d´enes moln´ar, purdue university 10th zim´anyi winter ... · d´enes moln´ar, purdue...
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From a viscous fluid to particles
Denes Molnar, Purdue University
10th Zimanyi Winter School on Heavy Ion Physics
Nov 29 - Dec 3, 2010, KFKI/RMKI, Budapest, Hungary
Outline
I. Hydro and identified particle observables
II. Converting fluid to particles
III. Summary
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 1
“Little bang”: plasma of quarks and gluons
- T ∼ 200 − 400 MeV- perturbative at high energies- electric AND magnetic screening
Relativistic Heavy Ion Collider Large Hadron Collider
↑1 km
↓ v = 0.99995 · c
since 2000-
Au+Au:√
s = 200 GeV/nucleon
already running!
Pb+Pb:√
s = 2700 GeV/nucl
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 2
RHIC collisions look largely thermalized → should be able to
measure equation of state and viscosity
e.g., efficient conversion of spatial eccentricity to momentum anisotropy
→ “elliptic flow”
ε ≡ 〈x2−y2〉〈x2+y2〉
v2 ≡ 〈p2x−p2y〉
〈p2x+p2y〉
≡ 〈cos 2φp〉
large energy loss, even for heavy quarks
RAA =measured yield
expected yield for dilute system
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 3
I. Hydro paradigm and particles
Identified particle observables are crucial
- help pin down initial conditions
- hold the key to equation of state
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 4
Hydrodynamicsdescribes systems near local equilibrium, in terms of macroscopic variables
e(x), p(x), nB(x) - energy density, pressure, baryon density
uµ(x) ≡ γ(v)(1, ~v(x)) - flow velocity
conservation laws: ∂µTµν(x) = 0, ∂µN
µB(x) = 0
Tµν: energy-momentum tensor, NµB: baryon current
medium given through equation of state p(e, nB)
ideal ≡ local equilibrium TµνLR = diag(e, p, p, p), NµB,LR = (nB, 0)
nonideal ≡ small deviations from local equilibrium → dissipation
Local Rest frame (comoving frame)ւ
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 5
∼ 2000-01: Ideal hydroAu+Au @ RHIC: spectral shapes work quite well Kolb & Heinz, nucl-th/0305084
0 1 2 3
10−2
100
102
pT (GeV)
1/ 2
π dN
/dyp
Tdp
T (
GeV
−2 ) π+ PHENIX
p PHENIXp STARπ+ hydrop hydro
most central
0 0.5 1 1.5 2 2.5
10−6
10−4
10−2
100
102
pT (GeV)
1/2π
dN
/dyp
Tdp
T (
GeV
−2 ) PHENIX
STARhydro
π−
0 − 5 %
5 − 15 %
15 − 30 %
30 − 60 %
60 − 92 %
centrality
0 1 2 3 410
−8
10−6
10−4
10−2
100 PHENIX
STARhydro
p
pT (GeV)
1/2π
dN
/dyp
Tdp
T (
GeV
−2 )
0 − 5 %
5 − 15 %
15 − 30 %
30 − 60 %60 − 92 %
centrality
0 0.5 1 1.5 2
10−6
10−4
10−2
100
pT (GeV)
1/2π
dN
/dyp
Tdp
T (
GeV
−2 )
PHENIXhydro
K+
0 − 5 %
5 − 15 %
15 − 30 %
30 − 60 %
60 − 92 %centrality
τ0 = 0.6 fm
ε0 = 25 GeV/fm3
T0 ≈ 360 MeV
Tfo ≈ 130 MeV
[75% wounded
+ 25% binary geom
bag EOS
s0 = 110 / fm3]
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 6
Elliptic flow (v2)
spatial anisotropy → final azimuthal momentum anisotropy
ε ≡ 〈x2−y2〉〈x2+y2〉 → v2 ≡ 〈p2
x−p2y〉
〈p2x+p2
y〉
- measure of early pressure gradients
- sensitive to interaction strength (degree of thermalization)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 7
also seen in ultra-cold atomic systems
dN(t)/dx dy dN(t)/dx dy
↔10−4m t ∼ 10−3s
droplet of 6Li
T ∼ 10−6 K
n ∼ 1019/m3
unitarity limit
(Feshbach resonance)
σ(k) = 4π/k2
O’Hara et al, Science 298 (’03)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 8
∼ 2000-01: Ideal hydrominimum-bias Au+Au at RHIC Kolb, Heinz, Huovinen et al (’01), nucl-th/0305084
0 0.25 0.5 0.75 10
5
10
pT (GeV)
v 2 (%
)π+ + π−
p + p
STARhydro EOS Qhydro EOS H
τtherm = 0.6 fm , 〈e〉init ∼ 5GeV/fm3, Tfreezeout = 120 MeV
pion-proton splitting favors QGP phase transition over pure hadron gas
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 9
Two recent refinements:
- realistic equation of state
- QGP viscosity
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 10
QCD equation of state (µB = 0)
QCD on a space-time lattice - Z =R
DψDψDAe−SQCDE
A. Bazavov et al, PRD80 (’09)
0
2
4
6
8
10
12
14
16
100 200 300 400 500 600 700
0.4 0.6 0.8 1 1.2 1.4 1.6
T [MeV]
Tr0 εSB/T4
ε/T4: Nτ=468
3p/T4: Nτ=468
eSBT 4
≈ d.o.f.
3
cross-over with a jump in effective degrees of freedom near Tc
quite robust results, though still evolving - Tc ≈ 170 − 180(−200) MeV
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 11
Ideal hydro + realistic EOSserious eyesore for hydro paradigm: realistic EOS gives same as hadron gas?!
Huovinen, NPA761, 296
(’05)
Q: bag model
qp: lattice fit(Tc = 170 MeV)
H: hadron gas
T: interpolated ε(T )between hadron gasand ε ∝ T 4 plasma
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 12
Hope: dissipation could help. 2005 PHENIX White Paper NPA757, 184 (’05): forbest agreement, must couple hydro to late-stage hadronic transport.
(Gev/c)Tp0 0.5 1 1.5 2 2.5 3 3.5
dy
TN
/dp
2)dπ
(1/2
10-3
10-2
10-1
1
10
102
, PHENIX sqrt(s)=200, 0-5%π QGP EOS+RQMD, Teaney et al.π QGP EOS +PCE, Hirano et al.π QGP EOS +PCE, Kolb et al.π QGP EOS, Huovinen et al.π RG+mixed EOS, Teaney et al.π RG EOS, Huovinen et al.π
(Gev/c)Tp0 0.5 1 1.5 2 2.5 3 3.5
dy
TN
/dp
2)dπ
(1/2
10-3
10-2
10-1
1
10
p PHENIX sqrt(s)=200, 0-5%
p QGP EOS+RQMD, Teaney et al.
(pbar/0.75) QGP EOS +PCE, Hirano et al.
(pbar/0.75) QGP EOS +PCE, Kolb et al.
p QGP EOS, Huovinen et al.
p RG+mixed EOS, Teaney et al.
p RG EOS, Huovinen et al.
(Gev/c)Tp0 0.5 1 1.5 2 2.5
ε/ 2v
00.10.20.30.40.50.60.70.80.9
1 PHENIX sqrt(s)=200, minBiasπ
STAR sqrt(s)=130, minBiasπ
QGP EOS+RQMD, Teaney et al.π
QGP EOS +PCE, Hirano et al.π
QGP EOS +PCE, Kolb et al.π
QGP EOS, Huovinen et al.π
RG+mixed EOS, Teaney et al.π
RG EOS, Huovinen et al.π
(Gev/c)Tp0 0.5 1 1.5 2 2.5
ε/ 2v
00.10.20.30.40.50.60.70.80.9
1p PHENIX sqrt(s)=200, minBias
p STAR sqrt(s)=130, minBias
p QGP EOS+RQMD, Teaney et al.
p QGP EOS +PCE, Hirano et al.
p QGP EOS +PCE, Kolb et al.
p QGP EOS, Huovinen et al.
p RG+mixed EOS, Teaney et al.
p RG EOS, Huovinen et al.
π spectra p spectra
π elliptic flow p elliptic flow
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 13
a contender paradigm (waiting for independent confirmation)
quark-gluon transport with elastic 2 → 2 AND radiative 3 ↔ 2
Xu & Greiner, (’08)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.000.020.040.060.080.100.120.140.160.180.20
PHOBOS Track-based (0-50%)
v2
pT (GeV/c)
s=0.3, ec=1 GeV fm-3
s=0.6, ec=1 GeV fm-3
s=0.3, ec=0.6 GeV fm-3
s=0.6, ec=0.6 GeV fm-3
BAMPS:
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 14
Viscous QGP(?)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 15
(Shear) viscosity
1687 - I. Newton (Principia)
Txy ≡FxA
= −η∂ux∂y
η: shear viscosityreduces velocity gradients
⇒ dissipation
1985 - Heisenberg ∆E · ∆t + kinetic theory: η/s ≥ h/15kBGyulassy & Danielewicz, PRD 31 (’85)
2004 - string theory AdS/CFT: η/s ≥ 1/4π·h/kBPolicastro, Son, Starinets, PRL87 (’02); Kovtun, Son, Starinets, PRL94 (’05)
revised to η/s ≥ 4h/(25π) Brigante et al, PRL101 (’08)
or even lower Camanho et al, arXiv:1010.1682
“minimal viscosity” or not - need to test it experimentally
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 16
Viscosity in QCD - not knownperturbation theory (T ≫ Tc): large η/s > O(1), small ζ/s ∼ 0.02α2
s ∼ 0Arnold, Moore, Yaffe, JHEP 0305 (’03); Arnold, Dogan, Moore, PRD74 (’06)
lattice QCD estimates: shear Nakamura & Sakai, NPA774 (’06) bulk Meyer, PRD76 (’07)
-0.5
0.0
0.5
1.0
1.5
2.0
1 1.5 2 2.5 3
243
8
163
8
s
KSS bound
Perturbative
Theory
T Tc
η
0.001
0.01
0.1
1
0.1 1
ζ/s
(ε-3P)/(ε+P)
1.02
1.24
1.652.22
3.22
L0/a=8 data
inversion problem - you compute G(τ) =∫ ∞
0dω σ(ω)K(ω, τ) at 8-12 points
τi, but you need dσ/dω at ω = 0 to obtain viscosities
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 17
viscosities at RHIC and LHC not known but could indeed be small
Schafer, PRA 76 (’07) Mueller et al, PRL103 (’09)
cold atoms
∼ 5 × 1
4π∼ 3 × 1
4π
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 18
What viscosity doesentropy production:
∂µSµ =
Π2
ζT− qµqµ
κT 2+
πµνπµν2ηT
> 0
sound damping:
ω(k) = csk − i
2k2Γs + O(k3) Γs ≡
43η + ζ
e + p
sound attenuation length
slower cooling:
dE = −pdV + TdS (dS > 0)
anisotropic pressure: e.g., longitudinal expansion, with shear only
Tµν = diag(e, p+πL/2, p + πL/2, p−πL)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 19
Dissipative frameworks
• causal relativistic hydrodynamics Israel, Stewart; ... Muronga, Rischke; Teaney et al; Romatschke
et al; Heinz et al, DM & Huovinen ... Niemi et al... Van et al..
∂µTµν = 0 (µB → 0)
Tµν = (e + p)uµuν − pgµν + πµν − Π∆µν
πµν = Fµν(e, u, π,Π) , Π = G(e, u, π,Π)
e.g. Israel-Stewart theory
• covariant transport Israel, de Groot,... Zhang, Gyulassy, DM, Pratt, Xu, Greiner...
pµ∂µf = C2→2[f ] + C2↔3[f ] + · · ·
fully causal and stable
near hydrodynamic limit, transport coefficients and relaxation times:
η ≈ 1.2T/σtr, τπ ≈ 1.2λtr
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 20
when viscosity is small, transport becomes viscous hydro
Au+Au at RHIC, b = 8 fm Huovinen & DM (’08)
pressure in the core, r⊥ < 1 fm
η/s ≈ 1/(4π) (σ ∝ τ2/3)
elliptic flow vs pT
• σ = const ∼ 47mb• η/s ≈ 1/(4π), i.e., σ ∝ τ2/3
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 21
Shear viscosity from RHIC dataRomatschke & Luzum, PRC78 (’08): Au+Au data vs 2+1D viscous hydrodynamics
0 1 2 3 4p
T [GeV]
0
5
10
15
20
25v 2
(per
cent
)STAR non-flow corrected (est.)
Glauber
η/s=10-4
η/s=0.08
η/s=0.16
although in the end gave conservative estimate η/s <∼ 0.5
many uncertainties: hydro validity, η/s(T ), initial conditions, decoupling...
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 22
Romatschke & Romatschke, PRL99 (’07)
0 1 2 3 4p
T [GeV]
0
5
10
15
20
25v 2
(per
cent
)
idealη/s=0.03η/s=0.08η/s=0.16STAR
Dusling & Teaney, PRC77
0
0.05
0.1
0.15
0.2
0 0.5 1 1.5 2
v 2
pT (GeV)
χ=4.5Ideal
η/s=0.05η/s=0.13
η/s=0.2
Huovinen & DM, JPG35 (’08): η/s ≈ 1/4π Song & Heinz, PRC78 (’08)
Consensus on ∼20−30% effects for η/s = 1/(4π) in Au+Au at RHIC.
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 23
ALSO - ballpark agreement with estimates based on kinetic theory, transversemomentum fluctuations, heavy-quark diffusion,... Zajc @ QM2009
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 24
II. From viscous fluid to particles
For viscous hydro calculations, identified
particle observables are challenging
(yet unsolved problem)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 25
Hydro → particles
heavy-ion applications in the end must match hydrodynamics to a particledescription
• in local equilibrium - one-to-one maping
TµνLR = diag(e, p, p, p) ⇔ feq,i = gi(2π)3
e−pµi uµ/T
• near local equilibrium - one-to-many
Tµν = Tµνideal + δTµν ⇐ fi = feq,i + δfi
corrections crucially affect basic observables - spectra, elliptic flow, ...
unavoidable whether we do pure hydro or hydro + transport
a separate issue: Cooper-Frye freezeout (“t 6= const”)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 26
Separate issue: Cooper-FryeCooper & Frye, PRD10 (’74)
fluid to a gas on a 3D hypersurface (e.g., T (x) = Tfo)
EdN
d3p= pµdσµ(x) fgas(T (x), µ(x), u(x), ~p)
dσµ: hypersurface normal at x
conversion at constant time is OK: dσµ = d3x (1,~0)
⇒ dN = fgas(T (x), µ(x), u(x), ~p) d3x d3p
but for arbitrary hypersurface, negative yield when pµdσµ(x) < 0
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 27
TWO effects: - dissipative corrections to hydro fields uµ, T, n- dissipative corrections to thermal distributions f → f0 + δf
Huovinen & DM (’08) η/s ≈ 1/(4π) (σ ∝ τ2/3)
Grad:
δf = f0
[
1 +pµpνπµν
8nT 6
]
most of the v2 reduction comes from phase space correction δf
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 28
Teaney et al, PRC81 (’10)
0
0.1
0.2
0.3
0.4
0.5
0 0.5 1 1.5 2 2.5 3 3.5 4
v 2 (
p T)
pT [GeV]
Ideal
Linear
Quadratic
Pure Glue
energy dependence of δf affects observables at higher pT
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 29
for one-component massless gas, with viscous shear only
δf ≡ feq × C(χ) πµνpµpνT 2
χ(p
T)
from Grad’s ansatz: χ ≡ 1
this is a starting point in deriving IS hydro from kinetic theory
from linear response: χ(x) ∼ xα with −1 <∼ α <∼ 0 Dusling, Teaney, Moore, (’09)
but δf blows up at large momenta ⇒ approximation breaks down
check these from nonequilibrium transport...
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 30
Test in 0+1D Bjorken → f = f(pT , ξ, τ), where ξ ≡ η − y
i) compute f from full nonequilibrium transport
ii) from f , determine Tµν and δf
iii) estimate δf from Tµν alone via various ansatzes for χ(p/T ) (e.g. Grad’s)
δf ∝ feqπL16p
(pTT
)α+2
[ch(2ξ) − 2]
iv) compare δfestimated with δfreal
For simplicity, compare integrated quantities dN(τ)/dp2Tdy|y=0 and dN(τ)/dξ
drive calculation by inverse Knudsen number K0 = τ/λtr ∝ (η/s)−1
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 31
DM (’09):
spectra in 0+1D Bjorken scenario at τ = 2τ0 and 5τ0
for η/s ∼ 0.1, local equilibrium initconds πµν(τ0) = 0
dN/dy d2pT
1e-06
1e-04
0.01
1
100
0 1 2 3 4 5 6 7 8 9
dN/d
y d2 p T
[G
eV-2
]
pT [GeV]
K0 = 3
τ = 2τ0τ = 5τ0
τ = τ0
transportidealviscous, α = 0viscous, α = -1
Grad ansatz (α = 0) works surprisingly well - on a log plot at least
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 32
ratio - transport spectra / Grad approximation, η/s ∼ 0.1
DM (’09):
0.94
0.96
0.98
1
1.02
1.04
1.06
0 1 2 3 4 5 6 7 8 9 10
(dN
/dy
d2 pT)tr
ansp
ort /
(dN
/dy
2 pT)G
rad
pT / T
K0 = 3
τ0 , Grad2τ0 , Grad3τ0 , Grad5τ0 , Grad10τ0 , Grad
Grad spectra are ≈ 1% accurate even at pT/T = 6(!)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 33
ratio - transport spectra / Grad approximation, η/s ∼ 0.3
DM (’09):
0.8
0.85
0.9
0.95
1
1.05
1.1
0 1 2 3 4 5 6 7 8 9 10
(dN
/dy
d2 pT)tr
ansp
ort /
(dN
/dy
2 pT)G
rad
pT / T
K0 = 1
τ0 , Grad2τ0 , Grad3τ0 , Grad5τ0 , Grad10τ0 , Grad
for higher viscosity, accuracy worsens - should affect other observables also
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 34
highlight viscous correction: spectra / ideal spectra
η/s ≈ 0.1 η/s ≈ 0.3
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 1 2 3 4 5 6 7
(∆dN
/dy
d2 pT)
/ (dN
/dy
2 pT)id
eal
pT / T
K0 = 3
Gradτ0 + ε2τ03τ05τ010τ0
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 1 2 3 4 5
(∆dN
/dy
d2 pT)
/ (dN
/dy
2 pT)id
eal
pT / T
K0 = 1
Gradτ0 + ε2τ03τ05τ010τ0
for higher viscosity, 20 − 40% error in viscous correction at low pT
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 35
Grad ansatz not as good for rapidity ξ ≡ η − y correlation
dN/dξ
dNideal/dξdistributions relative to IDEAL hydro, η/s ∼ 0.1 DM (’09-’10)
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
-3 -2 -1 0 1 2 3
(dN
/dξ)
/ (d
N/d
ξ)id
eal
ξ = η - y
K0 = 3
τ = 5τ0
τ = 2τ0transportidealviscous, α = 0viscous, α = -1
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 36
Interplay with pQCD jets
hydro accuracy is further limited by pQCD power-law tails
parton dN/dp2T dy - central Au + Au @ RHIC
1e-06
1e-05
1e-04
0.001
0.01
0.1
1
10
100
1000
0 1 2 3 4 5 6 7 8 9 10
dN/d
y d2 p T
[G
eV-2
]
pT [GeV]
pQCD jetsT = 0.7 GeV (τ0 = 0.1 fm)T = 0.385 GeV (τ0 = 0.6 fm)
study this in a two-component jet + bulk model
for jets: pQCD cross sections, for bulk: strong interactions (η/s ≈ 0.1)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 37
spectra vs hydro approximation DM (’09)-(’10)
1e-06
1e-04
0.01
1
100
0 1 2 3 4 5 6 7 8 9
dN/d
y d2 p T
[G
eV-2
]
pT [GeV]
K0 = 3, τ0 = 0.6 fm
σtr,0jet = 1.5 mb
τ = 2τ0
τ = 5τ0
transport + jetidealviscous, α = 0viscous, α = -1
[bulk: η/s ∼ 0.1, jets: σtr = T 20 /T 2 · 1.5 mb, T0 = 385GeV ]
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 38
ratio - transport spectra / Grad approximation
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5 4
(dN
/dy
d2 pT)tr
ansp
ort /
(dN
/dy
2 pT)G
rad
pT [GeV]
K0 = 3, τ0 = 0.6 fm
σtr,0jet = 1.5 mb
τ0 , Grad2τ0 , Grad3τ0 , Grad5τ0 , Grad10τ0 , Grad
[bulk: η/s ∼ 0.1, jets: σtr = T 20 /T 2 · 1.5 mb, T0 = 385GeV ]
jets spoil accuracy of Grad ansatz for pT >∼ 1.5 GeV
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 39
Hydro → gas mixture
must be tackled to address IDENTIFIED particle data
(!) from ONE set of viscous fields we need to obtain δfi for EVERY species
commonly used “democratic” prescription:
δfi ≡ feqi × πµν
2(e + p)
pµ,ipν,iT 2
(i = π, K, p, Λ, ...)
ignores equilibration dynamics
Ki ∼τ
λi∼ τ
∑
j
njσij
key drivers: relative Knudsen numbers between species Kj/Ki
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 40
Democratic vs 2 → 2 transport
2-component 0+1D Bjorken test DM (’10) - A equilibrates twice as fast as B
δfi = Ci (pT/T )2(sh2y − 1/2)feqi πL,i/pi = 8Ci
-0.4
-0.3
-0.2
-0.1
0
1 2 3 4 5 6 7 8 9 10
coef
fici
ent 8
C in
δf
τ / τ0
K0A = 2.25, K0
B = 1.125, nA = nBABdemocratic
“democratic” ansatz misses viscous effects by ∼ 20 − 25%
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 41
pressure evolution
0.495
0.5
0.505
1 2 3 4 5 6 7 8 9 10
p / p
tota
l
τ / τ0
AB(A+B)/2
VERY slightly more pressure work for species “A”
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 42
transport spectra / “democratic” Grad vs transport / dynamical Grad
DM (’10)
0.85
0.9
0.95
1
1.05
1.1
0 1 2 3 4 5 6 7 8(dN
/dy
d2 pT)tr
ansp
ort /
(dN
/dy
2 pT)de
moc
ratic
pT / T
K0A = 2, K0
B = 1, nA = nB
A, τ0 (+ε)A, 2τ0A, 3τ0A, 5τ0A, 10τ0
0.85
0.9
0.95
1
1.05
1.1
0 1 2 3 4 5 6 7(dN
/dy
d2 pT)tr
ansp
ort /
(dN
/dy
2 pT)dy
nam
ic
pT / T
K0A = 2, K0
B = 1, nA = nB
A, τ0 (+ε)A, 2τ0A, 3τ0A, 5τ0A, 10τ0
“democratic” Grad prescription not accurate
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 43
highlight dissipative correction: spectra for A / ideal ansatz
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 1 2 3 4 5 6 7
(∆dN
/dy
d2 pT)
/ (dN
/dy
2 pT)id
eal
pT / T
K0A = 2.3, K0
B = 1.1, nA = nB
A, τ0 (+ε)A, 2τ0A, 3τ0A, 5τ0A, 10τ0democratic (lines)
20-100% error in dissipative correction, depending on pT and time
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 44
So... we must obtain dynamically determined partial shear
stresses πµνi - BEFORE we can convert to particles
⇒ will a one-component viscous hydro be sufficient?
usual derivation of hydro from kinetic theory based on Grad ansatz givescoupled set of equations between πµνi
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 45
Denicol @ CATHIE-TECHQM, Dec 2009
MORE variables!
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 46
some hope: transport suggests universal sharing, at later times DM (’10)
δfi = Ci (pT/T )2(sh2y − 1/2)feqi πL,i/pi = 8Ci
1
1.25
1.5
1.75
1 2 3 4 5 6 7 8 9 10
CB
/ C
A
τ / τ0
democratic
KA / KB = 4.4 / 2.2 3.8 / 1.9 2.2 / 1.1 6 / 3.8 4.2 / 2.8
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 47
III. Summary
• Theory should strive to address identified particle data. In the hydroparadigm, this is key in order to constrain the equation of state and initialconditions.
• Converting a non-ideal fluid to particles is nontrivial, independently ofCooper-Frye freezeout assumptions.
Comparison with kinetic theory indicates that Grad’s quadratic ansatz isremarkably accurate (1-2%) up to pT /T ∼ 6, at least with 2 → 2interactions and for small shear viscosities η/s ≈ 0.1.
In the multicomponent case, kinetic theory indicates that per-speciesdissipative corrections are driven by the relative opacities. The commonlyassumed “democratic” sharing in viscous hydro calculations is unrealistic.
• Some open questions:
- whether we need to extend hydrodynamics with per-species viscous fields
- accuracy of linear response
- any simple shortcut to the end result
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 48
unique connections between different fields
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 49
Backup slides
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 50
“Bag” equation of state
EOS Q: simplified 1st-order phase transition parameterization
0 1 2 3 40
0.2
0.4
0.6
0.8
1
1.2
1.4
e (GeV/fm3)
p (G
eV/fm
3 )
n=0 fm−3
EOS I
EOS Q
EOS H
combines hadron resonance gas (EOS H) & plasma (EOS 1)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 51
Viscous hydrodynamics
Navier-Stokes: corrections linear in gradients [Landau]
TµνNS = T
µνideal + η(∇
µuν+ ∇
νuµ−
2
3∆µν∂αuα) + ζ∆
µν∂αuα
NνNS = N
νideal + κ
„
n
ε+ p
«2
∇ν
„
µ
T
«
[∆µν≡gµν−uµuν, ∇ν≡∆µν∂µ]
η, ζ: shear and bulk viscosity; κ: heat conductivity
unfortunately NS hydro is unstable and acausalMuller (’76), Israel & Stewart (’79), Hiscock & Lindblom, PRD31 (’85) ...
causal 2nd-order hydro: dynamical corrections
Tµν
≡ Tµνideal + π
µν− Π∆
µν, N
µ≡ N
µideal −
n
e+ pqµ
relaxation eqns for bulk pressure Π, shear stress πµν, heatflow qµ
e.g. Israel-Stewart theory, Ottinger-Grmela, conformal hydro, ...
Israel & Stewart, Ann.Phys 110&118; Ottinger & Grmela, PRE 56&57; Baier et al, JHEP04 (’08)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 52
Israel-Stewart theory - complete set of equations of motion
DΠ = − 1
τΠ(Π + ζ∇µu
µ) (1)
−1
2Π
(
∇µuµ + D ln
β0
T
)
+α0
β0∂µq
µ − a′0
β0qµDuµ
Dqµ = − 1
τq
[
qµ + κqT 2n
ε + p∇µ
(µ
T
)
]
− uµqνDuν (2)
−1
2qµ
(
∇λuλ + D ln
β1
T
)
− ωµλqλ
−α0
β1∇µΠ +
α1
β1(∂λπ
λµ + uµπλν∂λuν) +a0
β1ΠDuµ − a1
β1πλµDuλ
Dπµν = − 1
τπ
(
πµν − 2η∇〈µuν〉)
− (πλµuν + πλνuµ)Duλ (3)
−1
2πµν
(
∇λuλ + D ln
β2
T
)
− 2π〈µλ ων〉λ
−α1
β2∇〈µqν〉 +
a′1
β2q〈µDuν〉 .
where A〈µν〉 ≡ 12∆
µα∆νβ(Aαβ+Aβα)−13∆
µν∆αβAαβ, ωµν ≡ 1
2∆µα∆νβ(∂βuα−∂αuβ)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 53
Why does ideal hydrodynamics work in everyday life??
Ideal hydro is applicable when relative viscous corrections are smallδTµνviscous/Tµνideal ≪ 1
I.e., based on Navier-Stokes we need (with shear only)
η∇iuj
e + p∼ η∇iuj
T s∼ η
s× v
LT≪ 1
where L is the shortest length scale. In heavy ion physics,
v
LT∼ 1
τT∼ 1 → need
η
s≪ 1
In everyday problems v/(LT ) ≫ 1, compensating a large η/s.
What we are after is not ideal hydro behavior but systems where the viscosityis near its quantum (uncertainty) limit.
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 54
Realistically, η/s 6= const
interpolate between wQGP, sQGP, hadron gas + use τπ ≈ η/p
Denicol et al, JPG37 (’10)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 55
same elliptic flow as for low η/s ≈ 0.12(!)
Denicol et al, JPG37 (’10)
⇒ disentangling η and τπ will likely need good theory input
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 56
Validity of hydrodynamicstest against transport: IS hydro accurate for η/s ≈ 1/4π DM & Huovinen, JPG35(’08)
relevant condition: high-enough inverse Knudsen number Huovinen & DM, PRC79
(’08)K0 ≡ τ
λtr=
τexpτscatt
≈ 6τexp5τπ
> ∼ 2 − 3
in terms of shear viscosity η
s∼ 2.6
4πK0
<∼ 2 × 1
4π
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 57
Validity of Israel-Stewart hydro (0+1D Bjorken)Huovinen & DM, PRC79pressure anisotropy Tzz/Txx
Navier-StokestransportIS hydro
η/s ≈ const
ideal hydro
K0 = 1
K0 = 2K0 = 3
K0 = 6.67
τ/τ0
p L/p
T
1 2 3 4 5 6 7 8 9 10
1.2
1
0.8
0.6
0.4
0.2
0
IS is 10% accurate when K0 ≡ τ0/λtr,0 >∼ 2
average pressure
Navier-StokestransportIS hydro
2
1
η/s ≈ const
ideal hydro
1
2
3K0 = 6.67
τ/τ0
p/
pid
eal
1 2 3 4 5 6 7 8 910 20
1.6
1.4
1.2
1
entropy
Navier-StokestransportIS hydro
2
1
η/s ≈ const
ideal hydro
1
2
3K0 = 6.67
τ/τ0
(dS/dη)/
(dS
0/dη)
1 5 10 15 20
1.4
1.3
1.2
1.1
1
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 58
Connection to viscosity
K0 ≈ T0τ0
5
s0
ηs,0≈ 12.8 ×
(
T0
1 GeV
)
( τ0
1 fm
)
(
1/(4π)
η0/s0
)
For typical RHIC hydro initconds T0τ0 ∼ 1, therefore
K0 >∼ 2 − 3 ⇒ η
s<∼
1 − 2
4π(4)
I.e., shear viscosity cannot be many times more than the conjectured bound,for IS hydro to be applicable.
When IS hydro is accurate, dissipative corrections to pressure and entropydo not exceed 20% significantly (a necessary condition). This holds for awide range [0.476, 1.697] of initial pressure anisotropies.
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 59
(!) close to a noninteracting “resonance gas” at T <∼ 160 − 180 MeV
fi(p) =gi
(2π)31
exp[Ei(p) − µi]/T ± 1Huovinen & Petreczky, NPA837 (’10)
equation of state strangeness fluctuations
0
2
4
6
8
10
140 160 180 200 220 240
T [MeV]
(ε-3p)/T4
p4asqtad
0
0.2
0.4
0.6
0.8
1
140 160 180 200 220 240
T [MeV]
χS2
p4asqtad
important for late-stage dynamics (hadron transport)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 60
QGPHRG
bulk viscosity and relaxation time
Bulk viscosity:
Relaxation times: also peaks near Tc, ζτ ~Π
N-S initialization: )(0 u⋅∂−=Π ζ
Π
s/ζviscous hydro breaks down ( ) for larger 0<Π+p
viscous hydro is only valid with small small bulk viscous effects on V2
large near Tc keeps large negative value of in phase transition region Πτ
this plays an important role for bulk viscous dynamics
s/ζ
Song & Heinz (’09)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 61
Uncertainties from bulk viscosity
fm/c120)/( ⋅= sζτ π
1=C3.1=C
1=C100=C
N-S initialization Zero initialization
-with a critical slowing down , effects from bulk viscosity effects are much smaller than from shear viscosity
Πτ
bulk viscosity influences V2 ~5% (N-S initial.) <4% (zero initial.)
Song & Heinz, 0909
s/ηuncertainties to ~20% (N-S initial.) <15% (zero initial.)
fm/c120)/( ⋅= sζτ π
s/ζ s/ζ
Song Heinz (’09)
D. Molnar, Zimanyi School, Nov 29 - Dec 3, 2010 62