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**3-dim. QGP Fluid Dynamics and Flow Observables**László Csernai (Bergen Computational Physics Lab., Univ. of Bergen)**Introduction**• Strong flow is observed => • - Early, local eq., - EoS • nq scaling – QGP flows • no flow in hadronic matter > simultaneous hadronization and FO (HBT, high strangeness abundance)**Relativistic Fluid Dynamics**Eg.: from kinetic theory. BTE for the evolution of phase-space distribution: Then using microscopic conservation laws in the collision integral C: These conservation laws are valid for any, eq. or non-eq. distribution, f(x,p). These cannot be solved, more info is needed! Boltzmann H-theorem: (i) for arbitrary f, the entropy increases, (ii) for stationary, eq. solution the entropy is maximal, EoS P = P (e,n) Solvable for local equilibrium!**Relativistic Fluid Dynamics**For any EoS, P=P(e,n), and any energy-momentum tensor in LE(!): Not only for high v!**Two theoretical problems**• Initial state – • - Fitted initial states > moderate insight • Final Freeze Out • - Realistic Model, Continuos FO, ST layer, Non-eq. distribution**Stages of a Collision**Freeze Out >>> Detectors Hadronization, chemical FO, kinetic FO -------------- One fluid >>> E O S Fluid components, Friction Local Equilibration, Fluids Collective flow reveals the EoS ifwe have dominantly one fluid with local equilibrium in a substantial part of the space-time domain of the collision !!!**Heavy Colliding System**Initial state Idealizations FO Layer FO HS time QGP EoS One fluid Hadronization Chemical Freeze Out Kinetic Freeze Out**Fire streak picture - Only in 3 dimensions!**Myers, Gosset, Kapusta, Westfall**Initial state**3rd flow component**3-dim Hydro for RHIC Energies**Au+Au ECM=65 GeV/nucl. b=0.5 bmax Aσ=0.08 => σ~10 GeV/fm e [ GeV / fm3 ] T [ MeV] . . t=0.0 fm/c, Tmax= 420 MeV, emax= 20.0 GeV/fm3, Lx,y= 1.45 fm, Lz=0.145 fm 8.7 x 4.4 fm EoS: p= e/3 - B/3, B = 397 MeV/fm3**Au+Au ECM=65 GeV/nucl. b=0.5 bmax Aσ=0.08 => σ~10**GeV/fm e [ GeV / fm3 ] T [ MeV] . . t=9.1 fm/c, Tmax= 417 MeV, emax= 19.6 GeV/fm3, Lx,y= 1.45 fm, Lz=0.145 fm 20.3 x 5.8 fm**Au+Au ECM=65 GeV/nucl. b=0.5 bmax Aσ=0.08 => σ~10**GeV/fm e [ GeV / fm3 ] T [ MeV] . . t=18.2 fm/c, Tmax= 417 MeV, emax= 19.4 GeV/fm3, Lx,y= 1.45 fm, Lz=0.145 fm 34.8 x 8.7 fm**Directed Transverse flow**Global Flow patterns: 3rd flow component (anti - flow) X Z b Squeeze out + Spherical flow Elliptic flow**3rd flow component**Csernai & Röhrich [Phys.Lett.B458(99)454] Hydro [Csernai, HIPAGS’93]**Preliminary**“Wiggle”, Pb+Pb, Elab=40 and 158GeV [NA49] A. Wetzler v1< 0 158 GeV/A The “wiggle” is there!**Flow is a diagnostic tool**Impact par. Equilibrationtime Transparency – string tension Consequence:v1(y), v2(y), …**FOHS - Movies:**Freeze Out B=0, T-fo = 139MeV B=0.4, T-fo = 139MeV B=0, T-fo = 180MeV B=0.4, T-fo = 180MeV [Bernd Schlei, Los Alamos, LA-UR-03-3410]**(B) - Freeze out over FOHS- post FO distribution?= 1st.: n,**T, u, cons. Laws != 2nd.: non eq. f(x,p) !!! ->(C) • (Ci) Simple kinetic model • (Cii) Covariant, kinetic F.O. description • (Ciii) Freeze out form transport equation • Note: ABC together is too involved!B & C can be done separately -> f(x,p)**Freeze out is :**Stronglydirected process: Delocalized: Them.f.p. - reaches infinity Finite characteristic length Modified Boltzmann Transport Equation for Freeze Out description The Boltzmann Transport Equation and Freeze Out The change is not negligible in the FO direction**The invariant “ Escape” probability in finite layer**The escape form theint to free component • Not to collide, depends on remaining distance • If the particle momentum is not normal to the surface, the spatial distance increases Early models: 1**The invariant “ Escape” probability**A B C t’ x’ D E F [RFG] Escape probability factors for different points on FO hypersurface, in the RFG. Momentum values are in units of [mc]**Results – the cooling and retracting of the interacting**matter [RFF] [RFF] Space-Like FO Time-Like FO cooling retracting Cut-off factorflowvelocityNo Cut-off**[RFF]**[RFF] Results – the contour lines of the FO distribution, f(p) Space-Like FO Time-Like FO jump in [RFF] With different initial flow velocities**Recent open, flow related issues**• Is QGP a “perfect fluid” ? – • Small (?) viscosity, but strong interaction (?)- Laminar flow, not turbulent -> large viscosity- Cascades need high cross section to reproduce flow • Comprehensive flow assessment • - v1, v2, v3 … should be evaluated on equal footing - There is one reaction plane, , (not 123 … ) - y, , pT correlations are equally important (y ?) • Solution: Event by Event flow evaluation