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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: hs  p
pQCD and hadron gas: hs ~ 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  hs  0.3

				
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