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

2. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 th 2010 Introduction Hydrodynamic modeling of heavy ion collisions 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 Viscous hydro helps us … 1. Explain the space-time evolution of the QGP 2. Extract viscosities of the QGP 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), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 th 2010 Introduction Fourier analyses of particle spectra 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), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 th 2010 Introduction Fourier analyses of particle spectra 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), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 th 2010 Introduction Fourier analyses of particle spectra 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), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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 Important 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

9. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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 relativistic heavy ion collisions Cf. etc.

10. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 th 2010 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. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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,,,

12. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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:

13. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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

14. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 th 2010 First Order Theory Kinetic expressions with distribution function : The law of increasing entropy (1st order) : degeneracy : conserved charge number

15. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 th 2010 First Order Theory Linear response theory The cross terms are symmetric due to Onsager reciprocal relations Scalar Vector Tensor conventional termscross terms

16. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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)

17. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 th 2010 Second Order Theory Causality Issues Conventional formalism Not extendable for multi-component/conserved current systems one-component, elastic scattering Israel & Stewart (‘79) Dissipative currents (14),,,,, Moment equations (10) frame fixing, stability conditions (9) ? Linear response theory implies instantaneous propagation Relaxation effects are necessary for causality

18. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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

19. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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

20. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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

21. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 th 2010 Results (Cont’d) Energy current 1 st order terms 2 nd order terms Dufour effect relaxation

22. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 th 2010 Results (Cont’d) J-th charge current 1 st order terms 2 nd order terms Soret effect relaxation

23. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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

24. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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

25. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 th 2010 Discussion Comparison with Grad’s 14-moment approach 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 Betz et al. (‘09) Consistency with other approaches suggest our multi- component/conserved current formalism is a natural extension

26. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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 the Quark-gluon plasma at heavy ion collisions Cosmological fluid etc…

27. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 th 2010 The End Thank you for listening!

28. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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

29. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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

30. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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)

31. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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,

32. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 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,

33. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo), Strong and Electroweak Matter 2010, McGill University, Montreal, Canada, Jul. 29 th 2010 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” Discussion Only “ideal hydrodynamic” energy-momentum tensor is allowed: Viscous correction to the pressure (= bulk pressure ) is allowed