# Shear viscosity of nucleonic matter

Xian-Gai Deng  De-Qing Fang  Yu-Gang Ma<EMAIL_ADDRESS>Key Laboratory
of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern
Physics, Fudan University, Shanghai 200433, China  Shanghai Research Center
for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai 200438,
China

###### Abstract

The research status of the shear viscosity of nucleonic matter is reviewed.
Some methods to calculate the shear viscosity of nucleonic matter are
introduced, including mean free path, Green-Kubo, shear strain rate, Chapman-
Enskog and relaxation time approximation. Based on these methods, results for
infinite and finite nucleonic matter are discussed, which are attempts to
investigate the universality of the ratio of shear viscosity over entropy
density and transport characteristics like the liquid-gas phase transition in
nucleonic matter. In addition, shear viscosity is also briefly discussed for
the quantum chrodynamical matter produced in relativistic heavy-ion
collisions.

###### keywords:

shear viscosity, ratio of shear viscosity to entropy density , nucleonic
matter , KSS bound , Green-Kubo , Chapman-Enskog , relaxation time
approximation

††journal: Progress in Particle and Nuclear Physics

###### Contents

  1. 1 Introduction
  2. 2 Analysis methods of shear viscosity
    1. 2.1 The definition of shear viscosity
    2. 2.2 Green-Kubo method
    3. 2.3 Shear strain rate method
    4. 2.4 Chapman-Enskog method
    5. 2.5 Relaxation time method
  3. 3 Theoretical aspects of shear viscosity in nucleonic matter
    1. 3.1 Infinite nuclear matter
    2. 3.2 Finite nuclear matter
  4. 4 Experimental aspects of shear viscosity in nucleonic matter
    1. 4.1 Analysis method by experiment
    2. 4.2 Shear viscosities of finite nuclei in experiment
    3. 4.3 Others
  5. 5 $\displaystyle\eta$/s in relativistic heavy$\displaystyle-$ion collision
    1. 5.1 Ratio of shear viscosity to entropy density across the QCD phase diagram
    2. 5.2 The dependence of ratio of shear viscosity to entropy density on collision centrality
    3. 5.3 Magnetic field effect on ratio of shear viscosity to entropy density
    4. 5.4 Bayesian estimation of ratio of shear viscosity to entropy density
  6. 6 Summary and outlook
  7. A Appendices, if necessary

## 1 Introduction

In a non-perfect fluid, viscosity represents internal friction, which causes
an irreversible energy dissipation process, and it is a measure of the flow
resistance to the liquid, which means the higher value, and the more
resistance, and thus the fluid flows more slowly. Shear viscosity, which
describes momentum transfer ability in a flowing fluid, is one of the
important transport properties for lots of substances, such as macroscopic
matter as well as microscopic and quantum matter, e.g. hot dense quark matter,
etc. [1, 2]. Viscosity, which affects solute dynamics and reaction rates in a
solvent, is crucial for understanding biological processes and chemical
reactions [3, 4, 5]. In astrophysics, properties of transport coefficients
(bulk viscosity, shear viscosity, and thermal conductivity) have been studied
in different layers of neutron stars (NS), e.g. in the crust and in the core
[6, 7, 8]. The transport coefficients were described in the framework of
Brueckner-Hartree-Fock (BHF) theory [9, 10], and comparisons were made using
several equations of states (EoSs) [8]. For viscosity, one of the most
important features is the damping of oscillations. So it plays an important
role for $\displaystyle r$-modes in rotating pulsars [11] and affects the
emission and propagation of gravitational waves, which is important for the
planning of advanced gravitational wave experiments [12, 13]. The study on
transport coefficients could improve our understanding of the dynamical
behaviour of the merging of binary neutron stars through the signal of
gravitational waves, which was observed by LIGO and Virgo on August 17, 2017
(GW170817) [14]. The most common astrophysical manifestation of the NSs is the
radio pulsars, which are driven by the strong magnetic field [15] as high as
1012-1013 G [16]. With such high magnetic fields, it could impact the EoS and
have strong effects on the NS matter. How the magnetic field affects the
transport coefficients of the magnetized NS should be considered [12].

Viscous liquids like honey, syrup, and oil flow slowly, and less viscous
liquids like water and gasoline flow quickly. The Navier-Stokes (NS) equation
can be used to describe the motion of non-perfect fluid in fluid mechanics,
where the shear viscosity coefficient and the bulk viscosity coefficient are
included [17, 18]. At the microscopic level, viscosity is caused by the
potential and collisions of particles (or molecules). While macroscopically or
hydrodynamically, it is determined by a constitutive relationship as the ratio
of the shear stress to the shear rate [19, 20]. In equilibrium molecular
dynamics (EMD), the viscosity $\displaystyle\eta$ is given as an integral of
the shear-stress autocorrelation function determined during the simulation
[21], which is called Green-Kubo (GK) formula derived from linear response
theory [22] or given by Einstein relation [23]. In the framework of EMD
methods, for the convergence of shear-stress autocorrelation function, it
would take a long simulation time due to relatively large fluctuations in the
shear tensor. In molecular simulations for calculating viscosity, a
nonequilibrium molecular dynamics (NEMD) method with a planar Couette flow
field at strain rate $\displaystyle\dot{\gamma}=\partial u_{x}/\partial y$
becomes increasingly popular and has been used extensively to predict the
rheological properties of real fluids [21].

In 2005, Kovtun, Son, and Starinets (KSS) [24] studied the shear viscosity
with certain supersymmetric gauge theories with gravity duals and gave a lower
limit value for the ratio of shear viscosity to entropy density,
$\displaystyle\eta/s$ $\displaystyle\geqslant\hbar/(4\pi k_{B})$
($\displaystyle\hbar$ is the Planck’s constant divided by $\displaystyle 4\pi$
and $\displaystyle k_{B}$ is the Boltzmann constant), which is called the KSS
bound. It has been conjectured that this ratio is the lowest limit for a large
class of quantum field theories [24]. A new state of matter, the quark gluon
plasma (QGP), is created in ultra-relativistic heavy-ion collisions, which is
manifested by the results from the Relativistic Heavy Ion Collider (RHIC) at
Brookhaven [25, 26, 27, 28, 29, 30, 31, 32] and the Large Hadron Collider
(LHC) at CERN [33, 34, 35, 36, 37, 38]. Based on heavy-ion data from these two
colliders and theoretical modeling, lots of efforts are paid to uncover the
fundamental properties of the quark gluon plasma [39, 40, 41, 42, 43]. The
analysis of the elliptic flow $\displaystyle v_{2}$ in the ultra-relativistic
heavy-ion collisions shows that QGP behaves like a near perfect liquid with a
lower limit of $\displaystyle\eta/s$ [44]. How perfect is the fluid in
relativistic heavy-ion collisions [45]? In explaining the data from RHIC, the
results [46, 47] from the viscous hydrodynamics model suggest that the ratio
of shear viscosity to entropy density has a very small value. It indicates
that the quark-gluon plasma (QGP) is nearly an ideal fluid. Moreover, in the
analysis by Csernai et al. [48], it shows that the transition from hadrons to
quarks and gluons has behavior similar to helium, nitrogen, and water at and
near their phase transitions in the ratio $\displaystyle\eta/s$.

Near and below a critical temperature, $\displaystyle T_{c}$, the condensates
of bosons or fermion pairs behave like superfluids, which are frictionless
[49]. The behavior of shear viscosity versus temperature for bosonic and
fermionic system is different below $\displaystyle T_{c}$ [49]. For instance,
like the bosonic 4He, viscosity increases as the temperature approaches zero
below $\displaystyle T_{c}$ [50] and in the Fermionic 3He, the shear viscosity
decreases to zero with decreasing of temperature below $\displaystyle T_{c}$
[51, 52, 53]. The shear viscosity for 3He at the transition temperature
$\displaystyle T_{A}$ = 2.6 mK appears a rapid decrease [52]. The research on
shear viscosity and its ratio over entropy density as well as for a Fermions
system with the unitary Fermi gas has attracted lots of attention over the
past few years [54, 55, 56, 57, 58, 59, 60, 49]. The unitary Fermi gas where
particles interact at infinite scattering length is experimentally
investigated with ultracold atoms in a balanced mixture of, e.g., the two
lowest hyperfine levels of 6Li at a Feshbach resonance [56, 58, 59]. It is
found that the viscosity of a unitary Fermi gas is closer to that of fermionic
3He than to that of bosonic 4He [60, 49]. And both the viscosity and the ratio
($\displaystyle\eta$/s) of the unitary gas are predicted to exhibit a minimum,
somewhat below $\displaystyle T_{c}$ [60]. One found that a minimum value of
$\displaystyle\eta/s$$\displaystyle\simeq$ 0.44 (in units of
$\displaystyle\hbar$/ kB) at the temperature $\displaystyle
T/T_{F}$$\displaystyle\simeq$ 0.27, with $\displaystyle T_{F}$ the Fermi
temperature [59].

And it is also very interesting to see the ratio of shear viscosity to entropy
density for nuclear matter, which is composed of protons and neutrons.
Experimentally, some nuclear experimental facilities, such as FAIR (the
International Accelerator Facility for Antiproton and Ion Research) [61],
GANIL (Grand Acc$\displaystyle\acute{\rm e}$l$\displaystyle\acute{\rm
e}$rateur National d’Ions Lourds-National Large Heavy Ion Accelerator) [62] in
Europe, and High Intensity Accelerator Facility (HIAF) in China [63, 64, 65]
offer chances to explore phase diagram of nuclear matter in the region of high
baryon density as well as transport properties [66, 67, 68, 69, 70, 71, 72].
Theoretically, transport coefficients such as shear viscosity have been
studied for a few decades in nuclear physics and attracted lots of attentions
[73, 74, 75, 76, 24, 77, 44, 78, 79, 80, 81]. The equation of state and
transport coefficients are important quantities for the hot and dense hadron
gases in nuclear physics [76]. The behaviors of shear viscosity depend on the
nucleon-nucleon cross sections as well as the nuclear equation of state, etc.
For infinite nuclear matter, the effects of Pauli blocking, nucleon-nucleon
cross section, temperature, density, isospin asymmetry, etc. on shear
viscosity are discussed [74, 75, 82, 83, 84]. With the help of transport
models like Boltzmann-Uehling-Uhlenbeck model (BUU) and quantum molecular
dynamics model (QMD) [85, 86, 87, 88, 89, 90], the relation between shear
viscosity and the liquid-gas (LG) phase transitions, or collective flow, or
giant dipole resonance (GDR) has been investigated [78, 91, 79, 92].

Figure 1.1: (Color online) The QCD diagram. Picture taken from this (CSQCD
2017) conference poster.

The nuclear matter driven by the strong interaction acting like the Van der
Waals potential for water can undergo the liquid-gas phase transitions in
lower energy heavy-ion collisions [93, 94, 95, 96, 97, 98, 99, 100]. The QCD
diagram in Fig. 1.1, which presents the relationship among temperature, baryon
density (or baryon chemical potential), and nuclear asymmetry (nn-np), was
explored by the experimental facilities of LHC, RHIC, FAIR , and NICA, etc.
The understanding of the properties of the strong interaction phase transition
is still rather limited. The exploration of nuclear matter properties and QCD
phase diagram would need both new experimental data with extended detection
capabilities and theoretical calculations [101, 29, 102, 103, 104, 105]. As
seen from the QCD diagram, the phase transition is not only for nuclear matter
but also for the QGP matter. Compilation of data for different substances [48]
demonstrates a drop of the shear viscosity in the region of a phase
transition, when the entropy density is employed as a universal reference.
Furthermore, from the calculations of shear viscosity of QCD matter in the
hadronic phase by the coupled Boltzmann equations of pions and nucleons in low
temperatures and low baryon-number densities, one found that
$\displaystyle\eta/s$ shows a discontinuity in the bottom of the valley for a
first-order phase transition which is similar to water [106, 107]. The studies
on the specific shear viscosity near the liquid$\displaystyle-$gas phase
transition may shed new light on the nature of this transition in nucleonic
matter [78, 83].

## 2 Analysis methods of shear viscosity

Shear viscosity is a common property of substances, and there are some methods
to calculate it, such as the mean free path, Green-Kubo formula, shear strain
rate method, Chapman-Enskog method, and relaxation time method. In this
chapter, the details of these methods will be introduced.

### 2.1 The definition of shear viscosity

Figure 2.1: (Color online) The illustration of shear stress.

Considering two planes ‘1’ and ‘2’ with area $\displaystyle A$ as in Fig. 2.1,
a fluid with thickness $\displaystyle d$ is between them. When the plane ‘2’
is fixed, one drags plane ‘1’ with a force $\displaystyle\vec{F}$. Sometime
later, the fluid velocity will decrease from the value $\displaystyle u_{x}$
at plane ‘1’ to zero at the stationary plane ‘2’. The velocity gradient of the
fluid is determined by the shear force between flow layers. The shear stress
$\displaystyle F/A$ is proportional to $\displaystyle u_{x}/d$ which is called
Newton’s equation and is written as follows:

$\begin{split}\frac{F}{A}=-\eta\frac{u_{x}}{d},\end{split}$ (2.1)

where $\displaystyle\eta$ is called the friction coefficient or the
coefficient of shear viscosity. Rewritten Eq. (2.1) in a differential form,
one has

$\begin{split}\frac{F}{A}=-\eta\frac{\partial u_{x}}{\partial y}.\end{split}$
(2.2)

It is seen that the shear viscosity as internal friction causes the
irreversible transfer of momentum from points with large velocity to small
ones [18]. So the physical meaning of shear viscosity is the ability to
transfer momentum over a distance of the mean free path, which measures the
average distance traveled by a particle between collisions. Hence, a system
with lower values of $\displaystyle\eta$ indicates that the constituents of
the system interact strongly to transfer momentum easily. On the contrary, a
system with a large $\displaystyle\eta$ means that the constituents of the
system interact weekly, and it becomes strenuous for the momentum transfer
between the constituents [108].

Recalling the fluid mechanics [18], for the ideal fluid, we have,

$\begin{split}\frac{\partial p_{\alpha}}{\partial
t}=\frac{\partial\hat{\varrho}u_{\alpha}}{\partial t}=-\frac{\partial
T_{\alpha\beta}}{\partial r_{\beta}},\,\,\,\,\alpha,\beta=x,y,z,\end{split}$
(2.3)

where $\displaystyle p$, $\displaystyle\hat{\varrho}$, and $\displaystyle u$
are the local momentum density, particle mass density and local fluid
velocity, respectively. The tensor $\displaystyle T_{\alpha\beta}$ is called
momentum flux density tensor which is defined as (for ideal fluid),

$\begin{split}T_{\alpha\beta}=P\delta_{\alpha\beta}+\hat{\varrho}u_{\alpha}u_{\beta},\end{split}$
(2.4)

where $\displaystyle P$ is the pressure and
$\displaystyle\delta_{\alpha\beta}$ is equal to 1 (or 0) when
$\displaystyle\alpha=\beta$ ($\displaystyle\alpha\neq\beta$). For viscous
fluid, an irreversible viscous term-$\displaystyle\Pi_{\alpha\beta}$ is added
in momentum flux density tensor,

$\begin{split}T_{\alpha\beta}=\hat{P}\delta_{\alpha\beta}+\hat{\varrho}u_{\alpha}u_{\beta}+\Pi_{\alpha\beta}.\end{split}$
(2.5)

The form of the tensor $\displaystyle\Pi_{\alpha\beta}$ which is called
viscous stress tensor, can be written as follows,

$\begin{split}\Pi_{\alpha\beta}=-\eta(\frac{\partial u_{\alpha}}{\partial
r_{\beta}}+\frac{\partial u_{\beta}}{\partial
r_{\alpha}}-\frac{2}{3}\delta_{\alpha\beta}\frac{\partial u_{\gamma}}{\partial
r_{\gamma}})-\xi\delta_{\alpha\beta}\frac{\partial u_{\gamma}}{\partial
r_{\gamma}},\,\,\,\,\alpha,\beta,\gamma=x,y,z,\end{split}$ (2.6)

where $\displaystyle\eta$ and $\displaystyle\xi$ are shear viscosity and bulk
viscosity, respectively. In a flat-parallel flow of uniform fluid
$\displaystyle u=u_{x}$, the $\displaystyle x,y$ component of $\displaystyle
T_{\alpha\beta}$ can be obtained,

$\begin{split}T_{xy}=\Pi_{xy}=-\eta\frac{\partial u_{x}}{\partial
y}.\end{split}$ (2.7)

In this case, the momentum flux density tensor is equal to the viscous stress
tensor. And here the Eq. (2.7) corresponds to the Eq. (2.2).

Above, shear viscosity is given by a direct definition or from fluid
mechanics. In thermodynamics, a simple estimate of the shear viscosity for a
dilute gas can be obtained by the mean free path of particles, which reads as
[44, 109],

$\begin{split}\eta=\frac{1}{3}\rho m\bar{v}\lambda,\end{split}$ (2.8)

where $\displaystyle\rho$ is the density, $\displaystyle\bar{v}$ is the
average velocity of particles, and $\displaystyle\lambda$ is the mean free
path, which can be written as $\displaystyle\lambda$ = $\displaystyle
1/(\rho\sigma)$. Here $\displaystyle\sigma$ is the transport cross section.

### 2.2 Green-Kubo method

It is a very traditional way [110] to obtain shear viscosity (or other
transport coefficients) for a liquid from pressure or momentum fluctuations
with the Green-Kubo relation at equilibrium [111], which is related to the
linear response theory [22]. It can be found that the shear viscosity of a gas
or liquid system arises from the instantaneous fluctuations due to the thermal
motion in the stress tensor. In the classical limit [112], the standard Green-
Kubo formula for the calculation of the shear viscosity coefficient
$\displaystyle\eta$ can be expressed by [22, 113, 114, 115]:

$\begin{split}\eta=\frac{V}{T}\int_{t_{0}}^{\infty}\langle
T_{\alpha\beta}(t)T_{\alpha\beta}(t_{0})\rangle
dt;\,\,\alpha,\beta=x,y,z,\alpha\neq\beta,\end{split}$ (2.9)

where $\displaystyle V$ and $\displaystyle t_{0}$ are system volume and
equilibrium time, respectively. And the integral kernel $\displaystyle\langle
T_{\alpha\beta}(t)T_{\alpha\beta}(t_{0})\rangle$ is called the correlation
function $\displaystyle C(t)$. The bracket $\displaystyle\langle...\rangle$
denotes average over an equilibrium ensemble of events. In the Green-Kubo
method, the convergence of the correlation function $\displaystyle C(t)$ is
important. For the nuclear matter, $\displaystyle C(t)$ was checked in the
Improved Quantum Molecular Dynamics (ImQMD) model with mean field [115] as
shown in Fig. 2.2.

Figure 2.2: (Color online) The evolution of autocorrelation function for
different components at $\displaystyle T_{0}$ = 30 MeV and different densities
of $\displaystyle 0.4\rho_{0}$ (a) or $\displaystyle 1.0\rho_{0}$ (b) (here
$\displaystyle\rho_{0}$ = 0.16 fm-3 is the saturation density for nuclear
matter). It is worth to mention that the calculations for this figure is
without mean field [115]

.

One can see that $\displaystyle C(t)$ of nucleonic matter is convergent. At
densities of $\displaystyle 0.4\rho_{0}$, the correlation function
$\displaystyle C(t)$ tends to zero in a period of time $\displaystyle\sim$ 60
fm/c, and it decreases faster at a higher density of $\displaystyle
1.0\rho_{0}$. Since it is symmetric in the simulation box, correlation
functions for different components are almost the same.

In Eq. (2.9), $\displaystyle T_{\alpha\beta}(t)$ is an off-diagonal element of
the stress tensor, which can be given by,

$\begin{split}T_{\alpha\beta}(t)=\frac{1}{V}\int
d^{3}rT_{\alpha\beta}(\vec{r},t),\end{split}$ (2.10)

where $\displaystyle T_{\alpha\beta}(\vec{r},t)$ is local tensor which is
regarded as the momentum flux density tensor. Let us start from the momentum
conservation law [116, 117, 17]:

$\begin{split}\frac{\partial\textbf{p}(\vec{r},t)}{\partial
t}+\nabla_{r}\cdot\textbf{T}(\vec{r},t)=0,\end{split}$ (2.11)

where $\displaystyle\textbf{p}(\vec{r},t)$ and
$\displaystyle\textbf{T}(\vec{r},t)$ are momentum density and momentum flux
density, respectively. One can rewrite Eq. (2.11) as tensor form:

$\begin{split}\frac{\partial p_{\alpha}}{\partial t}+\frac{\partial}{\partial
r_{\beta}}T_{\alpha\beta}=0.\end{split}$ (2.12)

With a distribution function $\displaystyle f(\vec{r},\vec{p},t)$, the local
momentum density can be defined as [118]:

$\begin{split}\textbf{p}(\vec{r},t)=\int\vec{p}f(\vec{r},\vec{p},t)d^{3}p.\end{split}$
(2.13)

In statistical mechanics, a system is no longer considered as a continuous
fluid, but rather treated an ensemble consisted of particles with the number
$\displaystyle A$. In a many-particle system, the distribution function
$\displaystyle f(\vec{r},\vec{p},t)$ could have Dirac’s form [85]:

$\begin{split}f(\vec{r},\vec{p},t)=\sum_{i}^{A}\delta(\vec{r}-\vec{R}_{i}(t))\delta(\vec{p}-\vec{P}_{i}(t)),\end{split}$
(2.14)

where $\displaystyle\vec{R}_{i}(t)$ and $\displaystyle\vec{P}_{i}(t)$ are the
time-dependent coordinate and momentum of particle-$\displaystyle i$, or Gauss
form [86]:

$\begin{split}f(\vec{r},\vec{p},t)=\frac{1}{(\pi\hbar)^{3}}\sum_{i}^{A}{\rm
exp}\Big{[}-\frac{(\vec{r}-\vec{R}_{i}(t))^{2}}{2L}\Big{]}{\rm
exp}\Big{[}-\frac{2L(\vec{p}-\vec{P}_{i}(t))^{2}}{\hbar^{2}}\Big{]},\end{split}$
(2.15)

where $\displaystyle L$ is square of width of Gauss wave packet in unit of
$\displaystyle\rm fm^{2}$. With distribution function $\displaystyle
f(\vec{r},\vec{p},t)$, one can have following integral relations:

$\begin{split}\int f(\vec{r},\vec{p},t)d^{3}p=\rho(\vec{r})\,,\end{split}$
(2.16) $\begin{split}\int f(\vec{r},\vec{p},t)d^{3}r=g(\vec{p})\,.\end{split}$
(2.17)

where $\displaystyle\rho(\vec{r})$ and $\displaystyle g(\vec{p})$ are the
densities in coordinate and momentum spaces, respectively. Substituted
$\displaystyle f(\vec{r},\vec{p},t)$ into Eq.(2.13), one can get:

$\begin{split}\textbf{p}(\vec{r},t)=\sum_{i}^{A}\vec{P}_{i}\rho_{i},\end{split}$
(2.18)

where the $\displaystyle\rho_{i}$ is density of particle $\displaystyle i$
which can be expressed as:

$\begin{split}\rho_{i}(\vec{r})=\delta(\vec{r}-\vec{R}_{i}),\end{split}$
(2.19)

or

$\begin{split}\rho_{i}(\vec{r})=\frac{1}{(2\pi
L)^{3/2}}\exp\Big{[}-\frac{(\vec{r}-\vec{R}_{i})^{2}}{2L}\Big{]}\,.\end{split}$
(2.20)

Taking derivative of Eq. (2.18) with respect to time:

$\begin{split}\frac{\partial\textbf{p}}{\partial
t}=\sum_{i}^{A}\frac{\partial\vec{P}_{i}}{\partial
t}\rho_{i}+\sum_{i}^{A}\vec{P}_{i}\frac{\partial\rho_{i}}{\partial
t}.\end{split}$ (2.21)

For the derivative of $\displaystyle\rho_{i}$ with respect to time:

$\begin{split}\frac{\partial\rho_{i}}{\partial
t}=\frac{\partial\rho_{i}}{\partial\vec{R}_{i}}\frac{\partial\vec{R}_{i}}{\partial
t}=-\frac{\partial\rho_{i}}{\partial\vec{r}}\frac{\partial\vec{R}_{i}}{\partial
t}\,.\end{split}$ (2.22)

Thus the second term of right hand side of Eq. (2.21) can reach:

$\begin{split}\sum_{i}^{A}\frac{\partial\rho_{i}}{\partial
t}P_{i\alpha}=-\frac{\partial}{\partial
r_{\beta}}\sum_{i}^{A}\Big{(}\frac{P_{i\alpha}P_{i\beta}}{m}\Big{)}\rho_{i}\,.\end{split}$
(2.23)

And for the first term of right hand side of Eq. (2.21):

$\begin{split}\sum_{i}^{A}\frac{\partial\vec{P}_{i}}{\partial
t}\rho_{i}=\sum_{i}^{A}\vec{F}_{i}\rho_{i}\,.\end{split}$ (2.24)

If the interaction is only two-body, we have

$\begin{split}\vec{F}_{i}=\sum_{i\neq j}\vec{F}_{ij}\,.\end{split}$ (2.25)

Thus,

$\begin{split}\sum_{i}^{A}\vec{F}_{i}\rho_{i}=\frac{1}{2}\Big{(}\sum_{i}^{A}\vec{F}_{i}\rho_{i}+\sum_{j}^{A}\vec{F}_{j}\rho_{j}\Big{)}=\frac{1}{2}\Big{(}\sum_{i}\sum_{i\neq
j}\vec{F}_{ij}\rho_{i}+\sum_{j}\sum_{j\neq
i}\vec{F}_{ji}\rho_{j}\Big{)}\,,\end{split}$ (2.26)

where $\displaystyle\vec{F}_{ij}$ is force exerted on particle-$\displaystyle
i$ by particle-$\displaystyle j$. And $\displaystyle\vec{F}_{ij}$ =
$\displaystyle-\vec{F}_{ji}$, then

$\begin{split}\sum_{i}^{A}\vec{F}_{i}\rho_{i}=\frac{1}{2}\Big{[}\sum_{i}\sum_{i\neq
j}\vec{F}_{ij}\Big{(}\rho_{i}-\rho_{j}\Big{)}\Big{]}\,.\end{split}$ (2.27)

With Taylor series (due to $\displaystyle R_{ij\beta}=R_{i\beta}-R_{j\beta}$
which is relative position of particle $\displaystyle i$ and $\displaystyle j$
is finite):

$\begin{split}\rho_{i}(r_{\beta}-R_{i\beta})&=\rho_{i}[r_{\beta}-R_{j\beta}+(R_{j\beta}-R_{i\beta})]\\\
&=\rho_{i}[r_{\beta}-R_{j\beta}+R_{ji\beta}]\\\
&=\sum_{n=0}^{N}\frac{(R_{ij\beta})^{n}}{n!}\rho_{j}^{(n)}(r_{\beta}-R_{j\beta})\\\
&\approx\rho_{j}(r_{\beta}-R_{j\beta})+R_{ji\beta}\frac{\partial}{\partial
r_{\beta}}\rho_{j}(r_{\beta}-R_{j\beta})\,.\end{split}$ (2.28)

Then $\displaystyle R_{ij\beta}=-R_{ji\beta}$,

$\begin{split}\rho_{i}(r_{\beta}-R_{i\beta})-\rho_{j}(r_{\beta}-R_{j\beta})=-R_{ij\beta}\frac{\partial}{\partial
r_{\beta}}\rho_{j}(r_{\beta}-R_{j\beta}).\end{split}$ (2.29)

According to Eq. (2.24), Eq. (2.27), and Eq. (2.29),

$\begin{split}\sum_{i}^{A}\frac{\partial\vec{P}_{i}}{\partial
t}\rho_{i}=-\frac{1}{2}\sum_{i}\sum_{i\neq
j}F_{ij\alpha}R_{ij\beta}\frac{\partial}{\partial
r_{\beta}}\rho_{j}(r_{\beta}-R_{j\beta})=\frac{\partial}{\partial
r_{\beta}}\Big{(}-\frac{1}{2}\sum_{i}\sum_{i\neq
j}F_{ij\alpha}R_{ij\beta}\rho_{j}\Big{)}\,.\end{split}$ (2.30)

If interaction includes three-body component, in the same way, one can get

$\begin{split}\frac{1}{2}\sum_{i}\sum_{i\neq j}\sum_{i\neq j\neq
k}F_{ijk\alpha}\rho_{i}=\frac{1}{6}\sum_{i}\sum_{i\neq j}\sum_{i\neq j\neq
k}(F_{ijk\alpha}\rho_{i}+F_{jki\alpha}\rho_{j}+F_{kij\alpha}\rho_{k})\,.\end{split}$
(2.31)

With the condition,

$\begin{split}F_{ijk\alpha}+F_{jki\alpha}+F_{kij\alpha}=0\,.\end{split}$
(2.32)

Then we obtain,

$\begin{split}\frac{1}{2}\sum_{i}\sum_{i\neq j}\sum_{i\neq j\neq
k}F_{ijk\alpha}\rho_{i}\approx\frac{\partial}{\partial
r_{\beta}}\Big{[}-\frac{1}{6}\sum_{i}\sum_{i\neq j}\sum_{i\neq j\neq
k}(F_{ijk\alpha}R_{ik\beta}+F_{jki\alpha}R_{jk\beta})\rho_{k}\Big{]}\,.\end{split}$
(2.33)

Finally, according to Eqs.(2.12), (2.23), (2.24), (2.31), and (2.33), the
momentum flux density tensor can be given,

$\begin{split}&T_{\alpha\beta}(\vec{r},t)=\sum_{i}^{A}\frac{P_{{i}\alpha}P_{{i}\beta}}{m_{i}}\rho_{i}(\vec{r},t)\\\
&+\frac{1}{2}\sum_{i}^{A}\sum_{i\neq
j}^{A}F_{ij\alpha}R_{ij\beta}\rho_{j}(\vec{r},t)\\\
&+\frac{1}{6}\sum_{i}^{A}\sum_{i\neq j}^{A}\sum_{i\neq j\neq
k}^{A}(F_{ijk\alpha}R_{ik\beta}+F_{jki\alpha}R_{jk\beta})\rho_{k}(\vec{r},t)\\\
&+\cdots,\end{split}$ (2.34)

The first term on the right-hand side is the momentum term, the second is the
two-body interaction term, and the third is the three-body interaction term.

The $\displaystyle\eta$ in Eq. (2.9) is a normal (or traditional) form of the
Green-Kubo formula. In the last few decades, it has been extensively used in
molecular dynamics simulations. In Ref. [115], a more general form of the
Green-Kubo formula is given for the calculation of the shear viscosity for
both Fermi-Dirac and Boltzmann distribution systems, which is read as,

$\begin{split}\eta_{new}=\frac{mVA}{\langle\frac{1}{3}\sum_{i}^{A}p_{i}^{2}\rangle}\int_{t_{0}}^{\infty}\langle
T_{\alpha\beta}(t)T_{\alpha\beta}(t_{0})\rangle dt,\end{split}$ (2.35)

where $\displaystyle m$ is the particle mass. The derivation can be found in
the Ref. [115]. It is obvious, of course, in the classic limit, one has,

$\displaystyle\displaystyle\langle\frac{1}{3}\sum_{i}^{A}p_{i}^{2}\rangle=Amk_{B}T,$
(2.36)

where $\displaystyle k_{B}$ is the Boltzmann constant (here $\displaystyle
k_{B}$ set to 1). Substituting Eq. (2.36) into Eq. (2.35), we can get Eq.
(2.9). To calculate shear viscosity directly from the Green-Kubo relation, one
simply performs an equilibrium simulation of an $\displaystyle A$-particle
system.

### 2.3 Shear strain rate method

Figure 2.3: (Color online) Stress tensor as a function of time at different
temperatures and density of 0.4$\displaystyle\rho_{0}$
($\displaystyle\rho_{0}$ = 0.16 fm3 called saturation density of nuclear
matter) without the mean field [115].

Instead of measuring the intrinsic fluctuations of the system, the viscosity
can also be obtained using the non-equilibrium molecular dynamics (NEMD)
calculation by imposing a Couette flow. This method was proposed by Evans and
Morriss [21] and named the SLLOD algorithm, which has been extensively applied
to predict the rheological properties of real fluids [21]. The SLLOD algorithm
is related to the dynamics with strain rate $\displaystyle\dot{\gamma}$
associated with the Lees-Edwards periodic boundary conditions [119] and it is
for shear viscosity with a planar Couette flow field at shear rate
$\displaystyle\dot{\gamma}$ = $\displaystyle\partial v_{x}/\partial y$ which
is the change in streaming velocity $\displaystyle v_{x}$ in the
$\displaystyle x$-direction with vertical position $\displaystyle y$. We can
obtain shear viscosity by

$\begin{split}\eta=-\frac{\langle T_{xy}\rangle}{\dot{\gamma}}.\end{split}$
(2.37)

In Ref. [115], the strain rate was implanted in the framework of an improved
quantum molecular dynamic (ImQMD) model without mean field, which is the
cascade mode. As in Fig. 2.3 [115], it shows the stress tensor as a function
of time at different temperatures, which increases when the shear rate is
$\displaystyle\dot{\gamma}$ at 400 fm/c and reaches a constant value after a
time equals to the calculated shear viscosity.

Figure 2.4: (Color online) Deduced shear viscosity as a function of the shear
rate $\displaystyle\dot{\gamma}$ applied within the SLLOD algorithm, in the
system at the density of $\displaystyle 0.2\,\rho_{0}$ and temperature
$\displaystyle T=10\,\text{MeV}$, when no mean field is employed.

The effect of the shear rate on the shear viscosity has been checked. As in
Fig. 2.4, one can only notice in some ranges the shear viscosity keeps
constant. So in the simulations, the shear rate cannot be too large or small.
If it is small, flow fields cannot be implanted in the system. On the other
hand, when shear rate is too large, energy can not be kept conservative when
the dynamical equations of motion are solved with a finite time step.

By adding the shear rate, the dynamical equations of motion for the system can
be rewritten as

$\displaystyle\displaystyle\frac{d\vec{r}_{i}}{dt}=\frac{\vec{p}_{i}}{m_{i}}+\dot{\gamma}y_{i}\hat{x}$
(2.38)
$\displaystyle\displaystyle\frac{d\vec{p}_{i}}{dt}=\vec{F}_{i}-\dot{\gamma}p_{yi}\hat{x}-a\vec{p}_{i}.$
(2.39)

Eq. (2.38) and Eq. (2.39) are called the SLLOD equations. In order to keep the
kinetic energy conservation, one needs a ‘thermostat’. Thus a multiplier is
applied to motion equations. Considering conservation of kinetic energy, one
can get

$\begin{split}a=\frac{\sum_{i}(\vec{F}_{i}\cdot\vec{p}_{i}/m_{i}-\dot{\gamma}p_{xi}p_{yi}/m_{i})}{\sum_{i}p_{i}^{2}/m_{i}}.\end{split}$
(2.40)

### 2.4 Chapman-Enskog method

Chapman and Enskog (CE) have extracted the shear viscosity $\displaystyle\eta$
in a gas of nonrelativistic particles by using the Boltzmann kinetic equation
for the phase-space distribution function [120]. One should notice that
originally the CE method was for Maxwell-Boltzmann statistics, but here it is
for the Fermi system with the same strategy [75]. In the CE method, one can
solve the Boltzmann equation by applying a series expansion to the
distribution function [121]. In Ref. [74], one considered Boltzmann-Uehling-
Uhlenbeck equation without potential $\displaystyle U$ [85],

$\displaystyle\displaystyle\frac{\partial}{\partial
t}{f_{1}(\vec{r},\vec{p},t)}+\frac{\vec{p}}{m}\cdot\nabla_{\vec{r}}{f_{1}(\vec{r},\vec{p},t)}=I_{coll.}$
(2.41)

The right-hand side (rhs) of Eq.(2.41) is the collision term ($\displaystyle
I_{coll.}$), which denotes collision process of
1+2$\displaystyle\rightarrow$3+4, and it can be read as [87]:

$\displaystyle\displaystyle I_{coll.}=\big{(}\frac{\partial{f_{1}}}{\partial
t}\big{)}_{coll.}=\frac{\rm g}{(2\pi\hbar)^{3}}\int
d^{3}p_{2}d\Omega\frac{d\sigma_{NN}}{d\Omega}|\vec{\upsilon}_{1}-\vec{\upsilon}_{2}|{\times}\mathcal{P}{\times}\delta^{3}(\vec{p}_{1}+\vec{p}_{2}\\!-\\!\vec{p}_{3}\\!-\\!\vec{p}_{4}).$
(2.42)

where $\displaystyle\rm g=4$ is the spin and isospin degeneracy factor, and
$\displaystyle\sigma_{NN}$ is the nucleon-nucleon cross section. And the
$\displaystyle\delta^{3}(\vec{p}_{1}+\vec{p}_{2}\\!-\\!\vec{p}_{3}\\!-\\!\vec{p}_{4})$
term is for keeping momentum conservation in the collisions. In Eq.(2.42),
$\displaystyle\mathcal{P}$ denotes the Pauli blocking factor, which reads,

$\displaystyle\displaystyle\mathcal{P}=f_{3}f_{4}(1-f_{1})(1\\!-\\!f_{2})\\!-\\!f_{1}f_{2}(1\\!-\\!f_{3})(1\\!-\\!f_{4})$
(2.43)

where the first term of the right-hand side is the gain term
(3+4$\displaystyle\rightarrow$1+2) and the second term of the right-hand side
is the loss term (1+2$\displaystyle\rightarrow$3+4).

Allowing the system to be slightly out of equilibrium, the first order of
$\displaystyle f$ can be written as [74, 122, 123],

$\displaystyle\displaystyle f=f_{eq}+\delta f=f^{0}+f^{0}(1-f^{0})\Phi\,,$
(2.44)

where $\displaystyle f_{eq}=f^{0}$. Substituting Eq.(2.44) into the collision
integral Eq.(2.42) and neglecting terms of quadratic and higher order in
$\displaystyle\Phi$ [74, 123], one can obtain

$\displaystyle\displaystyle\frac{\partial}{\partial
t}{f_{1}^{0}}+\frac{\vec{p}}{m}\cdot\nabla_{\vec{r}}{f_{1}^{0}}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\frac{\rm
g}{(2\pi\hbar)^{3}}\int
d^{3}p_{2}d\Omega\frac{d\sigma_{NN}}{d\Omega}|\vec{\upsilon}_{1}-\vec{\upsilon}_{2}|{\times}f^{0}_{1}f^{0}_{2}(1-f_{3}^{0})(1-f^{0}_{4})$
(2.45) $\displaystyle\displaystyle{\times}$
$\displaystyle\displaystyle(\Phi_{3}+\Phi_{4}-\Phi_{1}-\Phi_{2}){\times}\delta^{3}(\vec{p}_{1}+\vec{p}_{2}\\!-\\!\vec{p}_{3}\\!-\\!\vec{p}_{4}).$

In the first-order approximation, the contribution of $\displaystyle\delta f$
on the left side can be neglected, and $\displaystyle f_{1}$ is replaced by
$\displaystyle f_{1}^{0}$. Considering the existence of a motion with a
slightly inhomogeneous velocity $\displaystyle\vec{u}$ in the liquid [116], we
can write the distribution function $\displaystyle f^{0}$ as,

$\displaystyle\displaystyle
f^{0}=\frac{1}{e^{(E-\vec{p}\cdot\vec{u}-\mu)/T}+1},$ (2.46)

where $\displaystyle\mu$ is chemical potential. By substituting (2.46) to the
left-hand side of (2.41), expanding in terms of gradients of temperature and
fluid velocity and assuming
$\displaystyle\bigtriangledown$$\displaystyle\cdot$$\displaystyle\vec{u}=0$
for an incompressible fluid that drives bulk viscosity and is neglected here
[116, 74, 124], we have

$\displaystyle\displaystyle\frac{\partial f^{0}}{\partial
t}+\frac{\vec{p}}{m}\cdot\nabla_{\vec{r}}{f^{0}}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{2mT}f^{0}f^{0}\Big{[}(p_{\alpha}p_{\beta}-\frac{1}{3}\delta_{\alpha\beta}p^{2})(\frac{\partial
u_{\alpha}}{\partial r_{\beta}}+\frac{\partial u_{\beta}}{\partial
r_{\alpha}}-\frac{2}{3}\delta_{\alpha\beta}\bigtriangledown\cdot\textbf{u})+\frac{1}{mT}(p^{2}-\frac{5}{3}\langle{p^{2}}\rangle)p_{\alpha}\frac{\partial{T}}{\partial{r_{\alpha}}}\Big{]}$
(2.47) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{2mT}f^{0}f^{0}\Big{[}(p_{\alpha}p_{\beta}-\frac{1}{3}\delta_{\alpha\beta}p^{2})(\frac{\partial
u_{\alpha}}{\partial r_{\beta}}+\frac{\partial u_{\beta}}{\partial
r_{\alpha}})+\frac{1}{mT}(p^{2}-\frac{5}{3}\langle{p^{2}}\rangle)p_{\alpha}\frac{\partial{T}}{\partial{r_{\alpha}}}\Big{]}.$

where $\displaystyle\langle{p^{2}}\rangle$ is an average over distribution. In
Eq.(2.47), we only pick up the terms which are relevant to shear viscosity and
heat conduction coefficient. For symmetry consideration between the right-hand
side of Eq.(2.45) and the right-hand side of Eq.(2.47), it can be found that
$\displaystyle\Phi$ has the form

$\displaystyle\displaystyle\Phi=q^{(1)}(p_{\alpha}p_{\beta}-\frac{1}{3}\delta_{\alpha\beta}p^{2})(\frac{\partial
u_{\alpha}}{\partial r_{\beta}}+\frac{\partial u_{\beta}}{\partial
r_{\alpha}})+q^{(2)}(p^{2}-\frac{5}{3}\langle{p^{2}}\rangle)p_{\alpha}\frac{\partial{T}}{\partial{r_{\alpha}}},$
(2.48)

where the $\displaystyle q^{(1)}$ and $\displaystyle q^{(2)}$ are constant
factors [116, 74]. And by multiplying both sides of Eq.(2.45) by
$\displaystyle\Phi_{1}$ and integrating over the momenta $\displaystyle
d^{3}p_{1}$, one can obtain the constants.

Figure 2.5: (Color online) Shear viscosity as a function of temperature at
different nuclear matter densities.

Shear viscosity ($\displaystyle\eta$) as the coefficient proportional to the
shear momentum flux tensor (one also can see from Eq. 2.6) without bulk
viscosity,

$\displaystyle\displaystyle\Pi_{\alpha\beta}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\eta(\frac{\partial u_{\alpha}}{\partial
r_{\beta}}+\frac{\partial u_{\beta}}{\partial
r_{\alpha}}-\frac{2}{3}\delta_{\alpha\beta}\bigtriangledown\cdot\vec{u})$
(2.49) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\eta(\frac{\partial u_{\alpha}}{\partial
r_{\beta}}+\frac{\partial u_{\beta}}{\partial r_{\alpha}})$

where the
$\displaystyle\bigtriangledown$$\displaystyle\cdot$$\displaystyle\vec{u}$ is
assumed to be zero. A slight out of equilibrium produces a change in the
dissipative stress tensor; at the lowest order in $\displaystyle\delta f$, we
can obtain,

$\displaystyle\displaystyle\Pi_{\alpha\beta}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{\rm g}{(2\pi\hbar)^{3}}\int
d^{3}p\frac{p_{\alpha}p_{\beta}}{m}f=\frac{\rm g}{(2\pi\hbar)^{3}}\int
d^{3}p\frac{p_{\alpha}p_{\beta}}{m}f_{0}+\frac{\rm g}{(2\pi\hbar)^{3}}\int
d^{3}p\frac{p_{\alpha}p_{\beta}}{m}\delta f$ (2.50)
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\frac{\rm
g}{(2\pi\hbar)^{3}}\int d^{3}p\frac{p_{\alpha}p_{\beta}}{m}\delta f$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\frac{\rm
g}{(2\pi\hbar)^{3}}\int
d^{3}p\frac{p_{\alpha}p_{\beta}}{m}f^{0}(1-f^{0})\Phi.$

The equilibrium term with $\displaystyle f_{0}$ in
$\displaystyle\Pi_{\alpha\beta}$ is zero. Substituting (2.48) into (2.50), one
can get the expression for shear viscosity [74, 125]:

$\begin{split}\eta&=\frac{5T}{9}\frac{\Big{(}\int
d^{3}p_{1}{\,\,}p^{2}_{1}f_{1}\Big{)}^{2}}{\displaystyle\int
d^{3}p_{1}d^{3}p_{2}d\Omega{\,}v_{12}q_{12}^{4}sin^{2}{\theta}\frac{d\sigma_{NN}}{d\Omega}f_{1}f_{2}\tilde{f}_{3}\tilde{f}_{4}},\end{split}$
(2.51)

where $\displaystyle v_{12}=|\vec{v}_{1}-\vec{v}_{2}|$, $\displaystyle
q_{12}=|\vec{p}_{1}-\vec{p}_{2}|/2$ and $\displaystyle\theta$ are relative
velocity, relative momentum, and scattering angle between the total momentum
$\displaystyle\vec{p}_{t}=\vec{p}_{1}+\vec{p}_{2}=\vec{p}_{3}+\vec{p}_{4}$ and
the relative momentum of the final state
$\displaystyle\vec{p}_{r}=\vec{p}_{3}-\vec{p}_{4}$, respectively.
$\displaystyle\sigma_{NN}$ is the total nucleon-nucleon cross section, and
$\displaystyle\tilde{f}_{3}$$\displaystyle\tilde{f}_{4}$ = [1-$\displaystyle
f(p_{3})$][1-$\displaystyle f(p_{4})$] is the Pauli blocking term. In
Eq.(2.51), the conservations of momentum and energy between $\displaystyle
p_{1},p_{2}$ and $\displaystyle p_{3},p_{4}$ are needed to be taken into
account in the collisions.

By solving Eq.(2.51) with the Monte-Carlo approach, one obtains the shear
viscosity as shown in Fig.2.5. Here the
$\displaystyle\frac{d\sigma_{NN}}{d\Omega}$ is set to 40
mb/4$\displaystyle\pi$. It is seen that the shear viscosity increases as
temperature increases for very low density (0.01$\displaystyle\rho_{0}$) which
can be treated as Fermion gas. For the Fermion gas, the Pauli blocking effect
is weak, and so the behavior for shear viscosity acts like that of classic
gas. For higher densities, however, shear viscosity decreases with increasing
of temperature in low-temperature region since Pauli blocking effect is
strong. As temperature increases again, particle velocity becomes dominative
leading to the increasing of shear viscosity.

### 2.5 Relaxation time method

In the relaxation time approximation method, one assumes that the effect of
collisions is always to bring the perturbed distribution function close to the
equilibrium distribution function [126, 127], $\displaystyle f=f_{eq}+\delta
f$ which is as Eq.(2.44). Then one writes the collision term exponentially
with a relaxation time of $\displaystyle\tau$,

$\displaystyle\displaystyle\mathcal{D}f=-\frac{f-f_{eq}}{\tau}=-\frac{\delta
f}{\tau}$ (2.52)

where $\displaystyle\mathcal{D}=\frac{\partial}{\partial
t}+\vec{v}\cdot\nabla_{\vec{r}}-\nabla_{\vec{r}}U\cdot\nabla_{\vec{p}}$. In
the relaxation time approximation, one starts from shear force between flow
layers per unit area and obtains the shear viscosity [82]

$\displaystyle\displaystyle\eta=-\sum_{\tau}\frac{\rm
g}{(2\pi)^{3}}\int{\tau}_{\tau}(p)\frac{p_{z}^{2}p_{x}^{2}}{pm^{\star}_{\tau}}\frac{df_{\tau}^{0}}{dp}d^{3}p\,,$
(2.53)

where the subscript $\displaystyle\tau$ is the isospin factor, which is set to
1/2 ($\displaystyle-$1/2) for neutrons (protons). So here the degeneracy
factor $\displaystyle\rm g$=2. And $\displaystyle m^{\star}_{\tau}$ is the
effective mass, which reads as,

$\displaystyle\displaystyle\frac{1}{m^{\star}_{\tau}}=\frac{1}{m}+\frac{1}{p}\frac{dU_{\tau}}{dp}\,,$
(2.54)

where $\displaystyle U_{\tau}$ is the single-particle potential for a nucleon
with momentum $\displaystyle\vec{p}$ and isospin $\displaystyle\tau$ in a
nuclear matter medium, which can be found in Ref. [82]. In Eq. (2.53),
$\displaystyle f_{\tau}$ is the local momentum distribution, which is written
as,

$\displaystyle\displaystyle f_{\tau}=\frac{1}{{\rm
exp}{\big{[}\frac{p^{2}}{2m}-U_{\tau}(\vec{p})-\mu_{\tau}\big{]}\big{/}T}+1}\,.$
(2.55)

And in Eq. (2.53), $\displaystyle\tau_{\tau}(p)$ is the relaxation time which
can be expressed as,

$\displaystyle\displaystyle\frac{1}{\tau_{\tau}(p_{1})}=\frac{1}{\tau_{\tau}^{\rm
same}(p_{1})}+\frac{1}{\tau_{\tau}^{\rm diff}(p_{1})}$ (2.56)

where $\displaystyle\tau_{\tau}^{\rm same(diff)}(p_{1})$ is the average
collision time for a nucleon with isospin $\displaystyle\tau$ and momentum
$\displaystyle p_{1}$ when colliding with other nucleons of same (different)
isospin, and they can be calculated from [82],

$\displaystyle\displaystyle\frac{1}{\tau_{\tau}^{\rm same}(p_{1})}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\big{(}{\rm
g}-\frac{1}{2}\big{)}\frac{1}{2\pi}\int p_{2}^{2}dp_{2}\,d{\rm
cos}\,\theta_{12}\,d\,{\rm
cos}\theta\frac{d\sigma_{\tau,\tau}}{d\Omega}\Big{|}\frac{\vec{p}_{1}}{m_{\tau}^{\star}}-\frac{{\vec{p}}_{2}}{m_{\tau}^{\star}}\Big{|}$
(2.57) $\displaystyle\displaystyle\times$
$\displaystyle\displaystyle\big{[}f^{0}_{\tau}(p_{2})-f^{0}_{\tau}(p_{2})f^{0}_{\tau}(p_{3})-f^{0}_{\tau}(p_{2})f^{0}_{\tau}(p_{4})+f^{0}_{\tau}(p_{3})f^{0}_{\tau}(p_{4})\big{]}\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle\displaystyle\frac{1}{\tau_{\tau}^{\rm diff}(p_{1})}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\frac{{\rm
g}}{2\pi}\int p_{2}^{2}dp_{2}\,d{\rm cos}\,\theta_{12}\,d\,{\rm
cos}\theta\frac{d\sigma_{\tau,\tau}}{d\Omega}\Big{|}\frac{\vec{p}_{1}}{m_{\tau}^{\star}}-\frac{{\vec{p}}_{2}}{m_{-\tau}^{\star}}\Big{|}$
(2.58) $\displaystyle\displaystyle\times$
$\displaystyle\displaystyle\big{[}f^{0}_{-\tau}(p_{2})-f^{0}_{-\tau}(p_{2})f^{0}_{\tau}(p_{3})-f^{0}_{-\tau}(p_{2})f^{0}_{-\tau}(p_{4})+f^{0}_{\tau}(p_{3})f^{0}_{-\tau}(p_{4})\big{]}\,,$

where $\displaystyle\theta_{12}$ is the angel between
$\displaystyle\vec{p}_{1}$ and $\displaystyle\vec{p}_{2}$.

In this chapter, the definition of shear viscosity is given from two planes,
and some methods to estimate shear viscosity are given. There are more methods
for calculating shear viscosity as in Refs. [128, 127, 129].

## 3 Theoretical aspects of shear viscosity in nucleonic matter

### 3.1 Infinite nuclear matter

Figure 3.1: (Color online) Different calculations of shear viscosity as a
function of temperature in cases of w/ or w/o Pauli blocking (PB) at 0.4
$\displaystyle\rho_{0}$ and 1.0 $\displaystyle\rho_{0}$ and the nucleon-
nucleon cross section is fixed at 40 mb [115].

In the last section, some approaches to extracting the shear viscosity of
nuclear matter or a Fermi system are introduced. In the previous work [115],
comparisons have been done among methods of Chapman-Enskog (for the Fermi
system), shear strain rate, and Greek-Kubo in a periodic box by using an
improved quantum molecular dynamic (ImQMD) model without mean field as
displayed in Fig. 3.1. It should be mentioned that ‘Boltzman’ denotes the
Chapman-Enskog method as the Eq. (2.51). The GK-nor and GK-new are the normal
Green-Kubo formula as Eq. (2.9) and new form of the Green-Kubo formula as Eq.
(2.35), respectively. As shown in Fig. 3.1, the Pauli blocking effect is
studied in the calculation. It is worth to mention that for a Fermi system
Pauli blocking should be taken into account. It can be seen that in Fig.
3.1(a) and Fig. 3.1(c) without the Pauli blocking, shear viscosity increases
with temperature at both densities of $\displaystyle 0.4\rho_{0}$ and
$\displaystyle 1.0\rho_{0}$. Seeing from Fig. 3.1 (b) and Fig. 3.1(d) with the
Pauli blocking, unlike Fig. 3.1(a) and Fig. 3.1(c) in the low-temperature
region, shear viscosity increases with the decreasing of temperature. It is
due to a stronger Pauli blocking effect in the low temperature region. At
lower temperature and with stronger Pauli blocking effect, it indicates that a
good Fermi sphere is formed and thus the elastic scattering rate is greatly
suppressed due to the Pauli blocking. Hence, the system in lower temperature
region has larger shear viscosity. However, an uncertainty in extracting the
value of the shear viscosity at T = 0 is given here for the small or finite
values of shear viscosity being given in the Refs. [130, 131, 132, 133, 134],
such an uncertainty is explained by different dissipation mechanisms [134]. As
temperature increases, the Pauli blocking effect becomes weaker and collision
increases which makes the momentum transfer easier and shear viscosity
decreasing. As the temperature increases again, even though the collision rate
increases, velocity becomes dominant. Thus, shear viscosity increases with
temperature. It is concluded that the naive kinetic approach is not valid for
the description of the unitary Fermi gas at low temperature, where the system
becomes strongly correlated [135]. For the dilute gas case as black dot-line
in Fig. 2.5, it increases with temperature since it is dominated by the term
of temperature (or particle velocity).

Figure 3.2: (Color online) Shear viscosity as a function of temperature
($\displaystyle T_{0}$) at densities 0.4$\displaystyle\rho_{0}$ (a) and
1.0$\displaystyle\rho_{0}$ (b) for different cross sections with Green-Kubo
formula as Eq. (2.35) in ImQMD model.

Besides, one can find that, among these three approaches, shear viscosity
determined by the Boltzmann type equation of Eq. (2.51) is lower than that
calculated by the shear rate and the Green-Kubo method. At very low densities
such as 0.05$\displaystyle\rho_{0}$, however, we found that the result by the
Boltzmann type equation (Eq. 2.51) is consistent with other methods, which
indicates that a low density approximation for Eq. 2.51 could be valid. Then
one would expect that shear viscosities determined by the shear rate and the
Green-Kubo approaches are the same. In Ref. [115], we re-derived and gave a
new form of the Green-Kubo formula for the calculation of shear viscosity as
Eq. (2.35). In the Boltzmann statistical analysis without Pauli blocking, we
found that shear viscosities from the shear rate method, the normal form of
the Green-Kubo formula, and new form of the Green-Kubo formula are close. As
the Pauli blocking is taken into account as shown in Fig. 3.1 (b) and Fig. 3.1
(d), however, one sees that in the low temperature region, the shear viscosity
from the normal form of the Green-Kubo formula (2.9) is higher than those from
both the shear strain rate method and the new form of the Green-Kubo formula.
However, shear viscosities from the shear rate method and the new form of the
Green-Kubo formula are almost the same. At high temperatures, shear
viscosities from the shear rate method, the normal form of the Green-Kubo
formula, and the new form of the Green-Kubo formula are to be the same since
particle collisions become dominant and Pauli blocking can be ignored as
temperature increases. So one should be careful when applying the normal form
of the Green-Kubo formula to extract shear viscosity of Fermi system at low-
temperature region.

From both the Chapman-Enskog method and relaxation time approximation method,
we can see that the collision has an important role for the shear viscosity.
For the results in Fig. 3.1, the nucleon-nucleon cross section is fixed at 40
mb. When one increases the cross section, the shear viscosity would reduce, as
shown in Fig. 3.2. It is obvious that as the cross-section increases, momentum
transfer becomes quite easier, which would lead to lower shear viscosity. It
can be seen that the shear viscosity is sensitive to the nucleon-nucleon cross
section. Thus, one adjusts the in-medium nucleon-nucleon (NN) cross section in
a transport model which is named pBUU [136, 137, 138, 139] to reproduce the
nuclear stopping data and use the fitted cross sections to calculate the shear
viscosity self-consistently [125]. The NN cross-section should also be
affected by the surrounding medium. The authors discussed three kinds of media
cross-sections with nuclear stopping. One is

$\displaystyle\displaystyle\sigma_{NN}^{\rm med}=\sigma_{0}{\rm
tanh}\Big{(}\frac{\sigma_{NN}^{\rm free}}{\sigma_{0}}\Big{)},$ (3.1)

where $\displaystyle\sigma_{0}=\nu\rho^{-2/3}$, $\displaystyle\nu$ and
$\displaystyle\rho$ are a parameter in order of 1 and medium of number
density, respectively. And $\displaystyle\sigma_{NN}^{\rm free}$ is free NN
cross section. A in-medium cross section carried out at the University of
Rostock [140] within a thermodynamic T-matrix approach is given by a
parametrization of the cross-section reduction [125],

$\displaystyle\displaystyle\sigma_{NN}^{\rm med}=\sigma_{NN}^{\rm free}{\rm
exp}\Big{(}-0.6\frac{\rho/\rho_{0}}{1+[T_{c.m.}/150\,{\rm MeV}]^{2}}\Big{)},$
(3.2)

where $\displaystyle T_{c.m.}$ is the total kinetic energy of the two
interacting particles in the frame where the local medium is at rest. The last
one is derived by Fuchs et al. [141] which is parameterized with,

$\displaystyle\displaystyle\sigma_{nn}^{\rm med}=\sigma_{nn}^{\rm free}{\rm
exp}\Big{(}-1.7\frac{\rho/\rho_{0}}{1+[T_{c.m.}/12\,{\rm MeV}]^{3/2}}\Big{)},$
(3.3) $\displaystyle\displaystyle\sigma_{np}^{\rm med}=\sigma_{np}^{\rm
free}{\rm exp}\Big{(}-1.4\frac{\rho/\rho_{0}}{1+[T_{c.m.}/33\,{\rm
MeV}]}\Big{)}.\quad\,\,\,$ (3.4)

With the help of the different cross-section reductions above, the authors can
compare predictions from pBUU to stopping data, as shown in Fig. 3.3. The
stopping observable $\displaystyle varxz$ is defined as [142],

$\displaystyle\displaystyle varxz=\frac{\triangle y_{x}}{\triangle y_{z}},$
(3.5)

where $\displaystyle\triangle y_{x}$ is the variance of particle rapidity
along a randomly chosen direction that is transverse to the beam and
$\displaystyle\triangle y_{z}$ is the variance of the standard particle
longitudinal rapidity. From Fig. 3.3, it is seen that around the peak region
in $\displaystyle varxz$ versus energy with the Rostock cross section or the
tempered cross section with $\displaystyle\nu$ = 0.8, the results from pBUU
can describe the data well. Then, by combining the constrained in-media cross-
sections $\displaystyle\sigma_{NN}^{\rm med}$ and the Eq. (2.51), one can
obtain shear viscosity as shown in Fig. 3.4. In such a way, the shear
viscosity is hopefully of greater generality than even the transport model
itself [125]. Shear viscosity versus temperature is sensitive to different
kinds of in-media cross-sections $\displaystyle\sigma_{NN}^{\rm med}$. But it
can be seen that a uniform factor could not describe the data for the whole
energy region, which indicates that the factor for in-media cross-sections
should be energy dependence. Here, the behavior of shear viscosity versus
temperature is no more different from that in Fig. 2.5. In Ref. [125], it only
considers the elastic collision cross section. Like comments in Ref. [125],
however, as collision energy or temperature increases to a high value, meson
effects and modifications of inelastic processes need to be explored in the
simulations.

Figure 3.3: (Color online) Stopping observable $\displaystyle varxz$ for
protons (a), deuterons (b), tritons (c), and particles with Z=1 (d) in Au + Au
collisions at different beam energies at bred $\displaystyle<$ 0.15. Lines
show the effects of different in-medium reductions of the NN cross section
[125]. The “tempered” cross-section reductions are marked with their tunable
parameter $\displaystyle\nu$. Symbols are experimental data from the FOPI
Collaboration [143]. Figure 3.4: (Color online) Shear viscosity for symmetric
nuclear matter at different densities and temperatures, deduced from Boltzmann
equation with in-medium cross sections [125]. Figure 3.5: (Color online) Shear
viscosity as a function of temperature at different isospin asymmetries.
Figure 3.6: (Color online) Shear viscosity $\displaystyle\eta$ and specific
viscosity $\displaystyle\eta/s$ (s is entropy density) as functions of isospin
asymmetry $\displaystyle\delta^{2}$ for different densities and temperatures.
The solid lines are from the linear fit [82].

In a more general sense, Shi and Danielewicz [75] solved the set of Boltzmann
equations for a binary system of fermions to find thermodynamic fluxes driven
by specific thermodynamic forces and to find general expressions for the shear
viscosity and other transport coefficients. The transport coefficients are
calculated numerically for nuclear matter using experimental nucleon-nucleon
cross sections. The numerical results for the coefficients have been obtained,

$\displaystyle\displaystyle\eta$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\Big{(}1+0.10\delta^{2}\Big{)}\Big{[}\frac{856}{T^{1.10}}(\frac{\rho}{\rho_{0}})^{1.81}-\frac{240.9}{T^{0.95}}(\frac{\rho}{\rho_{0}})^{2.12}+2.154T^{0.75}\Big{]},$
(3.6) $\displaystyle\displaystyle D_{I}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\Big{(}1-0.19\delta^{2}\Big{)}\Big{[}\frac{11.34}{T^{2.38}}(\frac{\rho}{\rho_{0}})^{1.54}+\frac{1.746}{T}(\frac{\rho}{\rho_{0}})^{0.56}+0.00585T^{0.913}(\frac{\rho}{\rho_{0}})\Big{]},$
(3.7) $\displaystyle\displaystyle\kappa$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\Big{(}1+0.10\delta^{2}\Big{)}\Big{[}\frac{0.235}{T^{0.755}}(\frac{\rho}{\rho_{0}})^{0.951}-0.0582(\frac{\rho}{\rho_{0}})^{0.0816}+0.0238T^{0.5627}(\frac{\rho}{\rho_{0}})^{0.0171}\Big{]}.$
(3.8)

where $\displaystyle\rm T$ is temperature in MeV, $\displaystyle\eta$ is shear
viscosity in MeV/(fm2c), $\displaystyle D_{I}$ is isospin diffusivity in
fm$\displaystyle\cdot$c, heat conductivity $\displaystyle\kappa$ is in c/fm2;
and $\displaystyle\delta=\frac{\rho_{n}-\rho_{p}}{\rho}$ is isospin asymmetry.
From these Eqs. (3.6)$\displaystyle-$(3.8), it is found that the shear
viscosity diffusion coefficient and heat-conduction coefficient weakly depend
on isospin asymmetry. One can also see from Fig. 3.5 in which shear viscosity
as a function of temperature at different isospin asymmetries is displayed.

On the other hand, one derives the shear viscosity of neutron-rich nuclear
matter from the relaxation time approximation approach using an isospin- and
momentum-dependent interaction and the nucleon-nucleon cross sections taken as
those from the experimental data modified by the in-medium effective masses as
used in the isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport
model calculations [82]. The expression for shear viscosity is shown by Eq.
(2.53).

As shown in Fig. 3.6, shear viscosity $\displaystyle\eta$ and specific
viscosity $\displaystyle\eta/s$ (s is entropy density) as functions of isospin
asymmetry $\displaystyle\delta^{2}$ for different densities and temperatures
are displayed. Here, the entropy density $\displaystyle s$ can be obtained by
[82],

$\displaystyle\displaystyle s=-\sum_{\tau}\frac{{\rm
g}}{(2\pi\hbar)^{3}}\int\Big{[}f_{\tau}{\rm\ln}f_{\tau}+(1-f_{\tau}){\rm\ln}(1-f_{\tau})\Big{]}d^{3}p.$
(3.9)

With $\displaystyle\delta^{2}=0$ for the symmetry nuclear matter in Fig. 3.6
(a), a simple comparison is shown for shear viscosity from the Boltzmann type
equation of Eq. (2.51), shear rate, and the Green-Kubo methods. We can see
that the shear viscosity by relaxation time approximation approach is close to
the one by Boltzmann type equation. However, it is less than those from the
shear rate and the Green-Kubo methods. And $\displaystyle\eta$ shows parabolic
approximation relation to isospin asymmetry $\displaystyle\delta^{2}$. It
presents that isospin asymmetry has a large effect on shear viscosity, which
could be different from the results obtained by solving the set of Boltzmann
equations as mentioned above. The reason here could be due to the isospin
interaction used in the calculations. The specific viscosity
$\displaystyle\eta/s$ which is a hot agenda and relates to a bound value of
$\displaystyle\hbar/4\pi$ is shown in Fig. 3.6. It can be seen that specific
viscosity $\displaystyle\eta/s$ also shows parabolic approximation relation to
isospin asymmetry $\displaystyle\delta^{2}$. From Fig. 3.6 (d), (e), and (f),
we can see that at T= 10 MeV and $\displaystyle\rho=\rho_{0}$,
$\displaystyle\eta/s$ is larger than 25 times of $\displaystyle\hbar/4\pi$.
When T=50 MeV, the specific viscosity is about 4$\displaystyle\sim$5
$\displaystyle\hbar/4\pi$, which is already close to KSS bound. At a higher
density at $\displaystyle\rho=2\rho_{0}$, specific viscosity becomes larger
with the increase in shear viscosity.

Figure 3.7: (Color online) Specific shear viscosity $\displaystyle\eta/s$ of
water as a function of temperature in different pressures [106].

For common matter like water, as shown in Fig. 3.7, it can be seen that below
the critical pressure of 22.06 MPa, $\displaystyle\eta/s$ shows a
discontinuity at the minimum. Above the critical pressure,
$\displaystyle\eta/s$ becomes smooth [106]. In Ref. [106], from the results
computed for the shear viscosity of QCD in the hadronic phase by the coupled
Boltzmann equations of pions and nucleons in low temperatures and low baryon-
number densities and combined common matter such as Ar, CO, CO2, H, He, H2S,
Kr, N, NH3, Ne, O, and Xe, the authors suspect that the general feature for a
first-order phase transition is that $\displaystyle\eta/s$ has a discontinuity
in the bottom of the $\displaystyle\eta/s$ valley. And the valley gradually
disappears when the phase transition turns into a crossover. Near the critical
point, there is a smooth valley in the crossover, which is similar to the
confinement-deconfinement crossover of QCD at $\displaystyle\mu=$ 0\. And as
the pressure is far away from the critical point, the valley shape will
disappear. The authors argue that these are general features of first-order
phase transitions which indicates that specific viscosity
$\displaystyle\eta/s$ may be used as a probe to study the phenomenon of
liquid$\displaystyle-$gas phase transition. Also, in Fig. 3.7, the
$\displaystyle\eta/s$ of water shows a positive jump across the phase
transition line from the gas to liquid phase. The sign of the
$\displaystyle\eta/s$ jump of water for first-order phase transitions is
similar to the materials including Ar, CO2, H, He, H2S, Kr, N, NH3, Ne, O, and
Xe. But it might not be universal, as the jump for QCD in the limit of a large
number of colors is negative from the low to high temperature phases [48] and
the behavior of nucleonic matter [83].

The specific viscosity ($\displaystyle\eta/s$) of neutron-rich nucleonic
matter near its liquid$\displaystyle-$gas phase transition was studied with
the relaxation time approximation approach [83]. In this framework, one treats
the phase coexistence region with a volume fraction $\displaystyle\xi$, and
the average number and entropy densities are given by,

$\displaystyle\displaystyle\rho$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\xi\rho_{l}+(1-\xi)\rho_{g},$ (3.10)
$\displaystyle\displaystyle s$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle{\xi}s_{l}+(1-\xi)s_{g}.$ (3.11)

where $\displaystyle\rho_{l(g)}$ and $\displaystyle s_{l(g)}$ are the number
and entropy densities of the liquid (gas) phase, respectively. To construct a
phase coexistence state for nuclear matter, the Gibbs conditions are used. For
phase coexistence, it is governed by the Gibbs conditions. For asymmetric
nuclear matter, two-phase coexistence equations are [144],

$\displaystyle\displaystyle\mu_{i}^{l}(T,\rho^{l}_{i})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mu_{i}^{g}(T,\rho^{g}_{i})\,\,\,\,(i=n\,{\rm
and}\,p),$ (3.12) $\displaystyle\displaystyle P_{i}^{l}(T,\rho^{l}_{i})$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
P_{i}^{g}(T,\rho^{g}_{i})\,\,\,\,(i=n\,{\rm or}\,p),$ (3.13)

where P is the pressure. To determine shear viscosity, one considers a
stationary flow field in the $\displaystyle z$ direction and the shear force
on the particles for a single phase of gas or liquid [82, 83],

$\displaystyle\displaystyle
F_{i}=\sum\langle(p_{z}-m_{\tau}u_{z})\rho_{\tau}v_{x}\rangle_{i},$ (3.14)

where $\displaystyle\tau$ refers to a neutron or proton, $\displaystyle i=l$
is for the liquid phase and $\displaystyle g$ is for the gas phase, and
$\displaystyle m_{\tau}$ is the nucleon mass. And
$\displaystyle\langle\cdots\rangle$ denotes the thermal average of the product
of the flux $\displaystyle\rho_{\tau}v_{x}-$ in the $\displaystyle x$
direction and the momentum transfer $\displaystyle p_{z}-m_{\tau}u_{z}$ in the
$\displaystyle z$ direction. The shear viscosity is determined by,

$\displaystyle\displaystyle F_{l(g)}=-\eta_{l(g)}\frac{\partial
u_{z}}{\partial x}.$ (3.15)

In the phase coexistence region, one can imagine gas bubbles in a liquid or
liquid droplets in a gas. Thus, finally, the average shear viscosity of the
mixed phase can be expressed as,

$\displaystyle\displaystyle\eta$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle{\xi}\eta_{l}+(1-\xi)\eta_{g}.$ (3.16)

Here, $\displaystyle\eta_{l}$ and $\displaystyle\eta_{g}$ can be separately
calculated owing to the density is uniform in each phase. And the strategy is
as in Ref.[82].

Figure 3.8: (Color online) Density (top panels), shear viscosity (middle
panels), and specific shear viscosity (bottom panels) as functions of
temperature at the fixed pressure of P = 0.1 MeV/fm3 for isospin symmetric
matter ($\displaystyle\delta$ = 0) (left column), neutron-rich matter
($\displaystyle\delta$ = 0.5) (middle column), and pure neutron matter
($\displaystyle\delta$ = 1) (right column) with the stiffer symmetry energy x
= 0. Solid lines are results including the liquid–gas phase transition with
‘L’, ‘M’, and ‘G’ denoting the liquid phase, the mixed phase, and the gas
phase, respectively. Dashed lines are results obtained by assuming the
liquid$\displaystyle-$gas phase transition does not happen inside the binodal
surface [83].

In Fig.3.8, density (top panels), shear viscosity (middle panels), and
specific shear viscosity (bottom panels) as functions of temperature at the
fixed pressure of P = 0.1 MeV/fm3 for different isospin symmetries with the
stiffer symmetry energy $\displaystyle{\rm x}$= 0 are shown. For a given
pressure, the shear viscosity of the liquid phase is greater than that of the
gas phase. And with $\displaystyle\delta$ = 0, there is a sudden jump for
density, shear viscosity, and specific shear viscosity at a temperature
T$\displaystyle\sim$12 MeV, which is due to liquid$\displaystyle-$gas phase
transition of first-order of symmetric nucleonic matter [83]. The behavior
here for shear viscosity versus temperature is similar to that of ${}^{3}He$
[52]. From Fig.3.8, it can be seen that the minimum of the specific shear
viscosity is exactly located at the critical temperature when a first-order
phase transition happens, while it is located at the boundary of the gas phase
if the phase transition is a second-order [83]. In a Fermi system, as we know,
there is competition between the Pauli blocking and collision among particles
with changes of temperature. So even for pure neutron matter without a liquid-
gas phase transition, it is also seen that there is still a valley-shape
around phase transition region. In Fig.3.8 (a), (b), (c), the jumps show a
negative sign and differ from water as in Fig. 3.7.

Figure 3.9: (Color online) The specific shear viscosity as functions of
temperature (upper panels) and density (lower panels) at different fixed
pressures and temperatures, respectively, in isospin symmetric matter
($\displaystyle\delta$ = 0), neutron-rich matter ($\displaystyle\delta$= 0.5),
and pure neutron matter ($\displaystyle\delta$= 1) for both symmetry energies
x = 0 and x = 1 [83].

Furthermore, one can get more results for specific shear viscosity as
functions of temperature and density from Ref. [83], as shown in Fig.3.9. The
results show that the specific shear viscosity of nucleonic matter is about
4$\displaystyle-$5 $\displaystyle\hbar/4\pi$ for isospin-symmetric nucleonic
matter and is generally smaller than that in neutron-rich nucleonic matter. It
is obvious that there are valley shapes for symmetric nuclear matter and pure
neutron matter. The symmetry energy effect is investigated as the solid and
dashed lines display. In the phase coexistence region, the softer (x = 1)
symmetry energy would give a larger specific shear viscosity than that with
the stiffer (x = 0) symmetry energy for the liquid-like region, as in Fig.3.9
(b) and Fig.3.9 (e). And for pure neutron matter without the liquid-gas phase
transition under a fixed pressure, specific shear viscosity for x = 1 is
similar to that for x = 0, as in Fig.3.9(c). At fixed temperature, specific
shear viscosities from different symmetry energies are the same for pure
neutron matter, as shown in Fig.3.9 (f).

Figure 3.10: (Color online) Shear viscosity as a function of temperature at
different densities in the calculations without (a) and with (b) mean field.
[84]. Figure 3.11: (Color online) Shear viscosity as a function of temperature
in the calculations without and with the Coulomb interaction [84].

From the above discussions, one can get that Pauli blocking, intermediate
cross-section, the asymmetry of nucleonic matters have an effect on the shear
viscosity. Both of the Pauli blocking and inter-mediate cross-section affect
nucleon-nucleon collisions and then the shear viscosity. A system with
different asymmetries of nucleonic matters would have different nucleon-
nucleon potentials and different nucleon-nucleon cross-sections which would
make shear viscosity different. However, one can notice that these
calculations are with uniform nucleonic matter. There are no fragment effects
considered in those calculations. The nuclear matter in the gas phase is
different from the gas of classical fluids (as the water vapour), which is
composed of not only single nucleons, neutrons, and protons, but also complex
particles like alpha-particles and light fragments depending on the
temperature conditions [145, 146]. In this case, one extracts the shear
viscosity coefficient by using the shear strain rate method and setting up a
periodic cubic box within the improved quantum molecular dynamic (ImQMD) model
[84]. In the framework of ImQMD, potential energy density [147, 148] without
the spin-orbit term can be written as,

$\begin{split}V_{loc}&=\frac{\alpha}{2}\frac{\rho^{2}}{\rho_{0}}+\frac{\beta}{\gamma+1}\frac{\rho^{\gamma+1}}{\rho_{0}^{\gamma}}+\frac{g_{sur}}{2\rho_{0}}\,(\bigtriangledown\rho)^{2}\\\
&+g_{\tau}\frac{\rho^{\eta+1}}{\rho_{0}^{\eta}}+\frac{g_{sur,iso}}{\rho_{0}}\,[\nabla(\rho_{n}-\rho_{p})]^{2}+\frac{C_{s}}{2\rho_{0}}\rho^{2}\,\delta^{2}\,,\end{split}$
(3.17)

where $\displaystyle\rho=\rho_{\text{n}}+\rho_{\text{p}}$ is the net nucleon
density, $\displaystyle\rho_{\text{n}}$ and $\displaystyle\rho_{\text{p}}$ are
the proton and neutron densities, respectively, and
$\displaystyle\delta=(\rho_{n}-\rho_{p})/\rho$ is the normalized asymmetry of
neutron and proton. The Skyrme density functional with interactions such as
Eq. (3.17), as well as other approaches for nuclear systems, yield the liquid-
gas phase transition at the subnormal densities for the near-symmetric nuclear
matter. By switching on and off the mean field and thus inducing the phase
transition, it can be able to observe the impact of clumping on viscosity in
the phase transition region. The calculations are only for the isospin-
symmetric nucleonic matters with the nucleon-nucleon cross section fixed at 40
mb.

In the case without mean field and in cascade mode, there is no phase
transition and fragment only when mean field is taken into account. However,
when the mean field is turned on, the nucleon potential and kinetic energy
will change due to the fragment formation which is one of the challenges for
simulations. To avoid the system being heated up, it needs to adjust this
transient state temperature to the initial temperature by introducing the
cooling factor ‘h’, which can be nominally positive and less than ‘1’.

Figure 3.12: (Color online) The specify shear viscosity over entropy density
$\displaystyle\eta/s$ as a function of temperature at different densities in
the calculations without (a) and with (b) mean field [84].

Without mean field and in cascade mode, the viscosity is that of a gas of
nucleons, as shown in Fig.3.10 (a) and it is similar to that in Fig.3.1 (b)
and (d). From the results with mean field in Ref. [84] as shown in Fig.3.10
(b), it is seen that the shear viscosities are similar to those without mean
field at high temperatures and densities. However, at densities below
$\displaystyle 0.4\,\rho_{0}$ and temperatures less than $\displaystyle
10\,\text{MeV}$, it is found that the shear viscosity has a strong reduction
when the mean field is switched on. The reasons could be understood in the
following way: in the region of the phase transition, separated fragments,
which move slowly, absorb most of the mass for the box system and stall the
transport of momentum. Since those fragments are moving slowly, they play only
a passive role in transporting the momentum. In the phase coexistence state,
except for liquid-like fragments, there are gas phase nucleons in the system,
which could become the primarily transported part. These gas phase nucleons
rather collide with the fragments than with each other, which shortens their
mean free path. In addition, with most mass in the fragments, the role of the
Pauli principle in the gas is reduced which decreases the path further. Thus,
in these cases, shear viscosity would be reduced. As density increases, the
fragments would connect to each other, and the low-density gas phase is
contained in voids in the liquid. Then momentum transport can progress, and
the Pauli principle can be restored.

Moreover, when the fragments form, the long-range Coulomb interaction may play
a role for shear viscosity [84]. It is worth to mentioning that in most of the
calculations for the infinite and symmetric nuclear-matter systems, the
Coulomb interactions are switched off, and the near-symmetric matter has
comparable numbers of neutrons and protons. In Ref. [84], the effect of
Coulomb interaction on the shear viscosity in coexistence nucleonic matter has
been investigated. From the results with and without Coulomb interaction, as
shown in Fig. 3.11, it is found that shear viscosity is reduced by the Coulomb
interaction in the region of low temperature. For an understanding of this
phenomena, one can imagine that two fluid layers pass from one to another.
However, the repulsive Coulomb interaction among protons will reduce momentum
transformation or momentum flux between two layers, which will decrease shear
viscosity. In low-temperature region, the Coulomb interaction could be
important in comparison with thermal motion. With the increasing of
temperature, the Coulomb interaction effect becomes smaller since nuclear
matter at higher temperature is more uniform and the effect of Coulomb
interaction becomes negligible. Based on the above arguments, we can explain
the temperature dependence of Coulomb correction on shear viscosity.

Table 3.1: $\displaystyle\eta/s$ of different atomic and molecular systems: Helium, Nitrogen and H2O in certain ranges of temperature and pressure [48, 84].  Systems [Ref.] |  T (K) |  Pressure (MPa) |  $\displaystyle\eta/s~{}(\hbar/4\pi)$
---|---|---|---
Helium [48] |  2$\displaystyle-$20 |  0.1$\displaystyle-$1 |  8.8$\displaystyle-$126
Nitrogen [48] |  50$\displaystyle-$600 |  0.1$\displaystyle-$10 |  11.9$\displaystyle-$7000
H2O [48] |  300$\displaystyle-$1200 |  10$\displaystyle-$100 |  25.5$\displaystyle-$377
Table 3.2: $\displaystyle\eta/s$ of different quark matter (QGP) and finite nucleonic matter (FNM) as well as infinite nucleonic matter (INM) in certain ranges of temperature and density. Here Tc$\displaystyle\approx$ 170 GeV is transition temperature from the hadronic phase to QGP phase [149, 84].  Systems [Refs.] |  T (MeV) |  $\displaystyle\rho$/$\displaystyle\rho_{0}$ |  $\displaystyle\eta/s~{}(\hbar/4\pi)$
---|---|---|---
QGP1 [149, 150, 151, 152] |  $\displaystyle\leqslant$$\displaystyle\rm T_{c}$ |  — |  3.77$\displaystyle-$25.1
QGP2 [149, 150, 151, 152, 153] |  $\displaystyle>$$\displaystyle\rm T_{c}$ |  — |  1.00$\displaystyle-$6.91
FNM1 [92, 80] |  0.8$\displaystyle-$2.1 |  $\displaystyle\sim$1.0 |  2.5$\displaystyle-$6.5
FNM2 [154] |  3.5$\displaystyle-$16 |  0.9$\displaystyle-$1.25 |  3.0$\displaystyle-$70.0
FNM3 [91] |  6$\displaystyle-$16 |  0.2$\displaystyle-$0.3 |  9.5$\displaystyle-$20.0
INM1 [83] |  8$\displaystyle-$14 |  0.01$\displaystyle-$0.3 |  4$\displaystyle-$30
INM2 [84] |  4$\displaystyle-$25 |  0.1$\displaystyle-$0.8 |  2$\displaystyle-$55

For the simulation system investigated in Ref. [84], it is easy to extract the
entropy density by Eq.(3.9). Then one can obtain the specific shear viscosity
$\displaystyle\eta/s$ which is an interesting topic for a bound value of
$\displaystyle\hbar/4\pi$, known as the Kovtun-Son-Starinets (KSS) bound [24]
as mentioned above. To briefly summarize the values of ratios of shear
viscosity over entropy density for different systems, including atomic and
molecular systems, QGP matter, finite nucleonic matter (FNM), as well as
infinite nuclear matter (INM), are summarized in Tables 3.1 and 3.2 (The
specific shear viscosities for QGP matter and finite nucleonic matter are
discussed in later sections). It is worth to mentioning that the FNM is matter
in a system with finite size like a single nucleus or a heavy-ion collision
system. The hot quark gluon plasma (QGP) has a very low $\displaystyle\eta/s$
which is very close to $\displaystyle\hbar/4\pi$ and behaves as a nearly
perfect fluid [44]. When the QGP cools down and coalesces into a relativistic
hadron gas state, specific shear viscosity $\displaystyle\eta/s$ increases and
becomes significantly higher than the KSS bound [155, 77]. When the
temperature goes down, we would get some hot finite nucleonic matter; the
$\displaystyle\eta/s$ is around 3.0 - 70 times of $\displaystyle\hbar/4\pi$
for the density of 0.2 - 1.25 $\displaystyle\rho_{0}$ [91, 154], in which the
minimum $\displaystyle\eta/s$ value corresponds to the liquid-gas phase
transition as discussed in the previous model calculations. And here the value
of $\displaystyle\eta/s$ is larger than 2 times of $\displaystyle\hbar/4\pi$
for the infinite nuclear matter, which is consistent with the results in Ref.
[83, 91, 78, 79]. For the finitely colder nuclear matter around the saturation
density at much lower temperature (0.8$\displaystyle-$2.1 MeV), it is
2.5$\displaystyle-$6.5 times of $\displaystyle\hbar/4\pi$ [80, 92]. From the
two tables, one can see that the QGP matter at around 170 MeV has the lowest
$\displaystyle\eta/s$ which is close to the KSS bound
($\displaystyle\hbar/4\pi$). However, the nuclear matter at around a few to a
few tens MeV, which is in a liquid gas phase transition range, has also
relative low $\displaystyle\eta/s$, i.e., about several times the KSS bound.
The above low $\displaystyle\eta/s$ may reflect the universal property of
strong interaction matter regardless of partonic and nucleonic level. On the
contrary, for atomic and molecular substances of He, Ni, and water, they have
relatively large $\displaystyle\eta/s$ which are dominated by electromagnetic
interaction [84].

### 3.2 Finite nuclear matter

A nucleus is a many-body quantum system in which the constituents with finite
nucleons are governed by strong interaction. A useful way to investigate the
shear viscosity of finite nuclear matter is through heavy-ion collisions. In
the past decades, different model-dependent calculations for shear viscosity,
especially for the specific shear viscosity $\displaystyle\eta/s$, have been
studied in intermediate-energy heavy-ion collisions.

Within the framework of the Boltzmann-Uehling-Uhlenbeck model, Li
$\displaystyle et.\,al.$ [78] calculated the shear viscosity in Au + Au
collisions below 100 MeV/A by using the Green-Kubo formula as Eq. (2.9). In
their calculations, an isolated volume is chosen from a spherical space with a
radius of 5 fm. For simplicity, the correlation functions can be rewritten in
exponential form [76],

$\begin{split}\langle
T_{\alpha\beta}(t_{0})T_{\alpha\beta}(t_{0}+t^{\prime})\rangle\propto{\rm
exp}(-\frac{t^{\prime}}{\tau_{\pi}});\,\,\alpha,\beta=x,y,z,\alpha\neq\beta.\end{split}$
(3.18)

The calculations of the stress tensor here only include the momentum part, and
the mean field part is neglected. And the shear viscosity coefficient can be
rewritten in a simple form:

$\begin{split}\eta=\frac{V}{T}\langle
T_{\alpha\beta}(t_{0})T_{\alpha\beta}(t_{0})\rangle{\tau_{\pi}};\,\,\alpha,\beta=x,y,z,\alpha\neq\beta.\end{split}$
(3.19)

where $\displaystyle\tau_{\pi}$ is the relaxation time of the shear flux. With
the phase space information, one could get the time evolution of the
correlation function. By fitting with $\displaystyle{\rm
exp}(-\frac{t^{\prime}}{\tau_{\pi}})$, $\displaystyle\tau_{\pi}$ and the shear
viscosity can be obtained. And the entropy density can be calculated by the
Gibbs formula,

$\begin{split}s=\frac{\epsilon+P-\mu_{n}\rho}{T},\end{split}$ (3.20)

where $\displaystyle\epsilon$ is energy density inside the volume with 5 fm
radius. And $\displaystyle\mu_{n}$ is the nucleon chemical potential,
$\displaystyle\rho$ is nucleon density of system within the given sphere.

Figure 3.13: (Color online) $\displaystyle\eta/s$ (here $\displaystyle\eta/s$
is divided by $\displaystyle\hbar$ and thus without unit) as a function of
beam energy for the central Au + Au collision in a spherical volume with
radius of 5 fm. The inset shows the derivative of $\displaystyle\eta/s$ versus
beam energy [78].

As shown in Fig.3.13, the results for $\displaystyle\eta/s$ show a rapid fall
with the increasing of incident energy up to E $\displaystyle<$ 70 MeV/A and
then drop slowly to a value close to 0.5 when E $\displaystyle>$ 70 MeV/A.
Here it is worth to mentioning that $\displaystyle\eta/s$ is a dimensionless
quantity since it has been divided by $\displaystyle\hbar$ in some figures
blow. The results show a strong dependence of $\displaystyle\eta/s$ on the
equation of state (EoS) and a harder EoS would give a larger
$\displaystyle\eta/s$. In the inset of Fig.3.13, there seems to be a turning
point around E$\displaystyle\sim$65 MeV/A. This turning point may indicate a
change in the dynamical behavior of the system that is not caused by a phase
transition. Since the BUU model is based on one body theory, there is no
fluctuation, correlation, or the phase transition behavior. Compared to the
KSS bound, here $\displaystyle\eta/s$ is larger than 6 times of $\displaystyle
1/4\pi$ (here $\displaystyle\hbar$ is set to ‘1’) in the energy region of E
$\displaystyle<$ 100 MeV/A.

With the same method of Green-Kubo formulas, Guo $\displaystyle etal.$
investigated the specific shear viscosity in low-energy 40Ca + 100Mo heavy-ion
collision by using an extended quantum molecular dynamics model [92]. As shown
in Fig. 3.14, the ratio of shear viscosity to entropy density of the nuclear
fireball shows a slight drop as temperature or beam energy increases. The
results are consistent with those in Ref. [156, 134]. Here the specific shear
viscosity shows 2 times $\displaystyle 1/4\pi$ ($\displaystyle\hbar$=1) which
is very close to the KSS limit.

Figure 3.14: (Color online) $\displaystyle\eta/s$ as a function of temperature
in a central region for head on 40Ca + 100Mo collisions [92]. The solid line
(Auerbach and Shlomo) is taken from Ref. [156], and the dashed line (Dinh
Dang) is an extrapolation of the phonon-damping model prediction for 208Pb in
Ref. [134].

Later, Zhou et al. [91], investigated the thermodynamic and transport
properties of nuclear fireballs created in the central region of heavy-ion
collisions below 400 MeV/A within the isospin-dependent quantum molecular
dynamic (IQMD) model. By fitting the results from Eq.(2.51), one can get shear
viscosity in a numerical form [74],

$\displaystyle\displaystyle\eta=\frac{1700}{T^{2}}\Big{(}\frac{\rho}{\rho_{0}}\Big{)}^{2}+\frac{2}{1+0.001T^{2}}\Big{(}\frac{\rho}{\rho_{0}}\Big{)}^{0.7}+\frac{5.8\sqrt{T}}{1+160T^{-2}}$
(3.21)

where $\displaystyle\eta$ is in unit of MeV/(fm2c), and $\displaystyle T$ is
in unit of MeV. Combined temperature and density in the collision process, one
can obtain shear viscosity. In the simulations, temperature and entropy
density are calculated by the generalized hot Thomas Fermi formalism (GHTFF)
[157].

Figure 3.15: (Color online) Time evolution of shear viscosity at different
beam energies and cross section factors [91]. Figure 3.16: (Color online) Time
evolution of specific shear viscosity at different beam energies and cross
section factors [91]. Figure 3.17: (Color online) Specific shear viscosity as
a function of temperature at different beam energies and cross section factors
[91].

In order to study the effect of cross section on shear viscosity, the nucleon-
nucleon (NN) cross section is multiplied by a coefficient $\displaystyle
C_{\sigma}$, $\displaystyle\sigma_{\rm NN}=C_{\sigma}\sigma_{\rm NN}^{free}$
[91]. In this case, the expression for shear viscosity changes to,

$\displaystyle\displaystyle\eta(T,\rho,C_{\sigma})=\frac{\eta(T,\rho)}{C_{\sigma}},$
(3.22)

The coefficient $\displaystyle C_{\sigma}$ is set to 0.5, 1.0, or 1.5 in the
simulations. As shown in Fig.3.15, shear viscosity reaches the maximum value
at the compression stage during the collision process. But specific shear
viscosity shows the minimum value as in Fig.3.17. As shown in Fig.3.17,
specific shear viscosity shows a transient minimal $\displaystyle\eta/s$ =
(5-10)1/4π in the largest compression stage. It can be noticed that the shear
viscosity of nuclear matter in heavy-ion collisions is time-dependent.

Figure 3.18: (Color online) The scaled elliptic flow as a function of the
average shear viscosity and specific viscosity at 45 MeV/u and 50 MeV/u (Here
MeV/u is the same with MeV/A) and corresponding impact parameters [158]. The
red-dash lines are linear fits.

In a fluid, shear viscosity plays a resistance role to the flow. As we know,
in non-central heavy-ion collisions, particles have collective behavior.
Indeed, in a heavy-ion collision, there are two distinct ways for shear
viscosity to influence the final emitted particles [159]. One is that it
increases the magnitude and decreases the anisotropies of the hydrodynamically
generated transverse flow which reflects the amount of shear viscosity over
the entire expansion history of the fireball. Another is that it causes a
deviation, $\displaystyle\delta f$, of the final phase-space distribution
$\displaystyle f(x,p)$ from its isotropic local equilibrium form,
$\displaystyle f(x,p)=f_{0}(x,p)+\delta f(x,p)$. In heavy-ion collisions,
anisotropic flow can be defined as the orders of harmonic coefficients of
Fourier expansion of the particle azimuthal distribution, such as the second-
order elliptic flow $\displaystyle v_{2}$,

$\displaystyle\displaystyle v_{2}=\langle{\rm
cos}(2\phi)\rangle=\Big{\langle}\frac{p_{x}^{2}+p_{y}^{2}}{p_{x}^{2}-p_{y}^{2}}\Big{\rangle},$
(3.23)

where $\displaystyle\phi$ is the azimuthal angle, $\displaystyle p_{x}$ and
$\displaystyle p_{y}$ are the momenta of the particle at the x and y axes,
respectively. For the further work from Zhou et al. [158], they consider the
correlation between the elliptic flow $\displaystyle v_{2}$ scaled by the
impact parameter b and the shear viscosity $\displaystyle\eta$ as well as the
specific viscosity $\displaystyle\eta$/s. Here $\displaystyle v_{2}$ is
reduced by impact parameter b for reducing influence from b. As shown in Fig.
3.17, it is seen that $\displaystyle v_{2}$/b decreases linearly with
increasing of average shear viscosity $\displaystyle\langle\eta\rangle$. It is
also found that $\displaystyle v_{2}$/b mostly increases with increasing of
average specific viscosity $\displaystyle\langle\eta/s\rangle$. So it presents
a strong correlation between the elliptic flow and the shear viscosity or
specific viscosity $\displaystyle\eta$/s. Somehow, larger shear viscosities
quickly reduce collective flow. In relativistic heavy-ion collisions, the
large elliptic flow observed at RHIC energies provides compelling evidence for
strongly interacting matter that behaves as an almost perfect liquid [25, 27,
26, 160, 161, 159]. Here for the $\displaystyle v_{2}$/b versus
$\displaystyle\langle\eta\rangle$ consistent with the findings in relativistic
heavy-ion collisions, and it is opposite for the $\displaystyle v_{2}$/b
versus specific viscosity $\displaystyle\eta$/s. For the later one, the
authors argued that it is due to the stronger dissipation in the hadronic
phase than in the partonic phase [158].

Figure 3.19: (Color online) Time evolution of specific shear viscosity at
different beam energies for the soft EoS in the central region of X[−5,5],
Y[−5,5], and Z[−5,5] [79]. Solid and dotted arrows indicate the compression
process and expansion process, respectively.

A similar work is done in Ref. [79] but with an extension model of Boltzmann-
Uehling-Uhlenbeck, namely the Van der Waals Boltzmann-Uehling-Uhlenbeck
(VdWBUU) model. In this model, its cross section within the nucleonic medium
is derived from the mean field, which is given,

$\displaystyle\displaystyle\sigma=(\frac{9\pi}{16})^{1/3}b^{\prime{2/3}},$
(3.24)

where $\displaystyle b^{\prime}$ denotes the proper volume of the constituent
particle, which can be related geometrically to its cross section for
interaction with other particles and reads as,

$\displaystyle\displaystyle a^{\prime}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-a,$ (3.25) $\displaystyle\displaystyle b^{\prime}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{b\kappa\rho^{\kappa}+2\gamma{a_{s}}(\frac{\rho}{\rho_{0}})^{1+\gamma}\tau_{z}I}{(\frac{f_{5/2}(z)}{f_{3/2}(z)})\rho{T}+b\kappa\rho^{1+\kappa}+2\gamma{a_{s}}\rho_{0}(\frac{\rho}{\rho_{0}})^{1+\gamma}\tau_{z}I}.$
(3.26)

Figure 3.20: (Color online) The same as in Fig. 3.19 but for the correlation
between specific shear viscosity (in units of $\displaystyle\hbar/4\pi$) and
temperature [79].

where $\displaystyle a^{\prime}$ is related to attractive interaction among
particles; $\displaystyle z=\mu/T$ is the fugacity value of nucleons and
$\displaystyle f_{5/2(3/2)}$ are Fermi integrals; $\displaystyle a_{s}$ is
coefficient of the symmetry energy term; $\displaystyle\gamma$ describes the
density dependence; and $\displaystyle a,b,$ and $\displaystyle\kappa$ are
parameters for the nuclear equation of state (EoS). The more detail can be
seen in Ref. [79]. In simulations, shear viscosity is calculated in a central
region which is defined as a $\displaystyle[-5,5]^{3}$ fm3 or
$\displaystyle[-3,3]^{3}$ fm3 box with its center located in the c.m. with a
numerical expression with isospin asymmetry taken into account as Eq. (3.6).
And temperature for shear viscosity calculation is extracted by a quantum
Fermion fluctuation approach, which is proposed in Ref. [162]. The entropy
density can be determined by the density and temperature [163]:

$\displaystyle\displaystyle
s=\frac{U-A}{T}\frac{1}{V}=[\frac{5}{2}\frac{f_{5/2}(z)}{f_{3/2}(z)}-\ln{z}]\rho.$
(3.27)

where $\displaystyle
f_{m}(z)=\frac{1}{\Gamma(m)}\int_{0}^{\infty}\frac{x^{m-1}}{z^{-1}e^{x}+1}dx$,
and $\displaystyle z=e^{\frac{\mu}{T}}$ is fugacity. $\displaystyle U$ is the
internal energy and $\displaystyle A$ is Helmholtz free energy [164].

From the results of Ref. [79] as shown in Fig. 3.19, it can be seen that in
the compression process, shear viscosity decreases with increasing
temperature. While in the expansion process, the behavior is inverted. It is
also noticed that in the classical liquid, as the temperature goes higher, the
particles become separated over larger distances, they lose local orientation
order, and their interactions become weaker, hence the decrease in shear
viscosity. A more microcosmic interpretation is the competition between Pauli
blocking and collision as for higher density cases as in Fig. 2.5. On the
contrary, the shear viscosity of gas increases with increasing of temperature
due to the momentum of nucleons increasing. Thus, it is assumed that nuclear
matter is in the liquid-like phase state during the compression process and in
a gas-like phase during the later process. Around the inflection point, it
should be in mixed-like phase [79]. But we cannot say there is a phase
transition since the one-body theory for BUU model used in that work [79]. In
Fig. 3.20, specific shear viscosity at a maximum compress stage tends to
remain constant as beam energy increases, as the red-dot arrow shows. It is
different from the shear viscosity versus T, as in Fig. 3.19.

Figure 3.21: (Color online) Specific shear viscosity as a function of
temperature at different densities [154]. Figure 3.22: (Color online)
Temperature dependence of the multiplicity of intermediate mass fragment (IMF)
after the freeze-out of the system for nuclear sources at different densities.
Inset is the time evolution of multiplicity of IMF at T$\displaystyle=$10 MeV
for $\displaystyle\rho=0.16$fm-3 [154].

In addition, Fang et al. [154] extracted shear viscosities of finite-size
nuclear sources by using the mean free path method within the framework of the
isospin-dependent quantum molecular dynamics model，and the entropy density is
from the Gibbs formula as Eq. (3.20). The classical kinetic theory relates
$\displaystyle\eta$ of the system with the mean free path of the particle, as
in Eq. (2.8). From the results as shown in Fig. 3.21, $\displaystyle\eta/s$ is
larger than 7 times KSS bound. And the authors found that a minimum of
$\displaystyle\eta/s$ is seen at around $\displaystyle T$ = 10 MeV for
$\displaystyle\rho\sim$1/2$\displaystyle\rho_{0}$. For higher densities,
$\displaystyle\rho$ = 0.16,0.11 fm-3, no minimum of $\displaystyle\eta/s$ is
seen, but instead $\displaystyle\eta/s$ approaches an asymptotical value at
higher temperatures. There may be a relation between the minimum of
$\displaystyle\eta/s$ as a function of $\displaystyle T$ and liquid-gas phase
transition of nuclear matter. And the authors have checked the multiplicity of
the intermediate mass fragment (IMF), which is a widely studied observable for
liquid-gas phase transition in heavy-ion collisions. As shown in Fig. 3.22,
when the density is around 1/2 $\displaystyle\rho_{0}$ and at the same
temperature, the multiplicity of the IMF of the system after freezing-out
shows a maximum value which could be the signal of a liquid-gas phase
transition [165]. However, for the low density of 0.05 fm-3, there is no
maximum value for multiplicity of IMF as shown in Fig. 3.22 but shows a
minimum of $\displaystyle\eta/s$ in Fig. 3.21. It may indicate that the
minimum of $\displaystyle\eta/s$ here is more than the liquid-gas phase
transition.

Figure 3.23: (Color online) Shear viscosity and specific shear viscosity as
functions of temperature with two different methods [166]. But the result for
specific shear viscosity in (b) has been modified.

Further, Liu et al. [166] investigated the thermal and transport properties of
central 129Xe + 119Sn collisions around the Fermi energy using the mean free
path method and numerical expression [74] in the framework of an isospin-
dependent quantum molecular dynamical model. As shown in Fig. 3.23 which has
been corrected for the value of specific shear viscosity, it is seen that
$\displaystyle\eta/s$ is larger than 8 times the KSS bound. And it decreases
as the beam energy increases, which is consistent with that in Refs.[78, 91].

In this section, shear viscosity is extracted in QMD or BUU type model with
numerical equations (3.19), (3.21), and (2.8). It can be seen that these
results are consistent. However, these formulas can not be utilized from the
experimental side and are hard to check with data.

To summarize this part of the discussion, the shear viscosity in infinite and
finite nucleonic matter, Pauli blocking and nucleon-nucleon cross section play
important roles. The values of specific shear viscosity by different methods
or models are several times of $\displaystyle\hbar/4\pi$ and they are not very
different from those of QGP matter.

## 4 Experimental aspects of shear viscosity in nucleonic matter

### 4.1 Analysis method by experiment

Figure 4.1: (Color online) The nuclear shear viscosity $\displaystyle\eta$
(a), entropy density s (b) and their ratio, ($\displaystyle\eta/s$) (c), in
units of $\displaystyle\hbar$/4$\displaystyle\pi{k_{B}}$, as functions of the
temperature, T. The values of the parameters used in the calculations are
$\displaystyle\epsilon_{F}$ = 40 MeV, $\displaystyle\rho$ = 0.16 fm3,
$\displaystyle\alpha$ = 9.2 MeV, and $\displaystyle\hbar\omega$ = 20 MeV
[156].

In the previous section, we reviewed some calculations for shear viscosity
($\displaystyle\eta$) and the ratio of shear viscosity to entropy density
($\displaystyle\eta/s$) in nucleonic matter. As mentioned above, the analysis
of the state of matter produced by the ultra-relativistic heavy-ion collisions
at the Relativistic Heavy Ion Collider (RHIC) shows the matter has very small
values of $\displaystyle\eta/s$ [44]. How is it for finite nucleonic matter or
infinite nuclear matter? The finite nucleonic matter like a finite nucleus,
with the constituent nucleons being a many-body quantum system, is governed by
strong interaction and shows highly correlated behavior [80]. It can be an
ideal system to search for near-perfect fluidity in matter. The search for the
universal characteristic of the strong interaction of the many-body quantum
systems with shear viscosity is significant.

Auerbach and Sholomo have done earlier theoretical work [156] regarding the
value of $\displaystyle\eta/s$ for finite nuclear matter in which the nucleons
are governed by strong interaction. The authors have extracted shear viscosity
within a generalized Fermi liquid drop model by employing a collision kinetic
equation, where the dissipative propagation of sound waves is taken into
account in finite nuclei and nuclear matter [156]. In the Fermi liquid drop
model, shear viscosity is expressed as,

$\displaystyle\displaystyle\eta=\frac{2}{5}\rho\epsilon_{F}\frac{\tau_{coll}}{1+(\omega\tau_{coll})^{2}},$
(4.1)

with

$\displaystyle\displaystyle\tau_{coll}=\frac{\tau_{0}}{1+\Big{(}\frac{\hbar\omega}{2{\pi}T}\Big{)}^{2}}.$
(4.2)

where $\displaystyle\hbar\omega$ is the excitation energy of a sound wave and
$\displaystyle\omega$-dependent terms are related to the dynamical distortion
of the particle momentum distribution due to memory effects [156]. In Eqs.
(4.1) and (4.2), $\displaystyle\tau_{coll}$ is the Landau approximation for
the collision relaxation time. And $\displaystyle\tau_{0}=\hbar\alpha/T^{2}$
and the value of $\displaystyle\alpha$ is sensitive to the in-medium-nucleon-
nucleon scattering cross section. $\displaystyle\alpha$ is taken to be 9.2 MeV
in Ref. [167]. As displayed in Fig.4.1, shear viscosity increases as
temperature increases at density of $\displaystyle\rho$=0.16 fm-3. Here the
unit 10-23 MeV$\displaystyle\cdot$s$\displaystyle\cdot$fm-3 is equal to 3
MeV/(fm2c), and c is speed of light. From Fig.4.1(c), the authors found that
the specific shear viscosity is not drastically different from the RHIC
result. So it could be a general characteristic feature in a many-body nuclear
system with strong interaction, not just for the state created in the
relativistic heavy-ion collisions [156].

It raises a lot of interest in both theory and experiment. In 2011, Dang [134]
proposed a formalism based on the Green-Kubo relation and the fluctuation
dissipation theorem (FDT). With the Green-Kubo formula [22], one can obtain
the expression for the linear transport coefficient of any system at a given
temperature T and density $\displaystyle\rho$ in terms of the time dependence
of equilibrium fluctuations in the conjugate flux [134]. The Green-Kubo
formula has been mentioned above. Here we write the shear viscosity
$\displaystyle\eta(T)$ as,

$\displaystyle\displaystyle\eta(T)=\lim_{\omega\rightarrow
0}\frac{1}{2\omega}\int dtd^{3}xe^{i\omega}\langle
T_{xy}(\mathbf{x},t)T_{xy}(\mathbf{x},t_{0})\rangle$ (4.3)

where $\displaystyle\langle...\rangle$ denotes average over an equilibrium
ensemble of events. From the fluctuation dissipation theorem, the right-hand
side of Eq. (4.3) can be written as proportion to the absorption cross section
$\displaystyle\sigma(\omega,T)$,

$\displaystyle\displaystyle\eta(T)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\lim_{\omega\rightarrow 0}\frac{1}{2\omega i}[{\rm
G}_{A}(\omega)-{\rm G}_{R}(\omega)]$ (4.4) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\lim_{\omega\rightarrow 0}\frac{{\rm Im}\,{\rm
G}_{R}(\omega)}{\omega}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\lim_{\omega\rightarrow
0}\frac{\sigma(\omega,T)}{C}$

where $\displaystyle{\rm G}_{A}$ and $\displaystyle{\rm G}_{R}$ are the
advanced and retarded Green functions, respectively, with

$\displaystyle\displaystyle G_{R}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-i\int dtd^{3}xe^{i\omega}\langle
T_{xy}(\mathbf{x},t)T_{xy}(\mathbf{x},t_{0})\rangle$ (4.5)

and $\displaystyle{\rm G}_{A}(\omega)$=$\displaystyle{\rm
G}_{R}^{\ast}(\omega)$. The $\displaystyle C$ in Eq. (4.4) is equal to
16$\displaystyle\pi$G with G being the ten-dimensional gravitational constant.
As in Ref. [134], one can obtain the final expression for the shear viscosity
at temperature T,

$\displaystyle\displaystyle\eta(T)=\eta(0)\frac{\Gamma(T)}{\Gamma(0)}\frac{E_{GDR}(0)^{2}+[\Gamma(0)/2]^{2}}{E_{GDR}(T)^{2}+[\Gamma(T)/2]^{2}},$
(4.6)

where $\displaystyle\Gamma$ is the full width at half maximum (FWHM) of the
giant dipole resonance (GDR), and $\displaystyle E_{GDR}$ is its peak energy.
The $\displaystyle\eta(0)$ is shear viscosity at T=0. It is worth to
mentioning that in the calculations, an uncertainty is in extracting the value
$\displaystyle\eta$(0) of the shear viscosity $\displaystyle\eta$(T) at T = 0
and it can be given different dissipation mechanisms [134]. By using Eq.
(4.6), one can calculate the shear viscosity $\displaystyle\eta$ of a finite
hot nucleus directly from the width and energy of the GDR of this nucleus in
the experimental data which gives model-independent estimation. The entropy
density (entropy per volume V) is calculated,

$\displaystyle\displaystyle s=\frac{S}{V}=\rho\frac{S}{A}.$ (4.7)

with the nuclear density ρ = 0.16 fm-3. In Ref. [134], two formulas in
different methods were used to determine the entropy. One is from phonon-
damping model (PDM) [168, 169, 170] and another is from the Fermi-gas formula,

$\displaystyle\displaystyle S=2a_{T}T$ (4.8)

where aT the temperature-dependent parametrization of the density parameter as
in Refs. [171, 172, 173]. This formula (4.8) is used in the FLDM and the
analysis of experimental data. Combined Eq. (4.7) and Eq. (4.8), one can
obtain the entropy density $\displaystyle s=2\rho\frac{a_{T}}{A}T$.
$\displaystyle T$ is the temperature corresponding to the final excitation
energy $\displaystyle E^{\ast}$ and is given by $\displaystyle
T=\sqrt{E^{\ast}/a_{T}}$, where $\displaystyle
E^{\ast}=E_{0}^{\ast}-E_{rot}-E_{GDR}-\triangle P$, $\displaystyle
E_{0}^{\ast}$, $\displaystyle E_{rot}$ and $\displaystyle\triangle P$ being
average excitation energy, the average rotational energy and the pairing
energy, respectively [80]. For a given hot nucleus, one can extract the
experimental temperature, the width, and the energy of the GDR. When
substituted these quantities into Eq. (4.6) and the equation for entropy
density $\displaystyle s=2\rho\frac{a_{T}}{A}T$, it can be given a model-
independent estimation of the ratio $\displaystyle\eta/s$.

Figure 4.2: (Color online) Shear viscosity $\displaystyle\eta$(T)
[(a)$\displaystyle-$(c)] and ratio of $\displaystyle\eta/s$
[(d)$\displaystyle-$(f)] as functions of T for nuclei in [(a) and (d)] copper,
[(b) and (e)] tin, and [(c) and (f)] lead regions. The gray areas are the PDM
predictions by using 0.6u$\displaystyle<$η(0)$\displaystyle<$1.2u [134]. Here
u is an unit 10-23 MeV s fm3. The experimental data for the calculations of
$\displaystyle\eta/s$ are from copper (59Cu [171] and 63Cu [172]), tin (by
Bracco et al. [174], Enders et al. [175], Baumann et al. [173], Heckmann et
al. [176], and Kelly et al. [177]), and lead (208Pb [173] and 200Pb [178]).

### 4.2 Shear viscosities of finite nuclei in experiment

The predictions for the shear viscosity $\displaystyle\eta$ and the ratio
$\displaystyle\eta/s$ by the phonon-damping model (PDM), the Fermi-liquid-drop
model (FLDM), the two versions of TSFM, namely, the adiabatic model (AM), and
the phenomenological parametrization of the two thermal-shape fluctuation
models (pTSFM) for 63Cu, 120Sn, and 208Pb are plotted as functions of
$\displaystyle T$ in comparison with the empirical results. The empirical
values for $\displaystyle\eta$ in Figs. 4.2(a)$\displaystyle-$(c) are
extracted from the experimental systematics for the GDR in copper, tin, and
lead regions [174, 171, 172, 175, 173, 176, 177, 178], making use of Eq.
(4.6). The theoretical results show agreement with that of experiment. And
shear viscosities for nuclei increase with temperature. The specific shear
viscosity shows a mild decrease with temperature and is larger than 3 times
the KSS bound.

Figure 4.3: (Color online) Experimental data (symbols) along with the
theoretical predictions (solid lines) for shear viscosity $\displaystyle\eta$
(upper panel), entropy density $\displaystyle s$ (middle panel), and specific
shear viscosity ($\displaystyle\eta/s$) (lower panel). Blue short-dashed line
(lower panel) is the KSS bound. The errors in $\displaystyle\eta$ and
$\displaystyle\eta/s$ include the statistical errors as well as the systematic
error due to the lower and upper bounds of $\displaystyle\eta$(0) [80]. Figure
4.4: (Color online) Shear viscosity (a), entropy density (b) and specific
shear viscosity (c) as functions of mass number ($\displaystyle A$) in the
different temperature regions [80].

Heavier nuclear systems from A$\displaystyle\sim$30 to A$\displaystyle\sim$208
are investigated in Ref. [80] for the shear viscosity and ratio of shear
viscosity over entropy density by using the formalism as Eq. (4.6). Shear
viscosity, entropy density, and the ratio of shear viscosity to entropy
density are well reproduced for the systems. Here, shear viscosity also shows
an increase as temperature increases. It is interpreted by the kinetic
theoretical calculations $\displaystyle\eta$ =
$\displaystyle\frac{1}{3}$$\displaystyle\rho mv\lambda$ as Eq. 2.8. For an
equilibrated nucleus, the mean free path $\displaystyle\lambda$
$\displaystyle\sim$ $\displaystyle v/N_{coll}$, where $\displaystyle N_{coll}$
is the collision number, and is not sensitive to temperature. Thus, shear
viscosity increases with temperature for both the mean free path
$\displaystyle\lambda$ and $\displaystyle v$ increasing with temperature with
$\displaystyle v\sim\sqrt{T}$. One can notice that the finite nucleus system
is not a dilute gas system but a fermion system with nucleon density reaching
0.16 fm-3 and Pauli blocking effect. However, the results for shear viscosity
of finite nucleus shown in Fig. 4.3 and Fig. 4.4 are different from those in
Fig. 2.5, Fig. 3.8 and Fig. 3.10. From the calculations with Green-Kubo,
Chapman-Enskog and relaxation time approximation methods, shear viscosity at
temperature $\displaystyle T$ = 0 is a large value. Here, $\displaystyle\eta$
at $\displaystyle T$ = 0 has a small finite value. Moreover, it is confined in
the range (2.5$\displaystyle-$6.5) $\displaystyle\hbar/(4\pi{k_{B}}$) for the
finite nuclear matter within the temperature range of
$\displaystyle\sim$(0.8$\displaystyle-$2.1) MeV [80]. For the dependence of
mass number, it is seen that $\displaystyle\eta$ increases with the mass
number at the highest available temperature for heavier nuclei, as in Fig. 4.4
(d). However, $\displaystyle\eta/s$ is still within
(5.1$\displaystyle-$6.5)$\displaystyle\hbar/(4\pi{k_{B}}$) and shows
independence with the nuclear size and the neutron-proton asymmetry at a given
temperature [80].

Figure 4.5: (Color online) Shear viscosity as a function of temperature for
different mass nuclei at different angular momenta ($\displaystyle J$) [179].

In other aspects, the effect of high angular momentum on shear viscosity is
discussed by Bhattacharya et al. [179]. With the help of the critical
temperature included fluctuation model (CTFM) and FLDM, the predictions for
shear viscosity at different angular momenta can be given, and comparison is
done with the experimental data by using experimental giant dipole resonance
(GDR) width ($\displaystyle\Gamma$) at high angular momenta ($\displaystyle J$
= 12$\displaystyle-$54 $\displaystyle\hbar$) and temperatures (T =
1.2$\displaystyle-$2.1 MeV) collected from the existing literature. As
displayed in Fig. 4.5, it can be seen that the shear viscosity
($\displaystyle\eta$) increases with T and $\displaystyle J$. The results from
the critical temperature included fluctuation model (CTFM) well describe
J-induced $\displaystyle\eta$. However, the shear viscosity from FLDM, which
is an extension of the liquid drop model (LDM) to quantum Fermi liquid,
displays a large deviation in the higher angular momentum region and is
failing to describe the experimental data. From Fig. 4.5, one could also learn
that the effect of J in $\displaystyle\eta$ is important for lighter mass
nuclei. The reason given by the author is due to smaller moment of inertia for
the lighter mass nuclei in comparison to that of the heavier mass nuclei.

Recently, Reichert et al. [81] estimated $\displaystyle\eta/s$ from data
provided by the HADES experiment at GSI on the flow coefficients
$\displaystyle v_{1}$,$\displaystyle\cdots$,$\displaystyle v_{4}$ for protons
in 197Au+197Au reactions at Elab = 1.23 AGeV (or $\displaystyle\sqrt{\rm
s_{NN}}$ =2.4 GeV). Through the coarse graining analysis of the UrQMD [180,
181, 182, 183] transport simulations of the flow harmonics in comparison to
the experimental data, it was obtained that
$\displaystyle\eta/s$$\displaystyle\approx$0.65$\displaystyle\pm$0.15 (or
(8$\displaystyle\pm$2) $\displaystyle 1/4\pi$) as shown by the red-square
point in Fig.4.6. It is seen that even in the higher collision energy,
$\displaystyle\eta/s$ shows a decrease as beam energy increases. Reichert et
al. pointed out that one could explore $\displaystyle\Lambda$,
$\displaystyle\bar{\Lambda}$ polarization from the spatial distribution of
$\displaystyle\eta$/s in the transverse plane. Also, one can get information
on the contributions of $\displaystyle\eta/s$ to the collective flow
components since the flow components were created at different times of the
collision. For instance, $\displaystyle v_{1}$ is mainly created during the
maximal compression phase, where $\displaystyle\eta$/s is lowest and
$\displaystyle v_{2}$ gains further contributions during the expansion phase,
where $\displaystyle\eta$/s moderately increases [81]. Moreover, one could
investigate the effects of shear viscosity on the Fourier-decomposition of
azimuthal particle distribution [81].

Figure 4.6: (Color online) Ratio of shear viscosity to entropy density
$\displaystyle\eta/s$ as a function of collision energy
$\displaystyle\sqrt{\rm s_{NN}}$ [81]. The blue triangles-up are extracted
from [184] at full overlap, while the black triangles-down are from [185] and
the green circles are from [186]. The orange star denotes an estimate from
[187], and the dotted black line shows the KSS bound [24].

### 4.3 Others

Fission is an important subject in the studies of nuclear physics, nuclear
astrophysics, and nuclear applications [188, 189, 190, 191]. Lots of work has
attempted to reproduce experimental data on fission in heavy ion collisions by
adjusting the properties of the viscosity of hot nuclear matter [188, 189,
192, 193, 194, 195]. It is well known that the standard statistical model
underestimates the number of measured pre-scission particles emitted in heavy-
ion reactions [193, 194, 195]. The main reason could be due to the nuclear
viscosity. For the fission, it has two main processes: one is the time needed
for transition from equilibrium compound state to saddle point called the
transient time ($\displaystyle\tau_{tr}$) and the other is the time of descent
from saddle to scission point, called $\displaystyle\tau_{ssc}$ [194]. And
such processes have been investigated and it is indicated that dissipative in
nature at high excitation energy is important [194]. In Ref. [195],
Eslamizadeh and Pirpour simulated the fission process of the excited compound
nucleus 251Es produced in 19F+232Th reaction with a stochastic approach based
on two-dimensional Langevin equations. Within the framework, it contains a
friction coefficient ($\displaystyle n$) as a free parameter which depends on
the nuclear viscosity. As shown in Fig. 4.7, it is shown that the results of
calculations are in good agreement with the experimental data by applying
values of the post-saddle friction equal to 13$\displaystyle-$15
$\displaystyle\times$ 1021 s-1 [195]. In Ref. [195], the author did not give
the value of the nuclear viscosity but the post-saddle friction, which
indicates that nuclear viscosity plays an important role in fission dynamics.

Figure 4.7: (Color online) The pre-scission fission neutron multiplicity for
the compound nucleus 251Es as a function of energy calculated with different
values of the post-saddle friction [195]. The experimental data with filled
symbols is from [196].

In this section, we have reviewed some results for shear viscosity
$\displaystyle\eta$ and specific shear viscosity $\displaystyle\eta/s$. Shear
viscosity in terms of the width and energy of GDR can help us extract it from
experiments. Shear viscosity of finite nuclei shows an increase as temperature
increases at low temperatures with a density $\displaystyle\rho$=0.16 fm-3.
However, for the shear viscosity of infinite nuclear matter as shown in Figs.
3.9 and 3.10, it can be seen that at density $\displaystyle\rho$=0.16 fm-3,
shear viscosity decreases as a function of temperature in the low temperature
region since there is a competition between Pauli blocking and nucleon-nucleon
collisions. Whether it is a finite-size effect or not should be checked in
future investigations. And from the experimental results for specific shear
viscosity $\displaystyle\eta/s$, it is not so different from that of QGP
matter produced in relativistic heavy-ion collisions. Also, we have presented
some previous results about the effects of mass and angular momentum on shear
viscosity and the effect of nuclear viscosity in fission dynamics studies.

## 5 $\displaystyle\eta$/s in relativistic heavy$\displaystyle-$ion collision

This review focuses mainly on the shear viscosity and ratio of shear viscosity
to the entropy density $\displaystyle\eta/s$ of nucleonic matter. However,
there is a strong connection between $\displaystyle\eta/s$ of nucleonic matter
and QGP in relativistic heavy-ion collisions. For completion, we will shortly
review $\displaystyle\eta/s$ in relativistic heavy-ion collisions. In heavy-
ion collision, the exact shear viscosity and ratio of shear viscosity to the
entropy density should be time dependence and depend on matter density, system
temperature, the presence of magnetic fields, and possibly other system
conditions since the collision system compression or expansion (in
relativistic heavy-ion collisions as the QGP expands and goes through a
transition into a hadron phase) [197].

### 5.1 Ratio of shear viscosity to entropy density across the QCD phase
diagram

After the KSS bound proposed by Kovtun, Son, and Starinets [24], lots of
efforts were paid to the ratio of shear viscosity to entropy density
$\displaystyle\eta/s$, the fluidity of QGP matter, the relation between
$\displaystyle\eta/s$ and phase diagram, etc. According to the results in Ref.
[48], the ratio of shear viscosity to entropy density near the phase
transition from hadrons to quarks and gluons shows a similar behavior to
helium, nitrogen, and water. As seen in Fig. 5.1, the experimental and
calculated $\displaystyle\eta/s$ values for nuclear matter show a minimum in
the vicinity of the critical temperature $\displaystyle T_{c}$ [198]. This
value of $\displaystyle\eta/s$ for QGP and that in relativistic heavy-ion
collision in RHIC is much lower than that obtained from helium, nitrogen, and
water. It has been interpreted that the QGP created in the early phase of RHIC
collisions is more strongly coupled than atomic and molecular substances. From
Fig. 5.1, $\displaystyle\eta/s$ for RHIC collisions at $\displaystyle T_{c}$,
is rather close to the conjectured lower bound of $\displaystyle
1/4\pi$(generally, in relativistic heavy-ion collisions
$\displaystyle\hbar$=1). The authors concluded that it is compatible with the
minimum expected if the hot and dense QCD matter produced in RHIC collisions
follows decay trajectories that are close to the critical end point (CEP)
[198].

Figure 5.1: (Color online) Ratio of shear viscosity to entropy density ($\displaystyle\eta/s$) as a function of reduce temperature (T-$\displaystyle T_{c}$)/$\displaystyle T_{c}$ for several substances [198]. The lattice QCD value critical temperature $\displaystyle T_{c}$ for nuclear matter is set at 170 MeV [199]. Table 5.1: Compilation table [197] for measured charged particle densities, dNch/d$\displaystyle\eta$, and longitudinal widths, $\displaystyle\sigma_{\bigtriangleup\eta}$, of the charge independent correlator G${}^{\rm CI}_{2}$, interpolated freezeout times, $\displaystyle\tau_{f}$, and computed values of $\displaystyle\eta$/s as a function of the centrality of Pb$\displaystyle+$Pb collisions at $\displaystyle\sqrt{s_{\rm NN}}$=2.76 TeV [200, 201, 202, 203] and Au$\displaystyle+$Au at $\displaystyle\sqrt{s_{\rm NN}}$= 200 GeV [204, 205]. Centrality |  dNch/d$\displaystyle\eta$ |  $\displaystyle\sigma_{\bigtriangleup\eta}$ of G${}^{\rm CI}_{2}$ |  $\displaystyle\tau_{f}$ (fm/c) |  $\displaystyle\eta$/s
---|---|---|---|---
LHC |  |  |  |
0-5% |  1601.00$\displaystyle\pm$60.00 |  0.68$\displaystyle\pm$0.01sta+0.11sys-0.03sys |  10.33$\displaystyle\pm$0.10$\displaystyle\pm$1.03fit |  0.05$\displaystyle\pm$0.00sta$\displaystyle\pm$0.04sys
5-10% |  1294.00$\displaystyle\pm$49.00 |  0.73$\displaystyle\pm$0.01sta+0.54sys-0.03sys |  9.62$\displaystyle\pm$0.09$\displaystyle\pm$0.91fit |  0.07$\displaystyle\pm$0.00sta$\displaystyle\pm$0.03sys
10-20% |  966.00$\displaystyle\pm$37.00 |  0.71$\displaystyle\pm$0.01sta+0.04sys-0.03sys |  8.73$\displaystyle\pm$0.08$\displaystyle\pm$0.75fit |  0.06$\displaystyle\pm$0.00sta$\displaystyle\pm$0.02sys
20-30% |  649.00$\displaystyle\pm$23.00 |  0.73$\displaystyle\pm$0.01sta+0.03sys-0.03sys |  7.64$\displaystyle\pm$0.07$\displaystyle\pm$0.56fit |  0.07$\displaystyle\pm$0.00sta$\displaystyle\pm$0.03sys
30-40% |  426.00$\displaystyle\pm$15.00 |  0.70$\displaystyle\pm$0.01sta+0.03sys-0.03sys |  6.64$\displaystyle\pm$0.06$\displaystyle\pm$0.39fit |  0.06$\displaystyle\pm$0.00sta$\displaystyle\pm$0.02sys
40-50% |  261.00$\displaystyle\pm$ 9.00 |  0.69$\displaystyle\pm$0.01sta+0.03sys-0.03sys |  5.64$\displaystyle\pm$0.05$\displaystyle\pm$0.23fit |  0.06$\displaystyle\pm$0.00sta$\displaystyle\pm$0.02sys
50-60% |  149.00$\displaystyle\pm$ 6.00 |  0.65$\displaystyle\pm$0.01sta+0.03sys-0.03sys |  4.68$\displaystyle\pm$0.04$\displaystyle\pm$0.13fit |  0.05$\displaystyle\pm$0.00sta$\displaystyle\pm$0.02sys
60-70% |  76.00$\displaystyle\pm$ 4.00 |  0.63$\displaystyle\pm$0.01sta+0.03sys-0.03sys |  3.74$\displaystyle\pm$0.04$\displaystyle\pm$0.20fit |  0.05$\displaystyle\pm$0.00sta$\displaystyle\pm$0.02sys
70-80% |  35.00$\displaystyle\pm$ 2.00 |  0.59$\displaystyle\pm$0.01sta+0.02sys-0.02sys |  2.89$\displaystyle\pm$0.03$\displaystyle\pm$0.33fit |  0.04$\displaystyle\pm$0.00sta$\displaystyle\pm$0.02sys
RHIC |  |  |  |
0-5% |  691.00$\displaystyle\pm$49.00 |  0.94$\displaystyle\pm$0.06sta$\displaystyle\pm$0.17sys |  7.24$\displaystyle\pm$0.07$\displaystyle\pm$0.50fit |  0.14$\displaystyle\pm$0.03sta$\displaystyle\pm$0.09sys
5-10% |  558.00$\displaystyle\pm$40.00 |  0.99$\displaystyle\pm$0.07sta$\displaystyle\pm$0.06sys |  6.81$\displaystyle\pm$0.06$\displaystyle\pm$0.41fit |  0.16$\displaystyle\pm$0.03sta$\displaystyle\pm$0.07sys
10-20% |  421.00$\displaystyle\pm$30.00 |  0.93$\displaystyle\pm$0.06sta$\displaystyle\pm$0.07sys |  6.27$\displaystyle\pm$0.05$\displaystyle\pm$0.30fit |  0.14$\displaystyle\pm$0.03sta$\displaystyle\pm$0.06sys
20-30% |  287.00$\displaystyle\pm$20.00 |  0.84$\displaystyle\pm$0.05sta$\displaystyle\pm$0.03sys |  5.62$\displaystyle\pm$0.04$\displaystyle\pm$0.17fit |  0.10$\displaystyle\pm$0.02sta$\displaystyle\pm$0.04sys
30-40% |  195.00$\displaystyle\pm$14.00 |  0.67$\displaystyle\pm$0.03sta$\displaystyle\pm$0.02sys |  5.05$\displaystyle\pm$0.04$\displaystyle\pm$0.12fit |  0.04$\displaystyle\pm$0.01sta$\displaystyle\pm$0.02sys
40-50% |  126.00$\displaystyle\pm$60.00 |  0.59$\displaystyle\pm$0.02sta$\displaystyle\pm$0.03sys |  4.48$\displaystyle\pm$0.05$\displaystyle\pm$0.17fit |  0.01$\displaystyle\pm$0.01sta$\displaystyle\pm$0.02sys
50-60% |  78.00$\displaystyle\pm$ 9.00 |  0.57$\displaystyle\pm$0.02sta$\displaystyle\pm$0.02sys |  3.95$\displaystyle\pm$0.06$\displaystyle\pm$0.27fit |  0.01$\displaystyle\pm$0.01sta$\displaystyle\pm$0.02sys
60-70% |  45.00$\displaystyle\pm$ 6.00 |  0.55$\displaystyle\pm$0.02sta$\displaystyle\pm$0.04sys |  3.43$\displaystyle\pm$0.08$\displaystyle\pm$0.38fit |  0.003$\displaystyle\pm$0.009sta$\displaystyle\pm$0.022sys

### 5.2 The dependence of ratio of shear viscosity to entropy density on
collision centrality

According to the work [197], the author gives the dependence of specific shear
viscosity on collision centrality in ultra-relativistic heavy-ion collisions
at RHIC and LHC by using the Gavin ansatz, which relates the longitudinal
broadening of transverse momentum two-particle correlators, $\displaystyle
G_{2}$. The $\displaystyle G_{2}$ correlator [206, 207] is designed to be
proportional to the covariance of momentum currents and is as such sensitive
to dissipative viscous forces at play during the transverse and longitudinal
expansion of the matter formed in A$\displaystyle+$A collisions. And these
forces lead to a longitudinal broadening of $\displaystyle G_{2}$ measured as
a function of the pseudorapidity difference of measured charged particles
[197]. Through this $\displaystyle G_{2}$ correlator, Gavin gave a clear image
for relation between $\displaystyle G_{2}$ and system expansion. As the matter
expands, neighboring fluid cells drag one another. Fast fluid cells tend to
slow down whereas slow fluid cells accelerate. This has the effect of
dampening the expansion and produces a progressive broadening of the
$\displaystyle G_{2}$ correlator with time. The longer the system lives, the
longer viscous effects play a role, and the broader the $\displaystyle G_{2}$
correlator becomes [197]. It means that one can get the difference in the
variance of the correlator among different collision centralities. For the
Gavin ansatz, one can write it as,

$\displaystyle\displaystyle\sigma_{c}^{2}-\sigma_{0}^{2}=\frac{4}{T_{c}}\frac{\eta}{s}\Big{(}\frac{1}{{\tau}_{0}}-\frac{1}{\tau_{c,f}}\Big{)}$
(5.1)

where $\displaystyle\sigma_{c}$ is the longitudinal width of the correlator
measured in most central collisions whereas $\displaystyle\sigma_{0}$ is the
longitudinal width of the correlator at formation time $\displaystyle\tau_{0}$
which is set 1.0$\displaystyle\pm$0.5 fm/c [208]. The critical temperature
$\displaystyle T_{c}$ is used at 160 MeV$\displaystyle\pm$5 MeV [208].
$\displaystyle\tau_{c,f}$ is the freeze-out time in most central collisions
which can be obtained from two pion Bose-Einstein measurements from AGS to LHC
energies [209]. With Eq. 5.1, it gives one way to compare model simulations
and experimental data. Based on RHIC and LHC data and with Gavin ansatz, the
estimates for specific shear viscosity are given as in table 5.1. Seeing from
table 5.1, $\displaystyle\eta$/s based on ALICE (A Large Ion Collider
Experiment) data exhibits very weak or no dependence on collision centrality
at LHC energy, while estimates obtained from STAR data hint that
$\displaystyle\eta$/s might be a function of collision centrality at top RHIC
energy. However, given the large systematic uncertainties of these data, one
cannot draw a strong conclusion about the dependence on collision centrality.
For the precision of estimates of the dependence of $\displaystyle\eta$/s, one
needs to improve the accuracy of the STAR measurement of $\displaystyle
G_{2}$. Additionally, it can be found that the mean values of
$\displaystyle\eta$/s at some collision centralities are less than the KSS
bound of 1/4$\displaystyle\pi$$\displaystyle\approx$0.08 and it is more
obvious from LHC results than from STAR data. If one neglects the systematic
uncertainties, it could raise the question of how the universal bound of
1/4$\displaystyle\pi$ is for QGP matter.

### 5.3 Magnetic field effect on ratio of shear viscosity to entropy density

In noncentral relativistic heavy-ion collisions, an extremely strong magnetic
field up to $\displaystyle\sim$$\displaystyle 10^{18}$ G can be created [210,
211, 212, 213]. As in Pb$\displaystyle+$Pb collisions at LHC, the computation
shows that the strength of the magnetic field can reach
eB$\displaystyle\sim$15$\displaystyle m_{\pi}^{2}$ [210], where $\displaystyle
m_{\pi}$=140 MeV is the mass of pion. With such a strong electromagnetic
field, it could induce some anomalous transport phenomena in hot quantum
chromodynamic (QCD) matter [214, 215], such as the chiral magnetic effect
(CME) [216, 217], chiral separation effect (CSE) [218], chiral electric
separation effect (CESE) [219], chiral magnetic waves (CMW) [220], chiral
vortical effect (CVE) [221] and particle polarization [222]. The magnetic
field created in noncentral collisions is perpendicular to the reaction plane
and Lorentz force can vanish in the direction parallel to the field. Lorentz
force lay down in the reaction plane which makes azimuthal anisotropy for
transverse flow of QGP [223]. And also, it would significantly influence on
the properties of strange quark matter. With strong magnetic field, the
condensates for light quarks/antiquarks (as u and d quark) degree of freedom
are affected [224]. If the condensates increase with the magnetic field at low
temperatures, it refers to magnetic catalysis (MC), and decrease for
temperatures close to TPC which refers to inverse magnetic catalysis (IMC)
[214, 225]. These effects would change the masses of the $\displaystyle u$ and
$\displaystyle d$ quarks which in turn play an important role to the transport
feature [226].

There is lots of work to investigate the influence of the magnetic field on
the QCD phase diagram. And also, the impact of magnetic field on the shear
viscosity of quark matter has been investigated in Refs. [223, 226, 227, 228].
The simulations are in the framework of
Nambu$\displaystyle-$Jona$\displaystyle-$Lasinio model (NJL) where
magnetohydrodynamics is considered. Under the relaxation time approximation,
one can obtain shear viscosity for the QGP system [226, 227],

$\displaystyle\displaystyle\eta=\frac{1}{15T}\sum_{a}{\rm
g_{a}}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{p^{4}}{E_{a}^{2}}\tau_{a}f_{a}^{0}(1-f_{a}^{0})$
(5.2)

where ‘a’ is index for quark and antiquark, $\displaystyle\tau_{a}^{c}$ is the
relaxation time of the quark generated by the collision mechanism and
$\displaystyle\rm g_{a}$ is degeneracy factor for quark and antiquark which is
equal to $\displaystyle N_{s}\times N_{f}\times
N_{c}$=2$\displaystyle\times$2$\displaystyle\times$3 where $\displaystyle
N_{s}$ is spin freedom; $\displaystyle N_{f}$ is flavor freedom [u and d, for
SU(2)]; $\displaystyle N_{c}$ is color freedom [226, 227, 229]. Here only two-
flavor (isospin-symmetric) quark matter is considered for a system medium
mainly with u and d quarks and their antiquarks. In Eq. (5.2), $\displaystyle
f^{0}$ is the equilibrium distribution function, which is written as covariant
form [226],

$\displaystyle\displaystyle f^{0}(x,p)=\frac{1}{{\rm
exp}\Big{[}\frac{u_{\mu}p^{\mu}\mp\mu}{T}\Big{]}+1}.$ (5.3)

where $\displaystyle u_{\mu}(x)=\gamma_{\mu}(1,\textbf{u}(x))$; $\displaystyle
u_{\mu}(x)$ is the four velocities of the fluid. The symbol $\displaystyle\mp$
is for quark (‘$\displaystyle-$’) and antiquark (‘$\displaystyle+$’). With
magnetic field, the shear viscosity coefficient of the dissipative fluid
system can be decomposed into five different components (n=0,1,2,3,4) [226].
The $\displaystyle n=0$ component remains undisturbed by magnetic field. The
other four components can be expressed [226],

$\displaystyle\displaystyle\eta_{2}=4\eta_{1}=\frac{1}{15T}\sum_{a}{\rm
g_{a}}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{p^{4}}{E_{a}^{2}}f_{a}^{0}(1-f_{a}^{0})\frac{(\tau_{a}^{B})^{2}}{\tau_{a}^{c}}$
(5.4)

and

$\displaystyle\displaystyle\eta_{4}=2\eta_{3}=\frac{1}{15T}\sum_{a}{\rm
g_{a}}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{p^{4}}{E_{a}^{2}}f_{a}^{0}(1-f_{a}^{0})\tau_{a}^{B}$
(5.5)

where $\displaystyle\tau_{a}^{B}$ is the relaxation time induced by the
magnetic field.

Figure 5.2: (Color online) The ratio of shear viscosity to entropy density
$\displaystyle\eta/s$ (a) and relaxation time $\displaystyle\tau_{c}$ (b) as
functions of temperature at eB=0 and eB=10 $\displaystyle m_{\pi}^{2}$ [227].

As shown in Fig. 5.2, without magnetic field, specific shear viscosity
decreases as temperature increases in the range of 100$\displaystyle-$300 MeV.
Here at eB=0, due to the increase in collisional frequency with temperature,
relaxation time decreases with T. Being proportional with the decreasing
function $\displaystyle\tau_{c}$ for eB=0, $\displaystyle\eta/s$ decreases