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Open issues in hydro and transport
Denes MolnarRIKEN/BNL Research Center & Purdue University
TIFR Program on “Initial conditions in heavy ion collisions”
September 16, 2008, International Centre, Dona Paula, Goa, India
Outline
• Region of validity of second-order schemes
• Properties of hot QCD matter
• Initial conditions / thermalization
• Decoupling (freezeout)
• Perturbative or s-QGP?
D. Molnar @ Goa, Sep 16, 2008 1
Hydrodynamics
• describes a system near local equilibrium
• long-wavelength, long-timescale dynamics, driven by conservation laws
• in heavy-ion physics: mainly used for the plasma stage of the collision
initial nuclei parton plasma hadronization hadron gas
<—— ∼ 10−23 sec = 10 yochto(!)-secs ——>
↑∼ 10−14 m
= 10 fermis
↓
nontrivial how hydrodynamics can be applicable at such microscopic scales
D. Molnar @ Goa, Sep 16, 2008 2
In heavy-ion collisions, timescales are short, gradients are large -corrections to ideal (Euler) hydrodynamics are relevant.
Two ways to study dissipative effects in heavy-ion collisions
- causal dissipative hydrodynamics
Israel, Stewart; ... Muronga, Rischke; Teaney et al; Romatschke et al; Heinz et al, DM & Huovinen
flexible in macroscopic properties
numerically cheaper
- covariant transport
Israel, de Groot,... Zhang, Gyulassy, DM, Pratt, Xu, Greiner...
completely causal and stable
fully nonequilibrium → interpolation to break-up stage
D. Molnar @ Goa, Sep 16, 2008 3
Dissipative hydrodynamics
relativistic Navier-Stokes hydro: small corrections linear in gradients [Landau]
TµνNS = Tµν
ideal + η(∇µuν + ∇νuµ −2
3∆µν∂αuα) + ζ∆µν∂αuα
NνNS = Nν
ideal + κ
(
n
ε + p
)2
∇ν(µ
T
)
where ∆µν ≡ uµuν − gµν, ∇µ = ∆µν∂ν
η, ζ shear and bulk viscosities, κ heat conductivity
Equation of motion: ∂µTµν = 0, ∂µNµ = 0
two problems:
parabolic equations → acausal Muller (’76), Israel & Stewart (’79) ...
instabilities Hiscock & Lindblom, PRD31, 725 (1985) ...
D. Molnar @ Goa, Sep 16, 2008 4
As an illustration, consider heat flow in a static, incompressible fluid (Fourier)
∂tT = κ∆T
parabolic eqns.
Greens function is acausal (allows ∆x > ∆t)
G(~x, t; ~x0, t0) =1
[4πκ(t − t0)]3/2exp
[
−(~x − ~x0)
2
4κ(t − t0)
]
—–
Adding a second-order time derivative makes it hyperbolic
τ∂2t T + ∂tT = κ∆T
Note, this is equivalent to a relaxing heat current
∂τT = ~∇~j , ∂t~j = −
~j − κ~∇T
τThe wave dispersion relation is ω2+ iω/τ = κk2/τ , i.e., now signals propagateat speeds cs =
√
κ/τ (at low frequencies), causal for not too small τ .
D. Molnar @ Goa, Sep 16, 2008 5
Causal dissipative hydroBulk pressure Π, shear stress πµν, heat flow qµ are dynamical quantities
Tµν ≡ Tµνideal + πµν − Π∆µν , Nµ ≡ Nµ
ideal −n
e + pqµ
Israel-Stewart: truncate entropy current at quadratic order [Ann.Phys 100 & 118]
Sµ = uµ
[
s −1
2T
(
β0Π2 − β1qνq
ν + β2πνλπνλ)
]
+qµ
T
(
µn
ε + p+ α0Π
)
−α1
Tπµνqν
(Landau frame). Imposing ∂µSµ ≥ 0 via a quadratic ansatz
T∂µSµ =Π2
ζ−
qµqµ
κqT+
πµνπµν
2ηs≥ 0
gives equations of motion for Π, qµ, πµν.
Also follows from covariant transport using Grad’s 14-moment approximation
f(x, p) ≈ [1 + Cαpα + Cαβpαpβ]feq(x, p)
and taking the “1”, pν, and pνpα moments of the Boltzmann equation.
D. Molnar @ Goa, Sep 16, 2008 6
Complete set of Israel-Stewart equations of motion
DΠ = −1
τΠ(Π + ζ∇µuµ) (1)
−1
2Π
(
∇µuµ + D lnβ0
T
)
+α0
β0∂µqµ −
a′0
β0qµDuµ
Dqµ = −1
τq
[
qµ + κqT 2n
ε + p∇µ
(µ
T
)
]
− uµqνDuν (2)
−1
2qµ
(
∇λuλ + D lnβ1
T
)
− ωµλqλ
−α0
β1∇µΠ +
α1
β1(∂λπλµ + uµπλν∂λuν) +
a0
β1ΠDuµ −
a1
β1πλµDuλ
Dπµν = −1
τπ
(
πµν − 2η∇〈µuν〉)
− (πλµuν + πλνuµ)Duλ (3)
−1
2πµν
(
∇λuλ + D lnβ2
T
)
− 2π〈µ
λ ων〉λ
−α1
β2∇〈µqν〉 +
a′1
β2q〈µDuν〉 .
where A〈µν〉 ≡ 12∆
µα∆νβ(Aαβ+Aβα)−13∆
µν∆αβAαβ, ωµν ≡ 12∆
µα∆νβ(∂βuα−∂αuβ)
D. Molnar @ Goa, Sep 16, 2008 7
We can solve these in 2+1D at present, 3+1D should also be doable.
There may be more terms - e.g., imposing ONLY conformal invariance Baier,
Romatschke, Son, JHEP04, 100 (’08)
πµν = · · · +λ1
η2πα〈µπ ν〉
α + λ3ωα〈µω ν〉
α (4)
No problem, we can solve with these too.
D. Molnar @ Goa, Sep 16, 2008 8
e.g.
Romatschke & Romatschke, arxiv:0706.1522
0 1 2 3 4p
T [GeV]
0
5
10
15
20
25
v 2 (p
erce
nt)
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, arXiv:0806.1367
D. Molnar @ Goa, Sep 16, 2008 9
Fries et al, arxiv:0807.4333 in 0+1D
0.1
0.2
0.3
0.4
0.5
0.6
/s,
/s
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
T (GeV)
/s/s scaled/s
(a)
0123456789
(-3
P)/
T4
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
T (GeV)
(b)
0.0
0.2
0.4
0.6
0.8
1.0
Rel
.Pre
ssu
re
5 1 2 5 10 2 5
(fm/c)
(a) Pz/P
0.0
0.05
0.1
0.15
0.2
0.25
0.3
Rel
.En
tro
py
c = 0c = 1, a = 1c = 2, a = 1
(b) S /Sf
c = 1, a = 2c = 2, a = 2
D. Molnar @ Goa, Sep 16, 2008 10
Applicability of IS hydro
In heavy ion physics applications, gradients ∂µuν/T , |∂µe|/(Te), |∂µn|/(Tn)at early times τ ∼ 1 fm are large ∼ O(1), and therefore cannot be ignored.
Hydrodynamics may still apply, if viscosities are unusually small η/s ∼ 0.1,ζ/s ∼ 0.1, where s is the entropy density in local equilibrium. In that case,pressure corrections from Navier-Stokes theory still moderate
δTµνNS
p≈
(
2ηs
s
∇〈µuν〉
T+
ζ
s
∇αuα
T
)
ε + p
p∼ O
(
8ηs
s,4ζ
s
)
. (5)
Heat flow effects can also be estimated based on
δNµNS
n≈
κqT
s
n
s
∇µ(µ/T )
T(6)
and should be very small at RHIC because µ/T ∼ 0.2, nB/s ∼ O(10−3)
D. Molnar @ Goa, Sep 16, 2008 11
Whereas Navier-Stokes comes from a rigorous expansion in small deviationsnear local equilibrium retaining all powers of momentum
f(x, ~p) = feq(x, ~p)[1 + φ(x, ~p)] (|φ| � 1 , |pµ∂µφ| � |pµ∂µfeq|/feq)
the quadratic truncation in Grad’s approach has no small control parameter.
In general, Navier-Stokes and Israel-Stewart are different
⇒ control against a nonequilibrium theory is crucial
D. Molnar @ Goa, Sep 16, 2008 12
IS hydro vs transport [Huovinen & DM, arXiv:0808.0953]
0+1D Bjorken scenario with massless e = 3p EOS, 2 → 2 interactions
πµνLR = diag(0,−πL
2 ,−πL2 , πL), Π ≡ 0, qµ ≡ 0 (reflection symmetry)
p +4p
3τ= −
πL
3τ(7)
πL +πL
τ
(
2K(τ)
3C+
4
3+
πL
3p
)
= −8p
9τ, (8)
where
K(τ) ≡τ
λtr(τ), C ≈
4
5. (9)
For σ = const: λtr = 1/nσtr ∝ τ ⇒ K = K0 = const, η/s ∼ Tλtr ∼ τ2/3
for η/s ≈ const: K = K0(τ/τ0)∼2/3 ∝ τ∼2/3
we kept the COMPLETE Israel-Stewart equations (every term)
D. Molnar @ Goa, Sep 16, 2008 13
Huovinen & DM, arXiv:0808.0953 pressure anisotropy Tzz/Txx for e = 3p EOS
σ = const η/s ≈ const
0
0.2
0.4
0.6
0.8
1
10 9 8 7 6 5 4 3 2 1
p_L
/p_T
tau/tau0
K0=20
K0=20/3
K0=3
K0=2
K0=1idealIS hydrotransportfree streaming
0
0.2
0.4
0.6
0.8
1
10 9 8 7 6 5 4 3 2 1
p_L/p
_T
tau/tau0
K0=20
K0=6.67K0=3
K0=2
K0=1
idealIS hydrosigma ~ tau^(2/3)
IS hydro applicable when K0 >∼ 2 − 3, i.e., λtr <
∼ 0.3 − 0.5 τ0
D. Molnar @ Goa, Sep 16, 2008 14
pressure evolution relative to ideal hydro
Navier-StokestransportIS hydro
21
σ = const
ideal hydro
1 2
3
6.67
K0 = 20
τ/τ0
p/
pid
eal
1 2 3 4 5 6 7 8 910 20
1.6
1.4
1.2
1
Navier-StokestransportIS hydro
2
1
η/s ≈ const
ideal hydro
1
2
3
K0 = 6.67
τ/τ0
p/
pid
eal
1 2 3 4 5 6 7 8 910 20
1.6
1.4
1.2
1
D. Molnar @ Goa, Sep 16, 2008 15
Connection to viscosity
K0 ≈T0τ0
5
s0
ηs,0≈ 12.8 ×
(
T0
1 GeV
)
( τ0
1 fm
)
(
1/(4π)
ηs0/s0
)
(10)
For typical RHIC hydro initconds T0τ0 ∼ 1, therefore
K0 >∼ 2 − 3 ⇒
η
s<∼
1 − 2
4π(11)
I.e., if the conjectured bound is accurate, hydro is marginally applicable.
Q1: is this true for a nonequilibrium theory other than covariant transport?
Q2: what is the viscosity of QCD matter at T ∼ 150 − 400 MeV?
D. Molnar @ Goa, Sep 16, 2008 16
Viscous hydro vs transport in 2+1DHuovinen & DM, arXiv:0806.1367
• excellent agreement when σ = const ∼ 47mb• good agreement for η/s ≈ 1/(4π), i.e., σ ∝ τ 2/3
D. Molnar @ Goa, Sep 16, 2008 17
Shear viscosity
1687 - I. Newton (Principia)
Txy = −η∂ux
∂y
acts to reduce velocity gradients
1985 - quantum mechanics: ∆E · ∆t ≥ h/2
+ kinetic theory: T · λMFP ≥ h/3 Gyulassy & Danielewicz, PRD 31 (’85)
η ≈ 4/5 · T/σtr , entropy s ≈ 4n
gives minimal viscosity: η/s = λtrT5 ≥ h/15
2004 - string theory AdS/CFT: η/s ≥ h/4πPolicastro, Son, Starinets, PRL87 (’02)
Kovtun, Son, Starinets, PRL94 (’05)
revised to 4h/(25π) Brigante et al, arXiv:0802.3318
D. Molnar @ Goa, Sep 16, 2008 18
Shear viscosity in QCDη = lim
ω→0
12ω
∫
dt d3x eiωt〈[Txy(x), Txy(0)]〉
perturbative QCD: η/s ∼ 1, lattice QCD: correlator very noisy
Nakamura & Sakai, NPA774, 775 (’06): Meyer, PRD76, 101701 (’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
PerturbativeTheory
T Tc
η
upper bounds:η/s(T=1.65Tc) < 0.96
η/s(T=1.24Tc) < 1.08
best estimate:η/s(T=1.65Tc) < 0.13±0.03
η/s(T=1.24Tc) < 0.10±0.05
many practitioners regardthese VERY preliminary
D. Molnar @ Goa, Sep 16, 2008 19
Bulk viscosity in QCDζ = lim
ω→0
118ω
∫
dt d3x eiωt 〈[Tµµ (~x, t), T µ
µ (0)]〉
perturbative QCD: ζ/s ∼ 0.02α2s is tiny Arnold, Dogan, Moore, PRD74 (’06)
from ε − 3p > 0: Kharzeev & Tuchin, arXiv:0705.4280v2 on lattice: Meyer, arXiv:0710.3717
1 2 3 4 5T�Tc0.0
0.2
0.4
0.6
0.8
Ζ�s
1.02 1.04 1.06 1.08 1.10 1.12 1.140.0
0.2
0.4
0.6
0.8
best estimates:η/s(T=1.65Tc) ∼ 0 − 0.015
η/s(T=1.24Tc) ∼ 0.06 − 0.1
η/s(T=1.02Tc) ∼ 0.2 − 2.7
many practitioners regardthese as well VERYpreliminary
D. Molnar @ Goa, Sep 16, 2008 20
TARGET
preequilibrium
QGP
freeze-outPROJE
CTILE
τfo
τ0
τh
hadronization
Time
first impact Position
hydro needs: initial conditionsboundary conditions = expansion to vacuum (ε = p = 0 outside)and a decoupling model (transition to free-streaming gas)
D. Molnar @ Goa, Sep 16, 2008 21
Initial conditionsHydro initial conditions should come from the very early collision dynamics.
Q3: when and how (if at all) does the matter thermalize in the collision?
In the meantime, resort to parameterizations:
- thermalization time τ0
- some shape of initial entropy or energy density profile (e.g. Glauber,binary, saturation model)
- assumptions about pressure anisotropies (need the full πµν, also Π, andin principle qµ too)
- baryon density profiles (e.g., through entropy ratio nB/s)
- Q4: any initial radial and/or elliptic flow? - typically set to zero
Initial condition parameters, and also freezeout parameters Tfo (εfo), are fitto data for central collisions. Noncentral collisions can then be “predicted”.
D. Molnar @ Goa, Sep 16, 2008 22
The main difference between the various profiles is in the centralitydependence (mainly affects dN(b)/dy) and the spatial eccentricity (mainlyaffects v2(b))
0 2 4 6 8 10 120
4
8
12
16
20
b (fm)
n WN a
nd n
BC
at (
x,y)
=(0
,0)
(in fm
−2 )
nBC
(0,0)
nWN
(0,0)
b (fm)0 2 4 6 8 10 12 14
ε0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8CGC
=0.85)αBGK (=1.0)α (partn=0.0)α (binaryn
Note that naive CGC results are shown - more realistic calculations giveeccentricities close to binary but results sensitive to how the edges arehandled (theory breaks down because Qs too low)
D. Molnar @ Goa, Sep 16, 2008 23
Thermalization and 2 → 2 transportDM & Gyulassy, NPA 697 (’02): v2(pT , χ) in Au+Au @ 130 GeV, b = 8 fm
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——————————perturbative value
perturbative 2 → 2 transport is lousy at maintaining local equilibrium
large v2 needs 15× perturbative opacities - σel × dNg/dη ≈ 45 mb ×1000
D. Molnar @ Goa, Sep 16, 2008 24
DM & Huovinen, PRL94
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even 50 mb does not give an ideal fluid
D. Molnar @ Goa, Sep 16, 2008 25
Radiative 3 ↔ 2 transporthigher-order processes also contribute to thermalization
+ ...
Greiner & Xu ’04: find thermalization time-scale τ ∼ 2 − 3 fm/c
2 → 2, 2 → 3 transport cross sections spectra vs. time
D. Molnar @ Goa, Sep 16, 2008 26
elliptic flow with ggg ↔ gg (minijet initconds, p0 = 1.4 GeV)
Q5: is this a perturbative alternative to the sQGP paradigm?
Q6: if yes, how do you hadronize out of equilibrium?
D. Molnar @ Goa, Sep 16, 2008 27
DecouplingBest state of the art: conversion to a gas of hadrons, which are then evolvedin a hadron transport model.
• Cooper-Frye formula: sudden transition from fluid to a gas on a 3Dhypersurface (typically T = const or ε = const)
E dN = pµdσµ(x) d3p fgas(x, ~p)
where dσµ is the hypersurfacenormal at x
Dissipative case - must include dissipative corrections to f
f → feq + δf , δfGradmassless = feq
πµνpµpν
8nT 3
D. Molnar @ Goa, Sep 16, 2008 28
DM & Huovinen, arXiv:0806.1367
RHIC Au+Au, b = 8 fm, η/s ≈ 1/(4π) (σ ∝ τ 2/3)
D. Molnar @ Goa, Sep 16, 2008 29
Thermodynamic consistency requires that matter properties - equation ofstate, transport coefficients - are the SAME for the fluid and the hadron gasat the switching point.
Q7: are hadron cascades, such as UrQMD and JAM, thermodynamicallyconsistent with QCD (e.g., the latest lattice EOS with Tc ≈ 190 MeV)?
Also conceptual problems:
a) for spacelike normal, pµdσµ < 0 possible, which gives NEGATIVE yields
ignored in practice because such contributions are small (less than a fewpercent) - slight violation of energy-momentum and charge conservation
b) hypersurface (T = const or ε = const) determined running hydro until thevery end, disregarding that parts of the system have frozen out already
Q8: how to dynamical interface hydro and transport?
D. Molnar @ Goa, Sep 16, 2008 30