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arXiv:1106.4210v1 [hep-th] 21 Jun 2011

Extraction of shear viscosity in stationary states of relativistic

particle systems

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

I.INTRODUCTION

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 [14]. In comparison to ideal hydrodynamic calculations [15], 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

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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. [16]. The Boltzmann-Vlasov equation and quasi-particle picture were recently em-

ployed to calculate the η/s ratio of a gluon gas in Ref. [17]. 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 [20] and the entropy principle

underlying the second-order Israel-Stewart hydrodynamics [21].

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

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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

πxz= −η∂vz(x)

∂x

. (1)

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?:

πµν= 2η∇<µuν>,(2)

where the projection

B?µν?≡

?1

2

?∆µ

α∆ν

β+ ∆ν

α∆µ

β

?−1

3∆µν∆αβ

?

Bαβ

(3)

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-

velocity [25]:

uµ=Tµνuν

e

,(4)

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where

Tµν=

?

d3p

(2π)3p0pµpνf(x,p) (5)

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

z.

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

A.Analytical Derivation

Instead of the hydrodynamic velocity vz(x) we address the position dependence of the

rapidity y(x), which is defined by

y(x) =1

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,

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y(x) is symmetric in x and thus, b = 0. If y(x) is continuous at the boundaries, we have

y(x) =2ywall

L

x. (9)

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

A(x,t) =

1

n(x,t)

?

dΓ1FA(p1)f(p1;x,t),(10)

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

1/p0

1we have the definition of particle four-flow A(x,t) =

Nµ(x,t); for FA(p1) =

1

2ln[(p0

1+ pz

1)/(p0

1− pz

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

by

˜f(p1;x,t) = θ(p1x)

?x

−∞

?∞

dx′wgain(p1;x′,t′)wfree(p1;x′,t′;x,t) +

θ(−p1x)

x

dx′wgain(p1;x′,t′)wfree(p1;x′,t′;x,t),(11)

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,

˜f(p1;x,t) =1

2

−∞

?∞

dx′wgain(p1;x′,t′)wfree(p1;x′,t′;x,t).(12)

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.

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Using the standard definition of cross section for binary collisions of identical particles

σ22=

1

4s

?

dΓ1

2p0

1

dΓ2

2p0

2

|M1′2′→12|2(2π)4δ(4)(p′

1+ p′

2− p1− p2),(13)

where M1′2′→12is the matrix element and s = (p1+ p2)2= (p′

1+ p′

2)2is the invariant mass,

we have

wgain(p1;x′,t′)dx′=1

n

?

dΓ′

1dΓ′

2f(p′

1;x′,t′)f(p′

2;x′,t′)vreldσ22

dΓ1dt′.(14)

vrel = s/(2p′0

1p′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′

< |p′

The probability wfree(p1;x′,t′;x,t) is a product of wfree(p1;x′′,t′′;x′′+dx′′,t′′+dt′′) over

x|/p′

0>−1and

x|/p′

0>= 1/2 in thermal equilibrium.

x′′from x′to x:

wfree(p1;x′,t′;x,t) =

x?

x′′=x′

wfree(p1;x′′,t′′;x′′+dx′′,t′′+dt′′) =

x?

x′′=x′

[1−wloss(p1;x′′,t′′)dx′′].

(15)

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

wloss(p1;x′′,t′′)dx′′=

?

dΓ2f(p2;x′′,t′′)vrelσ22dt′′,(16)

where vrel= s/(2p0

1p0

2) and dt′′= dx′′(px

1/p0

1)−1.

We now approximate wloss(p1;x′′,t′′) to be the averaged one over p1:

wloss(p1;x′′,t′′)dx′′≈

?

dΓ2f(p2;x′′,t′′)?vrelσ22??|px

1|

1

p0

?−1dx′′= 2n?vrelσ22?dx′′=2dx′′

λmfp

,

(17)

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

dx′′→0

?

1 −2dx′′

λmfp

?|x−x′|/dx′′

= exp

?

−2|x − x′|

λmfp

?

.(18)

Putting Eqs. (12), (14), and (18) into Eq. (10) gives

A(x,t) =

?∞

−∞

dx′e

−2|x−x′|

λmfp1

n

?

dΓ′

1dΓ′

2f(p′

1;x′,t′)f(p′

2;x′,t′)vrel

?

dΓ1FA(p1)dσ22

dΓ1

.(19)

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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′

1) +

FA(p′

2) = FA(p1) + FA(p2). We then have

A(x,t) =

?∞

−∞

?∞

−∞

1

λmfp

dx′e

−2|x−x′|

λmfp1

n

?

?

dΓ′

1dΓ′

2f(p′

1;x′,t′)f(p′

2;x′,t′)FA(p′

1)vrel

?

dΓ1dσ22

dΓ1

≈

dx′e

−2|x−x′|

λmfp1

n

dΓ′

1FA(p′

1)f(p′

1;x′,t′)

?

dΓ′

2f(p′

2;x′,t′)?vrelσ22?

=

?∞

−∞

dx′e

−2|x−x′|

λmfpA(x′,t′).(20)

The same approximation is made as for wlossin Eq. (17). Equation (20) resembles the one

derived in Ref. [22] 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

y(x) =

2ywall

L + λmfp

x.(21)

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.

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B.Numerical Confirmation

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

velocity px/E:

dNem

dtd3p∼px

Efwall(p)(22)

with

fwall(p) = g e−

pµuµ

wall

T

,(23)

where uµ

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

dNem

dt

=1

4Awallnwall,(24)

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

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-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-1 -0.5 0 0.5 1

y(x)

x [fm]

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 [22]

< 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,

t =L2

2η2D=L2e

≈

L2

0.82λmfp.(26)

For our setup we find t ≈ 24 fm/c, which is consistent with the numerical results shown in

Fig. 2.

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. [18]. 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

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-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-1-0.5 0 0.5 1

y(x)

x [fm]

BAMPS λ= 0.01fm

BAMPS λ= 0.1fm

BAMPS λ= 0.3fm

BAMPS λ= 1fm

BAMPS free-streaming

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

πxz= −ηdγvz(x)

dx

(27)

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

2ywall

.(28)

(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.

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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 [24]:

ηNS≈ 1.2654T

σ= 1.2654nT λmfp. (29)

Equation (29) serves as a benchmark to check the numerical methods applied to calculate

shear viscosity.

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

0

0.1

0.2

0

η [GeV/fm2]

x [fm]

BAMPS

mean value

analytic

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

1/√Nevents.

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-

consuming.

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

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0.1

1

10

0.01 0.1 1

η[GeV/fm2]

λmfp[fm]

BAMPS isotropic

analytic

FIG. 5: Shear viscosity as a function of mean free path.

proposed method for extracting the shear viscosity for relativistic systems from numerical

calculations.

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. [18].

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.

αs

0.010.03 0.10.20.3 0.5 0.6

η/s2→2

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

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0.1

1

10

100

0.01 0.03 0.1 0.2 0.3 0.4 0.6

η /s

αs

pQCD 2->2

pQCD 2->2; 2<->3

fit 22

fit 23

FIG. 6: Shear viscosity to entropy density ratio at various αs.

shear viscosity

to entropy density

η/s

0.1

1

10

100

0.1

1

10

100

pQCD coupling constant

αs

0.01 0.11 1

0.01 0.11 1

BAMPS:

Wesp

Reining

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. [20], El et al. [21] and Wesp et al.

[26]. 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. [20]. 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

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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. [26] 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. [26].)

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 [24]. The method proposed here to calcu-

late the shear viscosity coefficient is thus perfectly suitable for other microscopic transport

descriptions.

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

Page 15

15

our results with previously published results [20, 21] and also with a very recent work based

on the Kubo relation [26] and found a very good agreement.

Acknowledgements

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.

Appendix A

We calculate the number of particles ∆N with a thermal Maxwell-Boltzmann distribution,

passing through a wall of area A. The number of particles passing through a wall orthogonal

to the x-direction in a small timestep ∆t is equal to the number of all paricles with distance

∆x < −vx∆t from the wall:

∆N =

?

vx<0

d3p

(2π)3

?

A

dydz

?

0<x<−vx∆t

dxf(p) = −

?

vx<0

d3p

(2π)3Avx∆tf(p) (A1)

∆N

∆t

= −

?

vx<0

d3p

(2π)3Avxf(p)

= −A

?

vx<0

d3p

(2π)3Epxf(p) = −A

?

vx<0

d3p

(2π)3Epxge−uµpµ

T

= −

gA

(2π)3

?2π

π

dφ

?∞

0

ptdpt

?∞

−∞

dyptcos(φ)e−ptcosh(y+β)

T

(A2)

where uµ= (cosh(β),0,0,sinh(β)). After the transformation of variables y → y − β the

dependence on the boost velocity drops out:

∆N

∆t

= −

gA

(2π)3

?2π

π

dφ

?∞

0

ptdpt

?∞

−∞

dyptcos(φ)e−ptcosh(y)

T

=

2gA

(2π)3

?∞

0

dpt

?∞

−∞

dyp2

te−ptcosh(y)

T

Page 16

16

=

2gA

(2π)3

?∞

−∞

dy

2T3

cosh3(y)=gAT3

(2π)2=nA

4

where n = gT3/π2is the density in the local rest frame.

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