2+1 Relativistic hydrodynamics for heavy-ion collisions

1. 2+1 Relativistic hydrodynamics for heavy-ion collisions Mikołaj Chojnacki IFJ PAN NZ41

2. Outline • Angular asymmetry in non-central collisions • 2+1 Hydrodynamic equations • Boundary and initial conditions • Results • Conclusions 2+1 Relativistic hydrodynamics for heavy-ion collisions

3. Angular asymmetry in non-central collisions y x Space asymmetries transform to momentum space asymmetries Indirect proof that particle interactions take place 2+1 Relativistic hydrodynamics for heavy-ion collisions

4. Equations of relativistic hydrodynamics • Energy and momentum conservation law: • energy-momentum tensor • at midrapidity (y=0) for RHIC energies temperature is the only thermodynamic parameter • thermodynamic relations 2+1 Relativistic hydrodynamics for heavy-ion collisions

5. y v vR vT  z = 0 r  x Lorentz factor : System geometry Cylindrical coordinates ( r,  ) Boost – invariant symmetry Values of physical quantities at z ≠ 0 may be calculated by Lorentz transformation 2+1 Relativistic hydrodynamics for heavy-ion collisions

6. Equations in covariant form Non-covariant notation Dyrek + Florkowski, Acta Phys.Polon.B15 (1984) 653 2+1 Relativistic hydrodynamics for heavy-ion collisions

7. inverse function of Temperature dependent sound velocity cs(T) • Relation between T and s needed to close the set of three equations. TC = 170 [MeV] • Potential Φ Lattice QCD model by Mohanty and Alam Phys. Rev. C68 (2003) 064903 • Potential Φ dependent of T • Temperature T dependent of Φ 2+1 Relativistic hydrodynamics for heavy-ion collisions

8. where transverse rapidity Semifinal form of 2 + 1 hydrodynamic equations in the transverse direction • auxiliary functions: 2+1 Relativistic hydrodynamics for heavy-ion collisions

9. 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 • angular isotropy in initial conditions • potential Φ independent of  2+1 Relativistic hydrodynamics for heavy-ion collisions

10. Observables as functions of a± and  • solutions • velocity • potential Φ • sound velocity • temperature 2+1 Relativistic hydrodynamics for heavy-ion collisions

11. Boundary conditions • Single function a to describe a± a±, a,  a(r,,t) a+(r,,t) a-(r,,t) (r,,t) • Function  symmetrically extended to negative values of r (-r,,t) • Equal values at  = 0 and  = 2π r • Automatically fulfilled boundary conditions at r = 0 2+1 Relativistic hydrodynamics for heavy-ion collisions

12. Initial conditions - Temperature y • Initial temperature is connected with the number of participating nucleons A B b x Teaney,Lauret and Shuryak nucl-th/0110037 • Values of parameters 2+1 Relativistic hydrodynamics for heavy-ion collisions

13. Initial conditions – velocity field • Isotropic Hubble-like flow • Final form of the a± initial conditions 2+1 Relativistic hydrodynamics for heavy-ion collisions

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 = 0.01 [fm-1] 2+1 Relativistic hydrodynamics for heavy-ion collisions

15. Centrality class 0 - 20% b = 3.9 [fm] 2+1 Relativistic hydrodynamics for heavy-ion collisions

16. Centrality class 0 - 20% b = 3.9 [fm] 2+1 Relativistic hydrodynamics for heavy-ion collisions

17. Centrality class 20 - 40% b = 7.1 [fm] 2+1 Relativistic hydrodynamics for heavy-ion collisions

18. Centrality class 20 - 40% b = 7.1 [fm] 2+1 Relativistic hydrodynamics for heavy-ion collisions

19. Centrality class 40 - 60% b = 9.2 [fm] 2+1 Relativistic hydrodynamics for heavy-ion collisions

20. Centrality class 40 - 60% b = 9.2 [fm] 2+1 Relativistic hydrodynamics for heavy-ion collisions

21. 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). 2+1 Relativistic hydrodynamics for heavy-ion collisions

22. centrality class 0 - 100%, sound velocity: lattice QCD, H0 = 0.01, 2+1 Relativistic hydrodynamics for heavy-ion collisions

23. centrality class 0 - 100%, sound velocity: lattice QCD, H0 = 0.01, 2+1 Relativistic hydrodynamics for heavy-ion collisions

24. centrality class 40 - 60%, sound velocity: analytic, H0 = 0.01, 2+1 Relativistic hydrodynamics for heavy-ion collisions

25. centrality class 40 - 60%, sound velocity: analytic, H0 = 0.01, 2+1 Relativistic hydrodynamics for heavy-ion collisions

26. centrality class 40 - 60%, sound velocity: constant 3-1/2, H0 = 0.01, 2+1 Relativistic hydrodynamics for heavy-ion collisions

27. centrality class 40 - 60%, sound velocity: constant 3-1/2, H0 = 0.01, 2+1 Relativistic hydrodynamics for heavy-ion collisions

28. centrality class 40 - 60%, sound velocity: analytic, H0 = 0.25, 2+1 Relativistic hydrodynamics for heavy-ion collisions

29. centrality class 40 - 60%, sound velocity: analytic, H0 = 0.25, 2+1 Relativistic hydrodynamics for heavy-ion collisions

30. 2+1 Relativistic hydrodynamics for heavy-ion collisions