# Configuration entropy of a rotating quark-gluon plasma from holography

Nelson R. F. Bragaa<EMAIL_ADDRESS>a UFRJ — Universidade Federal do Rio de
Janeiro, Instituto de Física,
Caixa Postal 68528, Rio de Janeiro, Brasil Luiz F. Ferreirab
<EMAIL_ADDRESS>a UFRJ — Universidade Federal do Rio de Janeiro,
Instituto de Física,
Caixa Postal 68528, Rio de Janeiro, Brasil Octavio C. Junqueiraa
<EMAIL_ADDRESS>a UFRJ — Universidade Federal do Rio de Janeiro,
Instituto de Física,
Caixa Postal 68528, Rio de Janeiro, Brasil

###### Abstract

The configuration entropy (CE) provides a measure of the stability of physical
systems that are spatially localized. An increase in the CE is associated with
an increase in the instability of the system. In this work we apply a recently
developed holographic description of a rotating plasma, in order to
investigate the behaviour of the CE when the plasma has angular momentum.
Considering the holographic dual to the plasma, namely a rotating AdS black
hole, the CE is computed at different rotational speeds and temperatures. The
result obtained shows not only an increase with the rotational speed $v$ but,
in particular, a divergence of the CE as $v$ approaches the speed of light:
$\,v\to 1$. We discuss the results obtained showing that they are consistent
with the change in the geometry of the black hole caused by the rotation and
the corresponding variation of the volume of the dual plasma. We also connect
the results found here with those obtained in a recent work, where it was
shown that the complete dissociation of heavy mesons in a plasma is
represented by a positive singularity in the CE.

##  1 Introduction

In the last decades, one of the focus of the community devoted to the study of
strong interactions physics is the search for understanding the Quark-Gluon
Plasma (QGP), a state of matter formed by deconfined partons that interact
strongly[1]. The QGP is formed experimentally through ultra-relativistic
collisions of heavy nuclei produced in particle accelerators. There are many
properties that affect the behaviour of QGP, like temperature, density and the
presence of magnetic fields. Another, less studied, property is the rotation
of the plasma. When the collision is non-central, the resulting system
acquires not only a strong magnetic field but also angular momentum. A
fraction of this angular momentum can be transferred to the polarization of
the strange quarks in the QGP due to spin-orbit interaction, as discussed in
[2]. Until recently, no experimental signals of this effect had been detected.
Only in 2017, the global hyperon polarization has been observed in $Au+Au$
collisions at RHIC [3]. This discovery opened a new window to study the
properties of rotating Quantum Chromodynamics (QCD) matter. In particular, the
influence of the rotation on the QCD phase diagram have become a topic under
active investigations. Some phenomenological models have recently been applied
to the study of the effect of rotation in the QGP as, for example, Refs. [4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].

In particular, using the AdS/QCD approach [11, 17], it was found that the
critical temperature of the confinement/deconfinement transition decreases
with increasing angular velocity, which is in agreement with others
phenomenological models as the Nambu-Jona Lasinio (NJL). In contrast, the
simulations of relativistic rotation on the confinement/deconfinement phase
transition in gluodynamics lattice [13, 14] showed that the critical
temperature increases with increasing angular velocity.

Here we are interested in studying how rotation affects the stability of the
quark-gluon plasma. So, it is important to make it clear what does one mean by
stability in this case. There are two aspects to be considered. The first is
the fact that the plasma phase of QCD matter exists, or is stable, for
temperatures above some critical value $T_{c}$. For lower temperatures, QCD
matter is in the confined phase[21, 22]. The other instability is related to
the Hawking radiation. The higher is the temperature of the black hole dual to
the plasma, the stronger is the emission of radiation and the corresponding
loss of energy, resulting in an increase in the instability.

A rotating plasma with uniform rotational speed is described holographically
by a rotating black hole with cylindrical symmetry, like those studied in [23,
24]. The rotation is obtained by a coordinate transformation and the
holographic model obtained predicts that plasma rotation affects the critical
temperature of confinement/deconfinement transition [17]. This fact motivated
the present study concerning the stability of the rotating plasma, but now
considering the configuration entropy (CE) approach.

The idea of using the CE as an indicator of stability of physical systems
started in [25, 26, 27]. Afterwards many examples appeared in the literature
where the CE plays the role of representing stability in different physical
systems. For instance, in the context of AdS/QCD approach, the CE has provided
new results involving thermal behaviour of quarkonium in a medium [28, 29, 30,
31, 32], the mass spectra of several particles [33, 34, 35, 36, 37, 38, 39],
nuclear electromagnetic transitions [40] and in confinement/deconfinement
transition [41, 42, 43]. In particular, in ref. [32] it was shown how does the
CE marks the total dissociation of quasi-particles in a medium. Namely, the CE
diverges when the particles are completely dissociated. Other applications in
astrophysics, cosmology, AdS/CFT correspondence, nuclear physics and field
theory can be found, for example, in references [44, 45, 46, 47, 48, 49, 50,
51, 52, 53, 54, 55, 56, 57].

The CE is based on the Shannon information entropy [58], in the continuum
limit. It is in general defined in terms of the energy density of the physical
state in Fourier space. Assuming that the black hole is represented by the
grand canonical ensemble, which is consistent with the fact that the plasma is
rotating, one can find an expression for the rotating black hole energy
density as a function of the holographic coordinate, the temperature and the
rotational speed. The Fourier transform does not possess an analytical
solution, so that the results in this work were obtained through the
application of numerical methods.

The results obtained show an interesting nontrivial aspect. For a fixed
temperature, the CE diverges as the rotational speed of the black hole
approaches the asymptotic limit corresponding to the speed of light. As we
will discuss, there is a consistent interpretation for this behaviour.

This work is organized as follows: in Section 2, we describe the geometry of a
rotating cylindrical black hole. Section 3 is devoted to the construction of
the rotating plasma energy density, in the soft wall AdS/QCD model, assuming
that the system is represented by the grand canonical ensemble. In Section 4,
we compute the CE of the rotating plasma at different rotational velocities
and temperatures, and relate the results with the stability of the system.
Section 5 contains our conclusions and the appendix shows a table with
additional results for the CE as a function of the temperature and rotational
speed.

##  2 AdS black holes with cylindrical symmetry

At finite temperature, the anti-de Sitter space with radius $L$ possesses two
solutions given by the following metrics with compact time direction:

$ds^{2}=\frac{L^{2}}{z^{2}}\left(dt^{2}+d\overrightarrow{x}^{2}+dz^{2}\right)\;,$
(2.1)

and

$ds^{2}=\frac{L^{2}}{z^{2}}\left(f(z)dt^{2}+d\overrightarrow{x}^{2}+\frac{dz^{2}}{f(z)}\right)\;,$
(2.2)

with $f(z)=1-z^{4}/z_{h}^{4}$. The first geometry (2.1) corresponds to the
thermal AdS space, while the second one (2.2) to the AdS black hole (BH)
geometry, being $z_{h}$ the location of the horizon. Both geometris are
solutions of Einstein’s equations with negative cosmological constant
$\Lambda=-12/L^{2}$ and constant curvature $R=-20/L^{2}$.

The time coordinate of the BH geometry has a period $\beta$, related to the
horizon position and to the Hawking temperature, $T=1/\beta=1/(\pi z_{h})$
[59]. Requiring that the asymptotic limits of the two geometries at
$z=\epsilon$, with $\epsilon\to 0$, are the same, one finds that the period of
the time component of the thermal AdS space is $\beta^{\prime}=\pi
z_{h}\sqrt{f(\epsilon)}$.

###  2.1 Rotating cylindrical black hole

In order to represent a QGP rotating with homogeneous speed, we take the
metrics (2.1) and (2.2) in cylindrical coordinates

$ds^{2}=\frac{L^{2}}{z^{2}}\left(-dt^{2}+l^{2}d\phi^{2}+\sum_{i=1}^{2}dx_{i}^{2}+dz^{2}\right)\;,$
(2.3)

and

$ds^{2}=\frac{L^{2}}{z^{2}}\left(-f(z)dt^{2}+l^{2}d\phi^{2}+\sum_{i=1}^{2}dx_{i}^{2}+\frac{dz^{2}}{f(z)}\right)\;,$
(2.4)

where $l$ is the radius of a hyper-cylinder and $0\leq\phi\leq 2\pi$. Note
that these new metrics have a topology that is different from those of metrics
(2.1) and (2.2) since one of the spatial coordinates has been compactfied.
Then, we perform a coordinate transformation to an observer for which the
angular coordinate is varying uniformly with time around a cylinder with
radius $l$, namely,

$\displaystyle t$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{\sqrt{1-l^{2}\omega^{2}}}\left(t+l^{2}\omega\phi\right)\;,$
(2.5) $\displaystyle\phi$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{\sqrt{1-l^{2}\omega^{2}}}\left(\phi+\omega t\right)\;.$
(2.6)

After a straightforward calculation, one obtains the rotating cylindrical
black hole metric in canonical form:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-N(z)dt^{2}+\frac{L^{2}}{z^{2}}\frac{dz^{2}}{f(z)}+R(z)\left(d\phi+P(z)dt\right)^{2}+\frac{L^{2}}{z^{2}}\sum_{i-1}^{2}dx_{i}^{2}\;,$
(2.7)

with

$\displaystyle N(z)$ $\displaystyle=$
$\displaystyle\frac{L^{2}}{z^{2}}\frac{f(z)(1-\omega^{2}l^{2})}{1-f(z)\omega^{2}l^{2}}\;,$
(2.8) $\displaystyle R(z)$ $\displaystyle=$
$\displaystyle\frac{L^{2}}{z^{2}}\left(\gamma^{2}l^{2}-f(z)\gamma^{2}\omega^{2}l^{4}\right)\;,$
(2.9) $\displaystyle P(z)$ $\displaystyle=$
$\displaystyle\frac{\omega(1-f(z))}{1-f(z)\omega^{2}l^{2}}\;,$ (2.10)

where $\omega$ is the angular velocity of the rotating cylindrical black hole,
being $\gamma$ the Lorentz factor,

$\gamma(\omega l)=\frac{1}{\sqrt{1-l^{2}\omega^{2}}}\;.$ (2.11)

This metric represents, via holography, a plasma that rotates with the same
angular velocity $\omega$ of the rotating black hole, around a cylinder with
radius $l$. As shown in [17], the metric (2.7) is a solution of the same
Einstein equation satisfied by the AdS black hole metrics (2.2) and (2.4). For
the cylindrical thermal AdS space of eq. (2.3), the rotating version of the
metric has the same form of (2.7) but using $f(z)=1$ in eqs. (2.8)-(2.10).

It is important to remark that in the plasma formed in heavy ion collisions
the rotational speed is not uniform. The QGP formed in accelerators clearly
does not have a cylindrical form. What we are doing here is considering a
simpler situation where the speed is the same for all the parts of the plasma
in order to understand the qualitative effects of the plasma rotation. This
approach has some similarity with the one used for studying the effect of
magnetic fields on the plasma, where in general it is considered that the
field is uniform [60, 61, 62, 63, 32, 64, 65, 66, 67, 68, 69, 70], although
the actual fields acting on the QGP formed in non central heavy ion collisions
are not uniform.

Then, defining $h_{00}=-N(z)$, the Hawking temperature can be obtained from
the surface gravity formula [12]:

$\displaystyle T$ $\displaystyle=$
$\displaystyle|\frac{\kappa_{G}}{2\pi}|=\bigg{|}\frac{\lim_{z\rightarrow
z_{h}}-\frac{1}{2}\sqrt{\frac{g^{zz}}{-h_{00}(z)}}h_{00,z}}{2\pi}\bigg{|}=\frac{1}{\pi
z_{h}}\sqrt{1-\omega^{2}l^{2}}\;,$ (2.12)

where $\kappa_{G}$ is the surface gravity, and $g^{zz}$ the $zz$ component of
the inverse of the cylindrical black hole metric. The expression of Eq. (2.12)
shows that the rotation affects the relation between the Hawking temperature
$T$ of the black hole and its horizon position. This property will be
important to understand the relation between the dynamics of the rotating BH
geometry, the stability of the plasma and its configuration entropy at
different temperatures.

##  3 Energy density in the soft wall AdS/QCD model

The soft wall holographic AdS/QCD model [71] is built from the introduction in
the AdS geometry of a dilaton background $\Phi(z)=cz^{2}$, where $c$ is a
parameter with dimension of energy squared. In the gauge theory side of the
gauge/gravity duality $\sqrt{c}\,$ plays the role of an infrared energy (IR)
cutoff.

###  3.1 Regularized action density

One can write the five-dimensional gravitational action, at zero temperature,
in the general form[21, 22]

$I=-\frac{1}{2\kappa^{2}}\int_{0}^{z_{f}}dz\int
d^{4}x\sqrt{g}e^{-\Phi}\left(R-\Lambda\right)\;=\frac{4}{L^{2}\kappa^{2}}\int_{0}^{z_{f}}dz\int
d^{4}x\sqrt{g}e^{-cz^{2}},$ (3.1)

where $\kappa$ is the gravitational coupling associated with the Newton
constant, $\sqrt{c}$ is the IR energy parameter and $\Lambda$ the cosmological
constant, related to the curvature as
$\Lambda=\frac{3}{5}R=\frac{-12}{L^{2}}$. The integral in $z$ has an upper
limit, that we named as $z_{f}$, that is equal to $z_{h}$ for the black hole
case and to $z\to\infty$ for the thermal AdS case.

In order to compute the total energy density in the grand canonical ensemble
and perform the analysis of the Hawking-Page transition, we must determine the
on-shell regularized action. For the rotating metric (2.7), the determinant of
$g_{\mu\nu}$ is $g=\frac{L^{10}}{z^{10}}$, so that Eq. (3.1) becomes

$I_{\text{on-shell}}=\frac{4L^{3}}{\kappa^{2}}\int_{0}^{z_{f}}dz\int
d^{4}xz^{-5}e^{-cz^{2}}\;.$ (3.2)

Since the integration over the spatial bulk coordinates is trivial and only
contributes with a volume factor, $V_{3D}$, we now define an action density
$\mathcal{E}=\frac{1}{V_{3D}}I_{\text{on-shell}}$. For a compact time
direction in the Euclidean signature, we have $0\leq t<\beta_{s}$, where
$\beta_{s}$ depends on the space considered and the general form of the action
density is then given by the expression

$\mathcal{E}_{s}(\varepsilon)=\frac{4L^{3}}{\kappa^{2}}\int_{0}^{\beta_{s}}dt\int_{\varepsilon}^{z_{f}}dz\,z^{-5}e^{-cz^{2}}\;,$
(3.3)

where $\varepsilon$ is an ultraviolet (UV) regulator. As mentioned before, for
the black hole geometry, $\beta_{BH}=1/T$, while for the thermal AdS one,
$\beta_{AdS}=\sqrt{f(\epsilon)}\beta_{BH}=\sqrt{f(\epsilon)}/T$.

We must introduce the UV regulator because the black hole and thermal AdS
space possesses infinite action densities. In order to eliminate these
divergencies, one defines the regularized free energy density of the rotating
black hole as the difference between the energy densities of the two
geometries,

$\bigtriangleup\mathcal{E}(\varepsilon)=\lim_{\varepsilon\rightarrow
0}\left[\mathcal{E}_{BH}(\varepsilon)-\mathcal{E}_{AdS}(\varepsilon)\right]\;,$
(3.4)

where

$\displaystyle\mathcal{E}_{BH}(\varepsilon)$ $\displaystyle=$
$\displaystyle\frac{4L^{3}}{\kappa^{2}}\beta\int_{\varepsilon}^{z_{h}}dz\,z^{-5}e^{-cz^{2}}\;,$
(3.5) $\displaystyle\mathcal{E}_{AdS}(\varepsilon)$ $\displaystyle=$
$\displaystyle\frac{4L^{3}}{\kappa^{2}}\sqrt{f(\epsilon)}\,\beta\int_{\varepsilon}^{\infty}dz\,z^{-5}e^{-cz^{2}}\;.$
(3.6)

From the regularized action density (3.4), one can determine the energy
density of the rotating BH in the grand canonical ensemble, taking into
account the contribution of the angular momentum. From gauge/gravity duality,
it will correspond to the energy density of the rotating plasma with
cylindrical symmetry.

###  3.2 Total energy density in the grand canonical ensemble

The first step to compute the configuration entropy of the plasma at different
temperatures and angular velocities is to determine the total energy density.
First we consider the total energy $E$. Due to rotation, we should assume that
the system is represented by the grand canonical ensemble. In this case, the
total energy, with zero chemical potential, is defined by

$E=-\frac{\partial\log Z}{\partial\beta}+\omega J\;,$ (3.7)

where $Z$ is the partition function, and J, the angular momentum, which in
turn is defined by

$J=\frac{1}{\beta}\frac{\partial\log Z}{\partial\omega}\;.$ (3.8)

In the semiclassical Hawking-Page approach, $\log Z=-I$, such that

$E=\frac{\partial I}{\partial\beta}+\omega J\;,\quad\text{with}\quad
J=\frac{1}{\beta}\frac{\partial I}{\partial\omega}\;.$ (3.9)

As it was done in the previous subsection, we factorize the trivial three
dimensional volume in the bulk spatial variables $V_{3D}$. This corresponds to
replacing in the previous Eqs. the action $I$ by $\Delta\epsilon$. Explicitly:

$U\,\equiv\,\frac{E}{V_{3D}}=\frac{\partial\bigtriangleup\mathcal{E}}{\partial\beta}+\omega\,\frac{1}{\beta}\frac{\partial\bigtriangleup\mathcal{E}}{\partial\omega}\;.$
(3.10)

Now we define an energy density in the holographic coordinate $z\,$,
$\rho_{BH}(z,T,\omega)$, by:

$\int_{0}^{\infty}\rho_{BH}(z,T,\omega)\,dz=U(T,\omega)\;.$ (3.11)

Thus, using equations (3.4), (3.5) and (3.6), together with

$\beta\sqrt{f(\epsilon)}=\beta-\frac{\pi^{4}\epsilon^{4}}{2\beta^{3}(1-\omega^{2}l^{2})^{2}}\;,$
(3.12)

and introducing the coordinate $u=z/z_{h}$, one can write

$\displaystyle\bigtriangleup\mathcal{E}$ $\displaystyle=$
$\displaystyle\frac{4L^{3}}{\kappa^{2}}\left[\beta\int_{\epsilon\pi/(\beta\sqrt{1-\omega^{2}l^{2}})}^{1}du\,u^{-5}\left(\frac{\beta\sqrt{1-\omega^{2}l^{2}}}{\pi}\right)^{-4}e^{-c(\frac{u\beta\sqrt{1-\omega^{2}l^{2}}}{\pi})^{2}}\right.$
(3.13) $\displaystyle-$
$\displaystyle\left.\left(\beta-\frac{\pi^{4}\epsilon^{4}}{2\beta^{3}(1-\omega^{2}l^{2})^{2}}\right)\int_{\epsilon\pi/(\beta\sqrt{1-\omega^{2}l^{2}})}^{\infty}du\,u^{-5}\left(\frac{\beta\sqrt{1-\omega^{2}l^{2}}}{\pi}\right)^{-4}e^{-c(\frac{u\beta\sqrt{1-\omega^{2}l^{2}}}{\pi})^{2}}\right]\;.$

Taking the derivative of Eq. (3.13) with respect to $\beta$, and eliminating
terms $\mathcal{O(\epsilon)}$ by taking the ultraviolet limit
$\epsilon\rightarrow 0$, the first term of the total energy,
$U_{1}=\partial\bigtriangleup\mathcal{E}/\partial\beta$ in Eq. (3.10), reads

$\displaystyle U_{1}$ $\displaystyle=$
$\displaystyle\frac{4L^{3}}{\kappa^{2}}\left[-\frac{7\epsilon^{4}}{2z_{h}^{8}}\int_{\epsilon\pi/(\beta\sqrt{1-\omega^{2}l^{2}})}^{1}du\,u^{-5}e^{-c(\frac{u\beta\sqrt{1-\omega^{2}l^{2}}}{\pi})^{2}}+\frac{1}{2z_{h}^{4}}\right.$
(3.14) $\displaystyle+$
$\displaystyle\left.\frac{3}{z_{h}^{4}}\int_{1}^{\infty}du\,u^{-5}e^{-c(\frac{u\beta\sqrt{1-\omega^{2}l^{2}}}{\pi})^{2}}+\frac{2c}{z_{h}^{2}}\int_{1}^{\infty}du\,u^{-3}e^{-c(\frac{u\beta\sqrt{1-\omega^{2}l^{2}}}{\pi})^{2}}\right]\;.$

The first two terms of the equation above can be rewritten in the form

$-\frac{7\epsilon^{4}}{2z_{h}^{8}}\int_{\epsilon\pi/(\beta\sqrt{1-\omega^{2}l^{2}})}^{1}du\,u^{-5}e^{-c(\frac{u\beta\sqrt{1-\omega^{2}l^{2}}}{\pi})^{2}}+\frac{1}{2z_{h}^{4}}=-\frac{3}{8z_{h}^{4}}\;,$
(3.15)

then, using the following Gaussian integration,

$\lim_{\epsilon\rightarrow
0}\left[-\frac{3\sqrt{c}}{4z_{h}^{3}\sqrt{\pi}}\int_{\epsilon
z_{h}}^{\infty}e^{-c(uz_{h})^{2}}\,du\right]=-\frac{3}{8z_{h}^{4}}\;,$ (3.16)

and returning to the holographic $z$-variable, one obtains

$U_{1}=\lim_{\epsilon\to
0}\left(\int_{\epsilon}^{z_{h}}\rho^{(1)}_{1}(z)\,dz+\int_{z_{h}}^{\infty}\rho_{1}^{(2)}(z)\,dz\right)\;,$
(3.17)

with

$\displaystyle\rho^{(1)}_{1}(z)$ $\displaystyle=$
$\displaystyle-\frac{4L^{3}}{\kappa^{2}}\frac{3\sqrt{c}}{4z_{h}^{4}\sqrt{\pi}}e^{-cz^{2}}\;,\quad\,\,\,\,\;\;\,\quad\quad\quad\quad\quad\,\text{if}\quad\epsilon\leq
z\leq z_{h}\;,$ $\displaystyle\rho^{(2)}_{1}(z)$ $\displaystyle=$
$\displaystyle\frac{4L^{3}}{\kappa^{2}}\left(-\frac{3\sqrt{c}}{4z_{h}^{4}\sqrt{\pi}}+\frac{3}{z^{5}}+\frac{2c}{z^{3}}\right)e^{-cz^{2}}\;,\quad\text{if}\quad
z\geq z_{h}\;.$ (3.18)

In order to compute the second term of the total energy,
$U_{2}=\frac{1}{\beta}\frac{\partial\bigtriangleup\mathcal{E}}{\partial\omega}$,
we must work out the expression

$\displaystyle\frac{\partial\bigtriangleup\mathcal{E}}{\partial\omega}$
$\displaystyle=$
$\displaystyle-\frac{4L^{3}}{\kappa^{2}}\left[\frac{4\epsilon^{4}l^{2}\omega}{z_{h}^{8}(1-l^{2}\omega^{2})}\int_{\epsilon/z_{h}}^{1}du\,u^{-5}e^{-c(uz_{h})^{2}}-\frac{l^{2}\omega}{2z_{h}^{4}(1-l^{2}\omega^{2})}\right.$
(3.19) $\displaystyle-$
$\displaystyle\left.\frac{4l^{2}\omega}{z_{h}^{4}(1-l^{2}\omega^{2})}\int_{1}^{\infty}du\,u^{-5}e^{-c(uz_{h})^{2}}-\frac{2cl^{2}\omega}{z_{h}^{2}(1-l^{2}\omega^{2})}\int_{1}^{\infty}du\,u^{-3}e^{-c(uz_{h})^{2}}\right]\;.$

Similarly to the previous calculation to determine $U_{1}$, by using the
expression

$\frac{l^{2}\omega}{2z_{h}^{4}(1-l^{2}\omega^{2})}=\frac{l^{2}\omega\sqrt{c}}{z_{h}^{3}\sqrt{\pi}(1-l^{2}\omega^{2})}\int_{\epsilon/z_{h}}^{\infty}e^{-c(uz_{h})^{2}}du\;,$
(3.20)

and returning to $z$-variable, the contribution of the angular momentum to the
rotating black hole total energy yields

$U_{2}(\omega)=\lim_{\epsilon\to
0}\left(\int_{\epsilon}^{z_{h}}\rho^{(1)}_{2}(z,\omega)\,dz+\int_{z_{h}}^{\infty}\rho_{2}^{(2)}(z,\omega)\,dz\right)\;,$
(3.21)

whereby

$\displaystyle\rho^{(1)}_{2}(z,\omega)$ $\displaystyle=$
$\displaystyle-\frac{4L^{3}}{\kappa^{2}}\frac{l^{2}\omega^{2}\gamma^{2}\sqrt{c}}{z_{h}^{4}\sqrt{\pi}}e^{-cz^{2}}\;,\quad\,\,\,\,\;\;\;\;\,\;\;\,\,\quad\quad\quad\quad\text{if}\quad\epsilon\leq
z\leq z_{h}\;,$ $\displaystyle\rho^{(2)}_{2}(z,\omega)$ $\displaystyle=$
$\displaystyle\frac{4L^{3}}{\kappa^{2}}l^{2}\omega^{2}\gamma^{2}\left(-\frac{\sqrt{c}}{z_{h}^{4}\sqrt{\pi}}+\frac{4}{z^{5}}+\frac{2c}{z^{3}}\right)e^{-cz^{2}}\;,\quad\text{if}\quad
z\geq z_{h}\;.$ (3.22)

Finally, one concludes that the total energy density of the cylindrical
rotating BH in the grand canonical ensemble, obtained by adding the partial
densities (3.2) and (3.2) is given by the expression

$\rho^{BH}(z,\omega)=\begin{cases}-\frac{4L^{3}}{\kappa^{2}}\frac{\sqrt{c}}{z_{h}^{4}\sqrt{\pi}}\left(\frac{3}{4}+l^{2}\omega^{2}\gamma^{2}\right)e^{-cz^{2}}\;,&\quad\quad\quad\quad\,\,\,\;\;\;\;\;\;\;\,\text{if}\quad\epsilon\leq
z\leq z_{h}\;,\\\ \\\
\frac{4L^{3}}{\kappa^{2}}\Big{[}-\frac{\sqrt{c}}{z_{h}^{4}\sqrt{\pi}}\left(\frac{3}{4}+l^{2}\omega^{2}\gamma^{2}\right)+\frac{1}{z^{5}}\left(3+4l^{2}\omega^{2}\gamma^{2}\right)\\\
+\
\frac{2c}{z^{3}}\left(1+l^{2}\omega^{2}\gamma^{2}\right)\Big{]}e^{-cz^{2}}\;,&\quad\quad\quad\quad\quad\quad\;\;\text{if}\quad
z>z_{h}\;.\end{cases}$ (3.23)

##  4 Configuration entropy

The configuration entropy (CE) definition is motivated by information theory.
In particular in the Shannon information entropy [58] that for a discrete
random variable with probabilities $p_{n}$ of assuming one of $n$ possible
values is defined as:

$S=-\sum_{n}\,p_{n}\log p_{n}\;.$ (4.1)

The Shannon entropy represents a measure of the information content of the
random variable.

A definition for the CE was proposed in references [25, 26, 27] as a
continuous version of Eq. (4.1). For a one-dimensional system it reads:

$S_{C}[f]=-\int dk\,f({k})\ln f({k})\,,$ (4.2)

being $f(k)$ the so-called modal fraction, usually defined, for a localized
physical system, in terms of the energy density in momentum space, $\rho(k)$,
namely

$f(k)=\frac{|\rho(k)|^{2}}{|\rho(k)|^{2}_{max}}\,$ (4.3)

where $|\rho(k)|^{2}_{max}$ is the maximum value assumed by $|\rho(k)|^{2}$.
Instead of the maximum value of the energy density, one could eventually use
$\int|\rho(k)|^{2}dk$ in the denominator of Eq. (4.3). In this alternative
definition, the modal fraction appears as a normalized function, which would
be more similar to the Shannon entropy, but in the continuous case it could
lead to negative values for the CE.

It is now known through many examples, as those articles cited in the
introduction, that the CE works as a measure of the stability of physical
systems. In a few words, the current interpretation states that the CE
increases as the instability of the system increases.

###  4.1  CE definition for the rotating QGP

The Fourier transform of the BH energy density is given by:

$\displaystyle\widetilde{\rho}(k,\omega)=\frac{1}{2\pi}\lim_{\epsilon\to
0}\int_{\epsilon}^{\infty}dz\,\rho^{BH}(z,\omega)\,e^{ikz}\;,$ (4.4)

with $\rho^{BH}(z,\omega)$ defined by Eq. (3.23), where one notices that the
total energy density has two different expressions, separated by the horizon
position. The modal fraction, necessary for building the CE, is defined in
terms of the squared absolute value of $\widetilde{\rho}(k,\omega)$. Using
(4.4), it can be written as

$|\widetilde{\rho}(k,\omega)|^{2}=\left[\frac{1}{2\pi}\lim_{\epsilon\to
0}\int_{\epsilon}^{\infty}\rho^{BH}(z,\omega)\cos(kz)\,dz\right]^{2}+\left[\frac{1}{2\pi}\lim_{\epsilon\to
0}\int_{\epsilon}^{\infty}\rho^{BH}(z,\omega)\sin(kz)\,dz\right]^{2}\;.$ (4.5)

Eq. (4.5) above does possess an analytical solution. So we apply numerical
methods in order to determine the CE.

Using numerical integration, we plot in Figure 1
$|\widetilde{\rho}(k,\omega)|^{2}$ at $\bar{T}=0.6$, where
$\bar{T}=T/\sqrt{c}$. The IR parameter $\sqrt{c}$ is determined by hadronic
phenomenology – see, for instance, [21]. For other temperatures the pattern of
the curve is similar.

Figure 1:  Absolute value of the rotating BH energy density,
$|\widetilde{\rho}(k)|^{2}$, versus momentum at $\bar{T}=0.6$ and different
rotational velocities: $\omega l=0.1$ (blue), $\omega l=0.2$ (orange); and
$\omega l=0.3$ (green).

As one can see, $|\widetilde{\rho}(k,\omega)|^{2}$ first reaches the global
maximum $|\widetilde{\rho}(k)|^{2}_{max}$, then the curve begins to decrease
as the momentum increases, oscillating smoothly and tending to zero as
$k\rightarrow\infty$. Moreover, $|\widetilde{\rho}(k)|^{2}_{max}$ gets larger
as the rotational speed increases, with a small increase in the value of $k$
where these maxima occur. These global maxima are well-defined, and can be
easily computed by numerical methods. After determining the maxima of the
absolute value of the energy density in Fourier space (4.5), we are able to
evaluate the configuration entropy of the rotating plasma at different angular
velocities and temperatures, by using the definition (4.2) together with
(4.3).

###  4.2  Results obtained for the CE

Applying numerical methods, we computed the CE for different values of the
dimensionless temperature $\tilde{T}$ and of the rotational speed $\omega l$.
The values obtained are displayed in Table 1, in Appendix A. In order to show
the behaviour of the CE we plot, in Figure 2, the case of fixed temperature
$\bar{T}_{6}=0.6$ as a function of the rotational speed $\omega l$. One
notices that the CE increases monotonically with the speed, indicating that,
for a fixed temperature, the larger is the rotational speed, the more unstable
is the plasma. In particular, one also notices that as $\omega l$ approaches
the speed of light, $\omega l\rightarrow 1$, the CE diverges. This behaviour
is present for all the analyzed temperatures, as can be seen in Figure 3,
where we plot the CE for four different temperatures.

The asymptotic singular behaviour in the $\omega l\rightarrow 1$ limit can be
understood in a simple way. Looking at the expression (2.12) for the Hawking
temperature of a rotating black hole, one notices that, for a fixed
temperature, as the rotational speed increases, the horizon position $z_{h}$
decreases. The limit $\omega l\rightarrow 1$ corresponds to $z_{h}\rightarrow
0$. On the other hand, it is known that an increase in the CE is associated
with an increase in the instability of the physical system. In the present
case the instability corresponds to the contraction of the BH dimensions in
the limit $\omega l\rightarrow 1$.

From the point of view of AdS/QCD duality, this contraction of the BH
corresponds to the contraction of the quark-gluon plasma. In the limit $\omega
l\rightarrow 1$ the volume of plasma goes to zero. This is the reason why the
CE diverges. This result is consistent with a result found recently in [32],
where it was shown that the CE of bottomonium quasi-states becomes singular
when the temperature or the magnetic field reach values such that the quasi-
states completely dissociate in the medium. In other words, the disappearance
of the bottomnium in the medium (associated with the deconfinement of the
quarks) is translated by the CE as an infinite instability.

Figure 2: Configuration entropy of rotating QGP as a function of the
rotational speed ($\omega l$) at $\bar{T}_{6}=0.6$. Figure 3: CE of rotating
QGP as a function of $\omega l$, at different temperatures: $\bar{T}_{3}=0.3$
(blue), $\bar{T}_{4}=0.4$ (orange), $\bar{T}_{6}=0.6$ (green), and
$\bar{T}_{8}=0.8$ (red).

In order to have a clear understanding about the variation of the CE with
temperature and rotational speed, we plot in Fig. 4 the CE as a function of
$\tilde{T}$, at three different values of $\omega l$. One notices that for a
fixed rotational speed, the CE increases with the temperature. This increase
in the CE is associated to the increase in the thermodynamic instability
caused by Hawking radiation. Black holes at higher temperatures are subject to
a stronger loss of energy as a consequence of Hawking radiation. These results
are consistent with the interpretation of the CE as an indicator of the
stability of physical systems.

Figure 4: Configuration entropy of rotating QGP as a function of $\bar{T}$, at
different rotational velocities: $\omega l=0$ (blue), $\omega l=0.4$ (orange);
and $\omega l=0.6$ (green).

##  5 Conclusions

We investigated the dependence of the configuration entropy on the rotational
speed for a quark-gluon plasma with cylindrical symmetry using a holographic
model. The plasma was represented by the grand canonical ensemble (with null
chemical potential). In this scenario, we obtained an expression for the
energy density of the rotating AdS black hole, dual to the plasma, that was
applied to the calculation of the CE. The dependence of this quantity on the
rotational speed of the black hole was studied for different temperatures
using numerical methods.

The current interpretation of the configuration entropy states that it works
as a measure of the stability of physical systems. In short, the CE increases
as the instability increases [25, 26, 27]. The result found in section 4 – the
CE increases with the rotational speed $\omega l$ – is consistent with this
interpretation. Rotation of the plasma implies a Lorentz type of contraction.
So that the plasma becomes smaller in volume as $\omega l$ increases. The
limit $\omega l\rightarrow 1$ would correspond to a plasma of zero volume that
consistently corresponds to a positive singularity in the CE indicating a
“maximum instability”.

It is interesting to relate this result with a similar singular limit of the
CE recently found in [32]. In this article the CE for bottomonium quasi-states
was calculated and it was found that it becomes singular in the limits when
the temperature or the magnetic field approach values such that the quasi-
states completely dissociate in the medium. The disappearance of the
bottomonium quasi-states, associated with the deconfinement of the heavy
quarks is translated by the CE as an infinite instability, corresponding to a
positive infinite CE. Here we found the similar result that in the limit when
the volume of the plasma goes to zero, that would correspond to the
disappearance of the plasma, the CE goes to (positive) infinity.

The combined effect of temperature and rotation was already investigated. It
was found in [17] that, for a non-rotating plasma, the CE increases with the
temperature indicating the instability caused by the evaporation of the black
hole via Hawking radiation. Here we have shown that for a plasma with a fixed
non-vanishing rotational speed the CE also increases in a monotonic way with
temperature.

##  Appendix

##  Appendix A  CE$(T,\omega)$

X | $\bar{T}_{3}=0.3$ | $\bar{T}_{4}=0.4$ | $\bar{T}_{5}=0.5$ | $\bar{T}_{6}=0.6$ | $\bar{T}_{7}=0.7$ | $\bar{T}_{8}=0.8$ | $\bar{T}_{9}=0.9$
---|---|---|---|---|---|---|---
$\,\,\,\omega l=0.0\,\,\,$ | $\,\,\,17.3177\,\,\,$ | $\,\,\,27.1699\,\,\,$ | $\,\,\,35.7645\,\,\,$ | $\,\,\,43.5432\,\,\,$ | $\,\,\,50.9320\,\,\,$ | $\,\,\,57.9978\,\,\,$ | $\,\,\,64.8159\,\,\,$
$\,\,\,\omega l=0.1\,\,\,$ | $\,\,\,17.4737\,\,\,$ | $\,\,\,27.3514\,\,\,$ | $\,\,\,35.9698\,\,\,$ | $\,\,\,43.7748\,\,\,$ | $\,\,\,51.1908\,\,\,$ | $\,\,\,57.9978\,\,\,$ | $\,\,\,65.1263\,\,\,$
$\,\,\,\omega l=0.2\,\,\,$ | $\,\,\,17.9544\,\,\,$ | $27.9103$ | $\,\,\,36.6029\,\,\,$ | $\,\,\,44.4887\,\,\,$ | $51.9886$ | $\,\,\,59.1616\,\,\,$ | $\,\,\,66.0833\,\,\,$
$\,\,\,\omega l=0.3\,\,\,$ | $\,\,\,18.7997\,\,\,$ | $28.8953$ | $\,\,\,37.7188\,\,\,$ | $\,\,\,45.7479\,\,\,$ | $53.3956$ | $\,\,\,60.7113\,\,\,$ | $\,\,\,67.7703\,\,\,$
$\,\,\,\omega l=0.4\,\,\,$ | $\,\,\,20.0886\,\,\,$ | $30.3998$ | $\,\,\,39.4261\,\,\,$ | $\,\,\,47.6759\,\,\,$ | $55.5495$ | $\,\,\,63.0780\,\,\,$ | $\,\,\,70.3519\,\,\,$
$\,\,\,\omega l=0.5\,\,\,$ | $\,\,\,21.9659\,\,\,$ | $32.5914$ | $\,\,\,41.9220\,\,\,$ | $\,\,\,50.4976\,\,\,$ | $58.7014$ | $\,\,\,66.5529\,\,\,$ | $\,\,\,74.1494\,\,\,$
$\,\,\,\omega l=0.6\,\,\,$ | $\,\,\,24.7137\,\,\,$ | $35.7812$ | $\,\,\,45.5766\,\,\,$ | $\,\,\,54.6347\,\,\,$ | $63.3233$ | $\,\,\,71.6411\,\,\,$ | $\,\,\,79.6887\,\,\,$
$\,\,\,\omega l=0.7\,\,\,$ | $\,\,\,28.8948\,\,\,$ | $40.6155$ | $\,\,\,51.1591\,\,\,$ | $\,\,\,60.9631\,\,\,$ | $70.3972$ | $\,\,\,79.4415\,\,\,$ | $\,\,\,88.1499\,\,\,$
$\,\,\,\omega l=0.8\,\,\,$ | $\,\,\,35.8003\,\,\,$ | $48.7337$ | $\,\,\,60.6088\,\,\,$ | $\,\,\,71.6941\,\,\,$ | $82.3997$ | $\,\,\,92.6117\,\,\,$ | $\,\,\,102.292\,\,\,$
$\,\,\,\omega l=0.9\,\,\,$ | $\,\,\,50.3687\,\,\,$ | $66.3365$ | $\,\,\,81.2459\,\,\,$ | $\,\,\,95.0799\,\,\,$ | $108.213$ | $\,\,\,120.801\,\,\,$ | $\,\,\,133.186\,\,\,$
$\,\,\,\omega l=0.91\,\,\,$ | $\,\,\,52.9314\,\,\,$ | $69.4661$ | $\,\,\,84.9202\,\,\,$ | $\,\,\,99.1757\,\,\,$ | $112.752$ | $\,\,\,125.940\,\,\,$ | $\,\,\,138.910\,\,\,$
$\,\,\,\omega l=0.92\,\,\,$ | $\,\,\,55.9181\,\,\,$ | $73.1201$ | $\,\,\,89.2013\,\,\,$ | $\,\,\,103.917\,\,\,$ | $118.083$ | $\,\,\,131.964\,\,\,$ | $\,\,\,145.649\,\,\,$
$\,\,\,\omega l=0.93\,\,\,$ | $\,\,\,59.4664\,\,\,$ | $77.4686$ | $\,\,\,94.2683\,\,\,$ | $\,\,\,109.542\,\,\,$ | $124.557$ | $\,\,\,139.232\,\,\,$ | $\,\,\,153.798\,\,\,$
$\,\,\,\omega l=0.94\,\,\,$ | $\,\,\,63.7885\,\,\,$ | $82.7700$ | $\,\,\,100.376\,\,\,$ | $\,\,\,116.435\,\,\,$ | $132.489$ | $\,\,\,148.253\,\,\,$ | $\,\,\,163.931\,\,\,$
$\,\,\,\omega l=0.95\,\,\,$ | $\,\,\,69.2319\,\,\,$ | $89.4363$ | $\,\,\,107.993\,\,\,$ | $\,\,\,125.228\,\,\,$ | $142.681$ | $\,\,\,159.877\,\,\,$ | $\,\,\,177.005\,\,\,$
$\,\,\,\omega l=0.96\,\,\,$ | $\,\,\,76.4179\,\,\,$ | $98.1476$ | $\,\,\,118.090\,\,\,$ | $\,\,\,137.079\,\,\,$ | $156.494$ | $\,\,\,175.663\,\,\,$ | $\,\,\,194.774\,\,\,$
$\,\,\,\omega l=0.97\,\,\,$ | $\,\,\,86.5942\,\,\,$ | $110.284$ | $\,\,\,132.700\,\,\,$ | $\,\,\,154.408\,\,\,$ | $176.772$ | $\,\,\,198.872\,\,\,$ | $\,\,\,220.913\,\,\,$
$\,\,\,\omega l=0.98\,\,\,$ | $\,\,\,102.651\,\,\,$ | $129.994$ | $\,\,\,157.074\,\,\,$ | $\,\,\,237.976\,\,\,$ | $210.908$ | $\,\,\,237.976\,\,\,$ | $\,\,\,264.957\,\,\,$
$\,\,\,\omega l=0.99\,\,\,$ | $\,\,\,136.231\,\,\,$ | $173.956$ | $\,\,\,212.166\,\,\,$ | $\,\,\,249.391\,\,\,$ | $288.362$ | $\,\,\,326.702\,\,\,$ | $\,\,\,364.847\,\,\,$
$\,\,\,\omega l=0.995\,\,\,$ | $\,\,\,182.902\,\,\,$ | $236.174$ | $\,\,\,290.410\,\,\,$ | $\,\,\,342.816\,\,\,$ | $398.298$ | $\,\,\,452.576\,\,\,$ | $\,\,\,506.449\,\,\,$
Table 1: Configuration entropy of rotating QGP at different temperatures
($\bar{T}=T/\sqrt{c}$) and rotational velocities ($\omega l$).

Acknowledgments: The authors are supported by FAPERJ — Fundação Carlos Chagas
Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, CNPq - Conselho
Nacional de Desenvolvimento Científico e Tecnológico. This work received also
support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -
Brasil (CAPES) - Finance Code 001.

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