2. LP. Csernai, PASI'2002, Brazil1 Part I Relativistic Hydrodynamics For Modeling Ultra-Relativistic Heavy Ion Reactions

3. LP. Csernai, PASI'2002, Brazil2 Collaboration U of Bergen: C. Anderlik, L.P. Csernai, Ø Heggø- Hansen, Z. Lázár (U Cluj), V. Magas (U Lisbon), D. Molnár (Columbia U), A. Nyiri, K. Tamousiunas U of Oulu: A. Keranen, J. Manninen U of Sao Paulo: F. Grassi, Y. Hama U of Rio de Janeiro: T. Kodama U of Frankfurt: H. Stöcker, W. Greiner Los Alamos Nat. Lab.: D.D. Strottman 0.5 Tera-flop IBM e-series supercomputer, w/ 96 Power4 processors a’ 5.2 Giga-flop each (Bergen Computational Physics Lab. – EU Research Infrastructure) U of Rio de Janeiro: Proceedings - G. Grise, L. LimaSilviaPortugal, B. MattosTavares, D. dePaula

4. LP. Csernai, PASI'2002, Brazil3 FLOW  Is fluid dynamics applicable in relativistic nuclear physics?  Collective Nuclear Flow: Greiner – Koonin [1973 Balaton]  Transverse Flow Exp. Proof : [1984 Plastic Ball LBL] By now: Mc increases – close to macro continuous matter Local equilibrium  EoS / Phase Transition / QGP (during the middle part of the reaction, initial and final stages are out of equilibrium) Many flow-patterns are observed in nuclear collisions

5. LP. Csernai, PASI'2002, Brazil4 QGP: A new state of matter “The combined data coming from the seven experiments on CERN's Heavy Ion programme have given a clear picture of a new state of matter.... We now have evidence of a new state of matter where quarks and gluons are not confined. There is still an entirely new territory to be explored concerning the physical properties of quark-gluon matter.” [ L. Maiani]

6. LP. Csernai, PASI'2002, Brazil5 Water –Vapor Phase Transition - Discovery HeronHeron of Alexandria: “Pneumatica” steam-engine fl. AD. 62 –Roger Bacon: Heat, Kinetic theory 1220 – 1292 –Galileo Galilei: Temperature – Barothermoscope 1564 – 1642 –Anders Celsius: Temperature 1701 - 1744 –Joseph Black: Latent heat 1728 – 1799 –James Watt: Steam engine 1736 – 1819 –Sandi N.L. Carnot: Kinetic theory, Energy conservation 1796 – 1832 –Rudolf Clausius: 2 nd law of thermodynamics, entropy 1822 – 1888 –Ludwig Boltzmann: Kinetic theory 1844 – 1906 –Josiah W. Gibbs: Phase equilibrium, kinetic theory 1839 –1903 –Johannes D. Van der Waals: molecular interactions/ph.t.1837 - 1923 –Max Planck: Black body radiation 1858 –1947 Quantum Field Theory of phase transitions, mesoscopic dynamics Is there phase transition in a drop of water ?

7. LP. Csernai, PASI'2002, Brazil6 What did we see so far ? Phenomenology: –Strong stopping –Decreasing pressure Soft point ? –Flow, spherical, directed, elliptic, 3 rd component –Increased entropy –Chemical freeze-out at fixed T, fixed  –Strong strange baryon enhancement

8. LP. Csernai, PASI'2002, Brazil7 Stopping P+A: [Csernai, Kapusta, PRD31(1985)2795]:y=2.5 R. Stock [CERN (2000)] [W.Busza PASI 2002]

9. LP. Csernai, PASI'2002, Brazil8 Stopping at SPS / NA49

10. LP. Csernai, PASI'2002, Brazil9 Stopping at RHIC At RHIC y = 9.8 – 10.7 so Y-gap = 4-5 ! At RHIC there is also more stopping than expected. No sign of gap.

11. LP. Csernai, PASI'2002, Brazil10 45-55%35-45%25-35% 15-25%6-15%0-6%  dN ch /d  dN ch /d  vs. Centrality Octagon Rings [Peter Steinberg, QM 2001]

12. LP. Csernai, PASI'2002, Brazil11 [QM’2001] Stopping - RHIC

13. LP. Csernai, PASI'2002, Brazil12 Local equilibrium  Jüttner distr. (MB) Stationary solution of the BTE, and generalization of the MB distribution Lorentz Transformation Properties:

14. LP. Csernai, PASI'2002, Brazil13 Kinetic definition of density, energy, momentum These definitions are applicable for any, equilibrium or non-eq. situation!

15. LP. Csernai, PASI'2002, Brazil14 Normalization of Jüttner distribution From:  Similar expressions occur when we evaluate the EoS, energy density, e, pressure, P, and entropy density, s. [see also Takeshi Kodama PASI 2002]

16. LP. Csernai, PASI'2002, Brazil15

17. LP. Csernai, PASI'2002, Brazil16 Local equilibrium - Flow - LR frame ( Landau ) Def: Orthogonal proj. to flow Then: These definitions are applicable for any, equilibrium or non-equilibrium situation!

18. LP. Csernai, PASI'2002, Brazil17 Local equilibrium Large no. of degrees of freedom Strong Stopping Local equilibration  Equation of State (EoS) characterizes the equilibrium properties of matter Dynamics is well approximated by fluid dynamics (perfect, viscous, …) Model predictions become similar Multi Module Modeling

19. LP. Csernai, PASI'2002, Brazil18 EoS from the local eq. phase space distribution Eg.: From Jüttner  Ideal gas EoS & 2 nd law of thermodyn. (!) [see T.Kodama PASI’2002]

20. LP. Csernai, PASI'2002, Brazil19 Pressure – Soft Point? LBL, AGS, SPS: Collective flow – P-x vs. y Pressure sensitive Directed transverse flow decreases with increasing energy. [D. Rischke, 95] [E. Shuryak, 95] [Holme, et al., 89] But, does it recover at higher energies ?

21. LP. Csernai, PASI'2002, Brazil20 [F. Karsch, QM’2001]

22. LP. Csernai, PASI'2002, Brazil21

23. LP. Csernai, PASI'2002, Brazil22 [F. Karsch, PASI 2002]

24. LP. Csernai, PASI'2002, Brazil23 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!

25. LP. Csernai, PASI'2002, Brazil24 Relativistic Fluid Dynamics For any EoS, P=P(e,n), and any energy-momentum tensor in LE(!): Not only for high v!

26. LP. Csernai, PASI'2002, Brazil25 Multi Module Modeling 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)

27. LP. Csernai, PASI'2002, Brazil26 Initial State At low energies stopping in SHOCK or DETONATION waves (supersonic!) of width of 1-3fm (possible in Pb+Pb) [BEVALAC, GSI, AGS] Idealized: as discontinuity across a hyper- surface (or layer) in space time. Simple solutions of Rel. Fluid dynamics Generalized to other stationary processes: freeze-out, initial equilibration, phase trans.

28. LP. Csernai, PASI'2002, Brazil27 Matching Conditions  Conservation laws  Nondecreasing entropy Can be solved easily. Yields, via the “Taub adiabat” and “Rayleigh line”, the final state behind the hyper- surface. (See at freeze out.)

29. LP. Csernai, PASI'2002, Brazil28

30. LP. Csernai, PASI'2002, Brazil29 Fire streak picture - Only in 3 dimensions! Myers, Gosset, Kapusta, Westfall

31. LP. Csernai, PASI'2002, Brazil30 String rope --- Flux tube --- Coherent YM field

32. LP. Csernai, PASI'2002, Brazil31 Initial stage: Coherent Yang-Mills model [Magas, Csernai, Strottman, Pys. Rev. C ‘2001]

33. LP. Csernai, PASI'2002, Brazil32 Expanding string ropes – Full energy conservation

34. LP. Csernai, PASI'2002, Brazil33 Yo – Yo Dynamics

35. LP. Csernai, PASI'2002, Brazil34

36. LP. Csernai, PASI'2002, Brazil35 Initial state 3 rd flow component

37. LP. Csernai, PASI'2002, Brazil36 Multi Module Modeling 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)

38. LP. Csernai, PASI'2002, Brazil37 3-Dim Hydro for RHIC (PIC)

39. LP. Csernai, PASI'2002, Brazil38 3-dim Hydro for RHIC Energies Au+Au E CM =65 GeV/nucl. b=0.1 b max A σ =0.08 => σ~10 GeV/fm n / n 0 [ 1 ] e [ GeV / fm 3 ] T= 0.0 fm/c n max = 8.67 e max =32.46 GeV / fm 3 L x,y = 1.45 fm L z =0.145 fm.. 4.4 x 1.3 fm EoS: P = e/3

40. LP. Csernai, PASI'2002, Brazil39 3-dim Hydro for RHIC Energies Au+Au E CM =65 GeV/nucl. b=0.1 b max A σ =0.08 => σ~10 GeV/fm n / n 0 [ 1 ] e [ GeV / fm 3 ] T=1.9 fm/c n max = 8.66 e max = 31.82 GeV / fm 3 L x,y = 1.45 fm L z =0.145 fm..

41. LP. Csernai, PASI'2002, Brazil40 3-dim Hydro for RHIC Energies Au+Au E CM =65 GeV/nucl. b=0.1 b max A σ =0.08 => σ~10 GeV/fm n / n 0 [ 1 ] e [ GeV / fm 3 ] T= 3.8 fm/c n max = 7.77 e max = 27.22 GeV / fm 3 L x,y = 1.45 fm L z =0.145 fm... 4.4 x 1.3 fm

42. LP. Csernai, PASI'2002, Brazil41 3-dim Hydro for RHIC Energies Au+Au E CM =65 GeV/nucl. b=0.1 b max A σ =0.08 => σ~10 GeV/fm n / n 0 [ 1 ] e [ GeV / fm 3 ] T= 5.7 fm/c n max = 6.36 e max = 26.31 GeV / fm 3 L x,y = 1.45 fm L z =0.145 fm..

43. LP. Csernai, PASI'2002, Brazil42 3-dim Hydro for RHIC Energies Au+Au E CM =65 GeV/nucl. b=0.1 b max A σ =0.08 => σ~10 GeV/fm n / n 0 [ 1 ] e [ GeV / fm 3 ] T= 7.6 fm/c n max = 5.22 e max = 37.16 GeV / fm 3 L x,y = 1.45 fm L z =0.145 fm..

44. LP. Csernai, PASI'2002, Brazil43 3-dim Hydro for RHIC Energies Au+Au E CM =65 GeV/nucl. b=0.1 b max A σ =0.08 => σ~10 GeV/fm n / n 0 [ 1 ] e [ GeV / fm 3 ] T= 9.5 fm/c n max = 4.45 e max = 32.86 GeV / fm 3 L x,y = 1.45 fm L z =0.145 fm..

45. LP. Csernai, PASI'2002, Brazil44 Global Flow Directed Transverse flow Elliptic flow 3 rd flow component (anti - flow) 3 rd flow component (anti - flow) Squeeze out

46. LP. Csernai, PASI'2002, Brazil45 A=0.08 11.4 fm / c

47. LP. Csernai, PASI'2002, Brazil46 A=0.065 11.4 fm/c

48. LP. Csernai, PASI'2002, Brazil47 Aside: v 1 is not measured yet at RHIC!? (neither STAR nor PHENIX) As v 2 is measured, the reaction plane [x,z] is known, just the target/projectile side should be selected. This is not done due to the prejudice that the distribution of emitted particles is mirror-symmetric in CM: f CM ( p x, p y, p z ) = f CM ( p x, p y, -p z ) This is wrong (!) as the presented hydro calculations and SPS data show. At finite impact parameter (2-15%) there is a fwd / bwd asymmetry. Calculate event by event the Q-vector (a la [Danielevicz, Odyniecz, PL (1985)] ): Q k = S i k y CM p x For all particles, i, of type k. Only the sign is relevant, as the plane is known already. This Q-vector will select the same side (e.g. projectile) in each event. [Discussions with Art Poskanzer and Roy Lacey are gratefully acknowledged. ]

49. LP. Csernai, PASI'2002, Brazil48 NEXT: Flow experiments Modified Rel. Hydro with supercooling Freeze-out Discontinuities in hydro – Eq. => Eq. Freeze-out to non-eq. Kinetic freeze-out