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Norway

Norway. Relativistic Hydrodynamics and Freeze-out. L á szl ó Csernai (Bergen Computational Physics Lab., Univ. of Bergen). ?. Relativistic Fluid Dynamics. Eg.: from kinetic theory. BTE for the evolution of phase-space distribution:.

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Norway

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  1. Norway

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

  3. ?

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

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

  6. 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 !!!

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

  8. Small System FD Initial state, Pre-equilibrium, cascade, Multi Component Fluidno unique EoS time (One-Fluid) Hadronization & Freeze Out

  9. 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)

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

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

  12. Initial state 3rd flow component

  13. 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 - 4B/3, B = 397 MeV/fm3

  14. 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=2.3 fm/c, Tmax= 420 MeV, emax= 20.0 GeV/fm3, Lx,y= 1.45 fm, Lz=0.145 fm 11.6 x 4.6 fm

  15. Au+Au ECM=65 GeV/nucl. b=0.5 bmax Aσ=0.08 => σ~10 GeV/fm e [ GeV / fm3 ] T [ MeV] . . t=4.6 fm/c, Tmax= 419 MeV, emax= 19.9 GeV/fm3, Lx,y= 1.45 fm, Lz=0.145 fm 14.5 x 4.9 fm

  16. Au+Au ECM=65 GeV/nucl. b=0.5 bmax Aσ=0.08 => σ~10 GeV/fm e [ GeV / fm3 ] T [ MeV] . . t=6.9 fm/c, Tmax= 418 MeV, emax= 19.7 GeV/fm3, Lx,y= 1.45 fm, Lz=0.145 fm 17.4 x 5.5 fm

  17. 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

  18. Au+Au ECM=65 GeV/nucl. b=0.5 bmax Aσ=0.08 => σ~10 GeV/fm e [ GeV / fm3 ] T [ MeV] . . t=11.4 fm/c, Tmax= 416 MeV, emax= 19.5 GeV/fm3, Lx,y= 1.45 fm, Lz=0.145 fm 23.2 x 6.7 fm

  19. Au+Au ECM=65 GeV/nucl. b=0.5 bmax Aσ=0.08 => σ~10 GeV/fm e [ GeV / fm3 ] T [ MeV] . . t=13.7 fm/c, Tmax= 417 MeV, emax= 19.4 GeV/fm3, Lx,y= 1.45 fm, Lz=0.145 fm 26.1 x 7.3 fm

  20. Au+Au ECM=65 GeV/nucl. b=0.5 bmax Aσ=0.08 => σ~10 GeV/fm e [ GeV / fm3 ] T [ MeV] . . t=16.0 fm/c, Tmax= 417 MeV, emax= 19.4 GeV/fm3, Lx,y= 1.45 fm, Lz=0.145 fm 31.9 x 8.1 fm

  21. 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

  22. Directed Transverse flow Global Flow patterns: 3rd flow component (anti - flow) X Z b Squeeze out + Spherical flow Elliptic flow

  23. 3rd flow component Csernai & Röhrich [Phys.Lett.B458(99)454] Hydro [Csernai, HIPAGS’93]

  24. Preliminary “Wiggle”, Pb+Pb, Elab=40 and 158GeV [NA49] A. Wetzler v1< 0 158 GeV/A The “wiggle” is there!

  25. Flow is a diagnostic tool Impact par. Equilibrationtime Transparency – string tension Consequence:v1(y), v2(y), …

  26. What is Freeze Out (FO) ??? • 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.

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

  28. (A) – Movies: B=0.4, T-fo = 139MeV B=0, T-fo = 139MeV B=0, T-fo = 180MeV B=0.4, T-fo = 180MeV [Bernd Schlei, Los Alamos, LA-UR-03-3410]

  29. (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)

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

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

  32. 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:

  33. 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 Reference Frames p’x

  34. 2nd problem is not finished yet !!! • Reproduced the cut Juttner distribution as the post FO distribution, --- but: the other half of the distribution remained there, interacting for ever • 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 <vrel s> => FO without any collision beyond some radius !!!

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

  36. 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

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

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

  39. 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

  40. 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]

  41. Results – the cooling and retracting of the interacting matter [RFF] [RFF] Space-Like FO Time-Like FO cooling retracting  Cut-off factorflowvelocityNo Cut-off

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

  43. [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

  44. Time-like FO

  45. Conclusions • 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. Posters: 2/39 Bravina 2/56 Zschocke 4/125 Zabrodin 5/150 Manninen10/270 Magas 10/272 Molnar, E.

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