# Restricted Phase Space Thermodynamics of NED-AdS Black Holes

Mozib Bin Awal1<EMAIL_ADDRESS>Prabwal Phukon1,2<EMAIL_ADDRESS>1Department of Physics,Dibrugarh University, Dibrugarh,Assam,786004.
2Theoretical Physics Division, Centre for Atmospheric Studies, Dibrugarh
University, Dibrugarh,Assam,786004.

###### Abstract

We study the Restricted Phase Space Thermodynamics (RPST) of magnetically
charged by nonlinear electrodynamics (NED)-AdS black holes. The first law and
the corresponding Euler relation is examined using the scaling properties. The
Euler relation is found to hold automatically. We observe that the first order
homogeneity of mass and the zeroth order homogeneity of the intensive
variables is intact. We use numerical and graphical techniques to find the
critical points of the thermodynamic quantities $S,Q,T,F$. By utilizing the
re-scaling properties of the equation of states, we study the thermodynamic
processes using different pairs of variables. We also analyse the phase
transition behaviour of free energy and other thermodynamic variables. It is
observed that the thermodynamics in the RPS formalism of our concerned black
hole system is similar to RN-AdS, Kerr AdS and Kerr-Sen AdS black hole. This
suggests that there should be some underlying universality in the RPST
formalism. However, one particular $\mu-C$ process is found to differ from the
earlier studied black holes, calling for further studies on the $\mu-C$
processes to comment on their universality.

##  I Introduction

Ever since Hawking introduced quantum mechanical effects to black holes in his
celebrated paper [1], black hole thermodynamics has been an area of active
research in modern theoretical physics. But even before the discovery of
Hawking radiation, Hawking gave the area theorem [2] and Bekenstein showed
that the entropy of a black hole should be some finite multiple of it’s area
[3]. Then finally in 1973 Bardeen, Carter and Hawking gave the four laws of
black hole mechanics [4] which had clear analogies with the laws of
thermodynamics. The formalisms established at this stage may be referred to as
Traditional Black Hole Thermodynamics (TBHT). According to this formalism, the
mass of the black hole is interpreted as internal energy and the surface
gravity and area of event horizon is proportional to the temperature and
entropy of the black hole respectively. However, the nature of the microscopic
degrees of freedom that contribute to the entropy of the black holes remains
unexplained.

The thermodynamics of black holes is considerably enriched for asymptotically
Anti-de Sitter (AdS) black holes. It has been argued that the AdS black hole
state in the bulk is equivalent to the thermal state of the dual Conformal
Field Theory (CFT) via the AdS-CFT correspondence [5]. The duality is also
popularly known as the holographic principle [6, 7]. An important discovery of
the AdS black holes was that such black holes can be in thermal equilibrium
with their radiation field and exhibit what is called the Hawking-Page phase
transition [8]. Subsequently, the second major stage, called the Extended
Phase Space Thermodynamics (EPST), also known as Black Hole Chemistry was
developed. This formalism was initially developed by Kastor, Ray and Traschen
[9] and then further developed by many others. Refs. [10, 11, 12, 13, 14, 15,
16, 17, 18] are to cite a few in this respect. The technique here is the
inclusion of the negative cosmological constant $\Lambda$, which is related to
pressure via the relation $P=-\Lambda/8\pi G$, together with a thermodynamic
volume as new pair of conjugate variables. Using this formalism, the
thermodynamic behaviours, especially $P-v$ criticalities were studied.
Significant findings include phase transitions of normal type (e.g. similar to
Van der Waals liquid-gas system [13]) or of exotic type (e.g. with multiple
critical point allowing for reentrant and/or supercritical phase transitions
[16, 17, 18]). Additionally, including the $(P,V)$ variables made it possible
to consider AdS black holes as heat engines [19, 20]. Most recently, Visser
[21] developed another variant of the extended phase space formalism utilizing
the AdS/CFT correspondence. Here the first law of black hole thermodynamics
was further extended by including the central charge $C$ and the chemical
potential $\mu$ as a new pair of conjugate thermodynamic variable. It must be
noted that the idea of the introduction of chemical potential, conjugate to a
central charge was explored earlier [22]. Similar ideas had also been
introduced in [23, 24, 25, 26], but the conjugate variable in these analysis
is the square of the number of colours in the dual field theory, which is
related, but not always equal to the central charge [24, 25, 26, 27]. In
Visser’s formalism, the volume and the corresponding conjugate pressure are
transformed into that of the CFT side’s with the aid of the AdS/CFT
correspondence, i.e. $\mathcal{V}\sim L^{d-2}$, where $\mathcal{V}$ represents
the CFT volume and $L$ is the usual AdS length. The corresponding pressure
term, $\mathcal{P}$ can be determined using the CFT Equation of State (EOS),
$E=(d-2)\mathcal{P}\mathcal{V}$, $d$ being the bulk spacetime dimensions.
Visser’s formalism is different from the previous works in that the $(P,V)$
variables are transformed to $(\mathcal{P},\mathcal{V})$. So for a charged
rotating AdS black hole, the first law can be written as

$dE=TdS-\mathcal{P}d\mathcal{V}+\tilde{\Phi}d\tilde{Q}+\Omega dJ+\mu dC$

This in turn is accompanied by an Euler-like relation

$E=TS+\tilde{\Phi}\tilde{Q}+\Omega J+\mu C$ (1)

where $\tilde{Q}$ and $\tilde{\Phi}$ are the rescaled electric charge and its
conjugate potential. One of the issues of EPS formalism is the interpretation
of the mass as enthalpy rather than internal energy. Visser’s formalism gets
rid of this problem by replacing $(P,V)$ with $(\mathcal{P},\mathcal{V})$ and
the total mass of the black hole restores its interpretation as the internal
energy. Visser’s formalism gives a full description of the thermodynamics of a
black hole in the bulk, as well as in the CFT boundary. Fixing the central
charge C, Visser’s framework leads to a thermodynamic description for the CFT
which is holographically dual to the AdS black hole in the bulk [27, 28].
Another problem with the EPS formalism is the lack of homogeneity relation of
the internal energy and intensive variables. Visser’s framework also has to
suffer from what is called the issue of the “ensemble of theories” because in
this formalism $\mathcal{V}$ is defined in terms of the AdS radius $L$ and
varying $\mathcal{V}$ would mean varying the cosmological constant $\Lambda$.

To avoid such issues, an alternative version of Visser’s formalism has been
proposed called the Restricted Phase Space (RPS) formalism in [29] studying
the RN-AdS black hole in Einstein Maxwell theory. This formalism omits the
$(P,V)$ terms and hence is called Restricted Phase Space Thermodynamics
(RPST). RPST formalism however allows us to vary $G$. Unlike varying
$\Lambda$, varying $G$ does not change the geometry and so we do not have to
face the ensemble of theories issues. As a continuation of [29], the authors
also studied the case of Kerr-AdS black holes in [30]. Following these, quite
a few numbers of works have been done in the RPST formalism. Refs. [31, 32,
33, 34, 35, 36, 37] are to cite a few.

The organisation of the paper is as follows. In section II, we briefly review
the NED-AdS black hole, derive the various thermodynamic quantities and proof
the validity of the first law and the Euler relation. In section III, we re-
express the black hole mass in terms of intensive variables such as entropy
$S$, CFT charge $C$, magnetic charge $Q$, and AdS length $l$. In section IV we
study the various types of thermodynamic processes. Finally, in section V we
briefly summarize our results.

##  II NED-AdS black hole and the RPS formalism

We start with the action of the NED-AdS black holes,

$I=\int d^{4}x\sqrt{-g}\left(\frac{R-2\Lambda}{16\pi
G}+\mathcal{F}(\mathcal{L})\right)$ (2)

where $\Lambda=-3/l^{2}$ is the cosmological constant, $l$ is the AdS radius
and $G$ is the Newton’s constant. The NED lagrangian [38] is given by,

$\mathcal{L}(\text{{\hbox{\mathcal{F}}}})=-\frac{\mathcal{F}}{4\pi\text{cosh}^{2}\left(a\sqrt[4]{2\mathcal{|F|}}\right)}$
(3)

where $a$ is the coupling constant and $\mathcal{F}=F^{\mu\nu}F_{\mu\nu}/4$ is
the field invariant. We analyse the spacetime with spherical symmetry which
has the line element squared,

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (4)

We’ll exclusively examine magnetized black holes as being electrically charged
black holes, for NED which has Maxwell weak-field limit, leads to
singularities [38]. The metric function in equation (4) can be expressed as,

$f(r)=1-\frac{2MG}{r}+\frac{Q^{2}G}{br}\tanh\left(\frac{b}{r}\right)+\frac{r^{2}}{l^{2}}$
(5)

Here, for simplicity, the parameter $b=a\sqrt{q}$ is used. We can find the
mass of the black hole in terms of horizon radius by putting $f(r_{+})=0$.
This gives,

$M=\frac{Q^{2}\tanh\left(\frac{b}{r_{+}}\right)}{2b}+\frac{r_{+}^{3}}{2Gl^{2}}+\frac{r_{+}}{2G}$
(6)

Using the metric function, the Hawking temperature is calculated and it xomes
out to be,

$T=\frac{l^{2}r_{+}^{2}+3r_{+}^{4}-Gl^{2}Q^{2}\text{sech}^{2}\left(\frac{b}{r_{+}}\right)}{4\pi
l^{2}r_{+}^{3}}$ (7)

With temperature and mass, we can easily calculate the entropy of the black
hole which comes out to be a very simple expression,

$S=\frac{\pi r_{+}^{2}}{G}$ (8)

Additionally, the magnetic potential $\Phi$ is given by,

$\Phi=\frac{q}{b}\tanh\left(\frac{b}{r_{+}}\right)$ (9)

Similar to the works in [29, 30, 31], we introduce the extra pair of
thermodynamic variables for the RPST formalism,

$C=\frac{l^{2}}{G},\quad\mu=\frac{M-TS-\hat{\Phi}\hat{Q}}{C}$ (10)

where $\hat{Q}$ is the rescaled electric charge and $\hat{\Phi}$ is the
corresponding rescaled electric potential. These two quantities are expressed
as

$\hat{Q}=\frac{Ql}{\sqrt{G}},\quad\hat{\Phi}=\frac{\Phi\sqrt{G}}{l}$ (11)

We may pause here to realize that, all thermodynamic variables must always be
positive except for the chemical potential $\mu$. It is also to be mentioned
that the chemical potential can be defined as a quantity independent of the
other thermodynamic quantities using the of AdS/CFT dictionary,
$Z_{\text{CFT}}=Z_{\text{Gravity}}$, because the gibbs free energy $W=\mu
C=-T\log Z_{\text{CFT}}=T\log Z_{\text{Gravity}}$, where
$Z_{Gravity}=\exp(-\mathcal{A}_{E})$, where $\mathcal{A}_{E}$ represents the
euclidean action evaluated at the black hole configuration. One may refer to
[39, 40, 41] for details on this. As mentioned earlier, the RPST formalism is
different from Visser’s formalism in that, in the former we keep the AdS
length, $l$ fixed and so its variation vanishes. Keeping this in mind and
using the the thermodynamic quantities above, it can be easily observed that
the first law of thermodynamics holds good,

$dM=TdS+\hat{\Phi}d\hat{Q}+\mu dC$ (12)

which is clearly free from the $\mathcal{P}d\mathcal{V}$ term. So, equation
(10) follows the Euler-like relation,

$M=TS+\hat{\Phi}\hat{Q}+\mu C$ (13)

Equation (12) and (13) are fundamental in the RPS formalism. Additionally,
these two equations may inherently determine the probable Hawking-Page-like
phase transitions. As mentioned in [29, 30, 31], we should keep in mind that
although the $(\mu,C)$ variable pair is borrowed from the dual CFT, they can
actually be understood as the chemical potential and effective number,
$N_{\text{bulk}}$ of the microscopic degrees of freedom of black hole in the
bulk. Alternatively, we may introduce new the new rules in the AdS/CFT
dictionary, viz, $\mu_{\text{CFT}}=\mu_{\text{bulk}}$ and $C=N_{\text{bulk}}$.
However, for simplicity of notation, we keep the original $(\mu,C)$ instead of
$(\mu_{\text{bulk}},N_{\text{bulk}})$, bearing in mind that we study the
thermodynamics of the black hole in the bulk and not that of the dual CFT.
Another thing to draw our attention at is the fact that the $(\mu,C)$ pair is
followed by the re-scalings: $\mu\rightarrow\lambda^{-1}\mu$ and
$C\rightarrow\lambda C$, $\lambda$ being an arbitrary non-zero constant, while
noticing that the first law and the Euler-like relation stays intact.

##  III Equation of States and Homogeneity

In this section we write the expressions of the thermodynamic quantities such
as mass, temperature etc. in terms of the extensive variables $S,\hat{Q}$ and
$C$. For that, we first express $G$ and $r_{+}$ in terms of $C$, $S$. This can
be very easily done using equations (8) and (10). So, we have,

$G=\frac{l^{2}}{C},\quad r_{+}=\left(\frac{SG}{\pi}\right)^{1/2}$ (14)

Using these two equations and the rescaled electric charge and potential in
equation (11), we write the mass $M$ in equation (6) as,

$M(S,\hat{Q},C)=\frac{1}{2}\left[\frac{\hat{Q}^{2}\tanh\left(\frac{\sqrt{\pi}b}{\sqrt{\frac{l^{2}S}{C}}}\right)}{bC}+\frac{(\pi
C+S)\sqrt{\frac{l^{2}S}{C}}}{\pi^{3/2}l^{2}}\right]$ (15)

With the expression of mass in hand, and with the aid of the first law (12),
we can right away write the Equation of States (EOS) as partial derivatives of
$M$ as,

$T=\left(\frac{\partial M}{\partial
S}\right)_{\hat{Q},C}=\frac{l^{2}\left[S(\pi
C+3S)-\pi^{2}\hat{Q}^{2}\text{sech}^{2}\left(\frac{\sqrt{\pi}b}{\sqrt{\frac{l^{2}S}{C}}}\right)\right]}{4\pi^{3/2}C^{2}\left(\frac{l^{2}S}{C}\right)^{3/2}}$
(16) $\hat{\Phi}=\left(\frac{\partial
M}{\partial\hat{Q}}\right)_{S,C}=\frac{1}{bC}\left[\hat{Q}\tanh\left(\frac{\sqrt{\pi}b}{\sqrt{\frac{l^{2}S}{C}}}\right)\right]$
(17) $\mu=\left(\frac{\partial M}{\partial
C}\right)_{S,\hat{Q}}=\frac{\hat{Q}^{2}\left[\pi^{2}b\text{sech}^{2}\left(\frac{\sqrt{\pi}b}{\sqrt{\frac{l^{2}S}{C}}}\right)-\frac{2\pi^{3/2}l\sqrt{S}\tanh\left(\frac{\sqrt{\pi}b\sqrt{C}}{l\sqrt{S}}\right)}{\sqrt{C}}\right]+bS(\pi
C-S)}{4\pi^{3/2}bC^{2}\sqrt{\frac{l^{2}S}{C}}}$ (18)

Therefore, we have expressed the intensive variables in terms of the extensive
variables $S$, $\hat{Q}$ and $C$. We can observe that, with the rescaling
$S\rightarrow\lambda S$, $\hat{Q}\rightarrow\lambda\hat{Q}$ and
$C\rightarrow\lambda C$ the mass $M$ in equation (15) rescales as
$M\rightarrow\lambda M$. However, $T$,$\hat{\Phi}$ and $\mu$ in equation
(16,(17) and (18) does not get rescaled the same way proving the first order
homogeneity of $M$ and the zeroth homogeneity of $T$,$\hat{\Phi}$ and $\mu$ in
$S,\hat{Q},C$. At this point, we may remind that the zeroth order homogeneous
functions are, by definition intensive. With all these arguments and using the
first law (12) and the corresponding Euler-like relation (13), the Gibbs-Duhem
relation can be written as

$d\mu=-\mathcal{S}dT-\hat{\mathcal{Q}}d\hat{\Phi}$

where $\mathcal{S}=S/C$ and $\hat{\mathcal{Q}}=Q/C$ are homogeneous functions
of $S,\hat{Q}$ and $C$ of zeroth order. Any standard thermodynamic system
should follow the first law of black hole mechanics, the Euler-like relations
and the Gibbs-Duhem relation. Moreover, the homogeneity properties of the
intensive variables including the internal energy must also be obeyed. RPST
formalism satisfies all these relations and homogeneity properties which were
absent in the previous formalisms of black hole thermodynamics.

It is evident that equations (16) to (18) gives three algebraic relations for
six parameters in total. Therefore, a black hole macrostate would be
characterized by three of the six thermodynamic variables $(T,S)$,
$(\hat{\Phi},\hat{Q})$ and $(\mu,C)$. We will have to let any of the three
variables to vary for a macroscopic process to take place and hence to study
the system’s thermodynamic behaviour. We make one last comment before going
into the analysis of the thermodynamic processes. We notice that $\hat{Q}$
always appears as $\hat{Q}^{2}$ in equations (15), (16) and (18). Equation
(17) however indicates that $\hat{\Phi}$ and $\hat{Q}$ must always have the
same signature. So, we may conclude that considering only the cases with
$\hat{Q}\geq 0$ will be sufficient.

##  IV Thermodynamic Processes

In this section we aim to discuss some of the various thermodynamic processes
of the NED-AdS black hole in the RPS formalism. In the previous section we
discussed that the intensive variables depend on three extensive variables in
the equation of states. So, the analysis of all the permissible thermodynamic
processes can be extremely complicated. Therefore we will only focus on some
of the special kinds of processes along certain simple curves by keeping two
of the independent variables constant. We will consider simple, conventional
processes involving only one pair of canonically conjugate intensive-extensive
variables. However, as we shall see in the following sections, even the simple
processes can lead to complicated results and one must resort to alternative
methods such as numerical or graphical techniques for the solution of the
equations. For the Reissner-Nordström-AdS black hole [29], one gets the exact
analysis, for Kerr-AdS black holes, one has to take approximations in the slow
rotating limits to get the analytic results. For Kerr-Sen-Ads black holes
[31], the authors used numerical techniques for the analysis. In our case also
we adopt some graphical and numerical techniques to study the thermodynamic
processes.

###  IV.1 Critical Points and $T-S$ processes

The analysis of the $T-S$ curves is essential in order to decode the
information about first order phase transition below the critical points which
becomes second order at the critical points. This is true for both the EPS and
RPS fromalism. Obviously, the $T-S$ curves has to be analysed at fixed
$\hat{Q}$ (or $J$ in case of Kerr or Kerr-Sen-AdS black holes).

Now, in order to find the critical points on the $T-S$ curve at fixed
$\hat{Q}$, we need to solve for the inflection point. This can be done by
solving the following two equations

$\left(\frac{\partial T}{\partial
S}\right)_{\hat{Q},C}=0,\quad\left(\frac{\partial^{2}T}{\partial
S^{2}}\right)_{\hat{Q},C}=0$ (19)

Here we use the equation of state (16). This set of equation becomes extremely
complicated and is not possible to find an analytic solution. Therefore, we
use graphical and numerical technique. In doing so, we will get the
approximate critical points.

We know, by dimensional analysis, that we must have,

$S_{c}=p_{1}C,\quad\hat{Q}_{c}=p_{2}C$ (20)

where $p_{1}$ and $p_{2}$ are some coefficients. We need to keep in mind such
scalings while employing numerical or graphical techniques. After solving we
get the critical values as

$S_{c}\approx 0.4947C,\quad\hat{Q}_{c}\approx 0.1754C$

Accordingly, the critical temperature and the Helmholtz free energy, $F=M-TS$
comes out to be

$T_{c}\approx 0.2584l^{-l},\quad F_{c}\approx 0.1398l^{-l}C$

With these, and the critical points, we can now express the equation of state
(16) and free energy in terms of the reduced parameters,

$s=\frac{S}{S_{c}},\quad\tau=\frac{T}{T_{c}},\quad
q=\frac{\hat{Q}}{\hat{Q}_{c}}$

With the reduced parameters we can write the temperature and the free energy
as

$\tau(s,q)=\frac{(0.376107s+0.747786)s-0.123947q^{2}\text{sech}^{2}\left(\frac{0.00244227}{\sqrt{s}}\right)}{s^{3/2}}$
(21)
$f(s,q)=\frac{7.15308s(0.0992056\,-0.0156217s)+0.0193822q^{2}\text{sech}^{2}\left(\frac{0.252002}{\sqrt{s}}\right)}{\sqrt{s}}\\\
+\frac{0.153826q^{2}\sqrt{s}\tanh\left(\frac{0.252002}{\sqrt{s}}\right)}{\sqrt{s}}$
(22)

We need to mention here that we have used $b/l=0.1$ while writing these
equations. From the above equations, it is evident that in terms of the
reduced parameters, the EOS and the expression of free energy is totally
independent of the central charge $C$. So, black holes, even with different
central charges will show the same thermodynamic behaviour. This can be seen
in ordinary thermodynamics as well where the role of central is replaced by
the number of particles. This is known as the law of corresponding states.

(a)

(b)

Figure 1: (a) $T-S$ and (b) $F-T$ curves at fixed $\hat{Q}>0$ (iso-e-charge)

The plots of $T/T_{c}-S/S_{c}$ and $F/F_{c}-T/T_{c}$ corresponding to iso-e-
charge processes are given in Figure 1. We can observe from the structure of
the $T/T_{c}-S/S_{c}$ curves that below the critical point where
$\hat{Q}<\hat{Q}_{c}$, there is an oscillatory phase. At $T>T_{c}$, there are
three black hole states with the same charge and central charge but having
different entropies at some temperature $T=T_{0}$. This corresponds to a
first-order phase transition. Correspondingly, the $F/F_{c}-T/T_{c}$ curves
show a swallowtail behaviour indicating a Van der Waals like first-order phase
transition provided $0<\hat{Q}<\hat{Q}_{c}$. However when
$\hat{Q}>\hat{Q_{c}}$, the $F/F_{c}-T/T_{c}$ curves show continuous behaviour.
Similar phenomenon has been seen in RN-AdS, Kerr-AdS and Kerr-Sen-Ads black
holes.

We now consider the isovoltage processes by plotting the $T-S$ curves at fixed
$\hat{\Phi},C$. We start by replacing $\hat{Q}$ in equation (16) using
equation (17). We obtain,

$T_{\hat{\Phi}}=\frac{l^{2}\left[S(\pi
C+3S)-\pi^{2}b^{2}C^{2}\hat{\Phi}^{2}\text{csch}^{2}\left(\frac{\sqrt{\pi}b}{\sqrt{\frac{l^{2}S}{C}}}\right)\right]}{4\pi^{3/2}C^{2}\left(\frac{l^{2}S}{C}\right)^{3/2}}$

We have a bound for $\hat{\Phi}$ here in order for $T$ to be positive, which
is,

$\hat{\Phi}\leq\frac{\sqrt{S}\sqrt{\pi
C+3S}\sinh\left(\frac{\sqrt{\pi}b}{\sqrt{\frac{l^{2}S}{C}}}\right)}{\pi bC}$

We observe that $T=T|_{\hat{\Phi},C}$ does not have an inflection point but
rather has an extremum point. This point corresponds to the minimum of $T$
located at $S=S_{min}$. For NED-AdS black hole, we cannot find the minimum
temperature analytically. However, by plotting $T(S,\hat{\Phi},C)$ against
$S$, we can find $T_{min}$ located at $S=S_{min}$. We can rexpress the
temperature and free energy in terms of reduced parameters

$\hat{s}=\frac{S}{S_{min}},\quad\hat{\tau}=\frac{T}{T_{min}},\quad\hat{f}=\frac{F}{F_{min}}$

as $\hat{\tau}=\tau(\hat{s},\hat{Q},C)$ and $\hat{f}=f(\hat{s},\hat{Q},C)$

(a)

(b)

Figure 2: (a) $T-S$ and (b) $F-T$ curves at fixed $\hat{\Phi}$ (isovoltage)

The plots of $T/T_{min}-S/S_{min}$ and $F/F_{min}-T/T_{min}$ are shown in
Figure 2. We have to mention here that although it is not possible to find the
minimum temperature and entropy analytically, we observe that for all the
different values of $\hat{\Phi}$, the respective curves scale down to a single
curve after using the reduced parameters. In this case also, we have used
$b/l=0.1$. It is evident from the Figure 2 that for any $T>T_{min}$, there are
two black hole states having same $T$, $C$ and $\hat{\Phi}$ but with different
entropies. In accordance with the principle of maximum entropy, the black hole
state with the smaller entropy corresponds to the unstable state and the one
with the larger entropy corresponds to the stable state. Apparently, there are
no metastable state in this case in contrast to the earlier iso-e-charge
process. The plot of free energy versus temperature on the right panel does
not go to zero anywhere implying there are no Hawking-Page phase transition.
We will discuss more on the Hawking-Page transitions later.

We may also want to see the $\hat{\Phi}-\hat{Q}$ processes. However, from
equation (17) we can observe that $\hat{\Phi}$ is just proportional to
$\hat{Q}$ and therefore there will be no inflection point or any extremum
point on the $\hat{\Phi}-\hat{Q}$ curves.

###  IV.2 $\mu-C$ processes

In this section we aim to study the $\mu-C$ processes i.e. here we allow the
central charge $C$ to vary. Here $\mu$ has to be understood as the Gibbs free
energy divided by the central charge. We start with the expression for $\mu$
in equation (18). From this equation we find that there is a single extremum
on each of the $\mu-C$ curves at fixed $(S,\hat{Q})$ which corresponds to a
maximum of $\mu$ at $C=C_{max}$. It is not possible to solve for the maximum
values of $\mu$ and the corresponding $C$ for the NED-AdS black hole system.
However, as employed in the previous case, we shall use the reduced
parameters,

$c=\frac{C}{C_{max}},\quad m=\frac{\mu}{\mu_{max}}$

In doing so, we observe that all the $\mu-C$ curves scale down to a single
one. Note that the value of $\mu_{max}$ is always positive for any values of
$S$ and $\hat{Q}$.

Figure 3: $\mu-C$ curve in the adiabatic iso-e-charge process

Figure 3 shows the $\mu/\mu_{max}-C/C_{max}$ curves for NED-AdS black hole. In
this Figure, we can see that at a certain value of the central charge, the
chemical potential becomes zero. Such zeroes corresponds to the famous
Hawking-Page phase transition. Besides that, it is also important to mention
that similar behaviour has been observed in RN-AdS, Kerr-AdS and Kerr-Sen-AdS
black hole. This may not be just a coincidence but may indicate that there
lies some universality behind the $\mu-C$ processes at fixed $J$ (in case of
Kerr-AdS), $\hat{Q}$ (in case of RN-AdS and NED-AdS) and $J,\hat{Q}$ (in case
of Kerr-Sen-AdS).

We can also study the $\mu-C$ process at fixed $S,\hat{\Phi}$. In order to do
that we replace $\hat{Q}$ in equation (18) using equation (17). We refrain
ourselves from writing the cumbersome analytical expression. Howvever, using
similar techniques used in the previous process, we plot the $\mu-C$ curve.
Throughout this analysis we have to be careful while fixing $\hat{\Phi}$
because of it’s bound.

(a)

(b)

Figure 4: $\mu-C$ process at fixed $\Phi$.

In Figure 4, the $\mu-C$ curve at fixed $S,\hat{\Phi}$ is shown. It is found
that just like the previous case, $\mu-C$ curve at fixed $S,\hat{\Phi}$ also
has a maximum but this time the maximum is not always positive, in fact, the
maximum is always negative when $\hat{\Phi}$ is chosen according to it’s
bound. In the right panel we plot $\mu/\mu^{\prime}_{max}-C/C^{\prime}_{max}$
at fixed $\hat{\Phi}$. Since $\mu^{\prime}_{max}$ is negative, we see the plot
goes upside down. Clearly, nowhere does the chemical potential $\mu$ go to
zero, implying there are no Hawking-Page transition. This is consistent with
the result in the previous section where we saw that the free energy in Figure
2 never went to zero. This result is in contrast with the RN-AdS case, where
both the $\mu-C$ processes at fixed $\hat{Q}$ and $\hat{\Phi}$ are exactly the
same.

###  IV.3 $\mu-T$ process

For a better understanding of the Hawking-Page transition, we may also check
the $\mu-T$ process at fixed $S,\hat{\Phi}$. $T$, as we have discussed
earlier, has an extremum which corresponds to a minimum. The extremum of $\mu$
on the other hand corresponds to a maximum. $\mu$ is parametrically plotted
against $T$. Figure 5 shows the plot of $\mu/\mu_{ex}-T/T_{ex}$. It is again
observed that the chemical potential nowhere becomes zero indicating that
there are no Hawking-Page phase transition. This confirms our analysis of the
$\mu-C$ process at fixed $S,\hat{\Phi}$ in the previous section. Hence we may
conclude that the universality of the $\mu-C$ processes does not apply to
isovoltage processes. The $\mu-C$ processes, as it seems are not understood
very well and needs further studies.

Figure 5: $\mu-T$ at fixed $\hat{\Phi}$

##  V Conclusions

In this paper we studied the thermodynamics of NED-AdS black holes in the RPS
formalism. We kept the AdS radius fixed and included the central charge and
chemical potential as a new pair of canonically conjugate thermodynamic
variable. Meanwhile the Newton’s constant is allowed to vary. In the RPS
formalism, we saw that the first law conforms to the standard description of
ordinary thermodynamics where the mass is interpreted as the internal energy.
The Euler relation and the Gibbs-Duhem relation also hold perfectly. We then
calculated the approximate critical values of the thermodynamic quantities.
Using the critical points, we studied the iso-e-charge processes on the $T-S$
plane which shows a Van der Waals like first order phase transition for some
non-vanishing subcritical value, $0<\hat{Q}<\hat{Q_{c}}$. The isovoltage
process on the $T-S$ plane contains a single minimum and the free energy does
not go to zero anywhere suggesting the absence of a Hawking-Page phase
transition. The constant $(S,\hat{Q})$ process have a single maximum on the
$\mu-C$ plane and there exists one Hawking-Page transition point. While, no
Hawking-Page points are observed in the fixed $(S,\hat{\Phi})$ process, which
is confirmed by the $\mu-T$ process. This is in contrast with the universality
proposed for the $\mu-C$ process in the study of the RN-AdS and Kerr-AdS black
holes. The $\mu-C$ processes need further exploration for their full
understanding.

## References

  * [1] S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43 (1975), 199-220 [erratum: Commun. Math. Phys. 46 (1976), 206] doi:10.1007/BF02345020
  * [2] S. W. Hawking, “Gravitational radiation from colliding black holes,” Phys. Rev. Lett. 26 (1971), 1344-1346 doi:10.1103/PhysRevLett.26.1344
  * [3] J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D 7, 2333-2346 (1973) doi:10.1103/PhysRevD.7.2333
  * [4] J. M. Bardeen, B. Carter and S. W. Hawking, “The Four laws of black hole mechanics,” Commun. Math. Phys. 31, 161-170 (1973) doi:10.1007/BF01645742
  * [5] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2, 231-252 (1998) doi:10.4310/ATMP.1998.v2.n2.a1 [arXiv:hep-th/9711200 [hep-th]].
  * [6] G. ’t Hooft, “Dimensional reduction in quantum gravity,” Conf. Proc. C 930308, 284-296 (1993) [arXiv:gr-qc/9310026 [gr-qc]].
  * [7] L. Susskind, “The World as a hologram,” J. Math. Phys. 36, 6377-6396 (1995) doi:10.1063/1.531249 [arXiv:hep-th/9409089 [hep-th]].
  * [8] S. W. Hawking and D. N. Page, “Thermodynamics of Black Holes in anti-De Sitter Space,” Commun. Math. Phys. 87, 577 (1983) doi:10.1007/BF01208266
  * [9] D. Kastor, S. Ray and J. Traschen, “Enthalpy and the Mechanics of AdS Black Holes,” Class. Quant. Grav. 26, 195011 (2009) doi:10.1088/0264-9381/26/19/195011 [arXiv:0904.2765 [hep-th]].
  * [10] B. P. Dolan, “The cosmological constant and the black hole equation of state,” Class. Quant. Grav. 28, 125020 (2011) doi:10.1088/0264-9381/28/12/125020 [arXiv:1008.5023 [gr-qc]].
  * [11] B. P. Dolan, “Pressure and volume in the first law of black hole thermodynamics,” Class. Quant. Grav. 28, 235017 (2011) doi:10.1088/0264-9381/28/23/235017 [arXiv:1106.6260 [gr-qc]].
  * [12] B. P. Dolan, “Compressibility of rotating black holes,” Phys. Rev. D 84, 127503 (2011) doi:10.1103/PhysRevD.84.127503 [arXiv:1109.0198 [gr-qc]].
  * [13] D. Kubiznak and R. B. Mann, “P-V criticality of charged AdS black holes,” JHEP 07, 033 (2012) doi:10.1007/JHEP07(2012)033 [arXiv:1205.0559 [hep-th]].
  * [14] R. G. Cai, L. M. Cao, L. Li and R. Q. Yang, “P-V criticality in the extended phase space of Gauss-Bonnet black holes in AdS space,” JHEP 09, 005 (2013) doi:10.1007/JHEP09(2013)005 [arXiv:1306.6233 [gr-qc]].
  * [15] D. Kubiznak, R. B. Mann and M. Teo, “Black hole chemistry: thermodynamics with Lambda,” Class. Quant. Grav. 34, no.6, 063001 (2017) doi:10.1088/1361-6382/aa5c69 [arXiv:1608.06147 [hep-th]].
  * [16] W. Xu, H. Xu and L. Zhao, “Gauss-Bonnet coupling constant as a free thermodynamical variable and the associated criticality,” Eur. Phys. J. C 74, 2970 (2014) doi:10.1140/epjc/s10052-014-2970-8 [arXiv:1311.3053 [gr-qc]].
  * [17] W. Xu and L. Zhao, “Critical phenomena of static charged AdS black holes in conformal gravity,” Phys. Lett. B 736, 214-220 (2014) doi:10.1016/j.physletb.2014.07.019 [arXiv:1405.7665 [gr-qc]].
  * [18] M. Zhang, D. C. Zou and R. H. Yue, “Reentrant phase transitions and triple points of topological AdS black holes in Born-Infeld-massive gravity,” Adv. High Energy Phys. 2017, 3819246 (2017) doi:10.1155/2017/3819246 [arXiv:1707.04101 [hep-th]].
  * [19] C. V. Johnson, “Holographic Heat Engines,” Class. Quant. Grav. 31, 205002 (2014) doi:10.1088/0264-9381/31/20/205002 [arXiv:1404.5982 [hep-th]].
  * [20] H. Xu, Y. Sun and L. Zhao, “Black hole thermodynamics and heat engines in conformal gravity,” Int. J. Mod. Phys. D 26, no.13, 1750151 (2017) doi:10.1142/S0218271817501516 [arXiv:1706.06442 [gr-qc]].
  * [21] M. R. Visser, “Holographic thermodynamics requires a chemical potential for color,” Phys. Rev. D 105, no.10, 106014 (2022) doi:10.1103/PhysRevD.105.106014 [arXiv:2101.04145 [hep-th]].
  * [22] D. Kastor, S. Ray and J. Traschen, “Chemical Potential in the First Law for Holographic Entanglement Entropy,” JHEP 11, 120 (2014) doi:10.1007/JHEP11(2014)120 [arXiv:1409.3521 [hep-th]].
  * [23] J. L. Zhang, R. G. Cai and H. Yu, “Phase transition and thermodynamical geometry of Reissner-Nordström-AdS black holes in extended phase space,” Phys. Rev. D 91, no.4, 044028 (2015) doi:10.1103/PhysRevD.91.044028 [arXiv:1502.01428 [hep-th]].
  * [24] A. Karch and B. Robinson, “Holographic Black Hole Chemistry,” JHEP 12, 073 (2015) doi:10.1007/JHEP12(2015)073 [arXiv:1510.02472 [hep-th]].
  * [25] R. Maity, P. Roy and T. Sarkar, “Black Hole Phase Transitions and the Chemical Potential,” Phys. Lett. B 765, 386-394 (2017) doi:10.1016/j.physletb.2016.12.004 [arXiv:1512.05541 [hep-th]].
  * [26] S. W. Wei, B. Liang and Y. X. Liu, “Critical phenomena and chemical potential of a charged AdS black hole,” Phys. Rev. D 96, no.12, 124018 (2017) doi:10.1103/PhysRevD.96.124018 [arXiv:1705.08596 [gr-qc]].
  * [27] M. Rafiee, S. A. H. Mansoori, S. W. Wei and R. B. Mann, “Universal criticality of thermodynamic geometry for boundary conformal field theories in gauge/gravity duality,” Phys. Rev. D 105, no.2, 024058 (2022) doi:10.1103/PhysRevD.105.024058 [arXiv:2107.08883 [gr-qc]].
  * [28] W. Cong, D. Kubiznak and R. B. Mann, “Thermodynamics of AdS Black Holes: Critical Behavior of the Central Charge,” Phys. Rev. Lett. 127, no.9, 091301 (2021) doi:10.1103/PhysRevLett.127.091301 [arXiv:2105.02223 [hep-th]].
  * [29] G. Zeyuan and L. Zhao, “Restricted phase space thermodynamics for AdS black holes via holography,” Class. Quant. Grav. 39, no.7, 075019 (2022) doi:10.1088/1361-6382/ac566c [arXiv:2112.02386 [gr-qc]].
  * [30] Z. Gao, X. Kong and L. Zhao, “Thermodynamics of Kerr-AdS black holes in the restricted phase space,” Eur. Phys. J. C 82, no.2, 112 (2022) doi:10.1140/epjc/s10052-022-10080-y [arXiv:2112.08672 [hep-th]].
  * [31] M. S. Ali, S. G. Ghosh and A. Wang, “Thermodynamics of Kerr-Sen-AdS black holes in the restricted phase space,” Phys. Rev. D 108, no.4, 044045 (2023) doi:10.1103/PhysRevD.108.044045 [arXiv:2308.00489 [gr-qc]].
  * [32] Y. Ladghami, B. Asfour, A. Bouali, A. Errahmani and T. Ouali, “4D-EGB black holes in RPS thermodynamics,” Phys. Dark Univ. 41, 101261 (2023) doi:10.1016/j.dark.2023.101261 [arXiv:2303.00604 [hep-th]].
  * [33] X. Kong, Z. Zhang and L. Zhao, “Restricted phase space thermodynamics of charged AdS black holes in conformal gravity*,” Chin. Phys. C 47, no.9, 095105 (2023) doi:10.1088/1674-1137/ace9c2 [arXiv:2211.00963 [hep-th]].
  * [34] X. Kong, T. Wang, Z. Gao and L. Zhao, “Restricted Phased Space Thermodynamics for Black Holes in Higher Dimensions and Higher Curvature Gravities,” Entropy 24, no.8, 1131 (2022) doi:10.3390/e24081131 [arXiv:2208.07748 [hep-th]].
  * [35] Y. Z. Du, H. F. Li, Y. Zhang, X. N. Zhou and J. X. Zhao, “Restricted Phase Space Thermodynamics of Einstein-Power-Yang–Mills AdS Black Hole,” Entropy 25, no.4, 687 (2023) doi:10.3390/e25040687 [arXiv:2210.02006 [hep-th]].
  * [36] J. Sadeghi, M. Shokri, S. Gashti Noori and M. R. Alipour, “RPS thermodynamics of Taub–NUT AdS black holes in the presence of central charge and the weak gravity conjecture,” Gen. Rel. Grav. 54, no.10, 129 (2022) doi:10.1007/s10714-022-03024-0 [arXiv:2205.03648 [hep-th]].
  * [37] Y. Y. Bai, X. R. Chen, Z. M. Xu and B. Wu, “Revisit on thermodynamics of BTZ black hole with variable Newton constant,” [arXiv:2208.11859 [gr-qc]].
  * [38] K. A. Bronnikov, “Regular magnetic black holes and monopoles from nonlinear electrodynamics,” Phys. Rev. D 63, 044005 (2001) doi:10.1103/PhysRevD.63.044005 [arXiv:gr-qc/0006014 [gr-qc]].
  * [39] G. W. Gibbons and S. W. Hawking, “Action Integrals and Partition Functions in Quantum Gravity,” Phys. Rev. D 15, 2752-2756 (1977) doi:10.1103/PhysRevD.15.2752
  * [40] A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, “Charged AdS black holes and catastrophic holography,” Phys. Rev. D 60, 064018 (1999) doi:10.1103/PhysRevD.60.064018 [arXiv:hep-th/9902170 [hep-th]].
  * [41] G. W. Gibbons, M. J. Perry and C. N. Pope, “The First law of thermodynamics for Kerr-anti-de Sitter black holes,” Class. Quant. Grav. 22, 1503-1526 (2005) doi:10.1088/0264-9381/22/9/002 [arXiv:hep-th/0408217 [hep-th]].