viscous hydrodynamics and covariant transport
TRANSCRIPT
Viscous hydrodynamics and covariant
transport
Denes Molnar
RIKEN/BNL Research Center & Purdue University
(remote) Presentation at TECHQM II Workshop
Dec 15-17, 2008, LBNL, Berkeley, CA
• Viscosity, causal viscous hydrodynamics (Israel-Stewart)
• Region of validity for viscous hydrodynamics, shear viscosity at RHIC
• Implications of bulk viscosity
in collaboration with Pasi Huovinen
Thermalization at RHICefficient conversion of spatial eccentricity to momentum anisotropy
→ “elliptic flow”
ε ≡ 〈x2−y2〉〈x2+y2〉
v2 ≡〈p2
x−p2y〉
〈p2x+p2
y〉≡ 〈cos 2φp〉
very opaque - large energy loss, even forheavy quarks
RAA =measured yield
expected yield for dilute system
Ideal (nondissipative) hydrodynamics describes particle spectra and ellipticflow surprisingly well Kolb & Heinz, QGP Vol. 3 [nucl-th/0305084], Kolb, Heinz, Huovinen et al (’01), ...
D. Molnar @ TQM2, Dec 15-17, 2008 1
Hydrodynamics
• describes a system near local equilibrium
• long-wavelength, long-timescale dynamics, driven by conservation laws
∂µTµν(x) = 0, ∂µNµB = 0
• 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
↓
amazing that hydrodynamics can be applicable at such microscopic scales
D. Molnar @ TQM2, Dec 15-17, 2008 2
Shear viscosity in QCDshear viscosity: Txy = −η∂vx
∂y acts to reduce velocity gradients
largely unknown at T ∼ 200− 400 MeV relevant for RHIC
perturbatively: η/s ∼ O(1), lattice QCD: very preliminary
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
- no quarks (gluons only)- crude lattices
N = 4 super Yang-Mills ( 6= QCD): η/s ≥ h/4πPolicastro, Son, Starinets, PRL87 (’02)Kovtun, Son, Starinets, PRL94 (’05)
D. Molnar @ TQM2, Dec 15-17, 2008 3
In heavy-ion collisions, timescales are short, gradients are large -corrections to ideal (Euler) hydrodynamics are relevant.
Two ways to study dissipative effects:
- 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 @ TQM2, Dec 15-17, 2008 4
DM & Gyulassy, NPA 697 (’02): v2(pT , χ) DM & Huovinen, PRL94 (’05): hydro vs transport
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������
����F
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h� �hv � �vi � �ii � �
i � hi �v
i
——————————perturbative value
need 15× perturbative 2→ 2 opacities to reproduce elliptic flow at RHIC
but even at such opacities, 20− 30% dissipative corrections
D. Molnar @ TQM2, Dec 15-17, 2008 5
Snellings & Kolb et al, NPA 698, (’02): Teaney (’04): v2 vs Γs ≡4η3sT
[GeV/c]tp0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
) t(p 2v
0
0.05
0.1
0.15
0.2
0.25
0.3 Charged particlesHydro pionsHydro charged particlesHydro anti-protons
STAR Preliminary (Jan ’01)
(GeV)Tp0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
)T
(p2v
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
6.8 fm (16-24% Central)≈b
= 0.1oτ/sΓ
= 0.2oτ/sΓ
= 0oτ/sΓ
STAR Data
ideal hydro overshoots elliptic flow at high pT
but viscous corrections to phase space distributions at freezeout help δf ∝pµpν∇µuν
D. Molnar @ TQM2, Dec 15-17, 2008 6
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 [∂µ ≡ uµD + ∇µ]
η, ζ shear and bulk viscosities, κ heat conductivity
Equation of motion: ∂µTµν = 0, ∂µNµ = 0
two problems:
parabolic equations (e.g., heat flow) → acausal Muller (’76), Israel & Stewart (’79) ...
instabilities Hiscock & Lindblom, PRD31, 725 (1985) ...
D. Molnar @ TQM2, Dec 15-17, 2008 7
Causal dissipative hydro
Bulk pressure Π, shear stress πµν, heat flow qµ are dynamical quantities
Tµν ≡ Tµνideal + πµν −Π∆µν , Nµ ≡ Nµ
ideal −n
e + pqµ
Israel-Stewart theory [Ann.Phys 100 & 118]: relaxation equations
X = −1
τX(X −XNS) + X YX + ZX
⇒ alleviates the causality problem (τΠ = ζβ0, τq = κqTβ1, τπ = 2ηsβ2)
—
There may be more terms - e.g., imposing ONLY conformal invariance Baier,
Romatschke, Son, JHEP04, 100 (’08)
πµν = · · ·+λ1
η2πα〈µπ ν〉
α −λ2
ηπα〈µων〉
α + λ3ωα〈µω ν〉
α (1)
No problem, we can solve with these too, provided λ1,2,3 are known.
D. Molnar @ TQM2, Dec 15-17, 2008 8
Israel-Stewart theorystarting point - 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
in Landau frame (α0 = α1 = β0 = β1 = β2 = 0 gives Navier-Stokes).
Impose ∂µSµ ≥ 0 via a quadratic ansatz
T∂µSµ =Π2
ζ−
qµqµ
κqT+
πµνπµν
2ηs≥ 0
e.g., T∂µSµ = ΠX − qµXµ + πµνXµν will lead to equations of motion
Π = ζX , qµ = κT∆µνXν , πµν = 2ηsX〈µν〉
—
Also follows from covariant transport using Grad’s 14-moment approximation
f(x, p) ≈ [1 + Cαpα + Cαβpαpβ]feq(x, p)
and taking the lowest three momentum moments of the Boltzmann equation.
D. Molnar @ TQM2, Dec 15-17, 2008 9
Complete set of Israel-Stewart equations of motion
DΠ = −1
τΠ(Π + ζ∇µuµ) (2)
−1
2Π
(
∇µuµ + D lnβ0
T
)
+α0
β0∂µqµ −
a′0
β0qµDuµ
Dqµ = −1
τq
[
qµ + κqT 2n
ε + p∇µ
(µ
T
)
]
− uµqνDuν (3)
−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 (4)
−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 @ TQM2, Dec 15-17, 2008 10
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)
All groups find ∼ 20− 30% effects for η/s = 1/(4π) in Au+Au at RHIC.
D. Molnar @ TQM2, Dec 15-17, 2008 11
Applicability of IS hydro
Israel-Stewart hydrodynamics comes from a quadratic truncation (Grad’sapproach) that has no small control parameter.
⇒ crucial to test its validity against a nonequilibrium theory
HERE: test IS hydro against 2→ 2 covariant transport
massless e = 3p equation of state (ζ = 0)
ηs ≈4T
5σtr, τπ ≈ 1.2λtr ≡
1.2
nσtrσtr : transport cross section
Two scenarios: i) σ = const and ii) η/s ≈ const (set via σ ∝ time2/3).
in both cases, longitudinally expanding system vz = z/t (Bjorken flow)
D. Molnar @ TQM2, Dec 15-17, 2008 12
[Huovinen & DM, arXiv:0808.0953]
TµνLR =
ε 0 0 00 p− πL/2 0 00 0 p− πL/2 00 0 0 p + πL
p +4p
3τ= −
πL
3τ(5)
πL +πL
τ
(
2K(τ)
3C+
4
3+
πL
3p
)
= −8p
9τ(C ≈ 4/5) (6)
The most relevant parameter is the inverse Knudsen number
K(τ) ≡τ
λtr(τ)=
τexp
τscatt≈
6τexp
5τπ(7)
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
D. Molnar @ TQM2, Dec 15-17, 2008 13
Huovinen & DM, arXiv:0808.0953 - evolution of pressure anisotropy Tzz/Txx
free streamingNavier-StokestransportIS hydro
σ = const
ideal hydro
K0 = 1
K0 = 2
K0 = 3
K0 = 6.67
K0 = 20
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
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 hydro applicable when K0 >∼ 2− 3, i.e., λtr <
∼ 0.3− 0.5 τ0
D. Molnar @ TQM2, Dec 15-17, 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/τ0p
/p
idea
l
1 2 3 4 5 6 7 8 910 20
1.6
1.4
1.2
1
IS hydro applicable when K0 >∼ 2− 3, i.e., λtr <
∼ 0.3− 0.5 τ0
while Navier-Stokes needs K0 >∼ 6
D. Molnar @ TQM2, Dec 15-17, 2008 15
entropy produced relative to initial entropy
Navier-StokestransportIS hydro
21
σ = const
ideal hydro
1
2
3 6.67
K0 = 20
τ/τ0
(dS/dη)/
(dS
0/dη)
1 5 10 15 20
1.4
1.3
1.2
1.1
1
Navier-StokestransportIS hydro
2
1
η/s ≈ const
ideal hydro
1
2
3K0 = 6.67
τ/τ0(d
S/dη)/
(dS
0/dη)
1 5 10 15 20
1.4
1.3
1.2
1.1
1
same conclusions as before
D. Molnar @ TQM2, Dec 15-17, 2008 16
Connection to viscosity
K0 ≈T0τ0
5
s0
ηs,0≈ 12.8×
(
T0
1 GeV
)
( τ0
1 fm
)
(
1/(4π)
ηs0/s0
)
(8)
For typical RHIC hydro initconds T0τ0 ∼ 1, therefore
K0 >∼ 2− 3 ⇒
η
s<∼
1− 2
4π(9)
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.
D. Molnar @ TQM2, Dec 15-17, 2008 17
IS hydro validity test in 2DWe solve the full Israel-Stewart equations, including vorticity terms fromkinetic theory, in a 2+1D boost-invariant scenario. Shear stress only.
πµν+ 1τπ
πµν = 1β2∇〈µuν〉−1
2πµνDαuα−1
2πµν ˙
[ln β2T ]+2π
〈µλ ων〉λ−(uµπνα + uνπαµ)uα
Mimic a known reliable transport model:
• massless Boltzmann particles ⇒ ε = 3P• only 2↔ 2 processes, i.e. conserved particle number• η = 4T/(5σtr), β2 = 3/(4p)• either σ = const. = 47 mb (σtr = 14 mb) ← the simplest in transport
or σ ∝ τ2/3 ⇒ η/s ≈ 1/(4π)
“RHIC-like” initialization:
• τ0 = 0.6 fm/c• b = 8 fm• T0 = 385 MeV and dN/dη|b=0 = 1000
• freeze-out at constant n = 0.365 fm−3
D. Molnar @ TQM2, Dec 15-17, 2008 18
Pressure evolution in the coreT xx and T zz averaged over the core of the system, r⊥ < 1 fm:
η/s ≈ 1/(4π) (σ ∝ τ 2/3)
Huovinen & DM, arXiv:0806.1367
remarkable similarity!
D. Molnar @ TQM2, Dec 15-17, 2008 19
Viscous hydro elliptic flowTWO effects: - dissipative corrections to hydro fields uµ, T, n
- dissipative corrections in Cooper-Frye freezeout f → f0+δf
η/s ≈ 1/(4π) (σ ∝ τ 2/3)
Must use Grad’s quadratic correctionsin Cooper-Frye formula
EdN
d3p=
∫
pµdσµ(f0 + δf)
for massless ε = 3p, shear only
δf = f0
[
1 +pµpνπµν
8nT 3
]
calculation for σtr = const ∼ 15mb shows similar behavior
D. Molnar @ TQM2, Dec 15-17, 2008 20
Viscous hydro vs transport v2
Huovinen & 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 @ TQM2, Dec 15-17, 2008 21
Bulk viscosityacts against compression/dilution: Π = −ζ ~∇~v
largely unknown at T ∼ 200− 400 MeV relevant for RHIC
perturbatively tiny: ζ/s ∼ 0.02α2s, on lattice: very preliminary
Arnold, Dogan, Moore, PRD74 (’06) Meyer, arXiv:0710.3717
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
- no quarks (gluons only)- crude lattices
implies peak near Tc
0
0.05
0.1
0.15
0.2
0.25
0.1 0.2 0.3 0.4
zeta
/ s
T [GeV]
fitted zeta/s, Cheng et al EOSLorenzian fit
D. Molnar @ TQM2, Dec 15-17, 2008 22
lattice QCD equation of state Cheng et al, (’07)
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=46
3p/T4: Nτ=46
D. Molnar @ TQM2, Dec 15-17, 2008 23
Fries et al, PRC78 (’08): study in 0+1D Bjorken scenario
lattice EOS with η/s = 1/(4π), τ0 = 0.3 fm, vary width and height of ζ/s(T )
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
finds 10− 15% entropy production from bulk viscosity possible
however, study is thermodynamically inconsistent because it ignored terms
in equation of motion DΠ = − 1τΠ
(Π + ζ∇µuµ)−12Π
(
∇µuµD ln β0T
)
D. Molnar @ TQM2, Dec 15-17, 2008 24
Entropy density in Israel-Stewart theory
dS
dη= τAT
[
seq −1
2T
(
β0Π2 +
3
2β2π
2L
)]
and only complete IS equations ensure
d
dτ
(
dS
dη
)
= τAT
(
Π2
ζT+
3π2L
4ηsT
)
≡d
dτ
(
dS
dη
)bulk
+d
dτ
(
dS
dη
)shear
≥ 0
The “naive” approximation violates the 2nd law in general, and also
d
dτ
(
dS
dη
)
6=d
dτ
(
dS
dη
)bulk
+d
dτ
(
dS
dη
)shear
D. Molnar @ TQM2, Dec 15-17, 2008 25
Bulk viscosity produces extra entropy. If too much is generated, hydro itselfbecomes questionable.
one benchmark: relative entropy production
∆Sπ+Π
S0=
S(Tf = 180MeV )− S0
S0
<∼ 0.2− 0.25
depends on initial conditions (shear stress and bulk pressure), and initialtime.
0+1D exploration for “RHIC”-like conditions:
• most recent lattice EOS, continued over to a resonance gas at low T
• η/s = 1/(4π) fixed, vary width wζ and height hζ of ζ/s(T ), τΠ = τπ = 6η/(sT )
• ε0(τ0 = 0.6fm) = 15 GeV/fm3 - for other τ0, scale via s0τ0 = const (ideal)
• stop at beginning of hadron phase Tf = 180 MeV (τf ≈ 8 fm)
D. Molnar @ TQM2, Dec 15-17, 2008 26
entropy generation with shear viscosity only Huovinen & DM, (’08)
τf = 8 fmTf = 180 MeV
η = s/(4π)ζ = 0
0.20.3
0.6
τ0 = 1 fm
π0/p0
∆S/S
0
10.50-0.5-1
0.5
0.4
0.3
0.2
0.1
0
D. Molnar @ TQM2, Dec 15-17, 2008 27
naive vs complete BULK terms in equations Huovinen & DM, (’08)
naive bulk eqn.bulkshearcomplete IS, shear+bulk
τ0 = 0.6 fm
η = s/(4π)hζ = wζ = 1
τ [fm]
∆S/S
0
1086420
0.6
0.5
0.4
0.3
0.2
0.1
0
naive bulk eqn.bulkshearcomplete IS, shear+bulk
τ0 = 0.3 fm
η = s/(4π)hζ = wζ = 1
τ [fm]∆
S/S
0
1086420
0.6
0.5
0.4
0.3
0.2
0.1
0
correct IS equations produce somewhat less entropy
D. Molnar @ TQM2, Dec 15-17, 2008 28
dependence on height on width of ζ/s(T ) Huovinen & DM, (’08)
4, 1/43, 1/32, 1/21/2, 2default (hζ, wζ = 1, 1)
τ0 = 0.6 fm
τ [fm]
∆S/S
0
121086420
0.5
0.4
0.3
0.2
0.1
0
bulk effects significant for hζ · wζ ∼ O(1)
D. Molnar @ TQM2, Dec 15-17, 2008 29
Π0/p0
0.40.20-0.2-0.4
π0/p0
1
0.5
0
-0.5
-1
τ0 = 0.6
Π0/p0
0.40.20-0.2-0.4
π0/p0
1
0.5
0
-0.5
-1
τ0 = 0.6
Π0/p0
0.40.20-0.2-0.4
π0/p0
1
0.5
0
-0.5
-10.8
0.7
0.6
0.5
0.4
0.35
0.3
0.25τ0 = 0.6
Π0/p0
0.40.20-0.2-0.4
π0/p0
1
0.5
0
-0.5
-1
τ0 = 1
Π0/p0
0.40.20-0.2-0.4
π0/p0
1
0.5
0
-0.5
-1
τ0 = 1
Π0/p0
0.40.20-0.2-0.4
π0/p0
1
0.5
0
-0.5
-1
0.4
0.35
0.3
0.25
0.2
0.17τ0 = 1
Π0/p0
0.40.20-0.2-0.4
π0/p0
1
0.5
0
-0.5
-1
τ0 = 0.3
Π0/p0
0.40.20-0.2-0.4
π0/p0
1
0.5
0
-0.5
-1
τ0 = 0.3
Π0/p0
0.40.20-0.2-0.4
π0/p0
1
0.5
0
-0.5
-1
0.8
0.7
0.6
0.5
0.4
0.35
0.3τ0 = 0.3
requirement of modest ≤ 20 − 25%entropy production severely limits initialconditions if τ0 <
∼ 1 fm
even Navier-Stokes limit is hard toaccommodate
color glass π0 ∼ −p0 is marginal even atτ0 = 1 fm
D. Molnar @ TQM2, Dec 15-17, 2008 30
SummaryProgress so far:
• we can solve causal viscous hydro in 2+1D
• IS hydro is a good approximation, if shear viscosity is not too largeη/s <∼ few/(4π)
must use complete set of Israel-Stewart equations
• η/s ∼ O(1)/(4π) is compatible with RHIC v2(pT ) data (charged hadrons),
• if bulk viscosity is significant (Meyer’s central values), viscous hydro isapplicable only for very limited initial conditions. E.g., need τ0 >
∼ 0.5 fm.
Missing ingredients:
• establish IS hydro region of validity for realistic equation of state
• include bulk viscosity in 2+1D (in progress)
• couple hydro to hadron transport (proper freezeout)
D. Molnar @ TQM2, Dec 15-17, 2008 31