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LP. Csernai, NWE'2001, Bergen1 Part III Relativistic Hydrodynamics For Modeling Ultra-Relativistic Heavy Ion Reactions.

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Presentation on theme: "LP. Csernai, NWE'2001, Bergen1 Part III Relativistic Hydrodynamics For Modeling Ultra-Relativistic Heavy Ion Reactions."— Presentation transcript:

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2 LP. Csernai, NWE'2001, Bergen1 Part III Relativistic Hydrodynamics For Modeling Ultra-Relativistic Heavy Ion Reactions

3 LP. Csernai, NWE'2001, Bergen2 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)

4 LP. Csernai, NWE'2001, Bergen3 Matching Conditions  Conservation laws  Nondecreasing entropy

5 LP. Csernai, NWE'2001, Bergen4 Initial stage: Coherent Yang-Mills model [Magas, Csernai, Strottman, NEW’2001]

6 LP. Csernai, NWE'2001, Bergen5 Expanding string ropes – Full energy conservation

7 LP. Csernai, NWE'2001, Bergen6 Initial state 3 rd flow component

8 LP. Csernai, NWE'2001, Bergen7 Freeze out [L Bravina et al.]

9 LP. Csernai, NWE'2001, Bergen8 Hypersurface

10 LP. Csernai, NWE'2001, Bergen9 “Cooper-Frye” formula

11 LP. Csernai, NWE'2001, Bergen10 Conservation Laws across hypersurface

12 LP. Csernai, NWE'2001, Bergen11 Matching Conditions  Conservation laws  Nondecreasing entropy

13 LP. Csernai, NWE'2001, Bergen12 Consequences of conservation laws – Problem I  Non-decreasing entropy current across front!

14 LP. Csernai, NWE'2001, Bergen13 Aside: Taub-adiabat and Rayleigh line Perfect fluid on both sides of the front!

15 LP. Csernai, NWE'2001, Bergen14 Aside: Taub-adiabat and Rayleigh line

16 LP. Csernai, NWE'2001, Bergen15 Aside: Taub-adiabat and Rayleigh line

17 LP. Csernai, NWE'2001, Bergen16 Aside: Taub-adiabat and Rayleigh line Goal: scalar equations I: Parallel Projection

18 LP. Csernai, NWE'2001, Bergen17 Aside: Taub-adiabat and Rayleigh line Taub [‘48] missed the sign  was not applicable for freeze- out. The Rayleigh line is a straight line in the [P,X] plane. It gives the locus of final states “2” if the initial state “1” is known. The slope, j, is given by the current across the front.

19 LP. Csernai, NWE'2001, Bergen18 Aside: Taub-adiabat and Rayleigh line II: Orthogonal Projection  To obtain scalar …. Then to obtain scalar: (one of the cross terms and the last term cancel)

20 LP. Csernai, NWE'2001, Bergen19 Aside: Taub-adiabat and Rayleigh line Comparing the two equations for the current, j, : So, we obtain the Taub adiabat : The locus of the possible final states, “2”, lies on the Taub adiabat. If the initial state and the EoS of the final state is known the Taub adiabat with the Rayleigh line determine the final state. If the final state is out of equilibrium, I.e. not a perfect fluid, this is not applicable!

21 LP. Csernai, NWE'2001, Bergen20 Aside: Taub-adiabat and Rayleigh line Taub-adiabat for final states: E.g. Bag Model EoS: [P + (4B + P o )/3] (X – X o /3) = [w o – 4(B + P 0 )/3] X o /3 Eg. Ideal gas EoS: (P + 2P o /3) (X – 2X o /3) = (e o - 2P o /3) 2X o /3 P X “1” time-like space-like “2” Taub-adiabat Rayleigh-line Problem I is Solved

22 LP. Csernai, NWE'2001, Bergen21 Space-like hypersurface - Problem II

23 LP. Csernai, NWE'2001, Bergen22 Space-like hypersurface II

24 LP. Csernai, NWE'2001, Bergen23 Matching Conditions  Conservation laws  Nondecreasing entropy If the final state is out of Eq., the energy-momentum tensor has to be evaluated, and the above eqs. solved!!! [e.g.. Anderlik et al. Phys.Rev.C 59 (99) 3309]

25 LP. Csernai, NWE'2001, Bergen24 Post F.O. - Cut-Jüttner distribution [Bugaev, Nucl.Phys.A606(96)559] [Anderlik et al., Phys.Rev.C59(99)3309] Proposed by: Solved: p p x y Post F.O. distribution:  p m L m   f(p) V-parameter V-flow Matching conditions determine 5 parameters only. Ansatz in needed for final f(x,p) !

26 LP. Csernai, NWE'2001, Bergen25 Cut – Jüttner distribution: Pre FO velocity [Anderlik et al., Phys.Rev.C59(99)3309] [Bugaev, Nucl.Phys.A606(96)559] Θ(p.dσ) f(x,p) [Better non-eq. ansatz: Tamousiunas in pr.] Problem II is partly solved

27 LP. Csernai, NWE'2001, Bergen26 Kinetic freeze-out models  Kinetic approach  f (x,p) out of equilibrium  Asymmetry

28 LP. Csernai, NWE'2001, Bergen27 Freeze out model with rescattering [Anderlik et al., Phys.Rev.C59 (1999) 388-394]

29 LP. Csernai, NWE'2001, Bergen28 Freeze out distribution with rescattering V=0 [V. Magas, et al.,] Heavy Ion Phys.9:193-216,1999

30 LP. Csernai, NWE'2001, Bergen29 Freeze out model with rescattering V = 0.5 [V. Magas, et al. Heavy Ion Phys.9:193-216,1999 ]

31 LP. Csernai, NWE'2001, Bergen30 Change of the rest temperature in FO [V. Magas, et al., Heavy Ion Phys.9:193-216,1999 ]

32 LP. Csernai, NWE'2001, Bergen31 Change of the rest velocity during FO [V. Magas, et al., Heavy Ion Phys.9:193-216,1999]

33 LP. Csernai, NWE'2001, Bergen32 P-t distribution (T=130 MeV) [V. Magas et al., Phys.Lett.B459(99)33] Croonin effect ?

34 LP. Csernai, NWE'2001, Bergen33 Conclusion Hydro works amazingly well! Stronger and stronger hydro effects are observed!  Equilibrium and EoS exists ( in part of the reaction ) We have a good possibility to learn more and more about the EoS, with improved experimental and theoretical accuracy!

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