The Henryk Niewodniczański Institute of Nuclear Physics Polish Academy of Sciences Cracow, Poland

1. Characteristic form of 2+1 relativistic hydrodynamic equations Mikołaj Chojnacki The Henryk Niewodniczański Institute of Nuclear Physics Polish Academy of Sciences Cracow, Poland Based on paper M.Ch., W. Florkowski nucl-th/0603065 Cracow School of Theoretical Physics May 27 - June 5, 2006, Zakopane, POLAND

2. Outline • Angular asymmetry in non-central collisions • 2+1 Hydrodynamic equations • Boundary and initial conditions • Results from hydrodynamics • Freeze-out hypersurface and v2 • Conclusions

3. Angular asymmetry in non-central collisions y x Space asymmetries transform to momentum space asymmetries Indirect proof that particle interactions take place

4. energy-momentum tensor • at midrapidity (y=0) for RHIC energies temperature is the only thermodynamic parameter • thermodynamic relations Equations of relativistic hydrodynamics • Energy and momentum conservation law:

5. Cylindrical coordinates ( r,  ) y v vR vT  z = 0 r  x Boost – invariant symmetry Values of physical quantities at z ≠ 0 may be calculated by Lorentz transformation Lorentz factor : System geometry

6. Equations in covariant form Non-covariant notation Dyrek + Florkowski, Acta Phys.Polon.B15 (1984) 653

7. Temperature dependent sound velocity cs(T) TC = 170 [MeV] • Potential Φ Lattice QCD model by Mohanty and Alam Phys. Rev. C68 (2003) 064903 • Potential Φ dependent on T • Temperature T dependent on Φ inverse function of • Relation between T and s needed to close the set of three equations.

8. where transverse rapidity Semifinal form of 2 + 1 hydrodynamic equations in the transverse direction • auxiliary functions:

9. angular isotropy in initial conditions • potential Φ independent of  Generalization of 1+1 hydrodynamic equations by Baym, Friman, Blaizot, Soyeur, Czyz Nucl. Phys. A407 (1983) 541 2 + 1 hydrodynamic equations reduce to 1+1 case

10. Observables as functions of a± and  • solutions • velocity • potential Φ • sound velocity • temperature

11. Single function a to describe a± a(r,,t) • Function  symmetrically extended to negative values of r (-r,,t) • Equal values at  = 0 and  = 2π ( ) ( ) a f = = a f = p r , 0 , t r , 2 , t • Automatically fulfilled boundary conditions at r = 0 Boundary conditions a±,  a+(r,,t) a-(r,,t) (r,,t) r

12. Initial temperature is connected with the number of participating nucleons Teaney,Lauret and Shuryak nucl-th/0110037 • Values of parameters Initial conditions - Temperature y A B b x

13. Final form of the a± initial conditions Initial conditions – velocity field • Isotropic Hubble-like flow

14. Results • Impact parameter b and centrality classes • hydrodynamic evolution initial time t0 = 1 [fm] • sound velocity based on Lattice QCD calculations • initial central temperature T0 = 2 TC = 340 [MeV] • initial flow H0 = 0.001 [fm-1]

15. Centrality class 0 - 20% b = 3.9 [fm]

16. Centrality class 0 - 20% b = 3.9 [fm]

17. Centrality class 0 - 20% b = 3.9 [fm]

18. Centrality class 20 - 40% b = 7.1 [fm]

19. Centrality class 20 - 40% b = 7.1 [fm]

20. Centrality class 20 - 40% b = 7.1 [fm]

21. Centrality class 40 - 60% b = 9.2 [fm]

22. Centrality class 40 - 60% b = 9.2 [fm]

23. Centrality class 40 - 60% b = 9.2 [fm]

24. Freeze-out • Cooper-Frye formula • Hydro initial parameters • cS from Lattice QCD data • centrality: 0 - 80% • mean impact parameter • b = 7.6 fm • H0 = 0.001 fm-1 • T0 = 2.5 TC = 425 MeV • Freeze-out temperature • TFO = 165 MeV

25. Freeze-out hypersurface

26. Azimuthal flow of Ω _ Data points from STAR for Ω + Ω Phys. Rev. Lett. 95 (2005) 122301

27. Conclusions • New and elegant approach to old problem: we have generalized the equations of 1+1 hydrodynamics to the case of angular asymmetry using the method of Baym et al. (this is possible for the crossover phase transition, recently suggested by the lattice simulations of QCD, only 2 equations in the extended r-space, automatically fulfilled boundary conditions at r=0) • Velocity field is developed that tends to transform the initial almond shape to a cylindrically symmetric shape. As expected, the magnitude of the flow is greater in the in-plane direction than in the out-of-plane direction. The direction of the flow changes in time and helps the system to restore a cylindrically symmetric shape. • For most peripheral collisions the flow changes the central hot region to a pumpkin-like form – as the system cools down this effect vanishes. • Edge of the system preserves the almond shape but the relative asymmetry is decreasing with time as the system grows. • Presented results may be used to calculate the particle spectra and the v2 parameter when supplemented with the freeze-out model (THERMINATOR).