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![Page 1: Perfect Fluid: flow measurements are described by ideal hydro Problem: all fluids have some viscosity -- can we measure it? I. Radial flow fluctuations:](https://reader030.fdocuments.us/reader030/viewer/2022032606/56649eab5503460f94bb066c/html5/thumbnails/1.jpg)
Perfect Fluid: flow measurements are described by ideal hydro
Problem: all fluids have some viscosity -- can we measure it?
I. Radial flow fluctuations: Dissipated I. Radial flow fluctuations: Dissipated by shear viscosityby shear viscosity
II. Contribution to transverse momentum II. Contribution to transverse momentum correlationscorrelations
III. Implications of current fluctuation III. Implications of current fluctuation data data
Phys. Rev. Lett. 97162302 (2006)
Measuring Viscosity at RHIC Measuring Viscosity at RHIC
Sean Gavin Sean Gavin Wayne State University Wayne State University
Mohamed Abdel-AzizMohamed Abdel-Aziz Institut für Theoretische Physik, Institut für Theoretische Physik, FrankfurtFrankfurt
Measuring Viscosity at RHIC Measuring Viscosity at RHIC
Sean Gavin Sean Gavin Wayne State University Wayne State University
Mohamed Abdel-AzizMohamed Abdel-Aziz Institut für Theoretische Physik, Institut für Theoretische Physik, FrankfurtFrankfurt
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Measuring Shear ViscosityMeasuring Shear Viscosity
flow vx(z)
shear viscosity
€
⇒ Tzx = −η∂vx
∂z
Elliptic and radial flow suggest small shear viscosityElliptic and radial flow suggest small shear viscosity• Teaney; Kolb & Heinz; Huovinen & Ruuskanen
Problem: initial conditions & EOS unknown • CGC more flow? larger viscosity? Hirano et al. Kraznitz et al.; Lappi & Venugopalan; Adil et al.
Additional viscosity probes?
What does shear viscosity do? It resists shear flow.
Small viscosity or strong flow?
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Radial Flow FluctuationsRadial Flow Fluctuations
€
vr
neighboring fluid elements flow past one another viscous friction
small variations in radial flow in each event
shear viscosity shear viscosity drives velocity drives velocity toward the averagetoward the average
zvT rzr ∂∂−=
damping of radial flow fluctuations viscosity
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Evolution of Evolution of FluctuationsFluctuations
€
∂∂t
−ν∇ 2 ⎛
⎝ ⎜
⎞
⎠ ⎟gt = 0
diffusion equationdiffusion equation for momentum current
kinematic viscosity
€
ν = /Ts shear viscosity entropy density s, temperature T
r
z
€
gt ≡ T0r − T0r ≈ Ts u
€
Tzr ≈ −η∂u
∂z≈ −
η
Ts
∂gt
∂z
€
∂∂t
T0r +∂
∂zTzr = 0
momentum current momentum current for small fluctuations
momentum conservation
u
u(z,t) ≈ vr vr
shear stress
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viscous diffusion viscous diffusion drives fluctuation gt
V = 2ν/0
0 2 4 6
/0
€
∂gt
∂τ=
ν
τ 2
∂2gt
∂y 2
€
V = 2ν1
τ 0
−1
τ
⎛
⎝ ⎜
⎞
⎠ ⎟ V
random walk in random walk in rapidityrapidity y vs. proper time
kinematic viscosity ν Ts
formation at 0
Viscosity Broadens Rapidity Viscosity Broadens Rapidity DistributionDistribution
€
y€
gt
€
( )width2 ≡V rapidity broadeningrapidity broadening
increase
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Hydrodynamic Momentum CorrelationsHydrodynamic Momentum Correlations
€
rg = gt (x1)gt (x2) − gt (x1) gt (x2)
momentum flux density correlation functionmomentum flux density correlation function
rg rg r g,eq satisfies diffusion equation satisfies diffusion equation
Gardiner, Handbook of Stochastic Methods, (Springer, 2002)
fluctuations diffuse through volume, driving rg rg,eq
€
σ 2 = σ 02 + 2ΔV (τ f )
width in relative rapidity grows from initial value σ
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How Much Viscosity?How Much Viscosity?viscosity in collisions -- Hirano & Gyulassy
supersymmetric Yang-Mills: s pQCD and hadron gas: s ~ 1
viscous liquid νpQGP ~ ν HRG ~ 1 fmnearly perfect νsQGP ~ (4 Tc)-1 ~ 0.1 fm
Abdel-Aziz & S.G
€
ν = Ts
kinematic viscosity
effect on momentum diffusion:
limiting cases:
wanted:wanted: rapidity dependence of momentum correlation rapidity dependence of momentum correlation functionfunction
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observable:observable:
€
C = pt1pt 2 − pt
2
Transverse Momentum CovarianceTransverse Momentum Covariance
€
pt1pt 2 ≡1
N2 pti ptj
pairs i≠ j
∑
€
pt ≡1
Npti∑
propose:propose: measure C() to extract width σ2 of rg
measures momentum-density correlation function
€
C = 1
N2 Δrg dx1dx2∫
C depends on rapidity interval
C()
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4€
ν /τ 0 =1
€
ν /τ 0 = 0.1
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s ~ 0.08 ~ 1/4
Current Data?Current Data?
STAR measures rapidity width of pt fluctuationsSTAR measures rapidity width of pt fluctuations
• most peripheral σ* ~ 0.45 €
σ p t :n =1
Npti − pt( ) ptj − pt( )
i≠ j
∑
find width σ* increases in central collisions
• central σ* ~ 0.75
€
σ central2 −σ peripheral
2 = 4ν1
τ f ,p
−1
τ f ,c
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
naively identifynaively identify σ* with σ
freezeout f,p ~ 1 fm, f,c ~ 20 fm ν ~ 0.09 fm
at Tc ~ 170 MeV
J.Phys. G32 (2006) L37
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we want: 2
2
1t
jitjti ppp
NC −= ∑
≠
Uncertainty RangeUncertainty Range
STAR measures:
maybe σn 2σ* STAR, PRC 66, 044904 (2006)
uncertainty range σ* σ 2σ* 0.08 s 0.3
€
N Δσ p t :n = pti − pt( ) ptj − pt( )i≠ j
∑
€
= N2C − pt
2(density correlations)
density correlation functiondensity correlation function may differ from rg
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Current Data Current Data Viscosity Viscosity Hints!Hints!
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MeasureMeasure p ptt Covariance Covariance for Clearer Picturefor Clearer Picture
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current fluctuation data compatible with current fluctuation data compatible with
compare to other observables with different compare to other observables with different uncertaintiesuncertainties
Summary: small viscosity or strong Summary: small viscosity or strong flow?flow?
€
0.08 < η /s < 0.3
test the perfect liquid test the perfect liquid viscosity info viscosity info
• viscosity broadens momentum correlations in rapidity viscosity broadens momentum correlations in rapidity
• pptt covariance covariance measures these correlations measures these correlations
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0
1
2
3
4
0 5 10 15 20
sQGP + Hadronic CoronasQGP + Hadronic Coronaviscosity in collisions -- Hirano & Gyulassy
supersymmetric Yang-Mills: s pQCD and hadron gas: s ~ 1
€
ν =T−1(η /s)
Broadening from viscosity
€
/τ 0
€
σ 2 −σ 02
€
d
dτσ 2 ≈
4ν (τ )
τ 2,
H&G
Abdel-Aziz & S.G, in progress
QGP + mixed phase + hadrons T()
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Centrality DependenceCentrality Dependence
freezeout time modelTeaney, Lauret & Shuryak
rapidity width
€
centrality, 2nbin /npart
€
σ 2 −σ 02
( " )max
00.10.20.30.40.50.60.70.80.9
1
1 2 3 4 5 6
STAR, J.Phys. G32 (2006) L37
€
F ~ r2
part
centrality: mean parton path length
€
path length ~ 2nbin /npart
€
ν =const.
€
ν ∝ τ
€
d
dτσ 2 ≈
4ν (τ )
τ 2,
Abdel-Aziz & S.G, in progress
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covariance
€
C =1
N2 pti ptj
pairs i≠ j
∑ − pt
2
Covariance Covariance Momentum FluxMomentum Flux
unrestricted sum:
€
rgdx1dx2 =∫ pti ptj∑ − N2
pt
2= pti
2∑ + N2C
€
rg = gt (x1)gt (x2) − gt (x1) gt (x2)
€
C = 1
N2 (rg − rg, eq )dx1dx2∫
∫∑ = 2121,all
dndnpppp ttji
tjti
€
= dx1dx2 dp1 pt1 f1∫( )∫ dp2 pt 2 f2∫( )
€
gt (x) = dp pt Δf x, p( )∫
€
dn = f x, p( )dpdx
correlation function:
C =0 in equilibrium
∫→ 2121 )()( dxdxxgxg
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pptt Fluctuations Energy IndependentFluctuations Energy Independent
Au+Au, 5% most central collisions
STAR, Phys.Rev. C72 (2005) 044902
€
σ p t :n =1
Npti − pt( ) ptj − pt( )
i≠ j
∑ =N(N −1)
Nδpt1δpt 2
€
⇒ σ p t :n ≈ N δpt1δpt 2
![Page 18: Perfect Fluid: flow measurements are described by ideal hydro Problem: all fluids have some viscosity -- can we measure it? I. Radial flow fluctuations:](https://reader030.fdocuments.us/reader030/viewer/2022032606/56649eab5503460f94bb066c/html5/thumbnails/18.jpg)
€
Fp t
obs ∝N
obs
N=1− e− p t ,max /Teff 1+ pt,max /Teff( )
Flow ContributionFlow Contribution
0
1
2
3
4
0 0.5 1 1.5 2 2.5€
Fp t≈ N δpt1δpt 2 /2σ 2
σ 2 = pt2 pt 2
€
Fp t(%)
€
pt,max (GeV)
PHENIX
2/1
1
1⎟⎟⎠
⎞⎜⎜⎝
⎛
−+
≈r
reff v
vTT
explained by blueshift?explained by blueshift?
thermalization
flow added
dependence on dependence on maximummaximum pt, max
needed:needed: more realistic hydro + flow fluctuations more realistic hydro + flow fluctuations
![Page 19: Perfect Fluid: flow measurements are described by ideal hydro Problem: all fluids have some viscosity -- can we measure it? I. Radial flow fluctuations:](https://reader030.fdocuments.us/reader030/viewer/2022032606/56649eab5503460f94bb066c/html5/thumbnails/19.jpg)
want: ∫∑ −=−=≠
21,121 )(22 dxdxrrpppC eqggNt
jitjtiN
Covariance Covariance Momentum FluxMomentum Flux
STAR measures:
)()()()( 2121 xgxgxgxgr ttttg −=
two correlation functions:
€
σ p t :n =1
Npti − pt( ) ptj − pt( )
i≠ j
∑
€
=1
Npti ptj
i≠ j
∑ −N(N −1)
Npt
2
€
= N C − pt
2R( )
∫ −=−−
= 21,1
2
2
)()1(
2 dxdxrrN
NNNR eqnnN
)()()()( 2121 xnxnxnxnrn −=
where:
![Page 20: Perfect Fluid: flow measurements are described by ideal hydro Problem: all fluids have some viscosity -- can we measure it? I. Radial flow fluctuations:](https://reader030.fdocuments.us/reader030/viewer/2022032606/56649eab5503460f94bb066c/html5/thumbnails/20.jpg)
130 GeV
s1/2=200 GeV
blue-shift: • average increases
• enhances equilibrium contribution
€
∝Teff2 ∝
1+ vr
1− vr
thermalization
flow added
M. Abdel-Aziz & S.G.
participants
Thermalization Thermalization + Flow + Flow
20 GeV€
δpt1δpt 2 ∝ δT1δT2 r12∫
• blue-shift cancels in ratio
€
δpt1δpt 2
pt
2 ≈ constant
€
pt ∝Teff ∝1+ vr
1− vr
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 2
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Rapidity Dependence of Momentum Rapidity Dependence of Momentum Fluctuations Fluctuations
momentum correlation function near midrapiditymomentum correlation function near midrapidity
€
rg − rg,eq ∝ e−yr2 / 2σ 2
e−ya2 / 2Σ2
• relative rapidity yr = y1-y2
Abdel-Aziz, S.Gfluctuationsfluctuations in rapidity window
• weak dependence on ya = (y1+y2)/2
€
C ∝ (rg − rg,eq ) dy1dy2Δ
∫∫
take σo ~ 0.5, fo 2
![Page 22: Perfect Fluid: flow measurements are described by ideal hydro Problem: all fluids have some viscosity -- can we measure it? I. Radial flow fluctuations:](https://reader030.fdocuments.us/reader030/viewer/2022032606/56649eab5503460f94bb066c/html5/thumbnails/22.jpg)
we want: 2
2
1t
jitjti ppp
NC −= ∑
≠
Uncertainty RangeUncertainty Range
STAR measures:
density correlation functiondensity correlation function rn rn r n,eq may differ from rg
maybe σn 2σ* STAR, PRC 66, 044904 (2006)
uncertainty range σ* σ 2σ* 0.08 s 0.3
€
N Δσ p t :n = pti − pt( ) ptj − pt( )i≠ j
∑
momentum density momentum density correlationscorrelations
€
= dx1dx2 Δrg (x1, x2) − pt
2Δrn (x1, x2)[ ]∫
density density correlationscorrelations
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observable:observable:
€
C = pt1pt 2 − pt
2
Transverse Momentum CovarianceTransverse Momentum Covariance
€
pt1pt 2 ≡1
N2 pti ptj
pairs i≠ j
∑
€
pt ≡1
Npti∑
vanishes in equilibriumvanishes in equilibrium
€
N 2 − N2
= N ⇒
€
pti ptj
pairs i≠ j
∑eq
= N(N −1) pt
2= N
2pt
2
€
rg = gt (x1)gt (x2) − gt (x1) gt (x2)measures correlation functionmeasures correlation function
€
C = 1
N2 dx1dx2∫ rg − rg, eq( )
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viscous diffusionviscous diffusion
V = 2ν/0
0 2 4 6
/0
€
∂gt
∂τ=
ν
τ 2
∂2gt
∂y 2
€
V = 2ν1
τ 0
−1
τ
⎛
⎝ ⎜
⎞
⎠ ⎟ V
random walk in rapidityrandom walk in rapidity y vs. proper time
€
V = y 2∫ gt dy
€
d
dτΔV =
2ν
τ 2,
kinematic viscosity ν Ts
formation at 0
Viscosity Broadens Rapidity Viscosity Broadens Rapidity DistributionDistribution