Dénes Molnár, Purdue University 10th Zimányi Winter ... · Dénes Molnár, Purdue University 10th Zimányi Winter School on Heavy Ion Physics Nov 29 - Dec 3, 2010, KFKI/RMKI,

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


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


I. Hydro paradigm and particles

Identified particle observables are crucial

- help pin down initial conditions

- hold the key to equation of state


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


∼ 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]


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)


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)


∼ 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


Two recent refinements:

- realistic equation of state

- QGP viscosity


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


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


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


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:


Viscous QGP(?)


(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


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


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


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)


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


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


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...


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.


ALSO - ballpark agreement with estimates based on kinetic theory, transversemomentum fluctuations, heavy-quark diffusion,... Zajc @ QM2009


II. From viscous fluid to particles

For viscous hydro calculations, identified

particle observables are challenging

(yet unsolved problem)


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”)


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


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


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


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...


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


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


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(!)


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


for higher viscosity, accuracy worsens - should affect other observables also


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


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


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)


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 ]


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


[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


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


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%


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”


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


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


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


Denicol @ CATHIE-TECHQM, Dec 2009

MORE variables!


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


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


unique connections between different fields


Backup slides


“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)


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)


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)


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.


Realistically, η/s 6= const

interpolate between wQGP, sQGP, hadron gas + use τπ ≈ η/p

Denicol et al, JPG37 (’10)


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


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


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


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


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


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.


(!) 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)


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)


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)


Documents

Dénes Molnár, Purdue University 10th Zimányi Winter ... · Dénes Molnár, Purdue University 10th Zimányi Winter School on Heavy Ion Physics Nov 29 - Dec 3, 2010, KFKI/RMKI,