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

1. Reference: AM and T. Hirano, arXiv:1003:3087 Causal Viscous Hydrodynamics forRelativistic Systems with Multi-Components and Multi-Conserved Currents Akihiko Monnai Department of Physics, The University of Tokyo Collaborator: Tetsufumi Hirano Strong and Electroweak Matter 2010 June 29th 2010, McGill University, Montreal, Canada

2. Outline • Introduction Relativistic hydrodynamics and Heavy ion collisions • Formulation of Viscous Hydro Israel-Stewart theory for multi-component/conserved current systems • Results and Discussion Constitutive equations and their implications • 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 • Hydrodynamic modeling of heavy ion collisions ｔ particles t Freezeout surface Σ Hadronic cascade picture Hydro to particles QGPphase Hydrodynamic picture hadronic phase Pre-equilibrium Initial condition z CGC/glasma picture? Hydrodynamics works at the intermediate stage (~1-10 fm/c) Viscous hydro helps us … 1. Explain the space-time evolution of the QGP 2. Extract viscosities of the QGP from experimental data

5. Introduction • Fourier analyses of particle spectra from RHIC data Hirano et al. (‘09) Ideal hydro Viscosity Glauber Initial condition 1st order Eq. of state theoretical prediction experimental data Ideal hydro works well

6. Introduction • Fourier analyses of particle spectra from RHIC data Hirano et al. (‘09) Ideal hydro Viscosity Glauber Initial condition 1st order Lattice-based Eq. of state *EoS based on lattice QCD results theoretical prediction experimental data Ideal hydro works… maybe

7. Introduction • Fourier analyses of particle spectra from RHIC data Hirano et al. (‘09) Ideal hydro Viscosity CGC Glauber Initial condition Lattice-based Eq. of state *Gluons in fast nuclei may form color glass condensate (CGC) theoretical prediction experimental data Viscous hydro in QGP plays important role in reducing v2

8. Introduction • Formalism of viscous hydro is not settled yet: • Form of viscous hydro equations • Treatment of conserved currents • Treatment of multi-component systems Important in fine-tuning viscosity from experimental data Low-energy ion collisions are planned at FAIR (GSI) & NICA (JINR) Multiple conserved currents? # 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

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

10. Introduction • Relativistic hydrodynamics Macroscopic theory defined on (3+1)-D spacetime Flow (vector field) Temperature (scalar field) Chemical potentials (scalar fields) and are described by the macroscopic fields Gradient in the fields: thermodynamic force Response to the gradients: dissipative current

11. 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

12. 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

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

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

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

16. 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)

17. Second Order Theory • Causality Issues • Conventional formalism Linear response theory implies instantaneous propagation Relaxation effects are necessary for causality Israel & Stewart (‘79) Dissipative currents (14) (9) Moment equations (10) (9) ? , , , , , one-component, elastic scattering frame fixing, stability conditions Not extendable for multi-component/conserved current systems

18. 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

19. 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 10+4N unknowns , are determined in self-consistency conditions The entropy production is expressed in terms of and Viscous distortion tensor & vector Moment equations Dissipative currents , , , , , , Matching matricesfor dfi Semi-positive definite condition

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

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

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

23. 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)

24. Discussion • Comparison with AdS/CFT+phenomenological approach • Comparison with Renormalization group approach Baier et al. (‘08) • Our approach goes beyond the limit of conformal theory • Vorticity-vorticity terms do not appear in kinetic theory Tsumura & Kunihiro (‘09) • Consistent, but vorticity terms need further checking as some are added in their recent revision

25. Discussion • Comparison with Grad’s 14-moment approach Betz et al. (‘09) • The form of their equations are consistent with that of ours • Our method can deal with multiple conserved currents while Grad’s 14-moment method does not Consistency with other approaches suggest our multi-component/conserved current formalism is a natural extension

26. 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 ( 1st order theory) Justifies the moment expansion • Future prospects include applications to… • Hydrodynamic modeling of the Quark-gluon plasma at heavy ion collisions • Cosmological fluid etc…

27. The End • Thank you for listening!

28. The Law of Increasing Entropy • Linear response theory • Entropy production : dissipative current : thermodynamic force : transport coefficient matrix (symmetric; semi-positive definite) Theorem: Symmetric matrices can be diagonalized with orthogonal matrix

29. Thermodynamic Stability • Maximum entropy state condition - Stability condition (1st order) - Stability condition (2nd order) Preserved for any *Stability conditions are NOT the same as the law of increasing entropy

30. First Order Limits Equilibrium limit • 2nd order constitutive equations thermodynamic forces (Navier-Stokes) transport coefficients (symmetric) thermodynamics forces (2nd order) • Onsager reciprocal relations are satisfied • 1st order theory is recovered in the equilibrium limit

31. Distortion of distribution • Express in terms of dissipative currents 10+4N (macroscopic) self-consistency 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) Viscous distortion tensor & vector Dissipative currents , , , , , , : Matching matrices

32. 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

33. Discussion • Expanding universe Einstein equation with Robertson-Walker metric in flat space because the R-W metric assumes isotropic and homogeneous expansion. “Bulk viscous cosmological model” Only “ideal hydrodynamic” energy-momentum tensor is allowed: Viscous correction to the pressure (= bulk pressure ) is allowed