viscous hydrodynamics dpf 2009 huichao song the ohio state university supported by doe 07/30/2009...
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Viscous hydrodynamics
DPF 2009
Huichao Song
The Ohio State University
Supported by DOE
07/30/2009
July 27-July 31, Detroit, MI
with shear and bulk viscosity
in collaboration with Ulrich Heinz
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What is viscosity
the ability of momentum transfer
Shear viscosity –measures the resistance to flow
Bulk viscosity –measure the resistance to expansion
Determines the dynamics of
-volume viscosity
compressible fluid
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The QGP viscosity
-strongly coupled AdS/CFT prediction : D.T. Son et al. ‘01,’05
To extract the QGP viscosity from experimental data, we need viscous hydrodynamics
Shear viscosity: uncertainty principle requires a lower limit for -weakly coupled QCD:
)()]0()([lim20
1 4
0tTxTxed ijijti
)()]0()([lim18
1 4
0tTxTxed i
iii
ti
Kubo formulas:shear
viscosity: bulk
viscosity:
-lattice SU(3) gluon dynamics :
-weakly coupled QCD prediction:
-lattice SU(3) gluon dynamics :
-LET+ assum. of spectral fun. + Lattice data:
s/
Bulk viscosity: zero for classical massless particles, reaches a
peak near s/ cT
1~/ s
-strongly coupled AdS/CFT prediction:
5.0
0.2~ 73.0/
cTs Meyer, PRL08
08.0~4/1/ sMeyer, PRD 07
Kharzeev, et al. 07-088.0~/ ~ cTs
1/ sArnold, Moore & Yaffe, 00,03
)3/1(/2/ 2scss Buchel, 07
05.0~/ ~ cTs Gubser, et al . 0806 ..
1/ s Arnold, Dogan & Moore, PRD06
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Viscous hydro with shear & bulk viscosity
(2nd order shear-bulk -mixing term (Muronga, Rischke) not included.)
0)( xT
Conservation laws:
gpuupeT )()(
u
T
T
2
12
Evolution equations for shear pressure tensor and bulk presurre:
u
T
Tu
2
1)(
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Numerical Results
Bulk viscosity:
Shear viscosity: ,4/108.0/ s 0/or s
Relaxation times: (see later)
QGPHRG
Min. AdS/CFT prediction
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Shear viscosity vs. bulk viscosity (I)
-Shear viscosity: decelerate cooling process in early stage accelerate cooling process in middle and late stages
-Bulk viscosity: decelerate cooling process
Same initial & final conditionsideal hydro viscous hydro-shear only viscous hydro-bulk only
Local temperature
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Shear viscosity vs. bulk viscosity (II)
-shear viscosity: increases radial flow, results in flatter spectra
-bulk viscosity: decreases radial flow, results in steeper spectra
radial flow spectra
Same Initial & final conditionsideal hydro viscous hydro-shear only viscous hydro-bulk only
0,4/1 c1,0 c
0,0 css //
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Shear viscosity vs. bulk viscosity (III)
-v2 is sensitive to both shear and bulk viscosity
Elliptical flow v2
Same Initial & final conditionsideal hydro viscous hydro-shear only viscous hydro-bulk only
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Viscous v2 suppression: shear and bulk viscosity ideal hydro visc. hydro:
-at RHIC, 2 x min. bulk viscosity could result in ~50% additional v2 suppression
-when extracting the from RHIC data, bulk viscous effects cannot be neglected s/
20%30%
ss //
1,4/1 c0,4/1 c
2,4/1 c
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Viscous v2 suppression: shear and bulk viscosity ideal hydro visc. hydro:
-at RHIC, 2 x min. bulk viscosity could result in ~50% additional v2 suppression
-when extracting the from RHIC data, bulk viscous effects cannot be neglected s/
20%30%
ss //
1,4/1 c0,4/1 c
2,4/1 c
0
)0( bulk viscosity effects:
(a) Change the flow profile during hydro evolution (b) Additional spectra correction along freeze-out surfacef
Song & Heinz: v2 will decrease, flow corrections only (a), , at freeze-out Monnai & Hirano: v2 will increase, spectra corrections only(b), ideal hydro for evolution
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-relaxation time effects
Bulk Viscosity
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Bulk viscous v2 suppression:-- Smaller vs. larger relaxation time
-viscous effects from bulk viscosity strongly depend on relaxation time and the initialization for bulk pressure
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Effects from initialization of (I)Smaller relaxation time
-after (several relaxation times), viscous pressure loses memory of initial cond.
fm/c)5.0( is insensitive to different initializations of 2v-When is small ,
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Effects from initialization of (II)
-after (several relaxation times), viscous pressure loses memory of initial cond.
larger relaxation time
fm/c)5( is sensitive to different initializations of 2v-When is larger ,
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Effects from initialization of (III)
-viscous effects from bulk viscosity strongly depend on relaxation time and the initialization for bulk pressure
Smaller vs. larger relaxation time
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--Effects from system size
Multiplicity scaling of v2/ε
and collision energy
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Multiplicity scaling of v2/ε EOS I
- freeze-out condition introduces time scale, breaking scale invariance of id. hydro eqns. - Initial profiles for Cu+Cu and Au+Au systems are not identical after a rescaling
Ideal hydrodynamics: multiplicity scaling of v2/ε is weakly broken:
Viscous hydrodynamics: additional scale breaking by shear viscosity, resulting in fine structure of v2/ε:
Viscous effects are larger for smaller systems and lower collision energies
- for similar initial energy density, Cu+Cu curves are slightly below the Au+Au curves- at fixed , the curves are slightly above the ones
dy
dN
Sch1 3
0 GeV/fm15e 30 GeV/fm30e
full I-S eqn
ideal hydro v2 scaling line
Song & Heinz PRC 08
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good candidate to constrain)
Multiplicity scaling of v2/ε EOS L
- experimental data show qualitatively similar fine ordering as viscous hydro prediction
- to reproduce slope of v2/ε vs. (1/S)dN/dy, a better description of the highly viscoushadronic stage is needed: T-dependent , viscous hydro + hadron cascade
- the experimental v2/ε vs. (1/S)dN/dy scaling (slope and fine structure) is another
- this requires, however, experimental and theoretical improvements: reduced error bars, accounting for T-dependence of near Tc, modeling hadronic phase with realistic cascade
full I-S eqn
gasesdilute
hydro region
Song & Heinz PRC 08
s/
ss /,/
s/
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A Short Summary
-When extracting QGP viscosity from experimental data, bulk viscosity effects should not be neglected
-More theoretical inputs are needed for bulk viscosity: - relaxation time- initialization for bulk pressure - bulk viscosity of hadronic phase, etc
- multiplicity scaling of is a good candidate to extract the QGP viscosity: - larger viscous effects in smaller systems and at lower collision energies
/2v
- is sensitive to2v s/
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Thank You
20
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EOS
EOS
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Viscous hydro in 2+1-dimension
)())(())(()(1
pvTvTT yyy
yxy
xy
xy
xy y yy
xx xyy)())(())(()(
1pvTvTT xy
xyx
xx
x
x x
pTTT y
yx
x )()()(1 2
3+1 2+1coordinates),,,( yx
--the transport equations for energy momentum tensor are explicit written as:
u
T
TD
2
12
1
gpuupeT )()( 0)( xT
u
T
TuD
2
1)(
1
Bjorken approximation:
-shear tensor decelerate longitudinal expansion, but accelerate transverse expansion
-bulk pressure decelerates both longitudinal & transverse expansion (bulk pressure effectively softens the EoS near the QCD phase transition)
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Viscous hydro: a short summary for shear viscosity --shear viscosity only -2+1-d viscous hydro code individually developed by
different groups:
-v2 at RHIC is sensitive to even the minimum shear viscosity entropy ratio -v2 suppression from different groups ranges from 20% to 70%
-the above discrepancy was largely resolved by investigating effects from system size, EoS and different forms of I-S eqns. used
-Code checking within the TECHQM collaboration:
-The first attempt to extract QGP shear viscosity from RHIC data: Luzum & Romatschke, PRC 78 (2008)
TECHQM webpage
Song & Heinz, PRC 78 (2008)
Romatsche & Romatschke (INT), Song & Heinz (OSU), Dusling & Teaney (Stony) Huovinen & Molnar (Purdue), Chaudhuri (Kolkata, India)
70% 20%
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Effects from initialization of mn
- is insensitive to different initializations of mn2v
yyxx
-after ~1fm/c (several relaxation times), viscous pressure loses memory of initial cond.
Song & Heinz, PLB08 & PRC 77(2008)
mnmn 2 0mnvs.
-Effects on entropy production: ~20%
sT/3