# Shear viscosity expression for a graphene system in relaxation time
approximation

Cho Win Aung, Thandar Zaw Win, Gaurav Khandal, Sabyasachi Ghosh  Indian
Institute of Technology Bhilai, GEC Campus, Sejbahar, Raipur 492015,
Chhattisgarh, India

###### Abstract

We have gone through the detailed microscopic calculation of the shear
viscosity of a 2-dimensional graphene system in the relaxation time
approximation-based kinetic theory framework. After getting its final
expressions, we compared it with the shear viscosity expressions of other
possible 2-dimensional as well as 3-dimensional nonrelativistic and ultra-
relativistic fluid systems. The aim of the comparison is to reveal - how their
different one-body dispersion relations affect their many-body fluid
properties like shear viscosity and the viscosity to entropy density ratio. It
is also aimed to reveal the 3-dimension to the 2-dimension transformation of
their mathematical structures. We have numerically explored the differences in
their order of magnitude and dependence on thermodynamical parameters –
temperature and chemical potential. Marking two thermodynamical domains –
Dirac fluid and Fermi liquid – for a 2-dimensional graphene system, we have
noticed that shear viscosity, entropy density as well as their ratios decrease
toward saturated values when one goes from Fermi liquid to Dirac fluid domain.
When one shifts from mili-electron volt scales of temperature and chemical
potential in condensed matter physics location to their Mega-electron volt
scales in high energy physics location, then the same results may be expected
for hot quark matter case, where the transition from the neutron star to early
universe domains may be considered as Fermi liquid to Dirac fluid transition.

##  I Introduction

It is known that the mean free path of charge carriers in metal is generally
temperature dependent. The scattering between electron and lattice
imperfections (or “disorder”) normally dominates at low temperatures, while
electron-phonon scatterings dominate at high temperatures. Concerning these
two scattering mechanisms, another possible scattering is electron-electron
scattering processes, which are generally less effective in many conventional
metals. However, its opposite condition is possible in some specific systems
under specific conditions, where one can apply the electron hydrodynamic (eHD)
theory. For a long time, condensed matter physicists did not aware of such an
opposite phase in materials. Therefore, they used to give less attention to
the possibilities of the hydrodynamics behavior of electrons. After the
experimental observations of eHD in Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18] , the situation has drastically changed in recent
years. See Refs. [19, 20, 21] for recent reviews. It is graphene [1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14], which is identified as the best known such
material, where electron hydrodynamics can be observed. Apart from these
recently discovered hydrodynamic properties of electrons in graphene, it was
quite famous for its massless nature, concluded from the proportional relation
between its energy and momentum. Due to the proportional relation between
energy and momentum, electron motion in graphene will not be Galilean-
invariant. On the other hand, the relativistic effect of electrons can not
also be expected because its velocity ($v_{g}\approx 10^{6}$ m/s) is not very
close to the speed of light ($c\approx 3\times 10^{8}$ m/s). Hence, we cannot
claim the Lorentz-invariant property of electron motion. It opens
“unconventional” hydrodynamics [20] as neither nonrelativistic hydrodynamics
(NRHD) nor relativistic hydrodynamics (RHD) can be applicable. We may call
this “unconventional” hydrodynamics as Graphene hydrodynamics (GHD) by
imposing that the graphene (G) case has a unique dispersion or energy-momentum
relation and is different from the nonrelativistic (NR) and relativistic (R)
or ultra-relativistic (UR) cases. Now, whenever fluid dynamics or
hydrodynamics comes into the picture, then one dissipation coefficient like
the shear viscosity of that fluid, becomes a very important quantity, which is
not at all appeared in most of the metals or other condensed matter systems.
The present work is aimed at the microscopic calculation of the shear
viscosity of this electron fluid in a graphene system, which may be called in
short as graphene fluid (GF). When one microscopically calculates the
expression of the shear viscosity of GF, it will be different from its
standard expression for nonrelativistic fluid (NRF) as well as for
relativistic fluid (RF) or ultra-relativistic fluid (URF). So far, from the
best of our knowledge, experimental measurement of this shear viscosity
coefficient for GF is missing although experimental community [1, 9] observed
the Poiseuille’s flow pattern of electrons in graphene, which indirectly
reflects the existence of the non-zero viscosity. From the theoretical side,
we get only Refs. [22, 23, 24, 25, 26], where microscopic expressions of shear
viscosity have been addressed. In this context, one can get a long list of
Refs. [27, 28, 29, 30, 31, 32, 33, 34, 35, 36] (and references therein) for
microscopical estimations of shear viscosity for relativistic quark and
hadronic matter, expected in high energy heavy ion collision experiments.
Grossly two classes of frameworks - (1) Kinetic theory approach with
relaxation time approximation (RTA) [27, 28, 29, 30, 31, 32, 33] and Kubo
framework [34, 35, 36] \- are adopted by the heavy ion physics community. Both
frameworks have similar structure at the final level expressions for shear
viscosity coefficients with two main components. One carries interaction
information, called relaxation time, and the remaining part may be called as
thermodynamic phase-space of shear viscosity coefficient, which will be
function of temperature and chemical potential. If we analyze the shear
viscosity expression of graphene also from Ref. [22], then we can identify
these two components. Present work has zoomed in this structure via a
systematic calculation of this shear viscosity of GF in RTA methods and
compared with corresponding structures for NRF and URF. Here, one of our aims
is to compare the thermodynamic phase-space component of shear viscosity
coefficient for these three cases - G, NR, and UR. After knowing the lower
bound conjecture of shear viscosity to entropy density $(\eta/s)$ as
$\hbar/(4\pi k_{B})$ or $1/(4\pi)$ (in natural unit) [37], scientific
communities are curious to know those strongly coupled systems, which are
close to that bounds. Experimentally, the RF, like quark and hadronic matter,
produced in high energy heavy ion collision experiments and NRF like cold-atom
systems [38] are identified as those strongly coupled systems. On the other
hand, GF may also belong to that category according to the theoretical
prediction from Ref. [22], which is considered as reference point for tuning
our results. So, the present article will not intend to add any new content on
strongly coupled properties, rather its main goal will intend to find the
differences among GF, NRF, URF in terms of expressions and estimations of
shear viscosity. We believe that it was missing in the literature and very
important to address. The article is organized as follows. In the next section
Sec. (II), the RTA calculation of shear viscosity $\eta$ and entropy density
$s$ calculations of GF for 2D case is addressed in detail by mentioning the
other cases like 3D-GF, 3D-NR, 3D-UR, 2D-NR, 2D-UR. In Sec. (III), the
comparative results of $\eta$, $s$ and $\eta/s$ of different cases are
discussed. At the end, our findings are summarised in Sec. (IV) with some
conclusive bullet points.

##  II Formalism

Let us start our formalism from energy-momentum tensor ($T^{\mu\nu}$), as
practised for RF like quark and hadronic matter. Here, we will go for GF
calculation, so reader should have to be careful on some particular steps,
where it is different from the RF case. Showing these differences is one of
the core agenda of the present article. Although, reader can find similarities
between most of steps of GHD of GF and RHD of RF. The $T^{\mu\nu}$ has two
parts - the ideal part $T_{0}^{\mu\nu}$, related to the knowledge of
thermodynamics and the dissipative part $T_{D}^{\mu\nu}$, related to the
different dissipation processes. So,

$\displaystyle T^{\mu\nu}$ $\displaystyle=$ $\displaystyle
T_{0}^{\mu\nu}+T_{D}^{\mu\nu}.$ (1)

In this dynamic picture of fluid, ideal energy-momentum tensor and electron
number flow can be expressed in macroscopic form as,

$\displaystyle T_{0}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\epsilon\frac{u^{\mu}u^{\nu}}{v_{g}^{2}}-P\left(g^{\mu\nu}-\frac{u^{\mu}u^{\nu}}{v_{g}^{2}}\right),$
$\displaystyle N_{0}^{\mu}$ $\displaystyle=$ $\displaystyle
n\frac{u^{\mu}}{v_{g}},$ (2)

in terms of the building blocks - energy density $\epsilon$, pressure $P$,
number density $n$, fluid (element) velocity $u^{\mu}$ and metric tensor
$g^{\mu\nu}$. Here, four-velocity $u^{\mu}=\gamma_{g}(v_{g},\vec{u})$ for GHD
is designed by following the four velocity structure
$u^{\mu}=\gamma(c,\vec{u})$ for RHD as done in Ref. [20]. One can notice that
speed of light $c$ in RHD is replaced by graphene Fermi velocity $v_{g}$ in
GHD. So, Lorentz factor $\gamma=1/\sqrt{1-u^{2}/c^{2}}$ in RHD is also
converted into $\gamma_{g}=1/\sqrt{1-u^{2}/v_{g}^{2}}$ in GHD. In static limit
(${\vec{u}}\rightarrow 0$), four velocity,
$u^{\mu}=\gamma_{g}(v_{g},\vec{u})\rightarrow u^{\mu}=\gamma_{g}(v_{g},0)$ and
$\gamma_{g}=1/\sqrt{1-u^{2}/v_{g}^{2}}\rightarrow 1$. So, Eq. (2) provides a
static electron number flow $N^{\mu}_{0}\equiv n$, and static energy-momentum
tensor,

$T_{0}^{\mu\nu}\equiv\begin{pmatrix}\epsilon&0&0&0\\\ 0&P&0&0\\\ 0&0&P&0\\\
0&0&0&P\\\ \end{pmatrix},$ (3)

which reflects the standard static fluid aspect like Pascal’s law. The
macroscopic quantities $T_{0}^{\mu\nu}$ and $N^{\mu}_{0}$ can be expressed in
terms of the microscopic quantities; four-momentum ($p^{\mu}$) and four-
velocity ($v^{\mu}$) of electrons as,

$T_{0}^{\mu\nu}=N_{s}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}p^{\mu}v^{\nu}f_{0},$
(4)

and

$N_{0}^{\mu}=N_{s}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}v^{\mu}f_{0},$ (5)

where $N_{s}=2$ is spin degeneracy factor of electron and $f_{0}$ is its
Fermi-Dirac (FD) distribution function $f_{0}=1/\\{\exp{(\beta(E-\mu))}+1\\}$.
Here, $\beta=1/(k_{B}T)$ and $\mu$ are the thermodynamic parameter and the
chemical potential of the system, respectively. From these microscopic
expressions of the energy-momentum tensor and electron current, given in Eqs.
(4) and (5), we can write the energy density $\epsilon$, pressure $P$ and
number density $n$ for 2D graphene (G) case, which is addressed briefly in
next subsection. We follow natural unit $\hbar=c=k_{B}=1$ during the
calculation.

###  II.1 Entropy density in two-dimensional Graphene

For Graphene, the dispersion relation is given by

$E=pv_{g}~{}.$ (6)

The total number of fermions at any value of temperature is given by,

$N=\int_{0}^{\infty}D\left(E\right)dEf_{0},$ (7)

where $D\left(E\right)dE$ is number of energy states in energy range $E$ to
$E+dE$. After plugging the value of $D\left(E\right)dE$ (see Appendix A) in
the above Eq. (7) and $f_{0}$, the total number of electrons in graphene is,

$\displaystyle N=N_{s}\frac{2\pi
a}{\left(2\pi\right)^{2}v_{g}^{2}}\int_{0}^{\infty}\frac{E}{A^{-1}e^{\beta
E}+1}dE,$

where $A=\exp{(\beta\mu)}$ is the fugacity and $a$ is the area of the system,
respectively. After converting this integral into the Fermi integral function
(see Appendix B), we get the expression of number density as

$n_{g}^{2D}=\frac{N}{a}=\frac{N_{s}}{2\pi v_{g}^{2}}f_{2}\left(A\right)T^{2}.$
(8)

Now from the Eq. (4), the energy density for (2D) graphene system is,

$\epsilon_{g}^{2D}=T_{0}^{00}=N_{s}\int\frac{d^{2}p}{\left(2\pi\right)^{2}}\left(E\right)f_{0}.$
(9)

After using the graphene dispersion relation and plugging the value of
$f_{0}$, we get

$\epsilon_{g}^{2D}=\frac{1}{\pi
v_{g}^{2}}\int_{0}^{\infty}\frac{E^{2}}{A^{-1}e^{\beta E}+1}dE$ (10)

and after replacing this integral in terms of Fermi integral function, the
final expression of the energy density is given by

$\epsilon_{g}^{2D}=\frac{N_{s}}{\pi v_{g}^{2}}f_{3}\left(A\right)T^{3}.$ (11)

Now again from the Eq. (4), the pressure can be expressed as

$P_{g}^{2D}=T_{0}^{11}=N_{s}\int\frac{d^{2}p}{\left(2\pi\right)^{2}}\left(\frac{E}{2}\right)f_{0},$
(12)

since
$\vec{p}_{x}\vec{v}_{x}\approx\frac{|\vec{p}|}{\sqrt{2}}\frac{|\vec{v}_{g}|}{\sqrt{2}}=\frac{E}{2}$.
After solving this expression as similar to the energy density, we get

$P_{g}^{2D}=\frac{N_{s}}{2\pi v_{g}^{2}}f_{3}\left(A\right)T^{3}.$ (13)

In terms of number density, energy density, and pressure, we can write the
entropy density from the Euler thermodynamic relation in 2D graphene system:

$s=\frac{S}{a}=\frac{\epsilon+P-\mu n}{T}.$ (14)

After substituting the value of energy density ($\epsilon_{g}^{2D}$), pressure
($P_{g}^{2D}$), and number density ($n_{g}^{2D}$) in Eq. (14), we get

$s_{g}^{2D}=\frac{N_{s}}{2\pi
v_{g}^{2}}T^{2}\Big{[}3f_{3}\left(A\right)-\frac{\mu}{T}f_{2}\left(A\right)\Big{]}.$
(15)

###  II.2 Shear viscosity in two-dimensional graphene

Next, let us come to dissipative part of $T_{D}^{\mu\nu}$, where only shear
stress tensor $\pi^{\mu\nu}$ will be considered for calculating shear
viscosity coefficient ($\eta$). The detailed description of relativistic
hydrodynamics for calculating transport coefficients can be found in Refs.
[27, 29, 34], whose graphene or unconventional version (neither relativistic
nor nonrelativistic) is considered here. The dissipative term of energy-
momentum tensor includes shear stress $\pi^{\mu\nu}$ and bulk pressure $\Pi$
as

$T^{\mu\nu}_{D}=\pi^{\mu\nu}+\Pi\Delta^{\mu\nu}.$ (16)

Here, we have assumed Landau-Lifshitz definition of flow, where
$T^{\mu\nu}_{D}$ will be orthogonal to fluid velocity $u^{\mu}$, i.e.
$u_{\mu}T^{\mu\nu}_{D}=0$. These shear stress $\pi^{\mu\nu}$ and bulk pressure
$\Pi$ have proportional relations with fluid velocity gradient as

$\displaystyle\pi^{\mu\nu}$ $\displaystyle=$
$\displaystyle\eta\mathcal{U}^{\mu\nu},$ $\displaystyle\Pi$ $\displaystyle=$
$\displaystyle\zeta{\cal U}_{\zeta},$ (17)

where proportional constants are shear viscosity $\eta$ and bulk viscosity
$\zeta$. Their respective fluid velocity gradients are

$\mathcal{U}^{\mu\nu}_{\eta}=D^{\mu}u^{\nu}+D^{\nu}u^{\mu}-\frac{2}{3}\Delta^{\mu\nu}\partial_{\rho}u^{\rho}.$
(18)

and

${\cal U}_{\zeta}\equiv\partial_{\rho}u^{\rho},$ (19)

where $\Delta^{\mu\nu}=-g^{\mu\nu}+u^{\mu}u^{\nu}$ is projection tensor normal
to $u^{\mu}$ and $D^{\mu}=\partial^{\mu}-u^{\mu}u^{\rho}\partial_{\rho}$ is
derivative normal to $u^{\mu}$. They are designed such a way that we can get
$\Delta^{\mu\nu}\equiv\delta^{ij}$ and $D^{\mu}\equiv\partial^{i}$ in fluid
rest frame $u^{\mu}\equiv(1,{\vec{0}})$. Usually, greek index like
$\mu\equiv(0,i)$ takes values 0 for the temporal component and $i=1,2,3$ for
the spatial component for 3D system but here for 2D system, we will consider
$\mu\equiv(0,i=1,2)$ because z-component $i=3$ will not be considered. During
the transition from $\mu$-index to spatial component $i$, one can get
disspative part of energy momentum tensor as

$\displaystyle T^{ij}_{D}$ $\displaystyle=$
$\displaystyle\pi^{ij}+\Pi\delta^{ij}$ (20) $\displaystyle=$
$\displaystyle\eta(\partial^{i}u^{j}+\partial^{j}u^{i}-\frac{2}{3}\delta^{ij}\nabla\cdot
u)+\zeta\delta^{ij}\nabla\cdot u,$

which ensure that diagonal part ($i=j$) is linked with bulk viscosity $\zeta$
and off-diagonal part $(i\neq j)$ is linked with shear viscosity $\eta$.
Present work will focus only on shear viscosity coefficients, so we will not
proceed with discussion of bulk viscosity further.

The microscopic theory describes shear stress tensor in terms of particle
velocity $v$ and momentum $p$ as,

$\pi_{\mu\nu}=N_{s}\int\frac{d^{2}\vec{p}}{(2\pi)^{2}}p_{\mu}v_{\nu}\delta
f_{\eta},$ (21)

where we are assuming that equilibrium distribution function $f_{0}$ gets a
small deviation $\delta f$, which can be considered as first-order Taylor
series expansion equilibrium distribution function:

$\displaystyle\delta f$ $\displaystyle\propto$ $\displaystyle\frac{\partial
f_{0}}{\partial E}=\phi_{\eta}\frac{\partial f_{0}}{\partial E}$ (22)
$\displaystyle=$
$\displaystyle\mathcal{A}^{\mu\nu}\mathcal{U}_{\mu\nu}\frac{\partial
f_{0}}{\partial E}.$

Considering the relation $v_{\nu}=(E/p^{2})p_{\nu}$, macroscopic
$\pi_{\mu\nu}=\eta\mathcal{U}_{\mu\nu}$ and microscopic Eq. (21) can be
connected as,

$\displaystyle\pi_{\mu\nu}$ $\displaystyle=$
$\displaystyle\eta~{}\mathcal{U}_{\mu\nu}$ (23) $\displaystyle=$
$\displaystyle
N_{s}\int\frac{d^{2}\vec{p}}{(2\pi)^{2}}\left(\frac{E}{p^{2}}\right)p_{\mu}p_{\nu}\mathcal{A}^{\alpha\beta}\mathcal{U}_{\alpha\beta}\frac{\partial
f_{0}}{\partial E}.$

The four-momentum of an electron can be defined as $p^{\mu}=(E/v_{g},\vec{p})$
in unconventional notation. Considering energy as a static limit of
$p^{\nu}u_{\nu}$, we can write FD as,

$f_{0}=\frac{1}{exp{\left(\frac{p^{\nu}u_{\nu}-\mu(x)}{T(x)}\right)+1}}.$ (24)

Here, we have to consider the local thermalization concept, where
thermodynamical quantities $T(x)$, $\mu(x)$ as well as fluid velocity
$u^{\mu}(x)$ are assumed to be functions of $x\equiv x^{\mu}=(x^{0},x^{i})$.
To find the unknown coefficient $\mathcal{A}^{\alpha\beta}$, we will use
Boltzmann transport equation (BTE)

$\displaystyle\frac{\partial f}{\partial t}+v^{\mu}\frac{\partial f}{\partial
x^{\mu}}+F^{\mu}\frac{\partial f}{\partial p^{\mu}}=\left(\frac{\partial
f}{\partial t}\right)_{Col},$ (25)

where $\Big{(}\frac{\partial f}{\partial t}\Big{)}_{Col}$ is the collision
term that leads the system out of equilibrium. $F^{\mu}$ is represented as all
external forces, and $v^{\mu}$ is the velocity of the fluid particles. Using
velocity expression in terms of $E$ and $p$ for graphene,
$v^{\mu}=(\frac{E}{p^{2}})p^{\mu}$, we get:

$\displaystyle\left(\frac{E}{p^{2}}\right)p^{\mu}\partial_{\mu}f=\left(\frac{\partial
f}{\partial t}\right)_{Col},$ (26)

where we ignore $\frac{\partial f}{\partial t}$ and $F^{\mu}\frac{\partial
f}{\partial p^{\mu}}$ as they will not contribute in shear dissipation. Using
the Relaxation Time Approximation (RTA) method, the collision term can be
considered as,

$\left(\frac{\partial f}{\partial t}\right)_{Col}=-\frac{\delta f}{\tau_{c}},$
(27)

where $\tau_{c}$ is the relaxation time. Putting $f\approx f_{0}$ in the left-
hand side (lhs) of BTE,

$\displaystyle\left(\frac{E}{p^{2}}\right)p^{\mu}\partial_{\mu}f_{0}=-\frac{\delta
f}{\tau_{c}}.$ (28)

Using the Eq. (24), the lhs of the above Eq. (28) can be expanded as,

$\displaystyle\left(\frac{E}{p^{2}}\right)p^{\mu}\partial_{\mu}f_{0}$
$\displaystyle=-f_{0}(1-f_{0})\Big{[}\left(\frac{E}{p^{2}}\right)\frac{p^{\mu}p^{\nu}}{T}\partial_{\mu}u_{\nu}(x)\Big{]}$
$\displaystyle=-f_{0}(1-f_{0})\left(\frac{E}{p^{2}}\right)\frac{p^{\mu}p^{\nu}}{2T}(\partial_{\mu}u_{\nu}+\partial_{\nu}u_{\mu})$
$\displaystyle=-f_{0}(1-f_{0})\left(\frac{E}{p^{2}}\right)\frac{p^{\mu}p^{\nu}}{T}\mathcal{U}_{\mu\nu},$
(29)

and right hand side (rhs) of Eq. (28) can be written as

$\displaystyle-\frac{\delta f}{\tau_{c}}$ $\displaystyle=$
$\displaystyle\frac{f_{0}(1-f_{0})}{T}\frac{1}{\tau_{c}}\mathcal{A}^{\mu\nu}\mathcal{U}_{\mu\nu}.$
(30)

So, equating lhs and rhs of Eq. (28), we get
$\mathcal{A}^{\mu\nu}=(-\frac{E}{p^{2}}p^{\mu}p^{\nu}\tau_{c})$. Transforming
temporal+spatial to only spatial index, we can write the shear stress tensor
as

$\displaystyle\pi_{ij}$ $\displaystyle=$
$\displaystyle\eta\mathcal{U}_{ij}~{},$ (31) $\displaystyle=$ $\displaystyle
N_{s}\int\frac{d^{2}\vec{p}}{(2\pi)^{2}}\left(\frac{E}{p^{2}}\right)^{2}\tau_{c}(p_{i}p_{j}p_{k}p_{l})\mathcal{U}^{kl}\beta
f_{0}(1-f_{0})$ $\displaystyle=$
$\displaystyle\frac{N_{s}}{8}\int\frac{d^{2}\vec{p}}{(2\pi)^{2}}E^{2}\tau_{c}\beta
f_{0}(1-f_{0})\mathcal{U}_{ij},$

where, we used
$<p_{i}p_{j}p_{k}p_{l}>=\frac{\vec{p}^{4}}{8}\big{(}\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\big{)}$
(see Appendix C), and we have the two Eqs;

$\beta f_{0}\left(1-f_{0}\right)=-\frac{\partial f_{0}}{\partial E}$ (32)

and

$-\frac{\partial f_{0}}{\partial
E}=\beta\frac{e^{\beta\left(E-\mu\right)}}{\left(e^{\beta\left(E-\mu\right)}+1\right)^{2}}=\frac{\partial}{\partial\mu}\left(\frac{1}{e^{\beta\left(E-\mu\right)}+1}\right).$
(33)

Finally, the expression for shear viscosity is,

$\eta=\frac{N_{s}}{8}\int\frac{d^{2}\vec{p}}{(2\pi)^{2}}E^{2}\tau_{c}\beta
f_{0}(1-f_{0}).$ (34)

After using the Eqs. (32) and (33) and converting the momentum terms into
energy using dispersion relation (6), the Eq. (34) becomes

$\eta=\frac{N_{s}}{16\pi
v_{g}^{2}}\tau_{c}\frac{\partial}{\partial\mu}\int_{0}^{\infty}\frac{E^{3}}{A^{-1}e^{\beta
E}+1}dE.$ (35)

After solving this integration by using the identity of the Fermi integral
function, the expression of shear viscosity for 2D graphene (using subscript
and superscript notation to distinguish the expressions of different systems)
is

$\eta_{g}^{2D}=\frac{3N_{s}}{8\pi v_{g}^{2}}\tau_{c}f_{3}\left(A\right)T^{3}.$
(36)

Now, on taking the ratio of the shear viscosity (36) and entropy density (15),
we get

$\frac{\eta_{g}^{2D}}{s_{g}^{2D}}=\frac{3}{4}\tau_{c}\Bigg{[}3f_{3}\left(A\right)-\frac{\mu}{T}f_{2}\left(A\right)\Bigg{]}^{-1}f_{3}\left(A\right)T.$
(37)

After doing a similar way of calculation, the expressions of entropy density,
shear viscosity, and the ratio of shear viscosity and entropy density for a
nonrelativistic electron fluid (i.e. $E=p^{2}/(2m)$) 2-dimensional system are
given by

$\displaystyle
s_{NR}^{2D}=\frac{N_{s}mT}{2\pi}\Bigg{[}2f_{2}\left(A\right)-\frac{\mu}{T}f_{1}\left(A\right)\Bigg{]},$
(38)
$\displaystyle\eta_{NR}^{2D}=\frac{N_{s}m}{8\pi}\tau_{c}f_{2}\left(A\right)T^{2},$
(39)
$\displaystyle\frac{\eta_{NR}^{2D}}{s_{NR}^{2D}}=\frac{1}{4}\tau_{c}\Bigg{[}2f_{2}\left(A\right)-\frac{\mu}{T}f_{1}\left(A\right)\Bigg{]}^{-1}f_{2}\left(A\right)T.$
(40)

Most of the fluids or liquids (e.g., water) used in our daily life follow
nonrelativistic fluid dynamics, whose fluid constituent particles obey
$E=p^{2}/(2m)$ dispersion relation. However, for the purpose of comparing, we
may assume a hypothetical 2D NR system showing fluid behavior, which may be
difficult to be found in the real world. By this comparison (given details in
the results section), our aim is to encourage the scientific community to use
the expressions of G-case, given in Eqs. (36), (37) instead NR-case, given in
Eqs. (39), (40) when they are describing eHD in graphene system.

If we consider graphene as a 3-dimensional (3D) system, following the
dispersion relation $E=pv_{g}$, then it may be again a hypothetical example
but a good example for comparison purpose. Modifying our above calculation
with replacement of $\int d^{2}p\rightarrow\int d^{3}p$ and
$p_{x}v_{x}\approx\frac{pv_{g}}{3}=\frac{E}{3}$, we get the expressions of
entropy density, shear viscosity, and their ratio as

$\displaystyle
s_{g}^{3D}=\frac{N_{s}T^{3}}{\pi^{2}v_{g}^{3}}\Bigg{[}4f_{4}\left(A\right)-\frac{\mu}{T}f_{3}\left(A\right)\Bigg{]},$
(41)
$\displaystyle\eta_{g}^{3D}=\frac{4N_{s}}{5\pi^{2}v_{g}^{3}}\tau_{c}f_{4}\left(A\right)T^{4},$
(42)
$\displaystyle\frac{\eta_{g}^{3D}}{s_{g}^{3D}}=\frac{4}{5}\tau_{c}\Bigg{[}4f_{4}\left(A\right)-\frac{\mu}{T}f_{3}\left(A\right)\Bigg{]}^{-1}f_{4}\left(A\right)T.$
(43)

Now we have a 3-dimensional nonrelativistic (3D-NR) system of fermions. After
applying the same methodology to this system, we get all the expressions of
entropy density, shear viscosity, and the ratio of shear viscosity to entropy
density,

$\displaystyle
s_{NR}^{3D}=N_{s}\left(\frac{m}{2\pi}\right)^{\frac{3}{2}}T^{\frac{3}{2}}\Bigg{[}\frac{5}{2}f_{\frac{5}{2}}\left(A\right)-\frac{\mu}{T}f_{\frac{3}{2}}\left(A\right)\Bigg{]},$
(44)
$\displaystyle\eta_{NR}^{3D}=\frac{N_{s}}{4}\left(\frac{m}{2\pi}\right)^{\frac{3}{2}}\tau_{c}f_{\frac{5}{2}}\left(A\right)T^{\frac{5}{2}},$
(45)
$\displaystyle\frac{\eta_{NR}^{3D}}{s_{NR}^{3D}}=\frac{1}{4}\tau_{c}\Bigg{[}\frac{5}{2}f_{\frac{5}{2}}\left(A\right)-\frac{\mu}{T}f_{\frac{3}{2}}\left(A\right)\Bigg{]}^{-1}f_{\frac{5}{2}}\left(A\right)T.$
(46)

This 3D-NR system, showing fluid behavior, can be applicable for most of the
fluids or liquids (e.g. water) used in our daily life. One can consider the
above shear viscosity, entropy density, and their ratio for the water
molecule, where water molecule obeying NR dispersion relation, $E=p^{2}/(2m)$
with effective mass $m$, but that will be not a good example to compare with
the same expressions for eHD in graphene case. So, we can again consider a
hypothetical example - 3D eHD NR case.

We can also compare the above expressions for 2D, 3D eHD of G and NR cases
with the same for ultra-relativistic (UR) case. A good example is hot QGP,
where RHD can be applicable. According to the latest understanding [39], RHD
is quite successful in describing QGP phenomenology. Again to make our
comparison on equal footing, we will consider the hypothetical case of 2D,
3D-UR electron fluid. If the Fermi velocity of electrons in graphene $v_{g}$
is replaced by the factor $1$ (as $c=1$ in natural unit), then all expressions
of the G-case will be converted to corresponding expressions of UR case.

##  III Results

After addressing the final expressions of $\eta$, $s$, and its ratio for
different systems like 2D, 3D-NRF, GF and URF in the formalism section, here
we will discuss their numerical estimations through different graphs.

Figure 1: Location of condensed matter physics (CMP) domain and the high
energy physics (HEP) domain in $T$-$\mu$ diagram.

Let us first come to the entropy density results. In the early universe
scenario, a hot quark-gluon plasma (QGP) state around temperature $T=400$ MeV
or $T=700$ MeV and zero quark chemical potential ($\mu=0$) is expected just
after a few microseconds from the big bang. Due to the very high temperature
of the medium, the constituent particle average momenta become so large that
we can ignore its mass term, and it can be considered a UR case. UR case is
famous for photon gas or black body radiation example where internal energy
density or Intensity (they have connecting relation) follows $T^{4}$ law,
popularly known as Stefan-Boltzmann (SB) law. QGP thermodynamics at high
temperatures reaches that SB limits. Eq. (41) can be converted to UR case by
replacing $v_{g}=c=1$ and by putting $\mu=0$, we can get SB limit expression
for 3D case,

$s_{SB}=\frac{N_{s}T^{3}}{\pi^{2}}\Bigg{[}4\times\Big{\\{}\frac{7}{8}\zeta_{4}\Big{\\}}\Bigg{]},$
(47)

where the Fermi integral function can be converted into Riemann zeta function
$f_{4}=\frac{7}{8}\zeta_{4}$ for $\mu=0$ condition, following the general
relation $f_{n}=\Big{(}1-\frac{1}{2^{n-1}}\Big{)}\zeta_{n}$. For the QGP case,
quark degeneracy factors will have to be put into the $N_{s}$, and gluon
contribution must be added separately. Since two-flavor quark has a degeneracy
factor of 24 and gluon has a degeneracy factor of 16, so massless QGP entropy
density or SB limits of QGP will be

$s^{QGP}_{SB}=\frac{24T^{3}}{\pi^{2}}\Big{[}4\times\Big{\\{}\frac{7}{8}\zeta_{4}\Big{\\}}\Big{]}+\frac{16T^{3}}{\pi^{2}}\Big{[}4\times\zeta_{4}\Big{]}.$
(48)

When we plan to compare graphene entropy density with this SB limit, we have
to understand that the temperature range (few hundred mega electron volt
(MeV), which is equivalent to $10^{12}~{}^{0}$K ) of QGP is too much larger
than temperature range (1-23 milli-electron volt (meV), which is equivalent to
15-300 0K ) of graphene system. Fig. (1) has addressed nicely about these two
domains. It is basically $T$ vs. $\mu$ plots in log scale to cover a broad
band of $T$ and $\mu$ range. We marked the condensed matter physics (CMP)
domain, covering $T\approx 1-23$ meV and $\mu\approx 0-10$ eV. We know that
metal Fermi energy remains within the range $\mu=2-10$ eV, which is marked as
yellow. Unlike metal, graphene system Fermi energy can be changed via doping
methods, and its $\mu/T\ll 1$ and $\mu/T\gg 1$ domains are called Dirac fluid
(DF) or Dirac liquid (DL) and Fermi liquid (FL) domains, respectively, marked
by arrows in Fig. (1). Similar to DF and FL domains for electrons, we may call
early universe QGP as DF domain of quark and quark matter, expected in the
core of neutron star as FL domain of quark. A rectangular domain within
$T=1-400$ MeV and $\mu=0-1000$ MeV is marked as high energy physics (HEP)
domain for quark. Reader can easily visualize the gap between CMP and HEP
domains.

Figure 2: The ratio of entropy density in different domains to $s_{SB}$ with
$\mu/T$ (a) in the 3D case and (b) in the 2D case respectively

After realizing the scale gap in $T$-$\mu$ plane between URF of quark and GF
of electrons, one should understand that we must consider a hypothetical
electron URF to make an equal footing comparison. Within the temperature
($T=0-0.023$ eV) and chemical potential ($\mu=0-10$ eV) range, entropy density
of URF,

$s_{UR}^{3D}=\frac{N_{s}T^{3}}{\pi^{2}}\Bigg{[}4f_{4}\left(A\right)-\frac{\mu}{T}f_{3}\left(A\right)\Bigg{]},$
(49)

has to be plotted with a normalization by $s_{SB}$, given in Eq. (47). This
normalized estimation is sketched by the blue dotted line in the left panel of
Fig. (2), which shows that $s_{UR}^{3D}\approx s_{SB}$ in the domain $\mu/T\ll
1$, as expected. Interestingly, we noticed that the main $\mu/T$ dependence in
entropy density is coming beyond the $\mu/T=1$. Reader can understand that the
terms with Fermi integral function are the main source of $\mu/T$ dependence.
Next, using Eq. (41), the graphene entropy density for the Fermi velocity
$v_{g}=0.006$ is plotted (red solid). From Ref. [40], we can get knowledge
about a broad range of Fermi velocity $v_{g}=1-3\times 10^{6}$ m/s or
$v_{g}=0.003-0.01$ (in natural unit) in graphene system. As charge career
density or $\mu$ decreases, $v_{g}$ will increase and approach towards Dirac
fluid (DF) or strongly coupled electron-electron domain. We have considered
in-between constant values $v_{g}=0.006$. We can understand that the $\mu/T$
dependence of entropy density for URF and GF are the same but $GF\gg URF$ due
to the $1/v_{g}^{3}\approx 5\times 10^{6}$ term. Next, we use Eq. (44) to draw
the entropy density of NRF to plot (green dash line) in the left panel of Fig.
(2). Reader can understand its different trend of $\mu/T$ dependence for NRF
is because of the term
$\Big{[}\frac{5}{2}f_{\frac{5}{2}}\left(A\right)-\frac{\mu}{T}f_{\frac{3}{2}}\left(A\right)\Big{]}$.

A similar trend we can notice for 2D case with similar ranking URF $\ll$ GF
$\ll$ NRF. Only for the transition from 3D to 2D, their orders of magnitude
are shifted toward lower values.

Next, let us come to the shear viscosity results. Here also, we can expect SB
limit type simple expression for UR case at $\mu=0$:

$\eta_{SB}=\frac{4N_{s}}{5\pi^{2}}\tau_{c}\frac{7}{8}\zeta_{4}T^{4},$ (50)

from the general $\eta(T,\mu)$ expression for URF:

$\eta_{UR}^{3D}=\frac{4N_{s}}{5\pi^{2}}\tau_{c}f_{4}\left(A\right)T^{4},$ (51)

by putting $v_{g}=c=1$ in Eq. (42). For massless QGP at $\mu=0$ case, by
replacing degeneracy factors of quarks and gluons in $N_{s}$, we get

$\eta^{QGP}_{SB}=24\Big{[}\frac{4}{5\pi^{2}}\tau_{c}.\Big{\\{}\frac{7}{8}\zeta_{4}\Big{\\}}T^{4}\Big{]}+16\Big{[}\frac{4}{5\pi^{2}}\tau_{c}.\zeta_{4}T^{4}\Big{]}.$
(52)

Again, this QGP is a realistic example of URF but for comparison, we have to
consider electron URF. When we plan to compare the shear viscosity of URF, GF,
and NRF, then we should use Eqs. (51), (42), (45) and for SB limit, we will
use Eq. (50). Similar to normalized entropy density by its SB limit in Fig.
(2), we have plotted normalized shear viscosity by its SB limit in Fig. (3),
where 3D and 2D estimations are plotted in left and right panels respectively.
Shear viscosity expression carries two kinds of information. One is relaxation
time $\tau_{c}$, and another is the remaining thermodynamic phase-space part
as a function of $T$ and $\mu$. During the normalization, $\tau_{c}$
information is canceled, and we can only see their thermodynamic phase-space
part of shear viscosity. Interestingly, it follows a similar trend to other
thermodynamical quantities like entropy density - which shows two types of
$\mu/T$ dependence in the domain $\mu/T\ll 1$ and $\mu/T\gg 1$, which are
commonly assigned as DF and FL.

Figure 3: The ratio of shear viscosity of electron flow in different domains
to $\eta_{SB}$ with $\mu/T$ (a) in the 3D case and (b) in the 2D case
respectively

Now, let us come to the shear viscosity to entropy density ratio $\eta/s$,
which is a more important quantity than only $\eta$ to measure the fluidity of
the system. In the D.F domain, the extreme situation (mathematically) is
$\mu\rightarrow 0$. In this limit, $\eta$ and $s$ carry quite similar terms,
so when we take their ratio, we will get very simplified expressions:

$\displaystyle\frac{\eta}{s}$ $\displaystyle=$
$\displaystyle\frac{\tau_{c}T}{5}~{}{\rm for~{}3D~{}URF/GF},$ (53)
$\displaystyle\frac{\eta}{s}$ $\displaystyle=$
$\displaystyle\frac{\tau_{c}T}{10}~{}{\rm for~{}3D~{}NRF},$ (54)
$\displaystyle\frac{\eta}{s}$ $\displaystyle=$
$\displaystyle\frac{\tau_{c}T}{4}~{}{\rm for~{}2D~{}URF/GF},$ (55)
$\displaystyle\frac{\eta}{s}$ $\displaystyle=$
$\displaystyle\frac{\tau_{c}T}{8}~{}{\rm for~{}2D~{}NRF}.$ (56)

From the String theory-based calculation [37], it was conjectured that
$\eta/s$ has a lower bound, well known as KSS bound, which gives an inequality
$\frac{\eta}{s}\geq\frac{\hbar}{k_{B}}\frac{1}{4\pi}=\frac{1}{4\pi}$ (in
natural unit). Though classically one may expect $\tau_{c}\rightarrow
0\Rightarrow\frac{\eta}{s}\rightarrow 0$, but quantum mechanically, relaxation
or scattering time $\tau_{c}$ or mean free path $\lambda_{c}\approx v\tau_{c}$
can not be lower than de-Brogile range of time or wavelength scale. This
simple quantum mechanical concept also suggests a lower bound of
$\frac{\eta}{s}$, sometimes called a quantum lower bound. By imposing this
bound $\frac{\eta}{s}=\frac{1}{4\pi}$, we can get a rough expression lower
bound of $\tau_{c}$ as

$\displaystyle\tau_{c}$ $\displaystyle=$ $\displaystyle\frac{5}{4\pi T}~{}{\rm
for~{}3D~{}URF/GF},$ (57) $\displaystyle\tau_{c}$ $\displaystyle=$
$\displaystyle\frac{10}{4\pi T}~{}{\rm for~{}3D~{}NRF},$ (58)
$\displaystyle\tau_{c}$ $\displaystyle=$ $\displaystyle\frac{4}{4\pi T}~{}{\rm
for~{}2D~{}URF/GF},$ (59) $\displaystyle\tau_{c}$ $\displaystyle=$
$\displaystyle\frac{8}{4\pi T}~{}{\rm for~{}2D~{}NRF}.$ (60)

Figure 4: Shear viscosity to entropy density ratio; (a) with $T$ for $\mu=0$
(Undoped graphene) and (b) with $\mu/T$

This KSS bound conjecture [37] makes the scientific community curious to find
such fluid whose $\eta/s$ is close to this bound. In other words, if we write
$\eta/s=n/(4\pi)$, where $n\geq 1$, then fluid with $n=1-5$ may be considered
as those special fluids and may be called nearly or close to perfect fluid.
Empirically, QGP is the evidence of such perfect fluid ($n\approx 1-2$) in the
relativistic domain, while close to perfect fluid ($n\approx 5$) example for
NR case is cold atom systems [38]. According to Eqs. (60), we can expect gross
values of relaxation time for QGP and cold atom systems as
$\tau_{c}\approx\frac{5}{4\pi T}$-$\frac{10}{4\pi T}$ and
$\tau_{c}\approx\frac{50}{4\pi T}$ respectively. Similarly, according to the
theoretical prediction from Ref. [22], GF may also belong to this close-to-
perfect fluid category. So far, to the best of our knowledge, no experimental
measurement of $\eta/s$ vs. $T$ plot is available, so the theoretical plot of
$\eta/s$ vs. $T$ in Ref. [22] is considered as our reference to guess or tune
order of magnitude for $\tau_{c}\approx\frac{n}{\pi T}$. In the left panel of
Fig. (4), we can get guidance that $\tau_{c}\approx\frac{n}{\pi T}$ within
$n=3$-$5$ can cover the order of magnitude of $\eta/s$ in the temperature
range $T=35$-$150^{\circ}$K, predicted by Müller et al. [22]. By considering
an average value $\tau_{c}\approx\frac{4}{\pi T}$, we have plotted $\eta/s$ of
2D GF or URF (red solid line) and 2D-NRF (green dash line) against
$\mu/T$-axis in the right panel of Fig. (4). We notice that $\eta/s$ in DF
domain becomes lower than in the FL domain, mainly because of the
thermodynamical phase-space part of $\eta/s$. In terms of Fermi integral
function, this part for GF can be identified as
$\Big{[}3f_{3}\left(A\right)-\frac{\mu}{T}f_{2}\left(A\right)\Big{]}^{-1}f_{3}\left(A\right)$
from Eq. (37). We have put the NRF case for reference, but 2D-NRF for
electrons may be possible in a hypothetical situation. So present study
indicates that dropping $\eta/s$ values and saturating towards constant values
may be found during the transition from FL to DF domains in the graphene
system. Although we have a limitation in that we have considered
$\tau_{c}\propto 1/T$, which may be changed in the actual microscopic
calculation of $\tau_{c}$, and so the trend of $\eta/s$ may also be changed.
It demands more theoretical studies on these $\eta/s$ estimations as well as
the explicit measurement of this quantity from the experimental side.

The inversely proportional relation ($\tau_{c}=\frac{\alpha^{2}}{T}$, where
$\alpha$ is effective fine structure constant) between $\tau_{c}$ and $T$ can
also be noticed in the work of Müller et al. [22], which reflects that the KSS
bound inspired expression of relaxation time $\tau_{c}\propto\frac{1}{T}$ is
also in good agreement with microscopic calculations [22]. A proportional
relation $\tau_{c}\propto T$ is also estimated from the AdS/CMT based
calculations [41, 42, 43], whose $m$ of expression
$\tau_{c}=1/\Gamma=\frac{4\pi T}{m^{2}}$ is basically an effective mass
parameter for some of the metric fluctuations in the gravitational theory.
However, one may get $\tau_{c}=1/\Gamma=\frac{4\pi}{T}\propto\frac{1}{T}$
dependence in the condition $m\propto T$. In searching for the exact
$T$-dependence of relaxation time $\tau_{c}(T)$ of electrons for graphene
systems, more microscopic calculations in different directional (may be model
dependent) attempts may be needed for enriching the understanding of this
field.

##  IV Summary

We can summarize our investigation in the following steps. First, we introduce
a brief macroscopic description of electron fluid in graphene, then we focus
on its microscopic description. Our dealing quantity is considered as energy-
momentum tensor, whose ideal part represents energy density and pressure in
the static limit picture of fluid dynamics. Using those thermodynamical
quantities, our destination from the ideal part of the energy-momentum tensor
becomes entropy density, which will be used to be normalized with shear
viscosity. From the dissipative part of the energy-momentum tensor, shear
viscosity coefficients of electron fluid are calculated based on the kinetic
theory approach with relaxation time approximation. Temperature and chemical
potential-dependent general expressions of shear viscosity, entropy density,
and the shear viscosity to entropy density ratio has been calculated and
plotted for different cases of electron fluid like nonrelativistic, graphene
and ultra-relativistic cases. For completeness of comparison, we considered
both 3D and 2D systems. Analyzing the results of different cases, we get a
comparative understanding and conclusions, which are addressed briefly in
bullet points:

  1. 1.

The $\mu/T$ dependence of shear viscosity $\eta$ as well as entropy density
$s$ for URF and GF are exactly similar but a little different from the NRF.

  2. 2.

We notice a huge difference among URF, GF, and NRF in terms of the order of
magnitude of $\eta$ and $s$ with ranking URF $\ll$ GF $\ll$ NRF.

  3. 3.

During transiting from 3D to 2D, order of magnitude of $\eta$ and $s$ shift
towards lower values.

  4. 4.

When we go from Fermi Liquid ($\mu/T\gg 1$) to Dirac Liquid ($\mu/T\ll 1$)
domain, values of $\eta$, $s$ and $\eta/s$ ratio decrease towards a saturated
values.

  5. 5.

Interesting ranking for $\eta/s$ becomes URF $=$ GF $\geq$ NRF.

The present comparative study on the microscopic calculation of shear
viscosity may be considered as a good documentation of master formulas of
different cases from 3D-URF, GF, NRF to 2D-URF, GF and NRF. In the future, it
may be useful for actual graphene system estimation, where one should go with
some first principle or model-dependent calculation of relaxation time. Also,
one should deal with electron-hole plasma with appropriate degeneracy factor
in the Dirac Fluid domain for the actual graphene system but the present work
sticks with electron description only due to observing the estimations for its
different dispersion relations. Our immediate future plan is to concentrate on
the actual graphene phenomenology on the viscous aspects.

###### Acknowledgements.

This work was partly (CWA and TZW) supported by the Doctoral Fellowship in
India (DIA) program of the Ministry of Education, Government of India. The
authors thank the other members of eHD club - Sesha P. Vempati, Ashutosh
Dwibedi, Narayan Prasad, Bharat Kukkar, and Subhalaxmi Nayak.

##  Appendix A Density of States

The density of states is nothing but the total number of energy states per
unit energy interval. If the total number of energy states in energy range $E$
to $E+dE$ are $D\left(E\right)dE$, then the density of states will be

$g\left(E\right)=\frac{D\left(E\right)dE}{dE}~{}.$ (61)

Now, here are some expressions of the number of energy states for different-
different cases expressed as
Case:1. for 3D-Graphene,

$D\left(E\right)dE=N_{s}\frac{4\pi V}{h^{3}v_{g}^{3}}E^{2}dE,$ (62)

Case:2. for 3D nonelativistic,

$D\left(E\right)dE=N_{s}2\pi
V\left(\frac{2m}{h^{2}}\right)^{\frac{3}{2}}\sqrt{E}dE,$ (63)

Case:3. for 2D Graphene,

$D\left(E\right)dE=N_{s}\frac{2\pi a}{h^{2}v_{g}^{2}}E\,dE,$ (64)

Case:4. for 2D Nonrelativistic,

$D\left(E\right)dE=N_{s}\frac{2\pi a}{h^{2}}m\,dE.$ (65)

In the above expressions, $V$ represents the volume in position space.

##  Appendix B Fermi-Dirac Integral

We have the integral form as

$\displaystyle
f_{\nu}(A)=\frac{1}{\Gamma(\nu)}\int_{0}^{\infty}\frac{x^{\nu-1}}{A^{-1}e^{x}+1}dx,$
(66)

where $f_{\nu}(A)$ is known as the Fermi-Dirac integral and $x=\beta E$. And
the expression in terms of energy can be written in terms of $x$ as

$\displaystyle\int_{0}^{\infty}\frac{E^{\nu-1}}{A^{-1}e^{\beta E}+1}dE$
$\displaystyle=\frac{1}{\beta^{\nu}}\int_{0}^{\infty}\frac{x^{\nu-1}}{A^{-1}e^{x}+1}dx$
$\displaystyle=\frac{1}{\beta^{\nu}}\Gamma(\nu)f_{\nu}(A).$ (67)

##  Appendix C Average Angular Integral in 2D

We have the integral form as

$\displaystyle\int p_{i}p_{j}p_{k}p_{l}d^{2}p=pdp\int
p_{i}p_{j}p_{k}p_{l}d\theta.$ (68)

Since

$\vec{p}=p\hat{n},$

where

$\hat{n}=\cos{\theta}\,\hat{i}+\sin{\theta}\hat{j},$

and

$p_{i}=\vec{p}.\hat{e_{i}}=p\left(\hat{n}.\hat{e_{i}}\right)=pn_{i},$

the integral becomes

$\displaystyle\int p_{i}p_{j}p_{k}p_{l}d^{2}p=pdp,p^{4}\int
n_{i}n_{j}n_{k}n_{l}\,d\theta.$ (69)

Now, we have to calculate

$\int n_{i}n_{j}n_{k}n_{l}\,d\theta=?$

Case:1. The above integral becomes

$\displaystyle\int n_{1}^{2}n_{2}^{2}\,d\theta$
$\displaystyle=\int_{0}^{2\pi}\cos^{2}{\theta}\sin^{2}{\theta}\,d\theta$ (70)
$\displaystyle=4\int_{0}^{\frac{\pi}{2}}\cos^{2}{\theta}\sin^{2}{\theta}\,d\theta.$
(71)

Now, using the Beta function, we know that

$\displaystyle
B\left(u,v\right)=2\int_{0}^{\frac{\pi}{2}}\left(\cos{\theta}\right)^{2u-1}\left(\sin{\theta}\right)^{2v-1}\,d\theta$
(72) $\displaystyle\Rightarrow$ $\displaystyle B\left(u,v\right)=\frac{\Gamma
u\,\Gamma v}{\Gamma\left(u+v\right)}.$ (73)

Applying this, we get

$\int n_{1}^{2}n_{2}^{2}\,d\theta=\frac{2\pi}{8}.$ (74)

Case:2.

$\displaystyle\int n_{1}^{3}n_{2}\,d\theta$ $\displaystyle=\int
n_{1}n_{2}^{3}\,d\theta$ (75)
$\displaystyle=\int_{0}^{2\pi}\cos^{3}{\theta}\sin{\theta}\,d\theta=0.$ (76)

Case:3.

$\displaystyle\int n_{1}^{4}\,d\theta$ $\displaystyle=\int n_{2}^{4}\,d\theta$
(77) $\displaystyle=\int_{0}^{2\pi}\sin^{4}{\theta}\,d\theta$ (78)
$\displaystyle=2\,B\left(\frac{5}{2},\frac{1}{2}\right)=\frac{3\pi}{4}.$ (79)

Now, the above integral can be written as

$\displaystyle\int n_{1}^{4}\,d\theta$ $\displaystyle=\int
n_{2}^{4}\,d\theta=\frac{2\pi}{8}\times 3,$ (80)

Now, the integral can be expressed as

$\displaystyle\int
n_{i}n_{j}n_{k}n_{l}\,d\theta=\frac{2\pi}{8}\left(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right)$
(81)

and

$\displaystyle\int
p_{i}p_{j}p_{k}p_{l}\,d^{2}p=2\pi\,p\,dp\frac{p^{4}}{8}\left(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right).$
(82)

Now, the average final expression will be

$<p_{i}p_{j}p_{k}p_{l}>=\frac{p^{4}}{8}\left(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right).$
(83)

## References

  * Ku  _et al._ [2020] M. J. H. Ku _et al._ , Imaging viscous flow of the Dirac fluid in graphene, Nature (London) 583, 537 (2020).
  * Varnavides  _et al._ [2020] G. Varnavides, A. S. Jermyn, P. Anikeeva, C. Felser, and P. Narang, Electron hydrodynamics in anisotropic materials, Nat. Commun. 11, 4710 (2020).
  * Hasdeo  _et al._ [2021] E. H. Hasdeo, J. Ekström, E. G. Idrisov, and T. L. Schmidt, Electron hydrodynamics of two-dimensional anomalous hall materials, Phys. Rev. B 103, 125106 (2021).
  * Hui  _et al._ [2021] A. Hui, V. Oganesyan, and E.-A. Kim, Beyond ohm’s law: Bernoulli effect and streaming in electron hydrodynamics, Phys. Rev. B 103, 235152 (2021).
  * Huang and Lucas [2021] X. Huang and A. Lucas, Electron-phonon hydrodynamics, Phys. Rev. B 103, 155128 (2021).
  * Tavakol and Kim [2021] O. Tavakol and Y. B. Kim, Artificial electric field and electron hydrodynamics, Phys. Rev. Res. 3, 013290 (2021).
  * Di Sante  _et al._ [2020] D. Di Sante, J. Erdmenger, M. Greiter, I. Matthaiakakis, R. Meyer, D. Rodríguez Fernández, R. Thomale, E. van Loon, and T. Wehling, Turbulent hydrodynamics in strongly correlated Kagome metals, Nat. Commun. 11, 3997 (2020).
  * Narozhny  _et al._ [2021] B. N. Narozhny, I. V. Gornyi, and M. Titov, Hydrodynamic collective modes in graphene, Phys. Rev. B 103, 115402 (2021).
  * Sulpizio  _et al._ [2019] J. A. Sulpizio, L. Ella, A. Rozen, J. Birkbeck, D. J. Perello, D. Dutta, M. Ben-Shalom, T. Taniguchi, K. Watanabe, T. Holder, _et al._ , Visualizing poiseuille flow of hydrodynamic electrons, Nature (London) 576, 75 (2019).
  * Gallagher  _et al._ [2019] P. Gallagher, C.-S. Yang, T. Lyu, F. Tian, R. Kou, H. Zhang, K. Watanabe, T. Taniguchi, and F. Wang, Quantum-critical conductivity of the dirac fluid in graphene, Science 364, 158 (2019).
  * Berdyugin  _et al._ [2019] A. I. Berdyugin, S. G. Xu, F. M. D. Pellegrino, R. K. Kumar, A. Principi, I. Torre, M. B. Shalom, T. Taniguchi, K. Watanabe, I. V. Grigorieva, M. Polini, A. K. Geim, and D. A. Bandurin, Measuring hall viscosity of graphene’s electron fluid, Science 364, 162 (2019).
  * Ella  _et al._ [2019] L. Ella, A. Rozen, J. Birkbeck, M. Ben-Shalom, D. Perello, J. Zultak, T. Taniguchi, K. Watanabe, A. K. Geim, S. Ilani, _et al._ , Simultaneous voltage and current density imaging of flowing electrons in two dimensions, Nat. Nanotechnol. 14, 480 (2019).
  * Bandurin  _et al._ [2018] D. A. Bandurin, A. V. Shytov, L. S. Levitov, R. K. Kumar, A. I. Berdyugin, M. Ben Shalom, I. V. Grigorieva, A. K. Geim, and G. Falkovich, Fluidity onset in graphene, Nat. Commun. 9, 4533 (2018).
  * Bandurin  _et al._ [2016] D. A. Bandurin, I. Torre, R. K. Kumar, M. B. Shalom, A. Tomadin, A. Principi, G. H. Auton, E. Khestanova, K. S. Novoselov, I. V. Grigorieva, L. A. Ponomarenko, A. K. Geim, and M. Polini, Negative local resistance caused by viscous electron backflow in graphene, Science 351, 1055 (2016).
  * Braem  _et al._ [2018] B. A. Braem, F. M. D. Pellegrino, A. Principi, M. Röösli, C. Gold, S. Hennel, J. V. Koski, M. Berl, W. Dietsche, W. Wegscheider, M. Polini, T. Ihn, and K. Ensslin, Scanning gate microscopy in a viscous electron fluid, Phys. Rev. B 98, 241304(R) (2018).
  * Jaoui  _et al._ [2018] A. Jaoui, B. Fauque, C. W. Rischau, _et al._ , Departure from the Wiedemann-Franz law in ${WP_{2}}$ driven by mismatch in ${T}$-square resistivity prefactors, npj Quantum Mater. 3 64 (2018).
  * Moll  _et al._ [2016] P. J. W. Moll, P. Kushwaha, N. Nandi, B. Schmidt, and A. P. Mackenzie, Evidence for hydrodynamic electron flow in ${PdCoO_{2}}$, Science 351, 1061 (2016).
  * Crossno  _et al._ [2016] J. Crossno, J. K. Shi, K. Wang, X. Liu, A. Harzheim, A. Lucas, S. Sachdev, P. Kim, T. Taniguchi, K. Watanabe, T. A. Ohki, and K. C. Fong, Observation of the Dirac fluid and the breakdown of the Wiedemann-Franz law in graphene, Science 351, 1058 (2016).
  * Lucas and Fong [2018] A. Lucas and K. C. Fong, Hydrodynamics of electrons in graphene, J. of Phys.: Condens Matter 30, 053001 (2018).
  * Narozhny [2019] B. N. Narozhny, Electronic hydrodynamics in graphene, Ann. Phys. 411, 167979 (2019).
  * Narozhny [2022] B. N. Narozhny, Hydrodynamic approach to two-dimensional electron systems, Riv. Nuovo Cim. 45, 661 (2022).
  * Müller  _et al._ [2009] M. Müller, J. Schmalian, and L. Fritz, Graphene: A nearly perfect fluid, Phys. Rev. Lett. 103, 025301 (2009).
  * Principi  _et al._ [2016] A. Principi, G. Vignale, M. Carrega, and M. Polini, Bulk and shear viscosities of the two-dimensional electron liquid in a doped graphene sheet, Phys. Rev. B 93, 125410 (2016).
  * Svintsov [2018] D. Svintsov, Hydrodynamic-to-ballistic crossover in dirac materials, Phys. Rev. B 97, 121405 (2018).
  * Gall  _et al._ [2023] V. Gall, B. N. Narozhny, and I. V. Gornyi, Electronic viscosity and energy relaxation in neutral graphene, Phys. Rev. B 107, 045413 (2023).
  * Narozhny and Schütt [2019] B. N. Narozhny and M. Schütt, Magnetohydrodynamics in graphene: Shear and hall viscosities, Phys. Rev. B 100, 035125 (2019).
  * Gavin [1985] S. Gavin, Transport coefficients in ultrarelativistic heavy ion collisions, Nucl. Phys. A 435, 826 (1985).
  * Hosoya and Kajantie [1985] A. Hosoya and K. Kajantie, Transport Coefficients of QCD Matter, Nucl. Phys. B 250, 666 (1985).
  * Chakraborty and Kapusta [2011] P. Chakraborty and J. I. Kapusta, Quasiparticle theory of shear and bulk viscosities of hadronic matter, Phys. Rev. C 83, 014906 (2011).
  * Ghosh  _et al._ [2016] S. Ghosh, T. C. Peixoto, V. Roy, F. E. Serna, and G. a. Krein, Shear and bulk viscosities of quark matter from quark-meson fluctuations in the Nambu–Jona-Lasinio model, Phys. Rev. C 93, 045205 (2016).
  * Ghosh  _et al._ [2013] S. Ghosh, A. Lahiri, S. Majumder, R. Ray, and S. K. Ghosh, Shear viscosity due to Landau damping from the quark-pion interaction, Phys. Rev. C 88, 068201 (2013).
  * Arnold  _et al._ [2000] P. Arnold, G. D. Moore, and L. G. Yaffe, Transport coefficients in high temperature gauge theories (i): Leading-log results, J. High Energy Phys. 2000 (11), 001.
  * Itakura  _et al._ [2008] K. Itakura, O. Morimatsu, and H. Otomo, Shear viscosity of a hadronic gas mixture, Phys. Rev. D 77, 014014 (2008).
  * Fernandez-Fraile and Gomez Nicola [2009] D. Fernandez-Fraile and A. Gomez Nicola, Transport coefficients and resonances for a meson gas in Chiral Perturbation Theory, Eur. Phys. J. C 62, 37 (2009).
  * Lang  _et al._ [2012] R. Lang, N. Kaiser, and W. Weise, Shear Viscosity of a hot pion gas, Eur. Phys. J. A 48, 109 (2012).
  * Ghosh [2014] S. Ghosh, A real-time thermal field theoretical analysis of Kubo-type shear viscosity: Numerical understanding with simple examples, Int. J. Mod. Phys. A 29, 1450054 (2014).
  * Kovtun  _et al._ [2005] P. Kovtun, D. T. Son, and A. O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94, 111601 (2005).
  * Schäfer and Teaney [2009] T. Schäfer and D. Teaney, Nearly perfect fluidity: from cold atomic gases to hot quark gluon plasmas, Rep. Prog. Phys. 72, 126001 (2009).
  * Jaiswal and Roy [2016] A. Jaiswal and V. Roy, Relativistic hydrodynamics in heavy-ion collisions: general aspects and recent developments, Adv. High Energy Phys. 2016, 9623034 (2016).
  * Elias  _et al._ [2011] D. C. Elias, R. V. Gorbachev, A. S. Mayorov, S. V. Morozov, A. A. Zhukov, P. Blake, L. A. Ponomarenko, I. V. Grigorieva, K. S. Novoselov, F. Guinea, and A. K. Geim, Dirac cones reshaped by interaction effects in suspended graphene, Nat. Phys. 7, 701 (2011).
  * Davison [2013] R. A. Davison, Momentum relaxation in holographic massive gravity, Phys. Rev. D 88, 086003 (2013).
  * Andrade and Withers [2014] T. Andrade and B. Withers, A simple holographic model of momentum relaxation, J. High Energy Phys. 2014 (5), 101.
  * Davison and Goutéraux [2015] R. A. Davison and B. Goutéraux, Momentum dissipation and effective theories of coherent and incoherent transport, J. High Energy Phys. 2015 (1), 39.