# Restricted phase space thermodynamics for black holes in higher dimensions
and higher curvature gravities

Xiangqing Kong, Tao Wang, Zeyuan Gao and Liu Zhao
School of Physics, Nankai University, Tianjin 300071, China email:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>author, email<EMAIL_ADDRESS>
###### Abstract

The recently proposed restricted phase space thermodynamics is shown to be
applicable to a large class of higher dimensional higher curvature gravity
models coupled to Maxwell field, which are known as black hole scan models and
are labeled by the spacetime dimension $d$ and the highest order $k$ of the
Lanczos-Lovelock densities appearing in the action. Three typical example
cases with $(d,k)=(5,1),(5,2)$ and $(6,2)$ are chosen as example cases and
studied in some detail. These cases are representatives of Einstein-Hilbert,
Chern-Simons and Born-Infield like gravity models. Our study indicates that
the Einstein-Hilbert and Born-Infield like gravity models have similar
thermodynamic behaviors, e.g. the existence of isocharge $T-S$ phase
transitions with the same critical exponents, the existence of isovoltage
$T-S$ transitions and the Hawking-Page like transitions, and the similar high
temperature asymptotic behaviors for the isocharge heat capacities, etc.
However, the Chern-Simons like $(5,2)$-model behaves quite differently.
Neither isocharge nor isovoltage $T-S$ transitions could occur and no Hawking-
Page like transition is allowed. This seems to indicate that the Einstein-
Hilbert and Born-Infield like models belong to the same universality class
while the Chern-Simons like models do not.

##  1 Introduction

Understanding the thermodynamic properties of black holes is among the central
tasks of modern gravitational physics. Although black hole thermodynamics has
been studied for over half a century, this field of study is far from being
completed. Different proposals/formalisms emerge from time to time, which can
be briefly classified into two major classes, i.e. traditional black hole
thermodynamics (TBHT) [1, 2, 3, 4] and extended phase space thermodynamics
(EPST)[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] for AdS black holes. In
particular, Visser’s recent proposal for a holographic thermodynamics [16]
(when applied to black holes) can be regarded as a further development of
EPST, which also applies specifically to AdS black holes only [17, 18].

The resent interests in black hole thermodynamics has been largely triggered
by EPST because this formalism of black hole thermodynamics reveals some
possible nontrivial thermodynamic behaviors of black holes such as phase
transitions and critical phenomena. However, the EPST formalism is not free of
internal issues. For instance, well-known Smarr relations [19] for black hole
solutions always contain rational coefficients, which indicates the lack of
complete Euler homogeneity (see, e.g. [20] for standard description for the
importance of Euler homogeneity in standard thermodynamics) and constitutes a
stumbling stone in understanding the conditions for equilibrium conditions for
black holes; the variable cosmological constant causes the ensemble of
theories issue; the interpretation of black hole mass as enthalpy looks
unnatural and lacks physical interpretation; and the inability to extend to
non-AdS black holes seems to indicate that the EPST formalism is in short of
universality, etc.

In some recent works, we proposed yet another formalism for black hole
thermodynamics, i.e. the restricted phase space thermodynamics (RPST) [21, 22,
23, 24], which is a restricted version of Visser’s proposal [16] by removing
the $P,V$ variables. Unlike the previous formalisms, our formalism is free of
all the above mentioned issues because i) the cosmological constant is fixed
and removed from the list of thermodynamic variables, this choice removes the
ensemble of theories issue and brings back the interpretation of black hole
mass from enthalpy to internal energy; ii) the introduction of the effective
number of microscopic degrees of freedom $N=L^{D-2}/G$, together with its
conjugate, the chemical potential $\mu=GTI_{E}/L^{D-2}$, where $G$ is the
variable Newton constant, $I_{E}$ is the Euclidean action (which has been
extensively studies in various cases [25, 26, 27]) and $L$ is a constant
length scale, makes the Euler homogeneity hold perfectly; iii) the definition
of $N$ and $\mu$ as given above is independent of holographic duality,
therefore the RPST formalism is valid not only for AdS black holes but also
for non-AdS ones. Regarding the Euler homogeneity, let us remind that, there
exist some works which were able to realize the Euler homogeneity using the
EPST formalism [28, 29], however those works relies heavily on the holographic
duality and hence is not applicable to non-AdS cases.

Up till now, our study of RPST formalism is limited to black holes in Einstein
gravity in four and higher dimensions. It remains to check whether this
formalism works also for alternative theories of gravity, e.g. higher
curvature gravities [30, 31, 32]. The present work is a first step toward this
direction. In this work, we will extend the RPST formalism to higher curvature
gravity theories. Since the RPST formalism requires a fixed cosmological
constant (be it positive, negative or zero), the class of black hole scan
models [33] (which are a subclass of Lanczos-Lovelock models with some
particular choices of the coupling coefficients) is an ideal choice for such
studies, because such models admit a unique AdS solution with fixed
cosmological constant. Please be reminded that choosing the black hole scan
models as the first example for the application of the RPST formalism to
higher curvature gravity models does not imply that such models are specific.
It is simply a choice for convenience because this choice allows us to study
representatives for different types of gravity models starting from a single
unified action. To illustrate the strength of this formalism, we will consider
in detail the thermodynamic behaviors of three concrete cases, i.e. the
$d=5,k=1,2$ and $d=6,k=2$ cases, where $d$ is the spacetime dimension and
$1\leq k\leq\left[\frac{d-1}{2}\right]$. The reason for choosing these three
particular example cases lies in that these are the simplest representative
cases for Einstein-Hilbert (EI), Chern-Simons (CS) and Born-Infield (BI)
gravity models without degeneracy between different classes. For $d=3$, the EH
and CS theories degenerate, while for $d=4$, the EH and BI theories
degenerate. That’s why we start at $d=5$.

This paper is organized as follows. In Section 2, we briefly review the
gravity models to be analyzed and introduce the necessary thermodynamic
variables. It is shown that the RPST formalism applies perfectly to the whole
class of the so-called black hole scan models [33]. The Euler relation and the
first law hold simutaneously which implies the Gibbs-Duhem relation and Euler
homogeneity. Section 3 is devoted to the case studies. Three concrete models
will be taken as example cases, i.e. the $(d,k)=(5,1),(5,2)$ and $(6,2)$
models. Among these, the $(5,1)$-model will be studied in great detail, and
various thermodynamic behaviors are described carefully. The $(5,2)$\- and
$(6,2)$-models will be treated much more briefly, however without sacrifice in
the description of the major thermodynamic behaviors. In Section 4 we present
the summary of the work and some further discussions.

##  2 Review of the models and the RPST formalism

Throughout this article we work in units $k_{B}=\hbar=c=1$,
$\varepsilon_{0}=1/\mu_{0}=1/{A}_{d-2}$, where
${A}_{d-2}=\frac{2\pi^{(d-1)/2}}{\Gamma\left(\frac{d-1}{2}\right)}$ is the
area of the unit $(d-2)$-sphere, but leave the gravitational constant intact,
because the gravitational constant is not a constant but rather among the
thermodynamic variables in the RPST formalism.

The action for the black hole scan models coupled to the electromagnetic field
can be written in two parts,

$\displaystyle\mathcal{A}_{(d,k)}=I_{(d,k)}+I_{M},$ (1)

where $I_{(d,k)}$ denotes the gravitational action and $I_{M}$ is the Maxwell
action

$\displaystyle
I_{M}=-\frac{1}{4{A}_{d-2}}\int\sqrt{-g}F^{\mu\nu}F_{\mu\nu}{\rm d}^{d}x.$ (2)

The expression for $I_{(d,k)}$ reads

$\displaystyle I_{(d,k)}=\kappa_{(d,k)}\int\sum_{p=0}^{k}c_{p}^{k}L^{(p)},$
(3)

where $L^{(p)}$ is the $p$-th order Lanczos-Lovelock density which can be
written in terms of the Riemann curvature two form and the vielbein $e^{a}$ as

$L^{(p)}=\epsilon_{a_{1}\cdots a_{d}}R^{a_{1}a_{2}}\cdots
R^{a_{2p-1}a_{2p}}e^{a_{2p+1}}\cdots e^{a_{d}},$

where $R^{ab}={\rm d}\omega^{ab}+\omega^{a}{}_{c}\omega^{cd}$ and wedge
product between forms is understood. The Lanczos-Lovelock coupling
coefficients $c_{p}^{k}$ are taken to be

$\displaystyle
c_{p}^{k}=\begin{cases}\dfrac{\ell^{2(p-k)}}{(d-2p)}\begin{pmatrix}k\\\
p\end{pmatrix},&(p\leq k),\\\ 0,&(p>k),\end{cases}$ (4)

where the constants $\kappa_{(d,k)}$ and $\ell$ are related to the Newton
constant $G_{(d,k)}$ and the cosmological constant via

$\displaystyle\kappa_{(d,k)}$
$\displaystyle=\frac{1}{2(d-2)!A_{d-2}G_{(d,k)}},\quad\Lambda=-\frac{(d-1)(d-2)}{2\ell^{2}}.$
(5)

Notice that $c_{p}^{k}$ has dimension $[\text{length}]^{2(p-k)}$ and
$G_{(d,k)}$ has dimension $[\text{length}]^{d-2k}$, so that
$c_{p}^{k}/G_{(d,k)}$ has dimension $[\text{length}]^{2p-d}$ which is
independent of $k$. For $p=1$ this fixes the dimension of the coupling
coefficient in front of the Einstein-Hilbert action.

The above specific choices for the coupling coefficients makes the class of
black hole scan models distinguished from the generic Lanczos-Lovelock models.
The spacetime dimension $d$ and the highest order $k$ of the Lanczos-Lovelock
densities appearing in the action are used to label the concrete model and
hence the corresponding model will be referred to as the $(d,k)$-model. The
$(d,k)$-models with $d\leq 11$ are summarized in [33]. It should be emphasized
that each $(d,1)$-model is simply the Einstein-Hilbert theory in
$d$-dimensions (i.e. EH model). Moreover, each $(2k+1,k)$-model is a CS model,
and each $(2k+2,k)$-model is a BI model.

The merit for the particular choice (4) for the coupling coefficients lies in
that the corresponding charged spherically symmetric AdS black hole solutions
can be given in a unified form for any $d>3$ with a fixed cosmological
constant as given in eq.(5). The metric takes the form [33]

$\displaystyle{\rm d}s^{2}_{(d,k)}=-f_{(d,k)}(r){\rm d}t^{2}+\dfrac{{\rm
d}r^{2}}{f_{(d,k)}(r)}+r^{2}{\rm d}\Omega_{d-2}^{2},$ (6)

where ${\rm d}\Omega_{d-2}^{2}$ is the line element on a unit $(d-2)$-sphere
and

$f_{(d,k)}(r)=1+\frac{r^{2}}{\ell^{2}}-g_{(d,k)}(r),\quad
g_{(d,k)}(r)=\left(\frac{2G_{(d,k)}M+\delta_{d-2k,1}}{r^{d-2k-1}}-\frac{G_{(d,k)}}{(d-3)}\frac{Q^{2}}{r^{2(d-k-2)}}\right)^{1/k},$

wherein $M$ and $Q$ are integration constants which are interpreted as the
black hole mass and electric charge respectively. This metric is accompanied
by the static electromagnetic field $A_{\mu}=\Phi\delta_{\mu 0}$, where $\Phi$
represents the Coulomb potential

$\Phi=\frac{1}{(d-3)}\frac{Q}{r^{d-3}}.$

The case $d=3$ is exceptional, because in three dimensions the Coulomb
potential is not constant but rather logarithmic in $r$. The RPST formalism
for charged AdS black hole solution in this exceptional case is studied in
[34]. For $d>3$, $\Phi$ and $Q$ always have the same sign, therefore it
suffices to consider the cases with $Q>0,\Phi>0$ in this work.

The event horizon of the black hole is located at one of the zero $r_{h}$ of
the function $f_{(d,k)}(r)$. Thus the mass of the black hole can be solved
from the equation $f_{(d,k)}(r_{h})=0$, yielding

$\displaystyle
M(r_{h},G_{(d,k)},Q)=-\frac{\delta_{1,d-2k}}{2G_{(d,k)}}+\frac{r_{h}^{d-2k-1}\left(\frac{r_{h}^{2}}{\ell^{2}}+1\right)^{k}}{2G_{(d,k)}}+\frac{Q^{2}}{2(d-3)r_{h}^{d-3}}.$
(7)

The temperature of the black hole can be evaluated using the Euclidean period
method, which gives

$\displaystyle T$ $\displaystyle=\frac{1}{4\pi}\frac{{\rm d}f_{(d,k)}}{{\rm
d}r}\Big{|}_{r=r_{h}}$ $\displaystyle=\frac{1}{4\pi
kr_{h}^{2d-1}}\left(1+\frac{r_{h}^{2}}{\ell^{2}}\right)^{-(k-1)}\left[(d-2k-1)r_{h}^{2d}\left(1+\frac{r_{h}^{2}}{\ell^{2}}\right)^{k}-G_{(d,k)}Q^{2}r_{h}^{2k+4}\right]+\frac{2r_{h}}{4\pi\ell^{2}}.$
(8)

The entropy of the black hole is [33]

$S=S(r_{h},G_{(d,k)},Q)=\frac{2\pi
k}{G_{(d,k)}}\int_{0}^{r_{h}}r^{(d-2k-1)}\left(1+\frac{r^{2}}{\ell^{2}}\right)^{k-1}{\rm
d}r,$ (9)

which follows from the identity (c.f. eq.(64) in [33])

$\displaystyle I_{E}=T^{-1}M-S-T^{-1}\Phi Q,$ (10)

where $I_{E}$ is the Euclidean on-shell action. $I_{E}$ is presented in [33]
in a complicated integration form, which will not be made use of in this work,

$\displaystyle I_{E}$
$\displaystyle=-\beta\int_{r_{+}}^{\infty}\frac{N}{2}\left[\frac{{\rm d}}{{\rm
d}r}\left\\{\frac{r^{d-1}}{G_{k}}\left[F(r)+\frac{1}{l^{2}}\right]^{k}\right\\}-\frac{1}{\epsilon}r^{d-2}p^{2}\right]{\rm
d}r$
$\displaystyle\quad-\frac{1}{\epsilon}\beta\int_{r_{+}}^{\infty}\Phi\frac{{\rm
d}}{{\rm d}r}\left(r^{d-2}p\right){\rm d}r+B_{E}.$ (11)

For our purpose, it is important not to use the above identity for defining
the entropy. Rather, the entropy should be calculated using Bekenstein-Hawking
formula (when applicable) or Wald method [35, 36]. It turns out that the same
result (9) arises, at least in the concrete cases to be described in detail in
the next section.

In order to establish the RPST for the $(d,k)$-models, we need to introduce a
novel pair of thermodynamic variables, i.e. the effective number of
microscopic degrees of freedom $N$ and the corresponding chemical potential
$\mu$. These two quantities are defined as

$\displaystyle{\color[rgb]{0,0,0}N=N(G_{(d,k)})=\frac{L^{d-2k}}{G_{(d,k)}}},\quad\mu=\mu(r_{h},G_{(d,k)},Q)=\frac{TI_{E}}{N},$
(12)

where $L$ is a constant length scale which is introduced to make $N$
dimensionless. It should be noticed that $G_{(d,k)}$ is not the standard
Newton constant $G$ but rather differs from $G$ by a multiplicative constant
factor. This slight difference does not make any harm to the definitions of
$N,\mu$, because the multiplicative constant factor can be absorbed by a
simple redefinition of the arbitrary constant length scale $L$. However, when
evaluating the black hole entropy using Bekenstein-Hawking formula [4] (when
applicable), the multiplicative constant factor becomes relevant and needs to
be taken care of.

Another necessary step before establishing the RPST formalism is to make a
rescaling for the electric charge and the Coloumb potential, so that
$\kappa_{(d,k)}$ becomes an overall factor in front of the total action. The
rescaled charge and potential read

$\displaystyle\hat{Q}$
$\displaystyle=\hat{Q}(Q,G_{(d,k)})=\frac{QL^{(d-2k)/2}}{\sqrt{G_{(d,k)}}},$
(13) $\displaystyle\hat{\Phi}$
$\displaystyle=\hat{\Phi}(r_{h},G_{(d,k)},Q)=\frac{\Phi\sqrt{G_{(d,k)}}}{L^{(d-2k)/2}}=\frac{\sqrt{G_{(d,k)}}Q}{(d-3)L^{(d-2k)/2}r_{h}^{d-3}}.$
(14)

After the above preparation, it is now straightforward to recognize that
eq.(10) is nothing but the Euler relation

$\displaystyle M=TS+\hat{\Phi}\hat{Q}+\mu N,$ (15)

and consequently it is easy to check that total differential of the black hole
mass obeys the following first law of thermodynamics,

$\displaystyle{\rm d}M=T{\rm d}S+\hat{\Phi}{\rm d}\hat{Q}+\mu{\rm d}N.$ (16)

Eqs.(15) and (16) indicate that the RPST formalism is applicable to
$(d,k)$-models for any $d>3$ and any admissible $k$. It is worth mentioning
that the Euler relation (15) and the first law (16) take exactly the same form
for any admissible $(d,k)$-models. This is in contrast to the EPST formalism
in which the Euler relation is always absent and the number of thermodynamic
variables can be different in different models (e.g. the inclusion of coupling
coefficients in the set of thermodynamic variables). Moreover, eqs. (15) and
(16) imply that $M$ is a first order homogeneous function in the extensive
variables $S,\hat{Q},N$ while the intensive variables $T,\hat{\Phi},\mu$ are
zeroth order homogeneous functions in the extensive variables. Combining eqs.
(15) and (16) we can also obtain the Gibbs-Duhem relation

$S{\rm d}T+\hat{Q}{\rm d}\hat{\Phi}+N{\rm d}\mu=0,$

which is an important relationship between the intensive variables in order
that the thermodynamic system is consistent.

##  3 Case studies and thermodynamic behaviors

Although the form of the Euler relation and the first law remains the same for
any $(d,k)$-models, the thermodynamic behaviors can be drastically different
for different choices of $(d,k)$. In this section, three concrete cases will
be studied in detail, each corresponds to a different type of gravity models.
The chosen cases are, respectively, the $(5,1)$-model (EH gravity), the
$(5,2)$-model (CS gravity) and the $(6,2)$-model (BI gravity). Each model will
be described in a separate subsection.

###  3.1 $(5,1)$-model: EH gravity coupled to Maxwell field

####  3.1.1 Description of phase transitions

For all $(d,1)$-models the thermal and caloric equations of states for the
black holes can be worked out analytically by use of eqs.(7)-(14). This can be
achieved by solving $r_{h},G_{(d,k)},Q$ from the expressions
$S=S(r_{h},G_{(d,k)},Q),N=N(G_{(d,k)}),\hat{Q}=\hat{Q}(Q,G_{(d,k)})$ which
yields

$\displaystyle{\color[rgb]{0,0,0}r_{h}=(2\pi)^{-\frac{1}{d-2}}\left(\frac{(d-2)L^{d-2}S}{N}\right)^{\frac{1}{d-2}},\quad
G_{(d,1)}=\frac{L^{d-2}}{N},\quad Q=\frac{\hat{Q}}{\sqrt{N}}.}$ (17)

Substituting the above solution into eqs.(7) and (8), we get

$\displaystyle T(S,\hat{Q},N)$
$\displaystyle=\frac{1}{2}(2\pi)^{\frac{3-d}{d-2}}\tilde{S}\left(d-3+\frac{(d-1)(2\pi)^{-\frac{2}{d-2}}}{\ell^{2}}\tilde{S}^{2}-(2\pi)^{\frac{2(3-d)}{2-d}}L^{d-2}\frac{\tilde{S}^{2(3-d)}\hat{Q}^{2}}{N^{2}}\right),$
(18) $\displaystyle M(S,\hat{Q},N)$
$\displaystyle=\frac{1}{2}(2\pi)^{\frac{d-3}{2-d}}L^{2-d}N\tilde{S}^{d-3}\left(1+\frac{(2\pi)^{-\frac{2}{d-2}}}{\ell^{2}}\tilde{S}^{2}+\frac{(2\pi)^{\frac{2(3-d)}{2-d}}L^{d-2}}{d-3}\frac{\tilde{S}^{2(3-d)}\hat{Q}^{2}}{N^{2}}\right),$
(19)

where

$\displaystyle\tilde{S}=(d-2)^{\frac{1}{d-2}}L\left(\frac{S}{N}\right)^{\frac{1}{d-2}}.$
(20)

It is transparent that $M$ behaves as $M\to\lambda M$ when the extensive
variables $S,\hat{Q},N$ are rescaled as $S\to\lambda
S,\hat{Q}\to\lambda\hat{Q},N\to\lambda N$, while $T$ is kept unchanged under
the above rescaling. The zeroth order homogeneity of $T$ is very important in
understanding the condition for thermal equilibrium.

It can also be shown that for any $d>3$, the thermal equation of states (18)
always contains an inflection point with

$\displaystyle\left(\frac{\partial T}{\partial
S}\right)_{\hat{Q}}=\left(\frac{\partial^{2}T}{\partial
S^{2}}\right)_{\hat{Q}}=0,$ (21)

which indicates that the $T-S$ phase structure is similar for all EH models in
$d>3$. Therefore, it suffices to consider only the case $d=5$ as a
representative for all EH models with $d\geq 4$.

To be more explicit, let us rewrite the full action for the $(5,1)$-model as
follows,

$\displaystyle\mathcal{A}_{(d,k)}$
$\displaystyle=\frac{1}{12\pi^{2}G_{(5,1)}}\int\left(R+\frac{12}{\ell^{2}}\right)\sqrt{-g}{\rm
d}^{5}x-\frac{1}{8\pi^{2}}\int F_{\mu\nu}F^{\mu\nu}\sqrt{-g}{\rm d}^{5}x.$
(22)

This is nothing but the Einstein-Maxwell theory in five dimensions, with the
corresponding charged spherically symmetric black hole solution given by the
metric

$\displaystyle{\rm d}s^{2}_{(5,1)}=-f(r){\rm d}t^{2}+\frac{{\rm
d}r^{2}}{f(r)}+r^{2}{\rm d}\Omega_{3},$ (23)

wherein

$f(r)=\left(1+\frac{r^{2}}{\ell^{2}}+\frac{G_{(5,1)}Q^{2}}{\pi
r^{4}}-\frac{2G_{(5,1)}M}{r^{2}}\right).$

Since this is Einstein-Hilbert gravity, the entropy of the black hole can be
evaluated by simply employing the Bekenstein-Hawking formula $S=\frac{A}{4G}$,
where $G$ is the Newton constant which appear in the standard Einstein-Hilbert
action $\frac{1}{16\pi G}\int R\sqrt{-g}{\rm d}^{d}x$. It is evident that

$\frac{1}{16\pi G}=\frac{1}{12\pi^{2}G_{(5,1)}},\quad{\rm i.e.}\quad
G=\frac{3\pi G_{(5,1)}}{4},$

thus

$S=\frac{A_{3}r_{h}^{3}}{4G}=\frac{2\pi r_{h}^{3}}{3G_{(5,1)}}.$

This result coincide with what can be obtained by evaluating the integral in
eq.(9).

By solving the inflection point equation (21) in $d=5$, we get the critical
point parameters

$\displaystyle
S_{c}=\frac{2\pi\ell^{3}}{9\sqrt{3}L^{3}}N,\quad\hat{Q}_{c}=\frac{1}{3}\sqrt{\frac{2}{15}}\frac{\ell^{2}}{L^{3/2}}N.$
(24)

Inserting these parameters back into eqs.(18) and (14), we get

$\displaystyle
T_{c}=\frac{4\sqrt{3}}{5\pi\ell},\quad\hat{\Phi}_{c}=\frac{1}{\sqrt{30}L^{3/2}}.$
(25)

These critical parameters obey the identy

$\frac{S_{c}T_{c}}{\hat{\Phi}_{c}\hat{Q}_{c}}=8.$

Moreover, if we introduce the Helmholtz free-energy

$F=F(T,\hat{Q},N)=M-TS,$

then the critical value of the black hole mass and Helmholtz free energy read

$\displaystyle M_{c}=\frac{7\ell^{2}}{30L^{3}}N,\quad
F_{c}=\frac{\ell^{2}}{18L^{3}}N.$ (26)

Now we are in a position to study the thermodynamic behavior of the
$(5,1)$-model. For this purpose, it will be advantageous to adopt the relative
state parameters/functions

$\displaystyle s\equiv\frac{S}{S_{c}},\quad t\equiv\frac{T}{T_{c}},\quad
q\equiv\frac{Q}{Q_{c}},\quad\phi=\frac{\hat{\Phi}}{\hat{\Phi}_{c}},\quad
m\equiv\frac{M}{M_{c}},\quad f\equiv\frac{F}{F_{c}}$ (27)

which are independent of the scale of the black hole. Then the thermal,
caloric equations of states and the Helmholtz free energy can be written as

$\displaystyle t$ $\displaystyle=\frac{10s^{2}+15s^{4/3}-q^{2}}{24s^{5/3}},$
(28) $\displaystyle m$
$\displaystyle=\frac{5\left(s^{2}+3s^{4/3}\right)+q^{2}}{21s^{2/3}},$ (29)
$\displaystyle f$ $\displaystyle=\frac{-s^{2}+3s^{4/3}+q^{2}}{3s^{2/3}},$ (30)

where the dependence of $f$ upon $t$ is implicit via eq.(28). Notice that only
two relative parameters $s,q$ appear on the right hand side of the above
equations. The scale independence of the above equations is known as the law
of corresponding states in traditional thermodynamics for ordinary matter,
however in black hole thermodynamics, this law is only known to exist in the
RPST formalism. The $\hat{\Phi}-\hat{Q}$ equation of states can also be re-
expressed in terms of the relative parameters given above. The result reads

$\displaystyle\phi=\frac{q}{s^{3/2}}.$ (31)

As an immediate consequence of the law of corresponding states, the effective
number $N$ of microscopic degrees of freedom of the black hole is irrelevant
while analyzing the thermodynamic processes. As a bonus, we can also see that
$\ell$ and $L$ also disappear in eqs.(28)-(31), indicating that the choice of
values for these two constant parameters is also irrelevant to the analysis of
thermodynamic behaviors.

Figure 1: Bounds in the parameter space

Before analyzing the thermodynamic processes, it is important to realize that
there are some natural bounds in the parameter space which is triggered by the
requirement that $t\geq 0$ and that $q$ must be real-valued. It follows from
eq.(28) that $t\geq 0$ requires that

$\displaystyle q\leq\sqrt{5}\sqrt{3s^{4/3}+2s^{2}},$ (32)

or, after employing eq.(31), that

$\phi\leq\sqrt{5}\sqrt{2s^{2/3}+3}.$

These bounds are nothing but the well-known Bogomol’nyi bound [37] rewritten
in terms of the relative thermodynamic variables. In addition, from eq.(28),
one can solve $q$ in terms of $s,t$, which reads

$q=\sqrt{-24s^{5/3}t+15s^{4/3}+10s^{2}}.$

Thus the real-valuedness of $q$ poses a bound for $t$ versus $s$,

$t\leq\frac{5\left(2s^{2/3}+3\right)}{24\sqrt[3]{s}}.$

The allowed parameter space specified by the above bounds is shown as the dark
shaded area in Fig.1.

Now let us proceed to analyze some of the thermodynamic processes.

Figure 2: Isocharge $T-S$ and $F-T$ curves

First we consider the isocharge $T-S$ processes. The isocharge $T-S$ and $F-T$
curves are depicted in Fig.2 by use of eqs.(28) and (30). It can be seen that,
at $\hat{Q}<\hat{Q}_{c}$, every isocharge $T-S$ curve contains a segment on
which $T$ decreases as $S$ increases. This non-monotonic behavior implies a
first order phase transition which occurs at a constant temperature determined
by requiring minimization of the Helmholtz free energy, i.e.

$\left(\frac{\partial F}{\partial T}\right)_{\hat{Q}}=0.$

Effectively, this condition requires that the actual isocharge process does
not follow the analytical curve for $\hat{Q}<\hat{Q}_{c}$ but rather undergoes
an isocharge isothermal segment during the phase transition at an invariant
temperature $T^{\ast}>T_{c}$. Such phase transitions were also observed
previously in our works [21, 22] and are referred to as supercritical phase
transitions which occur only in black hole thermodynamics but not in
thermodynamics of ordinary matter. During this isocharge isothermal stage of
evolution, the Helmholtz free energy remains at a fixed value which
corresponds to the root of the swallow tail depicted in the right plots in
Fig.2. The isocharge isothermal segment on the $T-S$ curve corresponds to the
coexistence phase between the small and large black hole phases. The
coexistence zone on the $T-S$ phase plane is shown by the shaded area on the
left picture of Fig.2. At $T=T_{c}$, the $T-S$ phase transition becomes second
order which corresponds to the critical point.

We can also consider the isovoltage $T-S$ processes instead of the isocharge
processes. To do so, we need to carry out the following steps. First we re-
express $\hat{Q}$ in terms of $\hat{\Phi}$ and $r_{h}$ using eq.(14) and then
using the expression for $r_{h}$ given in eq.(17) to rewrite $\hat{Q}$ as a
function in $\hat{\Phi},S,N$, i.e.

$\hat{Q}=\frac{18^{1/3}L^{2}\hat{\Phi}S^{2/3}}{\pi^{2/3}{N}^{1/6}}.$

Inserting the above result into eq.(18) with $d=5$, we get

$T(S,\hat{\Phi},N)=\frac{\ell^{2}({3\pi^{2}N^{2}})^{1/3}\left(1-2L^{3}\hat{\Phi}^{2}\right)+3L^{2}(2S^{2})^{1/3}}{\ell^{2}L(36\pi^{4}NS)^{1/3}}.$

We wish to analyze the $T-S$ relation at fixed $\hat{\Phi}$. For this purpose
we first need to find out the extrema of the $T-S$ relation at constant
$\hat{\Phi},N$. It turns out that, for $\hat{\Phi}$ below the threshold value

$\displaystyle\hat{\Phi}_{\rm
thr}=\left(\frac{1}{2L^{3}}\right)^{1/2}=\sqrt{15}\hat{\Phi}_{c}>0,$ (33)

there is a single minimum which can be determined by
$\displaystyle\left(\frac{\partial T}{\partial S}\right)_{\Phi,N}=0$, with the
coordinate at the minimum given by

$\displaystyle S_{\rm
min}=\frac{\pi\ell^{3}N\left(1-2L^{3}\hat{\Phi}^{2}\right)^{3/2}}{3\sqrt{2}L^{3}},\quad
T_{\rm min}=\frac{\sqrt{2(1-2L^{3}\hat{\Phi}^{2})}}{\pi\ell}.$ (34)

Introducing the relative parameters $\tilde{s}=S/S_{\rm
min},\tilde{t}={T}/{T_{\rm min}}$, the isovoltage $T-S$ relation can be recast
in the form

$\tilde{t}=\frac{{\tilde{s}}^{2/3}+1}{2\sqrt[3]{\tilde{s}}}.$

It is remarkable that the rescaled isovoltage $T-S$ relation does not depend
on the potential $\hat{\Phi}$ at all provided $\hat{\Phi}<\hat{\Phi}_{\rm
thr}$.

For $\hat{\Phi}\geq\hat{\Phi}_{\rm thr}$, however, the minimum on the
isovoltage $T-S$ curve disappears and $T$ is monotonically increasing with
$S$, which starts at some nonvanishing zero point value which corresponds to
$T=0$. The nonvanishing zero point value of $S$ corresponds to an extremal
black hole remnant.

Similar considerations can also be applied in considering the isovoltage
$\mu-T$ relation, where the expression for $\mu$ can be inferred from the
Euler relation (15). It turns out that the isovoltage $\mu-T$ relation is
branched for $\hat{\Phi}<\hat{\Phi}_{\rm thr}$, and the chemical potential
$\mu_{\rm lower}$ in the lower branch becomes zero at the temperature

$T_{0}=\frac{3\sqrt{1-2L^{3}\hat{\Phi}^{2}}}{2\pi\ell},$

i.e. $\mu_{\rm lower}(T_{0})=0$. Meanwhile, at the temperature $T_{0}$, the
chemical potential $\mu_{\rm upper}$ in the upper branch takes the value

$\mu_{\rm
upper}(T_{0})=\frac{\ell^{2}\left(1-2L^{3}\hat{\Phi}^{2}\right)^{2}}{32L^{3}}.$

Therefore, we can take the liberty to define the rescaled temperature $\tau$
and chemical potential $m$ as

$\tau=\frac{T}{T_{0}},\quad\nu=\frac{\mu}{\mu_{\rm upper}(T_{0})}.$

Using these rescaled parameters, the isovoltage $\mu-T$ relations can be cast
in the following simple form,

$\displaystyle\nu$
$\displaystyle=\pm\frac{1}{8}\left(\sqrt{9\tau^{2}-8}-3\tau\right)^{2}\left[\sqrt{9\tau^{2}-8}\tau\mp(3\tau^{2}-4)\right],$
(35)

where the upper/lower signs correspond to the upper/lower branch of the
$\mu-T$ relation.

For $\hat{\Phi}\geq\hat{\Phi}_{\rm thr}$, the branched behavior for $\mu$
disappears and $\mu$ becomes identically negative and is monotonically
decreasing with $T$. Such behavior is similar to ordinary ideal gas, which
looks less interesting than the case $\hat{\Phi}<\hat{\Phi}_{\rm thr}$.

Figure 3: Isovoltage $T-S$ and $\mu-T$ curves

The isovoltage $T-S$ and $\mu-T$ curves at $\hat{\Phi}<\hat{\Phi}_{\rm thr}$
are shown in Fig.3. It can be seen from the plots that for each $T>T_{\rm
min}$, there are two black hole states, and the one with bigger entropy is
thermodynamically preferred, because only for the large black hole state with
bigger entropy the temperature and the entropy is positively correlated,
implying a positive heat capacity. Unlike the isocharge processes described
earlier, there is no phase equilibrium condition for the transition from the
unstable small black hole to the stable large black hole. Another point to be
noticed is that, in the absence of Coulomb potential, the zero $T_{0}$ for the
chemical potential corresponds to the famous Hawking-Page phase transition
[38], which is a phase transition from an AdS black hole to a thermal gas. The
Hawking-Page temperature is found to be

$T_{\rm HP}=T_{0}|_{\hat{\Phi}=0}=\frac{3}{2\pi\ell}.$

One can check that the radius of the black hole event horizon at the Hawking-
Page temperature is precisely the AdS radius, i.e. $r_{h}(T_{\rm HP})=\ell$.
In the presence of nontrivial Coulomb potential, the zero for the chemical
potential still appears, however it is not clear what physical state does the
zero $\mu_{\rm lower}(T_{0})=0$ correspond.

Figure 4: Adiabatic and isothermal $\hat{\Phi}-\hat{Q}$ curves

Next let us consider the $\hat{\Phi}-\hat{Q}$ processes. There are two
possible conditions for such processes, i.e. adiabatic and isothermal, with
the corresponding curves presented in Fig.4. The adiabatic
$\hat{\Phi}-\hat{Q}$ processes are quite simple. It can be seen from eq.(31)
that in the adiabatic processes, $\hat{\Phi}$ is always proportional to
$\hat{Q}$. The only thing to be noticed is that at any constant $S$, there is
an upper bound for $\hat{Q}$ according to eq.(32). The situation for the
isothermal $\hat{\Phi}-\hat{Q}$ processes is, however, more complicated. One
needs to consider $t$ as an implicit parameter in the equation of states (31)
and consequently the isothermal $\hat{\Phi}-\hat{Q}$ curves are not monotonic
and even not always single-valued. At sufficiently low temperatures the
isothermal $\hat{\Phi}-\hat{Q}$ relation is single-valued and possesses a
single minimum at nonzero $\hat{Q}$, which is given by the condition
$\left(\frac{\partial\hat{\Phi}}{\partial\hat{Q}}\right)_{T}=0$. In terms of
the relative parameters, the minimum is located at

$\displaystyle q_{\rm
min}=\frac{36\sqrt{15}t^{2}\sqrt{25-24t^{2}}}{125},\quad\phi_{\rm
min}=\dfrac{\sqrt{15}}{5}\sqrt{25-24\,t^{2}}.$ (36)

These can be easily translated into the original $\hat{Q},\hat{\Phi}$ values
by multiplication with $Q_{c}$ and $\hat{\Phi}_{c}$, i.e.

$\displaystyle\hat{Q}_{\rm min}=q_{\rm min}Q_{c},\quad\hat{\Phi}_{\rm
min}=\phi_{\rm min}\hat{\Phi}_{c}.$ (37)

When $t=1$, the isovoltage $\hat{\Phi}-\hat{Q}$ curve has a single vertical
tangent line. For $1<t<\bar{t}\equiv\left(\frac{25}{24}\right)^{1/2}$, the
$\hat{\Phi}-\hat{Q}$ relation becomes multivalued and each of the
corresponding curves possesses two vertical tangent lines. If $t\geq\bar{t}$,
all the $\hat{\Phi}-\hat{Q}$ curves are still multivalued and all such curved
will intersect at $(\hat{Q},\hat{\Phi})=(0,0)$. Another interesting feature of
the isovoltage $\hat{\Phi}-\hat{Q}$ curves which is worth to be noticed is
that, for any $T>0$, the corresponding curves always intersect at
$(\hat{Q},\hat{\Phi})=(0,\hat{\Phi}_{\rm thr})$, where $\hat{\Phi}_{\rm thr}$
is defined in eq.(33). The reason for the appearance of this intersection
point remains unclear to us.

It remains to consider the $\mu-N$ relationship in this concrete model. By use
of the Euler relation (15), the chemical potential can be written as

$\mu=\frac{\ell^{2}\left[3(12\pi^{2})^{1/3}LN^{2/3}S^{4/3}-2\pi^{2}\hat{Q}^{2}\right]-9L^{3}S^{2}}{12({18\pi^{4}})^{1/3}\ell^{2}L^{2}S^{2/3}N^{4/3}}.$

It can be shown that the $\mu-N$ curve with fixed $S,~{}\hat{Q}$ does not
contain any inflection point but rather has a maximum, with the parameters at
the maximum given by

$\displaystyle N_{\rm
max}=\frac{\sqrt{2}\left(2\pi^{2}\ell^{2}\hat{Q}^{2}+9L^{3}S^{2}\right)^{3/2}}{9\pi\ell^{3}L^{3/2}S^{2}},\quad\mu_{\rm
max}=\frac{3\ell^{2}S^{2}}{16\pi^{2}\ell^{2}\hat{Q}^{2}+72L^{3}S^{2}}.$ (38)

Therefore, in terms of the relative parameters $u=\mu/\mu_{\rm
max},~{}n=N/N_{\rm max}$, the $\mu-N$ equations of states can be written as

$\displaystyle u=\frac{2n^{2/3}-1}{n^{4/3}}.$ (39)

Once again, only a single relative parameter $n$ appears on the right hand
side, which can be understood as the law of corresponding states at an
enhanced level. The corresponding $\mu-N$ curve is depicted in Fig.5.

Figure 5: $\mu-N$ curve

Summarizing this subsection, we have found that the thermodynamic behavior for
the charged AdS black hole solution of the $(5,1)$-model in the RPST formalism
is quantitatively slightly different from that of the four dimensional RN-AdS
black hole, however the qualitative behaviors are basically identical. The
RPST behaviors for the black holes are similar to that of the ordinary matter
in some respects, e.g. the form of the first law and the Euler homogeneity
behaviors are similar, however there are also some significant differences,
e.g. the appearance of supercritical equilibrium phase transitions in the
isocharge processes, the non-equilibrium phase transitions in the isovoltage
processes and the existence of the yet-to-be-understood charged states with
vanishing chemical potential, etc. These behaviors seems to be universal for
different black hole solutions in Einstein gravity.

####  3.1.2 Process parameters

In the study of thermodynamic behaviors of ordinary matter, the parameters
that characterize certain physical processes are important objects. The
relevant parameters include the isochoric heat capacity $C_{V}$, the isobaric
expansion coefficient $\alpha_{P}$, the isochoric pressure coefficient
$\beta_{V}$ and the isothermal compression coefficient $\kappa_{T}$, etc.

In the case of charged black holes, we can introduce analogous concepts and
study their behaviors. Remember that, instead of $P-V-T$ systems, the charged
black holes are $\hat{\Phi}-\hat{Q}-T$ systems. Therefore, the analogy to
isochoric heat capacity $C_{V}$ is now changed into

$C_{\hat{Q}}\equiv\left(\frac{\partial M}{\partial
T}\right)_{\hat{Q}}=T\left(\frac{\partial S}{\partial T}\right)_{\hat{Q}},$

while the analogies to $\alpha_{P}$, $\beta_{V}$ and $\kappa_{T}$ are given,
respectively, as

$\displaystyle\alpha_{\hat{\Phi}}\equiv\frac{1}{{\hat{Q}}}\left(\frac{\partial\hat{Q}}{\partial
T}\right)_{\hat{\Phi}},\quad\beta_{\hat{Q}}\equiv\frac{1}{\hat{\Phi}}\left(\frac{\partial\hat{\Phi}}{\partial
T}\right)_{\hat{Q}},\quad\kappa_{T}\equiv-\frac{1}{{\hat{Q}}}\left(\frac{\partial\hat{Q}}{\partial\hat{\Phi}}\right)_{T}.$
(40)

Clearly, these parameters obey the following simple identity,

$\alpha_{\hat{\Phi}}=\hat{\Phi}\beta_{\hat{Q}}\kappa_{T}.$

Figure 6: Isocharge $c_{\hat{Q}}-T$ curves

We also introduce the isocharge specific heat capacity $c_{\hat{Q}}$ as

$c_{\hat{Q}}=C_{\hat{Q}}/N,$

because this object is scale independent and can be expressed in terms of the
parameters given in eq.(27) as

$c_{\hat{Q}}=\frac{2\pi\ell^{3}s\left(-q^{2}+15s^{4/3}+10s^{2}\right)}{15\sqrt{3}L^{3}\left(q^{2}-3s^{4/3}+2s^{2}\right)}.$

The dependence on $s$ is explicit, while dependence on $t$ is implicit by use
of eq. (28).

The $c_{\hat{Q}}-T$ behavior is shown in Fig.6. It can be seen that the
isocharge $c_{\hat{Q}}-T$ curves for $\hat{Q}<\hat{Q}_{c}$ always have two
divergent points, each corresponds to one of the two extrema on the $T-S$
curve as shown in Fig.2, and $c_{\hat{Q}}$ can take negative values between
the two singularities. This indicates that first order phase transitions can
happen between the two singularities. At $\hat{Q}=\hat{Q}_{c}$, the two
singularities merge together, the negative-valued branch of $c_{\hat{Q}}$
cease to appear, and the phase transition becomes critical. For
$\hat{Q}>\hat{Q}_{c}$, the isochargge heat capacity no longer diverge at
finite $T$, indicating that there is no $T-S$ phase transitions in such cases.
In the high temperature limit, $c_{\hat{Q}}$ becomes independent of $\hat{Q}$,
with asymptotic values given by

$\displaystyle\lim_{t\rightarrow\infty}c_{\hat{Q}}\sim\frac{384\sqrt{3}\pi}{125}\left(\frac{\ell}{L}\right)^{3}t^{3}.$
(41)

The parameter $\alpha_{\hat{\Phi}}$ is naturally associated with isovoltage
processes. We will be particularly interested in the cases with
$\hat{\Phi}<\hat{\Phi}_{\rm thr}$, in which the minimum on the isovoltage
$T-S$ curve exist and the chemical potential versus temperature curve is
branched. The analytical expression for $\alpha_{\hat{\Phi}}$ reads

$\displaystyle\alpha_{\hat{\Phi}}=\frac{20\sqrt{3}\pi\ell}{\phi^{2}+10s^{2/3}-15}=\frac{4\tilde{s}^{1/3}}{T_{\rm
min}(\tilde{s}^{2/3}-1)},$ (42)

where the first expression is applicable to any admissible $\hat{\Phi}$, while
the last expression is applicable only for $\hat{\Phi}<\hat{\Phi}_{\rm thr}$,
because otherwise $T_{\rm min}$ will be either zero or undefined. The
parameter $\beta_{\hat{Q}}$ is associated with isocharge processes. Therefore
we adopt the relative parameters $s,t$ to evaluate $\beta_{\hat{Q}}$,

$\displaystyle\beta_{\hat{Q}}$ $\displaystyle=-\frac{4\sqrt{3}\pi\ell
s^{5/3}}{q^{2}-3s^{4/3}+2s^{2}}.$ (43)

As for the parameter $\kappa_{T}$, it is naturally associated with isothermal
processes. $\kappa_{T}$ can be written as an explicit function in $(s,t)$,

$\displaystyle\kappa_{T}$
$\displaystyle=-\frac{15\sqrt{30}L^{3/2}\left(s^{2/3}-2s^{1/3}t+1\right)}{s^{1/3}\left(5s^{1/3}-6t\right)\sqrt{10s^{2/3}-24s^{1/3}t+15}}.$
(44)

However from the definition, it is better to think of $\kappa_{T}$ as a
function in $(t,\phi)$. Therefore the variable $s$ in the above equation
should be regarded as an implicit function in $(t,\phi)$ by use of eqs.(28)
and (31).

Figure 7: The $\alpha_{\hat{\Phi}}-T$ and $\beta_{\hat{Q}}-T$ curves

The plots of the parameters $\alpha_{\hat{\Phi}}$ and $\beta_{\hat{Q}}$ versus
$T$ are presented in Fig.7. The singularity of $\alpha_{\hat{\Phi}}$
corresponds to the minimum of the isovoltage $T-S$ curve. The two
singularities of $\beta_{\hat{Q}}$ at $\hat{Q}<\hat{Q}_{c}$ corresponds to the
two extrema on the isocharge $T-S$ curve, which merge together at
$\hat{Q}=\hat{Q}_{c}$.

Figure 8:  $\kappa_{T}-\hat{\Phi}$ curves for $t<\bar{t}$ and for
$t\geq\bar{t}$

The plots for the parameter $\kappa_{T}$ versus $\hat{\Phi}$ are more subtle,
and we present the result separately in Fig.8. The left picture in Fig.8
contains only the curves with $t<\bar{t}$, wherein we intentionally rescaled
$\hat{\Phi}$ with $\hat{\Phi}_{\rm min}$ in order to indicate that the
divergence of $\kappa_{T}$ in such cases corresponds to the minimum
$\hat{\Phi}_{\rm min}$ appeared in the isovoltage $\hat{\Phi}-\hat{Q}$ curves.
In such cases, the $\kappa_{T}$ values are branched, wherein the negative
branch corresponds to the part of the part of the isothermal
$\hat{\Phi}-\hat{Q}$ curve at $\hat{Q}<\hat{Q}_{\rm min}$, and the positive
branch corresponds to the part of the part of the isothermal
$\hat{\Phi}-\hat{Q}$ curve at $\hat{Q}>\hat{Q}_{\rm min}$. The right picture
in Fig.8 contains only the curves with $t\geq\bar{t}$, in which cases
$\hat{\Phi}_{\rm min}$ is either zero or does not exist. Therefore the curves
in these cases are created by scaling $\hat{\Phi}$ with $\hat{\Phi}_{c}$. It
should be noticed that, for $t\geq\bar{t}$, $\kappa_{T}$ does not have any
singularity at nonzero $\hat{\Phi}$. That the $\kappa_{T}-\hat{\Phi}$ curves
are still branched is due to the fact that the isothermal $\hat{\Phi}-\hat{Q}$
relation is already multivalued, as shown on the right plots in Fig.4.

####  3.1.3 Scaling properties near criticality and critical exponents

Scaling properties near the critical point is important in distinguishing
different universality classes in thermodynamics. In the present model, the
only critical point appears in the isocharge $T-S$ process. In order to
compare with the $P-V$ criticalities appearing in systems consisted of
ordinary matter, we need to make a list of parameter correspondences. The
parameters $P,V,T$ for ordinary matter corresponds to $T,S,\hat{Q}$ in the
present case in exactly the same order. This means that when considering
critical exponents, the isochoric heat capacity $C_{V}$ for ordinary matter
corresponds not to $C_{\hat{Q}}$ but rather to
$\kappa_{S}^{-1}\equiv-{\hat{Q}}\left(\frac{\partial\hat{\Phi}}{\partial{\hat{Q}}}\right)_{S}$,
where $\kappa_{S}$ is similar to $\kappa_{T}$ defined in eq.(40), but is
associated with adiabatic processes. Similarly, we expect that the scaling
behavior of the isothermal compression coefficient $\kappa_{T}$ (not the one
defined in eq.(40)) for ordinary matter corresponds to that of $C_{\hat{Q}}$
in our case, etc. Below we shall present the detailed studies.

Writing

$\displaystyle s=S/S_{c}\equiv 1+\delta s,\quad t=T/T_{c}\equiv 1+\delta
t,\quad q={\hat{Q}}/{\hat{Q}}_{c}\equiv 1+\delta q,$ (45)

we get from eq.(28) the following relation,

$\displaystyle\delta t=\frac{5}{162}(\delta s)^{3},$ (46)

which means

$\displaystyle|T-T_{c}|\sim|S-S_{c}|^{\delta},\quad\delta=3.$ (47)

Near the critical point [39], we have

$\displaystyle t(q,1+\delta s)=t(q,1-\delta s),$ (48)

which is implied by the fact that the critical phase transition is approached
via an isocharge isothermal process. The above identity implies

$\displaystyle\delta q=-\frac{2}{9}(\delta s)^{2},$ (49)

i.e.

$\displaystyle|S_{\rm large}-S_{\rm
small}|\sim\left|{\hat{Q}}-{\hat{Q}}_{c}\right|^{\beta},\quad\beta=\frac{1}{2}.$
(50)

Using the definition of the isocharge heat capacity, we get

$\displaystyle C_{\hat{Q}}$ $\displaystyle=T\left(\frac{\partial S}{\partial
T}\right)_{\hat{Q}}\propto t\left(\frac{\partial\delta s}{\partial\delta
t}\right)_{\delta q}\propto(\delta t)^{-{2}/{3}}\propto(\delta
s)^{-2}\propto-(\delta q)^{-1},$ (51)

When the charge approaches the critical value from below (i.e. $\delta q\to
0^{-}$), we have $C_{\hat{Q}}\rightarrow\infty$. Thus we get the third
critical exponent $\gamma=1$.

Now we turn to look at the behavior of
$\kappa_{S}^{-1}=-{\hat{Q}}\left(\frac{\partial\hat{\Phi}}{\partial{\hat{Q}}}\right)_{S}$.
Since $\hat{\Phi}=\left(\frac{\partial M}{\partial{\hat{Q}}}\right)_{S}$, we
have

$\displaystyle\kappa_{S}^{-1}=-{\hat{Q}}\left(\frac{\partial^{2}M}{\partial{\hat{Q}}^{2}}\right)_{S}=-\frac{M_{c}}{{\hat{Q}}_{c}}q\left(\frac{\partial^{2}m}{\partial
q^{2}}\right)_{s}.$ (52)

When $q\rightarrow 1^{+}$, we can make use of the caloric equation of states
(29) to get

$\displaystyle\kappa_{S+}^{-1}=-\frac{1}{\sqrt{30}L^{3/2}}\propto(\delta
q)^{0}.$ (53)

When $q\rightarrow 1^{-}$, the caloric equation of states (29) is no longer
reliable, so we adopt Maxwell’s linear interpolation [40]

$\displaystyle\bar{m}(s,q)=m(s_{\rm small},q)+(s-s_{\rm small})\frac{m(s_{\rm
large},q)-m(s_{\rm small},q)}{s_{\rm large}-s_{\rm small}},$ (54)

where $m(s,q)$ on the right hand side is given by the original caloric
equation of state (29). When approaching the critical point, we have

$\epsilon\equiv\frac{1}{2}(s_{\rm large}-s_{\rm small})=\delta
s=\frac{3\sqrt{2}}{2}(-\delta q)^{1/2}=\frac{3\sqrt{2}}{2}(1-q)^{1/2}.$

Thus we can expand the right hand side of eq.(54) into power series in
$\epsilon$ near the critical point and get

$\bar{m}(s,q)|_{s=1}=m(s,q)|_{s=1}+\frac{\epsilon^{2}}{2!}\left(\frac{\partial^{2}m}{\partial
s^{2}}\right)_{s=q=1}+\frac{\epsilon^{4}}{4!}\left(\frac{\partial^{4}m}{\partial
s^{4}}\right)_{s=q=1}+\cdots.$

Notice that the $q$ dependence of the second and the third terms are encoded
in $\epsilon$. Finally, using eq.(52), we get

$\displaystyle\kappa_{S-}^{-1}$
$\displaystyle=-\frac{M_{c}}{{\hat{Q}}_{c}}q\left(\frac{\partial^{2}\bar{m}}{\partial
q^{2}}\right)_{s=1}=-\frac{M_{c}}{{\hat{Q}}_{c}}q\left(\frac{\partial^{2}m}{\partial
q^{2}}\right)_{s=1}-\frac{27M_{c}}{16{\hat{Q}}_{c}}q\left(\frac{\partial^{4}m}{\partial
s^{4}}\right)_{s=q=1}$
$\displaystyle=-\frac{1}{\sqrt{30}L^{3/2}}-\frac{\sqrt{30}}{12L^{{3/2}}}\propto(\delta
q)^{0}.$ (55)

Combining eqs.(53) and (55) we find that

$\kappa^{-1}_{S\pm}\propto(\delta q)^{-\alpha_{\pm}},\quad\alpha_{\pm}=0,$

and there is a finite jump between $\kappa^{-1}_{S\pm}$:

$\kappa^{-1}_{S+}-\kappa^{-1}_{S-}=\frac{\sqrt{30}}{12L^{{3/2}}}.$

This jump arises because of the existence of a discontinuity in the second
order derivatives of $M$ with respect to $\hat{Q}$, i.e.
$\left(\frac{\partial^{2}M}{\partial{\hat{Q}}^{2}}\right)\big{|}_{{\hat{Q}}\rightarrow{\hat{Q}}_{c}^{-}}\neq\left(\frac{\partial^{2}M}{\partial{\hat{Q}}^{2}}\right)\big{|}_{{\hat{Q}}\rightarrow{\hat{Q}}_{c}^{+}}$,
which indicates that the critical point corresponds to a second order phase
transition in terms oh Ehrenfest classification.

It may be illustrative to summarize the above results about the critical
exponents in conjunction with a comparison with the corresponding scaling
relations for ordinary matter. Explicitly, we have found a perfect
correspondence between the scaling relations for ordinary matter and for the
charged AdS black hole,

$\left\\{\begin{array}[]{l}C_{V}\propto{\begin{array}[]{l}|T-T_{c}|^{-\alpha^{-}}\\\
|T-T_{c}|^{-\alpha^{+}}\end{array}},\\\
\rho_{l}-\rho_{g}\propto|T-T_{c}|^{\beta},\\\
\kappa_{T}\propto|T-T_{c}|^{-\gamma},\\\
|P-P_{c}|\propto|\rho-\rho_{c}|^{\delta},\end{array}\right.\Longleftrightarrow\left\\{\begin{array}[]{l}\kappa_{S}^{-1}\propto{\begin{array}[]{l}|{\hat{Q}}-{\hat{Q}}_{c}|^{-\alpha^{-}}\\\
|{\hat{Q}}-{\hat{Q}}_{c}|^{-\alpha^{+}}\end{array}},\\\ S_{\rm large}-S_{\rm
small}\propto|T-T_{c}|^{\beta},\\\
C_{\hat{Q}}\propto|{\hat{Q}}-{\hat{Q}}_{c}|^{-\gamma},\\\
|T-T_{c}|\propto|S_{\rm large}-S_{\rm small}|^{\delta},\end{array}\right.$

where $\alpha_{\pm}=0$, $\beta=\frac{1}{2}$, $\gamma=1$ and $\delta=3$.
Naturally, they obey the same scaling symmetry as in ordinary thermodynamic
systems [41],

$2-\alpha=2\beta+\gamma=\beta(\delta+1)=\gamma(\delta+1)/(\delta-1)$

###  3.2 $(5,2)$-model: CS gravity coupled to Maxwell field

The $(5,2)$-model is a CS like model for which the total action takes the form

$\displaystyle\mathcal{A}_{(5,2)}$
$\displaystyle=\frac{1}{6\pi^{2}\ell^{2}G_{(5,2)}}\int\left(R+\frac{6}{\ell^{2}}+\frac{\ell^{2}}{4}\mathcal{R}^{2}\right)\sqrt{-g}{\rm
d}^{5}x-\frac{1}{8\pi^{2}}\int F_{\mu\nu}F^{\mu\nu}\sqrt{-g}{\rm d}^{5}x,$
(56)

where

$\mathcal{R}^{2}=R_{\mu\nu\gamma\delta}R^{\mu\nu\gamma\delta}-4R_{\mu\nu}R^{\mu\nu}+R^{2}.$

The variation of the total action reads

$\delta\mathcal{A}_{(5,2)}=E^{g}_{\mu\nu}\delta g^{\mu\nu}+E^{\mu}_{A}\delta
A_{\mu}+{\rm d}\bm{\Theta},$

where

$\displaystyle
E^{g}_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-\frac{3}{\ell^{2}}g_{\mu\nu}+\frac{\ell^{2}}{4}H_{\mu\nu}-3\pi^{2}\ell^{2}G_{(5,2)}T_{\mu\nu},$
(57) $\displaystyle E^{\mu}_{A}=\nabla_{\nu}F^{\mu\nu},$ (58)

with

$T_{\mu\nu}=\frac{1}{2\pi^{2}}(F_{\mu\rho}F_{\nu}^{\rho}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}),$
$H_{\mu\nu}=2\left(R_{\mu\lambda\rho\sigma}R_{\nu}{}^{\lambda\rho\sigma}-2R_{\mu\rho\nu\sigma}R^{\rho\sigma}-2R_{\mu\sigma}R_{\nu}^{\sigma}+RR_{\mu\nu}\right)-\frac{1}{2}\left(R_{\mu\nu\gamma\delta}R^{\mu\nu\gamma\delta}-4R_{\mu\nu}R^{\mu\nu}+R^{2}\right)g_{\mu\nu}.$

$\bm{\Theta}$ is a $\delta g^{\mu\nu}$-dependent four form field which encodes
the on-shell Noether current if $\delta g^{\mu\nu}$ is replaced by the
transformation of $g^{\mu\nu}$ under diffeomorphism symmetries. The associated
Noether charge can be used to evaluate the Wald entropy, which reads

$S=\frac{4\pi r_{h}\left(3\sqrt{1+2G_{(5,2)}M-\frac{G_{(5,2)}Q^{2}}{\pi
r_{h}^{2}}}-\frac{2r_{h}^{2}}{\ell^{2}}\right)}{3G_{(5,2)}}.$

The $\hat{Q},\hat{\Phi}$ and $\mu,N$ variables can be inferred from the
generic definitions given in Sec.2 by inserting $d=5,k=2$ and the thermal and
caloric equations of states for this particular case are worked out to be

$\displaystyle T$
$\displaystyle=\dfrac{\pi\mathcal{Z}\left[\dfrac{\left(\left(\ell^{2}\mathcal{Z}\right)^{2/3}-4\pi^{2/3}\ell^{2}\right)^{4}}{4\pi^{4/3}\left(\ell^{2}\mathcal{Z}\right)^{4/3}}-\ell^{2}L\mathcal{Q}^{2}+\dfrac{\left(\left(\ell^{2}\mathcal{Z}\right)^{2/3}-4\pi^{2/3}\ell^{2}\right)^{6}}{16\pi^{2}\ell^{6}\mathcal{Z}^{2}}\right]}{\left[\left(\ell\mathcal{Z}\right)^{2/3}-4\pi^{2/3}\ell^{4/3}\right]^{3}\left[\frac{1}{4}\sqrt[3]{\pi}\left(\ell^{2}\mathcal{Z}\right)^{2/3}+\pi\ell^{2}\left(4\pi^{2/3}\mathcal{Z}^{-2/3}-1\right)\right]},$
(59) $\displaystyle M$
$\displaystyle=N\dfrac{(\pi\mathcal{Z})^{2/3}\left[\dfrac{\left(\left(\ell^{2}\mathcal{Z}\right)^{2/3}-4\pi^{2/3}\ell^{2}\right)^{6}}{\pi^{2}\ell^{4}\mathcal{Z}^{2}}+\dfrac{8\left(\left(\ell^{2}\mathcal{Z}\right)^{2/3}-4\pi^{2/3}\ell^{2}\right)^{4}}{\pi^{4/3}\ell^{2/3}\mathcal{Z}^{4/3}}+32\ell^{4}L\mathcal{Q}^{2}\right]}{32\ell^{16/3}L\left(\mathcal{Z}^{2/3}-4\pi^{2/3}\ell^{2/3}\right)^{2}},$
(60)

where

$\mathcal{Z}=\sqrt{64\pi^{2}\ell^{2}+9L^{2}\mathcal{S}^{2}}+3L\mathcal{S},\quad\mathcal{S}=S/N,\quad\mathcal{Q}=Q/N.$

These equations of states indicate that $T$ and $M$ are respectively zeroth
and first order homogeneous functions in the extensive variables. Moreover,
non-negativeness of the temperature gives rise to an upper bound for
$\hat{Q}$, which reads

$\hat{Q}\leq\frac{\left[(\ell^{2}N^{2}\mathcal{Z})^{2/3}-4\pi^{2/3}\ell^{2}N^{2}\right]\left[16\pi^{4/3}\ell^{2}N^{2}-4\pi^{2/3}(\ell
N)^{4/3}\mathcal{Z}^{2/3}+(\ell
N)^{2/3}\mathcal{Z}^{4/3}\right]^{1/2}}{4\pi\sqrt{L}\ell^{3}N^{3}\mathcal{Z}},$

which is to be understood as the Bogomol’nyi bound for the present case.

The overwhelming complicated form of eq.(59) prevents us from analyzing the
thermodynamic processes right away from the thermal equation of states. To
proceed, we choose to rewrite the thermal equation of states
$T=T(S,\hat{Q},N)$ in a parametric from

$\displaystyle T(r_{h},\hat{Q},N)$
$\displaystyle=\frac{-\frac{\ell^{2}L\hat{Q}^{2}}{N^{2}}+\frac{4r_{h}^{6}}{\ell^{2}}+4r_{h}^{4}}{8\pi
r_{h}^{3}\left(\ell^{2}+r_{h}^{2}\right)},$ (61) $\displaystyle S(r_{h},N)$
$\displaystyle=\frac{4\pi}{L}Nr_{h}\left(1+\frac{r_{h}^{2}}{3\ell^{2}}\right),$
(62)

where the radius $r_{h}$ of the black hole event horizon is taken as an
intermediate parameter, while the dependence on the thermodynamic parameters
$\hat{Q}$ and $N$ are made explicit by employing the simple relations

$G_{(5,2)}=\frac{L}{N}\qquad Q=\frac{\hat{Q}}{\sqrt{N}}.$

Notice that both $T(r_{h},\hat{Q},N)$ and $S(r_{h},\hat{Q},N)$ are single-
valued in $r_{h}$ at constant $\hat{Q}$ and $N$.

The above parametric form of the thermal equation of states allows us to
analyze the isocharge processes of the model. By straightforward calculations,
we can get

$\displaystyle\left(\frac{\partial T}{\partial
S}\right)_{\hat{Q}}=\frac{\displaystyle\left(\frac{\partial T}{\partial
r_{h}}\right)_{\hat{Q},N}}{\displaystyle\left(\frac{\partial S}{\partial
r_{h}}\right)_{\hat{Q},N}}=\frac{L\left[\ell^{4}\left(4N^{2}r_{h}^{4}+5L\hat{Q}^{2}r_{h}^{2}\right)+8N^{2}\ell^{2}r_{h}^{6}+4N^{2}r_{h}^{8}+3\ell^{6}L\hat{Q}^{2}\right]}{32\pi^{2}N^{3}r_{h}^{4}\left(\ell^{2}+r_{h}^{2}\right)^{3}}.$
(63)

Though the expression still looks complicated, it can be easily seen that
$\displaystyle\left(\frac{\partial T}{\partial S}\right)_{\hat{Q}}$ is
identically positive. This means that $\displaystyle
C_{\hat{Q}}=T\left(\frac{\partial S}{\partial T}\right)_{\hat{Q}}>0$, which
means that there is only a single stable AdS black hole phase in the isocharge
processes for the present model, no $T-S$ phase transition could occur. It can
be shown that at high temperatures, the asymptotic value of
$c_{\hat{Q}}=C_{\hat{Q}}/N$ for this model is

${\color[rgb]{0,0,0}\lim_{t\rightarrow\infty}c_{\hat{Q}}=\frac{32\pi^{4}\ell^{4}}{L}T^{3}}.$
(64)

In order to describe the isovoltage processes, we need to employ the relation
$\hat{Q}=2N\hat{\Phi}r_{h}^{2}$ to rearrange eq.(61) into the form

$\displaystyle T(r_{h},\hat{\Phi})$
$\displaystyle=r_{h}\frac{\ell^{2}+r_{h}^{2}-\ell^{4}L\hat{\Phi}^{2}}{2\pi\ell^{2}(\ell^{2}+r_{h}^{2})}.$
(65)

It is then easy to get

$\left(\frac{\partial T}{\partial
S}\right)_{\hat{\Phi}}=\frac{\displaystyle\left(\frac{\partial T}{\partial
r_{h}}\right)_{\hat{\Phi},N}}{\displaystyle\left(\frac{\partial S}{\partial
r_{h}}\right)_{\hat{\Phi},N}}=\frac{L\left[-\ell^{6}L\hat{\Phi}^{2}+\ell^{4}\left(L\hat{\Phi}^{2}r_{h}^{2}+1\right)+2\ell^{2}r_{h}^{2}+r_{h}^{4}\right]}{8\pi^{2}\hat{\Phi}\left(\ell^{2}+r_{h}^{2}\right)^{3}}.$

The equation $\displaystyle\left(\frac{\partial T}{\partial
S}\right)_{\hat{\Phi}}=0$ with the condition
$\displaystyle\left(\frac{\partial^{2}T}{\partial
S^{2}}\right)_{\hat{\Phi}}>0$ has a single solution

$r_{h}=r_{\rm
min}=\frac{\ell}{\sqrt{2}}\left[\ell\hat{\Phi}\left(\ell^{2}L^{2}\hat{\Phi}^{2}+8L\right)^{1/2}-\ell^{2}L\hat{\Phi}^{2}-2\right]^{1/2},$

however, the real-valuedness of $r_{\rm min}$ requires
$\hat{\Phi}\geq\frac{1}{\ell\sqrt{L}}$, which in turn implies that the minimum
at $r_{h}=r_{\rm min}$ corresponds to a non-positive temperature and is
physically irrelevant. For all states with $T>0$, it is not difficult to check
that $\displaystyle\left(\frac{\partial T}{\partial S}\right)_{\hat{\Phi}}$ is
always positive, therefore there is no isovoltage $T-S$ transitions in the
present model.

We can also write down the parametrized $\mu-T$ equation of states, which read

$\displaystyle\mu(r_{h},\hat{\Phi})$
$\displaystyle=-\frac{r_{h}^{2}\left[\ell^{4}\left(4L\hat{\Phi}^{2}r_{h}^{2}+6\pi\right)+7\pi\ell^{2}r_{h}^{2}+\pi
r_{h}^{4}-12\ell^{6}L\hat{\Phi}^{2}\right]}{6\pi\ell^{7}\left(\ell^{2}+r_{h}^{2}\right)},$
(66)

where implicit dependence on $T$ is implied by use of eq.(65). Then it is
straightforward to get

$\displaystyle\left(\frac{\partial\mu}{\partial
T}\right)_{\hat{\Phi}}=\frac{\displaystyle\left(\frac{\partial\mu}{\partial
r_{h}}\right)_{\hat{\Phi}}}{\displaystyle\left(\frac{\partial T}{\partial
r_{h}}\right)_{\hat{\Phi}}}=-\frac{4\pi\left(3\ell^{2}r_{h}+r_{h}^{3}\right)}{3\ell^{5}},$
(67)

which is identically negative at all $r_{h}>0$, which means that $\mu$ is
monotonically decreasing with $T$. From eq.(65) one can see that at any fixed
$T$, $\hat{\Phi}$ can be regarded as an implicit function in $r_{h}$. Thus one
can solve eq.(65) at $T=0$ as an equation for $\hat{\Phi}$ and substitute the
result into eq.(66) to get

$\mu|_{T=0}=-\frac{r_{h}^{2}\left[6(\pi-2)\ell^{2}+(4+\pi)r_{h}^{2}\right]}{6\pi\ell^{7}}<0.$

This result combined with the fact that
$\displaystyle\left(\frac{\partial\mu}{\partial T}\right)_{\hat{\Phi}}<0$
indicate that $\mu$ is always negative and thus has no zeros at positive $T$.
This excludes the possibility for the appearance of Hawking-Page like
transitions in the present model.

Summarizing this subsection, we have seen that the thermodynamic behavior of
the $(5,2)$-model is in sharp contrast to that of the $(5,1)$-model. No phase
transition of any type (be it isocharge, isovoltage or Hawking-Page like)
could occur in the $(5,2)$-model. We believe this is also the case in any CS
like or $(2k+1,k)$-models.

###  3.3 $(6,2)$-model: BI gravity coupled to Maxwell field

The full action for the $(6,2)$-model reads

$\displaystyle\mathcal{A}_{(6,2)}=\frac{3}{32\pi^{2}\ell^{2}G_{(6,2)}}\int\left(R+\frac{10}{\ell^{2}}+\frac{\ell^{2}}{12}\mathcal{R}^{2}\right)\sqrt{-g}{\rm
d}^{6}x-\frac{3}{32\pi^{2}}\int F_{\mu\nu}F^{\mu\nu}\sqrt{-g}{\rm d}^{6}x.$
(68)

Except for the different choices of coupling coefficients and the different
spacetime dimensions, the above action is quite similar to that of the
$(5,2)$-model.

The Wald entropy for the charged spherically symmetric AdS black hole solution
in this case reads

$S=\dfrac{\pi\left(\ell^{2}\sqrt{G_{(6,2)}\left(8Mr_{h}^{3}-\dfrac{2Q^{2}}{\pi}\right)}-r_{h}^{4}\right)}{G_{(6,2)}\ell^{2}},$

and the other thermodynamic parameters/functions can be inferred from the
general definition presented in Sec.2.

The thermal and caloric equations of states for this case can also be worked
out analytically, which appears to be much simpler than the case of
$(5,2)$-model,

$\displaystyle T$
$\displaystyle=\frac{5\mathcal{S}^{2}-14\pi^{1/2}\ell\mathcal{S}^{3/2}+13\pi\ell^{2}\mathcal{S}-4\pi^{3/2}\ell^{3}\mathcal{S}^{1/2}-\pi^{2}L^{2}\mathcal{Q}^{2}}{8\pi^{5/4}\ell^{3/2}\mathcal{S}^{1/2}(\mathcal{S}^{1/2}-\pi^{1/2}\ell)^{5/2}},$
(69) $\displaystyle M$
$\displaystyle=N\frac{\left(3\mathcal{S}^{2}-6\pi^{1/2}\ell\mathcal{S}^{3/2}+3\pi\ell^{2}\mathcal{S}+\pi^{2}L^{2}\mathcal{Q}^{2}\right)}{6\pi^{5/4}\ell^{3/2}L^{2}(\mathcal{S}^{1/2}-\pi^{1/2}\ell)^{3/2}},$
(70)

where $\mathcal{S}=L^{2}\frac{S}{N}+\pi\ell^{2}$ and
$\mathcal{Q}=\frac{\hat{Q}}{N}$. The isocharge $T-S$ curves may contain an
inflection point for $\hat{Q}$ below some critical value $\hat{Q}_{c}$, and
the values of various thermodynamic parameters/functions can be worked out
analytically. However, the analytical expressions for the critical parameters
are too complicated which do not deserve to be displayed here. As an
alternative, we present their approximate numerical values below,

$\displaystyle T_{c}$
$\displaystyle\approx\frac{0.1747}{\ell},\quad\quad~{}S_{c}\approx\frac{0.9058\ell^{2}}{L^{2}}N,$
(71) $\displaystyle\hat{\Phi}_{c}$ $\displaystyle\approx\frac{0.2406}{\ell
L},\quad\quad\,\displaystyle\hat{Q}_{c}\approx\frac{0.0358\ell^{2}}{L}N,$ (72)
$\displaystyle M_{c}$ $\displaystyle\approx\frac{0.2410\ell}{L^{2}}N,\quad
F_{c}\approx\frac{0.0828\ell}{L^{2}}N.$ (73)

Critical values of quantities which are intensive are simply constants and
those for extensive quantities are explicitly proportional to $N$, thus the
extensivity and Euler homogeneity are transparent.

The forthcoming analysis is in complete analogy to the case of $(5,1)$-model,
and we shall only present some of the plots for brevity.

Figure 9: Isocharge $T-S$ and $F-T$ curves for the $(6,2)$-model

The isocharge $T-S$ and $F-T$ curves are presented in Fig.9, and the isocharge
$c_{\hat{Q}}-T$ curves are presented in Fig.10. One can see that the curves
presented in Fig.9 are very similar to the curves shown in Fig.2 for the
$(5,1)$-model, and the curves in Fig.10 are very similar to those presented in
Fig.6. It is not surprising that the isocharge $T-S$ behavior for the
$(6,2)$-model is qualitatively identical to that of the $(5,1)$-model.

Figure 10: Isocharge $c_{\hat{Q}}-S$ and $c_{\hat{Q}}-T$ curves for the
$(6,2)$-model

At high temperatures, all $c_{\hat{Q}}-T$ curves in Fig.10 merge together,
with the asymptotic value

$\displaystyle\lim_{t\rightarrow\infty}c_{\hat{Q}}\approx\frac{7.4670\ell^{2}}{L^{2}}t^{4},$
(74)

where again $t=T/T_{c}$.

By use of numeric techniques, it is not difficult to see that the behavior of
the isovoltage $T-S$ processes in the $(6,2)$-model is also qualitatively
similar to the case of $(5,1)$-model and hence we omit the corresponding
plots.

We also studied the behaviors of the process parameters
$\alpha_{\hat{\Phi}},\beta_{\hat{Q}},\kappa_{T}$ and found that their
qualitative behaviors are similar to the case of $(5,1)$-model. However, due
to the more complicated form of the equations of states, the analytical
expressions for these parameters are more involved and does not deserve to be
displayed here. Even though, we found that the scaling properties of the
$(6,2)$-model coincide exactly with the case of $(5,1)$-model, in the sense
that the critical exponents are exactly the same. Therefore, we conclude that
the $(6,2)$-model and the $(5,1)$-model belong to the same universality class,
although they belong to different types of gravity models (BI and EH types
respectively).

##  4 Summary and discussions

We have shown that the RPST formalism is applicable to the large class of
$(d,k)$-models and yields the correct Euler and Gibbs-Duhem relations which
guarantee the extensivity of the relevant thermodynamic systems. The
applicability of the RPST formalism to the $(d,k)$-models makes us expect that
the formalism may be universally applicable to all models of gravity. One
reason to support this expectation lies in that, unlike the EPST formalism
which extends the space of macroscopic states for black holes by the inclusion
of cosmological constant or some other coupling coefficients which may be
either present or absent in different models of gravity, the RPST formalism
uses the Newton constant as an extra dimension in the space of macroscopic
states for black holes, and the Newton constant must be present in all models
of gravity and can be made as an overall factor of the total action by some
simple field rescaling (e.g. the rescaling made in eq.(13) can be understood
as a rescaling of the Maxwell field $A_{\mu}$.)

By choosing the $(5,1)$-, $(5,2)$\- and the $(6,2)$-models as representatives
of the classes of EH like, CS like and BI like gravity models coupled with
Maxwell field, we found that the $(5,1)$\- and the $(6,2)$-models have
basically the same thermodynamic behaviors, e.g. the appearance of the
supercritical isocharge $T-S$ phase transitions at temperatures above a
critical temperature, the existence of the isovoltage $T-S$ transitions above
some minimum temperature which do not subject to any equilibrium conditions,
the occurrence of Hawking-Page like transitions and the exactly same scaling
properties at the critical point, etc. These behaviors were also observed for
four dimensional RN black hole and Kerr-AdS black holes in our previous works
[21, 22], and it seems that the EH and BI like models belong to the same
universality class under the RPST formalism. The $(5,2)$-model, however,
behaves very differently. There is no phase transitions of any type in this
model, which indicates that the CS like models may belong to a different
universality class under the RPST formalism111The result of [34], which
studies the original Chern-Simons gravity, i.e. Einstein-Hilbert gravity in
(2+1)-dimensions, also supports this conclusion.. These results are a first
known application of the RPST formalism to the cases of higher curvature
gravity models. We expect that further applications of the RPST formalism to
other higher curvature gravity models may help to classify the possible
universality classes in all gravity models.

Regarding the possible phase transitions in the cases of EH and BI like
models, let us stress that, although there are similar near critical scaling
properties to the cases of Van der Waals liquid-gas phase transitions, the
$T-S$ phase transitions do not involve any changes in volume or pressure. The
occurrence of the $T-S$ transition is a simple consequence of the fact that
some of the thermodynamic parameters (e.g. the temperature) of the black holes
are not monotonic functions in the geometric parameters (e.g. the radius of
the event horizon). Therefore, if we characterize the black hole states using
the thermodynamic parameters, black holes of different geometric sizes may
appear to be in the same thermodynamic states, and that is why black hole
phase transitions could occur.

Among the various concrete behaviors of the example cases, the high
temperature asymptotic behavior of the isocharge heat capacity deserves a
special mention. In all three example cases, the high temperature asymptotic
values of $c_{\hat{Q}}$ behave as

$c_{\hat{Q}}\sim T^{d-2}.$

Remember that $d-2$ can be regarded as the “spatial dimension” of the black
hole event horizon regarded as a thermodynamic system. The high temperature
asymptotic behavior $c_{\hat{Q}}\sim T^{d-2}$ for black holes reminds us of
the low temperature asymptotic behavior of the isochoric specific heat
capacity $c_{V}$ for ordinary non-metallic solid matter, which is, according
to Debye’s theory, $c_{V}\sim T^{D}$, with $D$ being the spatial dimension of
the system. This similarity may not be a coincidence, there might be some deep
physics in behind.

## Acknowledgement

This work is supported by the National Natural Science Foundation of China
under the grant No. 11575088.

## References

  * [1] Bekenstein, J.D. “Black holes and the second law,” _Lett. Nuovo Cim._ 1972, 4(15):737–740.
  * [2] Bekenstein, J.D. “Black holes and entropy,” Phys. Rev. D 1973, 7(8):2333–2346.
  * [3] Bardeen, J.M.; Carter, B.; Hawking, S.W. “The four laws of black hole mechanics,” Comm. Math. Phys. 1973, 31(2):161–170.
  * [4] Hawking, S.W. “Particle creation by black holes,”  Comm. Math. Phys. 1975, 43:199–-220.
  * [5] Kastor, D.; Ray, S.; Traschen, J. “Enthalpy and the mechanics of AdS black holes,” Class. Quant. Grav. 2009, 26(19):195011, [arXiv:0904.2765].
  * [6] Dolan, B.P. “The cosmological constant and the black hole equation of state,” Class. Quant. Grav. 2010, 28(12):125020, [arXiv:1008.5023].
  * [7] Dolan, B.P. “Pressure and volume in the first law of black hole thermodynamics,” Class. Quant. Grav. 2011, 28(23):235017, [arXiv:1106.6260].
  * [8] Dolan, B.P. “Compressibility of rotating black holes,” Phys. Rev. D 2011, 84(12):127503, [arXiv:1109.0198].
  * [9] Kubizňák, D.; Mann, R.B. “P-V criticality of charged AdS black holes,” J. High Energy Phys. 2012, 7:1–25, [arXiv:1205.0559].
  * [10] Cai, R.G.; Cao, L.M.; Li, L; Yang, R.Q.; “PV criticality in the extended phase space of Gauss-Bonnet black holes in AdS space,” J. High Energy Phys. 2013, 9:1–22, [arXiv:1306.6233].
  * [11] Kubizňák, D; Mann, R.B.; Teo, M. “Black hole chemistry: thermodynamics with Lambda,” Class. Quant. Grav. 2017, 34(6):063001, [arXiv:1608.06147].
  * [12] Xu, W.; Zhao, L. “Critical phenomena of static charged AdS black holes in conformal gravity,” Phys. Lett. B 2014, 736:214–220, [arXiv:1405.7665].
  * [13] Xu, W.; Xu, H.; Zhao, L.; “Gauss–Bonnet coupling constant as a free thermodynamical variable and the associated criticality,” Euro. Phys. J. C 2014, 74(7):1–13, [arXiv:1311.3053].
  * [14] Lemos, J.P.S ; Zaslavskii, O.B. “Black hole thermodynamics with the cosmological constant as independent variable: Bridge between the enthalpy and the Euclidean path integral approaches,” Phys. Lett. B 2018, 786:296–299, [arXiv:1806.07910].
  * [15] Zhang, J.L.; Cai, R.G.; Yu, H.; “Phase transition and thermodynamical geometry of Reissner-Nordström-AdS black holes in extended phase space,” Phys. Rev. D 2015, 91(4):044028, [arXiv:1502.01428].
  * [16] Visser, M.R. “Holographic thermodynamics requires a chemical potential for color. Phys. Rev. D, 2022, 105(10):106014, [arXiv:2101.04145].
  * [17] Karch, A.; Robinson, B. “Holographic black hole chemistry,” J. High Energy Phys., 2015, 12:1–15, [arXiv:1510.02472].
  * [18] Cong, W.; Kubizňák, D.; Mann, R.B. “Thermodynamics of AdS black holes: critical behavior of the central charge’” Phys. Rev. Lett., 2021, 127(9):091301, [arXiv:2105.02223].
  * [19] Smarr, L. “Mass formula for Kerr black holes,” Phys. Rev. Lett., 1973, 30(2):71.
  * [20] Callen, H.B. “Thermodynamics and an Introduction to Thermostatistics,” 2nd Eds, John Willy & Sons 1985.
  * [21] Gao, Z.; Zhao, L. “Restricted phase space thermodynamics for AdS black holes via holography,” Class. Quant. Grav., 2021, 39:075019, [arXiv:2112.02386].
  * [22] Gao, Z.; Kong, X.; Zhao, L. “Thermodynamics of Kerr-AdS black holes in the restricted phase space,” Euro. Phys. J. C, 2022, 82(2):1–10, [arXiv:2112.08672].
  * [23] Wang, T.; Zhao, L. “Black hole thermodynamics is extensive with variable newton constant,” Phys. Lett. B, 2022, 827:136935, [arXiv:2112.11236].
  * [24] Zhao, L. “Thermodynamics for higher dimensional rotating black holes with variable Newton constant,” Chin. Phys. C, 2022, 46(5):055105, [arXiv:2201.00521].
  * [25] Gibbons, G.W.; Hawking, S.W. “Action integrals and partition functions in quantum gravity,” Phys. Rev. D, 1977, 15(10):2752–2756.
  * [26] York Jr, J.W. “Black-hole thermodynamics and the Euclidean Einstein action,” Phys. Rev. D, 1986, 33(8):2092.
  * [27] Gibbons, G.W.; Perry, M.J.; Pope, C.N. “The first law of thermodynamics for Kerr–anti-de Sitter black holes,” Class. Quant. Grav.,2005, 22(9):1503, [arXiv:hep-th/0408217].
  * [28] Y. Tian, X.-N. Wu, H. Zhang, “Holographic Entropy Production,”, _JHEP_ , 2014, 1410:170, [arXiv:1407.8273].
  * [29] Y. Tian, “A topological charge of black holes,” _Class. Quantum Grav._ 2019, 36:245001, [arXiv:1804.00249].
  * [30] Cai, R.G. “Gauss-bonnet black holes in AdS spaces,” Phys. Rev. D, 2002, 65(8):084014, [arXiv:hep-th/0109133].
  * [31] Lanczos, C. “Elektromagnetismus als natürliche eigenschaft der riemannschen geometrie,” Z. Phys., 1932, 73(3):147–168.
  * [32] Lovelock, D. “The Einstein tensor and its generalizations,” J. Math. Phys., 1971, 12(3):498–501.
  * [33] Crisostomo, J.; Troncoso, R.; Zanelli, J. “Black hole scan,” Phys. Rev. D, 2000, 62(8):084013, [arXiv:hep-th/0003271].
  * [34] Wu, B. “ Holographic thermodynamics of BTZ black hole,” talk presented on the 2022 Annual meeting of Chinese association of gravitation and astrophysics (paper version to appear).
  * [35] Wald, R.M. “Black hole entropy is the Noether charge,” Phys. Rev. D, 1993, 48(8):R3427, [arXiv:gr-qc/9307038].
  * [36] Iyer, V.; Wald, R.M. “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy,” Phys. Rev. D, 1994, 50(2):846–864, [arXiv:gr-qc/9403028].
  * [37] Bogomol’nyi, E.B. “The stability of classical solutions,” Sov. J. Nucl. Phys., 1976, 24(4).
  * [38] Hawking, S.W. ; Page, D.N. “Thermodynamics of black holes in anti-de Sitter space,” Commun. Math. Phys., 1983, 87(4):577–588.
  * [39] Tong, D. “Lectures on statistical physics,” Notes online from the author’s DAMTP teaching page (2012).
  * [40] Thompson, C.J. “Classical equilibrium statistical mechanics,” Oxford University Press, 1988.
  * [41] Pathria, R.K.; Beale, P.D. “Statistical mechanics,” 3rd Eds, Elsevier, 2011.