2. Norway

3. Relativistic Hydrodynamics and Freeze-out László Csernai (Bergen Computational Physics Lab., Univ. of Bergen)

4. ?

5. 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

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

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. Small System time Initial state, Pre-equilibrium, cascade, Multi Component Fluid no unique EoS Hadronization & Freeze Out (One-Fluid) FD

10. Multi Module Modeling A: Initial state - Fitted to measured data (?) B: Initial state - Pre-equilibrium: Parton Cascade; Coherent Yang-Mills [Magas] Local Equilibrium  Hydro, EoS Final Freeze-out: Kinetic models, measurables. - If QGP  Sudden and simultaneous hadronization and freeze out (indicated by HBT, Strangeness, Entropy puzzle) Landau (1953), Milekhin (1958), Cooper & Frye (1974)

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

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

13. Initial state 3 rd flow component

14. 3-Dim Hydro for RHIC (PIC)

15. 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 - 4B/3,B = 397 MeV/fm 3 8.7 x 4.4 fm

16. 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=2.3 fm/c, T max = 420 MeV, e max = 20.0 GeV/fm 3, L x,y = 1.45 fm, L z =0.145 fm.. 11.6 x 4.6 fm

17. 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=4.6 fm/c, T max = 419 MeV, e max = 19.9 GeV/fm 3, L x,y = 1.45 fm, L z =0.145 fm.. 14.5 x 4.9 fm

18. 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=6.9 fm/c, T max = 418 MeV, e max = 19.7 GeV/fm 3, L x,y = 1.45 fm, L z =0.145 fm.. 17.4 x 5.5 fm

19. 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

20. 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=11.4 fm/c, T max = 416 MeV, e max = 19.5 GeV/fm 3, L x,y = 1.45 fm, L z =0.145 fm.. 23.2 x 6.7 fm

21. 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=13.7 fm/c, T max = 417 MeV, e max = 19.4 GeV/fm 3, L x,y = 1.45 fm, L z =0.145 fm.. 26.1 x 7.3 fm

22. 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=16.0 fm/c, T max = 417 MeV, e max = 19.4 GeV/fm 3, L x,y = 1.45 fm, L z =0.145 fm.. 31.9 x 8.1 fm

23. 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

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

26. 3 rd flow component Hydro [Csernai, HIPAGS’93] [Phys.Lett.B458(99)454] Csernai & Röhrich

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

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

29. Kinetic freeze out: Strongly interacting matter becomes dilute and cold, the momentum distr. of particles in the absence of collisions freezes out and the particles propagate toward the detectors. Sometimes sequential FO is assumed: Chemical + Kinetic FO Now: Rapid, simultaneous FO and hadronization from super-cooled QGP in a thin layer (2-3 fm). The process of gradual FO is followed by kinetic description. What is Freeze Out (FO) ???

30. (A) (A)- identification of the Freeze Out Hyper-Surface (FOHS) [B. Schlei] (B) (B)- idealized FO over the FOHS (C) (C)- FO over a finite LAYER & described by kinetic theory, by the MBTE.

31. (A) – 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]

32. (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)

33. (A) Freeze out as a discontinuity Theory of discontinuities in relativistic flow (only space-like), [Taub, 1948] Generalization for both time-like and space- like discontinuities, [1987] 1 st problem: We must use the correct parameters of the matter in the Post FO distribution Matching conditions: Discontinuity = where the properties of matter change suddenly normal vector => n, T, u m

34. (A) Modified Cooper-Frye Formula The number of particles crossing, The kinetic definition of the particle four-flow Cooper-Frye formula, Cooper and Frye, 1975 2 nd : Problem of negative contributions, on the space-like part of the hyper-surface, Sinyukov, et al.  Sharp cut-off proposed, Bugaev, 1996 Solved, kinetic model, Anderlik et al. 1999, Magas et al. 1999 Not known form hydro !

35. Kinetic freeze out models, (a) just FO Assumptions: only short range interactions one dimensional geometry assume stationary flow, no explicit time dependence Simple kinetic model Kinetic model (b) with re-scattering, re-thermalization:

36. Reference Frames Pre FO – Local Rest frame of the gas, where the matter is at rest = Rest Frame of the Gas (RFG), moving with the peak of the Pre-FO distribution Post FO – Rest Frame of the Front, (RFF) attached to the FO front p’ x

37. (a)Reproduced the cut Juttner distribution as the post FO distribution, --- but: the other half of the distribution remained there, interacting for ever (b)Gradually all matter froze out, no negative contribution, result approximated by CJ distribution --- but: it took infinite time – unrealistic. Bondorf et al. [NP A296 (‘78) 320.]: Sph. Expansion -> increasing divergence & adiabatic cooling -> descreasing random thermal flux increasing divergence & adiabatic cooling -> descreasing random thermal flux => FO without any collision beyond some radius !!! 2 nd problem is not finished yet !!!

38. Feasibility: a realistic hydro model + a realistic FOHS model with Cooper Fry type FO is the only manageable model (B). However, a realistic post FO distribution should be used (!), and this should be investigated (C).

39. 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

40. The Boltzmann Transport Equation A nonlinear equation for dilute gasses, with the following assumptions: 1.Only binary collisions 2.The molecular chaos – no correlations: gives the number of collisions at the respective point 3.A smoothly varying function compared to the m.f.p. 1 2 3 4 Gain term Loss term

41. The Modified Boltzmann Transport Equation Introducing, and the FO probability – which feeds the free component FO probablity not included !!! re-thermalization term free component interacting component MBTE ->

42. Freeze out from conservation laws Freeze out happens in a directed way Total number of particles crossing the surface Through a four-volume element (layer), the particles are conserved Length element,FO direction Inside boundary Outside boundary

43. 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

44. 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]

45. Freeze out in a finite layer The corresponding equations for both space-like and time-like freeze out /wo rethermalization The solution: Space-like Time-like Problem !!!

46. Immediate re-thermalization limit We take into account re-scattering within the interacting component for the re-thermalization and re-equilibration of this component We simplify the dynamics by taking advantage of the relaxation time approach The re-thermalization happens simultaneously to freeze out: The change of the conserved currents caused by the particle transfer:

47. Immediate re-thermalization limit The change of proper particle density and energy density: The application to a baryonfree and massless gas, where EoS – Analytical results, for

48. 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]

49. Results – the momentum distribution Space-Like FO Time-Like FO asymmetric elongated in FO direction curved due to the FO process [RFF]

50. 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]

51. Time-like FO

52. Post FO distributions are non – thermal ! Conservation laws must be satisfied ! Post FO distributions must be calculated from transport theory, or can be approximated with adequate ansatz (Cancelling Juttner distribution) Note(!) BTE is not applicable, molecular chaos, and smoothness of the phase space distribution are not applicable for the FO process. Adequate MD models or MBTE models should be applied. Conclusions Posters: 2/39 Bravina 2/56 Zschocke 4/125 Zabrodin 5/150 Manninen 10/270 Magas 10/272 Molnar, E.