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Viscous hydrodynamics for relativistic heavy-ion collisions. Huichao Song. Department of Physics. The Ohio State University. 191 West Woodruff Avenue. Columbus, OH 43210. 2007 National Nuclear Summer School. FSU, FL. July 8 – July 21, 2007. Advisor: U. Heinz, Supported by DOE.

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Viscous hydrodynamics for relativistic heavy-ion collisions

Huichao Song

Department of Physics

The Ohio State University

191 West Woodruff Avenue

Columbus, OH 43210

2007 National Nuclear Summer School

FSU, FL. July 8 – July 21, 2007

Advisor: U. Heinz, Supported by DOE


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The QCD Phase Diagram

QGP


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The QGP Shear Viscosity

The QGP shear viscosity of is in hot theoretical debates recently:

-Weakly coupled QCD prediction:

P.Arnold,G.Moore,&Y.Yaffe, 00,03

-Strongly coupled AdS/CFT prediction:

D.T. Son, et,al. 03

The first estimation of the QGP shear viscosity from the experimental data (RHIC):

--blast wave model + the first order theory corrections

D. Teaney, PRC 68 (2003) 034913

Blast wave model analysis:

-only counts the viscous corrections to the

spectra during freeze-out procedure

-neglects shear viscosity effects on hydro dynamic evolution before freeze-out

Viscous hydrodynamics is needed!


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Input: “EOS”

Ideal hydrodynamics

hydro

S.Bass

Hydrodynamics:

-A macroscopic tool to describe the expansion of QGP or hadronic matter

Conservation laws

5 equ. 14 independent variables

-reduce # of independent variables (ideal hydro)

-Provide more equations? (viscous hydro)

Ideal hydrodynamics: reduce the independent variable from 14 to 6,


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Dissipative flows

charge flow

energy flow

bulk pressure

shear pressure tensor

entropy flow

Causal viscous hydrodynamics

Full tensor decomposition in frame of:

Besides these modified transport equations for, one need more equations

for these dissipative flows:

The first order theory formulism:

--- Eckart, Landau & Lifishitz around 1950’s

Problems in the first order theory formulism:

-Acuasl, infinite speed of signal propagation

-process instabilities


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Dissipative flows

charge flow

energy flow

bulk pressure

shear pressure tensor

entropy flow

Causal viscous hydrodynamics

Full tensor decomposition in frame of:

W. Israel, J.Stewart 79

The 2nd order Israel-Stewart formulation:

--expansion the entropy current to the 2nd order of dissipative flows,


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coordinates

-Bjorken approximation:

Causal viscous hydro in (2+1)-d

U. Heinz, H. Song & A. Chaudhuri, PRC 73(2006) 034904

3+1 2+1

-zero net baryon density, zero bulk pressure, zero heat conduction

Equivalent to the modified transport equations for

where, the transport equations for energy momentum tensor are explicit written as:


Viscous vs ideal hydro temperature entropy l.jpg

Initial & final conditions: (viscous & ideal)

Other viscous hydro parameter: Shear viscosity& relaxation time

Viscous vs. ideal hydro: temperature & entropy

-slowing down of cooling process due to decelerated longitudinal expansion initially ,

but faster cooling in middle and late stages due to stronger transverse expansion

-viscous effects are larger in early and middle stages, but neglectable in late stage


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Viscous vs. ideal hydro: radial flow & spectra

-More radial flow, flatter spectra

-the viscous effects to the hadron spectra could be absorbed by starting viscous

hydro later with lower initial energy density


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-Both the evolution corrections (viscous corrections to ) and spectra corrections

initial energy density distribution

(viscous corrections to ) have significant effects to , for low region evolution

correction dominant.

Viscous vs. ideal hydro: elliptical flow

For non-central collisions:

-Elliptical flow is very sensitive to even minimal shear viscosity.


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Summery and Conclusions

- Shear viscosityresults inslowing down of the cooling process in early stage due

to decelerated longitudinal expansion, faster cooling in middle and late stages due

to stronger transverse expansion.

-Shear viscosity leads to more radial flow generation and flatter spectra, the effects

of which could be absorbed by starting viscous hydro later with lower initial energy

density.

-elliptical flow is very sensitive to shear viscosity, even the minimum value from

AdS/CFT could result in a great suppression of .

  • in contrast with the Teaney’s blast wave model method, which only counts viscous

spectra corrections during freeze-out procedure, our results shows that evolution

  • corrections ( ) are the main factor to influence the low spectra elliptical flow.

-Do the results indicate that the QGP viscosity smaller than the AdS/CFT lowest

limit? Need more investigations in the at least flowing aspects:

a)Different initial conditions

b) more realistic descriptions of the hadronic stage


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Thank You


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vs. different relaxation time

1st theory vs. 2nd order theory

a)

b)

1st order theory vs. 2nd order theory

As the relaxation time reduces to zero, the 2nd order theory formulism returns to the 1st order one. First order theory , instable ?

a)calculated with the same velocity profile from the 2nd theory by

not the real 1st order Navier-Stokes hydrodynamics

b)Smaller relaxation time reduces the viscous effects on suppression of


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Blast wave model vs. real viscous hydrodynamics

Blast wave model

vs.

viscous hydrodynamics

+ the1st order theory corrections

(the 2 order theory Israel-Stewart formulation)

--with only spectra corrections

--with both evolution and spectra corrections

Compare the spectra corrections from blast wave model and from real hydrodynamics:

-Spectra corrections are sensitive to the details of flow pattern at freeze out, and are

not easily captured with blast-wave model parametrizations.


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viscous vs. ideal hydro: central Cu+Cu collisions:


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viscous vs. ideal hydro: central Cu+Cu collisions:


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Viscous vs. ideal hydro: momentum anisotropy


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Where ideal hydro works

Hydro: U Heinz & P.Kolb Data: STAR PHENIX

spectra for both central collision and noncentral collisions :


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Where ideal hydro works

STAR PRL04 PHENIX PRL03

STAR, PHENIX, PHOBOS

Elliptic flow coefficient at noncentral collisions:

as a function of centrality

for different identified hadrons

-Ideal hydro describes the data well at

-It also gives a correct mass splitting of at low region


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