1. Flow and Dissipation in Ultrarelativistic Heavy Ion Collisions September 16 th 2009, ECT* Italy Akihiko Monnai Department of Physics, The University of Tokyo Collaborator: Tetsufumi Hirano Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations AM and T. Hirano, arXiv:0903.4436; arXiv:0907.3078

2. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 1.Introduction 2.Distortion of Distribution 3.Numerical Estimation 4.Viscous Hydrodynamic Equations 5.Summary and Outlook Outlook 2

3. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 1. Introduction Next: 2. Distortion of Distribution 3

4. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Introduction 4 Success of ideal hydrodynamic models Development of viscous hydrodynamic models at relativistic heavy ion collisions (1) to understand the hot QCD matter itself and (2) to determine the transport coefficients from experimental data We put emphasize on bulk viscosity Paech & Pratt (‘06) Mizutani et al. (‘88) Kharzeev & Tuchin (’08) … “Large” bulk viscosity near Tc

5. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Viscous effects on the particle spectra Introduction 5 modification of the distributionvariation of the flow/hypersurface particles hadron resonance gas QGP freezeout hypersurface Σ Cooper & Frye (‘74) Determined in Grad’s method in a multi-component system (Sec. 2) Numerical estimation (Sec. 3) Hydrodynamic equations (Sec. 4) An interface from a hydrodynamic model to a cascade model needed

6. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 2. Distortion of Distribution Previous: 1. Introduction Next: 3. Numerical Results 6

7. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 for a single-component gas by Israel & Stewart (‘79) Tensor decomposition and the macroscopic variables: where and. Bulk pressure : Energy current: Charge current: Shear tensor: Distortion of Distribution 7

8. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Landau matching conditions:, = necessary conditions to ensure thermodynamic stability The Matching Conditions 8 etc. i.e.

9. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Kinetic definitions for a multi-component gas: where g i is the degeneracy and b i the baryon number. 14 equations (= kinetic definitions + matching conditions) = “Bridges” from the macroscopic variables to the microscopic distribution Relativistic Kinetic Theory 9 δf i Π, W μ, V μ, π μν GivenUnknown

10. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Distortion expressed with 14 (= 4+10) unknowns: where + for bosons and – for fermions. No scalar, but non-zero trace tensor. Grad’s 14-Moment Method 10 The trace part: The scalar term: particle species independent (macroscopic quantity) particle species dependent (mass dependent) NOT equivalent in a multi-component system!

11. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Why not the quadratic ansatz of ? thermodynamically unstable Why not ε μ i and ε μν i ? The number of macroscopic equations = 14 No room for additional unknowns Introducing more microscopic physics (i.e. cross sections)? Generality is lost to be consistent e.g. transport coefficients Comments 11 Dusling & Teaney (‘08)

12. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Must-items of the tensor decomposition: *Contributions are : [baryons] + [anti-baryons] + [mesons] : [baryons] – [anti-baryons] Decomposition of Moments 12

13. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Insert into the kinetic relations: where,, and. The unique form of the deviation is determined: Determination of Distortion 13 Scalar terms Vector terms Tensor term

14. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 3. Numerical Estimation Previous: 2. Distortion of Distribution Next: 4. Viscous Hydrodynamic Equations 14

15. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Equation of state: 16-component hadron resonance gas *mesons and baryons with mass up to under. Transport coefficients:, : sound velocity s: entropy density : free parameter Freezeout temperature: T f = 0.16(GeV) and ( ). EoS and Transport Coefficients 15 Weinberg (‘71) Kovtun et al.(‘05) …

16. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Profiles of the flow and the freezeout hypersurface : a (3+1)-dimensional ideal hydrodynamic simulation Estimation of dissipative currents: Navier-Stokes limit, Definition of elliptic flow coefficient v 2 (p T ): Flow and Dissipative Currents 16 Hirano et al.(‘06)

17. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Au+Au,, b = 7.2(fm), p T -spectra and v 2 (p T ) of Bulk Viscosity and Particle Spectra 17 p T -spectra suppressed v 2 (p T )enhanced *Possible overestimations for… (i) Navier-Stokes limit (no relaxation effects) (ii) ideal hydro flow (derivatives are larger)

18. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 p T -spectra and v 2 (p T ) of when Quadratic Ansatz 18 Effects of the bulk viscosity is underestimated in the quadratic ansatz.

19. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 4. Viscous Hydrodynamic Equations Previous: 3. Numerical Estimation Next: 5. Summary and Outlook 19

20. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Single component system Ideal hydrodynamics Viscous hydrodynamics 2 nd Order Israel-Stewart Theory 20 9 independent Eqs. for (1), (1), (1), (3) 5 Conservation Eqs. + 1 EoS 6 unknowns, and Constitutive Equations: (1), (3), (5) because does not count.

21. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Particle number conservation does not hold instead one has baryon number conservation The trace of the constitutive eqs. is also not 0 which means. In Multi-Component System 21 Two non-trivial consequences: i) Traceless is not allowed for a multi-component system ii) One more constitutive equation is obtained

22. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 The entropy current obeys 2 nd Law of Thermodynamics 22 is constrained If were traceless, so was, i.e., in the case of a multi-component system

23. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Multi-component system 10 Constitutive Eqs. (tensor-decomposed) Redundant Equation? 23 because 2 independent eqs. for 3 eqs. for 5 eqs. for The uncertainty can be removed by considering the correct linear combination of the constitutive eqs.

24. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Frame-independent constitutive equations (preliminary) Bulk pressure Heat current Shear stress tensor Viscous Hydrodynamic Equations 24 Cf. Betz et al. (‘08)

25. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 5. Summary and Outlook Previous: 4. Viscous Hydrodynamic Equations Next: Appendix 25

26. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Determination of in a multi-component system - Viscous correction has non-zero trace. Visible effects of on particle spectra - p T -spectra is suppressed; v 2 (p T ) is enhanced Derivation of constitutive eqs. in a multi-component system - Non-trivialities involved; post I-S terms reconfirmed Bulk viscosity can be important in constraining the transport coefficients from experimental data. Full Viscous hydrodynamics needs to be developed to see more realistic behavior of the particle spectra. Summary and Outlook 26

27. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Appendix Previous: 5. Summary and Outlook 27

28. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 p T -spectra and v 2 (p T ) of with shear viscous correction Shear Viscosity and Particle Spectra 28 Non-triviality of shear viscosity; both p T -spectra and v 2 (p T ) suppressed

29. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 p T -spectra and v 2 (p T ) of with corrections from shear and bulk viscosity Shear & Bulk Viscosity on Spectra 29 Accidental cancellation in viscous corrections in v 2 (p T )

30. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 p T -spectra and v 2 (p T ) of in Bjorken model with cylindrical geometry: Bjorken Model 30 Bulk viscosity suppresses p T -spectra Shear viscosity enhances p T -spectra

31. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 p T -spectra and v 2 (p T ) of Blast wave model 31 Shear viscosity enhances p T -spectra and suppresses v 2 (p T ).

32. Akihiko MonnaiViscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009 Distortion depends on all the components in the gas, i.e. Single vs. Multi-Component 32 because and in are “macroscopic” quantities.