1. Akihiko Monnai Department of Physics, The University of Tokyo Collaborator: Tetsufumi Hirano Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents Berkeley School of Collective Dynamics in High Energy Collisions June 10 th 2010, Lawrence Berkeley National Laboratory, USA Reference: AM and T. Hirano, arXiv:1003:3087

2. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 th 2010 Outline 1. Introduction Relativistic hydrodynamics and Heavy ion collisions 2. Relativistic Viscous Hydrodynamics Extended Israel-Stewart theory and Distortion of distribution 3. Results and Discussion Constitutive equations in multi-component/conserved current systems 4. Summary Summary and Outlook

3. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 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, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 th 2010 Introduction Elliptic flow coefficients from RHIC data Hirano et al. (‘06) Ideal hydro + CGC initial condition > experimental data Viscous hydro in QGP plays important role in reducing v 2 Hirano et al. (‘09) Ideal hydro + lattice EoS > experimental data

5. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 th 2010 Introduction Formalism of viscous hydro is not complete yet: 1. Form of viscous hydro equations 2. Treatment of conserved currents 3. Treatment of multi-component systems We need to construct a firm framework of viscous hydro Israel & Stewart (‘79)Muronga (‘02)Betz et al. (‘09) e.g.… Low-energy ion collisions are planned at FAIR (GSI) & NICA (JINR) Multi-conserved current systems are not supported # of conserved currents# of particle species baryon number, strangeness, etc.pion, proton, quarks, gluons, etc.

6. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 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.

7. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 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,,,

8. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 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:

9. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 th 2010 Relativistic Hydrodynamics Ideal hydrodynamics Viscous hydrodynamics (“perturbation” from equilibrium),,, Conservation laws （ 4+N ） + EoS(1)Unknowns （ 5+N ）, Additional unknowns （ 10+4N ） :,,,,, Constitutive equations are necessary the law of increasing entropy Irreversible processes 0 th order theory 1 st order theory 2 nd order theory ideal; no entropy production linear response; acausal relaxation effects; causal

10. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 th 2010 First Order Theory Kinetic expressions with distribution : The law of increasing entropy (1 st order) : degeneracy : conserved charge number

11. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 th 2010 First Order Theory Linear response theory The cross terms are symmetric due to Onsager reciprocal relations Scalar Vector Tensor conventional termscross terms

12. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 th 2010 First Order Theory Linear response theory 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)

13. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 th 2010 Second Order Theory Causality issues Conventional formalism Not extendable for multi-component/conserved current systems one-component, elastic scattering -> 9 constitutive eqs. frame fixing, stability conditions -> 9 unknowns Israel & Stewart (‘79) Linear response theory implies instantaneous propagation Relaxation effects are necessary for causality

14. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 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, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 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-consistent conditions,

16. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 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,

17. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 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

18. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 th 2010 Results (Cont’d) Energy current 1 st order terms 2 nd order terms Dufour effect relaxation

19. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 th 2010 Results (Cont’d) J-th charge current 1 st order terms 2 nd order terms Soret effect relaxation

20. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 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, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 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)

22. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 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)

23. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 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)

24. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 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…

25. A power point template created by Akihiko Monnai Akihiko Monnai, Viscous Hydrodynamics for Relativistic Systems with Multi-Components and Multiple Conserved Currents, Berkeley School 2010, Jul. 10 th 2010 The End Thank you for listening!