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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.