1. Norway

2. 3-dim. QGP Fluid Dynamics and Flow Observables László Csernai (Bergen Computational Physics Lab., Univ. of Bergen)

3. IntroductionIntroduction Strong flow is observed => - Early, local eq., - EoS - n q scaling – QGP flows - no flow in hadronic matter > simultaneous hadronization and FO (HBT, high strangeness abundance)

4. 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,   P = P (e,n) Solvable for local equilibrium! EoS

5. Relativistic Fluid Dynamics For any EoS, P=P(e,n), and any energy-momentum tensor in LE(!): Not only for high v!

6. Two theoretical problems Initial state – Initial state – - Fitted initial states > moderate insight Final Freeze Out Final Freeze Out - Realistic Model, Continuos FO, ST layer, Non-eq. distribution - Realistic Model, Continuos FO, ST layer, Non-eq. distribution

7. Local Equilibration, Fluids Fluid components, Friction E O S -------------- One fluid >>> E O S Hadronization, chemical FO, kinetic FO Freeze Out >>> Detectors Stages of a Collision Collective flow reveals the EoS if we have dominantly one fluid with local equilibrium in a substantial part of the space-time domain of the collision !!!

8. QGP EoS One fluid Hadronization Chemical Freeze Out Kinetic Freeze Out Initial state time Heavy Colliding System Idealizations FO Layer FO HS

9. Fire streak picture - Only in 3 dimensions! Myers, Gosset, Kapusta, Westfall

10. String rope --- Flux tube --- Coherent YM field

11. Initial state 3 rd flow component

12. 3-Dim Hydro for RHIC (PIC)

13. 3-dim Hydro for RHIC Energies Au+Au E CM =65 GeV/nucl. b=0.5 b max A σ =0.08 => σ~10 GeV/fm e [ GeV / fm 3 ] T [ MeV] t=0.0 fm/c, T max = 420 MeV, e max = 20.0 GeV/fm 3, L x,y = 1.45 fm, L z =0.145 fm.. EoS: p= e/3 - B/3, B = 397 MeV/fm 3 8.7 x 4.4 fm

14. Au+Au E CM =65 GeV/nucl. b=0.5 b max A σ =0.08 => σ~10 GeV/fm e [ GeV / fm 3 ] T [ MeV] t=9.1 fm/c, T max = 417 MeV, e max = 19.6 GeV/fm 3, L x,y = 1.45 fm, L z =0.145 fm.. 20.3 x 5.8 fm

15. Au+Au E CM =65 GeV/nucl. b=0.5 b max A σ =0.08 => σ~10 GeV/fm e [ GeV / fm 3 ] T [ MeV] t=18.2 fm/c, T max = 417 MeV, e max = 19.4 GeV/fm 3, L x,y = 1.45 fm, L z =0.145 fm.. 34.8 x 8.7 fm

16. Global Flow patterns: Directed Transverse flow Elliptic flow 3 rd flow component (anti - flow) Squeeze out

17. 3 rd flow component Hydro [Csernai, HIPAGS’93] [Phys.Lett.B458(99)454] Csernai & Röhrich

18. “Wiggle”, Pb+Pb, E lab =40 and 158GeV [NA49] A. Wetzler Preliminary 158 GeV/A The “wiggle” is there! v 1 < 0

19. Flow is a diagnostic tool Impact par. Transparency – string tension Equilibration time Consequence: v 1 (y), v 2 (y), …

20. FOHS - Movies: B=0, T-fo = 139MeV B=0, T-fo = 180MeV B=0.4, T-fo = 139MeV B=0.4, T-fo = 180MeV [Bernd Schlei, Los Alamos, [Bernd Schlei, Los Alamos, LA-UR-03-3410] Freeze Out

21. (B) - Freeze out over FOHS - post FO distribution? = 1 st.: n, T, u, cons. Laws ! = 2 nd.: 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)

22. The Boltzmann Transport Equation and Freeze Out Freeze out is : Strongly directed process: Delocalized: The m.f.p. - reaches infinity Finite characteristic length Modified Boltzmann Transport Equation for Freeze Out description The change is not negligible in the FO direction

23. The invariant “ Escape” probability in finite layer The escape form the int 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

24. The invariant “ Escape” probability Escape probability factors for different points on FO hypersurface, in the RFG. Momentum values are in units of [mc] AB C D EF t’ x’ [RFG]

25. Results – the cooling and retracting of the interacting matter Space-Like FO Time-Like FO cooling retracting  Cut-off factor flow velocity No Cut-off [RFF]

26. Results – the contour lines of the FO distribution, f(p) Space-Like FO Time-Like FO jump in [RFF] With different initial flow velocities [RFF]

27. Recent open, flow related issues Is QGP a “perfect fluid” ? – 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 Comprehensive flow assessment - v1, v2, v3 … should be evaluated on equal footing - There is one reaction plane, , (not  1  2  3 … ) - y, , pT correlations are equally important (y ?) - v1, v2, v3 … should be evaluated on equal footing - There is one reaction plane, , (not  1  2  3 … ) - y, , pT correlations are equally important (y ?) Solution: Solution: Event by Event flow evaluation