Relativistic Ideal and Viscous Hydrodynamics

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Relativistic Ideal and Viscous Hydrodynamics
Intensive Lecture YITP, December 10th, 2008

• Tetsufumi Hirano
• Department of Physics
• The University of Tokyo

TH, N. van der Kolk, A. Bilandzic,
arXiv0808.2684nucl-th to be published in
Springer Lecture Note in Physics.
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Plan of this Lecture

• 1st Day
• Hydrodynamics in heavy ion collisions
• Collective flow
• Dynamical modeling of heavy ion collisions
  (seminar)
• 2nd Day
• Formalism of relativistic ideal/viscous
  hydrodynamics
• Bjorkens scaling solution with viscosity
• Effect of viscosity on particle spectra
  (discussion)

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PART 1

• Hydrodynamics
• in Heavy Ion Collisions

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Why Hydrodynamics?

• Static
• Quark gluon plasma
• under equilibrium
• Equation of states
• Transport coefficients
• etc

Energy-momentum
Conserved number

• Dynamics
• Expansion, Flow
• Space-time evolution of
• thermodynamic variables

• Local thermalization
• Equation of states

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Longitudinal Expansion in Heavy Ion Collisions
Freezeout Re-confinement Expansion,
cooling Thermalization First contact (two
bunches of gluons)
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Strategy Bottom-Up Approach

• The first principle (QuantumChromo Dynamics)

• Inputs to phenomenology (lattice QCD)

• Complexity
• Non-linear interactions of gluons
• Strong coupling
• Dynamical many body system
• Color confinement

• Phenomenology (hydrodynamics)

• Experimental data
• _at_ Relativistic Heavy Ion Collider
• 200 papers from 4 collaborations
• since 2000

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Application of Hydro Results
Thermal radiation (photon/dilepton)
Jet quenching J/psi suppression Heavy quark
diffusion
Recombination Coalescence
Meson
J/psi
c
Baryon
c bar
Information along a path
Information on surface
Information inside medium
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Why Hydrodynamics?

• Goal To understand the hot QCD matter under
  equilibrium.
• Lattice QCD is not able to describe dynamics in
  heavy ion collisions.
• Analyze heavy ion reaction based on a model with
  an assumption of local equilibrium, and see what
  happens and whether it is consistent with data.
• If consistent, it would be a starting point of
  the physics of hot QCD matter under equilibrium.

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Plan of this Lecture

• 1st Day
• Hydrodynamics in Heavy Ion Collisions
• Collective flow
• Dynamical Modeling of heavy ion collisions
  (seminar)
• 2nd Day
• Formalism of relativistic ideal/viscous
  hydrodynamics
• Bjorkens scaling solution with viscosity
• Effect of viscosity on particle spectra
  (discussion)

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PART 2

• Collective Flow

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Sufficient Energy Density?
Bjorken(83)
Bjorken energy density
total energy (observables)
t proper time y rapidity R effective
transverse radius mT transverse mass
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Critical Energy Density from Lattice
Adopted from Karsch(PANIC05)
Note that recent results seem to be Tc190MeV.
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Centrality Dependence of Energy Density
Well above ec from lattice in central collision
at RHIC, if assuming t1fm/c.
ec from lattice
PHENIX(05) STAR(08)
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Caveats (I)

• Just a necessary condition in the sense that
  temperature (or pressure) is not measured.
• How to estimate tau?
• If the system is thermalized, the actual energy
  density is larger due to pdV work.
• Boost invariant?
• Averaged over transverse area. Effect of
  thickness? How to estimate area?

Gyulassy, Matsui(84) Ruuskanen(84)
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Matter in (Chemical) Equilibrium?
direct
Resonance decay
Two fitting parameters Tch, mB
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Amazing Fit!
T177MeV, mB 29 MeV
Close to Tc from lattice
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Caveats (II)

• Even ee- or pp data can be fitted well!
  See, e.g., BecattiniHeinz(97)
• What is the meaning of fitting parameters?
  See, e.g., Rischke(02),Koch(03)
• Why so close to Tc?
• No chemical eq. in hadron phase!?
• Essentially dynamical problem!

Expansion rate ?? Scattering rate
see, e.g., U.Heinz, nucl-th/0407067
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Recent example
Just a fitting parameter? Where is the region in
which we can believe these results as
temperature and chemical potential.
STAR, 0808.2041nucl-ex
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Matter in (Kinetic) Equilibrium?
Kinetically equilibrated matter at rest
Kinetically equilibrated matter at finite velocity
um
py
py
px
px
Lorentz-boosted distribution
Isotropic distribution
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Radial Flow
Kinetic equilibrium inside matter
Blast wave model (thermalboost)
e.g. Sollfrank et al.(93)
Pressure gradient ? Driving force of flow ? Flow
vector points to radial direction
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Spectral change is seen in AA!
Power law in pp dAu
Adopted from O.Barannikova, (QM05)
Convex to Power law in AuAu

• Consistent with thermal boost picture
• Large pressure could be built up in AA collisions

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Caveats (III)

• Flow reaches 50-60 of speed of light!?
• Radial flow even in pp?
• How does freezeout happen dynamically?

STAR, white paper(05)
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Basic Checks ? Necessary Conditions to Study the
QGP at RHIC

• Energy density can be well above ec.
• Thermalized?
• Temperature can be extracted.
• Why freezeout happens so close to Tc?
• High pressure can be built up.
• Completely equilibrated?

Importance of systematic study based on
dynamical framework
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Anisotropic Transverse Flow
y
f
z
x
x
Transverse Plane (perpendicular to collision
axis)
Reaction Plane
Poskanzer Voloshin (98)
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Directed and Elliptic Flow
The 1st mode, v1 directed flow coefficient
The 2nd mode, v2 elliptic flow coefficient
y
z
x
x

• Important in low
• energy collisions
• Vanish at midrapidity

• Important in high energy collisions

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What is Elliptic Flow?--How does the system
respond to spatial anisotropy?--
Ollitrault (92)
Hydro behavior
No secondary interaction
y
f
x
INPUT
Spatial Anisotropy
2v2
Interaction among produced particles
dN/df
dN/df
OUTPUT
Momentum Anisotropy
0
f
2p
f
0
2p
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Eccentricity Spatial Anisotropy
y
x
In hydrodynamics,
Energy density
or
Entropy density
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Eccentricity Fluctuation
Adopted from D.Hofman(PHOBOS), talk at QM2006
A sample event from Monte Carlo Glauber model
Interaction points of participants vary event by
event. ? Apparent reaction plane also varies. ?
The effect is relatively large for smaller system
such as CuCu collisions
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Elliptic Flow in Hydro
Kolb and Heinz (03)
Saturate in first several femto-meters v2 signal
is sensitive to initial stage.
Response of the system ( v2/e) is almost
constant. Pocket formulav20.2e
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Elliptic Flow in Kinetic Theory
Zhang et al.(99)
ideal hydro limit
v2
Ideal hydro
strongly interacting system
b 7.5fm
t(fm/c)
generated through secondary collisions
saturated in the early stage sensitive to cross
section (1/m.f.p.1/viscosity)
v2 is
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Discovery of Perfect Fluidity!?
Response(output)/(input)
Fine structure of elliptic flow
Data reaches hydro limit curve
Figures taken from STAR white paper(05)
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Several Remarks on the Discovery

1. Chemical Composition
2. Differential Elliptic Flow
3. Smallness of Transport Coefficients
4. Importance of Dynamics
5. Applicability of Boltzmann Eq.
6. Applicability of Blast Wave Model
7. Dependence of Initial Conditions

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Inputs for Hydrodynamic Simulations for perfect
fluids
Final stage Free streaming particles ? Need
decoupling prescription
t
Intermediate stage Hydrodynamics can be valid as
far as local thermalization is achieved. ? Need
EOS P(e,n)
z
0

• Initial stage
• Particle production,
• pre-thermalization?
• ? Instead, initial conditions
• for hydro simulations

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Main Ingredient Equation of State
Typical EOS in hydro models
EOS I Ideal massless free gas EOS H Hadron
resonance gas EOS Q QGP P(e-4B)/3 Hadron
Resonance gas
pe/3
P.Kolb and U.Heinz(03)
Latent heat
Note Chemically frozen hadronic EOS is needed to
reproduce heavy particle yields. (Hirano, Teaney,
Kolb, Grassi,)
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Interface 1 Initial Condition
Initial conditions (tuned to reproduce dNch/dh)
initial time, energy density, flow velocity
Energy density distribution
Reaction plane
Transverse plane
(Lorentz-contracted) nuclei
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Two Hydro Initial Conditions Which Clear the
First Hurdle
Centrality dependence
Rapidity dependence
1.Glauber model NpartNcoll 8515 2. CGC
model Matching I.C. via e(x,y,hs)
Kharzeev, Levin, and Nardi Implemented in hydro
by TH and Nara
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Interface 2 Freezeout --How to Convert Bulk to
Particles--
Cooper-Frye formula
Outputs from hydro in F.O. hypersurface
Contribution from resonance decays can be
treated with additional decay kinematics.
S
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Utilization of Hadron Transport Model for
Freezeout Process
(1) Sudden freezeout QGPhadron fluids
(2) Gradual freezeout QGP fluid hadron gas
At TTf, l0 (ideal fluid) ?linfinity (free
stream)
Automatically describe chemical and thermal
freezeouts
TTf
t
Hadron fluid
QGP fluid
QGP fluid
z
0
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Several Remarks on the Discovery

1. Chemical Composition
2. Differential Elliptic Flow
3. Smallness of Transport Coefficients
4. Importance of Dynamics
5. Applicability of Boltzmann Eq.
6. Applicability of Blast Wave Model
7. Dependence of Initial Conditions

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1. Data Properly Reproduced?
Final differential v2 depends on hadronic
chemical compositions.
100MeV
T.H. and K.Tsuda (02)
140MeV
0.8
1.0
0.4
0.6
0.2
0
0.6
0.8
0.2
0.4
0
transverse momentum (GeV/c)
CE chemical equilibrium (not consistent with
exp. yield) PCE partial chemical equilibrium
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Cancel between v2 and ltpTgt
Chemical Eq.
At hadronization
v2(pT)
v2(pT)
v2
v2
freezeout
pT
pT
ltpTgt
ltpTgt
Chemical F.O.
CE increase
v2(pT)
CFO decrease
v2
pT
ltpTgt
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Intuitive Picture
Chemical Freezeout
Mean ET decreases due to pdV work
MASS energy KINETIC energy
Chemical Equilibrium
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2. Is mass ordering for v2(pT) a signal of the
perfect QGP fluid?
Pion
20-30
Proton
Mass ordering comes from rescattering effect.
Interplay btw. radial and elliptic flows ?Not a
direct sign of the perfect QGP fluid
Two neglected effects in hydro chemical
freezeout and hadronic dissipation Mass
dependence is o.k. from hydrocascade.
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3. Is viscosity really small in QGP?

• 11D Bjorken flow Bjorken(83)
• Baym(84)Hosoya,Kajantie(85)Danielewicz,Gyulassy(
  85)Gavin(85)Akase et al.(89)Kouno et al.(90)

(Ideal)
(Viscous)
h shear viscosity (MeV/fm2), s entropy
density (1/fm3)
h/s is a good dimensionless measure (in the
natural unit) to see viscous effects.
Shear viscosity is small in comparison with
entropy density!
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Quiz Which has larger viscosityat room
temperature, water or air?
Pa N/m2
(Dynamical) Viscosity h 1.0x10-3 Pa s
(Water 20?) 1.8x10-5 Pa s (Air 20?)
Kinetic Viscosity nh/r 1.0x10-6 m2/s
(Water 20?) 1.5x10-5 m2/s (Air 20?)
hwater gt hair BUT nwater lt nair
Non-relativistic Navier-Stokes eq. (a simple form)
Neglecting external force and assuming
incompressibility.
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4. Is h/s enough?

• Reynolds number

Iso, Mori, Namiki (59)
Rgtgt1 ?Perfect fluid

• (11)D Bjorken solution

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5. Boltzmann at work?
MolnarGyulassy(00)
MolnarHuovinen(04)
25-30 reduction
gluonic fluid
s 15 spert !
Caveat 1 Where is the dilute approximation in
Boltzmann simulation? Is l0.1fm o.k. for the
Boltzmann description? Caveat 2 Differential v2
is tricky. dv2/dpTv2/ltpTgt. Difference of v2 is
amplified by the difference of ltpTgt. Caveat 3
Hadronization/Freezeout are different.
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6. Does v2(pT) really tell us smallness of h/s in
the QGP phase?
D.Teaney(03)

• Not a result from dynamical calculation, but a
  fitting to data.
• No QGP in the model
• t0 is not a initial time, but a freeze-out time.
• Gs/t0 is not equal to h/s, but to 3h/4sT0t0 (in
  11D).
• Being smaller T0 from pT dist., t0 should be
  larger (10fm/c).

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7. Initial condition is a unique?
Novel initial conditions from Color Glass
Condensate lead to large eccentricity.
Hirano and Nara(04), Hirano et al.(06) Kuhlman
et al.(06), Drescher et al.(06)
Need viscosity and/or softer EoS in the QGP!
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Summary Slide _at_ QM2004
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Summary So Far

• Interpretation of RHIC results involves many
  subtle issues in hydrodynamic modeling of
  reactions
• Three pillars Glauber initial condition Ideal
  QGP dissipative hadron gas
• Need to check each modeling to get conclusive
  interpretation
• Next task viscosity ? Lecture on the 2nd day