hydrodynamics in high-density scenarios assumes local thermal equilibrium (zero mean-free-path...
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Hydrodynamics in High-Density Scenarios Assumes local thermal equilibrium (zero mean-free-path limit) Assumes local thermal equilibrium (zero mean-free-path limit)
and solves equations of motion for fluid elements (not particles)and solves equations of motion for fluid elements (not particles) Equations given by continuity, conservation laws, and Equations given by continuity, conservation laws, and Equation of Equation of
State (EOS)State (EOS) EOS relates quantities like pressure, temperature, chemical EOS relates quantities like pressure, temperature, chemical
potential, volume = potential, volume = direct access to underlying physicsdirect access to underlying physics
Kolb, Sollfrank & Heinz,hep-ph/0006129
Hydromodels can describe mT (pT) spectra
• Good agreement with hydrodynamic prediction at RHIC & SPS (2d only)• RHIC: Tth~ 100 MeV, T ~ 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 hydrodynamicsHydrodynamic Equations
Energy-momentum conservation
Charge conservations (baryon, strangeness, etc…)
For perfect fluids (neglecting viscosity),
Energy density Pressure 4-velocity
Within ideal hydrodynamics, pressure gradient dP/dx is the drivingforce of collective flow. Collective flow is believed to reflect information about EoS! Phenomenon which connects 1st principle with experiment
Need equation of state(EoS)
P(e,nB)
to close the system of eqs. Hydro can be connecteddirectly with lattice QCD
TchemicalTchemical
Input for Hydrodynamic Simulations
Final stage:Hadronic interactions (cascade ?) Need decoupling prescription
Intermediate stage:Hydrodynamics can be appliedif thermalization is achieved. Need EoS (Lattice QCD ?)
Initial stage:Pre-equilibrium,Color Glass Condensate ?Instead parametrization () for hydro simulations
Caveats of the different stages
Initial stageInitial stage Recently a lot of interest (Hirano et al., Heinz et al.)Recently a lot of interest (Hirano et al., Heinz et al.) Presently parametrized through initial thermalization time Presently parametrized through initial thermalization time 00, ,
initial entropy density sinitial entropy density s00 and and parameter (pre-equilibrium parameter (pre-equilibrium
‘partonic wind’)‘partonic wind’) QGP stageQGP stage
Which EoS ? Maxwell construct with hadronic stage ?Which EoS ? Maxwell construct with hadronic stage ? Nobody uses Lattice QCD EoS. Why not ?Nobody uses Lattice QCD EoS. Why not ?
Hadronic stageHadronic stage Do we treat it as a separate entity with its own EoSDo we treat it as a separate entity with its own EoS Hadronic cascade allows to describe data without an Hadronic cascade allows to describe data without an
Interface 1: Initial ConditionInterface 1: Initial Condition
•Need initial conditions (energy density, flow velocity,…)
•Parametrize initialhydrodynamic field
Initial time 0 ~ thermalization time
Hira
no .
(’02)
•Take initial distributionfrom other calculations
e or s proportional to part, coll or apart + bcoll
Energy density from NeXus.(Left) Average over 30 events(Right) Event-by-event basis
x
x x
yy
Main Ingredient: Equation of State
Latent heat
One can test many kinds of EoS in hydrodynamics.
Typical EoS in hydro modelTypical EoS in hydro modelTypical EoS in hydro modelTypical EoS in hydro model
H: resonance gas(RG)
p=e/3
Q: QGP+RG
EoS with chemical freezeoutEoS with chemical freezeoutEoS with chemical freezeoutEoS with chemical freezeout
Kol
b an
d H
einz
(’0
3)
Hira
no a
nd T
suda
(’02)
PCE:partial chemical equilibliumCFO:chemical freeze outCE: chemical equilibrium
Interface 2: HadronizationInterface 2: Hadronization
Tc
QG
P p
has
eH
ad
r on
ph a
s e
PartialChemical
EquilibriumEOS
Hirano & Tsuda;Teaney;
Kolb & Rapp
Teaney, Lauret & Shuryak;
Bass & Dumitru
Tch
Tth
HadronicCascade
ChemicalEquilibrium
EOS
Tth
Kolb, Sollfrank,Huovinen & Heinz;
Hirano;…
Ideal hydrodynamics
The Three Pillars of Experimental Tests to Hydrodynamics Identified SpectraIdentified Spectra
Radial Flow in partonic and hadronic phaseRadial Flow in partonic and hadronic phase
Identified Elliptic Flow (v2)Identified Elliptic Flow (v2) Spatial to Momentum anisotropy, mostly in partonic phaseSpatial to Momentum anisotropy, mostly in partonic phase
HBT resultsHBT results Kinetic Freezeout SurfaceKinetic Freezeout Surface Lifetime of SourceLifetime of Source
Conclusions from hydroConclusions from hydro Early local thermalizationEarly local thermalization Viscosity, mean free pathViscosity, mean free path Coupling, CollectivityCoupling, Collectivity
π-, K-, p : reasonable agreement
BestBest agreement for : agreement for : TTdecdec= 100 MeV= 100 MeV
αα = 0.02 fm = 0.02 fm-1-1 αα ≠≠ 0 : importance of 0 : importance of
inital conditionsinital conditions Only at Only at low plow pTT
(p(pTT < 1.5 – 2 GeV/c) < 1.5 – 2 GeV/c)
FailingFailing at at higherhigher p pTT (> (>
2 GeV/c) expected:2 GeV/c) expected: Less rescatteringLess rescattering
NNAu+Au, s = 200 GeV
Tdec = 165 MeVTdec = 100 MeV
α : initial (at τ0) transverse velocity : vT(r)=tanh(αr)
Central Data
Thermalization validity limitP.F. Kolb and R. Rapp, Phys. Rev. C 67 (2003) 044903
π-, K-, p : apparent disagreement?
Predictions Predictions normalizednormalized to data to data LimitedLimited range of agreement range of agreement Hydro starts failing at 62 GeV?Hydro starts failing at 62 GeV? different feed-down treatment in different feed-down treatment in
data and hydro?data and hydro? DifferentDifferent initial / final initial / final
conditionsconditions than at 200 GeV ? than at 200 GeV ? Lower TLower Tdecdec at 62 GeV ? at 62 GeV ?
Larger Larger ττ00 at 62 GeV ? at 62 GeV ?
Increasing Increasing ττ00 gives much gives much betterbetter
agreement!agreement! TTdecdec = 100 MeV = 100 MeV
STAR preliminary data
NNAu+Au, s = 62.4 GeV
Conclusions from spectra
Central spectra well described either by including a pre-Central spectra well described either by including a pre-equilibrium transverse flow or by using a hadron cascade for equilibrium transverse flow or by using a hadron cascade for the hadronic phase.the hadronic phase.
Multistrange Baryons can be described with common Multistrange Baryons can be described with common decoupling temperature. Different result than blast wave fit. decoupling temperature. Different result than blast wave fit. Blast wave fit is always better.Blast wave fit is always better.
Centrality dependence poorly described by hydroCentrality dependence poorly described by hydro Energy dependence (62 to 200 GeV) indicates lower Energy dependence (62 to 200 GeV) indicates lower
decoupling temperature and longer initial thermalization decoupling temperature and longer initial thermalization time at lower energy. System thermalizes slower and stays time at lower energy. System thermalizes slower and stays together longer. together longer.
Elliptic Flow (in the transverse plane)
for a mid-peripheral collision
Dashed lines: hard sphere radii of nuclei
Reactionplane
In-planeOu
t-o
f-p
lan
e
Y
X
Re-interactions FLOW Re-interactions among what? Hadrons, partons or both?
In other words, what equation of state?
Flow
Flo
w
Anisotropic FlowAnisotropic Flow
A.Poskanzer & S.Voloshin (’98)
z
x
x
y
Transverse plane Reaction plane
0th: azimuthally averaged dist. radial flow1st harmonics: directed flow2nd harmonics: elliptic flow…
“Flow” is not a good terminologyespecially in high pT regions
due to jet quenching.
Large spatial anisotropy Large spatial anisotropy turns intoturns into
momentum anisotropy, IF momentum anisotropy, IF the particles interact the particles interact
collectively !collectively !
High pTprotons
Low pTprotons
How does the system respond to the How does the system respond to the initial spatial anisotropy ?initial spatial anisotropy ?
Ollitrault (’92) Hydrodynamic expansion
Initial spatial anisotropy
Final momentum anisotropy
INPUT
OUTPUT
Rescattering
dN/d
Free streaming
0 2dN
/d
0 2
2v2
x
y
Hydrodynamics describes the data
Hydrodynamics:strong coupling,small mean free path,lots of interactionsNOT plasma-like
Strong collective flow:elliptic and radial expansion withmass ordering
How strong is the coupling ?Navier-Stokes type calculationof viscosity – near perfect liquidViscous force ~ 0
Simple pQCD processes do not generate sufficient interaction strength (2 to 2 process = 3 mb)
v2
pT (GeV/c)
Remove your organic prejudicesRemove your organic prejudices Don’t Don’t equate viscous with “sticky” ! equate viscous with “sticky” !
Think instead of a not-quite-ideal fluid:Think instead of a not-quite-ideal fluid: ““not-quite-ideal” not-quite-ideal” “supports a shear stress” “supports a shear stress” Viscosity Viscosity
then defined asthen defined as
Dimensional Dimensional estimate:estimate:
ViscosityViscosityincreasesincreases with withtemperaturetemperature
LargeLarge cross sections cross sections smallsmall viscosity viscosity
Viscosity Primer
yv
AF xx
σmkT
η)(
σσ1
)()(η
gasidealnearlyafor
pn
pnmfppn
pathfreemeandensitymomentum
Ideal Hydrodynamics Why the interest in viscosity?Why the interest in viscosity?
A.) Its vanishing is associated with the applicability of ideal A.) Its vanishing is associated with the applicability of ideal hydrodynamics (Landau, 1955):hydrodynamics (Landau, 1955):
B.) Successes of ideal hydrodynamics applied to RHIC data B.) Successes of ideal hydrodynamics applied to RHIC data suggest that the fluid is “as perfect as it can be”, that is, it suggest that the fluid is “as perfect as it can be”, that is, it approaches the (conjectured) quantum mechanical limitapproaches the (conjectured) quantum mechanical limit
See “See “A Viscosity Bound ConjectureA Viscosity Bound Conjecture”, ”, P. P. KovtunKovtun, , D.T. SonD.T. Son, , A.O. A.O. StarinetsStarinets, , hep-th/0405231hep-th/0405231
11 so )(
1Forces DragForces Inertial
Number Reynolds Hydro Ideal
mfpL
mfpvLV
mfpv
LV
thermal
BULKthermal
BULK
s
4
)densityentropy (4
Consequences of a perfect liquid All “realistic” hydrodynamic calculations for RHIC fluids to date All “realistic” hydrodynamic calculations for RHIC fluids to date
have assumed zero viscosity have assumed zero viscosity = 0 = 0 “perfect fluid” “perfect fluid” But there is a (conjectured) quantum limitBut there is a (conjectured) quantum limit
Where do Where do “ordinary” “ordinary” fluids sit wrt fluids sit wrt this limit?this limit?
RHIC “fluid” RHIC “fluid” mightmightbe at ~2-3 on this be at ~2-3 on this scale (!)scale (!)
400 times less viscous than water,400 times less viscous than water,10 times less viscous than 10 times less viscous than superfluid helium !superfluid helium !
sDensityEntropy
4
)(4
T=10T=101212 KK
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)
Viscosity in Collisions Hirano & Gyulassy, Teaney, Moore, Yaffe, Gavin, etc.
supersymmetric Yang-Mills: s pQCD and hadron gas: s ~ 1
liquid ?
liquid
plasma
gas
d.o.f. in perfect liquid ? Bound states ?, constituent quarks ?, heavy resonances ?
Suggested Reading November, 2005 issue of Scientific November, 2005 issue of Scientific AmericanAmerican““The Illusion of Gravity” by J. The Illusion of Gravity” by J.
Maldacena Maldacena
A test of this prediction comes from the A test of this prediction comes from the Relativistic Heavy Ion Collider (RHIC) Relativistic Heavy Ion Collider (RHIC) at BrookhavenNational Laboratory, at BrookhavenNational Laboratory, which has been colliding gold nuclei at which has been colliding gold nuclei at very high energies. A preliminary very high energies. A preliminary analysis of these experiments indicates analysis of these experiments indicates the collisions are creating a fluid with the collisions are creating a fluid with very low viscosity. Even though Son and very low viscosity. Even though Son and his co-workers studied a simplified his co-workers studied a simplified version of chromodynamics, they seem to version of chromodynamics, they seem to have come up with a property that is have come up with a property that is shared by the real world. shared by the real world. Does this mean Does this mean that RHIC is creating small five-that RHIC is creating small five-dimensional black holes? It is really too dimensional black holes? It is really too early to tell,early to tell, both experimentally and both experimentally and theoretically. (Even if so, there is nothing theoretically. (Even if so, there is nothing to fear from these tiny black holes-they to fear from these tiny black holes-they evaporate almost as fast as they are evaporate almost as fast as they are formed, and they "live" in five formed, and they "live" in five dimensions, not in our own four-dimensions, not in our own four-dimensional world.)dimensional world.)
χ2 minimum resultD->e
2σ
4σ
1σ
Even charm flows strong elliptic flow of electrons from D strong elliptic flow of electrons from D
meson decays meson decays → v→ v22DD > 0 > 0
vv22cc of charm quarks? of charm quarks?
recombination Ansatz: recombination Ansatz: (Lin & Molnar, PRC 68 (2003) 044901)(Lin & Molnar, PRC 68 (2003) 044901)
universal vuniversal v22(p(pTT) for all quarks) for all quarks
simultaneous fit to simultaneous fit to , K, e v, K, e v22(p(pTT))
eT
D
cqT
D
uqT
D vpm
mbvp
m
mavpv 2222 )()()(
a = 1
b = 0.96
2/ndf: 22/27
within recombination model: charm within recombination model: charm flows like light quarks!flows like light quarks!
Constraining medium viscosity /s Simultaneous description of Simultaneous description of
STAR R(AA) and PHENIX v2STAR R(AA) and PHENIX v2for charm. for charm. (Rapp & Van Hees, PRC 71, 2005)(Rapp & Van Hees, PRC 71, 2005)
Ads/CFT == Ads/CFT == /s ~ 1/4/s ~ 1/4 ~ 0.08 ~ 0.08 Perturbative calculation of D (2Perturbative calculation of D (2t) ~6t) ~6
(Teaney & Moore, PRC 71, 2005) (Teaney & Moore, PRC 71, 2005) == == /s~1/s~1
transport models requiretransport models require small heavy quark small heavy quark
relaxation timerelaxation time small diffusion coefficient small diffusion coefficient
DDHQHQ x (2 x (2T) ~ 4-6T) ~ 4-6 this value constrains the this value constrains the
ratio viscosity/entropyratio viscosity/entropy /s ~ (1.3 – 2) / 4/s ~ (1.3 – 2) / 4 within a factor 2 of within a factor 2 of conjectured lower conjectured lower quantum boundquantum bound consistent with light hadron consistent with light hadron
vv22 analysis analysis electron Relectron RAAAA ~ ~ 00 R RAAAA at high p at high pTT - - is bottom suppressed as well?is bottom suppressed as well?
An alternate idea (Abdel-Aziz & Gavin)
viscous liquid pQGP ~ HRG ~ 1 fm
nearly perfect sQGP ~ (4 Tc)-1 ~ 0.1 fm
Abdel-Aziz & S.G
Ts
Level of viscosity will affect the diffusion of momentum correlationskinematic viscosity
effect on momentum diffusion:
limiting cases:
wanted:wanted: rapidity dependence of momentum correlation rapidity dependence of momentum correlation functionfunction
T 1( /s)
Broadening from viscosity
d
d 2
4 ( ) 2 ,
QGP + mixed phase + hadrons T()
= width of momentum covariance C in rapidity
we want: 2
2
1t
jitjti ppp
NC
STAR measurement
STAR measures:STAR measures:
maybe maybe n n 2 2**
STAR, PRC 66, 044904 (2006) STAR, PRC 66, 044904 (2006)
uncertainty range uncertainty range
** 2 2**
0.08 0.08 s s 0.3 0.3
N p t :n pti pt ptj pt ij
N2C pt
2(density correlations)
density correlation functiondensity correlation function may differ from rg