Advanced Studies Institute

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PHENIX, Phys. Rev. C69, 034909 (2004)
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data
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Input from lattice EoS of QCD Matter

• Old idea Quark Gluon Plasma
• Paradigm shift Liquid of quarks

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Notation for fluid dynamics

• Non-relativistic dynamics
• t time,
• r coordinate 3-vector, r (rx, ry, rz),
• m mass,
• (t,r) dependent variables
• n number density,
• ? entropy density,
• p pressure,
• ? energy density,
• T temperature,
• v velocity 3-vector, v (vx, vy, vz)

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Non-rel perfect fluid dynamics

• Equations of nonrelativistic hydro
• local conservation of
• charge continuity
• momentum Euler
• energy
• EoS needed
• Perfect fluid 2 equivalent definitions, term
  used by PDG
• 1 no bulk and shear viscosities, and no heat
  conduction.
• 2 T?? diag(e,-p,-p,-p) in the local rest
  frame.
• Ideal fluid ambiguously defined term,
  discouraged
• 1 keeps its volume, but conforms to the
  outline of its container
• 2 an inviscid fluid

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Dissipative, Navier-Stokes fluids

• Navier-Stokes equations dissipative,
  nonrelativistic
• EoS needed
• Shear and bulk viscosity, heat conductivity

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Parametric perfect hydro solutions

• Ansatz the density n (and T and ?) depend on
  coordinates
• only through a scale parameter s
• T. Cs. Acta Phys. Polonica B37 (2006),
  hep-ph/0111139
• Principal axis of ellipsoid
• (X,Y,Z) (X(t), Y(t), Z(t))
• Densityconst on ellipsoids. Directional
  Hubble flow.
• g(s) arbitrary scaling function. Notation n
  ?(s), T ?(s) etc.

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Family of perfect hydro solutions

• T. Cs. Acta Phys. Polonica B37 (2006)
  hep-ph/0111139
• Volume is V XYZ
• ? ?(T) exact solutions
• T. Cs, S.V. Akkelin, Y. Hama,
• B. Lukács, Yu. Sinyukov,
• hep-ph/0108067, Phys.Rev.C67
  034904,2003
• or see the sols of Navier-Stokes
  later.
• The dynamics is reduced to
• ordinary differential equations
• for the scales X,Y,Z
• PARAMETRIC solutions.
• Ti constant of integration
• Many hydro problems can be easily illustrated and
  understood on the
• equivalent problem a classical potential motion
  of a mass-point in (a shot)!
• Note temperature scaling function ?(s)
  arbitrary!

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The initial conditions and the EoS can covary so
that the freeze-out distributions are
unchanged (T/m 180/940)
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Family of viscous hydro solutions

• T. Cs.,Y. Hama in preparation
• Volume is V XYZ
• Similar to hep-ph/0108067
• The dynamics is reduced to
• non-conservative equations of motion
• for the parameters X,Y,Z
• n lt-gt s, m cancels from new terms
• depends on h/s and z/s

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Dissipative, heat conductive hydro solutions

• T. Cs. and Y. Hama, in preparation
• Introduction of kinematic heat conductivity
• Navier-Stokes, for small heat conduction,
• solved by the directional Hubble ansatz!
• Only new eq. from the energy equation
• Asymptotic (large t) role of heat conduction
• - same order of magnitude (1/t2) as bulk
  viscosity (1/t2)
• - shear viscosity term is one order of
  magnitude smaller (1/t3)
• - valid only for nearly constant densities,
• - destroys self-similarity of the solution (if
  hot spots)

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The initial conditions and the EoS can covary
even in viscous case so that exactly the same
freeze-out distributions (T/m 180/940, h/n
0.1 and z/n 0.1)
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Scaling predictions for (viscous) fluid dynamics
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• Buda-Lund hydro prediction Exact non-rel. hydro
  
• PHENIX data

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Universal hydro scaling of v2
hep-ph/0108067, nucl-th/0310040 nucl-th/0512078
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Hydro scaling of Bose-Einstein/HBT radii
Rside/Rout 1
Rside/Rlong 1
Rout/Rlong 1
1/R2side mt
1/R2out mt
1/R2long mt
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• Buda-Lund model
• Was a parameterization
• Is an interpolator btwn analytic, exact hydro
  solutions with
• Lattice QCD EOS
• Shear and Bulk viscosity (NR), heat conductivity
  (NR)
• Relativistic acceleration
• Scaling predictions of viscous hydrodynamics
• Scaling properties of slope parameters do not
  change
• Scaling properties of elliptic flow do not change
• Scaling properties of HBT radii are the same
• If shear and bulk viscosities are present in
  Navier-Stokes eqs.
• Asymptotic analysis
• Initially shear gt bulk gt perfect fluid effects
• At late times perfect fluid gt bulk gt shear
  effects

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• Back-up Slides

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Some analytic Buda-Lund results

• HBT radii widths
• Slopes, effective temperatures
• Flow coefficients are universal

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Role of initial temperature profiles
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2 Our last work inflation at RHIC
nucl-th/0206051
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1st milestone new phenomena
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2nd milestone new form of matter
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3rd milestone Top Physics Story 2005
http//arxiv.org/abs/nucl-ex/0410003 PHENIX White
Paper second most cited in nucl-ex during 2006
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4th Milestone A fluid of quarks
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Precision Probes

• This one figure encodes rigorous control of
  systematics
• in four different measurements over many orders
  of magnitude

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Motion Is Hydrodynamic

• When does thermalization occur?
• Strong evidence that final state bulk behavior
  reflects the initial state geometry
• Because the initial azimuthal asymmetry
  persists in the final state dn/d? 1 2
  v2(pT) cos (2??) ...

2v2
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The Flow Knows Quarks

• The fine structure v2(pT) for different mass
  particles shows good agreement with ideal
  (perfect fluid) hydrodynamics
• Scaling flow parameters by quark content nq
  resolves meson-baryon separation of final state
  hadrons

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Role of initial temperature profile

• Initial temperature profile arbitrary positive
  function
• Infinitly rich class of solutions
• Matching initial conditions for the density
  profile
• T. Cs. Acta Phys. Polonica B37 (2006) 1001,
  hep-ph/0111139
• Homogeneous temperature ? Gaussian density
• Buda-Lund profile
  Zimányi-Bondorf-Garpman profile

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Illustrations of exact hydro results

• Propagate the hydro solution in time numerically

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Principles for Buda-Lund hydro model

• Analytic expressions for all the observables
• 3d expansion, local thermal equilibrium, symmetry
• Goes back to known exact hydro solutions
• nonrel, Bjorken, and Hubble limits, 13 d
  ellipsoids
• but phenomenology, extrapolation for unsolved
  cases
• Separation of the Core and the Halo
• Core perfect fluid dynamical evolution
• Halo decay products of long-lived resonances
• Missing links phenomenology needed
• search for accelerating ellipsoidal rel.
  solutions
• first accelerating rel. solution nucl-th/0605070

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A useful analogy
Fireball at RHIC ? our Sun

• Core ? Sun
• Halo ? Solar wind
• T0,RHIC 210 MeV ? T0,SUN ? 16 million K
• Tsurface,RHIC 100 MeV ? Tsurface,SUN ? 6000
  K

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Buda-Lund hydro model
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The generalized Buda-Lund model

• The original model was for axial symmetry only,
  central coll.
• In its general hydrodynamical form
• Based on 3d relativistic and non-rel solutions of
  perfect fluid dynamics
• Have to assume special shapes
• Generalized Cooper-Frye prefactor
• Four-velocity distribution
• Temperature
• Fugacity

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Buda-Lund model is based on fluid dynamics
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Scaling predictions Buda-Lund hydro
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Exact scaling laws of NR hydro
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Hydro scaling of v2 and dependence
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Universal scaling and v2(centrality,?)
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Universal v2 scaling and PID dependence
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Universal scaling and fine structure of v2
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Solution of the HBT puzzle