar{r}_{2}$). Another phase is the intermediate black
hole branch, $\bar{r}_{1}<\bar{r}_{+}<\bar{r}_{2}$ with negative heat capacity
denoted as nBH. Moreover, there are two second-order phase transitions which
are the nBH-pBH1, and nBH-pBH2 transitions occurring at $\bar{r}_{1}$ and
$\bar{r}_{2}$, respectively. Interestingly, the pBH2 phase does not appear in
the standard GB statistics. Therefore, it could be an emergent phase within
the Rényi statistics. One has seen that the nBH phase, which has negative
slope in $\bar{T}_{\text{R}}$ or negative value in $\bar{C}_{\text{R}}$,
disappears when $\bar{q}\geq\bar{q}_{c}$, since the radii $\bar{r}_{1}$ and
$\bar{r}_{2}$ are identical when $\bar{q}=\bar{q}_{c}$ and then undefined when
$\bar{q}>\bar{q}_{c}$. In other words, the pBH1 and pBH2 phases are degenerate
for $\bar{q}>\bar{q}_{c}$. Hence, the phase structure is clearly dependent on
the values of $\bar{q}$. This result is a key difference between the
properties of the RN-AF described by the GB and Rényi statistics.

For the global stability, the Gibbs free energy profiles with different
$\bar{q}$ are illustrated in Fig. 5. Due to the appearance of the emergent
pBH2 phase, there exists another cusp corresponding to $\bar{r}_{2}$ for the
case $\bar{q}<\bar{q}_{c}$. The upper right panel in Fig. 5 is actually shown
the swallowtail behavior which is obviously different from the black hole via
GB statistics. Moreover, the first-order phase transition from pBH1 to pBH2
interestingly occurs for the Rényi case with $\bar{q}\leq\bar{q}_{c}$. This
transition is actually called the Hawking-Page phase transition (the
temperature at this transition is, then, called the Hawking-Page temperature
$\bar{T}_{\text{HP}}$) for the charged case. On the other hand, the uncharged
black hole, i.e., the Schwarzschild black hole, also has the Hawking-Page
transition which is the transition from the thermal radiation phase with zero
free energy to the pBH phase as shown in the upper left panel in Fig. 5. It is
noted that, for $0<\bar{q}\leq\bar{q}_{c}$, $\bar{T}_{\text{HP}}$ decreases as
the charge increasing. In addition, the Hawking-Page temperature is maximized
and minimized for the black hole with $\bar{q}=0$ and $\bar{q}=\bar{q}_{c}$,
respectively, as shown in Fig. 6.

---
Figure 5: The profiles of the free energy $\bar{\mathcal{G}}_{\text{R}}$ with
respect to the temperature $\bar{T}_{\text{R}}$ for various fixed values of
charge $\bar{q}$. Note that each of these with fixed value of charge
corresponds to an isobaric process.
---
Figure 6: The profile of the Hawking-Page temperature $\bar{T}_{\text{HP}}$
with respect to the charge $\bar{q}$.

We now move to analyze the mechanical stability by comparing to the thermal
one. The rescaled heat capacity in Eq. (42) can be expressed in terms of
$\bar{q}$ and $\bar{\phi}$ as

$\displaystyle\bar{C}_{\text{R}}=2\pi\bar{q}^{2}\left[\frac{(1-\bar{\phi}^{2})}{\bar{q}^{2}(1+\bar{\phi}^{2})-\bar{\phi}^{2}(1-3\bar{\phi}^{2})}\right].$
(46)

It is seen that the sign of this heat capacity depends directly on the sign of
the denominator because its numerator is always positive for the whole range
$0<\bar{\phi}<1$. For fixing $\bar{q}$, the denominator vanishes
($\bar{C}_{\text{R}}\rightarrow\infty$) at

$\displaystyle\bar{\phi}_{\pm}=\sqrt{\frac{1}{6}\left(1-\bar{q}^{2}\pm\sqrt{1-14\bar{q}^{2}+\bar{q}^{4}}\,\right)}.$
(47)

Straightforwardly, when $\bar{q}=\bar{q}_{c}=2-\sqrt{3}$, we found that the
two solutions (47) are degenerate,
$\bar{\phi}_{+}=\bar{\phi}_{-}=\bar{\phi}_{c}\approx 0.393$. For the large
charge case $\bar{q}>\bar{q}_{c}$, therefore, the range of negative value of
the denominator in Eq. (46) disappears, and hence $\bar{C}_{\text{R}}$ is
always positive. For the small charge case $\bar{q}<\bar{q}_{c}$, there are
three branches of charge black hole which are $0<\bar{\phi}<\bar{\phi}_{-}$
(positive $\bar{C}_{\text{R}}$), $\bar{\phi}_{-}<\bar{\phi}<\bar{\phi}_{+}$
(negative $\bar{C}_{\text{R}}$), and $\bar{\phi}_{+}<\bar{\phi}<1$ (positive
$\bar{C}_{\text{R}}$). This behavior is equivalent to the phase structure as
illustrated in the right panel in Fig. 4. We also conclude that, for
$\bar{q}<\bar{q}_{c}$, the regions $0<\bar{\phi}<\bar{\phi}_{-}$,
$\bar{\phi}_{-}<\bar{\phi}<\bar{\phi}_{+}$ and $\bar{\phi}_{+}<\bar{\phi}<1$
are the regions for the phases of pBH2, nBH and pBH1, respectively.

Using the rescaled version of Eq. (37), one can compute the rescaled
compressibility for the RN-AF in the Rényi case as follows:

$\displaystyle\bar{\kappa}_{T_{\text{R}}}$ $\displaystyle=$
$\displaystyle-\frac{L_{\lambda}}{\phi}\left(\frac{\partial\phi}{\partial
q}\right)_{T_{\text{R}}}=\frac{(\bar{\phi}^{2}-\bar{q}^{2})}{\bar{q}}\left[\frac{(1-\bar{\phi}^{2})}{\bar{q}^{2}(1+\bar{\phi}^{2})-\bar{\phi}^{2}(1-3\bar{\phi}^{2})}\right].$
(48)

The compressibility diverges at the same place as the heat capacity does,
since they have the same denominators, i.e.,
$\bar{q}^{2}(1+\bar{\phi}^{2})-\bar{\phi}^{2}(1-3\bar{\phi}^{2})$. For fixing
charge, the behaviour of the above compressibility is then analyzed as three
different cases which are $\bar{q}<\bar{q}_{c}$, $\bar{q}=\bar{q}_{c}$ and
$\bar{q}>\bar{q}_{c}$ as illustrated in the left, middle and right panels in
Fig. 7, respectively. The general features of the compressibility can be
listed as follows: (i) by comparing Eqs. (46) and (48), it is noticed that the
sign of the compressibility is opposite to that of the heat capacity when
$\bar{\phi}<\bar{q}$, and (ii) the compressibility vanishes when
$\bar{\phi}=\bar{q}$. For the small charge $\bar{q}<\bar{q}_{c}$ case, it is
found that, by using the Taylor series expansion around $\bar{q}=0$, the
potential $\bar{\phi}_{-}$ in Eq. (47) is approximated as

$\displaystyle\bar{\phi}_{-}\approx\bar{q}+2\bar{q}^{3}+12\bar{q}^{5}+\mathcal{O}(\bar{q}^{6}).$
(49)

This yields that $\bar{q}$ is always less than $\bar{\phi}_{-}$. Hence,
$\bar{\kappa}_{T_{\text{R}}}$ of pBH1 $(\bar{\phi}_{+}<\bar{\phi}<1)$ and nBH
$(\bar{\phi}_{-}<\bar{\phi}<\bar{\phi}_{+})$ phases have the same sign as
$C_{\text{R}}$. In other words, the pBH1 and nBH phases have the positive and
negative compressibility, respectively. It is very important to note that the
emergent pBH2 phase is locally stable for both $\bar{C}_{\text{R}}>0$ and
$\bar{\kappa}_{T_{\text{R}}}>0$ within the range
$\bar{q}>\bar{\phi}>\bar{\phi}_{-}$ as shown in the left panel in Fig. 7. For
the other two cases $\bar{q}\geq\bar{q}_{c}$, the black hole is mechanically
stable and unstable when $\bar{\phi}>\bar{q}$ and $\bar{\phi}<\bar{q}$,
respectively, as shown in the middle and right panel in Fig. 7.

---
Figure 7: The profiles of the heat capacity $\bar{C}_{\text{R}}$ (solid lines)
and the compressibility $\bar{\kappa}_{T_{\text{R}}}$ (dashed lines) with
respect to the electrostatic potential $\bar{\phi}$ for various values of
$\bar{q}$. For the left and middle panels, the blue, green and red lines are
the quantities for pBH2, nBH and pBH1, respectively. Note that the values of
$\bar{\phi}_{\pm}$ is for the case $\bar{q}=0.2$. For the right panel, the
quantities are shown only for the pBH phase.

We have investigated the thermodynamical properties of the black hole with the
effect of nonextensivity. The important differences between the system
described by the GB and Rényi statistics are pointed out. Especially, the
black hole can be thermodynamically stable using the Rényi statistics, while
it is not possible using the GB statistics. As will be discussed in the next
section, it is possible to construct the Maxwell equal area law for the
charged black holes similar to vdW fluid.

##  IV Maxwell equal area law

In this section, we are interested in finding the Maxwell equal area law on
the $\bar{q}-\bar{\phi}$ plane (similar to the $P-V$ plane for the vdW fluid).
By solving the rescaled version of Eq. (37) for $\bar{q}$, an equation of
state $\bar{q}=\bar{q}(\bar{\phi},\bar{T}_{\text{R}})$ is obtained in the form

$\displaystyle\bar{q}=\frac{\bar{\phi}\bar{T}_{\text{R}}}{1-\bar{\phi}^{2}}\left(1\pm\sqrt{1-\left(\frac{1-\bar{\phi}^{2}}{\bar{T}_{\text{R}}}\right)^{2}}\,\right).$
(50)

Let us define these two branches of $\bar{q}$ as follows:

$\displaystyle\bar{q}_{1}$ $\displaystyle=$
$\displaystyle\frac{\bar{\phi}\bar{T}_{\text{R}}}{1-\bar{\phi}^{2}}\left(1+\sqrt{1-\left(\frac{1-\bar{\phi}^{2}}{\bar{T}_{\text{R}}}\right)^{2}}\,\right),$
(51) $\displaystyle\bar{q}_{2}$ $\displaystyle=$
$\displaystyle\frac{\bar{\phi}\bar{T}_{\text{R}}}{1-\bar{\phi}^{2}}\left(1-\sqrt{1-\left(\frac{1-\bar{\phi}^{2}}{\bar{T}_{\text{R}}}\right)^{2}}\,\right).$
(52)

As discussed in Ref. Zhou and Wei (2019), we have split this consideration
into two cases, i.e. the cases of near and far from the critical point, where
the charge satisfying the Maxwell equal area law is denoted as
$\bar{q}^{\ast}$. In the near critical point case
$(\bar{T}_{\text{R}}\gtrsim\bar{T}_{c})$, the line of charge $\bar{q}^{\ast}$
intersects the isothermal curve of only the $\bar{q}_{2}$ branch as shown in
the left panel of Fig. 8. On the other hand, the line of charge
$\bar{q}^{\ast}$ intersects the curve of both $\bar{q}_{1}$ branch (dashed
curve) and $\bar{q}_{2}$ branch (solid curve) for the far from critical point
case $(\bar{T}_{\text{R}}\gg\bar{T}_{c})$ as illustrated in the right panel of
Fig. 8.

---
Figure 8: Equal area law in the $\bar{q}-\bar{\phi}$ plane where the
isothermal curves of $\bar{q}_{1}$ and $\bar{q}_{2}$ branches are represented
as the dashed and solid curves, respectively. The blue, green and red lines
denote the pBH2, nBH and pBH1 phases, respectively. Left: the case of near the
critical point with fixed $\bar{T}_{\text{R}}=0.91$. Right: the case of far
from the critical point with fixed $\bar{T}_{\text{R}}=0.97$.

It is found that the slope of the $\bar{q}_{1}$ branch for the case of far
from the critical point is positive corresponding to the negative
compressibility. Hence, this first-order phase transition is the transition
between the mechanically unstable phase and the stable phase. This feature is
actually called the novel equal area law Zhou and Wei (2019).

###  IV.1 Near the critical point

To satisfy the Maxwell equal area law, the line of constant charge
$\bar{q}^{\ast}$ intersects the isothermal curve of $\bar{q}_{2}(\phi)$ branch
at $\bar{\phi}_{1}$, $\bar{\phi}_{2}$ and $\bar{\phi}_{3}$ as shown in the
left panel in Fig. 8. For this case, the condition for finding
$\bar{q}^{\ast}$ is expressed as

$\displaystyle\bar{q}^{\ast}(\bar{\phi}_{3}-\bar{\phi}_{1})$ $\displaystyle=$
$\displaystyle\int_{\bar{\phi}_{1}}^{\bar{\phi}_{3}}\bar{q}_{2}d\bar{\phi}.$
(53)

As a result, one obtains

$\displaystyle\bar{q}^{\ast}$ $\displaystyle=$
$\displaystyle\frac{\bar{T}_{\text{R}}}{2(\bar{\phi}_{3}-\bar{\phi}_{1})}\left[\sqrt{1-\left(\frac{1-\bar{\phi}^{2}}{\bar{T}_{\text{R}}}\right)^{2}}-\ln\left\\{1+\sqrt{1-\left(\frac{1-\bar{\phi}^{2}}{\bar{T}_{\text{R}}}\right)^{2}}\,\right\\}\right]_{\bar{\phi}_{1}}^{\bar{\phi}_{3}}.$
(54)

In other conditions in which the line of charge $\bar{q}^{\ast}$ also
intersects the isothermal curve of the $\bar{q}_{2}$ branch at
$\bar{\phi}_{1}$ and $\bar{\phi}_{3}$ that is
$\bar{q}_{2}(\bar{\phi}_{1})=\bar{q}_{2}(\bar{\phi}_{3})=\bar{q}^{\ast}$, we
obtain

$\displaystyle\bar{q}^{\ast}$ $\displaystyle=$
$\displaystyle\frac{\bar{\phi}_{1}\bar{T}_{\text{R}}}{1-\bar{\phi}_{1}^{2}}\left[1-\sqrt{1-\left(\frac{1-\bar{\phi}_{1}^{2}}{\bar{T}_{\text{R}}}\right)^{2}}\,\right],$
(55) $\displaystyle\bar{q}^{\ast}$ $\displaystyle=$
$\displaystyle\frac{\bar{\phi}_{3}\bar{T}_{\text{R}}}{1-\bar{\phi}_{3}^{2}}\left[1-\sqrt{1-\left(\frac{1-\bar{\phi}_{3}^{2}}{\bar{T}_{\text{R}}}\right)^{2}}\,\right].$
(56)

It is obvious that there are three unknown variables with three equations. We
consider $[\text{Eq.}~{}\eqref{qs1}]=[\text{Eq.}~{}\eqref{qs2}]$ to eliminate
$\bar{\phi}_{1}$, and then, determine $\bar{\phi}_{3}$ from
$2[\text{Eq.}~{}\eqref{qs0}]=\text{Eq.}~{}\eqref{qs1}]+[\text{Eq.}~{}\eqref{qs2}]$.
Unfortunately, it is complicated to solve $\bar{\phi}_{3}$ analytically, since
we have to deal with the terms of logarithmic function. Instead, we used the
numerical calculation for each value of $\bar{T}_{\text{R}}$. The results for
the case of near the critical point are shown in Table. 1

Table 1: The values of $\bar{q}^{\ast}$, $\bar{\phi}_{1}$ and $\bar{\phi}_{3}$ at various values of $\bar{T}_{\text{R}}$ for the case of near critical point. $\bar{T}_{\text{R}}$ |  $\bar{q}^{\ast}$ |  $\bar{\phi}_{1}$ |  $\bar{\phi}_{3}$
---|---|---|---
0.91 | 0.266606 | 0.357255 | 0.430539
0.92 | 0.258548 | 0.300521 | 0.494178
0.93 | 0.250493 | 0.268988 | 0.532580
0.94 | 0.242440 | 0.245060 | 0.563344

###  IV.2 Far from the critical point

It is found that the minimum value of $\bar{\phi}$ (denoted as
$\bar{\phi}_{0}$) is actually the value that the branches in Eqs. (51) and
(52) merge together. Therefore, it is given by

$\displaystyle\bar{\phi}_{0}=\sqrt{1-\bar{T}_{\text{R}}}.$ (57)

Obviously, the temperature $\bar{T}_{\text{R}}$ cannot be greater than one for
the isothermal process. Therefore, the Maxwell construction of RN-AF on the
$\bar{q}-\bar{\phi}$ plane with the Rényi model will make sense in the region
$\bar{T}_{\text{R}}\leq 1$. Note that the bound in temperature does not hold
for the system evolving in other types of process. As seen in the right panel
in Fig. 8, the area satisfying the Maxwell equal area law for this case can be
expressed as

$\displaystyle\bar{q}^{\ast}(\bar{\phi}_{3}-\bar{\phi}_{1})$ $\displaystyle=$
$\displaystyle-\int_{\bar{\phi}_{0}}^{\bar{\phi}_{1}}\bar{q}_{1}d\bar{\phi}+\int_{\bar{\phi}_{0}}^{\bar{\phi}_{3}}q_{2}d\bar{\phi},$
(58)

where $\bar{\phi}_{1}$ is the value of $\bar{\phi}$ at which the line of
$\bar{q}^{\ast}$ intersects the curve of $\bar{q}_{1}(\bar{\phi})$ branch, and
$\bar{\phi}_{2}$ and $\bar{\phi}_{3}$ are the values of $\bar{\phi}$ at which
the line of $\bar{q}^{\ast}$ intersect the curve of $\bar{q}_{2}(\bar{\phi})$
branch. Eventually, one obtains

$\displaystyle\bar{q}^{\ast}$ $\displaystyle=$
$\displaystyle\frac{\bar{T}_{\text{R}}}{2(\bar{\phi}_{3}-\bar{\phi}_{1})}\Bigg{[}2\ln\left(\frac{1-\bar{\phi}_{1}^{2}}{\bar{T}_{\text{R}}}\right)+\sqrt{1-\left(\frac{1-\bar{\phi}_{1}^{2}}{\bar{T}_{\text{R}}}\right)^{2}}-\ln\Bigg{\\{}1+\sqrt{1-\left(\frac{1-\bar{\phi}_{1}^{2}}{\bar{T}_{\text{R}}}\right)^{2}}\,\Bigg{\\}}$
(59) $\displaystyle\hskip
71.13188pt+\sqrt{1-\left(\frac{1-\bar{\phi}_{3}^{2}}{\bar{T}_{\text{R}}}\right)^{2}}-\ln\Bigg{\\{}1+\sqrt{1-\left(\frac{1-\bar{\phi}_{3}^{2}}{\bar{T}_{\text{R}}}\right)^{2}}\,\Bigg{\\}}\Bigg{]}.$

The conditions from the line of $\bar{q}^{\ast}$ actually intersecting the
isothermal curves of the $\bar{q}_{1}$ and $\bar{q}_{2}$ branches at
$\bar{\phi}_{1}$ and $\bar{\phi}_{3}$, are, respectively written as

$\displaystyle\bar{q}^{\ast}$ $\displaystyle=$
$\displaystyle\frac{\bar{\phi}_{1}\bar{T}_{\text{R}}}{1-\bar{\phi}_{1}^{2}}\left[1+\sqrt{1-\left(\frac{1-\bar{\phi}_{1}^{2}}{\bar{T}_{\text{R}}}\right)^{2}}\,\right],$
(60) $\displaystyle\bar{q}^{\ast}$ $\displaystyle=$
$\displaystyle\frac{\bar{\phi}_{3}\bar{T}_{\text{R}}}{1-\bar{\phi}_{3}^{2}}\left[1-\sqrt{1-\left(\frac{1-\bar{\phi}_{3}^{2}}{\bar{T}_{\text{R}}}\right)^{2}}\,\right].$
(61)

Similar to the previous case, one can numerically determine the values of
$\bar{\phi}_{1}$, $\bar{\phi}_{3}$ and $\bar{q}^{\ast}$ by using Eqs.
(59)$-$(61). Eventually, the results are expressed in Table. 2.

Table 2: The values of $\bar{q}^{\ast}$, $\bar{\phi}_{1}$ and $\bar{\phi}_{3}$ at various values of $\bar{T}_{\text{R}}$ in the case of far from the critical point. $\bar{T}_{\text{R}}$ |  $\bar{q}^{\ast}$ |  $\bar{\phi}_{1}$ |  $\bar{\phi}_{3}$
---|---|---|---
0.95 | 0.234390 | 0.225272 | 0.589933
0.96 | 0.226342 | 0.208192 | 0.613779
0.97 | 0.218297 | 0.193062 | 0.635641
0.98 | 0.210254 | 0.179423 | 0.655978
0.99 | 0.202214 | 0.166972 | 0.675095
1.00 | 0.194176 | 0.155495 | 0.693204

Plotting in the $\bar{q}^{\ast}-\bar{\phi}$ plane, the points of
$(\bar{\phi}_{1},\bar{q}^{\ast})$ and $(\bar{\phi}_{3},\bar{q}^{\ast})$ can be
obtained from Tables 1 and 2, where the line connecting between them can form
a Maxwell curve as shown in the left panel in Fig. 9. The lowest points are
associated with the upper bound in temperature $\bar{T}_{\text{R}}=1$. At the
critical point, the charge $\bar{q}^{\ast}$ approaches to the critical charge
$\bar{q}_{c}$ while the potentials $\bar{\phi}_{1}$ and $\bar{\phi}_{3}$
merges together at the critical potential $\bar{\phi}_{c}$. It is also found
that the charge $\bar{q}^{\ast}$ increases as the temperature decreases, and
eventually ends at the critical values as shown in the right panel in Fig. 9.

---
Figure 9: Left: the behavior of $\bar{q}^{\ast}$ with respect to
$\bar{\phi}_{1}$ (as a red line) and $\bar{\phi}_{3}$ (as a blue line) also
called the Maxwell curve. Right: the Rényi temperature profile with respect to
$q^{\ast}$.

As mentioned that the first-order phase transition is not necessarily the
transition locally stable phases for the charged black holes. In other words,
the pBH2 phase with $\bar{q}^{\ast}$ and $\bar{\phi}_{1}$ at the transition
point has negative compressibility if the temperature is not small enough. By
numerical analysis, we have investigated the upper bound in temperature in
which the first-order phase transition is the transition between mechanically
stable pBH2 phase and pBH1 phase. As a result, this maximum temperature is
numerically evaluated as

$\displaystyle\bar{T}_{\text{R}}\approx 0.942.$ (62)

As seen in the Fig. 10, the value $\bar{\phi}_{1}$ is indeed the point that
the branches $\bar{q}_{1}$ and $\bar{q}_{2}$ merges, i.e., the point
$\bar{\phi}_{0}$. Moreover, the figure also illustrates that the pBH1 and pBH2
phases with the charge $\bar{q}^{\ast}$ have the same Gibbs free energy (see
right end point of the middle orange dotted line). The upper and lower orange
dotted lines show that the local extrema on the $\bar{q}-\bar{\phi}$ plane
correspond to the cusp in $\bar{q}-\bar{\mathcal{G}}_{\text{R}}$ plane in
which the second-order phase transitions occur.

---
Figure 10: The isothermal curves on the $\bar{q}-\bar{\phi}$ and
$\bar{q}-\bar{\mathcal{G}}_{\text{R}}$ diagrams with
$\bar{T}_{\text{R}}\approx 0.942$.

We show the phase structure of RN-AF with Rényi statistics on the
$\bar{q}-\bar{\phi}$ plane analogous to that of the vdW fluid on the $P-V$
plane in Fig. 11. The red, green and blue curves are the isothermal curves for
$\bar{T}_{\text{R}}<\bar{T}_{c}$, $\bar{T}_{\text{R}}=\bar{T}_{c}$ and
$\bar{T}_{\text{R}}>\bar{T}_{c}$, respectively. The dashed black curve and
dashed gray curve are the Maxwell curve or coexistence curve and the spinodal
curve, respectively. Each point along the spinodal curve has infinite heat
capacity or the point where the second-order phase transition between nBH and
pBH1 (or pBH2) occurs. The blue and white regions stand for the pure pBH2 and
pBH1 phases, respectively. The left and right red regions are the regions for
the metastable phases called the superheated pBH2 and supercooled pBH1,
respectively. In the yellow region, the phase is mixed between pBH1 and pBH2.
It is seen that the black hole can be in all phases when
$\bar{T}_{\text{R}}>\bar{T}_{c}$. This feature is similar to the vdW fluid
with the temperature being less than its critical value.

---
Figure 11: The phase structure of the RN-AF with Rényi statistics on the
$\bar{q}-\bar{\phi}$ plane is shown here, where the temperature of isotherm
lines increases from top to bottom.

The interesting properties in the aspect of the thermodynamical stability of
the charged black holes are discussed. The first-order phase transition
between pBH1 and pBH2 obeying the Maxwell equal area law has been
investigated. The phase structure for this black hole is also shown. In the
next section, the thermodynamical implication of the black hole is discussed
via the Ruppeiner geometry of the thermodynamical phase space.

##  V Black hole phase structure from Ruppeiner geometry

The thermodynamic geometry provides some understandings about the microscopic
behavior from the macroscopic quantities of thermodynamical systems.
Therefore, this approach is suitable for investigating the microscopic aspects
of black holes because their complete information are hidden from an observer
outside its event horizon. In the context of the Ruppeiner geometry, a line
element $\Delta l^{2}$ between two neighbouring fluctuation states can be
written in the form Ruppeiner (1979)

$\displaystyle\Delta l^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{k_{\text{B}}}g_{ij}\Delta x^{i}\Delta x^{j},$ (63)
$\displaystyle g_{ij}$ $\displaystyle=$
$\displaystyle-\frac{\partial^{2}S(x)}{\partial x^{i}\partial x^{j}},$ (64)

where $S(x^{i})$ is the entropy function of the phase space coordinates
$x^{i}$ chosen to be the thermodynamic variables, the metric tensor $g_{ij}$
is defined as the Ruppeiner metric, and $k_{\text{B}}$ is the Boltzmann
constant. In general relativity, it is well known that the gravitational
interaction is encoded in the form of the curvature of spacetime. In a similar
way, the scalar curvature of the Ruppeiner metric, as shown in Eq. (64) could
encode the microscopic interactions of a thermal system. It has been convinced
that the sign of the scalar curvature $R$ indicates the interaction between
two microstructures of the system, namely, $R<0$ and $R>0$ represent
attractive and repulsive interaction, respectively, while $R=0$ represents the
case of interaction-free Ruppeiner (2010). Moreover, the divergence of the
scalar curvature occurs at the second-order phase transition.

As suggested in Refs. Promsiri  _et al._ (2020, 2021) that the microstates of
a strongly self-gravitating system could be correlated in nonextensive
description, this nontrivial correlation may be encoded in the nonextensivity
parameter $\lambda$. In this section, we are interested in comparing between
the interactions among microstates of the RN-AF with $\lambda>0$ and those
with $\lambda=0$ through considering the values of scalar curvature $R$ in
some thermodynamical phase spaces.

###  V.1 RN-AF in GB statistics

Firstly, we review the thermodynamic geometry of RN-AF in the standard GB
statistics studied in Shen  _et al._ (2007); Wang  _et al._ (2020). Then we
proceed to consider the effects of nonextensivity parameter on the scalar
curvature and phase transition of RN-AF subsequently. The Ruppeiner metric for
the RN-AF can be defined as the second derivative of the Bekenstein-Hawking
entropy in Eq. (11) with respect to the phase space coordinates
$x^{i}=(u,\phi)$ as

$\displaystyle g_{ij}=-\frac{\partial^{2}}{\partial x^{i}\partial
x^{j}}S_{\text{BH}}(x).$ (65)

The internal energy $u$ has been defined in Eq. (19), which can be written in
the form

$\displaystyle u=\frac{r_{+}}{2}\left(1-\phi^{2}\right).$ (66)

Here, the coordinates $u$ and $\phi$ are assumed to be independent of each
other. Note that these phase space coordinates are chosen to correspond with
$S=S(u,\phi)$ (see Eq. (18)). Using Eq. (66), the entropy formula (11) can be
written in term of the coordinates $x^{i}$ as in the following:

$\displaystyle S_{\text{BH}}(u,\phi)=\frac{4\pi u^{2}}{(1-\phi^{2})^{2}}.$
(67)

As a result, the metric tensor in $u-\phi$ phase space is obtained as follows:

$\displaystyle g_{ij}(u,\phi)$ $\displaystyle=$
$\displaystyle-\begin{pmatrix}\partial_{u}\partial_{u}S_{\text{BH}}&\partial_{\phi}\partial_{u}S_{\text{BH}}\\\
\partial_{u}\partial_{\phi}S_{\text{BH}}&\partial_{\phi}\partial_{\phi}S_{\text{BH}}\end{pmatrix}=-\frac{8\pi}{(1-\phi^{2})^{2}}\begin{pmatrix}1&\frac{4u\phi}{1-\phi^{2}}\\\
\frac{4u\phi}{1-\phi^{2}}&\frac{2u^{2}(1+5\phi^{2})}{(1-\phi^{2})^{2}}\end{pmatrix}.$
(68)

Under the general coordinate transformation, $x^{i}\rightarrow x^{\prime i}$,
the metric $g_{ij}$ transform as

$\displaystyle g^{\prime}_{ij}=\frac{\partial x^{m}}{\partial x^{\prime
i}}\frac{\partial x^{n}}{\partial x^{\prime j}}g_{mn},$ (69)

which is just the metric tensor in the Riemannian geometry. Using Eqs. (14)
and (66), the metric tensor in the phase space coordinates $x^{\prime
i}=(r_{+},r_{-})$ can be written in the form

$\displaystyle
g_{ij}(r_{+},r_{-})=-\frac{\pi}{(r_{+}-r_{-})}\begin{pmatrix}2r_{+}-5r_{-}&&&r_{+}\\\
r_{+}&&&r^{2}_{+}/r_{-}\end{pmatrix}.$ (70)

Eventually, the Ricci curvature scalar $R$ is in the form

$\displaystyle R=-\frac{r_{+}-r_{-}}{\pi r_{+}(3r_{-}-r_{+})^{2}}.$ (71)

By comparing to the expression of heat capacity $C_{q}$ in Eq. (16), it is
interestingly found that both Ricci scalar $R$ and heat capacity $C_{q}$
vanish, and diverge at the same positions which are $r_{+}=r_{-}$, and
$r_{+}=r_{0}$, respectively. This implies that the thermodynamic geometry
approach can indicate an occurrence of the second-order phase transition
through the divergence of $R$. In the left panel in Fig. 12, the region of
negative value of $R$ for the RN-AF via the GB statistics, the regular one
with the condition $M^{2}>q^{2}$, in a parameter space of $r_{+}$ and $q$ is
shown in yellow, whereas the gray upper-left region corresponds to the non-
existence of charged black hole due to $M^{2}<q^{2}$. It is interesting to
note that the red line in the panel is the line of which $R\to-\infty$
corresponding to $|C_{q}|\to\infty$, representing the dividing line between
the nBH and pBH phases. The Ricci scalar profile versus the outer event
horizon $r_{+}$ and temperature $T_{\text{H}}$ are plotted in Fig. 13, whose
the left panel shows that the nBH-pBH transition occurs at the point $r_{0}$,
given by Eq. (15), at which $R$ approaches negatively infinite. Moreover, the
right panel shows that $R$ also turns out to be negatively infinite at
$T_{\text{max}}$, which corresponds to $r_{0}$. Obviously, the Ricci scalar
vanishes at both extremal and very large black hole limits. Note that, at the
same temperature, the Ricci scalar of pBH phase has more negative value than
the nBH phase as shown in the right panel of Fig. 13.

---
Figure 12: The region plots for the signs of the Ricci scalar of the Ruppeiner
geometry of black hole $R$ within the GB statistics (left) and the Rényi
statistics (right) are shown, where the regions of negative and positive
values of $R$ are in yellow and blue, respectively. Note that the black hole
solution does not exist in the gray region. The red lines are the lines
corresponding to $R\to-\infty$, and the point C represents the critical point.
---
Figure 13: The dimensionless Ricci scalar $Rq^{2}$ with respect to the
dimensionless outer horizon radius $r_{+}/q$ (left) and Hawking temperature
$T_{\text{H}}q$ (right).

According to Eq. (71), $R$ is always negative implying that the interactions
between microscopic constituents of the RN-AF with the GB thermodynamics is
attractive and becomes vanished at the extremal black holes. Hence, the
extremal limit could be regarded as an analogue of the limit to an ideal gas.
To illustrate this, let us discuss the competing roles between the mass $M$
and the charge $q$ terms in a charged black hole’s horizon function $f(r)$ in
Eq. (7). Even though we are working in the framework of general relativity,
$M$ and $q$ could be thought in some sense to be responsible to an attractive
gravitational interaction and the repulsive electrostatic interaction,
respectively. Comparing these two parameters in the geometric unit, the
condition $M^{2}>q^{2}$ may imply that the attractive gravitational force is
stronger than the repulsive electric force for this charged black hole system.
In the extremal black hole ($M^{2}=q^{2}$), the gravitational and
electrostatic interactions are balanced. In this way, the interactions between
the constituents of the black hole may be always attractive due to the
gravitational force domination, but the interactions becomes absent in the
extremal case. Since the solution does not contain a naked singularity obeying
the cosmic censorship conjecture Penrose (2002), the repulsive electrostatic
force cannot be stronger than the attractive gravitational force. Therefore,
the Ricci scalar curvature of the RN-AF in the GB statistics is less than or
equal to zero as shown in Fig. 13.

###  V.2 RN-AF in Rényi statistics

In the section III, we found that the thermodynamics of RN-AF with the Rényi
statistics has different thermal phase structure compared to the one with the
GB statistics, namely, the emergent pBH2 phase has occured in this case. It is
interesting to investigate some aspects of the connections between phase
structure and interaction among microscopic constituents of black hole in the
Rényi case via the thermodynamic geometry framework. In a similar way as
working with the GB statistics, we start by writing the Rényi entropy in terms
of $u$ and $\phi$ as follows:

$\displaystyle S_{\text{R}}=\pi
L^{2}_{\lambda}\ln\left[1+\frac{4u^{2}}{L^{2}_{\lambda}(1-\phi^{2})^{2}}\right].$
(72)

With the approach in a similar way as Eq. (65), the metric tensor in $u-\phi$
phase space can be computed as

$\displaystyle g_{ij}(u,\phi)$ $\displaystyle=$
$\displaystyle\frac{-8\pi}{\left(1-\phi^{2}\right)^{3}\left[1-\frac{4u^{2}}{L^{2}_{\lambda}\left(1-\phi^{2}\right)^{2}}\right]^{2}}\begin{pmatrix}1-\phi^{2}+\frac{4u^{2}}{L^{2}_{\lambda}(1-\phi^{2})}&4u\phi\\\
4u\phi&\frac{2u^{2}}{1-\phi^{2}}\left[1+5\phi^{2}+\frac{4u^{2}(1+\phi^{2})}{L^{2}_{\lambda}\left(1-\phi^{2}\right)^{2}}\right]\end{pmatrix}.$

In $r_{+}-r_{-}$ phase space, it becomes

$\displaystyle g_{ij}(r_{+},r_{-})=-\frac{\pi
r_{+}}{(r_{+}-r_{-})\left(1+\frac{r^{2}_{+}}{L^{2}_{\lambda}}\right)}\begin{pmatrix}\frac{\left[2r_{+}-5r_{-}-\frac{r^{2}_{+}}{L^{2}_{\lambda}}(2r_{+}+r_{-})\right]}{r_{+}\left(1+\frac{r^{2}_{+}}{L^{2}_{\lambda}}\right)}&&&1\\\
1&&&r_{+}/r_{-}\end{pmatrix}.$ (74)

The Ricci scalar corresponding to the Rényi entropy is given by

$\displaystyle R_{\text{R}}=-\frac{(r_{+}-r_{-})}{\pi
r_{+}\left[3r_{-}-r_{+}+\frac{r^{2}_{+}}{L^{2}_{\lambda}}(r_{+}+r_{-})\right]^{2}}\left[1-\frac{2r_{+}}{L^{2}_{\lambda}}(2r_{-}-r_{+})-\frac{r^{3}_{+}}{L^{4}_{\lambda}}(r_{+}+2r_{-})\right].$
(75)

By rescaling the parameters as introduced in (41), the curvature scalar in Eq.
(75) is transformed as $R_{\text{R}}\rightarrow\bar{R}_{\text{R}}=\pi
L^{2}_{\lambda}R_{\text{R}}$, and hence we obtain

$\displaystyle\bar{R}_{\text{R}}=-\frac{(\bar{q}^{2}-\bar{r}^{2}_{+})\big{[}\bar{r}^{4}_{+}-2\bar{r}^{2}_{+}(1-\bar{q}^{2})+4\bar{q}^{2}-1\big{]}}{\left[3\bar{q}^{2}-\bar{r}^{2}_{+}+\bar{r}^{2}_{+}(\bar{q}^{2}+\bar{r}^{2}_{+})\right]^{2}}.$
(76)

Using Eq. (76) to illustrate the phase structure in $(\bar{r}_{+},\bar{q})$
plane as shown in the right panel in Fig. 12, even though the black hole
solutions are allowed due to the condition $M^{2}>q^{2}$ in a similar fashion
as in the GB thermodynamics, the charged black holes can exist not only with
$\bar{R}_{\text{R}}<0$ (yellow region) but also with $\bar{R}_{\text{R}}>0$
(blue region) in the alternative Reńyi thermodynamics. The emergence of the
additional blue region in the right panel in Fig. 12 leads us to wonder what
is the nature of the critical phase transitions between these several phases,
which include the new emergent black hole phase with $\bar{R}_{\text{R}}>0$.
To explore some relations between critical phase transition and interactions
among black hole molecules, we will investigate $\bar{R}_{\text{R}}$ as a
function of $\bar{r}_{+}$ and $\bar{T}_{\text{R}}$. In Fig. 14, we plot
$\bar{R}_{\text{R}}$ versus $\bar{r}_{+}$ for $\bar{q}<\bar{q}_{c}$,
$\bar{q}=\bar{q}_{c}$ and $\bar{q}>\bar{q}_{c}$, respectively. For small
charge $\bar{q}<\bar{q}_{c}$, the $\bar{R}_{\text{R}}$ curve can be separated
into the curves for three branches of black holes, namely, pBH1 (red curve),
nBH (green curve) and pBH2 (blue curve) as shown in the left panel in Fig. 14.
Note that $\bar{R}_{\text{R}}\to-\infty$ at the horizon radius of the size
$\bar{r}_{1}$ and $\bar{r}_{2}$ corresponds to
$|\bar{C}_{\text{R}}|\to\infty$. The negativity of the Ricci scalar
$\bar{R}_{\text{R}}$ for both pBH1 and nBH phases implies attractive
interactions between black hole microstructures, in common with the
consideration using the GB statistics. Interestingly, $\bar{R}_{\text{R}}$ of
the emergent pBH2 phase has negative and positive values when
$\bar{r}_{2}<\bar{r}_{+}<\bar{r}_{z}$ and $\bar{r}_{+}>\bar{r}_{z}$,
respectively. This means that the interactions between microstructures of the
black hole turn out to be repulsive when the horizon size is sufficiently
larger than the nonextensivity length scale, namely
$\bar{r}^{2}_{+}\geq\bar{r}^{2}_{z}=1-\bar{q}^{2}+\sqrt{\bar{q}^{4}-6\bar{q}^{2}+2}$.
At the critical point ($\bar{q}=\bar{q}_{c}$), horizon radius of the sizes
$\bar{r}_{1}$ and $\bar{r}_{2}$ are degenerate with the value of
$\bar{r}_{0}$, such that the nBH phase disappears. As previously discussed,
apart from its sign, the value of $|\bar{R}_{R}|$ could be associated with the
strength of interactions or correlations among black hole constituents. In
other words, the larger the value of $|\bar{R}_{R}|$ is, the stronger the
interactions between black hole molecules are. As well known that the
correlation length $\xi$ of the system diverges near the critical point, the
infinity of $|\bar{R}_{R}|$ at this point as shown in the middle panel in Fig.
14 is therefore not so weird. For $\bar{q}>\bar{q}_{c}$, there exists only a
single phase with positive heat capacity where $\bar{R}_{R}$ is negative and
positive in the ranges $\bar{q}<\bar{r}_{+}<\bar{r}_{z}$ and
$\bar{r}_{+}>\bar{r}_{z}$, respectively (the right panel in Fig. 14).

---
Figure 14: The Ricci scalar $\bar{R}_{\text{R}}$ versus the outer horizon
radius $\bar{r}_{+}$ of the RN-AF from the Rényi statistics for fixed charges
$\bar{q}<\bar{q}_{c}$ (left), $\bar{q}=\bar{q}_{c}$ (middle) and
$\bar{q}>\bar{q}_{c}$ (right), respectively.

The phase structure can be explored by considering $\bar{R}_{R}$ as a function
of $\bar{T}_{\text{R}}$ which corresponds to the plot of Gibbs free energy in
Fig. 5. For small charge $\bar{q}<\bar{q}_{c}$, there are three branches of
charged black hole which are pBH1, nBH and pBH2 phases represented by red,
green and blue curves, respectively, as shown in the left panel in Fig. 15. At
the extremal limit corresponding to $\bar{T}_{\text{R}}=0$, we have found that
$\bar{R}_{\text{R}}=0$ in the same way as in the RN-AF with GB statistics.
However, it is different from the case of the Reissner-Nordström black hole in
asymptotically anti-de Sitter (RN-AdS) with GB statistics where the Ricci
scalar negatively diverges at the extremal black hole Shen  _et al._ (2007).
Since the heat capacity and the Ricci scalar diverge at the same value of
horizon radius, i.e. $\bar{r}_{1}$ and $\bar{r}_{2}$, the nBH-pBH1 and nBH-
pBH2 phase transitions occur at the temperatures denoted by $\bar{T}_{2}$ and
$\bar{T}_{1}(<\bar{T}_{2})$, respectively. One can split the temperature into
three ranges. As shown in the left panel in the figure, the pBH1 phase exists
at low temperature ($\bar{T}_{\text{R}}<\bar{T}_{1}$) as well as at a certain
temperature in the range $\bar{T}_{1}<\bar{T}_{\text{R}}<\bar{T}_{2}$, in
which the nBH and pBH2 phases simultaneously also appear corresponding to the
swallowtail behavior in the free energy diagram in the upper left panel in
Fig. 5. It is also seen that there exists the crossing between the Ricci
scalar for the pBH1 (red) and pBH2 (blue) phases. The temperature of this
crossing point is not the Hawking-Page temperature $T_{\text{HP}}$ (see the
orange dashed line). Unfortunately, the Ricci scalar for the Ruppeiner
geometry does not contain the information of the first-order phase transition.
When the temperature increases to $\bar{T}_{\text{R}}>\bar{T}_{2}$, the pBH1
and nBH phases disappear, while pBH2 phase persists for this range of high
temperature. This feature can be also found in the upper left panel in Fig. 5.
Moreover, the pBH1 and nBH phases have $\bar{R}_{\text{R}}<0$ at any
temperature. Therefore, the interactions between two black hole
microstructures are attractive. For pBH2 phase, we find that
$\bar{R}_{\text{R}}<0$ when $\bar{T}_{1}<\bar{T}_{\text{R}}<\bar{T}_{z}$.
Interestingly, at a sufficiently high temperature
$\bar{T}_{\text{R}}>\bar{T}_{z}$, the pBH2 phase begins to has
$\bar{R}_{\text{R}}>0$ and hence the interactions between two black hole
microstructures become repulsive as found in the RN-AdS with the GB statistics
Wei  _et al._ (2019b, a).

---
Figure 15: The Ricci scalar $\bar{R}_{\text{R}}$ versus the Rényi temperature
$\bar{T}_{\text{R}}$ of the RN-AF with the Rényi statistics for fixed charges
$\bar{q}<\bar{q}_{c}$ (left), $\bar{q}=\bar{q}_{c}$ (middle) and
$\bar{q}>\bar{q}_{c}$ (right), respectively.

Moreover, the temperatures $\bar{T}_{1}$ and $\bar{T}_{2}$ get closer as the
electric charge $\bar{q}(<\bar{q_{c}})$ increases. The temperatures then merge
together at $\bar{T}_{1}=\bar{T}_{2}=\bar{T}_{z}$ when the charge reaches its
critical value. This merger corresponds to the absence of the nBH phase. At
the critical point $\bar{q}=\bar{q}_{c}$, the crossing of $\bar{R}_{\text{R}}$
for pBH1 and pBH2 phases disappears resulting that $\bar{R}_{\text{R}}$
negatively diverges as shown in the middle panel in Fig. 15. For
$\bar{q}>\bar{q}_{c}$, there is only a single pBH phase of black hole
represented as the brown curve in the right panel in Fig. 15. Obviously, the
divergence in $\bar{R}_{\text{R}}$ becomes absent for a large amount of
charge.

It is very important to note that the pBH2 phase emerging within the framework
of the alternative Rényi statistics allows $\bar{R}_{\text{R}}$ to be positive
value, which is in the same way as the large black hole phase of the RN-AdS
from the GB statistics. This implies that the interactions between black hole
molecules are repulsive.

##  VI Critical Behavior of the RN-AF in Rényi statistics

It has been found that thermodynamic behaviors of matter near a critical point
of phase transition in different systems tend to have the property of
universality. Remarkably, this behavior does not depend on the types of
particles and of interactions between them. The scaling behavior of physical
quantities in the vicinity of critical phase transition can be described by a
set of critical exponents. For example, the scaling behavior of a vdW fluid
can be written as

$\displaystyle P-P_{c}$ $\displaystyle\sim$ $\displaystyle(V-V_{c})^{\delta},$
(77) $\displaystyle V_{g}-V_{l}$ $\displaystyle\sim$
$\displaystyle(T-T_{c})^{\beta},$ (78) $\displaystyle C_{V}$
$\displaystyle\sim$ $\displaystyle(T-T_{c})^{-\alpha},$ (79)
$\displaystyle\kappa_{\text{T}}$ $\displaystyle\sim$
$\displaystyle(T-T_{c})^{-\gamma},$ (80)

where the critical exponents are $\delta=3$, $\beta=1/2$, $\alpha=0$ and
$\gamma=1$. $V_{g}$ and $V_{l}$ are the volumes for the gas and liquid phases
of the vdW fluid. For different systems which are governed by the same set of
critical exponents, we can say that they are in the same universality class.
Interestingly, it has been found that the critical exponents of RN-AdS in the
canonical ensemble are identical with the vdW fluid, and hence these systems
are in the same universality class of phase transition Niu  _et al._ (2012);
Shen  _et al._ (2007); Wang  _et al._ (2020). As discussed in section III, the
phase transition of the RN-AF with nonextensive description and the vdW
liquid-gas system are very similar. There are the swallowtail behaviors of
free energy as shown in Fig. 5. There also exists the finite coexistence line
which terminates at the critical point as indicated in Fig. 6 and the
transitions between two phases, i.e., pBH1-pBH2 transition for the RN-AF with
Rényi statistics and liquid-gas transition for the vdW fluid, across the
coexistence line are the first-order type of phase transition. Notice that, in
both cases, these first-order phase transitions disappear when the systems
approach to the critical point at $T=T_{c}$. Moreover, by considering the
Maxwell construction in section IV, we have found that the pBH1 and pBH2
phases coexist on the isothermal curve at the charge $\bar{q}^{*}$ as shown in
Fig. 8. In other words, this value of charge can render the vapor pressure
analogue, where the pBH1-pBH2 coexistence can be associated with the dynamic
equilibrium between the liquid and gas phases at constant temperature on the
$P-V$ plane in the vdW fluid.

In this section, we are interested in categorizing the phase transition of the
RN-AF in Rényi description by considering the scaling behavior of physical
quantities. We identify the quantities $q$ and $\phi$ of the RN-AF as $P$ and
$V$ of the vdW fluid, respectively. Considering the system near the critical
point, Eq. (52) as an equation of state can be written in the form of power
series expansion in terms of a set of the variables $\tau$ and $\omega$ as

$\displaystyle\bar{q}$ $\displaystyle=$ $\displaystyle
a_{mn}\tau^{m}\omega^{n}$ (81) $\displaystyle=$ $\displaystyle
a_{00}+a_{10}\tau+a_{01}\omega+a_{11}\tau\omega+a_{20}\tau^{2}$
$\displaystyle+a_{02}\omega^{2}+a_{21}\tau^{2}\omega+a_{12}\tau\omega^{2}+a_{30}\tau^{3}+a_{03}\omega^{3}+\ldots,$

where $a_{mn}$ is the coefficient of the expansion, and the variables $\tau$
and $\omega$ are defined by

$\displaystyle\tau=\frac{\bar{T}_{\text{R}}}{\bar{T}_{c}}-1,\quad\omega=\frac{\bar{\phi}}{\bar{\phi}_{c}}-1.$
(82)

The power series in Eq. (81) can be numerically obtained by

$\displaystyle\bar{q}$ $\displaystyle=$
$\displaystyle\bar{q}_{c}-0.732\tau+1.732\tau\omega+3.732\tau^{2}$ (83)
$\displaystyle-24.12\tau^{2}\omega-5.598\tau\omega^{2}-27.856\tau^{3}-0.366\omega^{3}+\ldots,$

where $a_{00}=\bar{q}_{c}=0.268$ and $a_{01}=a_{02}=0$. Firstly, we consider
the change of $\bar{q}$ with $\bar{\phi}$ being near the critical point when
the temperature $\bar{T}_{R}$ is set to be $\bar{T}_{c}$, hence we have
$\tau=0$. Interestingly, Eq. (83) can be written in the form of scaling law as

$\displaystyle\bar{q}-\bar{q}_{c}\sim(\bar{\phi}-\bar{\phi}_{c})^{3}.$ (84)

Comparing this with Eq. (77), the critical exponent $\delta$ can be read off
as

$\displaystyle\delta=3.$ (85)

In section IV, we have shown the first-order pBH1-pBH2 phase transition at
$\bar{q}^{*}$, which can be associated with a jump between $\bar{\phi}_{3}$
and $\bar{\phi}_{1}$. To investigate this behavior near the critical point of
the second-order phase transition, we need to consider the Gibbs free energy.
The change of the Gibbs free energy can be written as

$\displaystyle
d\mathcal{\bar{G}}_{\text{R}}=-\bar{S}_{\text{R}}d\bar{T}_{\text{R}}+\bar{\phi}d\bar{q}.$
(86)

Since the pBH1 and pBH2 coexist, this implies that their Gibbs free energies
are identical, namely, we have
$\Delta\mathcal{\bar{G}}_{\text{R}}=\mathcal{\bar{G}}_{\text{pBH1}}-\mathcal{\bar{G}}_{\text{pBH2}}=0$.
Therefore, the values of $\bar{\phi}$ on either side of the coexistence curve
at fixed temperature can be found from two following conditions

$\displaystyle\Delta\mathcal{\bar{G}}_{\text{R}}=\int_{\bar{\phi}_{1}}^{\bar{\phi}_{3}}\bar{\phi}d\bar{q}=0,$
(87)

and

$\displaystyle\bar{q}^{*}=\bar{q}(\bar{\phi}_{1})=\bar{q}(\bar{\phi}_{3}).$
(88)

Differentiating Eq. (81) with respect to $\omega$, one obtains

$\displaystyle
d\bar{q}=(a_{11}\tau+a_{21}\tau^{2}+2a_{12}\tau\omega+3a_{03}\omega^{2})d\omega.$
(89)

Note that we have kept the dimensionless perturbation variables $\tau$ and
$\omega$ in the above formula up to the third order expansion. Let us define
$\bar{\phi}_{1}=\bar{\phi}_{c}(1-\omega_{1})$ and
$\bar{\phi}_{3}=\bar{\phi}_{c}(1+\omega_{3})$. Using Eqs. (82) and (89), the
conditions (87) and (88) can be, respectively, expressed in terms of the
expansion coefficients as follows:

$\displaystyle
a_{11}\tau(\omega_{3}+\omega_{1})+a_{21}\tau^{2}(\omega_{3}+\omega_{1})+\frac{1}{2}(a_{11}+2a_{12})\tau(\omega^{2}_{3}-\omega^{2}_{1})+a_{03}(\omega^{3}_{3}+\omega^{3}_{1})$
$\displaystyle=$ $\displaystyle 0,$ (90) $\displaystyle
a_{11}\tau(\omega_{3}+\omega_{1})+a_{21}\tau^{2}(\omega_{3}+\omega_{1})+a_{12}\tau(\omega^{2}_{3}-\omega^{2}_{1})+a_{03}(\omega^{3}_{3}+\omega^{3}_{1})$
$\displaystyle=$ $\displaystyle 0.$ (91)

When $\omega_{1}=\omega_{3}=\omega$, these two conditions have a non-trivial
solution which is given by

$\displaystyle\omega^{2}=-\frac{1}{a_{03}}(a_{11}\tau+a_{21}\tau^{2})\approx-\frac{a_{11}}{a_{03}}\tau=4.732\tau.$
(92)

Recalling Eq. (82), Eq. (92) can be written in the form of a scaling law
relating $\bar{\phi}$ with $\bar{T}_{R}$ near the critical point as

$\displaystyle\bar{\phi}-\bar{\phi}_{c}\sim(\bar{T}_{\text{R}}-\bar{T}_{c})^{\frac{1}{2}},$
(93)

where, comparing Eq. (78), we can read off the critical exponent

$\displaystyle\beta=\frac{1}{2}.$ (94)

The behavior of heat capacity at constant $\bar{\phi}$ near the critical point
can be described by the exponent $\alpha$. Since $C_{\phi}$ does not display a
singularity behavior at the critical point, the critical exponent therefore
vanishes, i.e. $\alpha=0$. Finally, we will calculate the critical exponent
$\gamma$ of the compressibility at constant temperature,

$\displaystyle\bar{\kappa}_{T_{\text{R}}}=-\frac{1}{\bar{\phi}}\left(\frac{\partial\bar{\phi}}{\partial\bar{q}}\right)_{\bar{T}_{\text{R}}}.$
(95)

By using Eq. (81), one can compute

$\displaystyle\left(\frac{\partial\bar{q}}{\partial\omega}\right)_{\tau}=a_{11}\tau+a_{21}\tau^{2}+2a_{12}\tau\omega+3a_{03}\omega^{2}.$
(96)

As the system approaches the critical point, we can use Eqs. (95) and (96) to
find that

$\displaystyle\bar{\kappa}_{T_{\text{R}}}\sim(\bar{T}_{\text{R}}-\bar{T}_{c})^{-1},$
(97)

which gives us the critical exponent

$\displaystyle\gamma=1.$ (98)

Interestingly, our studies have demonstrated that the critical exponents of
the RN-AF system from the Rényi statistics are identical to those of the vdW
fluid and of the RN-AdS from the GB statistics. Manifestly, these are in the
same universality class.

##  VII Conclusion and Discussion

In the present work, we have studied the thermodynamical properties of the
charged black hole in asymptotically flat spacetime using the nonextensive
Rényi statistics. Considering the phase space in which the electric charge and
electrostatic potential are treated as the thermodynamical pressure and
volume, respectively, we find that there exist the emergent pBH2 phase and the
critical behavior of the phase transition. These novel features due to the
nonextensive nature of the Rényi entropy do not appear in the case of the GB
statistics. Moreover, the swallowtail behavior appears in the free energy
diagrams. This behavior is reminiscent of the phase transition in the vdW
fluid. The local stability of the black hole can also be determined by the
positiveness of compressibility, in addition to that of heat capacity (see Eq.
(27)). Interestingly, the black hole in the emergent pBH2 phase is locally
stable when the conditions of both thermal and mechanical stabilities in the
range $\bar{q}<\bar{\phi}<\bar{\phi}_{-}$ are satisfied, where
$\bar{\phi}_{-}$ has been defined in Eq. (47). On the other hand, the nBH and
pBH1 are always locally unstable and locally stable, respectively, as
discussed in section III. For global stability, the sufficiently large black
hole in pBH2 phase is thermodynamically preferred than those in other phases,
since its Gibbs free energy is relatively lower. Remarkably, there is the
Hawking-Page phase transition by jumping from the pBH1 to pBH2 phases caused
by the presence of nonextensivity (see Fig. 5). The Hawking-Page temperature
decreases as charge increases and then terminates at its critical value (see
Fig. 6).

For the isothermal process, the RN-AF from the Rényi statistics is possible to
undergo the first-order phase transition by jumping between the pBH1 and pBH2
phases instead of passing the locally unstable nBH phase via the second-order
phase transitions along the isotherm. This jump can be described by the
Maxwell construction similar to those of the vdW fluid and the RN-AdS with the
GB statistics. In other words, we can find the certain charge $\bar{q}^{\ast}$
at which the Gibbs free energies (corresponding to the areas between the
isotherms and the horizontal line $\bar{q}=\bar{q}^{\ast}$ in the
$\bar{q}-\bar{\phi}$ plane) of the pBH1 and pBH2 phases are equal for
$\bar{T}_{\text{R}}>\bar{T}_{c}$. To study the Maxwell construction, we split
our consideration into two cases which are near the critical point
($\bar{T}_{\text{R}}\gtrsim\bar{T}_{c}$) and far away from the critical point
($\bar{T}_{\text{R}}\gg\bar{T}_{c}$). For the former (latter) case, the
aforementioned phase transition is the transition between the pBH1 phase and
mechanically stable (unstable) pBH2 phase. As discussed in section IV, the
pBH2 phases in the $\bar{q}_{2}(\bar{\phi})$ branch are mechanically stable
while those in the $\bar{q}_{1}(\bar{\phi})$ branch are mechanically unstable,
respectively. Interestingly, a cusp of isotherm at the origin
$\bar{q}=\bar{\phi}=0$ appears when $\bar{T}_{\text{R}}\geq 1$. Note that this
has been found in a similar way as in the RN-AdS case Chamblin  _et al._
(1999b); Zhou and Wei (2019). From the analysis with the Maxwell construction
in this work, the first-order phase transition seems to occur at very small
$\bar{q}$ and $\bar{\phi}$ with arbitrarily large $\bar{T}_{\text{R}}$.
Nevertheless, this tends to be inconsistent with the result from the approach
of comparing free energy, in which the phase transition between pBH1 and pBH2
phases in the RN-AF with small $\bar{q}$ and $\bar{\phi}$ from Rényi
statistics can occur only at the temperature below $T_{\text{HP}}$ of the
Schwarzschild black hole as seen in Fig. 6. To address this ambiguity, we
consider the merging point between $\bar{q}_{1}(\bar{\phi})$ and
$\bar{q}_{2}(\bar{\phi})$ branches. We have interestingly found that the
temperature is constrained as $\bar{T}_{\text{R}}\leq 1$ (see Eq. (57)). In
other words, the isotherms with $\bar{T}_{\text{R}}>1$ are undefined. It is
also noted that the black hole with $\bar{T}_{\text{R}}=1$ corresponds to the
Schwarzschild black hole ($\bar{q}=\bar{\phi}=0$). The maximum temperature of
occurrence the phase transition between mechanically stable pBH1 and pBH2
phases has been evaluated in Eq. (62). The isotherms at this temperature are
also illustrated in Fig. 10. From this figure, it can be seen that the first-
order phase transition in $\bar{q}-\bar{\phi}$ plane is directly related the
swallowtail behavior in the free energy diagram. The phase structure of the
black hole similar to those of the vdW fluid and of the RN-AdS from the GB
statistics can be obtained as shown in Fig. 11.

Some aspects of black hole phase structure can be seen further through the
approach of thermodynamic geometry. On the one hand, the second-order phase
transition in black hole system can be demonstrated through the divergence of
the Ricci scalar curvature in the Ruppeiner geometry, which corresponds to the
divergence of heat capacity. On the other hand, there is no information of the
first-order phase transition in the Ricci scalar. These results hold for the
black hole thermodynamics from both the GB statistics and the Rényi one.
However, a different result from these two framework can be seen in the sign
of the Ricci scalar curvature which may imply the types of the interaction of
the black hole microstructures, namely, either attractive or repulsive. It has
been found that the interaction can be repulsive in some range of pBH2 phase
in the Rényi statistics, whereas it would be only attractive in the GB case.

We then further interpret that, apart from the charge, the nonextensivity of
the black hole could be another contribution which lead to a repulsive
interaction between the constituents in microscopic structure of black hole.
Remarkably, the occurrence of repulsive interactions other than the electric
forces between charges of black hole has confirmed our previous suggestion
that some non-trivial correlations between the microstates of a self-
gravitating system could be emerged via the nonextensive nature of long-range
interaction systems Promsiri  _et al._ (2020, 2021). Interestingly, the
correlations could be manifested themselves as the nonextensive nature in
black hole thermodynamics, and vice versa. Recently, some peculiar phenomena
arising from the nonextensivity nature of correlated systems have been studied
in gravitational physics Komatsu (2017); Moradpour  _et al._ (2017, 2018);
Tavayef  _et al._ (2018); Jiulin (2004); Du (2006); Carvalho  _et al._ (2008);
Ourabah (2020). See also Tsallis (2009) for a good review of $q$ statistics,
together with its related thermodynamic aspects and its applications in a wide
range of physics.

Although, the phase structures of the RN-AdS and the RN-AF with Rényi
statistics in the fixed charge ensemble are very similar, their microscopic
aspects are quite different. By considering the thermodynamic geometry, our
study indicates that the value of the scalar curvature near the extremal limit
of the RN-AdS negatively diverges while that of the RN-AF with Rényi
statistics vanishes. This implies that the interaction among the
microstructures of the RN-AdS is highly attractive while the RN-AF with Rényi
description has no interaction between their microscopic constituents.
However, we have also investigated the behavior of the RN-AF in the vicinity
of the critical point of the second-order phase transition in the Rényi model.
The results show that the critical exponents, i.e., $\delta,\beta,\alpha$ and
$\gamma$, are identical with those in the vdW fluid and the RN-AdS from the GB
statistics. Therefore, these three systems could be said to be in the same
universality class. This is a further evidence that the behaviors of many
different physical systems near the critical point are independent of the type
of microscopic constituents and interaction between them.

One of the most intriguing discoveries is the connection between geometry and
thermodynamics of black holes, i.e., the surface gravity and the surface area
at event horizon are related to the Hawking temperature and the Bekenstein-
Hawking entropy, respectively. These relations are seem to be spoiled in the
Rényi description, and one may argue that the black hole thermodynamics with
this approach is not relevant, as discussed in Nojiri  _et al._ (2021).
However, the violation of the entropy area law can occur in many quantum
systems with the strong correlation or mutual information between microscopic
degrees of freedom Wolf (2006); Wolf  _et al._ (2008); Vitagliano  _et al._
(2010); Lai  _et al._ (2013). Moreover, a loophole in black hole
thermodynamics from the conventional approach can be seen from that the
Hawking temperature has to be derived based on the GB statistics Hawking
(1975, 1976), while its entropy implies the existence of nonextensive nature.
Hence, the entropic nature of a black hole can be debatable and a study on
black hole thermodynamics from an alternative statistics with the presence of
nonextensive nature cannot be said to make no sense. Based on one- or multi-
parameter deformation statistics including the Rényi one, various publications
study the systems of bosonic and fermionic fields Lavagno and Narayana Swamy
(2000, 2002, 2010); Dil (2017). This might be a suggestion to derive the
Hawking radiation of a black hole using an appropriate quantum field
description.

## Acknowledgement

We are grateful to Supakchai Ponglertsakul for helpful discussion. This
research project is supported by Thailand Science Research and Innovation
(TSRI) Basic Research Fund: Fiscal year 2021 under project number FRB640008.
RN is supported by National Research Council of Thailand (NRCT) :
NRCT5-RGJ63009-110. CP has been supported by the Petchra Pra Jom Klao Ph.D.
Research Scholarship from King Mongkut’s University of Technology Thonburi
(KMUTT).

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