# Comparative study of the heavy-quark dynamics with the Fokker-Planck
Equation and the Plastino-Plastino Equation

Eugenio Megías  Departamento de Física Atómica, Molecular y Nuclear and
Instituto Carlos I de Física Teórica y Computacional,
Universidad de Granada, Avenida de Fuente Nueva s/n, 18071 Granada, Spain
<EMAIL_ADDRESS>Airton Deppman  Instituto de Física - Universidade de São
Paulo, Rua do Matão 1371, São Paulo 05508-090, Brazil<EMAIL_ADDRESS>Roman
Pasechnik  Department of Physics, Lund University, Sölvegatan 14A, Lund
SE-22362, Sweden<EMAIL_ADDRESS>Constantino Tsallis  Centro
Brasileiro de Pesquisas Fisicas and National Institute of Science and
Technology of Complex Systems,
Rua Xavier Sigaud 150, Rio de Janeiro-RJ 22290-180, Brazil
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, 87501 NM, USA
Complexity Science Hub Vienna - Josefstädter Strasse 39, 1080 Vienna, Austria
<EMAIL_ADDRESS>
###### Abstract

The Fokker-Planck Equation (FPE) is a fundamental tool for the investigation
of kinematic aspects of a wide range of systems. For systems governed by the
non-additive entropy $S_{q}$, the Plastino-Plastino Equation (PPE) is the
correct generalization describing the kinematic evolution of such complex
systems. Both equations have been applied for investigations in many fields,
and in particular for the study of heavy quark evolution in the quark-gluon
plasma. In the present work, we use this particular problem to compare the
results obtained with the FPE and the PPE, and discuss the different aspects
of the dynamical evolution of the system according to the solutions for each
equation. The comparison is done in two steps, first considering the
modification that results from the use of a different partial derivative
equation with the same transport coefficients, and then investigating the
modifications by using the non-additive transport coefficients. We observe
clear differences in the solutions for all the cases studied here and discuss
possible experimental investigations that can indicate which of those
equations better describes the heavy-quark kinematics in the medium. The
results obtained here have implications in the study of anomalous diffusion in
porous and granular media, in Cosmology and Astrophysics. The obtained results
reinforce the validity of the relation
$(q-1)^{-1}=(11/3)N_{c}-(4/3)(N_{f}/2)$, where $N_{c}$ and $N_{f}$ are,
respectively, the number of colours and the effective number of flavours. This
equation was recently established in the context of a fractal approach to QCD
in the non-perturbative regime.

The interest in heavy-quark dynamics as a tool to probe the quark-gluon plasma
has increased with the High Energy Physics (HEP) experiments showing that the
interaction of those particles with the medium is relevant, as evidenced by
the observation of jet-quenching and of the collective flow observed through
hadrons with charm or heavier quarks. Those observations have placed the
Fokker-Planck Equation (FPE) at the centre of the theoretical investigations
on the heavy-quark dynamics in the medium [1]. This equation is a second-order
approximation of the Boltzmann Equation, but it is preferred since it avoids
the integral equation by the introduction of two transport coefficients, the
drift term and the diffusion term. A review of the applications of the FPE in
HEP can be found in Ref. [2].

The procedure to obtain the FPE from the BE is well known [3], and it leads to
the second-order differential equation

$\frac{\partial f}{\partial t}-\frac{\partial}{\partial
p_{i}}\left[A_{i}(\mathbf{p})f+\frac{\partial}{\partial
p_{j}}\left(B_{ij}(\mathbf{p})f\right)\right]=0\,.$ (1)

Here and below, we use natural units $\hbar=c=k_{B}=1$, unless noted
otherwise. The transport coefficients $A_{i}$ and $B_{ij}$ have been obtained
from microscopic calculations [4], establishing a clear path for studying the
heavy-quark dynamics. The solution $f\equiv f(\mathbf{p},t)$ is the momentum
distribution of the heavy quark.

The bulk matter formed at high-energy collisions presents evidence that
nonextensive statistical mechanics (based on the nonadditive entropy Sq) [5]
is the correct framework for studying the thermostatistical properties of the
quark-gluon plasma (QGP). Many analyses of the experimental data have been
produced in the use of the Sq with success to describe multiparticle
production that results from the decay of the QGP [6, 7]. The clearest
evidence of Sq is the heavy-tailed momentum distributions observed, which can
be accurately described by a $q$-exponential curve in a wide range of
transverse momentum and several decades of cross-section, showing the non-
extensive behaviour predicted by Sq [8]. This different statistical framework
imposes the use of the non-extensive thermodynamics [9] to describe the
thermal properties of the medium.

The non-extensive thermodynamics of the QGP has its impacts also in the heavy-
quark dynamics. In systems governed by Sq, the correct dynamical equation is
not the FPE, but the Plastino-Plastino Equation (PPE) [10], which has the form

$\frac{\partial f}{\partial t}-\frac{\partial}{\partial
p_{i}}\left[A_{i}(\mathbf{p})f+\frac{\partial}{\partial
p_{j}}\left(B_{ij}(\mathbf{p})f^{2-q}\right)\right]=0\,.$ (2)

This equation is a generalization of the FPE that reduces to the standard FPE
when the entropic index $q=1$, otherwise it differs from the standard
equation. In the present work, we compare the dynamics of the heavy quark in
the medium as obtained by the FPE and by the PPE. For this comparison, we
compute the transport coefficients based on microscopic calculations and use
the same coefficients to find the solutions of the FPE and PPE, comparing the
results in both cases.

Eqs. (1) and (2) correspond, in momentum space, to the Fokker-Planck and
Plastino-Plastino equations in the coordinate space. Observe that, in the
absence of external forces, the drift coefficient is $A_{i}=0$, and Eq. (1)
reduces to the heat equation, first proposed by Joseph Fourier in 1822. Under
the same conditions, Eq. (2) reduces to the porous media diffusion equation
[11], which has been investigated in Refs. [12, 13].

For the microscopic calculation of the transport coefficients we follow Ref.
[4], and define double-average of an undetermined function $F(p^{\prime})$ as

$\begin{split}\langle\langle
F(p^{\prime})\rangle\rangle=&\frac{1}{2\omega_{p}}\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}2\omega_{p^{\prime}}}\int\frac{d^{3}q}{(2\pi)^{3}2\omega_{q}}\int\frac{d^{3}q^{\prime}}{(2\pi)^{3}2\omega_{q^{\prime}}}|{\cal
M}|^{2}\times\\\
&\delta^{4}\left((\mathbf{p}^{\prime}-\mathbf{p})-(\mathbf{q}-\mathbf{q}^{\prime})\right)f(q)F(p^{\prime})\,,\end{split}$
(3)

where the matrix elements $|{\cal M}|^{2}$ are calculated in second-order
approximation. The coefficients can be calculated from the expression above by

$\begin{cases}A_{i}(\mathbf{p})=\langle\langle(p-p^{\prime})_{i}\rangle\rangle\\\
\\\
B_{ij}(\mathbf{p})=\frac{1}{2}\langle\langle(p-p^{\prime})_{i}(p-p^{\prime})_{j}\rangle\rangle\end{cases}\,.$
(4)

The calculation of the transport coefficients can be facilitated by defining
$A(p^{2})$, $B_{0}(p^{2})$ and $B_{1}(p^{2})$ such that

$A(p^{2})=\frac{p_{i}A_{i}(\mathbf{p})}{p^{2}}=\langle\langle
1\rangle\rangle-\frac{\langle\langle\mathbf{p}\cdot\mathbf{p}^{\prime}\rangle\rangle}{p^{2}}\,,$
(5)

and

$B_{0}(p^{2})=\frac{1}{2}\left[\delta_{ij}-\frac{p_{i}p_{j}}{p^{2}}\right]B_{ij}(\mathbf{p})\,,\quad
B_{1}(p^{2})=\frac{p_{i}p_{j}}{p^{2}}B_{ij}(\mathbf{p})\,,$ (6)

where $p^{2}\equiv|\mathbf{p}|^{2}$. Then, the coefficients may be decomposed
as

$\displaystyle A_{i}({\mathbf{p}})$ $\displaystyle=$ $\displaystyle
p_{i}\,A(p^{2})\,,$ (7) $\displaystyle B_{ij}({\mathbf{p}})$ $\displaystyle=$
$\displaystyle\left(\delta_{ij}-\frac{p_{i}p_{j}}{p^{2}}\right)B_{0}(p^{2})+\frac{p_{i}p_{j}}{p^{2}}B_{1}(p^{2})\,.$
(8)

For the computation of the coefficients using the matrix elements given in
Ref. [4], we need to make a Lorentz transformation to the centre-of-momentum
(CM) frame. This is done by using the transformation

$\begin{cases}\mathbf{\hat{p}}=\gamma_{cm}(\mathbf{p}-\mathbf{v}_{cm}E)\\\ \\\
\hat{E}=\gamma_{cm}(E-\mathbf{v}_{cm}\cdot\mathbf{p})\end{cases}\,,$ (9)

where $\gamma_{cm}\equiv 1/\sqrt{1-\mathbf{v}_{cm}^{2}}$. The first line in
Eq. (9) is valid in the approximation that $\mathbf{v}_{cm}$ is parallel to
$\mathbf{p}$.

The description above shows the main ingredients in the computation of the
transport coefficients. Notice that we have multiplied the gluons matrix
elements by a factor $-1/(4\pi)$, and those for quarks by $1/(2\pi)$. These
modifications take into account the information, in Ref. [4], of a
multiplicative factor in the calculation of the matrix elements.

The coefficients $B_{0}(p^{2})$ and $B_{1}(p^{2})$ obtained in the last
sections differ by a negligible amount, so we use the approximation
$B_{0}(p^{2})=B_{1}(p^{2})\equiv B(p^{2})$, and in this case we have
$B_{ij}({\mathbf{p}})=\delta_{ij}B(p^{2})$. We also notice that $A(p^{2})$ and
$B(p^{2})$ are slowly increasing with $p$ in the relevant region of the
present study, so we approximate both to constants $A$ and $B$, respectively.
These approximations were discussed in some detail in Ref. [14].

For the 3-dimensional solution of the FPE we use the ansatz

$f(\mathbf{p},t)=\frac{1}{\left(\sqrt{2\pi}\sigma(t)\right)^{3}}~{}~{}\exp\left[-\frac{(\mathbf{p}-\mathbf{p}_{M}(t))^{2}}{2\sigma(t)^{2}}\right]\,.$
(10)

This general function is a solution of the FPE if

$\begin{cases}\mathbf{p}_{M}(t)=\mathbf{p}_{o}\exp[-At]\\\ \\\
\sigma(t)=\sigma_{o}\sqrt{\left(1-\frac{B}{A\sigma_{0}^{2}}\right)\exp[-2At]+\frac{B}{A\sigma_{0}^{2}}}\end{cases}\,,$
(11)

where $\mathbf{p}_{o}$ is a constant vector corresponding to the heavy-quark
initial momentum, while $\sigma_{o}$ is the initial width of the
distribution111The second line of Eq. (11) can be written also in the form
$\sigma(t)=\sqrt{\frac{B}{A}\left(1-\exp[-2A(t+\lambda)]\right)}\,,$ (12)
where the constant $\lambda$ can be determined by the initial condition
$\sigma_{o}\equiv\sigma(t=0)=\sqrt{\frac{B}{A}\left(1-\exp[-2A\lambda]\right)}$..
The width in equilibrium is
$\sigma_{\infty}\equiv\lim_{t\to\infty}\sigma(t)=\sqrt{B/A}$. Notice that
$\sigma_{o}$ can have any positive value, even larger than $\sigma_{\infty}$.

For the 3-dimensional solution of the PPE we use the ansatz 222The
$q$-exponential function is defined as
$\exp_{q}[x]=\left[1-(q-1)x\right]^{-\frac{1}{q-1}}$.

$f(\mathbf{p},t)=\frac{1}{\left(\sqrt{2\pi\chi_{q}}\sigma(t)\right)^{3}}~{}~{}\exp_{q}\left[-\frac{(\mathbf{p}-\mathbf{p}_{M}(t))^{2}}{2\sigma(t)^{2}}\right]\,,$
(13)

which is a solution of the PPE if the parameter $\mathbf{p}_{M}(t)$ is given
by the first line of Eq. (11), while $\sigma(t)$ is

$\sigma(t)=\sigma_{o}\left[\left(1-\kappa\right)\exp\left[-(5-3q)At\right]+\kappa\right]^{\frac{1}{5-3q}}\,,$
(14)

with

$\kappa\equiv(2-q)\left(2\pi\chi_{q}\right)^{\frac{3}{2}(q-1)}\frac{B}{A}{\sigma_{o}^{3q-5}}\,,$
(15)

and

$\chi_{q}\equiv\frac{1}{q-1}\left(\frac{\Gamma\left(\frac{1}{q-1}-\frac{3}{2}\right)}{\Gamma\left(\frac{1}{q-1}\right)}\right)^{\frac{2}{3}}\,.$
(16)

The width of the distribution in equilibrium

$\sigma_{\infty}=\sigma_{0}\kappa^{\frac{1}{5-3q}}=\left[(2-q)\left(2\pi\chi_{q}\right)^{\frac{3}{2}(q-1)}\frac{B}{A}\right]^{\frac{1}{5-3q}}$
(17)

is independent of $\sigma_{o}$, and it diverges for $q\to 5/3$. It is
straightforward to see that the solution of the PPE given by Eq. (13) tends to
the solution of the FPE of Eq. (10) in the limit $q\to 1$, and in particular
$\chi_{q}\stackbin[q\to 1]{}{\to}1$. Both distributions are normalized as

$\int d^{3}pf(\mathbf{p},t)=1\,.$ (18)

We display in Fig. 1 (left) the distributions obtained with the FPE and the
PPE for two different values of the medium temperature: $T=200\,\textrm{MeV}$
(upper panels) and $500\,\textrm{MeV}$ (lower panels). The behaviours of the
widths as functions of time are displayed in the middle and right panels of
this figure for both $q=1.10$ (Sq) and $q=1$ (Boltzmann). The values of the
constants $A$ and $B$ have been computed, and they turn to be in agreement
with those shown in Ref. [4].

We note that the use of the same parameters $A$ and $B$ for the PPE is
inconsistent, since the distributions obtained in this case are
$q$-exponentials, while we used exponentials in the calculations of the
transport coefficients. The difference in the behaviour of the parameters when
the non-extensive momentum distributions are used can be observed in Fig. 2,
where we see that both $A(p^{2})$ and $B(p^{2})$ increase as $q$ increases
with respect to the value $q=1$, when the Boltzmann case is recovered.
However, the ratio $B/A$ increases, so the width of the distribution at the
stationary regime increases with $q$.

Comparing the results in Fig. 2, we observe that the coefficient $A$ decreases
with the quark mass, $m_{0}$, while the coefficient $B$ increases. Therefore,
the ratio $B/A$, and consequently the width at the stationary regime,
increases with the quark mass.

Figure 1: Left panels: Plots of the distribution functions integrated out in
transverse momenta,
$f(p_{z},t)\equiv\int_{-\infty}^{\infty}dp_{x}dp_{y}f(\mathbf{p},t)$, as a
function of $p_{z}$. We display the distributions for several values of $t$.
Middle panels: Widths of the distributions as a function of time. Right
panels: Widths of the distributions in equilibrium,
$\sigma_{\infty}\equiv\lim_{t\to\infty}\sigma(t)$, as a function of $q$. In
these panels, solid lines correspond to $q=1.10$ and dashed lines to $q=1$. We
have considered the initial momentum for the heavy quark to be
$p_{o}=0.47\,\textrm{GeV}$ in the $z$-direction, the initial width
$\sigma_{o}=0.2\,\textrm{GeV}$, while
$(A=0.0820\,\textrm{fm}^{-1},B=0.0216\,\textrm{GeV}^{2}/\textrm{fm})$ for
$T=200\,\textrm{MeV}$ (upper panels) and
$(A=0.381\,\textrm{fm}^{-1},B=0.228\,\textrm{GeV}^{2}/\textrm{fm})$ for
$T=500\,\textrm{MeV}$ (lower panels).

Figure 2: Plot of the drift coefficient $A(p=0)$ (left panels), diffusion
coefficient $B(p=0)$ (middle panels), and width of the distribution in
equilibrium $\sigma_{\infty}$ (right panels) as a function of the temperature.
We have displayed the results for $m_{0}=1.5\,\textrm{GeV}$ (upper panels) and
$m_{0}=3\,\textrm{GeV}$ (lower panels), and for $q=1,1.07$ and $1.14$. We have
considered $\alpha_{s}=0.6$ for the strong coupling constant, and $\mu=T$ for
a regulator introduced into the internal gluon propagator in the $t$-channel-
exchange diagrams to include the effects of Debye screening, cf. Ref. [4] for
details.

Figure 3: Plots of $p_{M}(t)/p_{o}$ and $\sigma(t)$ as functions of time. We
keep for $\sigma(t)$ the PPE solution with $q=1.07$, while comparing the
effects that produce in these parameters the $q$-dependence in $A$ and $B$
computed in Boltzmann-Gibbs statistics $(q=1)$ and in $q$-statistics $(q>1)$.
In the right panel, we have considered the initial width
$\sigma_{o}=0.2\,\textrm{GeV}$.

Observe that, while the FPE solution is a Gaussian with time-dependent
parameters, the solution for the PPE is a $q$-Gaussian. In both cases, the
system approaches asymptotically the equilibrium, and for
$t\rightarrow\infty$, the heavy-quark momentum occupies a 3D region around the
null momentum region, with a distribution that is isotropic and follows a
Gaussian or a $q$-Gaussian, in the cases of the FPE and of the PPE,
respectively.

Walton and Rafelski [14] had compared the FPE with a non-linear Fokker-Planck
Equation in order to introduce the Sq and found that the results by Svetitsky
for the FPE were well reproduced by a $q$-Gaussian with $q=1.1$. The results
in Fig. 1 clarify the agreement found by those authors: with that value for
the entropic index the Boltzmann distribution is reproduced also by the non-
extensive result.

The fact that the PPE leads to $q$-exponential distributions is in agreement
with the experimental data from HEP laboratories. This result shows that PPE
should be used instead of FPE. It is interesting to observe in Fig. 1 that for
temperatures around $T=200~{}\textrm{MeV}$ the PPE solution approaches the
stationary value more slowly, but the distribution width is asymptotically
similar to that of the FPE solution if $1<q<1.15$. For values of $q$ above
that range, the width of the non-extensive distribution increases fast as the
entropic index increases. For higher temperatures, though, the width of the
non-extensive distributions is always larger than the distributions obtained
from the FPE.

One interesting quantity is the length of the heavy-quark path in the medium,
which can be calculated by using

$m\,v_{M}(t)=p_{M}(t)=p_{o}e^{-At}\,,$ (19)

where $m=m_{0}/(1-v_{M}^{2}(t))^{1/2}$, and $m_{0}$ is the rest mass of the
heavy quark. The path length until equilibrium is given by

$\ell_{M}=\int_{0}^{\infty}dt\,v_{M}(t)\,,$ (20)

resulting in

$\ell_{M}=\frac{1}{A}{\textrm{arcsinh}}(p_{o}/m_{0})\,.$ (21)

Observe that for $p_{o}$ small compared to $m_{0}$, this result can be
approximated to

$\ell_{M}=\frac{p_{o}}{m_{0}}\frac{1}{A}\,,$ (22)

which corresponds to the non-relativistic result $\ell_{M}=v_{o}\tau_{M}$,
with $v_{o}$ being the initial velocity of the heavy quark, and $\tau_{M}=1/A$
– the relaxation time. The different evolution of the parameters of the
distribution for the different values of the entropic index, $q$, are
displayed in Fig. 3. Observe that, for any value of $q$, the behaviour of
$p_{M}(t)$ is always exponential, as given by Eq. (11). This results from the
fact that the behaviour of the parameter $p_{M}(t)$ in the solutions depends
only on the drift term of Eqs. (1) and (2), and not on the diffusion term.
Setting $B_{ij}=0$ in those equations results in the same equation for both
cases, and the general solution to the new equation is

$f(\mathbf{p},t)=\frac{1}{\left(\sqrt{2\pi}\sigma(t)\right)^{3}}~{}F\left[\frac{(\mathbf{p}-\mathbf{p}_{M}(t))^{2}}{2\sigma(t)^{2}}\right]\,,$
(23)

where $F$ is any analytical function, and the time-dependent parameters are

$\begin{cases}\mathbf{p}_{M}(t)=\mathbf{p}_{o}\exp[-At]\\\
\sigma(t)=\sigma_{o}\exp[-At]\end{cases}\,.$ (24)

The $q$-exponential and the exponential functions are particular solutions of
the simplified equation.

Hence, for the same initial momentum, the length will be proportional to
$A^{-1}$. Since, as shown above, the non-additive statistics leads to larger
values of $A$ as $q$ increases, the minimal size of the medium for allowing
the heavy quark to reach the stationary state diminishes with $q$. The
experimental data from high energy collisions indicate that the heavy quark
reaches the stationary state inside the medium before the freeze-out. These
indications are the momentum distributions of the heavy hadrons and the
collective flow observed through heavy hadrons, which are similar to those
observed for pions.

The results presented in the previous section indicate that the best situation
to compare the results associated with heavy quark time or distance travelled
to reach the stationary state, corresponds to relatively low-mass quarks as
this is better to distinguish between FPE or PPE. This is because $A$
decreases with the quark mass, so the length $\ell_{M}$ is larger in this
case.

Figure 4: Plot of the entropic index $q$ as a function of $N_{f}$ (solid blue
line), as given by Eq. (25) with $N_{c}=3$; for $N_{f}=2,3,6$ we respectively
obtain $q=32/29\simeq 1.10$, $q=10/9\simeq 1.11$ and $q=8/7\simeq 1.14$. The
experimental LHC/CERN value [15] is $q=1.14\pm 0.01$, while the Walton-
Rafelski value [14] is $q=1+1/8.8\simeq 1.114$ (red dashed line). We display
as a shaded (red) area the region corresponding to $q=1.14\pm 0.01$ and, as
horizontal lines, the values of $q$ for $N_{f}=2,3$ and $6$.

The investigation of those physical quantities associated with the width of
the distribution can be suitably studied by observing the properties of
heavier quarks. In particular, the difference in the stationary distribution
width as a function of the entropic index $q$ is enhanced for heavier quarks.
The results obtained above open an opportunity to test the number of degrees
of freedom ($n_{dof}$) relevant to the interaction of the heavy quark with the
medium. The number of colours, $N_{c}$, and the number of flavours, $N_{f}$,
are related to the number of degrees of freedom by [16]

$n_{dof}=\frac{1}{q-1}=\frac{11}{3}N_{c}-\frac{4}{3}\frac{N_{f}}{2}\,.$ (25)

Considering $N_{c}=3$ and $N_{f}=6$, we get $n_{dof}=7$ and $q=8/7\simeq
1.14$. However, the theory that leads to Eq. (25) assumes that the masses of
the quarks are negligible. In the case where the quark masses become relevant,
due to the $q$-exponential behaviour of the distributions, their contributions
to the thermalization of the quark can be neglected, which corresponds to a
reduction of the number of effective flavours in the process. If we use
$N_{f}=2$ in Eq. (25), we get $n_{dof}=29/3\sim 9.7$ and $q=32/29\simeq 1.10$.
If one considers $N_{f}=3$, then $q=10/9\simeq 1.11$. We display in Fig. 4 the
behaviour of the entropic index $q$ with $N_{f}$ as given by Eq. (25).

Walton and Rafelski [14] used a non-linear FPE associated with the non-
additive statistics to compare with Svetitsky’s calculations [4], and found
$q=1.1$, which is in agreement with the scenarios for $N_{f}=3$ and for
$N_{f}=2$. In Fig. 4 we plot the value for $q$ as a function of $N_{f}$ for
$N_{c}=3$. The shaded area corresponds to the interval for $q$ that results
from experimental data analysis by HEP experiments where the QGP deconfined
quark regime is attained. Calculations that consider only two or three
flavours, as those carried out here, lead to different values for $q$, in
accordance with the predicted in Eq. (25).

The meaning implied by Eq. (25) can be clearly understood in face of the
present results. That formula was obtained in Ref. [16], where the quark
masses are considered negligible. For processes at relatively low energy,
where the quark masses become relevant, the finite mass effects must be taken
into account, and one consequence is the dump of the heavy quark contributions
due to the $q$-exponential behaviour of the particle spectra in the QGP. A
phenomenological method for taking this effect into account is by reducing the
effective number of flavours for the process, as commonly used in QCD
calculations. The results obtained in the present work and that obtained in
Ref. [14] confirm the predictions given by that equation.

One of the main differences in the solutions of the FPE with respect to the
solutions of the PPE is the width of the distribution, which becomes larger
for the PPE than for the FPE as soon as the interaction between the heavy-
quark and the medium starts, and increases continuously until the stationary
regime is reached.

The heavy quark dynamic has implications in many aspects of the high-energy
collisions [17, 18]. The jet quenching [19, 20] and the heavy hadron
suppression [21] are phenomena that can benefit from the results obtained
here. Due to the different relaxation times and displacement in the medium,
the different behaviours of the additive and non-additive solutions can be
important for determining the dynamical effects in small and large systems.

Table 1: Table of physical quantities for $T=200$ MeV and $m_{0}=1.5$ GeV. We
have assumed $v_{o}=0.3c$ as the initial velocity of the heavy quark in the
computation of $\ell_{M}$. The quantities $\sigma_{\infty}$ and $\ell_{M}$ are
calculated according to Eqs. (17) and (21), respectively, and $\tau_{M}=1/A$.
Here, $A$ and $B$ are defined in Eqs. (7) and (8), and considered to be
approximately constant.
| FPE | PPE
---|---|---
| Eq. (1) | Eq. (2) with q=1.10
|  $A=0.0820$ fm-1 |  $A=0.0820$ fm-1
Eqs. (7), (8) |  $B=0.0216$ GeV2/fm |  $B=0.0216$ GeV2/fm
|  $\sigma_{\infty}=0.513$ GeV |  $\sigma_{\infty}=0.510$ GeV
with q=1 |  $\tau_{M}$ = 12.20 fm/c |  $\tau_{M}=12.20$ fm/c
|  $\ell_{M}=3.78$ fm |  $\ell_{M}=3.78$ fm
|  $A=0.129$ fm-1 |  $A=0.129$ fm-1
Eqs. (7), (8) |  $B=0.0486$ GeV2/fm |  $B=0.0486$ GeV2/fm
|  $\sigma_{\infty}$ = 0.613 GeV |  $\sigma_{\infty}$ = 0.629 GeV
with q=1.10 |  $\tau_{M}$ = 7.74 fm/c |  $\tau_{M}=7.74$ fm/c
|  $\ell_{M}=2.39$ fm |  $\ell_{M}=2.39$ fm

Table 1 summarizes the results obtained in the analysis performed here,
displaying the calculations of the physical parameters $\sigma_{\infty}$,
$\tau_{M}$ and $\ell_{M}$ corresponding, respectively, to the stationary
distribution width, the relaxation time and the characteristic distance
travelled by the heavy quark till the equilibrium. The results for FPE and for
PPE are shown. The top-left and the bottom-right panels show fully consistent
calculations: the first corresponds to additive coefficients used in the
additive FPE. The second corresponds to non-additive coefficients and non-
additive PPE.

In this work, we have studied the dynamical properties of heavy quarks in the
medium as predicted by the Fokker-Planck Equation and by its non-additive
counterpart, the Plastino-Plastino Equation. The transport coefficients were
obtained for both the additive and the non-additive statistics through a
microscopic second-order calculation of the heavy quark interaction with the
light quarks and gluons in the medium.

We observed different dynamical characteristics in the results of the additive
and non-additive equations. The main effect of introducing the Plastino-
Plastino equation is a broader distribution in the stationary regime, which is
obtained even when the transport coefficients calculations do not include the
non-additive features.

When the non-additive effects are included in the transport coefficients, by
considering the non-extensive distributions, the differences in the dynamics
of the quarks are more evident. Not only the width of the distribution is
broader, but also the relaxation time and the quark displacement until the
stationary regime is reached are smaller than the equivalent quantities in the
additive case. We emphasize that the only fully consistent calculations are
those with FPE and transport coefficients calculated with $q=1$, and the PPE
equations using the transport coefficients calculated with the same value for
$q\neq 1$. These cases correspond to the top-left and bottom-right cases in
Table 1. The comparison between the results obtained here and that performed
in Ref. [14] corroborates the findings of Ref. [16], where the fractal
approach to the non-perturbative QCD led to Eq. (25) relating the entropic
index $q$ to the number of colours, $N_{c}$ and the number of flavours,
$N_{f}$.

The results obtained here have implications in the study of several aspects of
high-energy collisions. Phenomena like jet quenching and heavy hadron
suppression depend on a careful description of the quark dynamics and can
benefit from these results. A possible way to distinguish experimentally
between the FPE and the PPE solutions is by observing the distance travelled
by a heavy quark in the medium. This can be done experimentally by observing
the ratio between di-jets carrying heavy quarks and non-correlated jets as a
function of collision centrality. For less central collisions, the relative
number of di-jets should increase, and the size of the system when this
happens should be related to the distance travelled by the heavy quark in the
medium. The study of the momentum distribution of jets is an alternative way
to investigate the phenomenon, and this will be calculated in future work.

Although the work performed here was motivated by the study of QGP and HEP,
the results can be of interest to researchers in many areas, such as solar
plasma [22], ionic diffusion [23] and neutron star [24, 25, 26, 27], for
instance. The micro-calculation of the transport coefficients for hadronic
matter has been under debate [27], with implications in hydrodynamical models.
The results obtained here imply modifications in both the form of the solution
for the dynamical equation and on the transport coefficients, and can be of
interest to researchers in those fields.

##  1 Acknowledgements

E.M. and A.D. thank E. Ruiz Arriola for discussions. A.D. would like to thank
the University of Granada, where part of this work has been done, for the
hospitality and financial support under a grant of the Visiting Scholars
Program of the Plan Propio de Investigación of the University of Granada. He
also acknowledges the hospitality at Carmen de la Victoria. The work of E.M.
is supported by the project PID2020-114767GB-I00 funded by
MCIN/AEI/10.13039/501100011033, by the FEDER/Junta de Andalucía-Consejería de
Economía y Conocimiento 2014-2020 Operational Program under Grant
A-FQM178-UGR18, and by Junta de Andalucía under Grant FQM-225. The research of
E.M. is also supported by the Ramón y Cajal Program of the Spanish MCIN under
Grant RYC-2016-20678. A.D. is supported by the Project INCT-FNA (Instituto
Nacional de Ciência e Tecnologia - Física Nuclear Aplicada) Proc. No.
464898/2014-5, by the Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq-Brazil), grant 304244/2018-0, by Project INCT- FNA Proc. No.
464 898/2014-5, and by FAPESP, Brazil grant 2016/17612-7. R.P. is supported in
part by the Swedish Research Council grants, contract numbers 621-2013-4287
and 2016-05996, as well as by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 668679). C.T. is partially supported by CNPq and Faperj
(Brazilian agencies).

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