相対論的流体模型を軸にした 重イオン衝突の理解

1. 相対論的流体模型を軸にした重イオン衝突の理解相対論的流体模型を軸にした重イオン衝突の理解 Kobayashi-Maskawa Institute Department of Physics, Nagoya University Chiho NONAKA June 23, 2013@Matsumoto 「RHIC-LHC 高エネルギー原子核反応の物理研究会」 -----  QGPの物理研究会 信州合宿  -----

2. Hydrodynamic Model hydro hadronization freezeout collisions thermalization observables strong elliptic flow @RHIC hydrodynamic model One of successful models for description of dynamics of QGP:

3. z Reaction plane y x Viscosity in Hydrodynamics Elliptic Flow RHIC Au+AuGeV Song et al, PRL106,192301(2011) 0.08 < h/s < 0.24 Elliptic Flow

4. Ridge Structure Long correlation in longitudinal direction 1+1 d viscous hydrodynamics

5. 1+1 d relativistic viscous hydrodynamics Fukuda

6. Perturvative calculation Fukuda

7. Perturbative Solution Fukuda F(1)の解：グリーン関数で構成される

8. Results Fukuda

9. z Reaction plane y x Viscosity in Hydrodynamics Elliptic Flow RHIC Au+AuGeV Song et al, PRL106,192301(2011) 0.08 < h/s < 0.24 Elliptic Flow

10. Higher Harmonics Mach-Cone-Like structure, Ridge structure Challenge to relativistic hydrodynamic model Viscosity effect from initial en to final vn Longitudinal structure (3+1) dimensional Higher harmonics high accuracy calculations • State-of-the-art numerical algorithm • Shock-wave treatment • Less numerical dissipation Higher harmonics and Ridge structure

11. Hydrodynamic Model hydro hadronization freezeout collisions thermalization observables higher harmonics strong elliptic flow @RHIC particle yields: PT distribution model fluctuating initial conditions hydrodynamic model final state interactions: hadron base event generators Viscosity, Shock wave One of successful models for description of dynamics of QGP:

12. Current Status of Hydro Ideal

13. Viscous Hydrodynamic Model • Relativistic viscous hydrodynamic equation • First order in gradient: acausality • Second order in gradient: • Israel-Stewart • Ottinger and Grmela • AdS/CFT • Grad’s 14-momentum expansion • Renomarization group • Numerical scheme • Shock-wave capturing schemes • Less numerical dissipation

14. Numerical Scheme • Lessons from wave equation • First order accuracy: large dissipation • Second order accuracy : numerical oscillation -> artificial viscosity, flux limiter • Hydrodynamic equation • Shock-wave capturing schemes: Riemann problem • Godunov scheme: analytical solution of Riemann problem, Our scheme • SHASTA: the first version of Flux Corrected Transport algorithm, Song, Heinz, Chaudhuri • Kurganov-Tadmor (KT) scheme, McGill

15. (COGNAC) Our Approach Takamoto and Inutsuka, arXiv:1106.1732 Akamatsu, Inutsuka, C.N., Takamoto,arXiv:1302.1665 • (ideal hydro) • 1. dissipative fluid dynamics = advection + dissipation exact solution for Riemann problem • Riemann solver: Godunov method Contact discontinuity Rarefaction wave t tt Shock wave Two shock approximation Mignone, Plewa and Bodo, Astrophys. J. S160, 199 (2005) Rarefaction wave shock wave • 2. relaxation equation = advection + stiff equation Israel-Stewart Theory

16. Riemann Problem Riemann problem Energy distribution shock wave: discontinuity surface Discretization

17. Riemann Problem Riemann problem Energy distribution example shock wave: discontinuity surface shock wave Initial Condition Discretization

18. Riemann Problem Riemann problem Energy distribution example shock wave: discontinuity surface Initial Condition Discretization

19. Riemann Problem Riemann problem Energy distribution example shock wave: discontinuity surface Initial Condition Discretization

20. Riemann Problem Riemann problem Energy distribution example shock wave: discontinuity surface shock wave rarefaction wave contact discontinuity shock wave Discretization

21. COGNAC COGite Numerical Analysis of heavy-ion Collisions Takamoto and Inutsuka, arXiv:1106.1732 Akamatsu, Inutsuka, C.N., Takamoto,arXiv:1302.1665 • (ideal hydro) • 1. dissipative fluid dynamics = advection + dissipation exact solution for Riemann problem • Riemann solver: Godunov method Contact discontinuity Rarefaction wave t tt Shock wave Two shock approximation Mignone, Plewa and Bodo, Astrophys. J. S160, 199 (2005) Rarefaction wave shock wave • 2. relaxation equation = advection + stiff equation Israel-Stewart Theory

22. Numerical Scheme Takamoto and Inutsuka, arXiv:1106.1732 1. Dissipative fluid equation 2. Relaxation equation + stiff equation advection I: second order terms Israel-Stewart Theory

23. Relaxation Equation Takamoto and Inutsuka, arXiv:1106.1732 + stiff equation advection • during Dt up wind method ~constant Piecewise exact solution fast numerical scheme Numerical scheme

24. Comparison • Analytical solution • Numerical schemes • SHASTA, KT, NT • Our scheme T=0.4 GeV v=0 EoS: ideal gas T=0.2 GeV v=0 10 0 Nx=100, dx=0.1 Shock Tube Test : Molnar, Niemi, Rischke, Eur.Phys.J.C65,615(2010)

25. Energy Density t=4.0 fm dt=0.04, 100 steps COGNAC analytic

26. Velocity t=4.0 fm dt=0.04, 100 steps COGNAC analytic

27. q t=4.0 fm dt=0.04, 100 steps COGNAC COGNAC analytic

28. Artificial and Physical Viscosities Molnar, Niemi, Rischke, Eur.Phys.J.C65,615(2010) Antidiffusionterms : artificial viscosity stability

29. Numerical Dissipation p fm-4 If numerical dissipationdoes not exist Cs0:sound velocity dp=0.1 fm-4 After one cycle: t=l/cs0 1000 Vs(x,t)=Vinit(x-cs0t) With finite numerical dissipation Vs(x,t)≠Vinit(x-cs0t) 0 2 -2 l=2 fm fm periodic boundary condition L1 norm after one cycle Sound wave propagation

30. Convergence Speed Space and time discretization Second order accuracy

31. Numerical Dissipation • numerical dissipation: • from fit of calculated data 1 1000

32. hnumvs Grid Size Numerical dissipation: Deviation from linear analyses (Llin) Ex. Heavy Ion Collisions l ~ 10 fm 0.1<h/s<1 Fluctuating initial condition T=500 MeV l ~ 1 fm Dx << 0.25 – 0.82 fm Dx << 0.8 – 2.6 fm

33. Viscosity in Hydrodynamics Elliptic Flow RHIC Au+AuGeV Song et al, PRL106,192301(2011) physical viscosity = input of hydro 0.08 < h/s < 0.24

34. Viscosity in Hydrodynamics Elliptic Flow RHIC Au+AuGeV Song et al, PRL106,192301(2011) With finite numerical dissipation 0.08 < h/s < 0.24 ? physical viscosity ≠ input of hydro physical viscosity = input of hydro + numerical dissipation Checking grid size dependence is important.

35. To Multi Dimension Operational split (C, S) Operational split and directional split

36. To Multi Dimension Operational split (C, S) 2d 3d Li : operation in i direction Operational split and directional split

37. Blast Wave Problems Velocity: |v|=0.9 Pressure distribution (0.2*vx, 0.2*vy) fm-4 1 Initial conditions

38. Blast Wave Problems

39. Higher Harmonics smoothed fluctuating • Initial conditions • Gluaber model

40. Higher Harmonics smoothed fluctuating t=10 fm t=10 fm • Initial conditions at mid rapidity • Gluaber model

41. 14 Viscosity Effect fm-4 initial Pressure distribution t~10 fm t~15 fm t~5 fm 1 Ideal 1 7 Viscosity 0.9 7 0.25

42. fm-4 Viscous Effect 20 initial Pressure distribution t~10 fm t~15 fm t~5 fm Ideal fm-4 0.25 1.2 9 Viscosity 0.3 1.2 9

43. Summary COGNAC with Duke and Texas A&M • We develop a state-of-the-art numerical scheme, COGNAC • Viscosity effect • Shock wave capturing scheme: Godunov method • Less numerical dissipation: crucial for viscosity analyses • Fast numerical scheme • Numerical dissipation • How to evaluate numerical dissipation • Physical viscosity  grid size • Work in progress • Analyses of high energy heavy ion collisions • Realistic Initial Conditions + COGNAC + UrQMD