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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,
hep-ph/0006129
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)
044903.
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
Hydrodynamic Equations
Energy-momentum conservation
Charge conservations (baryon, strangeness, etc…)
For perfect fluids (neglecting viscosity), Need equation of state
(EoS)
P(e,nB)
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
Tchemical Final stage:
Hadronic interactions (cascade ?)
Need decoupling prescription
Intermediate stage:
Hydrodynamics can be applied
if thermalization is achieved.
Need EoS (Lattice QCD ?)
Initial stage:
Pre-equilibrium,
Color Glass Condensate ?
Instead parametrization (a)
for hydro simulations
Caveats of the different stages
Initial stage
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
‘partonic wind’)
QGP stage
Which EoS ? Maxwell construct with hadronic stage ?
Nobody uses Lattice QCD EoS. Why not ?
Hadronic stage
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
y y
Hirano .(’02)
x x
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)
Q: QGP+RG
Hirano and Tsuda(’02)
Latent heat
PCE:partial chemical equiliblium
CFO:chemical freeze out
CE: chemical equilibrium
Interface 2: Hadronization
Kolb, Sollfrank, Hirano & Tsuda; Teaney, Lauret
Huovinen & Heinz; Teaney; & Shuryak;
QGP phase
Hirano;… Kolb & Rapp Bass & Dumitru
Ideal hydrodynamics
Tc
Tch Partial
Chemical Hadronic
Chemical
Equilibrium Cascade
Equilibrium
EOS
EOS
Tth Tth
t
The Three Pillars of Experimental
Tests to Hydrodynamics
Identified Spectra
Radial Flow in partonic and hadronic phase
Identified Elliptic Flow (v2)
Spatial to Momentum anisotropy, mostly in partonic phase
HBT results
Kinetic Freezeout Surface
Lifetime of Source
Conclusions from hydro
Early local thermalization
Viscosity, mean free path
Coupling, Collectivity
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
inital conditions
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
Thermalization
validity limit
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
agreement!
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
together longer.
Collective anisotropic flow
z
y
x
Elliptic Flow (in the transverse plane)
for a mid-peripheral collision
Flow
Out-of-plane
Y
Reaction Flow In-plane
plane X
Dashed lines: hard
sphere radii of nuclei
Re-interactions FLOW
Re-interactions among what? Hadrons, partons or both?
In other words, what equation of state?
Anisotropic Flow
y
f
x
z
x
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 !
High pT
protons
Low pT
protons
How does the system respond to the
initial spatial anisotropy ?
Free streaming Ollitrault (’92) Hydrodynamic expansion
y
f
x
INPUT
Initial spatial
anisotropy 2v2
dN/df
Rescattering
dN/df
OUTPUT
Final momentum
0 f 2p anisotropy
0 f 2p
Hydrodynamics describes the data
Strong collective flow:
elliptic and radial
expansion with
mass ordering
Hydrodynamics:
strong coupling,
small mean free path,
lots of interactions
NOT plasma-like
# III: The medium consists of constituent quarks ?
baryons
mesons
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)
v2
pT (GeV/c)
Viscosity Primer
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”
Viscosity h
then defined as
Fx v x
h
A
η ( momentum density ) mean free path )
y (
Dimensional 1 p
n p mfp n p
estimate: nσ σ
mkT
Viscosity for a( ideal gas η
nearly )
σ
increases with
temperature
Large cross sections small viscosity
Ideal Hydrodynamics
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)
s
4p 4p
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
4p 4p
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)
Where do
“ordinary”
fluids sit wrt
this limit?
RHIC “fluid” might
be at ~2-3 on this
scale (!)
400 times less viscous than water,
10 times less viscous than
superfluid helium !
T=1012 K
Viscosity in Collisions
Hirano & Gyulassy, Teaney, Moore, Yaffe, Gavin, etc.
supersymmetric Yang-Mills: hs p
pQCD and hadron gas: hs ~ 1
plasma
liquid ?
gas
liquid
d.o.f. in perfect liquid ? Bound states ?,
constituent quarks ?, heavy resonances ?
November, 2005 issue of Scientific
Suggested Reading
American
“The Illusion of Gravity” by J.
Maldacena
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-
dimensional world.)
Even charm flows
strong elliptic flow of electrons from D
meson decays → v2D > 0
v2c of charm quarks?
recombination Ansatz:
(Lin & Molnar, PRC 68 (2003) 044901)
m m
v2 ( pT ) av2 (
D q u
pT ) bv2 (
q c
pT ) v2
e
mD mD
χ2 minimum result
universal v2(pT) for all quarks D->e
simultaneous fit to p, K, e v2(pT)
a=1
b = 0.96
2σ
c2/ndf: 22/27
1σ
4σ within recombination model: charm
flows like light quarks!
Constraining medium viscosity h/s
Simultaneous description of
STAR R(AA) and PHENIX v2
for charm.
(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)
== h/s~1
transport models require
small heavy quark
relaxation time
small diffusion coefficient
DHQ x (2pT) ~ 4-6
this value constrains the
ratio viscosity/entropy
h/s ~ (1.3 – 2) / 4p
within a factor 2 of
conjectured lower
quantum bound
consistent with light hadron
v2 analysis
An alternate idea (Abdel-Aziz & Gavin)
Level of viscosity will affect the diffusion of momentum correlations
kinematic viscosity
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
dt t
Abdel-Aziz & S.G
effect on momentum diffusion:
limiting cases:
viscous liquid pQGP ~ HRG ~ 1 fm
nearly perfect sQGP ~ (4 pTc)-1 ~ 0.1 fm
wanted: rapidity dependence of momentum correlation function
STAR measurement
1
we want: C pti ptj pt
2
2
N i j
STAR N p t :n p ti pt ptj pt
i j
measures: 2 2
N C pt (density correlations)
density correlation function may differ from rg
maybe n 2*
STAR, PRC 66, 044904 (2006)
uncertainty range
* 2*
0.08 hs 0.3
