# Green-Kubo formula for Boltzmann and Fermi-Dirac statistics

X. G. Deng (邓先概)  Key Laboratory of Nuclear Physics and Ion-beam Application
(MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China
Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai
201800, China  University of Chinese Academy of Sciences, Beijing 100049,
China  Y. G. Ma (马余刚) 111Corresponding author<EMAIL_ADDRESS>Key
Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of
Modern Physics, Fudan University, Shanghai 200433, China  Shanghai Institute
of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China  Y. X.
Zhang (张英逊)  China Institute of Atomic Energy, Beijing 102413, China

###### Abstract

Shear viscosity of nuclear matter is extracted via the Green-Kubo formula and
the Gaussian thermostated SLLOD algorithm (the shear rate method) in a
periodic box by using an improved quantum molecular dynamic (ImQMD) model
without mean field, also it is calculated by a Boltzmann-type equation. Here a
new form of the Green-Kubo formula is put forward in the present work. For
classical limit at nuclear matter densities of $0.4\rho_{0}$ and
$1.0\rho_{0}$, shear viscosity by the traditional and new form of the Green-
Kubo formula as well as the SLLOD algorithm are coincident with each other.
However, for non-classical limit, shear viscosity by the traditional form of
the Green-Kubo formula is higher than those obtained by the new form of the
Green-Kubo formula as well as the SLLOD algorithm especially in low
temperature region. In addition, shear viscosity from the Boltzmann-type
equation is found to be less than that by the Green-Kubo method or the SLLOD
algorithm for both classical and non-classical limits.

##  I Introduction

Shear viscosity is a common transport property for lots of substances, such as
macroscopic matter, eg. water, oil, honey and air as well as microscopic and
quantum matter, eg. hot dense quark matter etc. Rev1 ; Rev2 ; Rev3 ; Gao ;
Tang ; Huang ; Shen ; Cao ; KEE15 , and it has been studied for a few decades
in nuclear physics GB78 ; PD84 ; LS03 ; AM04 ; KSS05 . An interesting behavior
was found that the ratios of shear viscosity over entropy density ($\eta/s$)
for many substances have minimum values at their corresponding critical
temperatures RAL07 . By using string theory method, Kovtun-Son-Starinets found
that the $\eta/s$ has a limiting value of $\hbar/4\pi$, which is called KSS
bound KSS05 . In relativistic heavy ion collisions, experimental data
indicated that the $\eta/s$ of the quark gluon plasma (QGP) is close to the
KSS bound, which means that QGP matter is almost perfect fluid BC05 ; PC05 ;
SC05 ; Heinz ; Song ; Shen ; Reining . So far there are a couple of approaches
to calculate shear viscosity Ma_book . From theoretical viewpoint, one can
obtain shear viscosity from the Chapman-Enskog and the relaxation time
approaches PD84 ; LS03 ; SP12 ; AW12 ; XJ13 . From the transport simulation
viewpoint, one can extract shear viscosity in the periodic box AM04 ; CJ08 ;
JA18 . Also the shear viscosity can be estimated by the mean free path of
nucleons DQ14 ; LiuHL . What’s more, one can extract shear viscosity from the
width and energy of the giant dipole resonance (GDR) NA09 ; ND11 ; GuoCQ ;
DM17 ; GDR2  or fragment production SP10  or fitting formula ZhouCL1 ; DXG16 .

There are two general methods, namely the Green-Kubo formula and SLLOD
algorithm to calculate shear viscosity, which were extensively used for the
molecular dynamics simulations GP05 ; MM12 ; ZY15 ; ZhouCL1 ; GuoCQ . One of
motivations of the present work is to give a new form of the Green-Kubo
formula. By a comparison among the traditional Green-Kubo formula, a new form
of the Green-Kubo formula is presented and its validity of those methods is
discussed by the SLLOD algorithm.

The paper is organized as follows: In Sec. II, we introduce the simulation
model and analysis methods. In Sec. III, we discuss the shear viscosity by
different approaches with and without the Pauli blocking at different
densities. Conclusion is given in Sec. IV.

##  II Nuclear system and analysis methods

###  II.1 ImQMD model

Generally there are two types of transport models, i.e. Boltzmann-Uehling-
Uhlenbeck type and quantum molecular dynamics type for describing heavy-ion
collisions at low and intermediate energies GF88 ; Aichelin ; LiBA ; Ono ;
XuJ2 ; XJ16 ; ZYX18  and had numerous applications LiBA2 ; Colonna ; Bonasera
; MaCW ; LiSX ; HeWB ; Wei ; Zhang ; Yu ; Yan ; WangSS ; HeYJ . In this work,
an improved quantum molecular dynamics (ImQMD) model is utilized ZYX06 . The
potential energy density without spin-orbit term in the ImQMD model reads
ZYX06 ; WN16 :

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

where $\rho,\rho_{n}$ and $\rho_{p}$ are the nucleon, neutron and proton
densities, respectively. Here the saturation density of nuclear matter is
$\rho_{0}\approx 0.16fm^{-3}$. And
$\delta=(\rho_{n}-\rho_{p})/(\rho_{n}+\rho_{p})$ is the isospin asymmetry. For
simplicity, we investigate the shear viscosity of infinite nuclear matter
without mean field. And a periodic box is constructed within the framework of
the ImQMD model as did in Ref. ZYX18 . The conditions for box initialization
are particle number $A$, density $\rho$ and temperature $T_{0}$. For the
simulations, the particle number $A$ is fixed at 600and the total nucleon-
nucleon cross section ($\sigma_{NN}$) is fixed at 40 mb. Then the box size
would be dependent on the nuclear density. Also we just consider the symmetric
nuclear matter (i.e., $\rho_{n}=\rho_{p}$) here. With the simulations by the
ImQMD model, shear viscosity is extracted by different approaches.

###  II.2 Pauli blocking effect on distribution of system

In the present work, we would consider the effects of Pauli blocking on shear
viscosity. However, one thing which is obviously affected by the Pauli
blocking is the momentum distribution of the system as shown in Fig. 1. The
Pauli blocking would determine what kind of distribution for a system, eg.
either Fermi-Dirac distribution or classical one (Boltzmann distribution).

Figure 1: (Color online) Time evolution of momentum distributions at density
1.0$\rho_{0}$ and temperature $T$ = 6 MeV. (a) without and (b) with the Pauli
blocking.

In this figure, we consider the cases without the Pauli blocking in Fig. 1 (a)
as well as with the Pauli blocking in Fig. 1 (b). Here the momenta of
particles are initialized by the Fermi-Dirac equation in a condition with a
given set of density ($\rho$), temperature ($T$) and chemical potential
($\mu$). The Fermi-Dirac distribution has the form of

$\begin{split}f(\epsilon)=\frac{1}{\exp(\frac{\epsilon-\mu}{T})+1},\end{split}$
(2)

where $\epsilon$ = $p^{2}/(2m)$ for non-relativistic case and while $\epsilon$
= $\sqrt{p^{2}+m^{2}}$ for relativistic case. The initial momentum
distributions are also shown with the red lines in these figures. In Fig. 1(a)
without the Pauli blocking, the shape of momentum distribution tends to the
Boltzmann one as time increases. It is worth to mention that in the standard
ImQMD model, the occupation probability in the Pauli blocking algorithm is
calculated by a Wigner density in phase space cell ZYX18 . This method
underestimates the occupation probability in the nuclear matter, and results
in a larger collision rate than the analytically evaluated one. In this work,
we calculate the occupation probability by using the Fermi-Dirac distribution
with the calculated density and temperature at each spatial point. As we
checked, the collision rate becomes reasonable comparing to the analytical
one.

###  II.3 Two forms of the Green-Kubo formula

One of approaches in this work to calculate shear viscosity is the Green-Kubo
formula which can be derived from the linear response theory. This method was
extensively discussed in molecular dynamics simulations GP05 ; MM12 ; ZY15 .
The Green-Kubo formula for the calculation of shear viscosity is by an
integral of the stress autocorrelation function (SACF) RK66 ; DJE08 ; AH84 :

$\begin{split}\eta_{nor}=\frac{V}{T}\int_{t_{0}}^{\infty}C(t)dt,\end{split}$
(3)

where $V$ and $t_{0}$ are system volume and equilibrium time, respectively.
The $\eta$ with subscript in Eq. 3 means a normal (or traditional) form of the
Green-Kubo formula. In last decades, it was extensively used in molecular
dynamics simulations. However, it was mostly used for a classical system, not
for the Fermi-Dirac distribution system. Thus in this work, we re-derive and
give a new form of the Green-Kubo formula for the calculation of shear
viscosity, i.e.

$\begin{split}\eta_{new}=\frac{VNm}{\langle\sum_{i}^{N}p_{ix}^{2}\rangle}\int_{t_{0}}^{\infty}C(t)dt,\end{split}$
(4)

where $N$ is total particle number and $m$ is particle mass. The derivation
can be found in the Appendix [A] and a more general form for shear viscosity
as in Eq. 34 . One can see that the new form the Green-Kubo formula does not
relate to the temperature but instead of the particle momenta. And in both Eq.
3 and Eq. 4, autocorrelation function $C(t)$ keeps the same and it reads,

$\begin{split}C(t)=\langle
P_{\alpha\beta}(t)P_{\alpha\beta}(t_{0})\rangle{\quad}\alpha,\beta=x,y,z,\end{split}$
(5)

where $P_{\alpha\beta}(t)$ is off-diagonal element of stress tensor. The
bracket $\langle...\rangle$ denotes average over equilibrium ensemble (average
by simulation events). Fig. 2 shows the autocorrelation function $C(t)$ as a
function of time for different tensor components at densities of $0.4\rho_{0}$
and $1.0\rho_{0}$ at $T_{0}$ = 30 MeV. Here temperature index $T_{0}$ means
the initial temperature we set for the box system.

Figure 2: (Color online) The evolution of autocorrelation function for
different components at $T_{0}$ = 30 MeV and different densities of
$0.4\rho_{0}$ (a) or $1.0\rho_{0}$ (b) without mean field.

One can see that $C(t)$ tends to zero with increasing time, which indicates
that $C(t)$ is convergent. However, $C(t)$ at the lower density has longer
correlation time in comparison with higher one, as shown in Fig. 2(a) and (b).

The macroscopic momentum flux in the volume $V$ is given by

$\begin{split}P_{\alpha\beta}(t)=\frac{1}{V}\int
d^{3}rP_{\alpha\beta}(\vec{r},t),\end{split}$ (6)

and the local stress tensor is defined as

$\begin{split}&P_{\alpha\beta}(\vec{r},t)=\sum_{i}^{N}\frac{p_{{i}\alpha}p_{{i}\beta}}{m_{i}}\rho_{i}(\vec{r},t)\\\
&+\frac{1}{2}\sum_{i}^{N}\sum_{i\neq
j}^{N}F_{ij\alpha}R_{ij\beta}\rho_{j}(\vec{r},t)\\\
&+\frac{1}{6}\sum_{i}^{N}\sum_{i\neq j}^{N}\sum_{i\neq j\neq
k}^{N}(F_{ijk\alpha}R_{ik\beta}+F_{jik\alpha}R_{jk\beta})\rho_{k}(\vec{r},t)\\\
&+\cdots,\end{split}$ (7)

where $F_{ij}$ and $\vec{R}_{ij}=\vec{r}_{j}-\vec{r}_{i}$ are interaction
force and relative position of particle $i$ and $j$. The first term of right-
hand side is momentum term, the second one is two-body interaction term, and
the third one is three-body interaction term. Actually, the mean field is not
considered for our simulations but we put the interaction terms here. Based on
the ImQMD model, the $i$-th particle density distribution is given by

$\begin{split}\rho_{i}(\vec{r})=\frac{1}{(2\pi\sigma^{2})^{3/2}}\exp[-\frac{(\vec{r}-\vec{r}_{i})^{2}}{2\sigma^{2}}],\end{split}$
(8)

where $\sigma$ is the wave-packet width which is taken as 2.0 $fm$ in the
present work.

###  II.4 The Gaussian thermostated SLLOD algorithm

Another approach is the SLLOD, which was named by Evans and Morriss GP06  and
related to the dynamics with (artificial) strain rate $\dot{\gamma}$,
algorithm for non-equilibrium molecular dynamics (NEMD) calculation and has
been extensively applied to predict the rheological properties of real fluids
GP06 . The SLLOD algorithm for shear viscosity was actually applied to a
planar Couette flow field at shear rate $\dot{\gamma}$ = $\partial
v_{x}/\partial y$ which is the change in streaming velocity $v_{x}$ in the
$x$-direction with vertical position $y$. One can get the shear viscosity at
shear rate

$\begin{split}\eta=-\frac{\langle P_{xy}\rangle}{\dot{\gamma}}.\end{split}$
(9)

With adding shear rate to the system, the dynamical equations of motion of the
system are rewritten as

$\displaystyle\frac{d\vec{r}_{i}}{dt}=\frac{\vec{p}_{i}}{m_{i}}+\dot{\gamma}y_{i}\hat{x}$
(10)
$\displaystyle\frac{d\vec{p}_{i}}{dt}=\vec{F}_{i}-\dot{\gamma}p_{yi}\hat{x}-h\vec{p}_{i}.$
(11)

Eq. 10 and Eq. 11 are called the SLLOD equations. In order to keep the kinetic
energy conservative, one needs a ‘thermostat’. Thus a multiplier is applied to
motion equations. Considering conservation of kinetic energy, one can get

$\begin{split}h=\frac{\sum_{i}(\vec{F}_{i}\cdot\vec{p}_{i}/m_{i}-\dot{\gamma}p_{xi}p_{yi}/m_{i})}{\sum_{i}p_{i}^{2}/m_{i}}.\end{split}$
(12)

One should notice that the shear rate can not be too large or small. If it is
small, flow field can not be implanted to the system.

Figure 3: (Color online) Stress tensor as a function of time at different
temperatures and density of 0.4$\rho_{0}$ without the mean field.

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. As in Fig. 3 shows, shear viscosity decreases with big shear rate
$\dot{\gamma}$ due to the energy loss with big values of $\dot{\gamma}$. So in
our simulations, different temperatures are correspondent to different values
of shear rate. Here $\dot{\gamma}$ = 0.0003 $c/fm$, $\dot{\gamma}$ = 0.0005
$c/fm$ and $\dot{\gamma}$ = 0.002 $c/fm$ are taken for $T_{0}$ = 4 MeV,
$T_{0}$ = 6 MeV and $T_{0}\geq$10 MeV, respectively.

Figure 4: (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
$\rho_{0}$ and 1.0$\rho_{0}$.  Figure 5: (Color online) The respective
relative difference between the shear rate method and the normal Green-Kubo
formula or and the new Green-Kubo formula as a function of temperature.

###  II.5 The linear Boltzmann equation set

How good are they for these methods mentioned above while they are applied to
the nuclear matter? As a comparison, a linear Boltzmann equation set which can
be used to calculate the shear viscosity of uniform nuclear matter is
presented here. One can get the expression by solving Boltzmann equation PD84
; LS03 ; BB19 :

$\begin{split}\eta&=\frac{5T}{9}\frac{\Big{(}\int
d^{3}p{\,\,}p^{2}f(p)\Big{)}^{2}}{\int
d^{3}p_{1}d^{3}p_{2}d\Omega{\,}v_{12}q_{12}^{4}sin^{2}{\theta}\frac{d\sigma_{NN}}{d\Omega}f_{1}f_{2}\tilde{f}_{3}\tilde{f}_{4}},\end{split}$
(13)

where $v_{12}=|\vec{v}_{1}-\vec{v}_{2}|$ and
$q_{12}=|\vec{p}_{1}-\vec{p}_{2}|/2$ are relative velocity and relative
momentum, respectively, $\sigma_{NN}$ is total nucleon-nucleon cross section,
and $\tilde{f}_{3}$$\tilde{f}_{4}$ = [1-$f(p_{3})$][1-$f(p_{4})$] is the Pauli
blocking term. In Eq. 13, the conservation of momentum and energy between
$p_{1},p_{2}$ and $p_{3},p_{4}$ are needed to be taken into account.

##  III Shear viscosity with different approaches

By using different approaches as we mentioned above, as shown in Fig.4, shear
viscosity is calculated at different temperatures and densities with and
without the Pauli blocking. One sees that in Fig. 4(a) and Fig. 4(c) without
the Pauli blocking, shear viscosity increases with increasing temperature at
both densities of $0.4\rho_{0}$ and $1.0\rho_{0}$. As temperature increases,
exchanging of momentum among the particles, a ‘stopping’ effect increases
between two flow layers. That indicates that shear viscosity would increase.
Seeing from Fig. 4 (b) and Fig. 4(d) with the Pauli blocking, unlike Fig. 4(a)
and Fig. 4(c) in the low temperature region, shear viscosity increases with
decreasing 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, also energy and
momentum transport becomes quite efficient YK16 . As the temperature
increases, the Pauli blocking effect reduces, and shear viscosity decreases.
And as temperature increases again, the collision number would increase, so
shear viscosity increases. Comparing among these three approaches, we find
that shear viscosity determined by the Boltzmann type equation is lower than
those which are calculated by the shear rate and the Green-Kubo method. At
very low densities such as 0.05$\rho_{0}$, however, we found that the result
by the Boltzmann type equation (Eq. 13) is consistent with other methods,
which indicates that low density approximation for Eq. 13 could be valid, then
one would expect that shear viscosities determined by the shear rate and the
Green-Kubo approaches are the same. As displayed in Fig. 4 (a) and Fig. 4 (c)
without the Pauli blocking, shear viscosities from the shear rate method (the
SLLOD algorithm), the normal form of the Green-Kubo formula, and the new form
of the Green-Kubo formula are consistent with each other. However, as the
Pauli blocking is taken into account as shown in Fig. 4 (b) and Fig. 4 (d),
one sees that in low temperature region, the shear viscosity from the normal
form of the Green-Kubo formula is higher than those from both the SLLOD
algorithm and the new form of the Green-Kubo formula. However, shear
viscosities from the SLLOD algorithm and the new form of the Green-Kubo
formula are almost the same. The relative difference is shown in Fig. 5. One
can find that there is more different between the normal form of the Green-
Kubo formula and the SLLOD algorithm or the new form of the Green-Kubo formula
at lower temperatures. It is obviously that the standard GK formula leads to
larger shear viscosity in fermionic system especially at low temperatures.

##  IV Conclusions

In summary, shear viscosities are obtained by the SLLOD algorithm and the
Green-Kubo formula in the framework of the ImQMD simulation and are compared
with the Boltzmann equation method in the present work. By comparisons among
different calculation methods, it is found that shear viscosity with the
Boltzmann equation method is less than those from the SLLOD algorithm and the
Green-Kubo formula. More interestingly, a new form of the Green-Kubo formula
(Eq. 4) is presented for shear viscosity calculation. By comparison with the
SLLOD algorithm, we found that the standard GK formula leads to larger shear
viscosity in fermionic systems especially at low temperatures. And the new
form of the Green-Kubo formula for shear viscosity is consistent with the
SLLOD algorithm for both classical and non-classical systems.

###### Acknowledgements.

Thanks for helpful discussions with P. Danielewicz and H. Lin. This work was
partially supported by the National Natural Science Foundation of China under
Contract Nos. 11947217, 11890710 and 11890714, China Postdoctoral Science
Foundation Grant No. 2019M661332, Postdoctoral Innovative Talent Program of
China No. BX20200098, the Strategic Priority Research Program of the CAS under
Grants No. XDB34000000, the Guangdong Major Project of Basic and Applied Basic
Research No. 2020B0301030008.

##  Appendix A The Green-Kubo formula for shear viscosity

The particle density can be written as (for simplicity, here we use
$\delta$-function to replace the Gaussian wave-packet). And the derivation of
the Green-Kubo formula for shear viscosity is tedious. Here we give a simple
introduction which is based on Ref. DJE08 . For more details one can find in
Refs. DJE96 ; DJE08 .

###  A.1 $\vec{\bf{r}}$ and $\vec{\bf{k}}$-space representations

The streaming velocity, mass density, momentum density and stress tensor at
position $\vec{\bf{r}}$ and time $t$ can be written:

$\displaystyle{\bf
u}(\vec{\bf{r}},t)=\frac{\sum_{i}^{N}m_{i}{\dot{\vec{\bf{r}}}_{\it{i}}}\delta(\vec{\bf{r}}-\vec{\bf{r}}_{i})}{\sum_{i}^{N}m_{i}\delta(\vec{\bf{r}}-\vec{\bf{r}}_{i})},$
(14)
$\displaystyle\rho(\vec{\bf{r}},t)=\sum_{i}^{N}m_{i}\delta(\vec{\bf{r}}-\vec{\bf{r}}_{i}),$
(15) $\displaystyle J(\vec{\bf{r}},t)=\rho(\vec{\bf{r}},t){\bf
u}(\vec{\bf{r}},t)=\sum_{i}^{N}m_{i}{\dot{\vec{\bf{r}}}_{\it{i}}}\delta(\vec{\bf{r}}-\vec{\bf{r}}_{i}),$
(16) $\displaystyle
P_{\alpha\beta}(\vec{\bf{r}},t)=\sum_{i}^{N}\frac{p_{{i}\alpha}p_{{i}\beta}}{m_{i}}\delta(\vec{\bf{r}}-\vec{\bf{r}}_{i})$
$\displaystyle+\frac{1}{2}\sum_{i}^{N}\sum_{i\neq
j}^{N}F_{ij\alpha}R_{ij\beta}\delta(\vec{\bf{r}}-\vec{\bf{r}}_{j})$ (17)
$\displaystyle+\cdots,$

where ‘$i$’ and ‘$j$’ are indexes of particles and $N$ is particle number. For
Eq. (A.1), the momentum density conservation law

$\displaystyle\frac{\partial J(\vec{\bf{r}},t)}{\partial t}=-\nabla_{\bf
r}\cdot\vec{P},$ (18)

is needed. Moreover it needs

$\displaystyle\frac{\partial}{\partial\vec{\bf{r}}_{\it{i}}}\delta(\vec{\bf{r}}-\vec{\bf{r}}_{i})=-\frac{\partial}{\partial\vec{\bf{r}}}\delta(\vec{\bf{r}}-\vec{\bf{r}}_{i}),$
(19)
$\displaystyle\delta(\vec{\bf{r}}-\vec{\bf{r}}_{i})-\delta(\vec{\bf{r}}-\vec{\bf{r}}_{j})\approx\vec{R}_{ij}\frac{\partial}{\partial\vec{\bf{r}}}\delta(\vec{\bf{r}}-\vec{\bf{r}}_{j}).$
(20)

One can define the Fourier transform and inverse in three-dimensions by

$\displaystyle f(\vec{\bf{k}})=\int d^{3}{\bf r}f(\vec{\bf{r}}){\rm exp}[{\rm
i}\vec{{\bf k}}\cdot\vec{\bf r}],$ (21) $\displaystyle
f(\vec{\bf{r}})=\frac{1}{(2\pi)^{3}}\int d^{3}{\bf k}f(\vec{\bf{k}}){\rm
exp}[-{\rm i}\vec{\bf{k}}\cdot\vec{\bf{r}}],$ (22)

where ‘${\rm i}$’ is the unit of imaginary. Then the mass density, momentum
density and stress tensor in $\vec{\bf{k}}$-space are given

$\displaystyle\rho(\vec{\bf{k}},t)={\sum_{i}^{N}}{m_{i}}{\rm exp}[{\rm
i}\vec{{\bf k}}\cdot\vec{\bf r}_{i}],$ (23) $\displaystyle
J(\vec{\bf{k}},t)=\sum_{i}^{N}m_{i}{\dot{\vec{\bf{r}}}_{\it{i}}}{\rm exp}[{\rm
i}\vec{{\bf k}}\cdot\vec{\bf r}_{i}],$ (24) $\displaystyle
P_{\alpha\beta}(\vec{\bf{k}},t)=\sum_{i}^{N}\frac{p_{{i}\alpha}p_{{i}\beta}}{m_{i}}{\rm
exp}[{\rm i}\vec{{\bf k}}\cdot\vec{\bf r}_{i}]$ (25)
$\displaystyle+\frac{1}{2}\sum_{i}^{N}\sum_{i\neq
j}^{N}F_{ij\alpha}R_{ij\beta}{\rm exp}[{\rm i}\vec{{\bf k}}\cdot\vec{\bf
r}_{j}]+\cdots\;.$

###  A.2 Shear viscosity and strain rate

The stress tensor corresponds to the strain rate (for simplicity, only the
$x-y$ component is considered) reads

$\begin{split}P_{xy}=-\eta\gamma(t),\end{split}$ (26)

where $\eta$ is static shear viscosity. Also one can see it in Eq. (9) but the
strain rate here is time dependent. The most general linear relation between
the strain rate and the shear stress can be written in the time domain as

$\begin{split}P_{xy}(t)=-\int_{0}^{t}ds\;\eta_{M}(t-s)\gamma(s),\end{split}$
(27)

where $\eta_{M}(t)$ is called the Maxwell memory function. The memory function
explains that the shear stress at time $t$ is not simply linearly proportional
to the strain rate at the current time $t$, but to the entire strain rate
process, over times $0\leqslant s\leqslant t$. For the frequency dependent
Maxwell viscosity is

$\begin{split}\tilde{\eta}_{M}(\omega)=\frac{\eta}{1+{\rm
i}\omega\tau_{M}},\end{split}$ (28)

where $\tau_{M}$ is the Maxwell relaxation time which controls the transition
frequency between low frequency viscous behaviour and high frequency elastic
behavior. In Eq.28, $\tilde{\eta}_{M}$ is the Fourier-Laplace transform which
is read as

$\begin{split}\tilde{\eta}(\omega)=\int_{0}^{\infty}dt\;{\rm exp}[-{\rm
i}\omega t]\eta(t).\end{split}$ (29)

###  A.3 Shear viscosity of the Green-Kubo formula

We separate vector-dependent momentum density into longitudinal (${\bf
J}^{||}$) and transverse (${\bf J}^{\bot}$) parts. Considering a transverse
momentum density ${\bf J}^{\bot}$($\vec{\bf k}$,t), for simplicity, we define
the coordinate system in which $\vec{\bf k}$ is in $y$-direction and ${\bf
J}^{\bot}$ is in the $x$-direction:

$\displaystyle J_{x}(k_{y},t)=\sum_{i}mv_{xi}(t){\rm exp}[{\rm
i}k_{y}y_{i}(t)].$ (30)

According to Eq. (25), one can get

$\displaystyle\dot{J}_{x}(k_{y},t)={\rm i}k_{y}P_{xy}(k_{y},t).$ (31)

In Ref. DJE08 , by Mori-Zwanzig formalism, one can get the shear viscosity
$\eta(t)$ which is time-dependent, i.e.

$\displaystyle\eta(t)=\frac{VNm}{\langle
J_{x}(k_{y})J_{x}^{\ast}(k_{y})\rangle}\langle P_{xy}(t)P_{xy}(0)\rangle.$
(32)

By the Fourier-Laplace transform of $\eta(t)$, one gets

$\displaystyle\tilde{\eta}(\omega)=\frac{VNm}{\langle
J_{x}(k_{y}=0)J_{x}^{\ast}(k_{y}=0)\rangle}$ (33)
$\displaystyle\times\int_{0}^{\infty}\langle P_{xy}(t)P_{xy}(0)\rangle{\rm
exp}[-{\rm i}\omega t]dt.$

As in Eq. (28), static shear viscosity needs $\omega\rightarrow$0\. Then one
gets

$\displaystyle\eta=\frac{VNm}{\langle
J_{x}(k_{y}=0)J_{x}^{\ast}(k_{y}=0)\rangle}\int_{0}^{\infty}\langle
P_{xy}(t)P_{xy}(0)\rangle dt.$ (34)

Here it should be noticed

$\displaystyle P_{xy}(t)=\lim_{k_{y}\rightarrow 0}\frac{P_{xy}(k_{y},t)}{V},$
(35) $\displaystyle P_{xy}(0)=\lim_{k_{y}\rightarrow
0}\frac{P_{xy}(k_{y},0)}{V},$ (36)

where $V$ is system volume. For the norm of the transverse current, one can
get

$\displaystyle\langle J_{x}(k_{y}=0)J_{x}^{\ast}(k_{y}=0)\rangle$ (37)
$\displaystyle=\langle\sum_{i}^{N}p_{xi}\sum_{j}^{N}p_{xj}\rangle$
$\displaystyle=\langle\sum_{i}^{N}p_{xi}^{2}\rangle+N(N-1)\langle
p_{1x}p_{2x}\rangle.$

At equilibrium, $p_{1x}$ is independent of $p_{2x}$, so the second term of
right-hand side of Eq. (37) is zero. Then we can obtain a new form of the
Green-Kubo formula for shear viscosity

$\displaystyle\eta_{new}=\frac{VNm}{\langle\sum_{i}^{N}p_{xi}^{2}\rangle}\int_{0}^{\infty}\langle
P_{xy}(t)P_{xy}(0)\rangle dt,$ (38)

where $\langle\cdots\rangle$ denotes ensemble average. For an equilibrium
system which obeys the Boltzmann distribution, one can get

$\displaystyle\langle\sum_{i}^{N}p_{xi}^{2}\rangle=\langle\frac{1}{3}\sum_{i}^{N}p_{i}^{2}\rangle=Nmk_{B}T,$
(39)

where $T$ is temperature and $k_{B}$ ($k_{B}$ = 1) is the Boltzmann constant.
Then the normal Green-Kubo formula for shear viscosity can be given

$\displaystyle\eta_{nor}=\frac{V}{T}\int_{0}^{\infty}\langle
P_{xy}(t)P_{xy}(0)\rangle dt.$ (40)

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