# Corrected Thermodynamics and Stability of Magnetic charged AdS Black Holes
surrounded by Quintessence

Dhruba Jyoti Gogoi  0000-0002-4776-8506<EMAIL_ADDRESS>Department of
Physics, Moran College, Moranhat, Charaideo 785670, Assam, India.  Theoretical
Physics Division, Centre for Atmospheric Studies, Dibrugarh University,
Dibrugarh 786004, Assam, India.  Yassine Sekhmani  0000-0001-7448-4579
<EMAIL_ADDRESS>Ratbay Myrzakulov Eurasian International Centre for
Theoretical Physics, Astana 010009, Kazakhstan.  Shyamalee Bora
<EMAIL_ADDRESS>Department of Physics, Dibrugarh University, Dibrugarh
786004, Assam, India.  Javlon Rayimbaev  0000-0001-9293-1838<EMAIL_ADDRESS>Institute of Fundamental and Applied Research, National Research University
TIIAME, Kori Niyoziy 39, Tashkent 100000, Uzbekistan  University of Tashkent
for Applied Sciences, Gavhar Str. 1, Tashkent 100149, Uzbekistan  Shahrisabz
State Pedagogical Institute, Shahrisabz Str. 10, Shahrisabz 181301, Uzbekistan
Tashkent State Technical University, Tashkent 100095, Uzbekistan  Jyatsnasree
Bora  0000-0001-9751-5614<EMAIL_ADDRESS>Department of Physics,
D.H.S.K. College, Dibrugarh, 786001, Assam, India  Ratbay Myrzakulov
0000-0002-5274-0815<EMAIL_ADDRESS>L. N. Gumilyov Eurasian National
University, Astana 010008, Kazakhstan.  Ratbay Myrzakulov Eurasian
International Centre for Theoretical Physics, Astana 010009, Kazakhstan.

###### Abstract

In this study, we explore the corrected thermodynamics of non-linear magnetic
charged anti-de Sitter (AdS) black holes surrounded by quintessence,
incorporating thermal fluctuations and deriving the corrected thermodynamic
potentials. We analyze the effects of corrections due to thermal fluctuations
on various thermodynamic potentials, including enthalpy, Helmholtz free
energy, and Gibbs free energy. Our results show significant impacts on smaller
black holes, with first-order corrections destabilizing them, while second-
order corrections enhance stability with increasing parameter values. The
specific heat analysis further elucidates the stability criteria, indicating
that the large black holes ensure stability against phase transitions.
However, the thermal fluctuations do not affect the physical limitation points
as well as the second-order phase transition points of the black hole. Our
findings highlight the intricate role of thermal fluctuations in black hole
thermodynamics and their influence on stability, providing deeper insights
into the behaviour of black holes under corrected thermodynamic conditions.

Black Hole Thermodynamics; Corrected Entropy; Thermal Fluctuations; Hawking
Temperature.

##  I Introduction

Black hole thermodynamics plays a crucial role in formulating any meaningful
theory of quantum gravity. This study area, where black holes are treated as
thermodynamic objects, was first started by J. D. Bekenstein in 1973. Anything
that goes inside a black hole is lost forever. This means that the entropy
carried by this object is also lost. If black holes have zero entropy, in that
case, the total entropy of the universe will decrease, which is a violation of
the second law of thermodynamics [1]. Therefore, he pointed out that black
holes should have entropy [1]. Later on, in 1974, S. Hawking showed that the
event horizon of a black hole emits thermal radiation, named Hawking radiation
[2]. Hawking also proved that the surface area of a black hole cannot decrease
in any process [3]. This is analogous to the second law of Thermodynamics,
which states that the entropy of a closed thermodynamic system always
increases. Hence, the laws of thermodynamics may apply to black holes, and
thermodynamic concepts like entropy, pressure, specific heat, etc., can be
associated with black holes.

The area law proposed by Bekenstein and Hawking establishes a relation between
entropy ($S$) and area ($A$) of the event horizon and is given by
$S=\frac{A}{4}$ [4, 3]. This means the maximum entropy of black holes is
proportional to the event horizon’s area. However, the size of black holes
decreases due to Hawking radiation. This brings out a need for quantum
corrections to the maximum entropy of small-sized black holes in the area law
due to thermal fluctuations around the equilibrium. This also led to the
modification of the Holographic principle. However, it was discovered that
quantum fluctuations do not affect the modified Hayward black hole’s stability
after studying logarithmic corrections to its thermodynamics [5]. Correction
to the entropy of a black hole results in modification in various
thermodynamical equations of states [6, 7, 8, 9, 10, 11, 12, 13, 14]. The
first-order approximation correction to the black hole entropy is the
logarithmic correction [15]. In the articles [16] and [17], the authors have
analyzed the first-order corrections to the dilatonic black hole and the
Schwarzchild Beltrami-de sitter black hole. The study of higher-order
corrections to the black hole entropy due to statistical fluctuations has
recently gained much attention.

It is well known that black hole thermodynamics is directly affected by the
dark energy component of the universe, “quintessence” being one of them.
Hence, how the thermodynamic quantities related to a black hole are affected
by the presence of quintessence is widely studied. For the charged Reissener-
Nordström black hole, Chamblin et al. have investigated the holography,
thermodynamics, and fluctuations [18]. Hawking and Page’s analysis suggests
that when the temperature hits a critical point, AdS Schwarzschild’s black
hole experiences a phase shift. The $P-V$ criticality of black holes is now
being studied as a consequence [19]. As a result of thermal fluctuations, the
first-order corrections to a dumb hole, the black hole’s counterpart, have
been examined in Ref. [20]. Additionally, Pourhassan et al. [21, 22]
investigated the thermodynamic effects of statistical thermal fluctuations on
a charged dilatonic black Saturn and single spinning Kerr-Ads black hole.
Nevertheless, it was discovered that quantum fluctuations did not affect the
modified Hayward black hole’s stability by analyzing logarithmic thermodynamic
adjustments [5]. The introduction of quantum fluctuations elevated from small
thermal fluctuations has occurred recently. Other noteworthy research
addressing various aspects of black hole characteristics in various gravity
theories can be found in articles [23, 24, 25, 26, 27, 28, 29]. In addition to
these works, there have been several investigations to explore different
thermal, optical and perturbative properties associated with black holes in
different modified theory of gravity frameworks [30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53]. These
works will contribute towards our understanding of black hole systems in
different modified gravity frameworks.

Thermodynamics of magnetically charged AdS black holes in an extended phase
space considering the coupling of nonlinear electrodynamics are studied in
Ref. [54]. In 2018, C. H. Nam derived a non-linear magnetic-charged black hole
surrounded by quintessence [55]. The author has studied its thermodynamic
stability. It reported that the black hole may undergo a thermal phase
transition, at some critical temperature, between a larger unstable black hole
and a smaller stable black hole [55]. Among the various techniques developed
to address the issue of singularity, ordinary black holes are one. The Bardeen
black hole is an example of a black hole that does not involve singularity.
Its event horizon satisfies the weak energy requirement. This was determined
by introducing a tensor representing energy and momentum, understood to be the
gravitational field of a non-linear magnetic monopole charge Q. This has led
to a growing interest in the geometrical and thermodynamic features of these
regular black holes among researchers. Recently, a study on the thermodynamic
features of a non-linear magnetic-charged AdS black hole surrounded by
quintessence in the background of perfect fluid dark matter is reported in
Ref. [56]. The corrected thermodynamics of a non-linear magnetic charged AdS
black hole surrounded by perfect fluid dark matter has been studied by R.
Ndongmo et al. and showed that small-sized black holes are affected by thermal
fluctuations as the second law of thermodynamics is violated by the corrected
entropy which in turn leads to a non-linear evolution and entropy decrease.
They also showed that the larger black holes are not affected by thermal
fluctuations [56].

With these motivations of the above-discussed work, we will study the higher-
order entropy corrections for a non-linear magnetic-charged AdS black hole
surrounded by a quintessence field. Further, we shall analyze other thermal
properties of this black hole system and derive corrected expressions for its
enthalpy, Helmholtz free energy, thermodynamic volume, specific heat, internal
energy, and Gibbs’ free energy. Our study focuses on modifying various
thermodynamic parameters of non-linear magnetic charged AdS black holes due to
thermal fluctuations around the equilibrium point.

Throughout this paper, we stick to the metric signature $(-,+,+,+)$ and
consider the geometrical units (fundamental constants) $c=G=\hbar=k_{B}=1$.
After this brief introduction, the rest of the paper is arranged as follows.
In section II, we briefly overview the geometry of non-linear magnetic-charged
AdS black holes. In section III, we briefly study the Hawking temperature and
basic thermodynamic variables. The effects of thermal fluctuations on entropy
have been studied in IV. The impacts of thermal fluctuations on thermodynamic
potentials have been investigated in section V. In section VI, we study the
stability of non-linear magnetic-charged AdS black holes under the effect of
thermal fluctuations. Finally, we summarize our results in section VII.

##  II Non-linear magnetic-charged AdS black hole surrounded by quintessence

This section will discuss the physics of non-linear magnetic charged AdS black
holes. For this, we will consider the action corresponding to the Einstein
gravity coupled to a non-linear electromagnetic field in the four-dimensional
AdS space-time and surrounded by a quintessence field as, [57, 58, 59, 60, 61,
62, 63]

$\displaystyle\begin{array}[]{r c l}S&=&\int
d^{4}x\sqrt{-g}\left[\frac{c^{4}}{16\pi
G}(R-2\Lambda)-(\mathcal{L}_{\textmd{charge}}-\mathcal{L}_{\textmd{quint}})\right]\end{array}$
(2)

In this action, $R$ represents the scalar curvature, $g$ is the determinant of
the metric tensor $g_{\mu\nu}$, $G$ is the Newton gravity constant, and $c$ is
the light speed. Also, $\Lambda$ is the cosmological constant, and
$\mathcal{L}_{\textmd{charge}}$ is the non-linear electrodynamic term. This
$\mathcal{L}_{\textmd{charge}}$ is a function of the invariant quantity
$F_{\mu\nu}F^{\mu\nu}/4\equiv F$, and $F_{\mu\nu}$ is the Faraday tensor of
electromagnetic field expressed as
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Where $A_{\nu}$ is
the gauge potential of the electromagnetic field. In Eq. (2), the non-linear
electrodynamic term $\mathcal{L}_{\textmd{charge}}$ and the quintessence term
$\mathcal{L}_{\textmd{quint}}$ can be defined respectively as follows [64, 65,
66, 67, 68, 69, 70],

$\mathcal{L}_{\textmd{charge}}=\frac{3M}{|Q|^{3}}\frac{(2Q^{2}F)^{3/2}}{\left[1+(2Q^{2}F)^{3/4}\right]^{2}}\,$
(3) $\mathcal{L}_{\textmd{quint}}=-\frac{1}{2}(\nabla\phi)^{2}-V(\phi).$ (4)

Here, the parameters $M$ and $Q$ are associated with the system’s mass and
magnetic charge. Also, $\phi$ is the quintessential scalar field, and
$V(\phi)$ represents the potential. Here, it can be noted that the magnetic
charge $Q$ is defined as [64]

$\frac{1}{4\pi}\int_{S_{2}^{\infty}}\textbf{{F}}=Q,$ (5)

where $S_{2}^{\infty}$ is a two-sphere at the infinity.

Upon application of the variational principle from Eq. (2), with respect to
the inverse of the metric $g^{\mu\nu}$, one can have,

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu\nu}}$ (6)
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{-g}}\left\\{\frac{c^{4}}{16\pi
G}\int d^{4}x\frac{\delta(\sqrt{-g}(R-2\Lambda))}{\delta
g^{\mu\nu}}-\left(\int
d^{4}x\frac{\delta(\sqrt{-g}\mathcal{L}_{\textmd{charge}})}{\delta
g^{\mu\nu}}-\int
d^{4}x\frac{\delta(\sqrt{-g}\mathcal{L}_{\textmd{quint}})}{\delta
g^{\mu\nu}}\right)\right\\}.$ (7)

This will lead to,

$\begin{array}[]{r c l}G_{\mu\nu}+\Lambda g_{\mu\nu}&=&\frac{8\pi
G}{c^{4}}\left(T_{\mu\nu}(\textmd{charge})-\mathcal{T}_{\mu\nu}\right),\end{array}$
(8)

here the different energy-momentum tensors are defined as [65, 71, 70],

$\displaystyle
T_{\mu\nu}(\textmd{charge})=\frac{2\delta(\sqrt{-g}\mathcal{L}_{\textmd{charge}})}{\delta
g^{\mu\nu}}=-\frac{\partial\mathcal{L}_{\textmd{charge}}}{\partial
F}F_{\mu}^{\beta}F_{\nu\beta}+\mathcal{L}_{\textmd{charge}}g_{\mu\nu},$ (9)
$\displaystyle\mathcal{T}_{\mu\nu}=\frac{2\delta(\sqrt{-g}\mathcal{L}_{\textmd{quint}})}{\delta
g^{\mu\nu}}=\left[\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\left((\nabla\phi)^{2}+2V(\phi)\right)\right].$
(10)

In this work we will consider the unit system where the Newton gravity
constant $G$, light speed $c$ and the number $\pi$ are such that $\frac{4\pi
G}{c^{4}}=1$. With this unit system, we finally arrive at the Einstein-Maxwell
equations of motion as follows:

$\displaystyle\begin{array}[]{r c
l}G^{\nu}_{\mu}+\Lambda\delta^{\nu}_{\mu}&=&2\left(\frac{\partial\mathcal{L}_{\textmd{charge}}}{\partial
F}F_{\mu\rho}F^{\nu\rho}-\delta^{\nu}_{\mu}\mathcal{L}_{\textmd{charge}}-\mathcal{T}^{\nu}_{\mu}\right),\end{array}$
(12)
$\displaystyle\nabla_{\mu}\left(\frac{\partial\mathcal{L}_{\textmd{charge}}}{\partial
F}F^{\nu\mu}\right)$ $\displaystyle=0,$ (13)
$\displaystyle\nabla_{\mu}*F^{\nu\mu}$ $\displaystyle=0.$ (14)

To find a spherically symmetric AdS black hole solution of the mass $M$ and
the magnetic charge $Q$ in the quintessence, the metric has to be written with
ansatz [64, 58, 72]

$\displaystyle\begin{array}[]{r c
l}ds^{2}&=&-e^{\nu}dt^{2}+e^{\lambda}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\\\
&=&-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),\\\ &\textnormal{and}&\ \
f(r)=1-\frac{2m(r)}{r}-\frac{\Lambda}{3}r^{2}.\end{array}$ (18)

One may note here that the term $Q$ (as defined in Eq.(5)) is the integral
constant that should be used to integrate Eqs. (13) and (14) to find the
expression of $F_{\mu\nu}$. Also, the term $M$ will allow us to find the
expression of $m(r)$. Now, taking into account Eqs. (13), (14) and (5), we can
reach the following expression [64],

$F=\frac{Q^{2}}{2r^{4}}.$ (19)

Thus from Eq. (3), the non-linear electrodynamic term can be rewritten as

$\mathcal{L}_{\textmd{charge}}=\frac{3MQ^{3}}{(r^{3}+Q^{3})^{2}}.$ (20)

Now using the time component of Eq. (12)

$\displaystyle\begin{array}[]{r c
l}G^{t}_{t}+\Lambda\delta^{t}_{t}&=&2\left(\frac{\partial\mathcal{L}_{\textmd{charge}}}{\partial
F}F_{t\rho}F^{t\rho}-\delta^{t}_{t}\mathcal{L}_{\textmd{charge}}-\mathcal{T}^{t}_{t}\right).\end{array}$
(22)

Moreover, as the time components of energy-momentum for quintessence are
related to their energy density, so we have [73, 74, 56],

$\displaystyle\mathcal{T}^{t}_{t}$ $\displaystyle=$
$\displaystyle-\frac{3\epsilon c_{q}}{2r^{3(\epsilon+1)}},$ (23)

here $c_{q}$ and $\epsilon$ are the quintessence parameters.

Therefore, from Eq. (22), one can arrive at the following expression

$G_{t}^{t}+\Lambda=-2\mathcal{L}_{\textmd{charge}}+\frac{3\epsilon
c_{q}}{r^{3(\epsilon+1)}}.$ (24)

To solve this Eq. (24), first we will find the component $G_{tt}$ of the
Einstein tensor as

$G_{tt}=e^{\nu}\left[\frac{1}{r^{2}}-e^{-\lambda}\left(\frac{1}{r^{2}}-\frac{\lambda^{\prime}}{r}\right)\right].$
(25)

Developing it, we get

$\displaystyle G_{tt}$ $\displaystyle=$ $\displaystyle
f(r)\left[\frac{1}{r^{2}}-f(r)\left(\frac{1}{r^{2}}-\frac{1}{r}\left(-\frac{f^{\prime}(r)}{f(r)}\right)\right)\right]$
$\displaystyle=$ $\displaystyle
f(r)\left[\frac{1}{r^{2}}-f(r)\left(\frac{1}{r^{2}}-\frac{1}{rf(r)}\left[2\left(\frac{1}{r}\frac{dm(r)}{dr}-\frac{m(r)}{r^{2}}+\frac{\Lambda}{3}r^{2}\right)\right]\right)\right]$
$\displaystyle=$ $\displaystyle
f(r)\left(\frac{2}{r^{2}}\frac{dm(r)}{dr}+\Lambda\right),$

with $\lambda=-\ln f(r)$.

Thus one can arrive at the component $G_{t}^{t}$ of Eq. (25) as

$G_{t}^{t}=g^{tt}G_{tt}=(-f(r))^{-1}f(r)\left(\frac{2}{r^{2}}\frac{dm(r)}{dr}+\Lambda\right),$
(26)

or,

$G_{t}^{t}=-\frac{2}{r^{2}}\frac{dm(r)}{dr}-\Lambda.$ (27)

Now, using this Eq. (27) in Eq. (24), we get

$\frac{dm(r)}{dr}=\frac{3MQ^{3}r^{2}}{(r^{3}+Q^{3})^{2}}-\frac{3\epsilon
c_{q}}{2r^{3\epsilon+1}}.$

Upon integration, we obtain

$m(r)=-\frac{MQ^{3}}{r^{3}+Q^{3}}+\frac{c_{q}}{2r^{3\epsilon}}+C_{0}.$ (28)

here the integral constant $C_{0}$ can be obtained using the boundary
condition [64, 73, 56]

$\displaystyle M$ $\displaystyle=$
$\displaystyle\lim\limits_{r\rightarrow\infty}\left\\{m(r)-\frac{c_{q}}{2r^{3\epsilon}}\right\\}$
$\displaystyle=$
$\displaystyle\lim\limits_{r\rightarrow\infty}\left\\{-\frac{MQ^{3}}{r^{3}+Q^{3}}+C_{0}\right\\},$

therefore

$C_{0}=M.$ (29)

The expression for the mass $m(r)$ and the function $f(r)$ can be given as

$m(r)=\frac{Mr^{3}}{r^{3}+Q^{3}}+\frac{c_{q}}{2r^{3\epsilon}},$ (30)
$f(r)=1-\frac{2Mr^{2}}{r^{3}+Q^{3}}-\frac{c_{q}}{r^{3\epsilon+1}}-\frac{\Lambda}{3}r^{2}.$
(31)

Finally, we reach the following expression for the spherically symmetric
solution for the action (2) as

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (32) $\textnormal{with}\
f(r)=1-\frac{2Mr^{2}}{r^{3}+Q^{3}}-\frac{c_{q}}{r^{3\epsilon+1}}-\frac{\Lambda}{3}r^{2}.$
(33)

For the quintessence field, we have chosen $\epsilon=-2/3$. When we put
quintessence parameter $c_{q}=0$ and the cosmological constant $\Lambda=0$
into Eq. (32), we retrieve the metric of a non-linear magnetic-charged black
hole.

##  III Temperature and Basic Thermodynamical Properties

In this section, the thermodynamical properties of non-linear magnetic charged
black holes will be studied, for which we will proceed through the discussion
about the event horizon. By solving the equation $f(r)=0$ at the event
horizon, we get the horizon radius $r_{+}$, which will give the volume of the
event horizon as:

$V=\frac{4\pi r_{+}^{3}}{3}.$ (34)

Using the given lapse function $f(r)$ i.e., Eq.(33), we can derive the Hawking
temperature $T_{H}$ as follows,

$T_{H}=\frac{f^{\prime}(r)}{4\pi}\Bigg{|}_{r\ =\
r_{+}}=-\frac{-Q^{3}r_{+}c_{q}+2r_{+}^{4}c_{q}+2Q^{3}+\Lambda
r_{+}^{5}-r_{+}^{3}}{4\pi Q^{3}r_{+}+4\pi r_{+}^{4}},$ (35)

where prime “$\prime$” represents the differentiation with respect to $r$, and
$r_{+}$ denotes the horizon radius of a black hole. However, we can observe
that

$\lim_{(\Lambda,c_{q},Q)\to(0,0,0)}T_{H}=\frac{1}{8\pi M}$ (36)

which is similar to the Hawking temperature calculated for the Schwarzschild
black hole (SBH).

Figure 1: Variation of the Hawking temperature with the black hole horizon
radius $r_{+}$.

We have shown the variation of the black hole temperature in Fig. 1. One can
see from the first panel of Fig. 1 that the magnetic charge has a noticeable
impact on the temperature of the black hole. For a small amount of charge $Q$,
the temperature is negative only for very small black holes, and as $r_{+}$
rises, the temperature becomes positive and slowly decreases as $r_{+}$
increases. With an increase in the value of $Q$, $T_{H}$ becomes negative even
for comparatively large black holes. The quintessence parameter $c_{q}$ also
impacts the temperature significantly. In the presence of the quintessence
field, the temperature of the black hole decreases, as shown in the second
panel of Fig. 1. Finally, the impact of the cosmological constant, which
mimics pressure in the black hole system on the temperature, is shown in the
third panel of Fig. 1. It also has a similar impact on $T_{H}$ as demonstrated
by the quintessence field.

The corresponding Bekenstein entropy can be calculated using the four
fundamental laws of Black hole thermodynamics, which leads us to the
uncorrected entropy:

$S_{0}=\frac{\pi\left(r_{+}^{3}-2Q^{3}\right)}{r_{+}}.$ (37)

The enthalpy energy of the system is calculated using the standard formula:

$H=\int T_{H}dS_{0}.$ (38)

This gives the enthalpy energy of the non-linear magnetic charged AdS black
hole system as:

$H=\frac{1}{2}\left(-\frac{Q^{3}c_{q}}{r_{+}}-r_{+}^{2}c_{q}+\frac{Q^{3}}{r_{+}^{2}}-\frac{\Lambda
r_{+}^{3}}{3}+r_{+}\right).$ (39)

The expression for pressure is given as:

$P=-\frac{\Lambda}{8\pi}.$ (40)

The above derived quantities are used to obtain the expressions for the other
system properties, such as internal energy $(U)$, Helmholtz Free energy $F$,
and specific heat $C$ for the system of Non-linear magnetic charged AdS black
holes. We use the familiar expression $U=H-PV$ to find the internal energy and
using the expressions from (40), (39) and (34) the internal energy is:

$U=-\frac{\left(Q^{3}+r_{+}^{3}\right)\left(r_{+}c_{q}-1\right)}{2r_{+}^{2}}.$
(41)

In the following sections, we shall discuss the effect of small stable
fluctuations near equilibrium of the thermodynamical properties of the non-
linear magnetic charged AdS black hole system.

##  IV Thermodynamic fluctuations: the second-order corrections to entropy

In this section, we calculate the effect of thermal fluctuations on the
entropy of the black hole. The seminal paper by Hawking and Page [75] shows
that black holes in asymptotically curved space-time can be described using a
canonical ensemble. Based on this, we consider the magnetically charged AdS
black holes system as a canonical ensemble consisting of $N$ particles with an
energy spectrum $E_{n}$. The statistical partition function of the system can
be written as:

$Z=\int_{0}^{\infty}dE\rho(E)e^{-\bar{\beta}_{\kappa}E},$ (42)

where $\bar{\beta}_{\kappa}$ is the inverse temperature in units of the
Boltzmann constant, and $\rho(E)$ is the canonical density of the system with
average energy $E$. Using the partition function $Z$ and Laplace inversion,
the density of states can be calculated as:

$\rho(E)=\frac{1}{2\pi
i}\int^{\bar{\beta}_{0\kappa}+i\infty}_{\bar{\beta}_{0\kappa}-i\infty}d\bar{\beta}_{\kappa}e^{S(\bar{\beta}_{\kappa})},$
(43)

where

$S=\bar{\beta}_{\kappa}E+\log Z.$ (44)

The entropy around the equilibrium temperature $\bar{\beta}_{0\kappa}$ is
obtained by eliminating all thermal fluctuations. However, in the presence of
thermal fluctuations, the corrected entropy can be expressed by a Taylor
expansion around $\bar{\beta}_{0\kappa}$ as:

$S=S_{0}+\frac{1}{2}(\bar{\beta}_{\kappa}-\bar{\beta}_{0\kappa})^{2}\left(\frac{\partial^{2}S(\bar{\beta}_{\kappa})}{\partial\bar{\beta}_{\kappa}^{2}}\right)_{\bar{\beta}_{\kappa}=\bar{\beta}_{0\kappa}}+\frac{1}{6}(\bar{\beta}_{\kappa}-\bar{\beta}_{0\kappa})^{3}\left(\frac{\partial^{3}S(\bar{\beta}_{\kappa})}{\partial\bar{\beta}_{\kappa}^{3}}\right)_{\bar{\beta}_{\kappa}=\bar{\beta}_{0\kappa}}+\cdots,$
(45)

where the dots denote higher-order corrections.

We should note that the first derivative of the entropy for
$\bar{\beta}_{\kappa}$ vanishes at the equilibrium temperature. Consequently,
the density of states can be expressed as:

$\rho(E)=\frac{e^{S_{0}}}{2\pi
i}\int^{\bar{\beta}_{0\kappa}+i\infty}_{\bar{\beta}_{0\kappa}-i\infty}d\bar{\beta}_{\kappa}\,\exp\left(\frac{(\bar{\beta}_{\kappa}-\bar{\beta}_{0\kappa})^{2}}{2}\left(\frac{\partial^{2}S(\bar{\beta}_{\kappa})}{\partial\bar{\beta}_{\kappa}^{2}}\right)_{\bar{\beta}_{\kappa}=\bar{\beta}_{0\kappa}}+\frac{(\bar{\beta}_{\kappa}-\bar{\beta}_{0\kappa})^{3}}{6}\left(\frac{\partial^{3}S(\bar{\beta}_{\kappa})}{\partial\bar{\beta}_{\kappa}^{3}}\right)_{\bar{\beta}_{\kappa}=\bar{\beta}_{0\kappa}}+\cdots\right).$
(46)

Following the approach in Ref. [76], we obtain:

$S=S_{0}-\frac{1}{2}\log{S_{0}T_{h}^{2}}+\frac{f(m,n)}{S_{0}}+\cdots,$ (47)

where $f(m,n)$ is considered a constant. A more general expression for the
corrected entropy is given by [77]:

$S=S_{0}-\frac{\beta_{1}}{2}\log(S_{0}T_{H}^{2})+\frac{\beta_{2}}{S_{0}}+\cdots,$
(48)

where the parameters $\beta_{1}$ and $\beta_{2}$ are introduced to track the
first-order and second-order corrected terms. When $\beta_{1}\rightarrow 0$
and $\beta_{2}\rightarrow 0$, the original results are recovered, and
$\beta_{1}=1$ and $\beta_{2}=0$ yield the usual corrections [76, 78].
Therefore, the first-order correction is logarithmic, while the second-order
correction is proportional to the inverse of the original entropy $S_{0}$.
These corrections can be considered quantum corrections to the black hole. For
large black holes, these corrections can be neglected. However, as the black
hole decreases in size due to Hawking radiation, the quantum fluctuations in
the black hole’s geometry increase. Thus, thermal fluctuations significantly
modify the thermodynamics of black holes [79], becoming more important as the
black holes reduce in size.

Figure 2: Variation of entropy with the black hole horizon radius.

Up to the second-order correction, the explicit form of the entropy for this
black hole is given by

$S_{c}=\frac{\pi\left(r_{+}^{3}-2Q^{3}\right)}{r_{+}}-\beta_{1}\log\left(\frac{\pi\left(r_{+}^{3}-2Q^{3}\right)\left(-Q^{3}r_{+}c_{q}+2r_{+}^{4}c_{q}+2Q^{3}+\Lambda
r_{+}^{5}-r_{+}^{3}\right){}^{2}}{r_{+}\left(4\pi Q^{3}r_{+}+4\pi
r_{+}^{4}\right){}^{2}}\right)+\frac{\beta_{2}r_{+}}{\pi\left(r_{+}^{3}-2Q^{3}\right)}.$
(49)

The variation of the corrected entropy of the black hole is shown in Fig. 2.
Here, we have demonstrated the variations concerning different parameters of
this study. With a comparatively small magnetic charge Q and zero quintessence
parameter $c_{q}$, the corrected entropy is a smooth increasing function of
$r_{+}$. It is shown in the first panel of the figure. However, we observe
that the nature of increasing entropy for large charge Q undergoes a sudden
rise in entropy. These peaks shift towards a large horizontal radius for large
Q values. In the second panel, considering $Q=0.3$, we investigate the entropy
variation for a set of quintessence parameter $c_{q}$. These variations in
entropy also suggest regular increasing values with some significant peaks.
However, for $c_{q}=0.1$, no such significant peaks are observed. In the last
two panels of this figure, corrected and uncorrected entropy variations are
shown for the first-order and second-order corrected terms $\beta_{1}$ and
$\beta_{2}$, taking the rest parameters to have fixed values. All these
variations only take positive values for all the cases considered.

In the next section, we will analyze the thermodynamic quantities of the black
hole with the corrected entropy.

##  V Impact of Thermal fluctuations: the Second order corrected thermodynamic
potentials

This section aims to evaluate various thermodynamic variables for the
magnetically charged AdS black hole in the presence of thermodynamic
fluctuations. As mentioned earlier, we shall consider the second-order entropy
corrections here. Using the second-order corrected entropy, it is possible to
derive corrected enthalpy energy $H_{c}$ as given by

$H_{c}=\int T_{H}dS_{c}+\int V_{c}dP,$ (50)

where $V_{c}$ is the corrected volume. However, as the second term on the RHS
of the above equation does not contribute to a constant value of $\lambda$,
using the expressions of $T_{H}$ and $S_{c}$ in the above equation, we can
have Corrected enthalpy

$H_{c}=-\frac{1}{144\pi^{2}}\left[\frac{72\pi\beta_{1}\left(3Q^{3}c_{q}+r_{+}\left(\Lambda
Q^{3}+3r_{+}\right)\right)}{Q^{3}+r_{+}^{3}}+\frac{12\beta_{2}r_{+}\left(3c_{q}+2\Lambda
r_{+}\right)}{r_{+}^{3}-2Q^{3}}+\frac{72\pi\left(\pi
Q^{3}c_{q}-3\beta_{1}\right)}{r_{+}}\right.\\\
\left.-\frac{12\log\left(r_{+}^{3}-2Q^{3}\right)\left(3\pi\beta_{1}Q^{3}c_{q}-\beta_{2}\right)}{Q^{3}}-\frac{36\log\left(r_{+}\right)\left(\beta_{2}+\pi\beta_{1}Q^{3}c_{q}\right)}{Q^{3}}\right.\\\
\left.+\frac{2^{2/3}\log\left(2Q^{2}+2^{2/3}Qr_{+}+\sqrt[3]{2}r_{+}^{2}\right)\left(3\
2^{2/3}\beta_{2}c_{q}+6\ 2^{2/3}\pi\beta_{1}\Lambda Q^{3}+4\beta_{2}\Lambda
Q\right)}{Q^{2}}\right.\\\ \left.-\frac{2\
2^{2/3}\log\left(2Q-2^{2/3}r_{+}\right)\left(3\ 2^{2/3}\beta_{2}c_{q}+6\
2^{2/3}\pi\beta_{1}\Lambda Q^{3}+4\beta_{2}\Lambda Q\right)}{Q^{2}}\right.\\\
\left.+\frac{2\
2^{2/3}\sqrt{3}\tan^{-1}\left(\frac{Q+2^{2/3}r_{+}}{\sqrt{3}Q}\right)\left(3\
2^{2/3}\beta_{2}c_{q}+6\ 2^{2/3}\pi\beta_{1}\Lambda Q^{3}-4\beta_{2}\Lambda
Q\right)}{Q^{2}}\right.\\\
\left.+72\pi^{2}r_{+}^{2}c_{q}-\frac{72\pi^{2}Q^{3}}{r_{+}^{2}}-72\pi
r_{+}\left(2\beta_{1}\Lambda+\pi\right)+24\pi^{2}\Lambda r_{+}^{3}\right].$
(51)

Figure 3: Variation of enthalpy with the black hole horizon radius.

Graphically, the corrected enthalpy $H_{c}$ is shown in Fig. 3 for different
values of the model parameters. An increase in the charge parameter $Q$
increases the enthalpy of the black hole. The variation is significant for
smaller black holes, while the variation is less significant for larger ones.
In the case of the parameter $c_{q}$, we observe a decrease in the enthalpy as
$c_{q}$ increases. The variations are negligible for smaller black holes and
significant for larger ones. It shows that the quintessence parameter $c_{q}$
signatures are prominent only for large black holes.

One can see that the behavior of enthalpy is different in the presence of
thermal fluctuations. For smaller black holes, thermal fluctuation corrections
of both first and second order significantly increase the enthalpy. In
comparison, the variations are not very significant for larger black holes.
Interestingly, for larger black holes, the presence of thermal fluctuations
results in a slight decrease in the enthalpy value. A discontinuity in the
graphs is present due to the thermal fluctuations.

Using the corrected enthalpy, one can find an expression for the corrected
volume given by

$V_{c}=\frac{dH_{c}}{dP}\Bigg{|}_{S_{c}=constant}$ (52)

Using the expressions for $H_{c}$ in the above relation, the explicit form of
corrected volume is found to be

$V_{c}=\frac{1}{18\pi}\left[\frac{72\pi\beta_{1}Q^{3}r_{+}}{Q^{3}+r_{+}^{3}}+\frac{24\beta_{2}r_{+}^{2}}{r_{+}^{3}-2Q^{3}}-\frac{4\
2^{2/3}\left(2\beta_{2}+3\
2^{2/3}\pi\beta_{1}Q^{2}\right)\log\left(2Q-2^{2/3}r_{+}\right)}{Q}\right.\\\
\left.+\frac{2\ 2^{2/3}\left(2\beta_{2}+3\
2^{2/3}\pi\beta_{1}Q^{2}\right)\log\left(2Q^{2}+2^{2/3}Qr_{+}+\sqrt[3]{2}r_{+}^{2}\right)}{Q}\right.\\\
\left.+\frac{4\ 2^{2/3}\sqrt{3}\left(3\
2^{2/3}\pi\beta_{1}Q^{2}-2\beta_{2}\right)\tan^{-1}\left(\frac{Q+2^{2/3}r_{+}}{\sqrt{3}Q}\right)}{Q}-144\pi\beta_{1}r_{+}+24\pi^{2}r_{+}^{3}\right].$
(53)

The corrected Helmholtz free energy can be calculated by using the standard
relation

$F_{c}=-\int S_{c}dT_{H}-\int PdV_{c}.$ (54)

Inserting the corresponding corrected parameters,

$F_{c}=\frac{1}{72\pi^{2}}\left[-\frac{36\pi^{2}Q^{3}}{r_{+}^{2}}+18\pi^{2}\Lambda
r_{+}^{3}+18\left(7\pi
c_{q}\beta_{1}+\frac{\beta_{2}}{Q^{3}}\right)\log\left(r_{+}\right)-72\pi\Lambda
r_{+}\beta_{1}+72\pi\log\left(Q^{3}+r_{+}^{3}\right)c_{q}\beta_{1}\right.\\\
\left.-72\pi\log\left(\left(2-r_{+}c_{q}\right)Q^{3}+r_{+}^{3}\left(\Lambda
r_{+}^{2}+2c_{q}r_{+}-1\right)\right)c_{q}\beta_{1}\right.\\\
\left.+\frac{18\pi\log\left(\frac{\left(r_{+}^{3}-2Q^{3}\right)\left(\left(2-r_{+}c_{q}\right)Q^{3}+r_{+}^{3}\left(\Lambda
r_{+}^{2}+2c_{q}r_{+}-1\right)\right){}^{2}}{16\pi
r_{+}^{3}\left(Q^{3}+r_{+}^{3}\right){}^{2}}\right)\left(-\Lambda
r_{+}^{5}+r_{+}^{3}+Q^{3}\left(3r_{+}c_{q}-2\right)\right)\beta_{1}}{r_{+}\left(Q^{3}+r_{+}^{3}\right)}\right.\\\
\left.+\frac{108\pi\beta_{1}}{r_{+}}+18\pi
r_{+}\left(4\Lambda\beta_{1}+\pi\right)+\frac{12\Lambda
r_{+}\left(3\pi\left(r_{+}^{3}-2Q^{3}\right)\beta_{1}Q^{3}+r_{+}\left(Q^{3}+r_{+}^{3}\right)\beta_{2}\right)}{-2Q^{6}-r_{+}^{3}Q^{3}+r_{+}^{6}}\right.\\\
\left.+\frac{2^{2/3}\log\left(2Q-2^{2/3}r_{+}\right)\left(6\
2^{2/3}\pi\Lambda\beta_{1}Q^{3}+4\Lambda\beta_{2}Q+3\
2^{2/3}c_{q}\beta_{2}\right)}{Q^{2}}\right.\\\
\left.+\frac{6\left(9\pi^{2}\left(\Lambda
Q^{3}+3r_{+}-3r_{+}^{2}c_{q}\right)Q^{3}-6\pi\left(3c_{q}Q^{3}+r_{+}\left(\Lambda
Q^{3}+3r_{+}\right)\right)\beta_{1}+\left(\Lambda
r_{+}^{2}+3c_{q}r_{+}-3\right)\beta_{2}\right)}{Q^{3}+r_{+}^{3}}\right.\\\
\left.-\frac{2^{2/3}\sqrt{3}\tan^{-1}\left(\frac{Q+2^{2/3}r_{+}}{\sqrt{3}Q}\right)\left(6\
2^{2/3}\pi\Lambda\beta_{1}Q^{3}-4\Lambda\beta_{2}Q+3\
2^{2/3}c_{q}\beta_{2}\right)}{Q^{2}}\right.\\\
\left.-\frac{\log\left(2Q^{2}+2^{2/3}r_{+}Q+\sqrt[3]{2}r_{+}^{2}\right)\left(6\
2^{2/3}\pi\Lambda\beta_{1}Q^{3}+4\Lambda\beta_{2}Q+3\
2^{2/3}c_{q}\beta_{2}\right)}{\sqrt[3]{2}Q^{2}}\right.\\\
\left.-\frac{6\log\left(r_{+}^{3}-2Q^{3}\right)\left(3\pi
c_{q}\beta_{1}Q^{3}+\beta_{2}\right)}{Q^{3}}-\frac{2\
2^{2/3}\Lambda\log\left(2Q-2^{2/3}r_{+}\right)\left(3\
2^{2/3}\pi\beta_{1}Q^{2}+2\beta_{2}\right)}{Q}\right.\\\
\left.+\frac{2^{2/3}\Lambda\log\left(2Q^{2}+2^{2/3}r_{+}Q+\sqrt[3]{2}r_{+}^{2}\right)\left(3\
2^{2/3}\pi\beta_{1}Q^{2}+2\beta_{2}\right)}{Q}\right.\\\ \left.+\frac{2\
2^{2/3}\Lambda\tan^{-1}\left(\frac{Q+2^{2/3}r_{+}}{\sqrt{3}Q}\right)\sqrt{3}\left(3\
2^{2/3}\pi Q^{2}\beta_{1}-2\beta_{2}\right)}{Q}\right],$ (55)

which is thermal fluctuation dependent. Without thermal fluctuation
correction, it is given by

$F=\frac{1}{4}\left(\frac{3Q^{3}\left(-3r_{+}^{2}c_{q}+\Lambda
Q^{3}+3r_{+}\right)}{Q^{3}+r_{+}^{3}}-\frac{2Q^{3}}{r_{+}^{2}}+\frac{\Lambda
r_{+}^{3}}{3}+r_{+}\right)+\frac{\Lambda r_{+}^{3}}{6}.$ (56)

Figure 4: Variation of Helmholtz free energy with the black hole horizon
radius.

Figure 5: Variation of Gibbs free energy with the black hole horizon radius.

We have shown the corrected Helmholtz free energy variation in Fig. 4. The
figures show that both the correction parameters have noticeable impacts on
the Helmholtz free energy. We observe that an increase in the charge $Q$
shifts the peak of the Helmholtz free energy curve towards larger black holes,
and the area under the curve increases gradually. However, an increase in the
quintessence parameter $c_{q}$ decreases the Helmholtz free energy. For the
first-order correction parameter $\beta_{1}$, an increase in its value
initially increases the Helmholtz free energy for small-sized black holes, but
after crossing a threshold event horizon radius, further increases in
$\beta_{1}$ decrease the Helmholtz free energy. Similarly, for the correction
parameter $\beta_{2}$, an increase in its value decreases the Helmholtz free
energy gradually for small-sized black holes. However, for small values of
$\beta_{2}$, the Helmholtz free energy remains greater than that of the
corresponding uncorrected black hole. A lower Helmholtz free energy indicates
that a black hole is in a more stable thermodynamic state, making states with
the lowest Helmholtz free energy the most stable. A decrease in Helmholtz’s
free energy implies that the system (black hole plus surroundings) is moving
towards a more stable equilibrium state. Furthermore, the Helmholtz free
energy represents the maximum amount of work that can be extracted from a
black hole system at constant temperature and volume, indicating that any
process resulting in a decrease in Helmholtz free energy can, in principle, be
harnessed to perform useful work. States with lower Helmholtz free energy are
exponentially more probable, and a black hole with a higher Helmholtz free
energy is more likely to radiate energy to move toward a more stable state.
Therefore, a black hole state with lower Helmholtz free energy achieves a
balance where the energy and entropy terms are optimized for stability at a
given temperature.

The corrected internal energy ($U_{c}$) can be obtained from the following
formula:

$U_{c}=H_{c}-PV_{c}.$ (57)

The explicit form is given by

$U_{c}=\frac{1}{24}\left(-\frac{36\beta_{1}Q^{3}c_{q}}{\pi\left(Q^{3}+r_{+}^{3}\right)}-\frac{6\beta_{2}r_{+}c_{q}}{\pi^{2}\left(r_{+}^{3}-2Q^{3}\right)}+\frac{6\beta_{1}c_{q}\log\left(r_{+}^{3}-2Q^{3}\right)}{\pi}+\frac{6\log\left(r_{+}\right)\left(\beta_{2}+\pi\beta_{1}Q^{3}c_{q}\right)}{\pi^{2}Q^{3}}\right.\\\
\left.-\frac{12Q^{3}c_{q}}{r_{+}}+\frac{2\sqrt[3]{2}\beta_{2}c_{q}\log\left(2Q-2^{2/3}r_{+}\right)}{\pi^{2}Q^{2}}-\frac{\sqrt[3]{2}\beta_{2}c_{q}\log\left(2Q^{2}+2^{2/3}Qr_{+}+\sqrt[3]{2}r_{+}^{2}\right)}{\pi^{2}Q^{2}}\right.\\\
\left.-\frac{2\sqrt[3]{2}\sqrt{3}\beta_{2}c_{q}\tan^{-1}\left(\frac{Q+2^{2/3}r_{+}}{\sqrt{3}Q}\right)}{\pi^{2}Q^{2}}-12r_{+}^{2}c_{q}-\frac{36\beta_{1}r_{+}^{2}}{\pi\left(Q^{3}+r_{+}^{3}\right)}-\frac{2\beta_{2}\log\left(r_{+}^{3}-2Q^{3}\right)}{\pi^{2}Q^{3}}+\frac{12Q^{3}}{r_{+}^{2}}+\frac{36\beta_{1}}{\pi
r_{+}}+12r_{+}\right).$ (58)

We shall now examine the effects of thermal fluctuations on Gibbs free energy.
In thermodynamics, Gibbs free energy quantifies the maximum amount of
mechanical work obtainable from a system. The following relation
mathematically defines it:

$G_{c}=F_{c}+PV_{c}.$ (59)

Using the corrected values of Helmholtz free energy and volume, we can get

$G_{c}=\frac{1}{144\pi^{2}}\Big{[}-144\pi\beta_{1}c_{q}\log\left(Q^{3}\left(2-r_{+}c_{q}\right)+r_{+}^{3}\left(2r_{+}c_{q}+\Lambda
r_{+}^{2}-1\right)\right)-\frac{72\pi\beta_{1}\Lambda
Q^{3}r_{+}}{Q^{3}+r_{+}^{3}}-\frac{24\beta_{2}\Lambda
r_{+}^{2}}{r_{+}^{3}-2Q^{3}}-\frac{72\pi^{2}Q^{3}}{r_{+}^{2}}\\\
+\frac{36\pi\beta_{1}\left(Q^{3}\left(3r_{+}c_{q}-2\right)-\Lambda
r_{+}^{5}+r_{+}^{3}\right)\log\left(\frac{\left(r_{+}^{3}-2Q^{3}\right)\left(Q^{3}\left(2-r_{+}c_{q}\right)+r_{+}^{3}\left(2r_{+}c_{q}+\Lambda
r_{+}^{2}-1\right)\right){}^{2}}{r_{+}^{3}\left(Q^{3}+r_{+}^{3}\right){}^{2}}\right)}{r_{+}\left(Q^{3}+r_{+}^{3}\right)}\\\
+\frac{12\left(-3\pi\beta_{1}(2+\log(16\pi))\left(3Q^{3}c_{q}+r_{+}\left(\Lambda
Q^{3}+3r_{+}\right)\right)+9\pi^{2}Q^{3}\left(-3r_{+}^{2}c_{q}+\Lambda
Q^{3}+3r_{+}\right)+\beta_{2}\left(3r_{+}c_{q}+\Lambda
r_{+}^{2}-3\right)\right)}{Q^{3}+r_{+}^{3}}\\\
+36\log\left(r_{+}\right)\left(7\pi\beta_{1}c_{q}+\frac{\beta_{2}}{Q^{3}}\right)+144\pi\beta_{1}c_{q}\log\left(Q^{3}+r_{+}^{3}\right)-\frac{12\log\left(r_{+}^{3}-2Q^{3}\right)\left(\beta_{2}+3\pi\beta_{1}Q^{3}c_{q}\right)}{Q^{3}}\\\
+\frac{2\ 2^{2/3}\log\left(2Q-2^{2/3}r_{+}\right)\left(3\
2^{2/3}\beta_{2}c_{q}+6\ 2^{2/3}\pi\beta_{1}\Lambda Q^{3}+4\beta_{2}\Lambda
Q\right)}{Q^{2}}+\frac{72\pi\beta_{1}(3+\log(16\pi))}{r_{+}}+12\pi^{2}\Lambda
r_{+}^{3}\\\ -\frac{2\
2^{2/3}\sqrt{3}\tan^{-1}\left(\frac{Q+2^{2/3}r_{+}}{\sqrt{3}Q}\right)\left(3\
2^{2/3}\beta_{2}c_{q}+6\ 2^{2/3}\pi\beta_{1}\Lambda Q^{3}-4\beta_{2}\Lambda
Q\right)}{Q^{2}}+\frac{24\Lambda
r_{+}\left(3\pi\beta_{1}Q^{3}\left(r_{+}^{3}-2Q^{3}\right)+\beta_{2}r_{+}\left(Q^{3}+r_{+}^{3}\right)\right)}{-2Q^{6}-Q^{3}r_{+}^{3}+r_{+}^{6}}\\\
+36\pi
r_{+}\left(\beta_{1}\Lambda(4+\log(16\pi))+\pi\right)-\frac{2^{2/3}\log\left(2Q^{2}+2^{2/3}Qr_{+}+\sqrt[3]{2}r_{+}^{2}\right)\left(3\
2^{2/3}\beta_{2}c_{q}+6\ 2^{2/3}\pi\beta_{1}\Lambda Q^{3}+4\beta_{2}\Lambda
Q\right)}{Q^{2}}\Big{]}.$ (60)

This shows that the thermal fluctuations also affect the Gibbs free energy. To
visualize the impacts of the first-order and second-order correction
parameters, we have plotted the corrected Gibbs free energy in Fig 5. In the
first two panels of Fig. 5, we showed the impacts of charge $Q$ and
quintessence parameter $c_{q}$ on the Gibbs free energy of the black hole
system. One can see that the corrected Gibbs free energy $G_{c}$ is negative
for smaller black holes. An increase in the value of $Q$ increases the
possibility of a negative $G_{c}$ for the small-sized black holes. After a
certain radius of the black hole, $G_{c}$ becomes positive. As shown in the
figure, the peak shifts towards larger black holes as $Q$ increases gradually.
However, towards larger values of $r_{+}$, $Q$ does not have a distinguishable
impact. On the other hand, with an increase in the value of the quintessence
parameter $c_{q}$, the peak of the curve does not shift horizontally but
instead of this, it decreases gradually. In this case, the impact of $c_{q}$
is not properly distinguishable for very large and very small black holes as
depicted in the figure.

The impacts of the correction parameters $\beta_{1}$ and $\beta_{2}$ are shown
in the last two panels of Fig. 5. One can see that both parameters have
noticeable impacts on the behavior of $G_{c}$. Initially, with an increase in
the correction parameter $\beta_{1}$ ($\beta_{2}$), $G_{c}$ increases
(decreases) significantly. In this scenario, Gibb’s free energy is greater
than the corresponding value of the uncorrected black hole. After crossing a
threshold value of the event horizon radius, the pattern becomes opposite
i.e., the uncorrected black hole attains greater Gibbs free energy. This
observation suggests that the thermal fluctuations decrease the stability of
the small-sized black holes and increase the stability of larger black holes.
The negative value of Gibbs free energy for very small-sized black holes
indicates that the black hole is thermodynamically stable. This means that
small perturbations or fluctuations will not lead to the decay or dramatic
changes in the black hole’s state. It also represents maximum energy that can
be extracted from the thermodynamical system of the black hole. A negative
Gibbs free energy suggests that the entropy increase (disorder) associated
with processes involving the black hole is favored, which aligns with the
second law of thermodynamics. This is in agreement with the outcomes depicted
by the black hole’s entropy variations.

##  VI  Stability in presence of Thermal fluctuations

Figure 6: Variation of specific heat with the black hole horizon radius.

To study the stability, we investigate the nature of its specific heat. We can
explore whether this black hole undergoes a phase transition from the specific
heat behavior. The positive value of specific heat ensures that the given
black hole system is stable against the phase transition, however, the
negative value of specific heat concludes the system’s instability. By
considering thermal fluctuation, we estimate the expression for specific heat,
which must subside to the uncorrected specific heat (when fluctuation is
switched off, i.e. $\beta=0$). From a classical thermodynamics point of view,
the specific heat ($C_{c}$) can be calculated with the help of the following
standard formula:

$C_{c}=T_{H}\frac{dS_{c}}{dT_{H}}.$ (61)

Or, in terms of parameter space, the specific heat can be identified in the
form of

$C_{c}=-\frac{1}{\pi
r_{+}\left(r_{+}^{3}-2Q^{3}\right){}^{2}\left(Q^{3}r_{+}^{3}\left(9r_{+}c_{q}+4\Lambda
r_{+}^{2}-10\right)-2Q^{6}+\Lambda
r_{+}^{8}+r_{+}^{6}\right)}\Big{[}2\Big{\\{}\beta_{2}r_{+}^{2}\left(Q^{3}+r_{+}^{3}\right){}^{2}\Big{(}Q^{3}\left(2-r_{+}c_{q}\right)\\\
+r_{+}^{3}\left(2r_{+}c_{q}+\Lambda
r_{+}^{2}-1\right)\Big{)}+\pi^{2}\left(2Q^{6}+Q^{3}r_{+}^{3}-r_{+}^{6}\right){}^{2}\left(Q^{3}\left(r_{+}c_{q}-2\right)-r_{+}^{3}\left(2r_{+}c_{q}+\Lambda
r_{+}^{2}-1\right)\right)\\\
+\pi\beta_{1}r_{+}\left(r_{+}^{3}-2Q^{3}\right)\left(Q^{9}\left(6-r_{+}c_{q}\right)+Q^{6}r_{+}^{3}\left(-18r_{+}c_{q}-7\Lambda
r_{+}^{2}+21\right)+4Q^{3}r_{+}^{6}\left(3r_{+}c_{q}+\Lambda
r_{+}^{2}-3\right)+2r_{+}^{10}\left(c_{q}+\Lambda
r_{+}\right)\right)\Big{\\}}\Big{]}.$ (62)

In absence of the corrections, uncorrected specific heat of the black hole is
given by

$C_{0}=-\frac{2\pi\left(Q^{3}+r_{+}^{3}\right){}^{2}\left(Q^{3}\left(r_{+}c_{q}-2\right)-r_{+}^{3}\left(2r_{+}c_{q}+\Lambda
r_{+}^{2}-1\right)\right)}{Q^{3}r_{+}^{4}\left(9r_{+}c_{q}+4\Lambda
r_{+}^{2}-10\right)-2Q^{6}r_{+}+\Lambda r_{+}^{9}+r_{+}^{7}}.$ (63)

Concretely, the obtained heat capacity is physically constrained within the
pair of corrected parameters $(\beta_{1},\beta_{2})$ as a corrected term and
operates the uncorrected term with the set of $(\beta_{1}=\beta_{2}=0)$. It is
worth noting that beyond certain limits, the heat capacity brings to such
uncorrected heat capacity like Schwarzschild-AdS by the parameter space
selection $(\beta_{1}=0,\beta_{2}=0,Q=0,c_{q}=0)$, while to Schwarzschild
across the selection $(\beta_{1}=0,\beta_{2}=0,Q=0,c_{q}=0,\Lambda=0)$. As a
matter of interest, the examination of the behaviour of the heat capacity
$(C_{c})$ as a function of the horizon radius $(r_{+})$ is based on certain
characteristics involving the pertinent Hawking temperature, the mass of the
black hole and the corrected entropy. Therefore, it is vital firstly to
identify a further feature about the variation in heat capacity involving two
specified points, namely the point of physical limitation and the point of
divergence representing critical phase transition points of the black hole.
More precisely, the subsequent algorithm is useful in the essence of the
corrected terms for displaying the specified points

$\displaystyle T_{H}=\left(\frac{\partial M}{\partial S_{c}}\right)$
$\displaystyle=$ $\displaystyle 0\hskip 11.38092pt\text{physical limitation
points},$ (64) $\displaystyle\left(\frac{\partial^{2}M}{\partial
S_{c}^{2}}\right)$ $\displaystyle=$ $\displaystyle 0\hskip
8.5359pt\text{second order phase transition}.$ (65)

Thus for our black hole scenario, the condition representing the physical
limitation points takes place,

$\left(2Q^{6}+Q^{3}r_{+}^{3}-r_{+}^{6}\right)\left[Q^{3}\left(r_{+}c_{q}-2\right)-r_{+}^{3}\left(2r_{+}c_{q}+\Lambda
r_{+}^{2}-1\right)\right]=0.$ (66)

The equation governing the second-order phase transitions is given by

$\left(2Q^{3}-r_{+}^{3}\right)\left(Q^{3}+r_{+}^{3}\right)\left[Q^{3}\left(r_{+}c_{q}-3\right)+r_{+}^{4}\left(c_{q}+\Lambda
r_{+}\right)\right]\left[Q^{3}\left(r_{+}c_{q}-2\right)-r_{+}^{3}\left(2r_{+}c_{q}+\Lambda
r_{+}^{2}-1\right)\right]=0.$ (67)

From the above two conditions, one can see that both the physical limitation
points and second-order phase transition points are independent of the thermal
fluctuation parameters $\beta_{1}$ and $\beta_{2}$.

The other key characteristic to consider is the sign of the heat capacity,
which is a crucial indicator of a system’s thermal stability. A positive heat
capacity implies that the system is thermally stable. This means that when
energy is added to the system, its temperature increases in a predictable and
controlled manner. Conversely, a negative heat capacity signifies thermal
instability. In this scenario, adding energy to the system leads to a decrease
in temperature, causing unpredictable and potentially unstable thermal
behavior. Therefore, the sign of the heat capacity serves as a fundamental
determinant of whether a system can maintain thermal equilibrium or is prone
to thermal fluctuations and instability. [80, 81]. Consequently, with these
elements in sight, examining heat capacity is accurately outlined and
furnishes significant evidence by analyzing Fig. 6. The first insight is taken
by varying the parameter $Q$ and keeping the other parameters fixed. This
configuration involves a set of sign-change phenomena, either by a continuous
process (physical limitation point) or discontinuous behavior (divergent
point). The sole feature that emerges here is where the parameter $Q$
increases, the position of physical limitation points grows, and the set of
divergent points is kept invariant. In addition, the system remains in a local
thermal state because of the positivity pertinent to the heat capacity. On the
other hand, the variation of the coupling constant $c_{q}$, in contrast to the
variation of the charge $Q$, does not lead to significant impacts on the
physical limitation points, even as the parameter value of $c_{q}$ increases.
This means that altering $c_{q}$ does not cause major changes in the critical
points where physical properties of the smaller black holes might otherwise be
expected to diverge or behave anomalously. Despite the changes in $c_{q}$, the
number of divergent points remains fixed, indicating a stable framework for
the system. Additionally, the thermal state of the black hole remains locally
stable under variations in $c_{q}$, suggesting that the system can maintain
thermal equilibrium without experiencing instability.

Moreover, the correction parameters $(\beta_{1},\beta_{2})$ have only minimal
impacts on the behavior of the heat capacity. These corrections introduce
slight variations, which are more significant for smaller black holes compared
to larger ones. For smaller black holes, the corrections might cause
noticeable changes in the heat capacity, whereas for larger black holes, the
effects are less pronounced. However, the crucial point to note is that these
corrections due to thermal fluctuations do not alter the physical limitation
points. They do not influence the locations where the system undergoes phase
transitions or exhibits critical behavior. Specifically, the corrections do
not affect the second-order phase transition of the black hole, meaning the
fundamental thermodynamic behavior and stability characteristics remain
unchanged. This highlights the robustness of the black hole’s thermodynamic
properties despite the presence of thermal fluctuations and parameter
variations.

##  VII Concluding Remarks

In this work, we have considered the corrected thermodynamics of non-linear
magnetic charged AdS black holes by considering thermal fluctuations and
calculating the corrected order terms of the thermodynamic potentials.
Initially, we derived the expression for the entropy $S$ of the system and
obtained a second-order corrected expression for the black hole’s entropy. The
entropy’s first-order correction term depends on the entropy and temperature
of the black hole. In contrast, the second-order corrected term depends
inversely on the black hole’s entropy. Subsequently, we obtained the corrected
thermodynamic terms of the black hole, incorporating the second-order entropy
correction. Our findings indicate that both the first and second-order
correction terms have distinct impacts on the thermodynamic potentials of the
black hole.

The variations in corrected entropy demonstrate that for small magnetic charge
$Q$ and quintessence parameter $c_{q}$, the corrected entropy is a smooth
increasing function of the horizon radius $r_{+}$. However, for larger $Q$,
entropy shows a sudden rise, with peaks shifting towards larger radii. The
quintessence parameter $c_{q}$ also influences entropy, leading to regular
increases with significant peaks, except for $c_{q}=0.1$, where no significant
peaks are observed. The corrected entropy remains positive across all cases
for both first-order and second-order corrections.

The analysis of enthalpy variations shows that increasing $Q$ raises the
enthalpy, particularly for smaller black holes, while $c_{q}$ reduces
enthalpy, especially for larger black holes. Thermal fluctuations
significantly increase enthalpy for smaller black holes and slightly decrease
it for larger ones, indicating a shift in stability due to these fluctuations.

The Helmholtz free energy analysis reveals that an increase in $Q$ shifts the
peak towards larger black holes and increases the area under the curve, while
$c_{q}$ decreases the Helmholtz free energy. Correction parameters $\beta_{1}$
and $\beta_{2}$ exhibit distinct behaviors: $\beta_{1}$ initially increases
the Helmholtz free energy for small black holes but decreases it beyond a
certain radius, whereas $\beta_{2}$ gradually decreases it for small black
holes.

The Gibbs free energy behavior indicates that smaller black holes exhibit
negative Gibbs free energy with increasing $Q$, signifying higher stability.
The quintessence parameter $c_{q}$ decreases the Gibbs free energy peak
without a significant horizontal shift. Correction parameters $\beta_{1}$ and
$\beta_{2}$ also play crucial roles, with $\beta_{1}$ increasing and
$\beta_{2}$ decreasing the Gibbs free energy initially but reversing their
effects beyond a threshold radius.

Finally, our investigation into specific heat reveals critical insights into
black hole stability. The specific heat calculations, influenced by thermal
fluctuation parameters, indicate that large black holes with positive specific
heat ensure stability against phase transitions. We identified conditions for
physical limitation points and second-order phase transitions, independent of
the correction parameters $\beta_{1}$ and $\beta_{2}$.

Overall, our results show that the thermal fluctuations significantly impact
the stability of small-sized black holes, making them less stable and
resulting in a shorter lifetime. The first-order thermodynamic corrections
tend to destabilize smaller black holes, while smaller values of second-order
correction parameters also lead to instability. However, increasing the
parameter values enhances the stability of smaller black holes. These findings
provide valuable insights into the complex nature of black hole thermodynamics
and the critical role of thermal fluctuations.

## Acknowledgment

DJG acknowledges the contribution of the COST Action CA21136 – “Addressing
observational tensions in cosmology with systematics and fundamental physics
(CosmoVerse)”.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this manuscript.

## Data availability

Data sharing does not apply to this article, as no data sets were analyzed or
generated during the current study.

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