Open issues in hydro and transporttheory.tifr.res.in/~qcdinit/talks/molnar.pdfGreens function is acausal (allows x > t) G(~x; t;~x0;t0) = 1 [4ˇ (t t0)]3=2 exp (~x ~x0)2 4 (t t0) |{Adding

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


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


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) ...


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 .


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.


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)


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.


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


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


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)


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


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)


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


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


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?


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


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


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


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


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)


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”.


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)


Thermalization and 2 → 2 transportDM & Gyulassy, NPA 697 (’02): v2(pT , χ) in Au+Au @ 130 GeV, b = 8 fm

��

�� ! " � � # � � $ � % �& ' � (

)* +, ) -./ 01 2 � � � �

3/ 465 7 8 9

3/ 4 5 : � 9

3/ 4 5 ; 9

3/ 4 2 �< = 9

>? @ ' � ( A

BDCFEHGJ

ILKMEHGJ

NOGJCQPRKOPS

NTGVUWPSNX

GZYZP\[^]`

_badc6efch

gjiWk

=87%:�� :

�� 8��

�� 8�

� �� 8

——————————perturbative value

perturbative 2 → 2 transport is lousy at maintaining local equilibrium

large v2 needs 15× perturbative opacities - σel × dNg/dη ≈ 45 mb ×1000


DM & Huovinen, PRL94

� � ��

� � � ��

� ��

��

!"#$&

%

'�( )�� ( )�� ( )�� ( '

� ( �� (�

�

even 50 mb does not give an ideal fluid


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


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?


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


DM & Huovinen, arXiv:0806.1367

RHIC Au+Au, b = 8 fm, η/s ≈ 1/(4π) (σ ∝ τ 2/3)


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?


Documents

Open issues in hydro and transporttheory.tifr.res.in/~qcdinit/talks/molnar.pdfGreens function is acausal (allows x > t) G(~x; t;~x0;t0) = 1 [4ˇ (t t0)]3=2 exp (~x ~x0)2 4 (t t0) |{Adding