# One-dimensional model of freely decaying two-dimensional turbulence

Leonardo Campanelli1
email<EMAIL_ADDRESS>1All Saints University, Asudom Academy
of Science, 5145 Steeles Ave., Toronto (ON), Canada

###### Abstract

ABSTRACT

We construct a discrete shell-model for two-dimensional turbulence that takes
into account local and nonlocal interactions between velocity modes in Fourier
space. In real space, its continuous limit is described by the one-dimensional
Burgers equation. We find a novel approximate scaling solution of such an
equation and show that it well describes the main characteristics of the
energy spectrum in fully developed, freely decaying two-dimensional
turbulence.

Keywords: Two-dimensional turbulence; shell models; analytical methods

##  I I. Introduction

Freely decaying, two-dimensional hydrodynamic (2HD) turbulence has been the
object of intensive studies in the last few decades (for a review of two-
dimensional turbulence see, e.g., Boffetta ). This is because, some three-
dimensional turbulent systems in nature, such as large-scale motions in the
atmosphere and oceans, are well approximated by two-dimensional hydrodynamical
models.

The main features of statistically homogeneous and isotropic turbulence are
encoded in the so-called kinetic energy spectrum $E(k,t)$, which defines the
energy contained in a given (Fourier) velocity mode of wavenumber $k$ at the
time $t$. Such a spectrum exhibits, in both direct numerical simulations and
laboratory experiments, peculiar “universal” properties that completely
characterize the evolution of fully developed turbulence. In particular, the
energy spectrum unveils the presence of three distinct regions, or ranges, of
turbulence with different characteristics: the large-scale range, the inertial
range, and the dissipative range.

The aim of this paper is to introduce and discuss a reduced dimensional model
of Navier-Stokes equation that governs two-dimensional turbulence. Such a
model is, under plausible assumptions, analytically solvable and seems to
describe well the main properties of freely decaying two-dimensional
turbulence, namely when external sources are not present in the turbulent
state.

The plan of the paper is as follows. In the following section, we review the
main features of the energy spectrum in freely decaying 2HD turbulence. In
Section III, we apply particular types of scaling arguments, first introduced
by Olesen in the study of freely decaying, three-dimensional
magnetohydrodynamic turbulence, to the hydrodynamical case in two-dimensions.
In Section IV, we introduce our one-dimensional model for two-dimensional
turbulence. In Section V, we find and discuss the solution of the equation
describing such a reduced dimensional model. In Section VI, we draw our
conclusion. Finally, in the Appendix, we derive the Saffman spectrum from the
Burgers equation.

##  II II. Hydrodynamics in two dimensions

The evolution of an incompressible fluid in two dimensions is described by the
two-dimensional Navier-Stokes equation,

$\frac{\partial\textbf{v}}{\partial
t}+(\textbf{v}\cdot\nabla)\textbf{v}=-\nabla p+\nu\nabla^{2}\textbf{v}$ (1)

and the incompressibility condition, $\nabla\cdot\textbf{v}=0$, where
$\textbf{v}(\textbf{x},t)=(v_{1},v_{2})$ is the velocity of bulk fluid motion,
$p(\textbf{x},t)$ is the pressure, and $\nu$ is the kinematic viscosity (see,
e.g., Biskamp ; DavidsonBook ). One can define the kinematic Reynolds number,
$\text{Re}=vl/\nu$, where $v$ and $l$ are the typical velocity and typical
length scale of the fluid motion. Hydrodynamic turbulence occurs when
$\text{Re}\gg 1$.

In Fourier space, 111For the Fourier transform, we use the convention
$\textbf{v}(\textbf{x})=\int\\!\frac{d^{n}k}{(2\pi)^{n/2}}\,e^{i\textbf{k}\textbf{x}}\,\textbf{u}(\textbf{k})$
and
$\textbf{u}(\textbf{k})=\int\\!\frac{d^{n}x}{(2\pi)^{n/2}}\,e^{-i\textbf{k}\textbf{x}}\,\textbf{v}(\textbf{x})$.
the two-dimensional hydrodynamic equations take the form

$\left(\frac{\partial}{\partial t}+\nu
k^{2}\\!\right)\\!u_{\alpha}(\textbf{k})=iP_{\alpha\beta\gamma}(\textbf{k})\int\\!d^{\,2}p\,u_{\beta}(\textbf{p})\,u_{\gamma}(\textbf{k}-\textbf{p})$
(2)

and $k_{\alpha}u_{\alpha}(\textbf{k},t)=0$, where $\textbf{u}(\textbf{k},t)$
is the velocity field in Fourier space,

$P_{\alpha\beta\gamma}(\textbf{k})=\frac{1}{2\pi}\left(\frac{k_{\alpha}k_{\beta}k_{\gamma}}{k^{2}}-\mbox{$\frac{1}{2}$}k_{\gamma}\delta_{\alpha\beta}-\mbox{$\frac{1}{2}$}k_{\beta}\delta_{\alpha\gamma}\right)\\!,$
(3)

and $k=|\textbf{k}|$. The pressure $p$ has been eliminated by use of the
incompressibility condition. Reality of the velocity field in real space
imposes that $\textbf{u}(-\textbf{k})=\textbf{u}^{*}(\textbf{k})$. (Greek
subscripts range from 1 to 2 and summation over repeated indices is implied.)

For isotropic fluids, two-dimensional hydrodynamics admits two invariants of
motion in the limit of null kinematic viscosity (the inviscid case): the
kinetic energy density (in a unitary two-dimensional volume),

$E(t)=\langle\frac{1}{2}\int\\!d^{\,2}x\,\textbf{v}^{2}\rangle=\int_{0}^{\infty}\\!\\!dkE(k,t),$
(4)

and the enstrophy,

$\Omega(t)=\langle\int\\!d^{\,2}x\,\omega^{2}\rangle=\int_{0}^{\infty}\\!\\!dkk^{2}E(k,t),$
(5)

where

$E(k,t)=\pi k\,\langle|\textbf{u}(\textbf{k})|^{2}\rangle$ (6)

is the kinetic energy density spectrum, and
$\omega=\epsilon_{\alpha\beta}\partial_{\alpha}v_{\beta}=\partial_{1}v_{2}-\partial_{2}v_{1}$
is the “scalar vorticity” field. Since we are interested in the evolution of
statistically homogeneous and isotropic velocity fields, an ensemble average
denoted by $\langle...\rangle$ has been introduced in the above definitions of
energy and enstrophy.

The main features of locally homogeneous and isotropic turbulence are
described by the kinetic energy spectrum to which we now turn our attention.
In freely decaying turbulence, the energy spectrum displays some universal
characteristics at both large and small scales. In particular, the results of
direct numerical simulations and laboratory experiments clearly show the
presence of three distinct regions or ranges: ($i$) the large-eddies range,
$k\ll k_{i}$, ($ii$) the enstrophy inertial range, $k_{i}\ll k\ll k_{diss}$,
and the dissipation range, $k\gg k_{diss}$. A sketch of the energy spectrum in
these ranges is shown in Fig. $1$. The existence of such three regions is not
surprising. Indeed, in freely decaying 2HD there are only two independent
length scales: the dissipation length $L_{diss}(t)=\sqrt{\nu t}$ and the
initial scale $L_{i}$, which we assume to be much greater than $L_{diss}$,
$L_{i}\gg L_{diss}$. The wavenumber corresponding to $L_{i}$ is
$k_{i}=1/L_{i}$, while the wavenumber corresponding to $L_{diss}$ is
$k_{diss}=1/L_{diss}$. These two length scales define the three ranges
introduced above.

Figure 1: Picture (log-log scale) of the expected energy spectrum $E(k,t)$ in
freely decaying two-dimensional turbulence as a function of the wavenumber
$k$. A time-dependent $k^{3}$ power-law spectrum is shown at large scales
($k\rightarrow 0$). Arrows indicate displacements in time.

Energy inertial range. – A physical homogeneous and isotropic field not
interacting with external systems is dependent only upon the initial
conditions which introduces a mean velocity $u_{i}$ and a length scale
$L_{i}$. This is the case, for example, of turbulence behind a grid (with
$u_{i}$ being the velocity field at the grid and $L_{i}$ being the spacing of
the bars of the grid perpendicular to the direction of the flow). In numerical
simulations, instead, it is assumed that the initial velocity field is
homogeneous and isotropic. In this case, the mean value of the velocity is
zero and $u_{i}$ in the following equations must be regarded as a (root-mean-
square) r.m.s value of the field. More generally, $L_{i}$ and $u_{i}$ can be
considered as typical scale values at the onset of fully developed turbulence.

By definition, an inertial range is a range where dissipation is negligible
and the dynamics is independent on the initial conditions. Dimensional
analysis, then, completely gives the shape of the energy spectrum in such a
range (see, e.g., Campanelli ),

$E(k,t)=ct^{-2}k^{-3},~{}~{}~{}k_{i}\ll k\ll k_{diss},$ (7)

where $c$ is a dimensionless constant. In forced turbulence, a time-
independent spectrum of the form $E(k,t)\propto k^{-3}$ is known as Batchelor
spectrum Batchelor .

Dissipative range. – At small scales, $k\gg k_{diss}$, the 2HD equations can
be exactly solved if one assumes that the rate of energy transfer due to
nonlinear interactions is completely negligible. Indeed, it can be shown that
the energy spectrum changes in time as $\partial E(k,t)/\partial t=T(k,t)-\nu
k^{2}E(k,t)$, where $T(k,t)$ is the energy change caused by nonlinear
interactions (see, e.g., Boffetta ). When this term is negligible with respect
to the viscous term, one obtains

$E(k,t)=c_{\infty}\nu^{2}ke^{-\nu k^{2}t},~{}~{}~{}k\gg k_{diss},$ (8)

where $c_{\infty}$ is a dimensionless constant.

The hypothesis that the rate of nonlinear energy transfer is negligible is
certainly true in the “mathematical” limit $k\rightarrow\infty$. However, such
a hypothesis is not justified in a region around the dissipation wavenumber.
Indeed, Tatsumi and Yanase Tatsumi , using the so-called “modified zero
fourth-order cumulant approximation”, found an analytic expression of the
energy spectrum in the dissipation range of the form

$E(k,t)\propto e^{-bk\sqrt{\nu t}},~{}~{}~{}k\gtrsim k_{diss},$ (9)

where $b$ is a dimensionless constant. The same authors found, by direct
numerical integration of 2HD equations, that
$E(k,t)\propto\exp{\\!\left[-b(k\sqrt{\nu t})^{s}\right]}$ with $s$ in the
range $[1.3,1.4]$. The discrepancy with Eq. (9) indicates, according to the
authors, that either the asymptotic behaviour $s=1$ is realized beyond the
numerical coverage or the numerical results are not accurate enough. In either
case, the asymptotic form of the spectrum is different from the purely viscous
spectrum (8) for $k\gtrsim k_{diss}$.

Large-eddies range. – In the limit $k\rightarrow 0$, a Maclaurin expansion of
the energy spectrum gives (see, e.g. DavidsonBook )

$E(k,t)=\left\\{\begin{array}[]{lll}4\pi\mathcal{L}k+\mathcal{O}(k^{3}),&~{}\mathcal{L}\neq
0,\\\ I(t)k^{3}+\mathcal{O}(k^{5}),&~{}\mathcal{L}=0.\end{array}\right.$ (10)

Here, $\mathcal{L}$ is known as the two-dimensional Saffman integral, and its
time-independence is a consequence of the conservation of linear momentum
DavidsonBook . Indeed, Davidson Davidson  has shown that the Saffman integral
in two dimensions can be written as

$\mathcal{L}=\langle\frac{1}{V}\\!\left(\,\int_{V}\\!d^{\,2}x\,\textbf{v}\right)^{\\!2}\rangle,$
(11)

where $V$ is some large two-dimensional volume. Accordingly, $\mathcal{L}$ is
nonzero whenever the turbulence has sufficient linear momentum. In this case,
then, the energy spectrum at large scales is linear in the wavenumber, a
result known as Saffman spectrum. If the total linear momentum is instead
negligible, a spectrum of the form $I(t)k^{3}$ develops at large scales where
$I(t)$, the so-called Loitsyansky integral in two dimensions, is generally
expected to be an increasing function of time DavidsonBook .

##  III III. Dimensional scaling

Olesen’s scaling. – The two-dimensional HD equations, under the scaling
transformations

$\textbf{x}\rightarrow\ell\,\textbf{x},~{}~{}t\rightarrow\ell^{1-h}\,t,$ (12)

admit solutions of the type

$\textbf{v}(\ell\,\textbf{x},\ell^{1-h}\,t)=\ell^{\,h}\,\textbf{v}(\textbf{x},t),$
(13)

provided that the dissipative parameter $\nu$ scales as

$\nu(\ell^{1-h}\,t)=\ell^{1+h}\,\nu(t).$ (14)

Here $\ell>0$ is the “scaling factor” and $h$ is a arbitrary real parameter.
Starting from the scaling relations, and following Olesen’s analysis Olesen ,
we obtain the energy spectrum

$E(k,t)=kt^{2p}\Psi(kt^{(1+p)/2}),$ (15)

where $p=(1+h)/(1-h)$ and $\Psi$ is a arbitrary scaling-invariant function of
its argument. We have two cases:

($i$) If dissipation is negligible, as it happens at large scales and, by
definition, in the inertial range, then $h$ (and in turns $p$) is completely
arbitrary, to wit, it is not fixed by scaling arguments.

($ii$) If dissipation is important, as in the dissipative range (at very small
scales), then $h$ is fixed by the scaling properties of $\nu$. Indeed, if we
differentiate Eq. (14) with respect to $\ell$, and put $\ell=1$ afterwards, we
get $\nu\propto t^{p}$. In particular, as we will assume in this paper, if
$\nu$ is constant then we must take to $p=0$ and, accordingly,

$E(k,t)=k\psi(kt^{1/2}).$ (16)

If dissipation is not negligible and $\nu$ does not evolve in time as a power
law, then Eq. (13) is not a solution of the 2HD equations and in turn the
energy spectrum do not exhibit a scaling of the type given by Eq. (15).

Dimensional analysis. – For the sake of later convenience let us re-write Eqs.
(16) and (15) in a form such that the scaling functions $\Psi$ and $\psi$ are
dimensionless. It is easy to check that they must be written as

$E(k,t)=\nu^{2}k\psi(k\sqrt{\nu t})$ (17)

and

$E(k,t)=E_{i}\kappa\tau^{2p}\Psi(\kappa\tau^{(1+p)/2}),$ (18)

respectively. Here, $E_{i}=u_{i}^{2}L_{i}$, $\kappa=k/k_{i}$, $\tau=t/t_{i}$,
and $t_{i}=L_{i}/u_{i}$ is the so-called initial “eddy turnover time”.

It is interesting to observe that form of the energy spectrum in the inertial
range is completely fixed by scaling considerations only. Indeed, since in
this range the dynamics is completely independent on dissipation (and on the
initial conditions), the only possible form for the scaling function $\psi$
such that the energy spectrum does not depend on the kinematic viscosity is
$\psi(x)=cx^{-4}$. This gives the Batchelor spectrum in Eq. (7) with $c$ being
a “universal” dimensionless constant.

##  IV IV. One-dimensional model

In this Section, we propose a one-dimensional, continuous (toy) model for two-
dimensional hydrodynamics. We start by generalizing the well-known discrete
shell model shell  for HD turbulence by including all possible local and
nonlocal interaction terms in Fourier space. Next, we consider the continuous
limit of such a discrete shell model. The resulting model will be analyzed in
the next section.

Shell model with nonlocal interactions. – Let us consider a shell model
constructed in a discrete wavenumber space which is approximated by $2N$
shells,

$\\{k_{n}\,|\,k_{n}=\\!Kn,\;n=\\!-N,-N+1,...,0,...,N-1,N\\},$ (19)

where $K$ gives the spacing between two consecutive shells. The velocity is
represented by a set of complex variables $\\{u_{n}\\}$, where $u_{n}$ stands
for the velocity components whose scalar wavenumber $\vec{k}\in\mathbb{R}$
satisfies $k_{n}\leq\vec{k}\leq k_{n+1}$. We now construct the Navier-Stokes
equation in this scalar model in the following way,

$\left(\frac{d}{dt}+\nu
k_{n}^{2}\right)\\!u_{n}=iP_{n}\\!\sum_{j=-N}^{N}\\!u_{j}\,u_{n-j},$ (20)

where $P_{n}$ is a real vector and $u_{-n}=u_{n}^{*}$. The sum on the left-
hand side of (20) includes all possible local and nonlocal interaction terms
between different Fourier modes.

In the shell model, the kinetic energy density is
$E(t)=\sum_{n=0}^{N}E_{|n|}$, where $E_{|n|}(t)=\pi k_{|n|}|u_{n}|^{2}$ is the
energy spectrum (for a justification of the form of the energy spectrum, see
below.) One can check that a sufficient condition for the conservation of
energy in the inviscid limit is $P_{-n}=-P_{n}$. Mimiking Eq. (3), we can
assume assume that $P_{n}$ is linear in $n$, $P_{n}\propto k_{n}$. With this
$P_{n}$, it is easy to check that the quantity
$\sum_{n=0}^{N}f(|k_{n}|)\,|u_{n}|^{2}$, with $f(x)$ any function such that
$f(0)=0$, is conserved in the limit of vanishing viscosity. In particular, for
$f(x)=x^{3}$ the above expression is the analogue of the enstrophy.

Continuous limit. – The continuous limit of the above shell model corresponds
to take $N\rightarrow\infty$ and $K\rightarrow 0$. In this case, $k_{n}$ is
replaced by $\vec{k}$, the discrete set $u_{n}(t)$ is replaced by a continuous
complex variable $u(\vec{k},t)$, the real vector $P_{n}$ by a real function of
$P(\vec{k})$, and sums are replaced by integrals, e.g.,
$K\sum_{j=-N}^{+N}\rightarrow\int_{-\infty}^{+\infty}d\vec{k}$. The continuous
limit of the discrete shell model (20) is then given by

$\left(\frac{\partial}{\partial t}+\nu
k^{2}\right)\\!u(\vec{k})=iP(\vec{k})\\!\int_{-\infty}^{+\infty}\\!\\!d\vec{q}\,u(\vec{q})\,u(\vec{k}-\vec{q}),$
(21)

where $k=|\vec{k}|$, $P(\vec{k})\propto\vec{k}$, and
$u(-\vec{k})=u^{*}(\vec{k})$. The energy spectrum is

$E(k,t)=\pi k\,|u(\vec{k})|^{2}$ (22)

and the energy is $E(t)=\int_{0}^{\infty}\\!dkE(k,t)$. The energy, and in the
general the quantity $\int_{0}^{\infty}\\!dkg(k)\,|u(\vec{k})|^{2}$ with
$g(k)$ being any real function such that the previous integral exists, is
conserved in the limit of vanishing viscosity.

In the following, we take

$P(\vec{k})=-\vec{k}/2\sqrt{2\pi}.$ (23)

A possible multiplicative constant in the above definition of $P(\vec{k})$ can
be reabsorbed in the definition of time and kinematic viscosity, and then will
be neglected in the following.

Burgers equation. – The factor $-1/2\sqrt{2\pi}$ in Eq. (23) has been
introduced for convenience. In this case, indeed, the inverse Fourier
transform of Eq. (21) is the well-known viscid Burgers equation (see Burgers
and references therein)

$\frac{\partial v}{\partial t}+v\frac{\partial
v}{\partial\zeta}=\nu\frac{\partial^{2}v}{\partial\zeta^{2}},$ (24)

where
$v(\zeta,t)=\int\\!\frac{d\vec{k}}{\sqrt{2\pi}}\,e^{i\vec{k}\zeta}u(\vec{k},t)$.

It is assumed in the literature that the viscid Burgers equation cannot
describe the main features of hydrodynamic turbulence Burgers . This is
because an exact solution of such an equation can be given that does not share
an important property of the Navier–Stokes equation, namely the sensitivity to
small changes in the initial conditions (at sufficiently high Reynolds
numbers) that trigger the spontaneous arise of randomness by chaotic dynamics.
Such an exact solution is found by applying a Cole-Hopf transformation defined
by introducing the function $f(\zeta,t)$ as

$v=-\frac{2\nu}{f}\,\frac{\partial f}{\partial\zeta}\,.$ (25)

Inserting the above equation in the viscid Burgers equation, one gets

$H\,\frac{\partial f}{\partial\zeta}=f\frac{\partial H}{\partial\zeta}\,,$
(26)

where

$H=\frac{\partial f}{\partial
t}-\nu\frac{\partial^{2}f}{\partial\zeta^{2}}\,.$ (27)

Consequently, if $f$ solves the “heat equation” $H=0$ than $v$, given the
Cole-Hopf transformation, solves the viscid Burgers equation. Since the heat
equation can be exactly solved, the exact form of $f$ is known. An inverse
Cole-Hopf transformation then gives the analytical expression for $v$ Burgers
. Moreover, it can be shown that such a solution for $v$ is unique if the
initial condition

$v(\zeta,0)=v_{0}(\zeta)$ (28)

is given. The existence of such a solution, then, excludes the possibility
that the Burger equation could describe the transition from an initial state
to a state of fully developed hydrodynamic turbulence.

However, the above arguments do not exclude the possibility that the Burger
equation could describe the evolution of the system after a turbulent state
has been reached. Indeed, our claim is that this is the case, in the sense
that Burgers equation does describe the main features of the energy spectrum
in fully developed, freely decaying two-dimensional turbulence. In order this
to be possible one has to assume that the Burgers equation is a model of
turbulence only at times $t\gg t_{i}$, namely only when turbulence is already
fully developed. In particular, the initial condition (24) is inapplicable in
this case, which means that the velocity field must be not defined for $t=0$
(or, in general, for $t=t_{0}$, where $t_{0}$ is the initial time).

A possible solution of the Burgers equation that satisfies the above condition
and that will correspond, as discussed in Section V, to the case of large
wavenumbers can be obtained by writing $v(\zeta,t)$ as

$v(\zeta,t)=\sqrt{\frac{\nu}{t}}\,\hat{\phi}\\!\left(\\!-\frac{\zeta}{\sqrt{\nu
t}}\right)\\!,$ (29)

with $\hat{\phi}(x)$ a “scaling function” to be determined. Observe that this
solution is not defined for $t=0$.

Inserting into Eq. (24) we get

$2\hat{\phi}^{\prime\prime}+2\hat{\phi}\hat{\phi}^{\prime}+\omega\hat{\phi}^{\prime}+\hat{\phi}=0,$
(30)

where a prime indicates a derivative with respect to $\omega=-\zeta/\sqrt{\nu
t}$. The solution of the above equation is

$\hat{\phi}(\omega)=\frac{2c_{1}c_{2}H_{c_{1}-1}\\!\\!\left(\frac{\omega}{2}\right)-\omega\\!\left[c_{2}H_{c_{1}}\\!\\!\left(\frac{\omega}{2}\right)+\,_{1}F_{1}\\!\\!\left(\\!-\frac{c_{1}}{2},\frac{1}{2};\frac{\omega^{2}}{2}\right)+c_{1}\,{}_{1}F_{1}\\!\\!\left(1-\frac{c_{1}}{2},\frac{3}{2};\frac{\omega^{2}}{2}\right)\right]}{c_{2}H_{c_{1}-1}\\!\\!\left(\frac{\omega}{2}\right)+\,_{1}F_{1}\\!\\!\left(\\!-\frac{c_{1}}{2},\frac{1}{2};\frac{\omega^{2}}{2}\right)},$
(31)

where $c_{1}$ and $c_{2}$ are real constants of integration, $H_{n}(x)$ is the
Hermite polynomial of degree $n$, and ${}_{1}F_{1}(a,b;x)$ is the Kummer
confluent hypergeometric function Abramowitz .

As we will discuss later, the solution (24) describes the system in a state of
fully developed HD turbulence. The constants $c_{1}$ and $c_{2}$ are then to
be considered “universal” and cannot be determined by initial conditions.
Indeed, they will corresponds to the constants $C$ and $c_{\infty}$ defined in
Section V.

Let us finally observe that the function $f(\zeta,t)$ defined by the Cole-Hopf
transformation does not satisfy the heat equation. Indeed, writing

$f(\zeta,t)=\sqrt{\frac{\nu}{t}}\,\hat{\varphi}\\!\left(\\!-\frac{\zeta}{\sqrt{\nu
t}}\right)\\!,$ (32)

where $\hat{\varphi}(x)$ is an arbitrary function of its argument, and
inserting into the Cole-Hopf transformation (25), we get
$\hat{\phi}=2\hat{\varphi}^{\prime}/\hat{\varphi}$, from which

$\hat{\varphi}(\omega)=e^{\frac{1}{2}\\!\int\\!d\omega\hat{\phi}},$ (33)

with $\hat{\phi}$ given by Eq. (31). It is easy to check now, by direct
inspection, that $f(\zeta,t)$ as given by Eqs. (32) and (33) does not satisfy
the heat equation.

In the next paragraph we show that, under particular conditions, the Burger
equation describes the dynamics of a two-dimensional fluid. Moreover, the
results of this analysis will provide a justification of the definition of
energy spectrum introduced above. In the next section, instead, we will
discuss in some detail the above “scaling solution” of Burger equation. For
the sake of convenience, we will work in Fourier space and then consider Eq.
(21).

Compressible fluid with negligible pressure. – Let us consider a compressible
fluid with negligible pressure. We introduce the “complex velocity field”
$V(x,y)$ as

$V(x,y)=v_{x}+iv_{y}.$ (34)

Using the Navier-Stokes equation, we find that the $V$ satisfies the equation

$\frac{\partial V}{\partial t}+\left(\mbox{Re}[V]\frac{\partial}{\partial
x}+\mbox{Im}[V]\frac{\partial}{\partial y}\right)\\!V=\nu\nabla^{2}V.$ (35)

Taking the complex conjugate of the above equation, it is easy to see that a
possible solution of it is such that $V^{*}=e^{-i\varphi}V$, with
$\varphi\in\mathbb{R}$.

Let us introduce the new real quantity $\upsilon(x,y)$ as

$\upsilon(x,y)=e^{-i\varphi/2}V=\frac{1}{2}\left[\sec(\varphi/2)v_{x}+\csc(\varphi/2)v_{y}\right].$
(36)

It satisfies the equation

$\frac{\partial\upsilon}{\partial
t}+\upsilon\left[\cos(\varphi/2)\frac{\partial}{\partial
x}+\sin(\varphi/2)\frac{\partial}{\partial
y}\right]\\!\upsilon=\nu\nabla^{2}\upsilon.$ (37)

We can diagonalize the operator in square parenthesis in the above equation by
performing a rotation of an angle $\varphi$ of the coordinate system,

$\displaystyle\xi=\cos(\varphi/2)x+\sin(\varphi/2)y,$ (38)
$\displaystyle\eta=-\sin(\varphi/2)x+\cos(\varphi/2)y.$ (39)

The function $\upsilon$ in the new coordinate system, $\upsilon(\xi,\eta)$,
satisfies the equation

$\frac{\partial\upsilon}{\partial
t}+\upsilon\frac{\partial\upsilon}{\partial\xi}=\nu\nabla^{2}\upsilon,$ (40)

where $\nabla^{2}=\partial_{\xi}^{2}+\partial_{\eta}^{2}$. The r.m.s. value of
the velocity field, $v_{rms}$, is given by

$v_{rms}^{2}(t)=\int\\!d^{\,2}x\,\textbf{v}^{2}(\textbf{x})=\int\\!d^{\,2}\bm{\zeta}\upsilon^{2}(\bm{\zeta})=\int\\!dk\,w_{rms}^{2}(k),$
(41)

where $\bm{\zeta}=(\xi,\eta)$ and

$w_{rms}^{2}(k,t)=2\pi kw(\textbf{k})\,w^{*}(\textbf{k}),$ (42)

with $w_{rms}$ being the spectrum of the r.m.s. velocity. Here,
$w(\textbf{k})$ is the Fourier transform of $\upsilon(\bm{\zeta})$,
$\textbf{k}=(k_{\xi},k_{\eta})$, and $k=|\textbf{k}|$.

Let us now assume that $\upsilon(\xi,\eta)$ depends only on $\xi$,
$\upsilon(\xi,\eta)=\upsilon(\xi)$. Equation (40) reduces then to the Burgers
equation. The r.m.s. velocity is still given by Eq. (41), while its spectrum
reads

$w_{rms}^{2}(k,t)=2\pi k|w(k_{\xi},t)|^{2},$ (43)

where $k=|k_{\xi}|$. For constant density (case that is not realized in
compressible, pressurless 2HD turbulence), we have $E(t)=v_{rms}^{2}/2$ (with
density equal to one), so that the energy spectrum is
$E(k,t)=w_{rms}^{2}(k,t)/2=\pi k|w(k_{\xi},t)|^{2}$. This result justifies the
definition of energy spectrum introduced above.

##  V V. Results

As discussed in Section II, the state of a turbulent isotropic fluid depends
entirely on the initial conditions, $L_{i}$ and $u_{i}$, and on the
dissipation parameter $\nu$. The energy contained in a given wavenumber $k$
evolves in time with different characteristics in the three different ranges
defined by $k\ll k_{i}$, $k_{i}\ll k\ll k_{diss}$, and $k\gg k_{diss}$.
Indeed, while in the large-scale range the energy spectrum generally depends
on the initial conditions but not on dissipation, in the dissipative range it
depends on $\nu$ but not on $L_{i}$ and $u_{i}$. The inertial range, that in
wavenumber space is placed in between the two above ranges, is instead
characterized by a universal energy spectrum not dependent on either initial
conditions or viscosity. Based on this peculiarities of the energy spectrum,
it is plausible to assume that the wavenumbers defined by $k\ll k_{i}$ and
$k\gg k_{i}$ “do not communicate”, in the sense that nonlocal interactions
between modes in this different ranges are negligible. We will assume, in the
following, that this is the case. Accordingly, and roughly speaking, the
integral in Eq. (21) can be split in two parts

$\int_{-\infty}^{+\infty}\\!\\!d\vec{q}\,u(\vec{q})\,u(\vec{k}-\vec{q})=\theta(k_{i}-k)\\!\int_{q<k_{i}}\\!\\!d\vec{q}\,u(\vec{q})\,u(\vec{k}-\vec{q})+\theta(k-k_{i})\\!\int_{q>k_{i}}\\!\\!d\vec{q}\,u(\vec{q})\,u(\vec{k}-\vec{q}),$
(44)

where $\theta(x)$ is the Heaviside step function.

Enstrophy inertial range. – In this range, the energy spectrum has the form
(17). Accordingly, we look for solutions of Eq. (21) of the form

$u(\vec{k},t)=\nu\phi(\vec{k}\sqrt{\nu t}).$ (45)

Inserting into Eq. (21) and taking into account Eq. (44), we find that a
possible solution is defined through the scaling function $\phi(x)$ that, in
turns, satisfies the equation

$x\frac{d\phi(x)}{dx}+2x^{2}\phi(x)=-\frac{i}{\sqrt{2\pi}}\,x\int_{|y|>k_{i}/k_{diss}}\\!\\!dy\phi(y)\phi(x-y),$
(46)

where $x=\vec{k}\sqrt{\nu t}=\vec{k}/k_{diss}$. In the inertial range, the
viscous term in not effective and can be neglected. Also, and for the sake of
simplicity, we will take the limit $k_{i}/k_{diss}\\!\rightarrow\\!0$ in the
bound of the integral in (46). In this case, the Fourier transform of Eq. (46)
gives

$2\hat{\phi}\hat{\phi}^{\prime}+\omega\hat{\phi}^{\prime}+\hat{\phi}=0,$ (47)

where $\hat{\phi}(\omega)$ is the Fourier transform of $\phi(x)$ and a prime
indicates the derivative with respect to $\omega$. Observe that
$\hat{\phi}(\omega)$ is exactly the scaling function introduced in Section IV
[see Eq. (29)] and Eq. (47) corresponds to Eq. (30) when the dissipative term
(proportional to $\hat{\phi}^{\prime\prime}$) is neglected.

Equation. (47) admits two solutions

$\hat{\phi}(\omega)=-\frac{1}{2}\left(\omega\pm\sqrt{\omega^{2}+C^{2}}\,\right)$
(48)

specified by the $\pm$ sign, where $C\in\mathbb{R}$ is a constant of
integration. We can neglect the first term in Eq. (48) since its inverse
Fourier transforms is proportional to the derivative of the $\delta(x)$
function and we are assuming that $|x|=k/k_{diss}>k_{i}/k_{diss}\neq 0$. In
this case, taking the inverse Fourier transform of Eq. (48) gives

$\phi(x)=\pm\frac{C}{\sqrt{2\pi}|x|}\,K_{1}(C|x|),$ (49)

where $K_{\nu}(x)$ is the modified Bessel function of the second kind
Abramowitz .

For small values of $|x|$, $C|x|\ll 1$, the asymptotic expansion of $\phi(x)$
is

$\phi(x)=\pm\frac{1}{\sqrt{2\pi}x^{2}}$ (50)

at the leading order in $x$. This gives the Batchelor spectrum (7) with
$c=1/2$.

Dissipative range. – This range is defined by the condition $k\gtrsim
k_{diss}$. We can distinguish two “sub-dissipative ranges”: In the “pre-
viscous damping range”, the rate of energy transfer due to nonlinear
interactions is not negligible. In this case, we expect a “slower” decay of
the energy than the “fast” decay in the purely viscous case. In the “viscous
damping range”, mathematically defined by $k/k_{diss}\rightarrow\infty$,
instead, the rate of energy transfer due to nonlinear interactions is absent
and the dynamics is dominated by viscosity.

Pre-viscous damping range. – At the leading order, the asymptotic expansion of
$\phi(x)$ for large $|x|$, $C|x|\gg 1$, is

$\phi(x)=\pm\frac{\sqrt{C}}{2|x|^{3/2}}\,e^{-C|x|}.$ (51)

This gives the energy spectrum

$E(k,t)=\frac{\pi C\nu^{1/2}}{4k^{2}t^{3/2}}\,e^{-2Ck\sqrt{\nu t}}.$ (52)

The above result is compatible with the results of Tatsumi and Yanase
discussed in Section II.

Viscous damping range. – In this range, we can neglect the right-hand side of
Eq. (46). Accordingly, we get $\phi(x)=\sqrt{c_{\infty}/\pi}\,e^{-x^{2}}$,
where $c_{\infty}>0$ is a constant. This gives the energy spectrum (8) in the
purely viscous case.

The viscous term is negligible when the second term in Eq. (46) is negligible
with respect to the first one, $\left|d\ln\phi/dx^{2}\right|\gg 1$. Inserting
Eq. (49), this is the case when $K_{1}(C|x|)/K_{2}(C|x|)\ll C/2|x|$ or, using
the asymptotic expansion of $K_{\nu}(x)$ for large and small arguments, when
$|x|\lesssim 1$ if $C|x|\lesssim 1$ or $|x|\lesssim C/2$ if $C|x|\gtrsim 1$.
This in turns gives two possibilities for the scaling function
$\psi(|x|)=\pi|\phi(x)|^{2}$:

$\displaystyle\psi(|x|)=\left\\{\begin{array}[]{lll}\frac{1}{2x^{4}}\,,&~{}|x|\lesssim\mbox{min}(1/C,1),\\\
c_{\infty}e^{-2x^{2}},&~{}|x|\gtrsim\mbox{min}(1/C,1),\end{array}\right.$ (55)

if $C\lesssim\sqrt{2}$ or

$\psi(|x|)=\left\\{\begin{array}[]{lll}\frac{1}{2x^{4}}\,,&~{}|x|\lesssim
1/C,\\\ \frac{\pi C}{4|x|^{3}}\,e^{-2C|x|},&~{}1/C\lesssim|x|\lesssim C/2,\\\
c_{\infty}e^{-2x^{2}},&~{}|x|\gtrsim C/2,\end{array}\right.$ (56)

if $C\gtrsim\sqrt{2}$. We can get an estimate of $c_{\infty}$ if we impose the
continuity of $\psi(|x|)$ at $|x|=\mbox{min}(1/C,1)$ for the first case
($C\lesssim\sqrt{2}$) and at $|x|=C/2$ for the second case
($C\gtrsim\sqrt{2}$). Roughly speaking, we get that $c_{\infty}$ is of order
unity if $C\ll\sqrt{2}$, and exponentially suppressed, $c_{\infty}\sim
e^{-C^{2}/2}$, if $C\gg\sqrt{2}$. Accordingly, the exponential tail
$c_{\infty}e^{-2x^{2}}$ in the spectrum (56) is exponentially small when $C$
is large, and then difficult to see both in experiments and direct numerical
simulations of two-dimensional turbulence.

Large scales. – In the limit $k\rightarrow 0$, dissipation is not effective.
Taking into account the Olesen’s arguments and Eq. (44), we can search for
solutions of Eq. (21) of the form

$u(\vec{k},t)=u_{i}L_{i}\tau^{p}\Phi(\vec{\kappa}\tau^{q}),$ (57)

where $\vec{\kappa}=\vec{k}/k_{i}$, $q=(1+p)/2$, and $\Phi(x)$ is a
dimensionless function of its argument. Inserting the above equation in Eq.
(21) and neglecting the term proportional to $\nu$, we find

$qz\,\frac{d\Phi(z)}{dz}+p\Phi(z)=-\frac{i}{2\sqrt{2\pi}}\,z\int_{-\tau^{q}}^{+\tau^{q}}\\!\\!dz^{\prime}\Phi(z^{\prime})\Phi(z-z^{\prime}),$
(58)

where $z=\vec{\kappa}\tau^{q}$. In the following, and for the sake of
simplicity, we will take the limit $\tau=t/t_{i}\rightarrow\infty$ in the
bounds of the integral in Eq. (58) and we will assume that $q>0$, namely

$p>-1.$ (59)

For $z\rightarrow 0$ (corresponding to $k\rightarrow 0$), we can write
$\Phi(z-z^{\prime})\simeq\Phi(-z^{\prime})$. The last assumption is true if
$|\Phi(z)|^{2}$ is integrable at $z=0$. Accordingly, to the leading order in
$z\rightarrow 0$, the right-hand side of Eq. (58) can be approximated by
$i\Gamma_{0}z$, where
$\Gamma_{0}=-(1/2\sqrt{2}\pi)\int_{-\infty}^{+\infty}\\!dz|\Phi(z)|^{2}$
(assuming that the integral exists). We then find, to the leading order in
$z\rightarrow 0$,

$\Phi(z)=\left\\{\begin{array}[]{lll}3i\Gamma_{0}z\left(C_{-1/3}-i\frac{\pi}{2}+\ln
z\right),&~{}p=-1/3,\\\
\frac{i\Gamma_{0}z}{p+q}+C_{p}z^{-p/q},&~{}p\neq-1/3,\end{array}\right.$ (60)

where, $C_{-1/3}$ and $C_{p}$ and are dimensionless constants of integration.
The condition $\Phi(-z)=\Phi^{*}(z)$ implies that $C_{-1/3}\in\mathbb{R}$, and
that $C_{p}\in\mathbb{R}$ if $p/2q\in\mathbb{Z}$, $C_{p}$ is purely imaginary
if $(p/q-1)/2\in\mathbb{Z}$, and $\mbox{Im}[C_{p}]/\mbox{Re}[C_{p}]=\tan(\pi
p/2q)$, otherwise.

For $p=-1/3$, the energy spectrum is given, at the leading order, by

$E(k,t)=c_{-1/3}\,k^{3}\ln^{2}\\!\\!\left[\frac{k}{k_{i}}\\!\left(\frac{t}{t_{i}}\right)^{\\!1/3}\right],$
(61)

where $c_{-1/3}=9\pi u_{i}^{2}L_{i}^{4}\Gamma_{0}^{2}$.

For $p\neq-1/3$, we have two cases:

($i$) If $C_{p}=0$, the energy spectrum is given by

$E(k,t)=c_{p}t^{1+3p}k^{3},$ (62)

where $c_{p}=\pi u_{i}^{3(1+p)}L_{i}^{3(1-p)}\Gamma_{0}^{2}/(1+3p)^{2}$. The
cases $E(k,t)\propto t^{\gamma}k^{3}$ with $\gamma=1,2,2.5,4$ found in the
literature (see DavidsonBook ), then correspond to $p=0,1/3,1/2,1$,
respectively. Observe, also, that the energy spectrum at large scales is
increasing in time for $p>-1/3$ and decreasing in time for $p<-1/3$. Finally,
observe that for $p=1$ the energy spectrum does not depend on $L_{i}$. In this
case $E(k,t)\propto t^{4}k^{3}$. For $p=-1$, instead, the energy spectrum does
not depend on $u_{i}$. In this case $E(k,t)\propto t^{-2}k^{3}$.

($ii$) If $C_{p}\neq 0$, the integrability condition of $|\Phi(z)|^{2}$ at
$z=0$ implies that $-1<p<1/3$. This in turns gives two cases:

($a$) For $-1<p<-1/3$, we have $-p/q>1$, so that $\Phi(z)=i\Gamma_{0}z/(p+q)$
at the leading order. The energy spectrum is then given by Eq. (62).

($b$) For $-1/3<p<1/3$, instead, $\Phi(z)=C_{p}z^{-p/q}$ at the leading order.
The energy spectrum is then given by

$E(k,t)=c^{\prime}_{p}k^{\alpha_{p}},$ (63)

where $c^{\prime}_{p}=\pi u_{i}^{2}L_{i}^{2(1-p)/(1+p)}|C_{p}|^{2}$ and
$\alpha_{p}=(1-3p)/(1+p)$. Imposing that the Saffman integral is finite we get
$\alpha_{p}\geq 1$, corresponding to $p\leq 0$. Accordingly,
$1\leq\alpha_{p}<3$.

Particularly important is the case $p=0$ (see the Appendix for a different
derivation of the expression of the energy spectrum in this case). This is the
only case where $\Phi(0)$ is finite and different form zero. In this case, we
have the Saffman spectrum

$E(k,t)=c_{0}k,$ (64)

where $c_{0}=\pi u_{i}^{2}L_{i}^{2}|\Phi(0)|^{2}$.

Analytical case. – If one assumes that $\Phi(z)$ is analytical at $z=0$,
namely it admits a Maclaurin expansion, then it is easy to see that the form
of energy spectrum is that given in Eq. (10) with $\mathcal{L}=c_{0}/4\pi$ and
$I(t)=c_{p}t^{1+3p}$, where $p\neq-1/3$, and $c_{p}$ and $c_{0}$ are the same
as those previously defined.

Absolute state. – Let us conclude this section by observing that, in the
inviscid limit, Eq. (21) admits a static solution of the form

$u(\vec{k},t)=\frac{e^{i\vartheta\vec{k}}}{\sqrt{a+bk^{2}}}\,,$ (65)

where $\vartheta\in\mathbb{R}$, $a>0$, and $b>0$ are constants. This solution
corresponds to the static energy spectrum

$E(k,t)=\frac{\pi k}{a+bk^{2}}\,.$ (66)

The above spectrum has, correctly, the form of the “canonical” distribution
for two-dimensional turbulence first discussed by Kraichnan Kraichnan2  (see,
also, Biskamp ), with $a$ and $b$ related to the inviscid invariants $E$ and
$\Omega$.

##  VI VI. Conclusions

In this paper, we have proposed a (continuous) one-dimensional, toy model for
two-dimensional turbulence. Starting by the well-known (discrete) shell model
of hydrodynamics, we have constructed such a model by adding all possible
nonlocal interaction terms between velocity modes in Fourier space, and then
by taking its continuous limit. In real space, such a model corresponds to the
one-dimensional Burgers equation.

As it is well-known in the literature, Burgers equation cannot describe the
transition from an initial state to a state of fully developed hydrodynamic
turbulence. This is because the solution of Burgers equation with given
Dirichlet boundary conditions is unable to describe the spontaneous arise of
randomness by chaotic dynamics.

Nevertheless, we have shown that particular asymptotic scaling solutions of
Burgers equation, which do not satisfy Dirichlet boundary conditions, do
indeed well describe the behaviour of a two-dimensional fluid in a turbulent
state, after the transition to chaotic dynamics has happened. Such scaling
solutions successfully explain the main characteristic of the kinetic energy
spectrum at both small, intermediate, and large scales, as seen in both direct
numerical simulations and laboratory experiments. In particular:

($i$) If the velocity field admits a Maclaurin expansion at small wavenumbers,
then the energy spectrum is that given in Eq. (10) as theoretically predicted,
under particular assumptions, in the exact two-dimensional theory. However,
our model also admits the possible existence of a time-independent, power-law
spectrum of the form given in Eq. (63).

($ii$) The model predicts the existence of an inertial range, where the
dynamics in independent on initial conditions and viscosity. Moreover, the
energy spectrum is correctly given by the Batchelor spectrum (7).

($iii$) At small scales, corresponding to the dissipative range, the model
admits the existence of two different sub-dissipative ranges. In the “pre-
viscous damping range” [see Eq. (52)], where we expect the rate of energy
transfer due to nonlinear interactions to be not negligible, the energy decays
slower than in the “viscous damping range” [see Eq. (8)], where dynamics is
dominated by viscosity.

To our knowledge, the existence of a pre-viscous damping range has been
discussed in the literature only once, in 1981 by Tatsumi and Yanase Tatsumi .
Needless to say, further investigations are needed to confirm the existence
and characteristics of such a peculiar range in freely-decaying two-
dimensional turbulence.

##  VII Appendix: Saffman spectrum from Burgers equation

For $p=0$ ($q=1/2$) Eq. (57) reads

$u(\vec{k},t)=u_{i}L_{i}\Phi(\vec{k}\sqrt{u_{i}L_{i}t})$ (67)

and, in turns, Eq. (58) becomes

$z\,\frac{d\Phi(z)}{dz}=-\frac{i}{\sqrt{2\pi}}\,z\int_{-\infty}^{+\infty}\\!\\!dz^{\prime}\Phi(z^{\prime})\Phi(z-z^{\prime}),$
(68)

where $z=\vec{k}\sqrt{u_{i}L_{i}t}$. Taking the Fourier transform of the above
equation, we find

$2\hat{\Phi}\hat{\Phi}^{\prime}+\varpi\hat{\Phi}^{\prime}+\hat{\Phi}=0,$ (69)

where $\hat{\Phi}(\varpi)$ is the Fourier transform of $\Phi(z)$ and a prime
indicates the derivative with respect to $\varpi$. Equation (69) admits two
solutions

$\hat{\Phi}(\varpi)=-\frac{1}{2}\left(\varpi\pm\sqrt{\varpi^{2}+c^{2}}\,\right)$
(70)

specified by the $\pm$ sign, where $c\in\mathbb{R}$ is a constant of
integration. Since the limit $\vec{k}\rightarrow 0$ corresponds to
$\varpi\rightarrow\pm\infty$ (large scales), we must take the minus sign in
Eq. (70) in order to have a finite velocity field and then a finite energy
spectrum. In this case, we have

$\hat{\Phi}(\varpi)=\frac{c^{2}}{4\varpi}+\mathcal{O}(\varpi^{-3}).$ (71)

The inverse Fourier transform of the above equation reads

$\hat{\Phi}(z)=-\frac{i\sqrt{\pi}c^{2}}{4\sqrt{2}}\,\mbox{sgn}(z)+\mathcal{O}(z^{2}),$
(72)

where $\mbox{sgn}(x)$ is the sign function. The leading term of the energy
spectrum at large scales is then given by Eq. (64) with
$c_{0}=\pi^{2}c^{4}u_{i}^{2}L_{i}^{2}/32$. Comparing this expression for
$c_{0}$ with the one after Eq. (64), we find that $c$ is related to $\Phi(0)$
by $c=2(2/\pi)^{1/4}|\Phi(0)|^{1/2}$.

Declarations of interest: none

## References

  * (1) G. Boffetta and R. E. Ecke, Annu. Rev. Fluid Mech. 44, 427 (2012).
  * (2) D. Biskamp, Magnetohydrodynamic Turbulence, (Cambridge University Press, Cambridge, 2003).
  * (3) P. A. Davidson, Turbulence, (Oxford University Press, New York, 2004).
  * (4) L. Campanelli, Mod. Phys. Lett. B 3, no. 19, 1950218 (2019).
  * (5) G. K. Batchelor, Physics of Fluids Supp. II, 233 (1969).
  * (6) T. Tatsumi and S. Yanase, J. Fluid Mech. 110, 476-496 (1981).
  * (7) P. A. Davidson, J. Fluid Mech. 580, 431 (2007).
  * (8) P. Olesen, Phys. Lett. B398, 321 (1997).
  * (9) P. D. Ditlevsen, Turbulence and Shell Models (Cambridge University Press, Cambridge, UK, 2011).
  * (10) U. Frisch and J. Bec, in Les Houches 2000: New Trends in Turbulence; M. Lesieur, A.Yaglom and F. David, eds., pp. 341-383, Springer EDP-Sciences (2001), arXiv:nlin/0012033 [nlin.CD].
  * (11) M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, New York, 1972).
  * (12) R. H. Kraichnan, J. Fluid Mech. 67, 155, 1975.


More Exact Thermodynamics of Nonlinear Charged AdS Black Holes in 4D
Critical Gravity
Prosenjit<EMAIL_ADDRESS>Indian Institute of Engineering Science and Technology (IIEST), Shibpur-711103, WB, India
Sudhaker Upadhyay[Corresponding author][Visiting Associate, IUCAA Pune, Maharashtra-411007<EMAIL_ADDRESS>Department of Physics, K.L.S. College, Magadh University, Nawada, Bihar 805110, India
School of Physics, Damghan University, Damghan, 3671641167, Iran
Yerlan<EMAIL_ADDRESS>Department of General & Theoretical Physics,
LN Gumilyov Eurasian National University, Astana, 010008, Kazakhstan
Dharm Veer SinghVisiting Associate, IUCAA Pune, Maharashtra-411007, India<EMAIL_ADDRESS>Department of Physics, Institute of Applied Sciences and Humanities,
GLA University, Mathura 281406, Uttar Pradesh, India.
Kairat<EMAIL_ADDRESS>Department of General & Theoretical Physics,
LN Gumilyov Eurasian National University, Astana, 010008, Kazakhstan

In this paper, we investigate nonlinearly charged AdS black holes in four-dimensional critical gravity and study more exact black hole thermodynamics under the effect of small statistical fluctuations. We compute the correction to the thermodynamics of nonlinearly charged AdS black hole up to the leading order. We discuss the stability of  black holes under the circumstances of fluctuation and find that fluctuation causes instability in the black holes. Moreover, both the isothermal  and adiabatic
compressibilities are also derived.
Finally, we estimate the role of small fluctuations on the equation of states and  study the $P-v$ diagram of nonlinearly charged AdS black hole.

§ OVERVIEW AND MOTIVATION

The history of black hole thermodynamics is quite long. In 1972, Bekenstein conjectured that black
holes possess an entropy [1]. Later, in 1973, a relationship between black hole entropy and horizon area is established by Bekenstein [2]. There it was found that the   black hole entropy is proportional to the area of the event
horizon. In 1973, Bardeen, Carter, and Hawking proposed the four-laws of black hole mechanics [3] following the analogy with the four-laws of thermodynamics. In 1974, Hawking [4, 5] proposed that black holes emit thermal radiation and the temperature of the radiation is inversely proportional to black hole mass. Hawking and Page found a black hole solution in asymptotically AdS space [6] which possesses the thermodynamics properties like entropy, temperature, etc.

Black hole physics has been a fascinating subject of study for several decades and their thermodynamics has been an active area of research. Jacob Bekenstein proposed the so-called “entropy-area" law, which suggested that the entropy of a black hole is proportional to the area of its event horizon. Subsequent research in the 1990s and 2000s focused on understanding the quantum corrections to black hole entropy that arise due to thermal fluctuations and other quantum effects. The Cardy formula
[7, 8], introduced by John Cardy in 1986, stands out as one of the most significant breakthroughs in this domain. The Cardy formula provides a powerful tool for studying the connection between black hole thermodynamics and conformal field theory and has been used extensively to study black hole entropy. Another important development was the discovery of the AdS/CFT correspondence in 1997 by   Maldacena. This correspondence provides a way to understand the behavior of quantum systems in curved space-time, such as that near a black hole, in terms of a dual quantum field theory living on the boundary of space-time. This has led to important insights into the nature of black hole entropy and the holographic principle, which suggests that the properties of a system can be understood in terms of its boundary degrees of freedom.   Other interesting  black hole solutions are also studied [9, 10, 11, 12].

In 2002, Das computed [13] logarithmic corrections to the entropy. In recent years, there has been growing interest in studying the  logarithmic corrections to the entropy of black holes due to small statistical fluctuations around black hole equilibrium. Assuming that a black hole behaves as a thermodynamic system and this system should follow the equilibrium with thermal radiation. However, the logarithmic corrections to thermodynamic entropy arise for all thermodynamic systems when small statistical fluctuations around equilibrium are taken into account [13]. A nontrivial multiplicative factor to the expression for the density of states arises due to the small statistical fluctuations and the logarithm of these multiplicative factors leads to the corrections to the entropy. Thus, Bekenstein-Hawking entropy can be modified by logarithmic corrections that result from thermal fluctuations of the black hole around its state of equilibrium. These logarithm corrections to the entropy of the black hole are universal and apply to all kinds of black hole spacetime irrespective of whether they arise in Einstein's gravity or any higher-order theories of gravity. The inclusion of logarithmic corrections to the Bekenstein-Hawking entropy can be understood as the thermal fluctuations experienced by the black hole as it deviates from its stable state.

Recent studies have demonstrated that thermal fluctuations play a crucial role in understanding the behaviour of charged anti-de Sitter black holes [14], leading to corrections in their thermodynamic properties. In fact, a thorough analysis of black holes has revealed that the quantum approach to their thermodynamics at small scales is essential, resulting in a variety of corrections to thermodynamic quantities. Investigations into the effects of such corrections have been conducted for a range of black holes, including the Godel black hole [15], quasitopological black holes [16] and the Schwarzschild–Beltrami–de Sitter black hole [17], charged rotating black holes in AdS space [18],  Horava-Lifshitz black holes [19, 20], charged black holes in gravity rainbow [21], black holes in $f(R)$ gravity [22], rotating and charged BTZ black hole [23] and
Schwarzschild black hole
immersed in holographic quintessence [24] etc. The pioneering research of Frolov [25] has provided significant insights into the quantum corrections to black hole thermodynamics. The logarithmic corrections to the entropy of black holes have important consequences. One of the most significant is that they violate the area law of black hole entropy. The area law states that the entropy of a black hole is proportional to the area of its event horizon. However, the logarithmic corrections introduce additional terms in the entropy that are not proportional to the area of the event horizon.

The theory of nonlinear electrodynamics was first proposed [26, 27, 28, 29, 30] by Born and Infeld to remove the singularity of electromagnetic fields due to point particle. After Born-Infeld electrodynamics, a new model of nonlinear electrodynamics was proposed by Plebanski using antisymmetric conjugate tensor $P^{\mu \nu}$ (known as Plebanski tensor) and a structure-function $\mathcal{H} = \mathcal{H}(P, Q)$, where $P$  and $Q$ are the invariants formed with the antisymmetric conjugate Plebanski tensor. The structure-function $\mathcal{H}(P,Q)$ is related to the Lagrangian $\mathcal{L}(F,G)$ by the relation

\begin{equation}
\mathcal{H}(P,Q)= 2F\mathcal{L}_{F}(F,G) - \mathcal{L},
\end{equation}

and the Lagrangian is dependent on the invariant formed with
the Maxwell tensor $F^{\mu \nu}$.  Plebanski nonlinear
electrodynamics has been used to study nonlinear optics and
condensed matter physics. Plebanski theory has been studied
extensively in the context of gravitational theories and
regular nonrotating black hole solution has been obtained
[31, 32, 33, 34, 35]. Very
recently, a charged rotating black hole solution using Plebanski
the theory is also obtained in Refs.
[36, 37, 38]. Using
Plebanski nonlinear electrodynamics formalism is an interesting
model of nonlinearly charged AdS black hole was obtained in 4D
critical gravity [39] and its
logarithmic corrections to thermodynamics are not studied.
This provides us with an opportunity to fill this gap.

This work considers an AdS black hole in four-dimensional critical gravity coupled with nonlinear electrodynamics and discusses their thermal properties. Furthermore, we study the effects of thermal fluctuations on the thermodynamics of this black hole. In this regard, we compute first-order correction to the entropy of nonlinearly
charged AdS black holes. Next, we plot the entropy as a function of horizon radius for both cases with
and without considering thermal fluctuations. Here, we find that the thermal fluctuations affect the entropy of
small black holes significantly and for large black holes their impacts are negligible. Moreover, we compute the
corrected mass (enthalpy) of the system using the Hawking temperature and corrected entropy. The pressure
can be expressed in terms of the cosmological constant. So, the conjugate (corrected) thermodynamic volume of
a black hole is calculated using the expression of corrected mass. This can be done based on the fact that the
system must satisfy the first-law of thermodynamics. Once we have expressions of the corrected mass, volume, and entropy, it is a matter of calculation to compute corrected Helmholtz and Gibbs free energy. Here, we find
that the Helmholtz free energy decreases with horizon radius and the thermal fluctuations do not change the
nature of Helmholtz free energy. Thermal fluctuation decreases the Helmholtz free energy a bit. In
the case of Gibbs free energy, we find that for large black holes, Gibbs free energy takes a negative value. Also, we
notice that the effects of thermal fluctuation are significant for small black holes.

The stability of this black hole system is also studied. For this, we calculate the specific heat. The positive
value of specific heat suggests that the black hole is the stable state for the system in equilibrium. However,
due to small statistical fluctuation, the system undergoes to an unstable state for the small black holes. The thermal fluctuations do not affect the stability of large black holes. Next, we compute the effects of
thermal fluctuations on specific heat. The effects of thermal fluctuations on isothermal compressibility are also
computed. A phase transition occurs for the corrected isothermal compressibility from a positive to a negative
value at a critical horizon radius. We also compute the speed of sound for this black hole whose value ranges
from zero to one. Finally, we consider the system as a Van der Waals fluid and it is observed that the pressure is
discontinuous with respect to a specific volume of the black hole.

The main aim of this paper is to study correction on various thermodynamics parameters of nonlinearly
charged AdS black holes in 4D critical gravity when small statistical fluctuations around its equilibrium are
taken into account. In section <ref>, we study the black hole solution in critical gravity coupled with nonlinear electrodynamics. Within this section, we study uncorrected electric charge, electric potential, Hawking temperature, Wald entropy, the mass of the black hole, the thermodynamic volume of the black hole, free energy, and specific
heat. In section <ref>, we compute the effects of thermal fluctuations on various thermodynamic parameters. In
section <ref>, we study the charged AdS black hole in nonlinear electrodynamics as a Van der Waals fluid. Finally,
in section <ref>, we summarize our results.

§ THE METRIC AND THERMODYNAMICS

In this section, we recapitulate some of the known facts about nonlinear electrodynamics in critical gravity.
Let us begin by writing an action describing the theory of critical gravity coupled with nonlinear electrodynamics [39]

\begin{equation}\label{eq:2}
S[g_{\mu \nu} , A_{\mu}, P^{\mu \nu}]= \int d^{4}x \sqrt{-g} [\mathcal{L}_{CG} + \mathcal{L}_{NLE}] ,
\end{equation}

where $\mathcal{L}_{CG}$ and $\mathcal{L}_{NLE}$ are the Lagrangian of the critical gravity and nonlinear electrodynamics, respectively. Here, $\mathcal{L}_{CG}$ has the following form:

\begin{equation}\label{eq:3}
\mathcal{L}_{CG}= \frac{1}{2 \kappa} \Bigl( R - 2 \Lambda + \beta_{1} R^{2} +\beta_{2} R_{\mu \nu} R^{\mu \nu}  \Bigl) ,
\end{equation}

where $\kappa$ is the surface gravity, $R$ and $R_{\mu \nu}$ are the Ricci scalar and Ricci tensor, $\Lambda$ is the cosmological constant, $\beta_1$
and $\beta_2$ are the coupling constants. Critical gravity allows for the massive spin-zero fields to vanish if the coupling constants $\beta_1$ and $\beta_2$ are restricted to obey the relations [39]

\begin{equation}\label{eq:4}
\begin{split}
\beta_{2} &= -2\beta_1, \\
\beta_1 &=-\frac{1}{2\Lambda}.
\end{split}
\end{equation}

The expression for the Lagrangian describing the nonlinear electrodynamics is given by

\begin{equation}\label{eq:5}
\mathcal{L}_{NLE} = -\frac{1}{2} P^{\mu \nu} F_{\mu \nu} + \mathcal{H}(P,Q),
\end{equation}

where $P^{\mu \nu}$ is conjugate antisymmetric tensor known as Plebanski tensor and  structure-function $\mathcal{H}(P,Q)$,  where $P$ and
$Q$ are the invariants formed with the antisymmetric conjugate Plebanski tensor. The field strength tensor $F_{\mu \nu}$ is defined in terms of vector field $A_{\mu}$ as: $F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$. $\mathcal{H}(P)$ is  a structure-function depending on the invariant formed with the conjugated antisymmetric tensor.

Variation of the action (<ref>)  gives the following field equations:

\begin{equation}\label{eq:6}
\begin{split}
G_{\mu \nu} + \Lambda g_{\mu \nu} + \chi_{\mu \nu}^{CG} &=\kappa T_{\mu \nu}^{NLE}, \\
\nabla_{\mu} P^{\mu \nu} &=0,
\end{split}
\end{equation}

\begin{eqnarray}
\chi_{\mu \nu}^{CG}&= &2 \beta_{2} \Bigl( R_{\mu \rho} R_{\nu}^{ \rho} - \frac{1}{4} R^{\rho \sigma} R_{\rho \sigma} g_{\mu \nu} \Bigl) + 2 \beta_{1} R \Bigl( R_{\mu \nu} - \frac{1}{4} R g_{\mu \nu}  \Bigl) + \beta_{2} \Bigl( \Box R_{\mu \nu} \nonumber\\
& +& \frac{1}{2} \Box  g_{\mu \nu} -2 \nabla_{\rho} \nabla_{( \mu} R_{\nu )}^{\rho} \Bigl) +2 \beta_{1} \Bigl( g_{\mu \nu} \Box R - \nabla_{\mu} \nabla_{\nu} R \Bigl) ,\label{eq:7} \\
T_{\mu \nu}^{NLE} &=& \mathcal{H}_{p} P_{\mu \lambda} P_{\nu}^{\lambda} - g_{\mu \nu} \bigl( 2 P \mathcal{H}_{p} -\mathcal{H}  \bigl),\label{eq:8}
\end{eqnarray}

where $\beta_1$ and $\beta_2$ are given in equation (<ref>) and    $
\mathcal{H}_{P}   =\frac{\partial{\mathcal{H}}}{\partial{P}}$.
The  asymptotically AdS black hole metric is given by [39]

\begin{equation}\label{eq:10}
ds^{2}= - \frac{r^{2}}{l^{2}} f(r) dt^{2} + \frac{l^{2}}{r^{2}} \frac{ dr^{2}}{f(r)} + \frac{r^{2}}{l^{2}} d{\Omega_{2}}^{2},
\end{equation}

where cosmological constant $\Lambda= - \frac{3}{l^2}$, with the  asymptotic condition

\begin{equation}\label{eq:11}
\lim_{r \to \infty} f(r) =1.
\end{equation}

The nonlinear source is described by the structure-function $\mathcal{H}$ is real. Here we choose the structure-function depends only $P$, because we are interested in static configurations and $\mathcal{H}=\mathcal{H}(P)$ [39]

\begin{equation}\label{eq:12}
\mathcal{H}(P) = \frac{(\alpha_{2}^{2} - 3 \alpha_{1} \alpha_{3} ) l^{2} P}{3 \kappa} - \frac{2 \alpha_{2} (-2P)^{\frac{1}{4}}}{l \kappa} + \frac{\alpha_{2} \sqrt{-2P}}{\kappa},
\end{equation}

where $\alpha_{1}$, $\alpha_{2}$, and $\alpha_{3}$ are coupling constant. From second equation of (<ref>) one can obtain

\begin{equation}\label{eq:13}
\end{equation}

where $M$ is  an integration constant. Therefore, the
structure-function $\mathcal{H}$ in equation (<ref>) is real. Finally, using field equation (<ref>) one can obtain the function $f(r)$ as

\begin{equation}\label{eq:14}
f(r) = 1 - \alpha_{1} \sqrt{M} \frac{l}{r} +  \alpha_{2} {M} \frac{l^2}{r^2} - \alpha_{3} {M}^{\frac{3}{2}} \frac{l^3}{r^3}.
\end{equation}

It is shown in Ref. [39] that the structural coupling constants have a significant role in the characterization of
the solutions.

§ THERMODYNAMICS

In this section, we study the thermodynamics of nonlinearly charged AdS black holes in critical gravity. The metric of such a black hole is given in equation (<ref>). The electric charge of the black hole is calculated by [39]

\begin{equation}\label{eq:15}
Q= \frac{\Omega_{2} r_{h}^{2}}{\zeta^{2} l^4},
\end{equation}

where $r_{h}$ is the position of the horizon and  can be expressed as $ r_{h} = \zeta \sqrt{M} l$, where $\zeta$ is the roots of the  polynomial
$\zeta^{3} - \alpha_{1} \zeta^{2} + \alpha_{2} \zeta - \alpha_{3} =0$. The electric potential is

\begin{equation}\label{eq:16}
\Phi= \frac{r_{h}}{\kappa} \bigl( \alpha_{2} + \alpha_{1}^{2}  - \frac{3}{2} \alpha_{1} \zeta - \frac{\alpha_{1} \alpha_{2}}{\zeta} +\frac{\alpha_{2}^{2}}{3 \zeta^{2}}   \bigl).
\end{equation}

The Hawking temperature  due to surface gravity is calculated by

\begin{equation}\label{eq:17}
T_{H} = \frac{r_{h}}{4 \pi l^2}  \bigl( 3 - \frac{2 \alpha_{1}}{\zeta} + \frac{\alpha_{2}}{\zeta^{2}}  \bigl).
\end{equation}

The Wald entropy is given by [40, 39]

\begin{equation}\label{eq:18}
S_{0}= \frac{2 \Omega_{2} \pi}{\kappa} \Bigl( \frac{r_{h}}{l} \Bigl)^{2} \biggl( \frac{\alpha_{1}}{\zeta} - \frac{2 \alpha_{2}}{3 \zeta^{2}} \biggl),
\end{equation}

where $\Omega_2$ refers to the finite volume of the compact planar manifold.
The mass of the black hole in central gravity with nonlinear electrodynamics has the following expression [39]:

\begin{equation}\label{eq:19}
\mathcal{M} = \frac{\alpha_{1} \alpha_{2} \Omega_{2} r_{h}^{3}}{9 \kappa \zeta^{3} l^4}=\frac{64 \alpha_{1} \alpha_{2} \Omega_{2} \pi^{2} P^{2} r_{h}^{3}}{81 \kappa \zeta^{3}},
\end{equation}

This is equivalent to the enthalpy of the system. It is well known that the cosmological constant is responsible for pressure in AdS space, $P=-\Lambda/8 \pi= 3/8 \pi l^2$. Now, the mass of the black hole in terms of Wald entropy and charge written by

\begin{equation}\label{eq:20}
\mathcal{M}(S_0,Q) =\frac{\sqrt{6 \kappa} S_{0}^{3/2} \bigl(  3\zeta^{2} -2 \alpha_{1} \zeta + \alpha_{2} \bigl)}{12 \sqrt{\Omega_{2}} \pi^{3/2} \sqrt{3 \alpha_{1} \zeta -2 \alpha_{2}} l \zeta}  + \frac{Q^{3/2} l^2 \zeta \Psi }{9 \sqrt{\Omega_{2}} \kappa},
\end{equation}

where $\Psi$ is given by

\begin{equation}\label{eq:21}
\Psi= 6 \alpha_{2} +6 \alpha_1^2 -9\alpha_1 \zeta - \frac{6 \alpha_1 \alpha_2}{\zeta} + \frac{2 \alpha_2^2}{\zeta}.
\end{equation}

The conjugate volume of the black hole is [41]

\begin{equation}\label{eq:22}
V=\frac{\partial{M}}{\partial{P}}= \frac{128  \alpha_{1} \alpha_{2} \Omega_{2} \pi^{2} P r_{h}^{3}}{81 \kappa \zeta^{3}}.
\end{equation}

With the above-mentioned thermodynamical quantities, we can compute further properties of the black hole such as internal energy $(U)$, Helmholtz free energy $(F)$, and Gibbs free energy $(G)$. The internal energy is
calculated as

\begin{equation}\label{eq:23}
U= \mathcal{M} - PV= -\frac{\alpha_{1} \alpha_{1} \Omega_{2} r_{h}^{3} }{9 \kappa \zeta^{3} l^4}.
\end{equation}

Using the standard definition of Helmholtz free energy, we obtain

\begin{equation}\label{eq:24}
F=U-T_{H}S_0 = -\frac{\Omega_{2} r_{h}^{3} (27 \zeta^{3} \alpha_{1} -18 \zeta^{2} \alpha_{1}^{2}-18 \zeta^{2} \alpha_{2} +23 \alpha_{1} \alpha_{2} \zeta -6 \alpha_{2}^{2})}{18 \zeta^{4} \kappa  l^{4}} .
\end{equation}

Now, it is a matter of calculation to obtain the expression of the Gibbs free energy $ G=\mathcal{M} - T_HS_0-\Phi Q$. This reads

\begin{equation}\label{eq:26}
G= -\frac{3 \Omega_{2} r_{h}^{2}}{2 \kappa \zeta^{4} l^4}  \biggl( \alpha_{1} (r_{h} - 1) \zeta^{3} - \frac{2(\alpha_{1}^{2} +\alpha_{2} ) (r_{h} - 1) \zeta^{2}}{3}   +\frac{19 (r_{h} - \frac{18}{19}) \alpha_{1} \alpha_{2} \zeta}{27} - \frac{2 \alpha_{2}^{2} (r_{h} + 1)}{9}
\biggl).
\end{equation}

The specific heat of a black hole plays an important role in the stability of the system. Now, we calculate the specific heat at constant electric potential as

\begin{equation}\label{eq:27}
C_{\Phi} = T_{H}  \biggl(\frac{\partial{S_{0}}}{\partial{T}} \biggl)_{\Phi} ,
\end{equation}

\begin{equation}\label{eq:28}
C_{\Phi} = \frac{4 \Omega_{2} \pi  r_{h}^{2} (3 \alpha_{1} \zeta -2 \alpha_{2} )}{3 \kappa \zeta^{2}   l^{2}}.
\end{equation}

The temperature and specific heat will be positive if and only if

\begin{equation*}
\Psi_1= (3\zeta^{2} -2 \alpha_{1} \zeta + \alpha_{2}) \geq 0
\end{equation*}

\begin{equation}\label{eq:29}
\Psi_2=(3 \alpha_{1} \zeta -2 \alpha_{2} ) \geq 0.
\end{equation}

If $T_{H},\Phi \geq 0$, then above equation and
$     \Psi= \frac{\alpha_1 \alpha_2}{\zeta} - \frac{\Psi_1 \Psi_2}{\zeta^2} \geq 0,$
hold, where $\Psi$ is defined in equation (<ref>).

$\Psi_1$ denoted by red line, $\Psi_2$ denoted by blue line and $\Psi$ denoted by black line with $\zeta=1$.

The possible solutions for $\alpha_1$ and $\alpha_2$ are shown in Fig. <ref>. The region bounded by the red and blue curve in the first octant is the possible solution, except the region bounded by the black curve.

§  THERMODYNAMICS WITH FIRST-ORDER CORRECTION

In this section, we study the effect of thermal fluctuations on various thermodynamic parameters of black holes up to the first order. To the first order, correction to entropy was first studied in Ref. [13], which is logarithmic in nature. The logarithmic corrections to thermodynamic entropy arise when small stable fluctuations around equilibrium are taken into account. The logarithmic correction to the entropy of BTZ black hole, Schwarzschild AdS black hole & Reissner-Nordstrom black hole studied in Ref. [13, 23]. The partition function of a thermodynamics system is

\begin{equation}\label{eq:31}
Z(\beta) = \int_{0}^{\infty} \rho(E) e^{-\beta E} dE,
\end{equation}

where $\beta=1/T_{H}$. The density of states for fixed energy can be obtained from the above equation using inverse Laplace transformation

\begin{equation}\label{eq:32}
\rho= \frac{1}{2 \pi i} \int_{c - \textbf{i} \infty}^{c + \textbf{i} \infty} e^{\mathcal{S}(\beta)} d{\beta}.
\end{equation}

The above complex integral can be computed using the steepest descent method at saddle point $\beta_{0}$, and we obtain

\begin{equation}\label{eq:33}
\mathcal{S}(\beta)= S_{0} + \frac{1}{2} (\beta - \beta_0)^2 \biggl( \frac{\partial^2 \mathcal{S}_{0}}{\partial \beta^2} \biggl)_{\beta_{0}} + \cdots,
\end{equation}

where $\mathcal{S}(\beta)$ and $S_0$  is the exact entropy and zeroth order entropy. Substituting equation (<ref>) into equation (<ref>) we have

\begin{equation}\label{eq:34}
\rho= \frac{e^{S_0}}{2 \pi i} \int_{c - \textbf{i} \infty}^{c + \textbf{i} \infty} e^{\frac{1}{2} (\beta - \beta_0)^2 ( \frac{\partial^2 \mathcal{S}_{0}}{\partial \beta^2} )_{\beta_{0}}} d{\beta}.
\end{equation}

Finally, the above integral gives [13]

\begin{equation}\label{eq:35}
\rho(E)= \frac{e^{S_0}}{\sqrt{2 \pi \mathcal{S}^{\prime \prime}(\beta_{0})}}.
\end{equation}

Therefore, the exact entropy due to thermal fluctuation is

\begin{equation}\label{eq:36}
S= \ln{\rho}= S_0 - \frac{1}{2} \ln{\left( \frac{\partial^2 \mathcal{S}_{0}}{\partial \beta^2} \right)}.
\end{equation}

From [13] one can write the above entropy as

\begin{equation}\label{eq:37}
S = S_{0} - \frac{1}{2} \ln{(S_{0} T_{H}^{2})}.
\end{equation}

Therefore, entropy received correction due to thermal fluctuations. Now, to identify the effect of this correction term on other thermodynamical quantities, we  label the $1/2$ factor in the R.H.S of equation (<ref>) by $\gamma$. Fianlly, the corrected entropy becomes

\begin{equation}\label{eq:38}
S = S_{0} - \gamma \ln{(S_{0} T_{H}^{2})},
\end{equation}

where $\gamma=0$ refers to uncorrected entropy $S_0$ and $\gamma=1/2$ refers to corrected entropy in equation (<ref>). Therefore, the correction coefficient $\gamma$ can only take two values, i.e. $\gamma=0$ or $0.5$ & $\gamma$ is a dimensionless quantity. The correction coefficients $\gamma$ arise due to the thermal fluctuation in the equilibrium thermodynamics of black holes. This thermal nature leads to a prefactor in the expression for the density of states of the system, which in turn modifies the entropy of the black hole.

§.§ Corrected entropy

Using the relations    (<ref>), (<ref>) and  (<ref>),  we obtain

\begin{equation}\label{eq:39}
S = \frac{2 \Omega_{2} \pi  r_{h}^{2} (\alpha_{1} \zeta -\frac{2 \alpha_{2}}{3})}{\zeta^{2} \kappa  l^{2}}-\gamma  \ln  \Biggr[ \frac{\Omega_{2} r_{h}^{4} (\alpha_{1} \zeta -\frac{2 \alpha_{2}}{3}) (3 \zeta^{2}-2 \alpha_{1} \zeta +\alpha_{2} )^{2}}{8 \pi  \zeta^{6} \kappa  l^{6}}\Biggr].
\end{equation}

The effects of thermal fluctuation on entropy are depicted in Fig. <ref>. From the diagram <ref>, we observe that entropy increases with the horizon radius. The thermal fluctuation increases the entropy of the black hole significantly for small-sized black holes with horizon radius $r_h < 0.2$. Therefore, quantum effects significantly dominate for smaller-sized black holes with $r_h < 0.2$. However, for larger black holes with $r_h>0.2$, the effects of thermal fluctuations on the entropy are negligible.

Entropy vs. black hole horizon with $\zeta=1$, $\alpha_{1} = \alpha_{2}=2$ and $l=2$. Here $\gamma=0$ is denoted by a black line and $\gamma=0.5$ is denoted by a blue dash-dot line.

§.§ Corrected mass

Now, we analyze the effect of thermal fluctuation on the total mass (enthalpy) of the black holes. The corrected mass can be evaluated with the help of the following definition:

\begin{equation}\label{eq:40}
\mathcal{M}_{c} = \int T_{H} dS.
\end{equation}

Here, we have introduced the corrected entropy in place of equilibrium entropy. Substituting the value of Hawking temperature and corrected entropy from equations (<ref>) and (<ref>), respectively, into equation (<ref>), we have

\begin{equation}\label{eq:41}
\mathcal{M}_{c} = \frac{(3 \zeta^{2}-2 \alpha_{1} \zeta +\alpha_{2} ) r_{h} (3 \pi  \zeta  r_{h}^{2} \Omega_{2} \alpha_{1} -9 \gamma  \zeta^{2} \kappa  l^{2}-2 \pi  r_{h}^{2} \Omega_{2} \alpha_{2} )}{9 l^{4} \zeta^{4} \pi  \kappa}.
\end{equation}

Mass(enthalpy) Vs.  black hole horizon with $\zeta=1$, $\alpha_{1} = \alpha_{2}=2$ and $l=2$. Here $\gamma=0$ is denoted by a black line and $\gamma=0.5$ is denoted by a blue dash-dot   line.

Now, to do a comparative analysis, we plot the corrected mass and equilibrium mass with respect to the horizon radius in Fig. <ref>. From Fig. <ref>, we see that the mass is an increasing function of the horizon radius. Interestingly, we find that the thermal fluctuations decrease the mass a bit but do not change the behavior of the mass.

§.§ Corrected thermodynamic volume of black hole

In this subsection, we study the corrected thermodynamic volume of the black hole as a function of pressure and horizon radius. The thermodynamic volume of an asymptotically AdS black hole is defined as [41]

\begin{equation}\label{eq:42}
V=\biggl( \frac{\partial{\mathcal{M}_{c}}}{\partial{P}} \biggl)_{S_{0},Q},
\end{equation}

To compute this equation, we first write the corrected mass in terms of pressure. Since pressure depends on the cosmological constant. So, the corrected mass  (<ref>) can be expressed in terms of pressure as follows:

\begin{equation}\label{eq:43}
\mathcal{M}_{c} = \frac{64 (3 \zeta^{2}-2 \alpha_{1} \zeta +\alpha_{2} ) r_{h} (3 \pi  \zeta  r_{h}^{2} \Omega_{2} \alpha_{1} -\frac{27 \gamma  \zeta^{2} \kappa}{8 \pi  P}-2 \pi  r_{h}^{2} \Omega_{2} \alpha_{2} ) \pi  P^{2}}{81 \zeta^{4} \kappa}.
\end{equation}

Substituting the value of (<ref>) in Eq. (<ref>), we obtain the corrected thermodynamic volume of black hole as

\begin{equation}\label{eq:44}
V_{c}=\frac{8 (3 \zeta^{2}-2 \alpha_{1} \zeta +\alpha_{2} ) r_{h} (48 P \pi^{2} \zeta  r_{h}^{2} \Omega_{2} \alpha_{1} -32 P \pi^{2} r_{h}^{2} \Omega_{2} \alpha_{2} -27 \gamma  \zeta^{2} \kappa )}{81 \zeta^{4} \kappa},
\end{equation}

This further simplifies to

\begin{equation}\label{eq:45}
V_{c}=\frac{8 (3 \zeta^{2}-2 \alpha_{1} \zeta +\alpha_{2} ) r_{h} (6 \pi  \zeta  r_{h}^{2} \Omega_{2} \alpha_{1} -9 \gamma  \zeta^{2} \kappa  l^{2}-4 \pi  r_{h}^{2} \Omega_{2} \alpha_{2} )}{27 l^{2} \zeta^{4} \kappa}.
\end{equation}

To study the behavior of thermodynamic volume and their dependency on thermal fluctuations, we plot Fig. <ref>. We find that the thermodynamic volume of the black hole increases with the horizon radius. The thermal fluctuation decreases the volume of the black holes which becomes significant in larger black holes.

Thermodynamic volume of the black hole Vs.  black hole horizon with $\zeta=1$, $\alpha_{1} = \alpha_{2}=2$ and $l=2$. Here $\gamma=0$ is denoted by the black line, and $\gamma=0.5$ is denoted by the blue dash-dot line.

§.§ Corrected Helmholtz free energy

In this subsection, we study the corrected Helmholtz free energy of the AdS black hole due to thermal fluctuation. The Helmholtz free energy is defined by

\begin{equation}\label{eq:46}
F_{c}= U - T_{H} S.
\end{equation}

By plugging the values from Eqs. (<ref>), (<ref>) and (<ref>), the above expression leads to

\begin{equation}\label{eq:47}
F_{c}= \frac{r_{h} (3 \zeta^{2}-2 \alpha_{1} \zeta +\alpha_{2} )  \biggr[9 \gamma  \ln  \biggl(\frac{\Omega_{2} r_{h}^{4} (3 \alpha_{1} \zeta -2 \alpha_{2} ) (3 \zeta^{2}-2 \alpha_{1} \zeta +\alpha_{2} )^{2}}{24 \pi  \zeta^{6} \kappa  l^{6}}\biggl) \zeta^{2} \kappa  l^{2}-30 \pi  \zeta  r_{h}^{2} \Omega_{2} \alpha_{1} +20 \pi  r_{h}^{2} \Omega_{2} \alpha_{2}  \biggr]}{36 l^{4} \zeta^{4} \pi  \kappa}.
\end{equation}

Helmholtz free energy  Vs.  black hole horizon with $\zeta=1$, $\alpha_{1} = \alpha_{2}=2$ and $l=2$. Here $\gamma=0$ is denoted by a black line and $\gamma=0.5$ is denoted by a blue dash-dot line.

To study the behavior of Helmholtz free energy and their dependencies on thermal fluctuation, we plot Fig. <ref>. From the figure, it is evident that the Helmholtz free energy decreases with the horizon radius. The thermal fluctuation does not change the nature of Helmholtz free energy. Thermal fluctuation decreases the Helmholtz free energy a bit.

§.§ Corrected Gibbs free energy

Another important thermal quantity that plays important role in the discussion of the stability of black holes is Gibbs free energy. The Gibbs free energy is defined by

\begin{equation}\label{eq:48}
G_{c} = \mathcal{M}_{c} -T_{H}S_{c} - {\Phi} Q.
\end{equation}

Substituting the expression of $\mathcal{M}_{c}$, $T_{H}$, $S$, $\Phi$ and $Q$ from equations (<ref>), (<ref>), (<ref>), (<ref>) and  (<ref>) into above equation, we have

\begin{eqnarray}
G_{c} &=& -\frac{\biggl(-9 r_{h} \alpha_{1} \zeta^{3}+6 r_{h} (\alpha_{1}^{2}+\alpha_{2} ) \zeta^{2}-6 r_{h} \alpha_{1} \alpha_{2} \zeta -2 r_{h} \alpha_{2}^{2}\biggl) \Omega_{2} r_{h}}{6 \kappa  \zeta^{4} l^{4}}\nonumber\\
&-&\frac{ \biggl(3 \zeta^{2}-2 \alpha_{1} \zeta +\alpha_{2} \biggl) \Biggr[\frac{2 \Omega_{2} \pi  r_{h}^{2} (\alpha_{1} \zeta -\frac{2 \alpha_{2}}{3})}{\zeta^{2} \kappa  l^{2}}-\gamma  \ln  \biggl(\frac{\Omega_{2} r_{h}^{4} (\alpha_{1} \zeta -\frac{2 \alpha_{2}}{3}) (3 \zeta^{2}-2 \alpha_{1} \zeta +\alpha_{2} )^{2}}{8 \pi  \zeta^{6} \kappa  l^{6}}\biggl)\Biggr] r_{h}}{4 l^{2} \zeta^{2} \pi}\nonumber\\
& +&\frac{\biggl(3 \zeta^{2}-2 \alpha_{1} \zeta +\alpha_{2} \biggl)  \biggl(3 \pi  \zeta  r_{h}^{2} \Omega_{2} \alpha_{1} -9 \zeta^{2} \gamma  \kappa  l^{2}-2 \pi  r_{h}^{2} \Omega_{2} \alpha_{2} \biggl) r_{h} }{9 l^{4} \zeta^{4} \pi  \kappa}.\label{eq:49}
\end{eqnarray}

[][]Gibbs free energy  Vs.  black hole horizon with $\zeta=1$, $\alpha_{1} = \alpha_{2}=2$ and $l=2$. Here $\gamma=0$ is denoted by a black line and $\gamma=0.5$ is denoted by a blue dash-dot line.

To do a comparative analysis of corrected Gibbs free energy with its equilibrium value, we plot a diagram <ref>. The effect of the correction terms
is significant for small black holes which are depicted in Fig. <ref> (a). However, the behavior of Gibbs free energy for the large horizon radius is denoted in
Fig. <ref> (b).
From the diagram, it is obvious that Gibbs free energy starts from zero and takes the maximum positive corrected Gibbs free energy with $\gamma=0.5$ value and then starts decreasing towards a negative value. It means that for larger black holes uncorrected Gibbs free energy takes a negative value.

§.§ Stability and specific heat

In this subsection, we check the stability of the black hole by estimating the specific heat of the black hole. The specific heat of the black hole is defined by

\begin{equation}\label{eq:50}
\Bigl( C_{\Phi} \Bigl)_{c} = T_{H} \frac{\partial{S}}{\partial{T_{H}}}=T_{H} \frac{\partial{S}/\partial r_{h}}{\partial{T_{H}}/ \partial r_{h}}.
\end{equation}

Using equations (<ref>) and (<ref>)  Specific heat takes the following form:

\begin{equation}\label{eq:51}
\Bigl( C_{\Phi} \Bigl)_{c} = \biggr[\frac{4 \Omega_{2} \pi  r_{h} (\alpha_{1} \zeta -\frac{2 \alpha_{2}}{3})}{\zeta^{2} \kappa  l^{2}}-\frac{4 \gamma}{r_{h}}\biggr]r_{h}.
\end{equation}

The stability of a black hole is determined by the condition $C_{\Phi} \geq 0 $. Now, we plot the specific heat with respect to the horizon radius to see the signature. From Fig. <ref>, we observe that the black hole is stable for $\gamma=0$. However, the thermal fluctuation causes instability to the small ($r_h<0.13$) black holes, i.e. black holes with small horizon radii are thermodynamically locally unstable. A transition from negative specific heat to positive one occurs at $r_h=0.13$ Therefore, black holes with horizon radius $r_h>0.13$ are thermodynamically stable. As the horizon increases thermal fluctuation becomes ineffective to the specific heat, both corrected & uncorrected specific heat coincide for $r_h >>0.13$.

Specific heat Vs.  black hole horizon with $\zeta=1$, $\alpha_{1} = \alpha_{2}=2$ and $l=2$. Here $\gamma=0$ is denoted by a black line and $\gamma=0.5$ is denoted by a blue dash-dot line.

§.§ Corrected isothermal compressibility

In this subsection, we study the effects of thermal fluctuation on isothermal compressibility and adiabatic compressibility. Let us first define the isothermal compressibility of black hole [42]

\begin{equation}\label{eq:52}
\beta_{T} = -\frac{1}{V_{c}} \biggl( \frac{\partial{V{c}}}{\partial{P}} \biggl)_{T}.
\end{equation}

Using equations (<ref>) and (<ref>) isothermal compressibility takes the following form:

\begin{equation}\label{eq:53}
\beta_{T} = -\frac{48 \pi^{2} \zeta  r_{h}^{2} \Omega_{2} \alpha_{1} -32 \pi^{2} r_{h}^{2} \Omega_{2} \alpha_{2}}{48 P \pi^{2} \zeta  r_{h}^{2} \Omega_{2} \alpha_{1} -32 P \pi^{2} r_{h}^{2} \Omega_{2} \alpha_{2} -27 \gamma  \zeta^{2} \kappa}.
\end{equation}

Isothermal compressibility Vs.  black hole horizon with $\zeta=1$, $\alpha_{1} = \alpha_{2}=2$ and $l=2$. Here $\gamma=0$ is denoted by the black line and $\gamma=0.5$ is denoted by a blue dash-dot line.

Now, we plot the isothermal compressibility of nonlinearly charged AdS black hole in fourth dimensions critical gravity in Fig. <ref>. Here, we find that for $\gamma=0$ (equilibrium) isothermal compressibility takes constant
negative value. However, for $\gamma=0.5=0$ (considering thermal fluctuations), a phase transition for the isothermal compressibility occurs which takes a positive value for small-sized black holes and a negative value for massive black holes.

The adiabatic compressibility of a black hole is defined as [42]

\begin{equation}\label{eq:54}
\beta_{S} = -\frac{1}{V_{c}} \biggl( \frac{\partial{V{c}}}{\partial{P}} \biggl)_{S}=0.
\end{equation}

Here, we find that the adiabatic compressibility is zero.
A speed of sound can be calculated for the black hole from the given formula

\begin{equation}\label{eq:55}
v_{S}^{-2}= \biggl( \frac{\partial{\rho}}{\partial{P}} \biggl)_{S},
\end{equation}

where $\rho$ refers to the density of the black hole. This simplifies to

\begin{equation}\label{eq:56}
v_{s}^{-2}= \frac{1152 P^{2} r_{h}^{4} (\alpha_{1} \zeta -\frac{2 \alpha_{2}}{3})^{2} \pi^{4} \Omega_{2}^{2}-1296 \zeta^{2} P r_{h}^{2} (\alpha_{1} \zeta -\frac{2 \alpha_{2}}{3}) \gamma  \pi^{2} \kappa  \Omega_{2} +729 \zeta^{4} \gamma^{2} \kappa^{2}}{2304 (P r_{h}^{2} (\alpha_{1} \zeta -\frac{2 \alpha_{2}}{3}) \pi^{2} \Omega_{2} -\frac{9 \gamma  \zeta^{2} \kappa}{16})^{2}},
\end{equation}

where value of $v_{s}^{2}$ ranges from zero to one. From Fig. <ref>, we find that the thermal fluctuation increases the speed of sound for smaller black holes. However, for large black holes ($r_h >>1$) speed is constant and the effects of thermal fluctuation are not significant.

$ v_{s}^{-2}$ Vs.  black hole horizon with $\zeta=1$, $\alpha_{1} = \alpha_{2}=2$ and $l=2$. Here $\gamma=0$ is denoted by a black line and $\gamma=0.5$ is denoted by a blue dash-dot line.

§ VAN DER WAALS BLACK HOLES

In this section, we study the behavior of charged AdS black holes in nonlinear electrodynamics as a Van der Waals fluid. The Van der Waals equation of state describes real fluids and modifies the ideal gas equation of states as

\begin{equation}\label{eq:57}
\Bigl( P+\frac{a}{v^2} \Bigl) (v - b) = T,
\end{equation}

where $ v = \frac{V}{N} $ is the specific volume of the fluid. In the case of a black hole,  $N = A/l_{p}^{2}$ denotes the number of degrees of freedom associated with the black hole horizon. Constant represents the interaction between the molecules of a given fluid and constant b represents the nonzero size of molecules. The specific volume of black hole [43] is given by

\begin{equation}\label{eq:58}
v = 6 \frac{V_{c}}{N}.
\end{equation}

This simplifies to

\begin{equation}\label{eq:59}
v = \frac{(3 \zeta^{2}-2 \alpha_{1} \zeta +\alpha_{2} ) (48 \pi^{2} P  \zeta  r_{h}^{2} \Omega_{2} \alpha_{1} -32 \pi^{2} P  r_{h}^{2} \Omega_{2} \alpha_{2} -27 \gamma  \zeta^{2} \kappa )}{36 r_{h} \zeta^{2} \pi^{2} P \Omega_{2} (\alpha_{1} \zeta -\frac{2 \alpha_{2}}{3})}.
\end{equation}

The above equations yield the pressure as

\begin{equation}\label{eq:60}
P = -\frac{3 \gamma  \zeta^{2} \kappa  (3 \zeta^{2}-2 \alpha_{1} \zeta +\alpha_{2} )}{4 r_{h} \Omega_{2} \pi^{2} \Bigl\{ (v -4 r_{h} ) \zeta^{2}+\frac{8 \zeta  r_{h} \alpha_{1}}{3}-\frac{4 r_{h} \alpha_{2}}{3} \Bigl\}  (\alpha_{1} \zeta -\frac{2 \alpha_{2}}{3}) }.
\end{equation}

[][]Pressure Vs. specific volume of  black hole with $\zeta=1$, $\alpha_{1} = \alpha_{2}=2$, $r_{h} =1$ and $l=2$.  Left panel: $r_h = 1$. Right panel:
$r_h = 10$ Here $\gamma=0.5$ is denoted by blue dash-dot line.

We plot $P-v$ diagram with $r_h = 1$ and $r_h = 10$ as depicted in Fig. <ref>. We find that when the thermal fluctuations are not taken into account the pressure remains zero, but in the presence of thermal fluctuations a phase transition of the pressure of the black hole occurs from a negative value to a positive value. In Fig. <ref>(a) pressure is depicted with $r_h =1$ when specific volume $v \leq 1$ pressure takes a negative value, a phase transition occurs when $v > 1$ and finally pressure takes a positive value. A similar kind of behaviour is shown in Fig. <ref>(b) with $r_h=10$.

The equation (<ref>) leads to the pressure as

\begin{equation}\label{eq:61}
P = \frac{3 T \xi^{2}}{2 r_{h} (3 \zeta^{2}-2 \alpha_{1} \zeta +\alpha_{2} )}.
\end{equation}

Comparing equations (<ref>) and (<ref>) we can conclude that $a=0$.   Black hole mimicking the ideal gas behaviour.

§ CONCLUSIONS

In this paper, we have studied the effects of small statistical fluctuation on the equilibrium thermodynamics of nonlinearly charged AdS black holes in four-dimensional critical gravity. To do so, we computed the Hawking temperature for the black holes. The entropy of the black hole has an additional term at first order due to the thermal fluctuations. With the help of  Hawking temperature and corrected entropy, we have computed the
more exact Helmholtz free energy of the black hole. The corrected Helmholtz free energy of the black hole takes a negative value. The equilibrium Gibbs free energy of the black hole is positive for small-sized black holes and for larger black holes, in contrast, it takes negative values. The corrected Gibbs free energy of the black hole is positive. We have found that
the black hole is stable in absence of thermal fluctuations. However, in presence of  thermal fluctuations,   the
small-sized black hole becomes unstable and the large-sized black hole remains stable. Incidentally, the internal energy has not found any correction at the leading order.

On the other hand, we have also computed the corrected isothermal compressibility for the black hole. The equilibrium isothermal compressibility has found a constant negative value. However, when thermal fluctuations are taken into account, the isothermal compressibility takes a positive value for small-sized black holes and a negative for large black holes.
Finally, we have studied the $P-v$ diagram of a black hole and found that  the thermodynamic pressure vanishes for the system in equilibrium. However, the thermodynamic pressure is negative/positive for small/large-sized (with respect to specific volume) black holes with non-vanishing correction parameters.

§ ACKNOWLEDGEMENTS

This research was funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP19674521).   D.V.S.  thanks University Grant Commission for the Start-Up Grant No. 30-600/2021(BSR)/1630.

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