1. Akihiko Monnai Department of Physics, The University of Tokyo Collaborator: Tetsufumi Hirano Relativistic Viscous Hydrodynamics for Multi-Component Systems with Multiple Conserved Currents Hot Quarks 2010 June 25 th 2010, La Londe-Les-Maures, France Reference: AM and T. Hirano, arXiv:1003:3087

2. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Outline 1. Introduction Relativistic hydrodynamics and Heavy ion collisions 2. Formulation of Viscous Hydro Israel-Stewart theory for multi-component/conserved current systems 3. Results and Discussion Constitutive equations and their implications 4. Summary Summary and Outlook

3. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Introduction Quark-Gluon Plasma (QGP) at Relativistic Heavy Ion Collisions RHIC experiments (2000-) LHC experiments (2009-) T (GeV) Tc ~ 0.2 Hadron phaseQGP phase “Small” discrepancies; non-equilibrium effects? Relativistic viscous hydrodynamic models are the key Well-described in relativistic ideal hydrodynamic models Asymptotic freedom -> Less strongly-coupled QGP?

4. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Introduction Hydrodynamic modeling of heavy ion collisions for RHIC particles hadronic phase QGP phase Freezeout surface Σ Pre- equilibrium Hydrodynamic picture CGC/glasma picture? Hadronic cascade picture Initial condition Hydro to particles ｔ z t Purpose of Viscous Hydro: 1. Explaining the space-time evolution of the QGP 2. Constraining the parameters (viscosities, etc.) from experimental data Hydrodynamics works at the intermediate stage (~1-10 fm/c)

5. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Introduction Elliptic flow coefficients from RHIC data Hirano et al. (‘09) Ideal hydro Glauber 1 st order Viscosity Eq. of state theoretical prediction experimental data Ideal hydro works well Initial condition

6. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Introduction Elliptic flow coefficients from RHIC data Hirano et al. (‘09) Ideal hydro Glauber 1 st order Viscosity Eq. of state theoretical prediction experimental data Lattice-based Ideal hydro works… maybe Initial condition *EoS based on lattice QCD results

7. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Introduction Elliptic flow coefficients from RHIC data Viscous hydro in QGP plays important role in reducing v 2 Hirano et al. (‘09) Ideal hydro Glauber Viscosity Eq. of state Initial condition theoretical prediction experimental data CGC Lattice-based *Gluons in fast nuclei may form color glass condensate (CGC)

8. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Introduction Formalism of viscous hydro is not settled yet: 1. Form of viscous hydro equations 2. Treatment of conserved currents 3. Treatment of multi-component systems Fixing the equations is essential in fine- tuning viscosity from experimental data # of conserved currents# of particle species baryon number, strangeness, etc.pion, proton, quarks, gluons, etc. Low-energy ion collisions are planned at FAIR (GSI) & NICA (JINR) Multiple conserved currents? We need to construct a firm framework of viscous hydro Song & Heinz (‘08)

9. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Introduction Categorization of relativistic systems Number of components Types of interactions Single component with binary collisions Israel & Stewart (‘79), etc… Multi-components with binary collisions Prakash et al. (‘91) Single component with inelastic scatterings (-) Multi-components with inelastic scatterings Monnai & Hirano (‘10) QGP/hadronic gas at heavy ion collisions Cf. etc.

10. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Overview Energy-momentum conservation Charge conservations Law of increasing entropy START GOAL (constitutive eqs.) Onsager reciprocal relations: satisfied Moment equations, Generalized Grad’s moment method,,,

11. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Thermodynamic Quantities Tensor decompositions by flow where is the projection operator 10+4N dissipative currents 2+N equilibrium quantities *Stability conditions should be considered afterward Energy density deviation: Bulk pressure: Energy current: Shear stress tensor: J-th charge density dev.: J-th charge current: Energy density: Hydrostatic pressure: J-th charge density:

12. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Relativistic Hydrodynamics Ideal hydrodynamics Viscous hydrodynamics,,, Conservation laws （ 4+N ） + EoS(1)Unknowns （ 5+N ）, We derive the equations from the law of increasing entropy 0 th order theory 1 st order theory 2 nd order theory ideal; no entropy production linear response; acausal relaxation effects; causal Additional unknowns （ 10+4N ）,,,,, !? Constitutive equations “perturbation” from equilibrium

13. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Second Order Theory Kinetic expressions with distribution function : Conventional formalism Not extendable for multi-component/conserved current systems one-component, elastic scattering Israel & Stewart (‘79) : degeneracy : conserved charge number Dissipative currents (14),,,,, Moment equations (10) frame fixing, stability conditions (9) ?

14. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Extended Second Order Theory Moment equations Expressions of and Determined through the 2 nd law of thermodynamics Unknowns (10+4N), Moment eqs. (10+4N), New eqs. introduced All viscous quantities determined in arbitrary frame where Off-equilibrium distribution is needed

15. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Extended Second Order Theory Moment expansion *Grad’s 14-moment method extended for multi-conserved current systems so that it is consistent with Onsager reciprocal relations Dissipative currents Viscous distortion tensor & vector Moment equations,,,,,, Semi-positive definite condition Matching matrices for dfi 10+4N unknowns, are determined in self-consistency conditions The entropy production is expressed in terms of and

16. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Results 2 nd order constitutive equations for systems with multi-components and multi-conserved currents Bulk pressure 1 st order terms : relaxation times, : 1 st, 2 nd order transport coefficients 2 nd order terms relaxation

17. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Results (Cont’d) Energy current 1 st order terms 2 nd order terms relaxation

18. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Results (Cont’d) J-th charge current 1 st order terms 2 nd order terms relaxation

19. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Onsager Cross Effects 1 st order terms Vector Dufour effect Soret effect potato Thermal gradient Permeation of ingredients soup Chemical diffusion caused by thermal gradient (Soret effect) Cool down once – for cooking tasty oden (Japanese pot-au-feu) It may well be important in our relativistic soup of quarks and gluons

20. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Results (Cont’d) Shear stress tensor Our results in the limit of single component/conserved current 1 st order terms 2 nd order terms Consistent with other results based on AdS/CFT approach Renormalization group method Grad’s 14-moment method Betz et al. (‘09) Baier et al. (‘08) Tsumura and Kunihiro (‘09) relaxation

21. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Discussion Comparison with AdS/CFT+phenomenological approach Comparison with Renormalization group approach Our approach goes beyond the limit of conformal theory Vorticity-vorticity terms do not appear in kinetic theory Baier et al. (‘08) Tsumura & Kunihiro (‘09) Consistent, but vorticity terms need further checking as some are added in their recent revision

22. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Discussion Comparison with Grad’s 14-moment approach The form of their equations are consistent with that of ours Multiple conserved currents are not supported in 14-moment method Betz et al. (‘09) Consistency with other approaches suggest our multi- component/conserved current formalism is a natural extension

23. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Summary and Outlook We formulated generalized 2 nd order theory from the entropy production w/o violating causality 1. Multi-component systems with multiple conserved currents Inelastic scattering (e.g. pair creation/annihilation) implied 2. Frame independent Independent equations for energy and charge currents 3. Onsager reciprocal relations ( 1 st order theory) Justifies the moment expansion Future prospects include applications to… Hydrodynamic modeling of Quark-gluon plasma at relativistic heavy ion collisions Cosmological fluid etc…

24. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 The End Thank you for listening!

25. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 The Law of Increasing Entropy Linear response theory Entropy production : transport coefficient matrix (symmetric; semi-positive definite) : dissipative current: thermodynamic force Theorem: Symmetric matrices can be diagonalized with orthogonal matrix

26. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Thermodynamic Stability Maximum entropy state condition - Stability condition (1 st order) - Stability condition (2 nd order)Preserved for any *Stability conditions are NOT the same as the law of increasing entropy

27. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 First Order Limits 2 nd order constitutive equations transport coefficients (symmetric) thermodynamics forces (2 nd order) 1 st order theory is recovered in the equilibrium limit Onsager reciprocal relations are satisfied Equilibrium limit thermodynamic forces (Navier-Stokes)

28. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Introduction Hydrodynamic modeling of heavy ion collisions for RHIC particles hadronic phase QGP phase Freezeout surface Σ Pre- equilibrium Hydrodynamic picture CGC/glasma picture? Hadronic cascade picture Initial condition Hydro to particles ｔ z t Initial condition Hydrodynamic model requires Equation of state Viscosity Hadronization 1 st order phase transition, crossover(lattice), etc. Ideal hydro, viscous hydro JAM, UrQMD, etc. Glauber, color glass condensate, etc.

29. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Introduction Relativistic hydrodynamics Macroscopic theory defined on (3+1)-D spacetime Flow (vector field) Temperature (scalar field) Chemical potentials (scalar fields) Physics should be described by the macroscopic fields only Gradient in the fields: thermodynamic force Response to the gradients: dissipative current

30. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Distortion of distribution Express in terms of dissipative currents *Grad’s 14-moment method extended for multi-conserved current systems (Consistent with Onsager reciprocal relations) Fix and through matching Viscous distortion tensor & vector Dissipative currents,,,,,, : Matching matrices Moment expansion with 10+4N unknowns, 10+4N (macroscopic) self-consistency conditions,

31. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Second Order Equations Entropy production Constitutive equations Semi-positive definite condition : symmetric, semi-positive definite matrices Dissipative currents Viscous distortion tensor & vector Moment equations,,,,,, Semi-positive definite condition Matching matrices for dfi Viscous distortion tensor & vector Moment equations,

32. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Discussion Comparison with AdS/CFT+phenomenological approach Our approach goes beyond the limit of conformal theory Vorticity-vorticity terms do not appear in kinetic theory Shear stress tensor in conformal limit, no charge current Mostly consistent w ideal hydro relation Baier et al. (‘08) (Our equations)

33. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Discussion Comparison with Renormalization group approach in energy frame, in single component/conserved current system Form of the equations agrees with our equations in the single component & conserved current limit w/o vorticity Tsumura & Kunihiro (‘09) Note: Vorticity terms added to their equations in recent revision (Our equations)

34. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25 th 2010 Discussion Comparison with Grad’s 14-momemt approach Form of the equations agrees with ours in the single component & conserved current limit Betz et al. (‘09) *Ideal hydro relations in use for comparison Consistency with other approaches suggest our multi- component/conserved current formalism is a natural extension in energy frame, in single component/conserved current system (Our equations)