Hydrodynamics in High-Density Scenarios
Assumes local thermal equilibrium (zero mean-free-path limit)
and solves equations of motion for fluid elements (not particles)
Equations given by continuity, conservation laws, and Equation of
State (EOS) Kolb, Sollfrank
EOS relates quantities like pressure, temperature, chemical & Heinz,
potential, volume = direct access to underlying physics
Hydromodels can describe mT (pT) spectra
• Good agreement with hydrodynamic prediction at RHIC & SPS (2d only)
• RHIC: Tth~ 100 MeV, bT ~ 0.55 c
Blastwave vs. Hydrodynamics
Tdec = 100 MeV
Kolb and Rapp,
PRC 67 (2003)
Mike Lisa (QM04): Use it don’t abuse it ! Only use a static
freeze-out parametrization when the dynamic model doesn’t work !!
Basics of hydrodynamics
Charge conservations (baryon, strangeness, etc…)
For perfect fluids (neglecting viscosity), Need equation of state
to close the system of eqs.
Energy density Pressure 4-velocity Hydro can be connected
directly with lattice QCD
Within ideal hydrodynamics, pressure gradient dP/dx is the driving
force of collective flow.
Collective flow is believed to reflect information about EoS!
Phenomenon which connects 1st principle with experiment
Input for Hydrodynamic Simulations
Tchemical Final stage:
Hadronic interactions (cascade ?)
Need decoupling prescription
Hydrodynamics can be applied
if thermalization is achieved.
Need EoS (Lattice QCD ?)
Color Glass Condensate ?
Instead parametrization (a)
for hydro simulations
Caveats of the different stages
Recently a lot of interest (Hirano et al., Heinz et al.)
Presently parametrized through initial thermalization time t0,
initial entropy density s0 and a parameter (pre-equilibrium
Which EoS ? Maxwell construct with hadronic stage ?
Nobody uses Lattice QCD EoS. Why not ?
Do we treat it as a separate entity with its own EoS
Hadronic cascade allows to describe data without an a
Interface 1: Initial Condition
•Need initial conditions (energy density, flow velocity,…)
Initial time t0 ~ thermalization time
•Parametrize initial •Take initial distribution
hydrodynamic field from other calculations
x Energy density from NeXus.
e or s proportional to (Left) Average over 30 events
rpart, rcoll or arpart + brcoll (Right) Event-by-event basis
Main Ingredient: Equation of State
One can test many kinds of EoS in hydrodynamics.
EoS with chemical freezeout
Typical EoS in hydro model
H: resonance gas(RG)
Kolb and Heinz (’03)
Hirano and Tsuda(’02)
PCE:partial chemical equiliblium
CFO:chemical freeze out
CE: chemical equilibrium
Interface 2: Hadronization
Kolb, Sollfrank, Hirano & Tsuda; Teaney, Lauret
Huovinen & Heinz; Teaney; & Shuryak;
Hirano;… Kolb & Rapp Bass & Dumitru
The Three Pillars of Experimental
Tests to Hydrodynamics
Radial Flow in partonic and hadronic phase
Identified Elliptic Flow (v2)
Spatial to Momentum anisotropy, mostly in partonic phase
Kinetic Freezeout Surface
Lifetime of Source
Conclusions from hydro
Early local thermalization
Viscosity, mean free path
π-, K-, p : reasonable agreement
Au+Au, sNN = 200 GeV
Best agreement for :
Tdec= 100 MeV
α = 0.02 fm-1
α ≠ 0 : importance of
Central Data Only at low pT
(pT < 1.5 – 2 GeV/c)
Failing at higher pT (>
2 GeV/c) expected:
Tdec = 165 MeV Less rescattering
Tdec = 100 MeV
P.F. Kolb and R. Rapp, Phys. Rev. C 67 (2003) 044903
α : initial (at τ0) transverse velocity : vT(r)=tanh(αr)
π-, K-, p : apparent disagreement?
Au+Au, sNN = 62.4 GeV
Predictions normalized to data
STAR preliminary data
Limited range of agreement
Hydro starts failing at 62 GeV?
different feed-down treatment in
data and hydro?
Different initial / final conditions
than at 200 GeV ?
Lower Tdec at 62 GeV ?
Larger τ0 at 62 GeV ?
Increasing τ0 gives much better
Tdec = 100 MeV
PHENIX proton and pion spectra vs. hydro
Conclusions from spectra
Central spectra well described either by including a pre-
equilibrium transverse flow or by using a hadron cascade
for the hadronic phase.
Multistrange Baryons can be described with common
decoupling temperature. Different result than blast wave
fit. Blast wave fit is always better.
Centrality dependence poorly described by hydro
Energy dependence (62 to 200 GeV) indicates lower
decoupling temperature and longer initial thermalization
time at lower energy. System thermalizes slower and stays
Collective anisotropic flow
Elliptic Flow (in the transverse plane)
for a mid-peripheral collision
Reaction Flow In-plane
Dashed lines: hard
sphere radii of nuclei
Re-interactions among what? Hadrons, partons or both?
In other words, what equation of state?
Transverse plane Reaction plane
A.Poskanzer & S.Voloshin (’98)
0th: azimuthally averaged dist. radial flow “Flow” is not a good terminology
1st harmonics: directed flow especially in high pT regions
2nd harmonics: elliptic flow due to jet quenching.
Large spatial anisotropy turns into
momentum anisotropy, IF the
particles interact collectively !
How does the system respond to the
initial spatial anisotropy ?
Free streaming Ollitrault (’92) Hydrodynamic expansion
0 f 2p anisotropy
0 f 2p
Hydrodynamics describes the data
Strong collective flow:
elliptic and radial
small mean free path,
lots of interactions
# III: The medium consists of constituent quarks ?
Ideal liquid dynamics –
reached at RHIC for the 1st time
How strong is the coupling ?
Navier-Stokes type calculation Simple pQCD processes do not
of viscosity – near perfect liquid generate sufficient interaction
Viscous force ~ 0 strength (2 to 2 process = 3 mb)
Remove your organic prejudices
Don’t equate viscous with “sticky” !
Think instead of a not-quite-ideal fluid:
“not-quite-ideal” “supports a shear stress”
then defined as
Fx v x
η ( momentum density ) mean free path )
Dimensional 1 p
n p mfp n p
estimate: nσ σ
Viscosity for a( ideal gas η
Large cross sections small viscosity
Why the interest in viscosity?
A.) Its vanishing is associated with the applicability of ideal
hydrodynamics (Landau, 1955):
Inertial Forces rV BU LK L
Ideal Hydro Reynolds Number 1
Drag Forces h
rV BU LK L L
h r v t herm al(mfp )so 1 1
rv t herm al mfp mfp
B.) Successes of ideal hydrodynamics applied to RHIC data
suggest that the fluid is “as perfect as it can be”, that is, it
approaches the (conjectured) quantum mechanical limit
h (entropy density)
See “A Viscosity Bound Conjecture”,
P. Kovtun, D.T. Son, A.O. Starinets, hep-th/0405231
Consequences of a perfect liquid
All “realistic” hydrodynamic calculations for RHIC fluids to date have
assumed zero viscosity h = 0 “perfect fluid”
But there is a (conjectured) quantum limit h ( Entropy Density ) s
Motivated by calculation of lower viscosity bound in black hole via supersymmetric N=4
Yang Mills theory in AdS (Anti deSitter) space (conformal field theory)
fluids sit wrt
RHIC “fluid” might
be at ~2-3 on this
400 times less viscous than water,
10 times less viscous than
superfluid helium !
Viscosity in Collisions
Hirano & Gyulassy, Teaney, Moore, Yaffe, Gavin, etc.
supersymmetric Yang-Mills: hs p
pQCD and hadron gas: hs ~ 1
d.o.f. in perfect liquid ? Bound states ?,
constituent quarks ?, heavy resonances ?
November, 2005 issue of Scientific
“The Illusion of Gravity” by J.
A test of this prediction comes from the
Relativistic Heavy Ion Collider (RHIC)
at BrookhavenNational Laboratory,
which has been colliding gold nuclei at
very high energies. A preliminary
analysis of these experiments indicates
the collisions are creating a fluid with
very low viscosity. Even though Son and
his co-workers studied a simplified
version of chromodynamics, they seem to
have come up with a property that is
shared by the real world. Does this mean
that RHIC is creating small five-
dimensional black holes? It is really too
early to tell, both experimentally and
theoretically. (Even if so, there is nothing
to fear from these tiny black holes-they
evaporate almost as fast as they are
formed, and they "live" in five
dimensions, not in our own four-
Even charm flows
strong elliptic flow of electrons from D
meson decays → v2D > 0
v2c of charm quarks?
(Lin & Molnar, PRC 68 (2003) 044901)
v2 ( pT ) av2 (
D q u
pT ) bv2 (
pT ) v2
χ2 minimum result
universal v2(pT) for all quarks D->e
simultaneous fit to p, K, e v2(pT)
b = 0.96
4σ within recombination model: charm
flows like light quarks!
Constraining medium viscosity h/s
Simultaneous description of
STAR R(AA) and PHENIX v2
(Rapp & Van Hees, PRC 71, 2005)
Ads/CFT == h/s ~ 1/4p ~ 0.08
Perturbative calculation of D (2pt) ~6
(Teaney & Moore, PRC 71, 2005)
transport models require
small heavy quark
small diffusion coefficient
DHQ x (2pT) ~ 4-6
this value constrains the
h/s ~ (1.3 – 2) / 4p
within a factor 2 of
consistent with light hadron
An alternate idea (Abdel-Aziz & Gavin)
Level of viscosity will affect the diffusion of momentum correlations
h Ts QGP + mixed phase + hadrons
T 1 (h /s) T(t)
Broadening from viscosity
d 2 4 (t )
2 , = width of momentum covariance C in rapidity
Abdel-Aziz & S.G
effect on momentum diffusion:
viscous liquid pQGP ~ HRG ~ 1 fm
nearly perfect sQGP ~ (4 pTc)-1 ~ 0.1 fm
wanted: rapidity dependence of momentum correlation function
we want: C pti ptj pt
N i j
STAR N p t :n p ti pt ptj pt
measures: 2 2
N C pt (density correlations)
density correlation function may differ from rg
maybe n 2*
STAR, PRC 66, 044904 (2006)
0.08 hs 0.3