Relativistic Viscous Hydrodynamics for Multi-Component Systems with Multiple Conserved Currents

1. Reference: AM and T. Hirano, arXiv:1003:3087 Relativistic Viscous Hydrodynamics for Multi-Component Systems with Multiple Conserved Currents Akihiko Monnai Department of Physics, The University of Tokyo Collaborator: Tetsufumi Hirano • Hot Quarks 2010 • June25th 2010, La Londe-Les-Maures, France

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 for RHIC 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) Purpose of Viscous Hydro: 1. Explaining the space-time evolution of the QGP 2. Constraining the parameters (viscosities, etc.) from experimental data

5. Introduction • Elliptic flow coefficients 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 • Elliptic flow coefficients 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 • Elliptic flow coefficients 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 Song & Heinz (‘08) Fixing the equations is essential 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 heavy ion collisions Cf. etc.

10. 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

11. 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

12. 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

13. Second Order Theory • Kinetic expressions with distribution function : • Conventional formalism : degeneracy : conserved charge number 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

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. 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

16. 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

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

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

19. Onsager Cross Effects • 1st order terms 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) It may well be important in our relativistic soup of quarks and gluons

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 • 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

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

23. 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 Quark-gluon plasma at relativistic heavy ion collisions • Cosmological fluid etc…

24. The End • Thank you for listening!

25. 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

26. 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

27. 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

28. Introduction • Hydrodynamic modeling of heavy ion collisions for RHIC 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? Hydrodynamic model requires Initial condition Glauber, color glass condensate, etc. Equation of state 1st order phase transition, crossover(lattice), etc. Viscosity Ideal hydro, viscous hydro Hadronization JAM, UrQMD, etc.

29. 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. 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) Dissipative currents Viscous distortion tensor & vector , , , , , , : Matching matrices

31. 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

32. 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

33. 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

34. 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