Akihiko Monnai Department of Physics, The University of Tokyo Collaborator: Tetsufumi Hirano

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

2. Outline • Introduction Relativistic hydrodynamics and Heavy ion collisions • Relativistic Viscous Hydrodynamics Extended Israel-Stewart theory and Distortion of distribution • Results and Discussion Constitutive equations in multi-component/conserved current systems • Summary Summary and Outlook

3. Introduction • Quark-Gluon Plasma (QGP) at Relativistic Heavy Ion Collisions • RHIC experiments (2000-) • LHC experiments (2009-) Hadron phase QGPphase Tc ~0.2 T (GeV) Well-described in relativistic ideal hydrodynamic models “Small” discrepancies; non-equilibrium effects? Asymptotic freedom -> Less strongly-coupled QGP? Relativistic viscous hydrodynamic models are the key

4. Introduction • Elliptic flow coefficients from RHIC data Hirano et al. (‘09) Hirano et al. (‘06) Ideal hydro + CGC initial condition > experimental data Ideal hydro + lattice EoS > experimental data Viscous hydro in QGP plays important role in reducing v2

5. Introduction • Formalism of viscous hydro is not complete yet: • Form of viscous hydro equations • Treatment of conserved currents • Treatment of multi-component systems e.g. … Israel & Stewart (‘79) Muronga (‘02) Betz et al. (‘09) 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. We need to construct a firm framework of viscous hydro

6. Introduction • Categorization of relativistic systems QGP/hadronic gas at heavy ion collisions Cf. etc.

7. Overview START Energy-momentum conservation Charge conservations Law of increasing entropy Generalized Grad’s moment method Moment equations , GOAL (constitutive eqs.) , , , Onsager reciprocal relations: satisfied

8. Thermodynamic Quantities • Tensor decompositions by flow where is the projection operator 2+N equilibrium quantities 10+4N dissipative currents Energy density: Hydrostatic pressure: J-th charge density: Energy density deviation: Bulk pressure: Energy current: Shear stress tensor: J-th charge density dev.: J-th charge current: *Stability conditions should be considered afterward

9. Relativistic Hydrodynamics • Ideal hydrodynamics • Viscous hydrodynamics (“perturbation” from equilibrium) Unknowns（5+N） Conservation laws（4+N）+ EoS(1) , , , , , Additional unknowns（10+4N）: , , , , , Constitutive equations are necessary the law of increasing entropy Irreversible processes 0th order theory 1st order theory 2nd order theory ideal; no entropy production linear response; acausal relaxation effects; causal

10. First Order Theory • Kinetic expressions with distribution : • The law of increasing entropy (1st order) : degeneracy : conserved charge number

11. First Order Theory • Linear response theory conventional terms cross terms Scalar Vector Tensor The cross termsare symmetric due to Onsager reciprocal relations

12. First Order Theory • Linear response theory Dufour effect Soret effect • Cool down once – for cooking tasty oden (Japanese pot-au-feu) Vector Permeation of ingredients potato soup Thermal gradient Chemical diffusion caused by thermal gradient (Soret effect)

13. Second Order Theory • Causality issues • Conventional formalism Linear response theory implies instantaneous propagation Relaxation effects are necessary for causality Israel & Stewart (‘79) one-component, elastic scattering -> 9 constitutive eqs. frame fixing, stability conditions -> 9 unknowns Not extendable for multi-component/conserved current systems

14. Extended Second Order Theory • Moment equations • Expressions of and Determined through the 2nd law of thermodynamics New eqs. introduced Unknowns (10+4N) Moment eqs. (10+4N) , , All viscous quantities determined in arbitrary frame where Off-equilibrium distribution is needed

15. Distortion of distribution • Express in terms of dissipative currents 10+4N (macroscopic) self-consistent conditions , Fixandthrough matching Moment expansion with 10+4N unknowns , *Grad’s 14-moment method extended for multi-conserved current systems (Consistent with Onsager reciprocal relations) Dissipative currents Viscous distortion tensor & vector , , , , , , : Matching matrices

16. Second Order Equations • Entropy production • Constitutive equations Semi-positive definite condition Viscous distortion tensor & vector Moment equations , :symmetric, semi-positive definite matrices Viscous distortion tensor & vector Moment equations Dissipative currents , , , , , , Matching matricesfor dfi Semi-positive definite condition

17. Results • 2nd order constitutive equations for systems with multi-components andmulti-conserved currents Bulk pressure 1st order terms 2nd order terms relaxation : relaxation times , : 1st, 2nd order transport coefficients

18. Results • (Cont’d) Energy current Dufour effect 1st order terms 2nd order terms relaxation

19. Results • (Cont’d) J-th charge current Soret effect 1st order terms 2nd order terms relaxation

20. Results • (Cont’d) Shear stress tensor • Our results in the limit of single component/conserved current 1st order terms 2nd order terms relaxation Consistent with other results based on AdS/CFT approach Baier et al. (‘08) Renormalization group method Tsumura and Kunihiro (‘09) Grad’s 14-moment method Betz et al. (‘09)

21. Discussion • Comparison with AdS/CFT+phenomenological approach Shear stress tensor in conformal limit, no charge current Baier et al. (‘08) (Our equations) Mostly consistent w ideal hydro relation • Our approach goes beyond the limit of conformal theory • Vorticity-vorticity terms do not appear in kinetic theory

22. Discussion • Comparison with Renormalization group approach in energy frame, in single component/conserved current system Tsumura & Kunihiro (‘09) (Our equations) Form of the equations agrees with our equations in the single component & conserved current limit w/o vorticity Note: Vorticity terms added to their equations in recent revision

23. Discussion • Comparison with Grad’s 14-momemt approach in energy frame, in single component/conserved current system Betz et al. (‘09) *Ideal hydro relations in use for comparison Form of the equations agrees with ours in the single component & conserved current limit (Our equations) Consistency with other approaches suggest our multi-component/conserved current formalism is a natural extension

24. Summary and Outlook • We formulated generalized 2nd order theory from the entropy production w/o violating causality • Multi-component systems with multiple conserved currents Inelastic scattering (e.g. pair creation/annihilation) implied • Frame independent Independent equations for energy and charge currents • Onsagerreciprocal relations ( 1storder 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. The End • Thank you for listening!