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Intensive Lecture
YITP, December 10th, 2008
Relativistic Ideal and
Viscous Hydrodynamics
Tetsufumi Hirano
Department of Physics
The University of Tokyo
TH, N. van der Kolk, A. Bilandzic, arXiv:0808.2684[nucl-th];
to be published in “Springer Lecture Note in Physics”.
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
• Bjorken’s scaling solution with viscosity
• Effect of viscosity on particle spectra
(discussion)
PART 1
Hydrodynamics
in Heavy Ion Collisions
Why Hydrodynamics?
Energy-momentum:
Static
•Quark gluon plasma
under equilibrium
•Equation of states
•Transport coefficients
•etc
Conserved number:
Dynamics
•Expansion, Flow
•Space-time evolution of
thermodynamic variables
•Local thermalization
•Equation of states
Longitudinal Expansion
in Heavy Ion Collisions
Freezeout
“Re-confinement”
Expansion, cooling
Thermalization
First contact
(two bunches of gluons)
Strategy: Bottom-Up
Approach
•The first principle (QuantumChromo Dynamics)
•Inputs to phenomenology (lattice QCD)
Complexity
•Non-linear interactions
of gluons
•Phenomenology
(hydrodynamics)
•Strong coupling
•Dynamical many body system
•Color confinement
•Experimental data
@ Relativistic Heavy Ion Collider
~200 papers from 4 collaborations
since 2000
Application of Hydro Results
Jet quenching
Recombination
J/psi suppression
Coalescence
Heavy quark diffusion
Thermal
radiation
(photon/dilepton)
Meson
J/psi
c
Baryon
c bar
Information
along a path
Information
on surface
Information
inside medium
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.
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
• Bjorken’s scaling solution with viscosity
• Effect of viscosity on particle spectra
(discussion)
PART 2
Collective Flow
Sufficient Energy Density?
Bjorken(’83)
Bjorken energy density
total energy
(observables)
t: proper time
y: rapidity
R: effective transverse radius
mT: transverse mass
Critical Energy Density from Lattice
Adopted from
Karsch(PANIC05)
Note that recent results seem to be Tc~190MeV.
Centrality Dependence of Energy
Density
Well above ec
from lattice in
central collision
at RHIC, if
assuming
t=1fm/c.
ec from lattice
PHENIX(’05)
STAR(’08)
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.
Gyulassy, Matsui(’84) Ruuskanen(’84)
• Boost invariant?
• Averaged over transverse area. Effect of
thickness? How to estimate area?
Matter in (Chemical) Equilibrium?
direct
Resonance decay
Two fitting parameters: Tch, mB
Amazing Fit!
T=177MeV, mB = 29 MeV
Close to Tc from lattice
Caveats (II)
• Even e+e- or pp data can be fitted
well!
See, e.g., Becattini&Heinz(’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
Recent example
Just a fitting parameter?
Where is the region in which we can believe
these results as “temperature” and “chemical
potential”.
STAR, 0808.2041[nucl-ex]
Matter in (Kinetic) Equilibrium?
Kinetically equilibrated
matter at rest
Kinetically equilibrated
matter at finite velocity
um
py
py
px
Isotropic distribution
px
Lorentz-boosted distribution
Radial Flow
Kinetic equilibrium
inside matter
Blast wave model
(thermal+boost)
e.g. Sollfrank et al.(’93)
Pressure gradient
 Driving force of flow
 Flow vector points to
radial direction
Spectral change is seen in AA!
Adopted from
O.Barannikova,
(QM05)
Power law in pp & dAu
Convex to Power law
in Au+Au
•“Consistent” with
thermal + boost
picture
•Large pressure
could be built up in
AA collisions
Caveats (III)
•Flow reaches 50-60% of speed of light!?
•Radial flow even in pp?
•How does freezeout happen dynamically?
STAR, white paper(’05)
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
Anisotropic Transverse Flow
y
f
z
x
x
Reaction Plane
Transverse Plane
(perpendicular to collision axis)
Poskanzer & Voloshin (’98)
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
Ollitrault (’92)
What is Elliptic Flow?
--How does the system respond to spatial anisotropy?-No secondary interaction
Hydro behavior
y
f
x
INPUT
Spatial Anisotropy
OUTPUT
dN/df
dN/df
Interaction among
produced particles
2v2
Momentum Anisotropy
0
f
2p
0
f
2p
Eccentricity:
Spatial Anisotropy
y
x
In hydrodynamics,
:Energy density
or
:Entropy density
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 Cu+Cu collisions
Kolb and Heinz (’03)
Elliptic Flow in Hydro
Saturate in first several
femto-meters
v2 signal is sensitive to
initial stage.
Response of the system
(= v2/e) is almost
constant.
Pocket formula:v2~0.2e
Elliptic Flow in Kinetic Theory
ideal hydro limit
v2
Zhang et al.(’99)
: Ideal hydro
b = 7.5fm
: strongly
interacting
system
t(fm/c)
generated through secondary collisions
v2 is saturated in the early stage
sensitive to cross section (~1/m.f.p.~1/viscosity)
Response=(output)/(input)
Discovery of Perfect Fluidity!?
Fine structure of
elliptic flow
Data reaches
hydro limit curve
Figures taken from
STAR white paper(’05)
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
Inputs for Hydrodynamic
Simulations for perfect fluids
Final stage:
Free streaming particles
 Need decoupling prescription
t
z
0
Intermediate stage:
Hydrodynamics can be valid
as far as local thermalization is
achieved.  Need EOS P(e,n)
Initial stage:
Particle production,
pre-thermalization?
 Instead, initial conditions
for hydro simulations
Main Ingredient: Equation of State
P.Kolb and U.Heinz(’03)
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
Latent heat
Note: Chemically frozen hadronic EOS is
needed to reproduce heavy particle yields.
(Hirano, Teaney, Kolb, Grassi,…)
Interface 1: Initial Condition
Initial conditions (tuned to reproduce dNch/dh):
initial time, energy density, flow velocity
Energy density distribution
Transverse plane Reaction plane
(Lorentz-contracted)
nuclei
Two Hydro Initial Conditions
Which Clear the “First Hurdle”
Centrality dependence
Rapidity dependence
1.Glauber model
Npart:Ncoll = 85%:15%
2. CGC model
Matching I.C. via e(x,y,hs)
Kharzeev, Levin, and Nardi
Implemented in hydro
by TH and Nara
Interface 2: Freezeout
--How to Convert Bulk to Particles-Cooper-Frye formula
S
Outputs from hydro
in F.O. hypersurface
Contribution from
resonance decays can
be treated with additional
decay kinematics.
Utilization of Hadron Transport
Model for Freezeout Process
(1) Sudden freezeout:
QGP+hadron fluids
At T=Tf, l=0 (ideal fluid)
l=infinity (free stream)
T=Tf
t
(2) Gradual freezeout:
QGP fluid + hadron gas
Automatically describe
chemical and thermal
freezeouts
t
Hadron fluid
QGP fluid
QGP fluid
z
z
0
0
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
1. Data Properly Reproduced?
Final differential v2 depends on hadronic
chemical compositions.
140MeV
0
0.2
0.4
0.2 0.4 0.6 0.8
0
0.6 0.8
transverse momentum (GeV/c)
1.0
CE: chemical equilibrium (not consistent with exp. yield)
PCE: partial chemical equilibrium
T.H. and K.Tsuda (’02)
100MeV
v2(pT)
Chemical Eq.
v2
At hadronization
<pT>
pT
Chemical F.O.
v2
<pT>
pT
freezeout
v2(pT)
v2 v2(pT)
Cancel between v2 and <pT>
<pT>
pT
Intuitive Picture
Chemical
Freezeout
Mean ET
decreases
due to pdV
work
Chemical
Equilibrium
MASS energy
KINETIC
energy
2. Is mass ordering for v2(pT) a signal
of the perfect QGP fluid?
Pion
20-30%
Proton
Two neglected effects in hydro:
chemical freezeout and hadronic
dissipation
Mass dependence is o.k. from
hydro+cascade.
Mass ordering comes from
rescattering effect. Interplay btw.
radial and elliptic flows
Not a direct sign of the perfect
QGP fluid
3. Is viscosity really small in QGP?
•1+1D 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!
Quiz: Which has larger viscosity
at 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 n=h/r:
~1.0x10-6 [m2/s] (Water 20℃)
~1.5x10-5 [m2/s] (Air 20℃)
hwater > hair BUT nwater < nair
Non-relativistic Navier-Stokes eq. (a simple form)
Neglecting external force and assuming incompressibility.
4. Is h/s enough?
•Reynolds number
Iso, Mori, Namiki (’59)
R>>1
Perfect fluid
•(1+1)D Bjorken solution
5. Boltzmann at work?
Molnar&Gyulassy(’00)
Molnar&Huovinen(’04)
gluonic
fluid
25-30%
reduction
s ~ 15 * spert !
Caveat 1: Where is the “dilute” approximation in Boltzmann
simulation? Is l~0.1fm o.k. for the Boltzmann description?
Caveat 2: Differential v2 is tricky. dv2/dpT~v2/<pT>.
Difference of v2 is amplified by the difference of <pT>.
Caveat 3: Hadronization/Freezeout are different.
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 1+1D).
Being smaller T0 from pT dist., t0 should be larger (~10fm/c).
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!
Summary Slide @ QM2004
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