Viscous hydrodynamics for relativistic heavy-ion collisions ...

Viscous hydrodynamics for relativistic heavy-ion collisions ∗
Ulrich Heinz
Department of Physics
The Ohio State University
191 West Woodruff Avenue
Columbus, OH 43210
presented at
Workshop on Particle Correlations and Femtoscopy (WPCF 2007)
Santa Rosa, California, August 1 – 3, 2007
Collaborators:
Asis Chaudhuri, Huichao Song
∗ Supported by the U.S. Department of Energy (DOE)

Collective flow tests the Equation of State:
Hydrodynamic equations, ideal fluid limit:
( ˙ f = time derivative in local rest frame, ∂·u = local expansion rate)
ṅ B = −n B (∂·u)
˙ε = −(ε + p) (∂·u)
˙u µ = ∇µ p
ε + p =
c2 s
1+c 2 s
∇ µ ε
ε
• flow driven by pressure gradients ∇ µ p
• acceleration ∇µ p
ε+p
speed of sound c 2 s = ∂p
∂ε
closely related to
Karsch+Laermann, hep-lat/0305025
0.40
2
0.35 v s =dp/dε
Chojnacki et al., nucl-th/0410036
0.30
0.25
0.20
0.15
0.10
n f =2, m=0.65
n f =2, m=0.75
n f =2, m=0.85
n f =2, m=0.95
SU(3)
0.05
T / T c
0.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
“Softest point” near T = T cr .
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 1(26)

Successes of hydrodynamics at RHIC:
1/ 2π dN/dyp T
dp T
(GeV −2 )
1/2π dN/dyp T
dp T
(GeV −2 )
Single particle spectra from central and peripheral
Au+Au @ 130 A GeV (STAR, PHENIX):
π + PHENIX
10 2 p PHENIX
p STAR
π + hydro
10 0
p hydro
10 −2
most central
0 1 2 3
p (GeV) T
10 0 PHENIX
STAR
hydro
10 −2
10 −4
10 −6
10 −8
p
60 − 92 %
centrality
0 − 5 %
5 − 15 %
15 − 30 %
30 − 60 %
0 1 2 3 4
p (GeV) T
1/2π dN/dyp T
dp T
(GeV −2 )
1/2π dN/dyp T
dp T
(GeV −2 )
PHENIX
10 2 STAR
hydro
10 0
centrality
0 − 5 %
10 −2
5 − 15 %
15 − 30 %
10 −4
30 − 60 %
10 −6
π − 60 − 92 %
0 0.5 1 1.5 2 2.5
p (GeV) T
0 − 5 %
PHENIX
hydro
10 0 5 − 15 %
10 −2
15 − 30 %
30 − 60 %
10 −4
60 − 92 %
centrality
10 −6
K +
0 0.5 1 1.5 2
p (GeV) T
Model parameters fixed with π, ¯p spectra at b = 0;
all other spectra predicted (UH & P.Kolb, hep-ph/0204061).
v 2
(%)
v 2
(%)
Centrality and momentum
dependence of elliptic flow v 2
(STAR, PHENIX, PHOBOS):
10
8
6
4
25
20
15
10
eWN
sWN
eBC
sBC
STAR
2
h +/−
0
0 0.25 0.5 0.75 1
n ch
/n max
5
STAR
PHENIX
EOS Q
EOS H
0
0 1 2 3 4
p (GeV) T
v 2 = 〈cos(2φ)〉
h +/−
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 2(26)

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Breakdown of hydrodynamics at high p ⊥ :
upper limits for the QGP viscosity
D. Molnár and M. Gyulassy, NPA 697 (2002) 495
D. Teaney, PRC 68 (2003) 034913
./0
¨§¨
2 0
v 2
(p T
)
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
b ≈ 6.8 fm (16-24% Central)
STAR Data
Γ s /τ o = 0
Γ s /τ o = 0.1
Γ s /τ o = 0.2
=%¦
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
p T
(GeV)
Γ s = 4 3η/(T ·s)
• For sufficiently (very) large σ el , v 2 (p ⊥ ) from covariant parton transport model MPC follows
hydrodynamic curve at low p ⊥ and reproduces observed saturation at high p ⊥
• Similar pattern is seen in viscous hydrodynamics: viscous corrections increase ∼ p 2 ⊥
• v 2 data suggest Γ s
τ
< 0.1, close to minimum viscosity η s = ¯h
4π
(Son et al. 2002)
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 3(26)

v 2
/ε
Limits of ideal fluid dynamics: smaller, less dense systems
STAR, PRC 66 (’02) 034904; NA49, PRC 68 (’03) 034903
0.25
0.2
HYDRO limits
(3+1)-d hydrodynamics:
T. Hirano, PRC 65 (’02) 011901; 66 (’02) 054905
0.08
0.06
STAR
PHOBOS
PCE
CE
0.15
0.1
0.05
E lab
/A=11.8 GeV, E877
E lab
/A=40 GeV, NA49
E lab
/A=158 GeV, NA49
s NN
=130 GeV, STAR
v 2
0.04
0.02
s NN
=200 GeV, STAR Prelim.
0
0 5 10 15 20 25 30 35
(1/S) dN ch
/dy
0
−6
−4 −2 0 2 4 6
η
• vmeasured 2
v hydro
2
scales with 1 S
dN ch
dy
∝ s init
• e init > 10 GeV/fm 3 needed for v 2 to saturate before hadronization and exhaust ideal hydro
limit!
• hydrodynamics predicts non-monotonic v 2 /ɛ: between AGS and RHIC it decreases, due to
softening of EOS by quark-hadron transition (Kolb, Sollfrank, UH, PRC 62 (2000) 054909)
• data show instead monotonous increase of v 2 /ɛ with √ s!?
What’s going on??
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 4(26)

Hadronic dissipation reduces elliptic flow
at forward rapidity:
T. Hirano, U. Heinz, D. Kharzeev, R. Lacey, Y. Nara, PLB 636 (2006) 299
0.12
0.1
0.08
b=4.0fm
hydro+JAM
th
T =100MeV
th
T =169MeV
PHOBOS 3-15%
0.12
0.1
0.08
b=6.3fm
hydro+JAM
th
T =100MeV
th
T =169MeV
PHOBOS 15-25%
0.12
0.1
0.08
b=8.5fm
hydro+JAM
th
T =100MeV
th
T =169MeV
PHOBOS 25-50%
v 2
0.06
v 2
0.06
v 2
0.06
0.04
0.04
0.04
0.02
0.02
0.02
0
-6 -4 -2 0 2 4 6
η
0
-6 -4 -2 0 2 4 6
η
0
-6 -4 -2 0 2 4 6
η
[Glauber model initial conditions (85% soft/15% hard)]
• Not enough elliptic flow from perfect QGP fluid – some hadronic contribution
to v 2 is required
• Treating the hadronic stage as ideal fluid overpredicts v 2 in peripheral collisions
and at forward rapidities
• Dissipation in hadronic cascade brings theory in line with data (except for small
b – excess in data due to event-by-event geometry fluctuations (Miller & Snellings, PHOBOS))
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 5(26)

But is it enough?
3D Hydro+Cascade Model: Ideal fluid dynamics for QGP above T c , hadronic cascade with
realistic cross sections (JAM) below T c
v 2
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
Hirano et al., PLB 636 (2006) 299
0
0 50 100 150 200 250 300 350 400
N part
CGC, T dec
=100MeV
BGK, T dec
=100MeV
CGC, hydro+cascade
BGK, hydro+cascade
PHOBOS(hit)
PHOBOS(track)
ε
0.6
0.5
0.4
0.3
0.2
0.1
Lappi & Venugopalan, PRC 74 (2006) 054905
CYM, m = 0.2 GeV
CYM, m = 0.5 GeV
Glauber, N part
Glauber, N coll
KLN, Q s
2
~ Npart
0
0 2 4 6 8 10
b [fm]
• Hadronic dissipation reduces elliptic flow buildup in peripheral collisions
• Color Glass Condensate (CGC-KLN) model (McLerran & Venugopalan 1994; Kharzeev, Levin, Nardi 2001) produces
steeper edge of initial distribution, resulting in larger eccentricities ɛ than in Glauber model
• Ideal hydrodynamics turns larger spatial eccentricity ɛ into larger elliptic flow v 2
• For Glauber model initial conditions, hadronic dissipation fully explains the data;
for CGC/KLN initial conditions hadronic dissipation not enough – need additional QGP viscosity!
=⇒ Need better control over initial conditions!
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 6(26)

Relativistic hydrodynamics
for viscous fluids
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 7(26)

Viscous relativistic hydrodynamics (Israel & Stewart 1979)
Include shear viscosity η, neglect bulk viscosity (massless partons) and heat conduction
(µ B ≈ 0); solve
with modified energy momentum tensor
∂ µ T µν = 0
T µν (x) = ( e(x)+p(x) ) u µ (x)u ν (x) − g µν p(x) + π µν .
π µν = traceless viscous pressure tensor which relaxes locally to 2η times the shear
tensor σ µν ≡ ∇ 〈µ u ν〉 on a microscopic kinetic time scale τ π :
Dπ µν = − 1
τ π
(
π µν − 2η∇ 〈µ u ν〉) − ( u µ π νλ +u ν π µλ) Du λ
where D ≡ u µ ∂ µ is the time derivative in the local rest frame.
Kinetic theory relates η and τ π , but for a strongly coupled QGP neither η nor
this relation are known =⇒ treat η and τ π as independent phenomenological
parameters. For consistency: τ π θ ≪ 1 (θ = ∂ µ u µ = local expansion rate).
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 8(26)

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(1+1)-d viscous hydrodynamic equations
(Muronga & Rischke 2004, Chaudhuri & Heinz 2005)
Azimuthally symmetric transverse dynamics with long. boost invariance:
Use (τ, r, φ, η) coordinates and solve
• hydrodynamic equations for T ττ = (e+P)γ 2 r −P, T τr = (e+P)γ 2 r v r
(with “effective pressure” P = p−r 2 π φφ −τ 2 π ηη ) together with
• kinetic relaxation equations for π φφ , π ηη :
1
τ ∂ τ
1
τ ∂ τ
τT ττ
τT τr
∂ τ + v r ∂ r
∂ τ + v r ∂ r
+ 1 r ∂ r r(T ττ + P)v r
= − p + τ 2 π ηη
,
τ
+ 1 r ∂ r r(T τr v r + P) = + p + r2 π φφ
,
r
π ηη = − 1 π ηη − 2η θ
γ r τ π τ 2 3 − γ r
,
τ
π φφ = − 1
γ r τ π
π φφ − 2η
r 2
θ
3 − γ rv r
r
Close equations with EOS p(e) where e = T ττ −v r T τr and v r = T τr /(T ττ +P).
.
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 9(26)

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(2+1)-d viscous hydrodynamic equations
Heinz, Song & Chaudhuri, PRC 73 (2006) 034904
Transverse dynamics w/o azimuthal symmetry, but with long. boost invariance:
Use (τ, x, y, η) coordinates and solve
• hydrodynamic equations for T ττ = (e+p)γ 2 r −p+πττ , T τx = (e+p)γ 2 ⊥ v x+π τx ,
1
τ ∂ τ
1
τ ∂ τ
1
τ ∂ τ
τT ττ
τT τx
τT τy
+ ∂ x v x T ττ
+ ∂ x v x T τx
+ ∂ x v x T τy
T τy = (e+p)γ 2 ⊥ v y+π τy :
+ ∂ y v y T ττ
+ ∂ y v y T τx
+ ∂ y v y T τy
= ScT ττ [v x , v y , π ηη , π ττ , π τx , π τy ]
= ScT τx [v x , v y , π xx , π xy , π τx ]
= ScT τy [v x , v y , π yy , π xy , π τy ]
• kinetic relaxation equations for π ττ , π τx , π τy , and π ηη (4, not 3!).
Close equations with EOS p(e) where e = M 0 − v ⊥ M and v ⊥ = M/(M 0 +p(e))
(again one implicit scalar equation!), with the definitions √
(M 0 , M x , M y ) ≡ (T ττ −π ττ , T τx −π τx , T τy −π τy ) and M = Mx+M 2 y,
2
and the relations v x = M x /M, v y = M y /M.
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 10(26)

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(2+1)-d viscous hydro: less longitudinal work, more radial flow
Huichao Song
Cu+Cu @ b = 0, EOS Q
τ 0 = 0.6 fm c , e 0 = 30 GeV
fm 3 , η s = 1
4π , τ π = 0.24
200 MeV
T
fm
c , T dec = 130 MeV
s(fm -3 )
100
10
1
Entropy density vs. time
r=0 fm
r=3fm
viscous hydro 1+1d
ideal hydro 1+1d
viscous hydro 1d Bj
ideal hydro 1d Bj ~ τ −1
Cu+Cu, b=0
τ −1 τ −3
1 10
τ−τ 0
(fm/c)
T (MeV)
300
200
100
r=0 fm
r=9 fm
Temperature vs. time
Cu+Cu, b=0
viscous hydro (1+1d)
ideal hydro (1+1d)
viscous hydro 1d Bj
ideal hydro 1d Bj
0
0 2 4 6 8 10 12
τ−τ 0
(fm/c)
• Radial flow develops much faster, expansion turns 3-dimensional more abruptly
• Shear viscosity initially reduces the cooling due to longitudinal work, but then leads to faster
cooling in the fireball center than for ideal fluid later, due to stronger radial flow
(seen also by Teaney 2004, Chaudhuri 2006,2007; Romatschke et al. 2006,2007)
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 11(26)

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Central Cu+Cu (b=0): ideal vs. viscous hydro (I)
Huichao Song
τ 0 = 0.6 fm c , e 0 = 30 GeV
fm 3 , η s = 1
4π , τ π = 0.24
200 MeV
T
fm
c , T dec = 130 MeV
15
T=130 MeV
Cu+Cu, b=0 fm (ideal hydro)
0.1 0.2 0.3 0.4 0.5
free hadrons
0.6
15
T=130 MeV
Cu+Cu, b=0 fm (viscous hydro)
0.1 0.2 0.3 0.4 0.5
free hadrons
0.6
τ (fm/c)
10
T=150 MeV
HRG
HRG
0.7
τ (fm/c)
10
T=150 MeV HRG
HRG
0.7
5
MP
5
MP
QGP
QGP
0 2 4 6 8
r (fm)
0 2 4 6 8
r (fm)
• Viscous hydro smoothes out phase transition structures
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 12(26)

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Central Cu+Cu (b=0): ideal vs. viscous hydro (II)
Huichao Song
τ 0 = 0.6 fm c , e 0 = 30 GeV
fm 3 , η s = 1
4π , τ π = 0.24
200 MeV
T
fm
c , T dec = 130 MeV
15
Cu+Cu, b=0 fm (ideal hydro v.s. viscous hydro)
T=130 MeV
free hadrons
viscous hydro
ideal hydro
15
Cu+Cu, b=0 fm (ideal hydro v.s. viscous hydro)
T=130 MeV
0.1 0.2 0.3 0.4 0.5
free hadrons
viscous hydro
ideal hydro
τ (fm/c)
10
T=150 MeV
HRG
HRG
τ (fm/c)
10
T=150 MeV
HRG
HRG
0.6
0.7
5
MP
5
MP
QGP
QGP
0 2 4 6 8
r (fm)
0 2 4 6 8
r (fm)
• Viscous hydro cools more slowly than ideal hydro, except for the center where
cooling is accelerated by faster radial expansion in the viscous case
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 13(26)

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(2+1)-d viscous hydro: more radial flow, flatter spectra
Cu+Cu @ b = 0, EOS Q
τ 0 = 0.6 fm c , e 0 = 30 GeV
fm 3 , η s = 1
4π , τ π = 0.24
200 MeV
T
fm
c , T dec = 130 MeV
radial velocity vs. time
hadron p T -spectra
〈V r
〉
0.5
0.4
0.3
0.2
0.1
Cu+Cu, b=0 fm
EOS Q
0
0 2 4 6 8 10
τ-τ 0
(fm/c)
(1/2π)dN/dyp T
dp T
(GeV -2 )
10 2
10 0
10 -2
10 -4
10 -6
π −
K +
P
b=0
Viscous hydro (evolution+spectra correction)
viscoushydro (evolution part)
ideal hydro
0 1 2 3
P T
(GeV)
• For identical initial and freeze-out conditions, viscous evolution yields more radial flow and flatter
spectra (as previously seen by Chaudhuri 2006,2007; Romatschke 2007)
• Effect on b = 0 spectra can be largely absorbed by starting viscous hydro later with lower initial
density (Romatschke et al., 2006,2007)
E dN
d 3 p =
Σ
p·d 3 σ(x)
(2π) 3
f eq (x, p) + δf(x, p)
=
Σ
p·d 3 σ(x)
(2π) 3 f eq (x, p)
1 + 1 p α p β π αβ (x)
2 T 2 (x) (e+p)(x)
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 14(26)

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Non-central collisions (Cu+Cu @ b=7 fm)
Huichao Song
τ 0 = 0.6 fm c , e 0 = 30 GeV
fm 3 , η s = 1
4π , τ π = 0.24
200 MeV
T
fm
c , T dec = 130 MeV
τ(fm/c)
Cu+Cu, b=7 (ideal hydro)
Cu+Cu, b=7 fm (viscous hydro)
0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.1 0.2 0.3 0.4 0.5 0.6 0.7
10 10
y
T=130MeV
T=150MeV
free hadrons
HRG
5 HRG
5
x
τ(fm/c)
0.7
0.6
0.5 0.3
10 10
free hadrons
y
0.4
0.2
T=130MeV
0.1
T=150MeV
HRG
0.1 0.2 0.3 0.4 0.5 0.6 0.7
5 HRG
5
x
MP
MP
QGP
QGP
Y
-5 0 5
r (fm)
X
Y
-5 0 5
r (fm)
X
• Viscous hydro smoothes out phase transition structures
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 15(26)

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Non-central collisions (Cu+Cu @ b=7 fm)
Huichao Song
τ 0 = 0.6 fm c , e 0 = 30 GeV
fm 3 , η s = 1
4π , τ π = 0.24
200 MeV
T
fm
c , T dec = 130 MeV
Cu+Cu, b=7 fm (along x axis)
Cu+Cu, b=7fm (along y axix)
10
0.7
0.6
0.5 0.4 0.3
T=130MeV
0.2
0.1
free hadrons
0.1 0.20.3 0.4 0.5 0.6 0.7
0.7 0.6 0.5 0.4 0.3 0.2
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.1
10 10
T=130 MeV
free hadrons
τ(fm/c)
5
viscous
T=150 MeV
HRG
HRG
ideal
τ(fm/c)
viscous
T=150 MeV
HRG
ideal
5 HRG
5
MP
MP
QGP
QGP
X
-5 0 5
r (fm)
X
Y
-5 0 5
r (fm)
Y
• Viscous hydro smoothes out phase transition structures
• Other than that, viscous fluid cools more slowly than ideal fluid (less radial flow than b=0)
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 16(26)

(2+1)-d viscous hydro: less momentum anisotropy
Cu+Cu @ b = 7 fm, EOS Q, same initial and final conditions
0.06
0.05
0.04
0.03
0.02
0.01
flow anisotropy vs. time
Cu+Cu, b=7 fm
EOS Q
0
0 2 4 6 8
τ−τ 0
spatial eccentricity and momentum anisotropy
ε x
0.25
0.2
0.15
0.1
0.05
0
Cu+Cu, b=7 fm
viscous hydro
ideal hydro
0 2 4 6 8
τ−τ 0
(fm/c)
0.25
0.2
0.15
0.1
0.05
0
viscous hydro
ideal hydro
0 2 4 6 8
τ−τ 0
ε p
0.08
µν
viscous hydro (ideal T 0 )
0.06
0.04
0.02
0
viscous hydro (full T µν )
ideal hydro
0 2 4 6 8
τ−τ 0
(fm/c)
• Flow anisotropy develops faster initially, but stalls earlier than for ideal fluids;
• Source eccentricity decays initially faster, but more slowly later;
• Total momentum anisotropy is reduced by almost 50% relative to ideal fluid;
viscous pressure components contribute very strongly to this reduction during the first 3-4 fm/c
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 17(26)

Elliptic flow from (2+1)-dim. viscous hydrodynamics (I)
Cu+Cu @ b = 7 fm
(Initial conditions adjusted such that at b=0 same π, ¯p spectra as in ideal hydro)
v 2
0.4
η/s = 1/4π, τ π = 0.24(200 MeV/T ) fm/c
Viscous hydro (evolution + spectra correction)
Viscous hydro (evolution part)
ideal hydro
0.3
0.2
Cu+Cu, b=7 fm
π −
K +
P
0.1
0
0 0.5 1 1.5 2 2.5 3
P T
(GeV)
• Elliptic flow very sensitive to even minimal shear viscosity!
• Viscous corrections to equilibrium distribution fct. have significant effect on v 2 (Teaney 2003),
but at low p T the effects from the reduced hydrodynamic flow anisotropy are larger
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 18(26)

Elliptic flow from (2+1)-dim. viscous hydrodynamics (II)
Cu+Cu @ b = 7 fm
(Initial conditions adjusted such that at b=0 same π, ¯p spectra as in ideal hydro)
v 2
0.4
Viscous hydro (evolution + spectra correction)
Viscous hydro (evolution part)
ideal hydro
0.3
0.2
Cu+Cu, b=7 fm
η/s=0.08, τ π
=1.5η/sT
η/s=0.08, τ π
=3η/sT
π −
0.1
0
0 1 2 3
P T
(GeV)
• Faster kinetic relaxation at fixed η/s reduces viscous effects −→ Janik 2007
Ulrich Heinz Viscous hydrodynamics (WPCF, 08/01/2007) 19(26)

The limits of viscous hydrodynamics
At sufficiently large p T , viscous corrections become large even if η/s is small.
|δN(p)| > 1 2 |N 0(p)| indicates breakdown of the assumptions:
10
Where viscous hydro breaks down
Cu+Cu
p T
(GeV)
from viscous hydro calculation b=0 fm
from viscous hydro calculation b=7 fm
1
0 50 100 150 200
e 0
(GeV/fm 3 )
• For larger initial energy densities, p T -range of applicability of viscous hydro to describe
hadron spectra increases.
Viscous hydrodynamics (WPCF, 08/01/2007) 20(26)

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¢
£
(GeV/fm 3 )
0
-1
Which viscous pressure components dominate?
Cu+Cu @ b = 7 fm, EOS Q
τ 0 = 0.6 fm c , e 0 = 30 GeV
fm 3 , η s = 1
4π , τ π = 0.24
τ 2
0.025
(GeV/fm 3 )
0
Cu+Cu, b=7 fm
0 2 4 6 8
τ−τ 0
(fm/c)
-2
0 2 4 6 8
τ−τ 0
(fm/c)
(GeV/fm 3)
0
-1
-2
200 MeV
T
fm
c , T dec = 130 MeV
Cu+Cu, b=7 fm
-p-τ∂ x
(pv x
)-τ∂ y
(pv y
)-τ 2 π ηη
-τ∂ x
p-τ∂ x
π xx
-τ∂ y
p-τ∂ y
π yy
full viscous
full viscous
full viscous
0 2 4 6 8
τ−τ 0
(fm/c)
• For constant η/s, viscous pressure effects (relative to p) are largest at early times
• Largest viscous pressure components are π ηη , π xx , π yy .
• At late times, all viscous pressure components become much smaller than thermodynamic pressure
• Still, effects on spectra are large at high p T , due to factor (p T /T ) 2
E dN
d 3 p =
Σ
p·d 3 σ(x)
(2π) 3
f eq (x, p) + δf(x, p)
=
Σ
p·d 3 σ(x)
(2π) 3 f eq (x, p)
1 + 1 p α p β π αβ (x)
2 T 2 (x) (e+p)(x)
Viscous hydrodynamics (WPCF, 08/01/2007) 21(26)

Sensitivity to initial values for viscous pressure tensor (I)
Romatschke & Romatschke 2007 seem to find much smaller viscous effects than we do. But they
initialize their evolution with π mn = 0. Could this be the origin of the discrepancy? No!
0.06
0.05
0.04
0.03
0.02
0.01
flow anisotropy vs. time
Cu+Cu, b=7 fm
EOS Q
0
0 2 4 6 8
τ−τ 0
spatial eccentricity and momentum anisotropy
ε x
0.3
0.25
0.2
0.15
0.1
0.05
0
ideal hydro
viscous hydro, initialized by π mn =2ησ mn
viscous hydro, initialized by π mn =0
0 2 4 6 8
τ−τ 0
(fm/c)
0.1
0.25
0.2
0.15
0.1
0.05
0
0 2 4 6 8
τ−τ 0
ε p
0.08
µν
viscous hydro (ideal T 0 )
0.06
0.04
0.02
0
viscous hydro (full T µν )
ideal hydro
0 2 4 6 8
τ−τ 0
(fm/c)
Green lines show results for π mn
0
= 0, with otherwise identical parameters
=⇒ weak sensitivity to initial conditions for viscous pressure tensor.
Viscous hydrodynamics (WPCF, 08/01/2007) 22(26)

¡
Sensitivity to initial values for viscous pressure tensor (II)
Green lines: π mn
0 = 0; Other lines: π mn
0 = 2ησ mn ≡ 2η∇ 〈m u n〉 .
τ 0 = 0.6 fm c , e 0 = 30 GeV
fm 3 , η s = 1
4π , τ π = 0.24
200 MeV
T
fm
c , T dec = 130 MeV
v 2
0.3
0.2
0.1
pion elliptic flow
Cu+Cu, b=7 fm, EOS Q
ideal hydro
viscous hydro, initialized by π mn =0
viscous hydro, initialized by π mn =2ησ mn
Viscous hydro (evolution + spectra correction)
Viscous hydro (evolution part)
0
0 1 2 3
P T
(GeV)
π −
(GeV/fm 3 )
largest viscous pressure components vs. time
0
-1
τ 2
(GeV/fm 3 )
0
Cu+Cu, b=7 fm
EOS Q
0 2 4 6 8
τ−τ 0
(fm/c)
0 2 4 6 8
τ−τ 0
(fm/c)
• After τ ∼ 1 fm/c ∼ 4τ π , viscous pressure tensor has lost all memory of initial
conditions!
• Effects on final v 2 small – discrepancy with (Romatschke) 2 unresolved
Viscous hydrodynamics (WPCF, 08/01/2007) 23(26)

Other tests of the viscous hydro code
• Setting η = 0, code accurately reproduces results obtained with P. Kolb’s ideal
fluid AZHYDRO code.
• For transversally homogeneous initial conditions (infinite slab, no transverse
flow), fixed c 2 s (p = e/3) and τ π → 0 (Navier-Stokes limit), code accurately
reproduces known analytic solution for e(τ).
• For small η and τ π , code based on 2nd-order Israel-Stewart approach accurately
reproduces results from a 1st-order Navier-Stokes code (this tests the kinetic
evolution algorithm for π mn ):
Viscous hydrodynamics (WPCF, 08/01/2007) 24(26)

η/s = 0.01 fm/T, τ π = 0.03 fm/c, ideal gas EoS w/o phase transition
v 2
0.5
0.4
0.3
0.2
0.1
ideal hydro
1st order viscous hydro (evolution+spectra corrections)
1st order viscous hydro (evolution corrections)
2nd order viscous hydro (evolution+ spectra corrections)
2nd order viscous hydro (evolution corrections)
gluons
Cu+Cu, b=7 fm
EOS I
0 1 2 3 4
P T
(GeV)
• Still need to check that σ mn is correctly computed.
Viscous hydrodynamics (WPCF, 08/01/2007) 25(26)

Summary
• Shear viscosity reduces the longitudinal pressure but increases the transverse
pressure in heavy ion collision
=⇒ slower cooling by longitudinal work initially, but faster cooling by stronger
transverse expansion later
• While viscous pressure effects on angle-averaged p T -spectra (radial flow) can be
largely absorbed by changing the initial conditions (starting the transverse expansion
later and with lower initial energy density), this increases the destructive
effects of shear viscosity on the buildup of elliptic flow.
• The effects of shear viscosity on elliptic flow are shockingly large; even Son’s
minimal viscosity η/s = 1/4π seems incompatible with RHIC data =⇒ needs
more checking.
• Shorter kinetic relaxation times for the viscous pressure tensor seem to reduce
the effects from shear viscosity.
• Viscous corrections to the momentum spectra are sensitive to details of the
flow pattern at freeze-out and not easily captured with blast-wave model
parametrizations.
Viscous hydrodynamics (WPCF, 08/01/2007) 26(26)

Supplements
Viscous hydrodynamics (WPCF, 08/01/2007) 27(26)

Rest mass dependence of differential elliptic flow
(the “fine structure”)
STAR Coll., PRL 87, 182301 (2001) and PRL 92, 052302 (2004); PHENIX Coll., PRL 91, 182301 (2003)
Anisotropy Parameter v 2
0.3
0.2
0.1
0
Hydro model
π
K
p
Λ
PHENIX Data
π + +π -
+ -
K +K
p+p
STAR Data
0
K S
Λ+Λ
0 2 4 6
Transverse Momentum p T
(GeV/c)
v 2
(%)
10
5
STAR
hydro EOS Q
hydro EOS H
π + + π −
p + p
0
0 0.25 0.5 0.75 1
p (GeV) T
Data follow the hydrodynamically predicted rest mass dependence of v 2 (p ⊥ ) out
to p ⊥ ∼ 1.5 GeV for mesons and out to p ⊥ ∼ 2.3 GeV for baryons
=⇒ bulk of matter (> 99% of all particles) behaves hydrodynamically!
Note: mass-splitting of v 2 (“fine structure”) sensitive to EOS!
Viscous hydrodynamics (WPCF, 08/01/2007) 28(26)

Radial flow profiles for b=0 Cu+Cu:
η/s = 1/4π, τ π = 0.24(200 MeV/T ) fm/c
Cu+Cu, b=0 fm
v r
0.8
0.6
e dec
=1.65GeV/fm 3
v r
0.8
0.6
e dec
=0.45GeV/fm 3
viscous hydro
ideal hydro
0.4
QGP-MP
0.4
MP-HRG
0.2
0.2
0
0
0 2 4 6 r (fm) 0 2 4 6 r (fm)
v r
v r
0.8
0.8
0.6
T dec
=150 MeV
0.6
T dec
=130 MeV
0.4
0.4
0.2
0.2
0
0 2 4 6
r (fm)
0
0 2 4 6
r (fm)
Viscous hydrodynamics (WPCF, 08/01/2007) 29(26)

Israel-Stewart or Navier-Stokes?
Deviations of the dynamical viscous pressure components from their relaxed values
are significant:
2
2
2(〈π 1st 〉-〈π 2nd 〉)/(〈π 1st 〉+〈π 2nd 〉)
1.5
1
0.5
0
Cu+Cu, b=0 fm
η/s=0.08, τ π
=3η/sT
η/s=0.08, τ π
=1.5η/sT
2(〈π 1st 〉-〈π 2nd 〉)/(〈π 1st 〉+〈π 2nd 〉)
1
0
-1
Cu+Cu, b=0 fm
π ηη π φφ 0 5 10
η/s=0.08, τ π
=3η/sT
η/s=0.08, τ π
=1.5η/sT
τ 2 π ηη
π xx (π yy )
-0.5
0 5 10
τ−τ 0
(fm/c)
• AdS/CFT approach (Janik 2007) suggests τ π = τ π Boltz
30
, but
-2
τ−τ 0
(fm/c)
• τ π → 0 =⇒ Navier-Stokes =⇒ problems with acausal signal propagation!
How to recognize numerical problems with causality?!
Viscous hydrodynamics (WPCF, 08/01/2007) 30(26)

Viscous corrections to p T -spectra:
Blast wave model vs. real hydrodynamics
Using a blast-wave model parametrization of the freeze-out distribution, Teaney
(2003) found opposite sign of viscous correction effects on p T -spectra:
Vr
1
0.5
Blast Wave model
u r
=(0.5r/R 0
)θ(R 0
-r)
R 0
=6fm
Vr
1
0.5
ideal hydro EOS I
Cu+Cu *
6
b=0fm
4
2
1
*
*
*
Vr
1
0.5
along freeze-out surface
ideal hydro EOSI
Cu+Cu, b=0fm
0.005
π rr (GeV/fm 3 )
0
0 2 4 6 8
0
0
π rr =2η∇ 〈r u r〉
τ f
=4.1fm/c
2 4 6 8
r (fm)
r (fm)
0
0 2 4 6 8
0
π rr (GeV/fm 3 )0.5
-0.5
0
x1
x3
x10
x50
π rr =2η∇ 〈r u r〉
*
*
*
2 4 6 8
r (fm)
τ (fm/c)
1
2
4
6
r (fm)
π rr (GeV/fm 3 )
0
0 2 4 6
0.02
0.01
0
-0.01
-0.02
π rr =2η∇ 〈r u r〉
-0.03
0 2 4 6
• Velocity gradients near surface dominate viscous corrections to spectra!
• Blast wave model too unrealistic.
r(fm)
r(fm)
Viscous hydrodynamics (WPCF, 08/01/2007) 31(26)