2010-4-18 DaLian university and technology

The effects of viscosity on hydrodynamical evolution of QGP

苏中乾大连理工大学

Dalian University of Technology

Contents

• I. Formalization of viscous hydrodynamics

• II. Equation of state (EOS) and initial conditions

• III. Numerical results

• IV. Summery

The effects of viscosity on hydrodynamical evolution of QGP

Viscous hydrodynamics

• Why viscous?

• The ideal fluid description works well in almost central Au+Au collisions near midrapidity at top RHIC energy, but gradually breaks down in more peripheral collisions, at forward rapidity, or at lower collision energies, indicating the onset of dissipative effects. And viscous fluid is a real fluid.

Viscous hydrodynamics

Energy-momentum tensor decomposition:

( ) 13

( ) 12

( )

( )

u T u

p

T

a a

a a a

Energy density in fluid rest frame

Pressure in fluid rest frame,

Bulk viscous pressure

Shear stress tensor,

Symmetrized tensor

Symmetrized, traceless spatial projection

13p T

0, 0u u

( )T u u p

0S

0N Charge conservation

0T Energy-momentum conservation

The second law of thermodynamics

Starting point : conservation laws

Viscous hydrodynamics

( )T u u p 0T

Energy-momentum tensor conservation: Thinking about zero baryon system, we can ignore the effects of vorticity ( ) and heat conductivity of flow ( )0 0q

Relaxation equations for shear stress tensor and bulk pressure

122 ( )u

Tu T

12 ( )u

Tu T

Second-order Israel-Stewart equation

Navier-Stokes (NS) equations: simple, but: unstable and acausal equations of motion

Viscous hydrodynamics

With , viscous hydrodynamics reduces to ideal hydrodynamics

• Use Bjorken picture and thinking about azimuthally symmetric systems:

, 0

/zv z t (Boost invariant)

( , , , )r

Notice: once we know and , velocity and the temperature at will be Obtained by Lorentz boost:

viscous hydrodynamics equations

Close equations with EOS ( )p p

( , , , )r

Boost-invariant and azimuthally symmetric systems:

Use coordinates and solve:

hydrodynamic equations for

with

kinetic relaxation equations for shear viscous tensor and bulk viscous pressure

shear

bulk

Energy-momentum

Equation of state and initial condition

with

for

the MIT bag equation of state with bag constant

First: QCD lattice results (Cross over EOS: entropy density of the system is function of temperature)

•Input: EOS (influence on the dynamic of evolution)( )p p

Second： Equation of state for massless quarks and gluons gas

where (the number of quark flovers)

A. Muronga & D. Rischke; nucl-th/0407114v2

The result equation of state

EOS results

0.1T Tc

/ 3.6r dQ dH

The Solid line:

The dashed line: quark and gluon gas equation of state.

The initial conditions

The components of the shear tensor and bulk pressure

0 maximum energy density for central collisions at initial time

initial transverse radius of source.0 6R fm

the initial temperature at the center of source and initial proper time

Initial conditions: Initial energy density distribution in the transverse plane use Woods-saxon parameterization

Viscous vs. ideal hydrodynamics——Temperature and velocity

For ideal hydrodynamic equation the temperature cool faster thanthat the shear viscosity are included, this means viscous effect lead to slow down of expansion of system.

1/ 2

4s

Viscous vs. ideal hydrodynamics——shear and bulk viscous

The profiles of shear tensor and bulk pressure components in unit of at different times as function of

0p

0/r R

Our results VS A.Muronga & D. Rischke

A. Muronga & D. Rischkenucl-th/0407114v2

Ideal quark &gluon gasEOS

Cross over EOSand woods-saxondistribution

Viscous vs. ideal hydrodynamics——Space-time diagrams of freeze-out

At a fixed temperature, a cross over transition equation of state lead to a larger space-time of freeze-out and a little larger while we take the effect of viscosity into account.

The source with considering of viscosity, reaches to Tf have a larger space until a certain time and then it will be a littlesmaller.

The reason : at binging due to effect of viscosity, source will expand slowly, however after a while, this effect will be dissipated.R.Baier & P.Romatschke: Eur. Phys. J. C 51, 677 (2007).D. A. Teaney: J.Phys. G 30, S1247 (2004).A. K. Chaudhuri: Phys. Rev. C 74, 044904 (2006).

• I. For ideal fluid the temperature cool faster than that the viscosity are included, this means viscous effect lead to slow down of the temperature decrease and the expansion of system.

II. A cross over transition equation of state lead to a larger space-time of freeze-out than no phase transition and a little larger while we take the effect of viscosity into account.

Summery

Thank you!