Cooling and heating regions of Joule-Thomson expansion for AdS black holes:
Maxwell-power-Yang-Mills and Kerr Sen black holes
Jafar Sadeghi ⋆111Email<EMAIL_ADDRESS>Mohammad Reza Alipour ⋆222Email:
<EMAIL_ADDRESS>Saeed Noori Gashti†,⋆333Email:
<EMAIL_ADDRESS>
Mohammad Ali S. Afshar ⋆444Email<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences,
University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran
†School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran

###### Abstract

In this paper, we explore the Joule-Thomson expansion (JTE) process for the
Einstein-Power-Young-Mills (EPYM) and the AdS Kerr Sen (AKS) black holes. We
study the effect of free parameters on the Joule-Thomson coefficient (JTC),
the inversion curve, and the $T_{i}^{min}/T_{c}$. The isenthalpic curves of
the AKS black hole show cooling or heating behavior depending on the inversion
curve, which is affected by the mass and the parameters $b$ and $a$ of the
black hole. If we assume the parameter $b$ to be zero, the results reduce to
the Kerr–AdS black holes[1]. In [2, 3], for the Einstein–Power–Yang–Mills AdS
black hole with $q>1$ and $n=2$, the $T_{i}^{min}/T_{c}$ is $1/2$. But in this
paper, for the AdS-Maxwell-power-Yang-Mills black hole, when $q>1$, the
$T_{i}^{min}/T_{c}$ is almost equal to $1/2$ for the increase of Maxwell’s
charge $C$, and when $q=1/2$, the $T_{i}^{min}/T_{c}$ is equal to $1/2$ for
all values of $C$. Also, when $1/2<q<1$, the $T_{i}^{min}/T_{c}$ is close to
the value of $1/2$, and finally when $0<q<1/2$, the values of
$T_{i}^{min}/T_{c}$ move away from the value of $1/2$, that is, they become
smaller. For the AKS black hole, we found that for free parameters $a=0.00951$
and $b=0.00475$, the $T_{i}^{min}/T_{c}$ is almost $1/2$. Finally, we compare
our findings with others in the literature and summarize our results in Tables
1-5.

Keywords: JTE, AdS-Maxwell-power-Yang-Mills black holes, AdS Kerr Sen black
holes

###### Contents

  1. 1 Introduction
  2. 2 Joule-Thomson expansion
  3. 3 Case I: AdS-Maxwell-power-Yang-Mills black holes
  4. 4 Case II: AdS-Kerr-Sen black hole
  5. 5 Discussion and results
    1. 5.1 Case I
    2. 5.2 Case II
  6. 6 Conclusion

##  1 Introduction

The insights provided by astronomy, astrophysics, and experimental cosmology
suggest that cosmic structures adhere to the same principles and foundations
observed on Earth, albeit with minor variations and necessary simplifications.
This structural and fractal similarity is a crucial tool for human ideation
and cognition, enabling us to decode more aspects of this global pattern
daily. Researchers leverage this understanding in hypothesizing theoretical
models, which may not yet be empirically verifiable. They construct various
models based on mathematical logic and physical laws, exploring all potential
scenarios to predict the most plausible ideas consistent with the formation of
the enigmatic, infinite universe. The existing black hole models, are actually
one of the most obvious examples of this form of inference. Despite being one
of the most elusive cosmic entities, they are studied based on this pattern.
The comparison of a black hole’s gravitational behavior with a thermodynamic
ensemble in the 1970s gave rise to a significant branch of black hole physics,
namely black hole thermodynamics. For instance, the four laws of black hole
mechanics bear a striking resemblance to the laws of thermodynamics [4, 5].
Similarly, a black hole’s surface area is analogous to entropy in
thermodynamics, and its surface gravity is comparable to temperature[4, 5].
The introduction of Maldacena duality, also known as the AdS/CFT
correspondence, established a deeper connection between black hole
thermodynamics and this duality, offering profound insights into quantum
gravity. In the context of AdS/CFT, a black hole’s entropy in the bulk AdS
space correlates with the entropy of the corresponding CFT on the boundary,
known as the Bekenstein-Hawking entropy [6]. Moreover, the black hole’s
temperature is related to the CFT’s temperature[6, 7, 8, 9].

This correspondence provides a powerful tool to study the quantum aspects of
gravity and black holes using the methods of quantum field theory. In AdS
space, there is a Hawking-Page phase transition between a stable large black
hole and a thermal gas[10]. This phase transition is a first-order transition
that occurs when the temperature of the system reaches a critical value, where
the free energy of the black hole becomes lower than that of the thermal
gas[10]. This phase transition can be interpreted as a confinement
deconfinement phase transition of a gauge field[11]. The Hawking-Page phase
transition can be seen as a transition from a deconfined phase in the thermal
gas to a confined phase in the black hole[11]. When the AdS black holes have
electric charge, they exhibit rich phase structures that were studied by
Chamblin et al[12, 13]. They found that the phase transition behavior of
charged AdS black holes resembles the liquid-gas phase transition in a van der
Waals system[14]. In the extended phase space where the cosmological constant
is treated as pressure[15], the P-V critical behavior of charged AdS black
holes was investigated and it was shown that they have a similar analogy to
the van der Waals liquid-gas system. In addition to the phase transition and
critical phenomena[16, 17, 18, 19, 20, 21, 22, 23], the analogy between the
black holes and the van der Waals system was also creatively applied to the
well-known JTE process[24] recently. This means that the JTC can be used to
study the thermodynamics of black holes as well. For example, the isenthalpic
expansion process is the analogue of the JTE process for black holes, where
the black hole mass is constant while the black hole pressure and volume are
changed. The black hole pressure is related to the cosmological constant, and
the black hole volume is related to the horizon radius.In this case, the
inversion curve is the curve that separates the regions where the black hole
temperature increases or decreases as the pressure decreases.One of the
intriguing features of the inversion curves for black hole systems is that
they have only positive slopes, unlike the van der Waals system, which has
both positive and negative slopes. For example, for charged AdS black holes
[24] and Kerr-AdS black holes [1], the isenthalpic expansion process and the
inversion curve have been analyzed and observed that the inversion curve was
found to have a positive slope, meaning that the black hole temperature always
decreases as the pressure decreases. Then the analysis was generalized to
other types of AdS black holes, such as quintessence charged AdS black
holes[25], holographic superfluids [26], charged AdS black holes in f(R)
gravity [27], AdS black holes with a global monopole [28], and AdS black holes
in Lovelock gravity [29]. For further study, you can see also [30, 31, 32, 33,
34, 35, 36, 37, 2, 38].

All the results showed that the inversion curves for all these black hole
systems have only positive slopes. We are interested in exploring whether this
feature is universal for all black hole systems, or whether there are other
effects that can alter it. To this end, we will focus on two types of AdS
black holes: AdS-Maxwell-power-Yang-Mills and AdS Kerr Sen. These black holes
have additional parameters that can affect their thermodynamic behavior and
phase transitions. AdS-Maxwell-power-Yang-Mills black holes are black holes
that have a non-linear electromagnetic field, which is described by a power-
law function of the field strength[39]. This field can be seen as a
generalization of the Maxwell field, which is the standard model of
electromagnetism. AdS Kerr Sen black holes are black holes that have electric
charge and angular momentum in a low-energy limit of heterotic string
theory[40]. This theory is a type of string theory that combines the features
of bosonic and supersymmetric strings. These black holes have different
properties and characteristics than the standard AdS black holes, such as the
existence of a dilaton field, which is a scalar field that couples to the
electromagnetic field and the curvature. We will investigate how these
parameters affect the inversion curves and the JTE process for these black
holes, and compare them with the previous results for other black hole
systems.

The structure of this paper is as follows. In Sec.II, we give a brief overview
of the JTE. In Sec.III and Sec.IV, We introduce briefly AdS-Maxwell-Power-
Yang-Mills and the AdS Kerr Sen black holes and their thermodynamic
properties.In Sec.V, we study the JTE process for these black holes, derive an
explicit expression for the JTC, and analyze and discuss the effect of the
parameters of each model on the inversion curves. In Sec.VI, we present our
conclusion and discussion.

##  2 Joule-Thomson expansion

In classical thermodynamics, a throttling process or the JTE process,
discovered in 1852, is a method of cooling or heating a system by changing its
pressure and volume, without adding or removing heat. In this process, the
high-pressure gas passes through a porous plug into a region with a low
pressure, while keeping the enthalpy constant. Since it is a constant-enthalpy
process, it can be used to experimentally measure the lines of constant
enthalpy (isenthalps) on the (p, T) diagram of a system. Combined with the
specific heat capacity at constant pressure, it allows the complete
measurement of the thermodynamic potential for the gas[41].

In this method, The main goal is to investigate the behavior of the
coefficient that describes the temperature change during the expansion or
compression of a system at constant enthalpy, which is denoted by $\mu$ and is
known as the JTC,

$\mu=\big{(}\frac{\partial T}{\partial
P}\big{)}_{H}=\frac{1}{C_{P}}\big{[}T(\frac{\partial V}{\partial
T}\big{)}_{P}-V\big{]}.$

If the above coefficient is positive, as a result of pressure reduction, the
temperature will decrease. In other words, the expansion of the gas causes
cooling and the compression of the gas under investigation causes it to heat
up. In other words, the positive JTC indicates the same direction of
temperature and pressure. Whereas, if the JTC is negative, a decrease in
pressure causes an increase in temperature.
The JT inversion temperature,which determined by setting $\mu=0$, is the
temperature at which the sign of the JTC changes. Most real gases have an
inversion point. The temperature of this point depends on the gas pressure
before expansion.

$T_{i}=V\big{(}\frac{\partial T}{\partial V}\big{)}_{P}.$

The important point is that if you plot the JTC on the (p,T) diagram, a closed
parabolic curve is created. In simpler terms, the inversion temperature is
placed on the boundary of the curve of temperature changes in terms of
pressure. At this temperature, the JTC changes from negative to positive. At a
given pressure, the isopressure-temperature line intersects the drawn curve at
two different points. These two points are called low temperature and high
temperature of inversion. In fact, at temperatures higher and lower than these
two temperatures, the sign of the JTC is negative and between these two
temperatures, the sign of the JTC is positive. According to the above, the
interesting phenomenon in this process is that the (p, T) diagram has two
regions: one where the gas cools down and one where the gas heats up. These
regions are separated by the inversion curve,the curve that shows the points
where the system temperature does not change during the expansion process,
that divides the graph into two regions: the cooling region and the heating
region. The cooling region is where the gas temperature decreases as the
pressure decreases, and the heating region is where the gas temperature
increases as the pressure decreases. The inversion curve depends on the type
of the gas and its initial conditions. If with respect to zero coefficient for
ideal gases, we choose the van der Waals system, which is a more realistic
model than the ideal gas, and takes into account the finite size and the
attractive forces of the molecules, we find that for the van der Waals system,
the inversion curves have both positive and negative slopes, forming a circle
in the pressure axis. The inversion curve for the van der Waals system has a
negative slope in the low-pressure region, where the attractive forces
dominate, and a positive slope in the high-pressure region, where the
repulsive forces dominate.

##  3 Case I: AdS-Maxwell-power-Yang-Mills black holes

The AMPYM black holes are rooted in the study of black hole solutions in the
context of supergravity theories, especially in anti-de Sitter (AdS) space.
The study of black holes in AdS space is important for various reasons,
including its relevance to string theory and the AdS/CFT correspondence, which
is a duality between gravitational theories in AdS space and field theories
defined on its boundary. These black holes are solutions to the equations of
motion of supergravity theories with additional matter fields, such as Maxwell
fields and power-Yang-Mills fields, within the context of AdS space that
arises from a generalization of the Einstein-Maxwell theory with a negative
cosmological constant and a non-Abelian gauge field.The history of AMPYM black
holes can be traced back to the discovery of the first black holes in
Einstein-Yang-Mills theory, which were considered in the works of Yasskin [42]
and Kasuya [43]. It should be noted that in the non-Abelian case there are
various gauge groups, but to obtain black hole solutions it is necessary to
choose a specific form for the gauge potential. One of the simplest forms that
nevertheless allowed to derive interesting and important results is the so-
called Wu-Yang ansatz, which leads to magnetic-type solutions. [44, 45, 46,
47]. Of course, an interesting point that can be mentioned is that the primary
black holes with non-Abelian fields, which were studied in the late 80s [48,
49, 50, 51], were found to be unstable in the case of asymptotically flat
geometry [52, 53], but later black hole solutions were obtained in the AdS
case, which were shown to be stable [54, 55, 56, 57]. The AMPYM black holes
have been extensively studied in the context of theoretical physics,
particularly in the context of holography and its applications to
understanding strongly coupled field theories. In recent years, the study of
AMPYM black holes has continued to be an active area of research, with a focus
on understanding their thermodynamic properties, phase transitions, and
connections to gauge/gravity duality. These black holes have been studied as
models for strongly coupled systems in the dual field theories, providing
valuable insights into nonperturbative phenomena in quantum field theories.
Researchers have investigated various aspects of AMPYM black holes, such as
their stability, entropy, and critical behavior. Their thermodynamic
properties have been of particular interest, as they exhibit rich phase
structures and can undergo phase transitions similar to those observed in
condensed matter systems. Furthermore, the holographic interpretation of these
black holes has led to fruitful connections between gravitational physics and
field theory, shedding light on fundamental questions in quantum gravity and
strongly coupled systems[58, 59, 60]. This section provides a summary of the
thermodynamics of N-dimensional Einstein-Maxwell-power-Yang-Mills gravity with
a cosmological constant $\Lambda$. This theory of gravity is based on the
following action, which we will explain in more details. So we will have,

$\begin{split}I=\frac{1}{2}\int
dx^{N}\sqrt{-g}\bigg{(}R-\frac{(N-1)(N-2)\Lambda}{3}-F_{\mu\nu}F^{\mu\nu}-(Tr(F_{\mu\nu}^{(a)}F^{(a)\mu\nu}))^{q}\bigg{)}.\end{split}$
(1)

The trace element is represented by $Tr(.)=\Sigma_{a=1}^{(N-1)(N-2)/2}(.)$,
with $R$ as the Ricci scalar and $q$ as a real positive parameter. The
Yang–Mills and Maxwell fields are defined accordingly [4],

$\begin{split}F_{\mu\nu}^{(a)}=\partial_{\mu}A_{\nu}^{(a)}-\partial_{\nu}A_{\mu}^{(a)}+\frac{1}{2\sigma}C^{(a)}_{(b)(c)}A^{b}_{\mu}A^{c}_{\nu},\end{split}$
(2)
$\begin{split}F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},\end{split}$
(3)

$C^{(a)}_{(b)(c)}$ represents the structure constants of the $(N-1)(N-1)/2$
parameter Lie group $G$, while $\sigma$ denotes the coupling
constant.$A^{(a)}_{\mu}$ refers to the $SO(N-1)$ gauge group Yang-Mills
potentials, with $A_{\mu}$ representing the conventional Maxwell potential
[39]. The metric solution corresponding to the $N$-dimensional spherically
symmetric line element is as follows [40],

$\begin{split}ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Omega_{n}^{2}.\end{split}$
(4)

The $d\Omega_{n}^{2}$ denotes the volume of the unit $n$-sphere. In this
study, we will direct our attention to the Einstein-Maxwell-power-Yang-Mills
theory ($EMPYM$) with $N(=n+2)\geq 4$ and $q\neq(n+1)/4$. The solution to the
$N$-dimensional $EMPYM$ black hole with a negative cosmological constant under
the condition of $q\neq\frac{n+1}{4}$ is provided [39],

$\begin{split}&f(r)=1-\frac{2m}{r^{n-1}}-\frac{\Lambda
r^{2}}{3}+\frac{2(n-1)C^{2}}{nr^{2(n-1)}}+\frac{Q}{r^{4q-2}}\\\
&Q=\frac{[n(n-1)Q_{1}^{2}]^{q}}{n(4q-n-1)}.\end{split}$ (5)

It is important to note that the parameter $m$ represents the mass of the
black hole, while $C$ and $Q_{1}$ denote the charges of the Maxwell field and
Yang-Mills field, respectively. In the extended phase space, the cosmological
constant is considered as a thermodynamic pressure $P=-\frac{\Lambda}{8\pi}$,
in this case, the Hawking temperature, mass and entropy of the black hole are
obtained as follows,

$\begin{split}T=-\frac{C^{2}(n-1)^{2}}{2\pi
nr_{+}^{2n-1}}+\frac{2}{3}(n+1)Pr_{+}-\frac{Q(-n+4q-1)}{4\pi
r_{+}^{4q-1}}+\frac{n-1}{4\pi r_{+}},\end{split}$ (6)
$\begin{split}M=\frac{n\omega_{n}}{48\pi}\bigg{(}\frac{6C^{2}(n-1)}{nr_{+}^{n-1}}+8\pi
Pr_{+}^{n+1}+3Qr_{+}^{n-4q+1}+3r_{+}^{n-1}\bigg{)},\end{split}$ (7)
$\begin{split}S=\frac{\omega_{n}r_{+}^{n}}{4},\qquad\omega_{n}=\frac{2\pi^{\frac{n+1}{2}}}{\Gamma\left(\frac{n+1}{2}\right)},\end{split}$
(8)

where $r_{+}$ and $\omega_{n}$ are the horizon radius and the volume of the
unit $n$-sphere respectively.

##  4 Case II: AdS-Kerr-Sen black hole

The Kerr-Sen-AdS black hole is a complex and fascinating topic in the field of
theoretical physics. It is a type of rotating, charged black hole that emerges
from heterotic string theory. It is a generalization of the Kerr-Newman-AdS
black hole, which is a solution of the Einstein-Maxwell equations with a
negative cosmological constant. The Kerr-Sen-AdS black hole also involves a
dilaton and an axion field, which are scalar and pseudoscalar fields that
appear in string theory [61]. The first exact solution of the Einstein field
equations, known as the Schwarzschild solution, was discovered by Karl
Schwarzschild in 1916. This solution describes a simple, non-rotating,
uncharged black hole. The next major advancement came in 1963, when Roy P.
Kerr found a solution to the Einstein field equations that describes a
rotating black hole. This was a significant development, as it is believed
that most black holes in the universe are rotating. In 1965, Ezra Newman and
his collaborators found a solution that describes a rotating, charged black
hole, known as the Kerr-Newman black hole. This solution was later extended by
Ashoke Sen to include a dilaton field and an axion field, resulting in the
Kerr-Sen black hole. The AdS form of this black hole was first derived by
Ashoke Sen in 1992, by applying a series of duality transformations to the
Kerr-Newman-AdS black hole. Sen showed that the Kerr-Sen-AdS black hole
retains some of the symmetries and properties of the Kerr-Newman-AdS black
hole, such as the existence of an event horizon, an ergosphere, and a Penrose
process. However, the Kerr-Sen-AdS black hole also has some unique features,
such as the dependence of the mass and angular momentum on the dilaton charge,
and the violation of the cosmic censorship conjecture for some values of the
parameters [61]. The Kerr-Sen-AdS black hole solution is a specific example of
a rotating black hole with additional fields, which is of particular interest
due to the role of angular momentum and the presence of nontrivial matter
content in the spacetime geometry and is a solution to the equations of motion
of supergravity theories with additional matter fields, such as the Sen-type
dilaton and antisymmetric tensor fields, within the context of AdS space.
The properties and thermodynamics of Kerr Sen-AdS black holes have been the
subject of intense research, given their relevance to understanding the
behavior of rotating black holes in the presence of nontrivial matter content
and their implications for the AdS/CFT correspondence[62, 63, 64, 65, 66, 67,
68, 69, 70, 71, 72]. The study of Kerr Sen-AdS black holes has been motivated
by theoretical developments in string theory, quantum gravity, and holography,
as well as by their potential implications for gravity dualities and the
behavior of strongly coupled systems in the dual field theories. This black
hole has been studied from various perspectives, including its thermodynamic
properties, stability, and connections to the dynamics of dual field theories.
In recent years, there has been growing interest in exploring the dynamical
behavior of Kerr Sen-AdS black holes, including their evolution, instability,
and the connections to chaos and information loss puzzles. Researchers have
investigated the behavior of these black holes under various perturbations and
have sought to understand their implications for fundamental questions about
black hole thermodynamics and the fate of information in quantum gravity.
Moreover, the holographic interpretation of Kerr Sen-AdS black holes has led
to fruitful connections between gravitational physics and field theory,
shedding light on fundamental questions in quantum gravity and strongly
coupled systems[62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72]. In summary, the
history and ongoing research on Kerr Sen-AdS black holes represent a rich and
interdisciplinary area of study that has fruitful connections to diverse
fields of theoretical physics, including string theory, quantum gravity,
holography, and nonlinear dynamics. The exploration of these black holes
continues to be an exciting frontier for probing the fundamental nature of
spacetime and its connections to quantum theory. In this section provides a
short overview of the Kerr-Sen black hole and its generalization to the anti-
de Sitter spacetimes. Sen [61] found a solution of the low-energy effective
action of the heterotic string theory, which describes a charged rotating
black hole, known as the Kerr-Sen black hole.The action is a modification of
the general relativity action with additional fields from the heterotic string
theory, given by[62, 78],

$\begin{split}S=\int
d^{4}x\sqrt{-\widetilde{g}}e^{-\Phi}\left[\mathcal{R}+(\nabla\Phi)^{2}-\frac{1}{8}F^{2}-\frac{1}{12}H^{2}\right],\end{split}$
(9)

where $\tilde{g}$ is the determinant of the metric tensor $g_{\mu\nu}$,
$\mathcal{R}$ is the Ricci scalar, $F=F_{\mu\nu}F^{\mu\nu}$ with $F_{\mu\nu}$
being the $U(1)$ Maxwell field strength tensor, $\Phi$ is a scalar dilaton
field, and $H=H_{\mu\nu\rho}H^{\mu\nu\rho}$ is the field strength for the
axion field. The action can be transformed to the Einstein frame by a
conformal transformation of the metric:

$\begin{split}ds^{2}_{E}=e^{-\Phi}d\widetilde{s}^{2}.\end{split}$ (10)

The action in the Einstein frame is given by:

$\begin{split}S=\int
d^{4}x\sqrt{-g}\left[R-\frac{1}{2}(\nabla\Phi)^{2}-\frac{e^{-\Phi}}{8}F^{2}-\frac{e^{-2\Phi}}{12}H^{2}\right].\end{split}$
(11)

The metric of a Kerr-Sen-AdS black hole in Boyer-Lindquist coordinates is
given by[62, 78]:

$\begin{split}ds^{2}=-\frac{\Delta_{r}}{\rho^{2}}(dt-\frac{a\sin^{2}\theta}{\Xi}d\phi)^{2}+\frac{\rho^{2}}{\Delta_{r}}dr^{2}+\frac{\rho^{2}}{\Delta_{\theta}}d\theta^{2}+\frac{\sin^{2}\theta\Delta_{\theta}}{\rho^{2}}[-\frac{(r^{2}+2br+a^{2})}{\Xi}d\phi+adt]^{2},\end{split}$
(12)

where

$\begin{split}&\rho^{2}=r^{2}+a^{2}\cos^{2}\theta+2br,\\\
&\Delta=(r^{2}+2br+a^{2})(1+\frac{r^{2}+2br}{\ell^{2}})-2mr,\\\
&\Xi=1-\frac{a^{2}}{\ell^{2}},\\\
&\Delta_{\theta}=1-\frac{a^{2}}{\ell^{2}}\cos^{2}\theta.\end{split}$ (13)

The parameter b is the dyonic charge of the black holes and is expressed as
$b=q^{2}/(2m)$, where q is the electric charge and m is the mass of the black
holes. In the limit of $\ell\to\infty$, the metric reduces to the usual Kerr-
Sen black holes. The non-rotating case ($a=0$) reduces to the Gibbons-Maeda-
Garfinkle-Horowitz-Strominger (GMGHS) solution. Gibbons and Maeda [73, 74]
found the black hole and black brane solutions in the dilaton theory, and
Garfinkle-Horowitz-Strominger obtained their charged version [75]. It is worth
mentioning that the Kerr-Sen-AdS black holes in four dimensions have been
explored from various aspects, such as the black hole shadows [76] and the
phase space thermodynamics in the extended phase space[77]. The ADM mass M,
the angular momentum J, and the charge Q in AdS spacetimes are related by,

$\begin{split}&M=\frac{m}{\Xi^{2}},\\\ &J=\frac{ma}{\Xi^{2}},\\\
&Q=\frac{q}{\Xi}.\end{split}$ (14)

Also, the entropy is given by,

$\begin{split}S=\frac{A}{4}=\frac{\pi(r_{+}^{2}+2br+a^{2})}{\Xi},\end{split}$
(15)

where $r_{+}$ is the radius of the event horizon, which is the largest root of
$\Delta=0$. When b=0, the thermodynamic quantities become the same as those of
Kerr-AdS black holes. Kerr-AdS black holes are solutions of the Einstein field
equations in four dimensions with negative cosmological constant and rotation.

##  5 Discussion and results

In this section, we investigate the JTE process for AdS black holes: AMPYM and
AdS Kerr Sen and derive a clear formula for the JTC. We also examine how the
charge and the parameters of AMPYM and Kerr-Sen theories affect the inversion
curves. We also compare the inversion curves for different scenarios

###  5.1 Case I

In this section, we explore the JTE of black hole systems in the extended
phase space, where the black hole mass M is the same as the enthalpy H. We
compare this process with the JT process of van der Waals gases with fixed
particle number, and we use the fixed charge Q for the black hole systems. We
also assume that the other parameters are constant. Using [2, 45] as a
reference, and considering the mass of black holes, we can write the pressure
P as a function of M and $r_{+}$, the mass and the horizon radius of a black
hole. By substituting this expression in the temperature formula, we can
obtain the temperature as a function of M and $r_{+}$ as well. Furthermore, we
can express the mass M and the temperature T in terms of the pressure P and
the radius $r_{+}$ of a black hole. Therefore, we get,

$P(M,r_{+})=-\frac{3\left(\frac{2C^{2}(n-1)r_{+}^{-2n}}{n}+Qr_{+}^{-4q}+\frac{1}{r_{+}^{2}}\right)}{8\pi}+\frac{6Mr_{+}^{-n-1}}{n\omega_{n}}.$
(16)

Also, we have T in terms of M and $r_{+}$ as follows,

$T(M,r_{+})=-\frac{2C^{2}(n-1)r_{+}^{2-2n}+2qQr_{+}^{2-4q}+1}{2\pi
r_{+}}+\frac{4M(n+1)r_{+}^{-n}}{n\omega_{n}}.$ (17)

By solving the above equations, one can obtain the function T(M, P), which is
lengthy and will not be shown here. For particular free parameters for this
model, the T(M, P) curve can be shown. According to the definition of the JTC
$\mu=\left(\frac{\partial T}{\partial P}\right)_{M}$, the inversion pressure
and temperature between the cooling and heating regions are
($\tilde{P}$-$\tilde{T}$), which are determined by $\mu=0$. Therefore, the
most important thing is to find the function expression of $\mu$. By setting
$\mu=0$, one can obtain the inversion points ($\tilde{P}$, $\tilde{T}$) for
different fixed enthalpy M. With respect to[24], the JTC is given by,

$\begin{split}\mu=\big{(}\frac{\partial T}{\partial
P}\big{)}_{M}=\frac{1}{C_{P}}\big{[}T(\frac{\partial V}{\partial
T}\big{)}_{P}-V\big{]}.\end{split}$ (18)

This approach is elegant. However, in this paper, we will use more
straightforward methods by applying only mass and temperature to derive the
JTC $\mu$. we can see that temperature is a function of pressure and radius,
and radius is a function of pressure and mass. So, the JTC is given by,

$\begin{split}\mu=\big{(}\frac{\partial T}{\partial
P}\big{)}_{M}=\frac{\partial_{,r_{+}}T}{\partial_{,r_{+}}P}.\end{split}$ (19)

Now, using the relationship $\mu=\frac{\partial T}{\partial
P}|_{M}=\frac{\frac{\partial T}{\partial r_{+}}}{\frac{\partial P}{\partial
r_{+}}}$, we calculate Joule Thomson coefficient as follows,

$\begin{split}&\mu=\frac{2nr_{+}\left(\omega_{n}\left(r_{+}^{4q}\left(2C^{2}\left(2n^{2}-3n+1\right)r_{+}^{2}+r_{+}^{2n}\right)+8q^{2}Qr^{2n+2}-2qQr_{+}^{2n+2}\right)-8\pi
M(n+1)r_{+}^{n+4q+1}\right)}{3\left(n\omega_{n}\left(r_{+}^{4q}\left(2C^{2}(n-1)r_{+}^{2}+r_{+}^{2n}\right)+2qQr_{+}^{2n+2}\right)-8\pi
M(n+1)r^{n+4q+1}\right)}.\end{split}$ (20)

When the value of the coefficient $\mu$ is positive during the expansion, it
means that the temperature decreases and therefore it is called a cooling
phenomenon. However, when $\mu$ is negative, the temperature increases, and
this is called a heating process.Using various equations, we can write the
mass M and the temperature T as functions of the pressure P and the radius
$r_{+}$, which are the properties of a black hole. For $\mu=0$, we can obtain
the inversion temperature, in which the process of the temperature changes
reverses. It can be obtained by the formula,

$T_{i}=V\big{(}\frac{\partial T}{\partial V}\big{)}_{P},$ (21)
$\begin{split}V=\frac{\partial M}{\partial
p}=\frac{1}{6}nr_{+}^{n+1}\omega_{n}.\end{split}$ (22)

At the inversion temperature, the value of $\mu$ is 0, and the inversion
temperature is determined by the following equation:

$\begin{split}T_{i}=V\frac{\partial T}{\partial V}=V\frac{\frac{\partial
T}{\partial r_{+}}}{\frac{\partial V}{\partial r_{+}}}.\end{split}$ (23)

This is beneficial for identifying the areas of heating and cooling in the
$T-P$ plane. We calculate t using the equations (17), (20), (21), (22) and
(23),

$\begin{split}T_{i}=\frac{r_{+}\left(\frac{6C^{2}(n-1)^{2}(2n-1)r_{+}^{-2n}}{n}+8\pi(n+1)P_{i}+3(1-4q)Q(n-4q+1)r_{+}^{-4q}+\frac{3-3n}{r_{+}^{2}}\right)}{12\pi(n+1)}.\end{split}$
(24)

We can also have from relation (17),

$\begin{split}T_{i}=\frac{-\frac{6C^{2}(n-1)^{2}r_{+}^{1-2n}}{n}+8\pi(n+1)r_{+}P_{i}+3Q(n-4q+1)r_{+}^{1-4q}+\frac{3(n-1)}{r_{+}}}{12\pi}.\end{split}$
(25)

Figure 1 displays the Hawking temperature as a function of the horizon. We
keep the free parameters constant for each plot. In each subfigure, we can
observe some zero points for different free parameters. These zero points
correspond to the divergence points of the JTC, as we can easily see in figure
1. According to the radius of the horizon and the values of free parameters of
a black hole, for small values for the radius of the event horizon, our
structural behavior is completely distinct, and for larger radii, the figures
almost converge.

Figure 1: It shows the plot of $(T-r_{+})$ for the AMPYM black hole with
respect to free parameters that are determined in each plot

We have plotted the isenthalpic curves and the inversion curve of the AMPYM
black hole for various values of the free parameter in each plot of Figure 2.
In every subfigure, two isenthalpic curves with different values of M, along
with the corresponding inversion curve that occurs at the highest point of the
isenthalpic curves. We denote the inversion temperature and pressure of each
isenthalpic curve as $T_{i}$ and $P_{i}$, respectively. The isenthalpic curves
are divided into two regions by the inversion curve: for $P<P_{i}$, the
isenthalpic curve has a positive slope, indicating that the black hole
undergoes cooling during the expansion process. However, for $P>P_{i}$, the
isenthalpic curve has a negative slope, implying that the black hole
experiences heating during the expansion process. This behavior is consistent
for different values of the free parameter in Figures (2a-2b).

Figure 2: Isenthalpic curves and inversion curve of the AMPYM black hole with
respect to mentioned free parameters that determine in each plot.

We do not present the explicit expressions of the solutions here, as they are
too long and complex. However, we can illustrate the relation between the
inversion temperature $T_{i}$ and the inversion pressure $P_{i}$ by using the
equations given above. Figure 3 shows the inversion curves for different free
parameters. The inversion curve has only one branch. The inversion temperature
rises steadily with the inversion pressure, but the slope of the inversion
curves becomes smaller. We can also notice some fine structures in the cases
of subfigures 3. For low pressure, the inversion temperature varies with the
free parameters that are specified in each plot. It was interesting for us to
study the effect of parameters and we observed that the slope of the inversion
curve increases with the changes in the various values of each of the
parameters.

Figure 3: It shows the plot of The inversion curve for the AMPYM black hole
with respect to free parameters that are determined in each plot

Before the end of this section, if we want to talk purely about the effect of
Maxwell charges as a specific result of our work, we should say that in [3],
the authors found that for $q>1$ and $n=2$, the ratio $T_{i}^{min}/T_{c}$ is
fully established, but in the present work, by adding the Maxwell charge to
the mentioned black hole, we found that for $C\gg 1$ the Yang–Mills parameter
has no effect on the ratio, and in any case, the $T_{i}^{min}/T_{c}$ is equal
to $1/2$, which has the same behavior as a four-dimensional charged black
hole. For $q>1$, when $C$ increases, the $T_{min}/T_{c}$ approaches the value
of $1/2$. However, for $q<1$, the conditions are slightly different from the
previous situation, because when $q=1/2$, $n=2$, and $Q_{1}=1$, the
$T_{i}^{min}/T_{c}$ is equal to $1/2$ for all values of ”C”, which is the same
as a charged black hole in four dimensions. Also, according to Table 1 to 3,
it can be inferred that when $1/2<q<1$, with the increase of Maxwell charge,
i.e. $C$, the $T_{i}^{min}/T_{c}$ only approaches the value of $1/2$ and
exhibits a more similar behavior to a four-dimensional charged black hole. If
for $0<q<1/2$, the $T_{i}^{min}/T_{c}$ becomes further or smaller than the
value of $1/2$, which is contrary to a charged black hole in four dimensions.
In general, when $q>1$, for the increase of Maxwell’s charge $C$, the
$T_{i}^{min}/T_{c}$ is almost equal to $1/2$ and when $q=1/2$, for all values
of the Maxwell’s charge, the $T_{i}^{min}/T_{c}$ is equal to $1/2$. Also, when
$1<q<1/2$, it is close to the value of $1/2$ and finally when $1/2<q<0$, the
values of the $T_{i}^{min}/T_{c}$ become smaller than of $1/2$.

###  5.2 Case II

With respect to [62, 78] and the Mass of black holes, We can express pressure
P in terms of M and $r_{+}$ and replace it in the formula for the temperature,
which will also become in terms of M and $r_{+}$. We also rewrite the mass M
and temperature T as a function of the pressure P and the radius $r_{+}$ of a
black hole. So we will have,

$P(M,r_{+})=-\frac{3\left(a^{2}+r_{+}\left(2b-2\Xi^{2}M+r_{+}\right)\right)}{8\pi
r_{+}(2b+r_{+})\left(a^{2}+r_{+}(2b+r_{+})\right)}.$ (26)

Also, the temperature in terms of M and $r_{+}$ is as follows[62, 78],

$\begin{split}&\mathcal{X}=a^{4}(b+r_{+})+a^{2}r_{+}(4b^{2}+6br_{+}+r_{+}(2r_{+}-\Xi^{2}M))+r_{+}^{2}(2b+r_{+})(3r_{+}(b-\Xi^{2}M)+2b(b-\Xi^{2}M)+r_{+}^{2})\\\
&\mathcal{Y}=2\pi
r_{+}(2b+r_{+})(a^{2}+2br_{+}+r_{+})(a^{2}+r_{+}(2b+r_{+}))\\\
&T(M,r_{+})=-\frac{\mathcal{X}}{\mathcal{Y}}.\end{split}$ (27)

Therefore, by using the equation (19) on can obtain,

$\begin{split}&\mathcal{A}=a^{8}(2b^{2}+2br_{+}+r_{+}^{2})+a^{6}r_{+}\big{(}16b^{3}+2b^{2}(9r_{+}+2)+br_{+}(2e^{2}M+8r_{+}+5)+r_{+}^{2}(r_{+}+2)\big{)}+a^{4}r_{+}^{2}\\\
&\bigg{[}48b^{4}+4b^{3}(2\Xi^{2}M+16r_{+}+5)+4b^{2}r_{+}(8\Xi^{2}M+7r_{+}+8)+br_{+}^{2}(20\Xi^{2}M+2r_{+}+19)\\\
&-r_{+}^{2}\big{(}\Xi^{2}(M-4Mr_{+})+(r_{+}-4)r_{+}\big{)}\bigg{]}+a^{2}r_{+}^{3}(2b+r_{+})\bigg{[}32b^{4}+4b^{3}(9r_{+}+4)+2b^{2}r_{+}(8\Xi^{2}M+5r_{+}+11)\\\
&+br_{+}\big{(}4\Xi^{2}M(r_{+}+1)+r_{+}(11-2r_{+})\big{)}-(r_{+}-2)r_{+}^{3}\bigg{]}\\\
&+(2b+1)r_{+}^{4}(2b+r_{+})^{2}\bigg{[}4b^{2}(b-\Xi^{2}M)+r_{+}^{2}(b-3\Xi^{2}M)+4br_{+}(b-\Xi^{2}M)\bigg{]}\\\
&\mathcal{B}=\bigg{[}2\pi
r_{+}^{2}(2b+r_{+})^{2}(a^{2}+r_{+}+2br_{+})^{2}(a^{2}+r_{+}(2b+r_{+}))^{2}\bigg{]}\\\
&\mathcal{C}=3\bigg{[}a^{4}(b+r_{+})+a^{2}r_{+}\big{(}4b^{2}+6br_{+}+r_{+}(2r_{+}-\Xi^{2}M)\big{)}+r_{+}^{2}(2b+r_{+})\big{(}3r_{+}(b-\Xi^{2}M)+2b(b-\Xi^{2}M)+r_{+}^{2}\big{)}\bigg{]}\\\
&\mathcal{D}=4\pi
r_{+}^{2}(2b+r_{+})^{2}\big{(}a^{2}+r_{+}(2b+r_{+})\big{)}^{2}\\\
&\mu=\frac{\mathcal{A}/\mathcal{B}}{\mathcal{C}/\mathcal{D}}.\end{split}$ (28)

We express the mass M and temperature T in terms of the pressure P and the
radius $r_{+}$ of a black hole, using different equations,

$M(P,r_{+})=\frac{\left(a^{2}+r_{+}(2b+r_{+})\right)(8\pi
pr_{+}(2b+r_{+})+3)}{6\Xi^{2}r_{+}},$ (29) $T(P,r_{+})=\frac{a^{2}\left(8\pi
pr_{+}^{2}-3\right)+r_{+}^{2}(8\pi p(2b+r_{+})(2b+3r_{+})+3)}{12\pi
r_{+}\left(a^{2}+2br_{+}+r_{+}\right)}.$ (30)

The $V(P)$ for the AKS black hole is calculates as follows,

$V(P)=\frac{4\pi(2b+r_{+})\left(a^{2}+r_{+}(2b+r_{+})\right)}{3\Xi^{2}}.$ (31)

So, by using the equation (19), (21), (22) and (23) one can obtain,

$\begin{split}&T_{i}=(2b+r_{+})(a^{2}+r_{+}(2b+r_{+}))\\\
&\times\bigg{[}a^{4}(8\pi P_{i}r_{+}^{2}+3)+a^{2}r_{+}\bigg{(}32\pi
b^{2}P_{i}r_{+}+4b(32\pi P_{i}r_{+}^{2}+3)+3(24\pi
P_{i}r_{+}^{3}+r_{+}+2)\bigg{)}\\\
&+16\pi(2b+1)P_{i}r_{+}^{4}(4b+3r_{+})\bigg{]}\\\ &\bigg{/}12\pi
r_{+}^{2}(a^{2}+2br_{+}+r_{+})^{2}\big{(}a^{2}+(2b+r_{+})(2b+3r_{+})\big{)}\end{split}$
(32)

Also, with respect to equation (30) we will have,

$\begin{split}T_{i}=\frac{a^{2}\left(8\pi
r_{+}^{2}P_{i}-3\right)+r_{+}^{2}(8\pi P(2b+r_{+})(2b+3r_{+})+3)}{12\pi
r_{+}\left(a^{2}+2br_{+}+r_{+}\right)}.\end{split}$ (33)

We have drawn the isenthalpic curves and the inversion curve of the AdS Kerr
Sen black hole for various free parameter values in each plot of Figure 4. In
every subfigure, three isenthalpic curves with different M are visible, along
with the corresponding inversion curve that occurs at the highest point of the
isenthalpic curves. We denote each isenthalpic curve’s inversion temperature
and pressure as $T_{i}$ and $P_{i}$, respectively. The isenthalpic curves are
divided into two regions by the inversion curve: for $P<P_{i}$, the
isenthalpic curve has a positive slope, indicating that the black hole
undergoes cooling during the expansion process. With the changes in the
parameters of the black hole, we found that the slope of the figures will
change, and these changes are visible in each subfigure. Because an analytical
solution is difficult to obtain, we used a numerical method to draw graphs. If
we assume the parameter $b$ to be zero, our equations and graphs will simplify
to the Kerr–AdS black holes, whose results are thoroughly discussed in [1].

Figure 4: Isenthalpic curves and inversion curve of the AdS-Ker-Sen black hole
with respect to mentioned free parameters that determine in each plot.

Due to the complexity of the calculations, we do not show the explicit
expressions here. However, we can demonstrate the relation between the
inversion temperature $T_{i}$ and the inversion pressure $P_{i}$ by using the
equations given above. Figure (5a-5d) displays the inversion curves for
different free parameters of the AKS black holes. The inversion curve has only
one branch. The inversion temperature increases gradually with the inversion
pressure, but the slope of the inversion curves becomes smaller. We can also
observe some fine structures in the cases of subfigures 5. Inversion curves
change with the free parameters that are specified in each plot. It was very
difficult to find the analytical solution for drawing the graphs, so we used
numerical solutions to draw them. Moreover, we note that due to the huge
difference in the values of the horizontal and vertical graphs corresponding
to the change of the free parameters, we have drawn the figures separately.

Figure 5: It shows the plot of The inversion curve for the AdS-Ker-Sen black
hole with respect to free parameters that are determined in each plot with
M=20

In Figure 6, we plot the Hawking temperature as a function of the horizon,
which is the boundary of the black hole. The horizon can be affected by the
presence of other fields or dimensions, which are represented by the free
parameters in our model. We consider the free parameters constant for each
plot, but we vary them across different plots to see how they influence the
Hawking temperature. In each subfigure, we can observe some zero points, where
the Hawking temperature becomes zero for different free parameters. This means
that the black hole became extremal at these points. These zero points
correspond to the divergence points of the JTC, which is a quantity that
describes the temperature change when a gas expands or compresses at constant
enthalpy.

Figure 6: It shows the plot of $(T-r_{+})$ for the AdS-Ker-Sen black hole with
respect to free parameters that are determined in each plot

In the study of Kerr black hole[1], it was shown that the rotation parameter
”a” does not have much effect on the Joule-Thomson representation. For
example, it is practically eliminated in the ratio of the $T_{i}^{min}/T_{C}$,
and the value of this ratio is always a constant value of 1/2. In this paper,
we found that the parameters related to AKS black hole, i.e. a and b, can play
a vital role in the representation of the value of the $T_{i}^{min}/T_{C}$.
The value of this ratio increases with the reduction of these parameters. For
this purpose, due to the structural complexities of this black hole, we could
not use the analytical method to obtain the critical points of the black hole,
so we resorted to the approximate method and the numerical calculations. In
the [77], the value of critical temperature and critical pressure has been
calculated. We calculated some values for this black hole according to Table
4. As evident, regarding various values of free parameters, the
$T_{i}^{min}/T_{C}$ is obtained, which shows the obvious effect of parameters
a and b.

##  6 Conclusion

In this article, we investigated the behavior of two structurally different
black holes under the JTE thermodynamic process. The JTE is a process that
cools or heats a system by changing its pressure and volume at constant
enthalpy, and its main goal is to study the behavior of the $\mu$ coefficient,
which describes the change in temperature during the expansion or compression
of a system. But before presenting the results, we must first explain the
motivations that led us to choose these particular black holes. Recently,
Einstein-Power-Young-Mills black hole with AdS structure under JTE process was
investigated in a research paper[2, 3]. This raised the question in our mind
that if a fields in the Maxwellian form are added to the elements in the
action of the above article, can this change and addition of a special form of
the field cause a different thermodynamic behavior in their JT representation?
Similarly, in the past, the behavior of various forms of rotating black holes
was also investigated in this thermodynamic process. But what effect can the
addition of Sen-type dilaton fields and antisymmetric tensor fields have on
its thermodynamic properties and JT representation? This was a question that
could be a good motivation for our investigation. For this purpose, we plotted
the isenthalpic curves and the inversion curves for each of the two black
holes, for different values of the free parameters, which can be referred to
the following results for each. For the AMPYM black hole, the Hawking
temperature has zero points that depend on the free parameters and the horizon
radius, and the JTC diverges at these points. The isenthalpic curves are
divided into two regions by the inversion curve: for $P<P_{i}$, the
isenthalpic curve has a positive slope, indicating that the black hole is
cooling during the expansion process, and for $P>P_{i}$, the isenthalpic curve
has a negative slope, which means that the black hole experiences heating
during the expansion process. The inversion curves for different free
parameters show that the inversion curve has only one branch and the inversion
temperature increases steadily with the inversion pressure, but the slope of
the inversion curves becomes smaller. Also, the graphs showed that for low
pressure, the inversion temperature varies with the free parameters specified
in each graph. Studying the effect of parameters was interesting for us
because we observed that the slope of the inversion curve increases with
changes in different values of each parameter. But if we want to talk about
the effect of Maxwell’s charges as a special result of our work, we must state
that, we have found that the ratio of minimum to critical temperature for a
4-dimensional black hole with Yang-Mills hair depends on the values of $q$,
$M$, and $C$. For $q>1$, the ratio is almost equal to $1/2$ regardless of the
Yang-Mills parameter when $"C\gg 1"$. For $q=1/2$, the ratio is exactly equal
to $1/2$ for all values of $C$. For $1/2<q<1$, the ratio approaches $1/2$ as
$C$ increases. For $0<q<1/2$, the ratio deviates from $1/2$ as $C$ increases.
These results show that the Maxwell charge can affect the thermodynamic
behavior of the black hole and make it more or less similar to a 4-dimensional
charged black hole. Also, for the AKS black hole, we found that its
parameters, i.e. a and b, can play a vital role in the representation of the
value of the ratio of $T_{i}^{min}/T_{C}$, so that by reducing these
parameters, the value of the ratio increases. The results are shown in Figure
7 and are summarized in Tables 1 to 5. For the AdS Kerr Sen black, It was very
difficult to find the analytical solution for drawing the graphs, so we used
numerical solutions to draw them. For $P<P_{i}$, the isenthalpic curve has a
positive slope, indicating that the black hole is cooling during the expansion
process, but with the increases of the black hole parameters, we found that
the slope of the figures will change. Also, if we assume the parameter b (
dyonic charge) to be zero, our equations and graphs will simplify to the Kerr-
AdS black holes, whose results are thoroughly discussed in[1]. In general, it
can be said about this black hole the isenthalpic curves of the AKS black hole
show cooling or heating behavior depending on the inversion curve, which is
affected by the mass and the parameter $b,a$ of the black hole. The inversion
curve has a single branch with a positive slope that decreases with the free
parameters, and the inversion temperature and pressure are related by some
equations that are numerically solved. The Hawking temperature of the AKS
black hole has zero points that depend on the free parameters and the horizon
radius and lead the JTC to reflect divergence behavior.

Figure 7: It shows the plot of $(\frac{T_{i}^{min}}{T_{c}})$ in terms of $C$ with respect to free parameters mentioned in the plot.  $q=0.3,n=2,Q_{1}=1$ |  $T_{i}^{min}$ |  $T_{c}$ |  $T_{i}^{min}/T_{c}$
---|---|---|---
$C=0.1$ |  $0.171532$ |  $0.355378$ |  $0.482674$
$C=0.2$ |  $0.068796$ |  $0.148927$ |  $0.461947$
$C=0.3$ |  $0.035473$ |  $0.082056$ |  $0.432299$
$C=1$ |  Not Exist |  Not Exist |  Not Exist
$C=5$ |  Not Exist |  Not Exist |  Not Exist
$C=10$ |  Not Exist |  Not Exist |  Not Exist
Table 1: Summary of the results for the AMPYM black hole for $q=0.3$ $q=0.5,n=2,Q_{1}=1$ |  $T_{i}^{min}$ |  $T_{c}$ |  $T_{i}^{min}/T_{c}$
---|---|---|---
$C=0.1$ |  $0.034331$ |  $0.068662$ |  $0.5$
$C=0.2$ |  $0.017165$ |  $0.034330$ |  $0.5$
$C=0.3$ |  $0.011443$ |  $0.022886$ |  $0.5$
$C=1$ |  $0.003433$ |  $0.006866$ |  $0.5$
$C=5$ |  $0.000686$ |  $0.001372$ |  $0.5$
$C=10$ |  $0.000343$ |  $0.000686$ |  $0.5$
Table 2: Summary of the results for the AMPYM black hole for $q=0.5$ $q=0.9,n=2,Q_{1}=1$ |  $T_{i}^{min}$ |  $T_{c}$ |  $T_{i}^{min}/T_{c}$
---|---|---|---
$C=0.1$ |  $0.019146$ |  $0.03874$ |  $0.493657$
$C=0.2$ |  $0.018873$ |  $0.038208$ |  $0493962$
$C=0.3$ |  $0.018446$ |  $0.037309$ |  $0.494405$
$C=1$ |  $0.013846$ |  $0.027832$ |  $0497502$
$C=5$ |  $0.004141$ |  $0.008286$ |  $0.499762$
$C=10$ |  $0.002133$ |  $0.004267$ |  $0.499921$
Table 3: Summary of the results for the AMPYM black hole for $q=0.9$ ($a,b$) |  $T_{i}^{min}$ |  $T_{c}$ |  $T_{i}^{min}/T_{c}$
---|---|---|---
($0.0099$, $0.00499$) |  $0.0787687$ |  $0.335030$ |  $0.235109$
($0.00972$, $0.00486$) |  $0.0782007$ |  $0.2302899$ |  $0.33957$
($0.00951$, $0.00475$) |  $0.078821$ |  $0.156982$ |  $0.5$
($0.00937$, $0.00468$) |  $0.0788340$ |  $0.116006$ |  $0.67379$
($0.00892$, $0.00442$) |  $0.776375$ |  $0.0454247$ |  $1.709$
Table 4: Summary of the results for the AKS black hole AMPYM BH |  $q>1$, $C\gg 1$ |  $T_{min}/T_{c}=\frac{1}{2}$ |  This paper
---|---|---|---
AMPYM BH |  $q=0.5$, all values of $"C"$ |  $T_{min}/T_{c}\simeq\frac{1}{2}$ |  This paper
AMPYM BH |  $0.5<q\leq 1$ , $C\gg 1$ |  $T_{min}/T_{c}\simeq\frac{1}{2}$ |  This paper
AKS BH |  ($a=0.00951$, $b=0.00475$) |  $T_{min}/T_{c}\simeq\frac{1}{2}$ |  This paper
Van der Waals fluid |  Exist |  $T_{min}/T_{c}=\frac{3}{4}$ |  [24]
RN-AdS BH |  Exist |  $T_{min}/T_{c}=\frac{1}{2}$ |  [25]
d-dimensional AdS BH |  Exist |  $T_{min}/T_{c}<\frac{1}{2}$ |  [79]
Gauss-Bonnet BH |  Exist |  $T_{min}/T_{c}=0.4765$ |  [80]
torus-like BH |  Not Exist |  $T_{min}/T_{c}$= Not Exist |  [81]
BTZ BH |  Not Exist |  $T_{min}/T_{c}$= Not Exist |  [82]
Table 5: Summary of the results for the AMPYM and AKS black holes compare with
other results

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