arXiv:1106.4210v1 [hep-th] 21 Jun 2011
Extraction of shear viscosity in stationary states of relativistic
F. Reining,1I. Bouras,1A. El,1C. Wesp,1Z. Xu,1,2and C. Greiner1
1Institut f¨ ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨ at,
Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany
2Frankfurt Institute for Advanced Studies,
Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany
(Dated: June 22, 2011)
Starting from a classical picture of shear viscosity we construct a stationary veloc-
ity gradient in a microscopic parton cascade. Employing the Navier-Stokes ansatz we
extract the shear viscosity coefficient η. For elastic isotropic scatterings we find an
excellent agreement with the analytic values. This confirms the applicability of this
method. Furthermore for both elastic and inelastic scatterings with pQCD based
cross sections we extract the shear viscosity coefficient η for a pure gluonic system
and find a good agreement with already published calculations.
PACS numbers: 47.75.+f, 12.38.Mh, 25.75.-q, 66.20.-d
Recent results of the Relativistic Heavy Ion Collider (RHIC) and of the Large Hadron
Collider (LHC) indicate the formation of a new state of matter, the quark-gluon plasma
(QGP), in relativistic heavy-ion collisions. The large value of the elliptic flow coefficient
v2observed in these experiments [1–4] leads to the indication that the QGP behaves like
a nearly perfect fluid. This has been confirmed by calculations of viscous hydrodynamics
[5–11] and microscopic transport calculations [12, 13]. However, the shear viscosity coeffi-
cient η has a finite value, possibly close to the conjectured lower bound η/s = 1/4π from
the correspondence between conformal field theory and string theory in an Anti-de-Sitter
space . In comparison to ideal hydrodynamic calculations , dissipative hydrodynamic
formalisms with finite η/s ratio [5–11] demonstrate a better agreement of the differential
elliptic flow v2(pt) with experimental data. The shear viscosity is therefore an important
parameter in viscous hydrodynamics but needs to be calculated from microscopic theory.
The η/s ratio was obtained in a full leading order pertubative QCD calculation in
Ref. . The Boltzmann-Vlasov equation and quasi-particle picture were recently em-
ployed to calculate the η/s ratio of a gluon gas in Ref. . The shear viscosity coefficient
has also been extracted from microscopic transport calculations with BAMPS (Boltzmann
Approach of Multi Parton Scatterings) simulations [18, 19] using expressions based on a
first-order gradient expansion of the Boltzmann Equation  and the entropy principle
underlying the second-order Israel-Stewart hydrodynamics .
The goal of this work is to extract the shear viscosity coefficient η numerically from
microscopic calculations using a standard setup motivated by the classical textbook picture
[22, 23]. In Fig. 1 we introduce a particle system embedded between two plates. The two
FIG. 1: The classical definition of shear viscosity. Two plates moving in opposite directions with
velocity ±vwall. A flow gradient is established between the plates. The viscosity is proportional to
the frictional force.
plates move in opposite directions each with velocity vwallin z-direction. The moving walls
are supplemented by two thermal reservoirs with ±vwall. In x-direction the system has an
extension of size L. In y- and z-direction the system is homogeneous and can be of infinite
size. The mean free path of the particles should be very small compared to the system size,
i.e. λmfp<< L. On a sufficiently long time-scale a stationary velocity field vz(x) should be
established. In the non-relativistic limit the velocity field is linear. With the Navier-Stokes-
ansatz the shear stress tensor πµνis proportional to the gradient of the velocity
The proportionality factor is defined to be the shear viscosity coefficient η.In Sec. II
we give basic definitions and information on the numerical model we use. In Sec. III we
demonstrate that Eq. (1) does not hold in general for the relativistic case, where the gradient
is not necessarily linear and we discuss the shape of an ideal relativistic velocity gradient.
Furthermore we will discuss the effect of viscosity and finite size effects on the velocity profile
in Sec. III, where an analytical formulation for the shape of the velocity profile is derived. We
employ BAMPS to reproduce the velocity gradient as discussed in this chapter. In Sec. IV
we compare the numerical results for the shear viscosity coefficient η to an analytical value
in order to confirm the applicability of our method. Finally we present the results on
shear viscosity to entropy density ratio obtained from BAMPS with cross sections based on
pertubative quantum chromodynamics (pQCD) and compare them to existing calculations.
We close with a summary.
II. BASIC IDEA AND DEFINITIONS
When systems are in stationary states, the first-order Navier-Stokes formulation of rela-
tivistic viscous hydrodynamics can be used to calculate the shear viscosity η, which is the
proportionality factor between the shear tensor πµν= T?µν?and the velocity gradient ∇?µuν?:
where the projection
denotes the symmetric traceless part of the tensor Bµν. ∆µν= gµν− uµuνis the transverse
projection operator and the metric is gµν= diag(1,−1,−1,−1).
Some definitions are in order. We use the Landau’s definition of the hydrodynamic four-
is the energy-momentum tensor and the local energy density is defined as
e = uµTµνuν. (6)
The shear tensor πµνis the difference of Tµνto its equilibrium value. For the geometry
depicted in Fig. 1 uµ= γ(1,0,0,vz) with γ = 1/?1 − v2
We will build up stationary states of particle systems via numerical simulations, which
are realized by employing the microscopic transport model BAMPS, which solves the Boltz-
mann equations for on-shell particles within a stochastic model [18, 19]. In principle any
microscopic transport model can be used for this purpose.
Local values of πµνand uµcan be easily extracted from the numerical simulations by
averaging over all particles contained in a bin of size ∆x. However, to obtain the gradient
of uµone has to take values from neighbouring local cells, which would cause additional
numerical errors. To avoid such numerical problem we will first derive the analytical form
of vz(x) for the given setup in Fig. 1. Then we use this form and the numerically extracted
πµνto calculate the shear viscosity.
III.VELOCITY, RAPIDITY AND FINITE SIZE EFFECT
Instead of the hydrodynamic velocity vz(x) we address the position dependence of the
rapidity y(x), which is defined by
2ln1 + vz(x)
1 − vz(x).(7)
Thus, vz(x) = tanhy(x). In the non-relativistic limit, where vz(x) is small, we have vz(x) ≈
y(x). The advantage of y(x) is that it gets a shift by a Lorentz-boost e.g. with vz(xA)
Λvz(xA)[y(x)] = y(x) − y(xA).(8)
Demanding boost-invariance, i.e., Λvz(xA)[y(x)] = y(x − xA), we obtain the solution y(x) =
ax + b, where a und b are constant. Due to the boundary condition y(x = ±L/2) = ±ywall,
y(x) is symmetric in x and thus, b = 0. If y(x) is continuous at the boundaries, we have
In the following we will convince ourselves from relativistic kinetic theory that Eq. (9)
is only valid if the particle mean free path vanishes, or the distance L between two plates
is infinitely long. For a non-vanishing mean free path and a finite distance L we will see
discontinuities of y(x) at the boundaries. This is referred to as a finite size effect.
We consider a general local observable A(x,t) with the definition
where dΓ1 = d3p1/(2π)3and n(x,t) =
?dΓ1f(p1;x,t) is the particle number density. p
denotes the particle four-momentum. In our case n does not depend on position and time.
In particular, for FA(p1) = npµ
1we have the definition of particle four-flow A(x,t) =
Nµ(x,t); for FA(p1) =
1)] we obtain the rapidity A(x,t) = y(x,t) as
given in Eq. (7), when using the Landau definition of the hydrodynamic four-velocity. For
stationary states A(x,t) and the particle distribution function f(p;x,t) are constant in time.
We define˜f(p;x,t) = f(p;x,t)/n(x,t), which is the probability density for the occurrence
of a particle with momentum p around dΓ at (x,t). One obtains˜f(p;x,t) by summing
probabilities for such events that a collision at (x′,t′) makes a particle having the momentum
p and this particle travels to x at t without further collisions. It is mathematically expressed
˜f(p1;x,t) = θ(p1x)
where wgain(p1;x′,t′) denotes the probability density that a particle with momentum p1is
created via a collision at (x′,t′), and wfree(p1;x′,t′;x,t) the probability that this particle
travels from (x′,t′) to (x,t) without further collisions. Because˜f(p;x,t) is invariant under
the transformation p → −p, the two integrals in Eq. (11) are equal. Thus,
Our goal is to find the relation between A(x,t) and A(x′,t′), which then can be used to solve
A(x,t) analytically when the boundary conditions are given.
Using the standard definition of cross section for binary collisions of identical particles
2− p1− p2),(13)
where M1′2′→12is the matrix element and s = (p1+ p2)2= (p′
2)2is the invariant mass,
vrel = s/(2p′0
2) denotes the relative velocity for massless particles. dt′is the average
time interval, during which a particle travels through dx′: dt′= dx′< |p′
The probability wfree(p1;x′,t′;x,t) is a product of wfree(p1;x′′,t′′;x′′+dx′′,t′′+dt′′) over
0>= 1/2 in thermal equilibrium.
x′′from x′to x:
wloss(p1;x′′,t′′) denotes the probability density that a particle with momentum p1is destroyed
via a collision at (x′′,t′′) and is expressed by
where vrel= s/(2p0
2) and dt′′= dx′′(px
We now approximate wloss(p1;x′′,t′′) to be the averaged one over p1:
where λmfpdenotes the particle mean free path. This approximation applies for isotropic
cross sections. In general, if the angular distribution is non-isotropic λmfphas to be replaced
by an effective length scale, which is calculated as an average of the differential cross section.
With Eq. (17) we obtain the obvious expression
wfree(p1;x′,t′;x,t) = lim
−2|x − x′|
Putting Eqs. (12), (14), and (18) into Eq. (10) gives
It is clear that replacing FA(p1) by FA(p1)+ FA(p2) will leads to 2A(x,t). We now consider
particular observables A(x,t) such that FA is conserved in each collision, i.e., FA(p′
2) = FA(p1) + FA(p2). We then have
The same approximation is made as for wlossin Eq. (17). Equation (20) resembles the one
derived in Ref.  using ”path integral method” in non-relativistic cases.
We emphasize that Eq. (20) holds only if the total FA is conserved in collisions. For
instance, the total particle velocity p1/E1+ p2/E2 is not conserved except in case the
energy of all particles is same, whereas the total particle momentum rapidity is conserved.
Therefore, the rapidity y(x) defined by Eq. (7) obeys Eq. (20), but the hydrodynamic velovity
vz(x) does not. However, the total particle momentum rapidity is not conserved in 2 → 3
or 3 → 2 processes. In this case one has to take detailed balance into account and the sum
of the total rapidity of a 2 → 3 and its back reaction is conserved on average. If y(x) is
conserved in collisional processes, it obeys Eq. (20).
Equation (20) represents a homogeneous first-order integral equation for A(x). It can
easily be shown that the second derivative of A(x) vanishes, which leads to the solution
A(x) = ax + b, where a and b are constant. We choose the boundary conditions to be
A(x) = −ywall for x < −L/2 and A(x) = ywall for x > L/2 to reproduce the scenario
indroduced in Sec. I. Since this scenario is symetric in x we have b = 0. To determine a we
insert A(x) = ax into Eq. (20) and obtain a = 2ywall/(L + λmfp). Finally the rapidity has
the following form
L + λmfp
We recognize the discontinuities of y(x) at the boundaries, which disappear only for vanishing
mean free path λmfp→ 0 or long distance L → ∞. Equation (21) is a new finding and
accounts for finite size effects which must be taken into accout, if for numerical reasons
λmfp/L cannot be made sufficiently small.
In this subsection we will confirm our finding Eq. (21) by performing numerical transport
calculations. We employ the parton cascade BAMPS. Details of numerical operations can
be found in Refs. [18, 19]. One important feature of BAMPS is that the model can simulate
multiplication and annihilation processes such as the gluon bremsstrahlung process and its
back reaction gg ↔ ggg with full detailed balance. In order to verify the analytic findings
we will first employ isotropic cross sections in BAMPS in the following.
The numerical realization of the boundary conditions is as follows. Particles that reach
the boundaries x = ±L/2 are removed, which simulates the particle absorption by the
plates. Independent of the absorption, the plates emit particles, which pick up the velocities
±vwall of the plates. Here we treat the plates as thermal reservoirs of particles with the
same temperature as those between the plates. The momentum distribution for emitting
particles is proportional to the equilibrium Boltzmann distribution fwall(p) and the particle
fwall(p) = g e−
wall= γwall(1,0,0,vwall), γwall= 1/?1 − v2
gluons in SU(3), and T is the temperature. In the distribution (23) we neglect the quantum
wall, g = 16 is the degeneracy factor for
statistic factor for bosons and fermions. The rate of emissions can be calculated analytically
(see App. A) and is
where Awallis the transverse area of the plates and nwallis the particle density. In the xy-
and xz-plane the boundary conditions are periodic.
Particles between the two plates are initially distributed by the equilibrium form like
Eq. (23) with zero velocity. Figure 2 shows the buildup of the rapidity profile. We have
chosen L = 2 fm, vwall= 0.5 (ywall= 0.55), and T = 0.4 GeV. Only binary collisions with
a constant cross section are considered. The mean free path is set to be λmfp = 0.2 fm.
Collision angles of the binary scatterings are distributed isotropically.
The timescale for the buildup of the rapidity (or velocity) profile can be estimated as the
mean diffusion time of particles traveling from one plate to another. For a gaussian diffusion
-1 -0.5 0 0.5 1
BAMPS t= 0.0fm
BAMPS t= 2.0fm
BAMPS t= 4.0fm
BAMPS t= 8.0fm
BAMPS t= 16.0fm
BAMPS t= 20.0fm
FIG. 2: Build-up with time of the rapidity profile. Results are obtained by averaging 500 events.
process one has 
< x2>= 2Dt, (25)
where in the non-relativistic limit the diffusion constant D is the ratio of the shear viscosity
η to the mass density ρ. For relativistic case we replace ρ by the energy density e. As we
will see in the next section, η ≈ 1.2654nTλmfp= 0.42eλmfp(see Eq. (29)), where e = 3nT
is used. Thus,
For our setup we find t ≈ 24 fm/c, which is consistent with the numerical results shown in
Figure 3 shows the final rapidity profiles at sufficient long times. Calculations are per-
formed for several mean free paths, in order to demonstrate the finite size effect. The
differential cross sections are momentum-independent, i.e. isotropic. The lines present the
analytical results given via Eq. (21), while the symbols show the numerical values. One
can see an excellent agreement, although approximations are made to obtain Eq. (21). This
indicates the validity of the approximations for using isotropic cross sections.
On the contrary, when using the pQCD cross sections for gluons, which strongly depend
on the invariant mass s one will have deviations from Eq. (21). The elastic and gluon
bremsstrahlung process and its back reaction implemented in BAMPS are based on pQCD
matrix elements given in Ref. . Although the numerically extracted rapidity profile is
different from the analytical form Eq. (21), it is still linear in x. Replacing λmfpin Eq. (21)
with an effective scale λeffone obtains the general formula. Using pQCD cross sections λeff
-1-0.5 0 0.5 1
BAMPS λ= 0.01fm
BAMPS λ= 0.1fm
BAMPS λ= 0.3fm
BAMPS λ= 1fm
FIG. 3: Rapidity profiles for different mean free paths, λmfp= 0.02,0.2,2,∞ fm. Constant cross
sections and isotropic distribution of the collision angle are considered. The numerical results from
BAMPS (symbols) are compared with the analytical ones (lines) given by Eq. (21).
has to be extracted numerically. Qualitatively, λefffor pQCD interactions should be larger
than λmfp, since pQCD-based processes prefer small angle scatterings and thus, are not as
efficient for momentum transport as scatterings with isotropic angular distribution.
IV. EXTRACTION OF SHEAR VISCOSITY
In stationary states we can use the Navier-Stockes’s formula Eq. (2) to calculate the shear
viscosity η. For the particular setup shown in Fig. 1, Eq. (2) is simplified to
with γ = 1/?1 − v2
z(x). Using Eq. (21) for vz(x) = tanhy(x) we obtain
η = −πxz?
1 − v2
z(x)L + λeff
(Here λmfpreplaced by λeff.) πxzand vz(x) are extracted from BAMPS (the results of vz(x)
are already shown in the previous section), whereas λeffis obtained by fitting y(x).
Results for isotropic constant cross sections are presented in Sec. IVA. Section IVB
contains results for full pQCD interactions.
A.Elastic isotropic constant cross sections
Elastic isotropic constant cross sections are meant that cross sections for elastic binary
collisions are constant and the distribution of collision angle is isotropic. In this case the
shear viscosity of an ultrarelativistic Maxwell-Boltzmann gas is well known :
σ= 1.2654nT λmfp. (29)
Equation (29) serves as a benchmark to check the numerical methods applied to calculate
Setups for numerical calculations are L = 2 fm, vwall= 0.5, and T = 0.4 GeV. Results
are averaged over Nevents= 2000 events. Figure 4 shows the numerically extracted shear
viscosity from Eq. (28) in each bin of size ∆x = 0.2 fm. The mean value is in good agreement
FIG. 4: Shear viscosity extracted from BAMPS and compared to the analytical result from Eq. (29)
for λmfp= 0.02 fm.
with the analytical one from Eq. (29) within the standard deviation, which decreases with
Of couse this method for the extraction of shear viscosity can only be applied, when the
particle system has relaxed to a stationary state. The relaxation time can be estimated
according to Eq. (26). The relaxation time is inversely proportional to the mean free path
and thus the extraction of shear viscosity for nearly perfect fluids with high Neventsis time-
In Fig. 5 we show the mean shear viscosity with the standard deviation as a function of
mean free path. The agreement with the analytical results (line) is perfect. This confirms the
0.01 0.1 1
FIG. 5: Shear viscosity as a function of mean free path.
proposed method for extracting the shear viscosity for relativistic systems from numerical
B. pQCD interactions
In this subsection the results on the shear viscosity are presented for a system of gluons.
For gluon interactions elastic gg → gg and inelastic gg ↔ ggg in leading-order pQCD based
processes are included. For a more detailed discussion refere to Ref. .
Setups for this case are L = 40 fm, vwall= 0.5, and T = 0.4 GeV. L has to be chosen
appropriately as the mean free path increases with decreasing coupling constant αs. Running
coupling is not implemented in the presented BAMPS calculations.
The extracted mean values of the shear viscosity to entropy ratio η/s are given in Table I
and also shown in Fig. 7 with the resulting standard deviations of the simulations. The
entropy density is taken by its equilibrium value s = 4n.
TABLE I: η/s at various αs.
0.010.03 0.10.20.3 0.5 0.6
192.5 ± 23 32.6 ± 3.49 5.76 ± 0.63 2.25 ± 0.2 1.35 ± 0.14 0.64 ± 0.064 0.55 ± 0.06
η/s2→2,2↔3 43.6 ± 5.2 8.22 ± 0.66 0.87 ± 0.09 0.26 ± 0.03 0.17 ± 0.01 0.1 ± 0.01 0.08 ± 0.01
0.01 0.03 0.1 0.2 0.3 0.4 0.6
pQCD 2->2; 2<->3
FIG. 6: Shear viscosity to entropy density ratio at various αs.
to entropy density
pQCD coupling constant
0.01 0.11 1
0.01 0.11 1
Xu [PRL (2008) 100:172301]
El [PRC (2009) 79044914]
FIG. 7: Shear viscosity to entropy density ratio at various αs. Comparisons with other calculations
using the same matrix elements for gluon interactions are made (more in text).
The new results are compared to values by Xu et al. , El et al.  and Wesp et al.
. Xu et al. identified the shear viscosity coeffient from the Navier-Stokes equation and
used a gradient expansion in the Boltzmann equation. Calculating the second moment of the
Boltzmann equation they obtained the shear viscosity coefficient in terms of the transport
collision rate defined in Ref. . El et al. derive the shear viscosity coefficient from the
entropy principle, which also can be applied to derive the Israel-Stewart equations. For
their derivation El et al. used Grad’s approximation for the off-equilibrium distribution
function and obtained an expression for the shear viscosity similar to the one introduced by
Xu et al., but with a sligthly different definition of transport collision rate. The results from
Wesp et al.  originate from equilibrium fluctuations. Here the Green-Kubo relations are
employed to extract the shear viscosity. We see a good agreement with all three calculations
for αs > 0.2. For αs < 0.2 our data is in exellent agreement with the data from El et
al. as well es Wesp et al. We observe the 1/(α2
slog(1/αs)) scaling behavior expected from
Ref. [16, 27].
Applicability of the methods by Xu et al. and El et al. crucially depends on the chosen
parametrisations for the off-equilibrium distibution functions. In particular the momentum
dependence of the off-equilibrium correction to the equilibrium distribution was chosen dif-
ferently by these authors, which might explain the deviations between their results (Compare
also the discussion in Ref. .)
V.CONCLUSIONS AND OUTLOOK
In this work we started with the classical picture of shear viscosity and shear flow. We
demonstrated that the classical picture of a linear velocity field does not apply to relativistic
systems. Rather, we found that velocity fields have a non-linear form, whereas the rapidity
increases in fact linearly.
We also derived an analytical expression for the rapidity and velocity profiles in systems
where the mean-free path is non-zero. With an increasing mean-free path to system size ratio
the slope of the rapidity profile decreases and finite size effects are not negligible anymore.
We employed the numerical transport model BAMPS to create the velocity and rapidity
profiles, compared the numerical results to our theoretical findings and observed an almost
perfect agreement. The stationary gradient allows us to apply the relativistic Navier-Stokes
equation to calculate the shear viscosity coefficient η. We found again a perfect agreement
to the analytical value derived from kinetic theory . The method proposed here to calcu-
late the shear viscosity coefficient is thus perfectly suitable for other microscopic transport
Furthermore we have then used this setup to calculate the shear viscosity to entropy den-
sity ratio in a numerical simulation with elastic and inelastic pQCD processes implemented
in BAMPS for fixed coupling constant αs, which is varied from = 0.01 to 0.6. We compared
our results with previously published results [20, 21] and also with a very recent work based
on the Kubo relation  and found a very good agreement.
The authors are grateful to the Center for the Scientific Computing (CSC) at Frankfurt
for the computing resources. FR, CW and IB are grateful to “Helmhotz Graduate School
for Heavy Ion research”. AE and FR acknowledge support by BMBF. This work was sup-
ported by the Helmholtz International Center for FAIR within the framework of the LOEWE
program launched by the State of Hesse.
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