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

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

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

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

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

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

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

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

9. 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 

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

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

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

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

14. 2+1 Relativistic hydrodynamics for heavy-ion collisions 14 Results  Impact parameter b and centrality classes  hydrodynamic evolution initial timet 0 = 1 [fm] sound velocity based on Lattice QCD calculations  sound velocity based on Lattice QCD calculations  initial central temperatureT 0 = 2 T C = 340 [MeV] nitial flowH 0 = 0.01 [fm -1 ]  initial flowH 0 = 0.01 [fm -1 ]

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

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

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

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

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

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

21. 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 v 2 parameter when supplemented with the freeze-out model (THERMINATOR).

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

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

24. 2+1 Relativistic hydrodynamics for heavy-ion collisions 24 centrality class 40 - 60%,sound velocity: analytic, H 0 = 0.01,

25. 2+1 Relativistic hydrodynamics for heavy-ion collisions 25 centrality class 40 - 60%,sound velocity: analytic, H 0 = 0.01,

26. 2+1 Relativistic hydrodynamics for heavy-ion collisions 26 centrality class 40 - 60%,sound velocity: constant 3 -1/2, H 0 = 0.01,

27. 2+1 Relativistic hydrodynamics for heavy-ion collisions 27 centrality class 40 - 60%,sound velocity: constant 3 -1/2, H 0 = 0.01,

28. 2+1 Relativistic hydrodynamics for heavy-ion collisions 28 centrality class 40 - 60%,sound velocity: analytic, H 0 = 0.25,

29. 2+1 Relativistic hydrodynamics for heavy-ion collisions 29 centrality class 40 - 60%,sound velocity: analytic, H 0 = 0.25,

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