1. Akihiko Monnai Department of Physics, The University of Tokyo Collaborator: Tetsufumi Hirano Viscous Hydrodynamics Heavy Ion Meeting 2010-12 December 10 th 2010, Yonsei University, Korea AM and T. Hirano, Nucl. Phys. A 847 (2010) 283

2. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Outline 1. Introduction Relativistic hydrodynamics and heavy ion collisions 2. Relativistic Dissipative Hydrodynamics Extended Israel-Stewart theory from law of increasing entropy 3. Results and Discussion Constitutive equations in multi-component/conserved current systems 4. Summary Summary and Outlook Introduction

3. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Introduction Quark-gluon plasma (QGP) at relativistic heavy ion collisions Hadron phase QGP phase (crossover) sQGP(wQGP?) Discovery of QGP as nearly perfect fluid RHIC experiments (2000-) Thermodynamics is at work in QGP at RHIC energies Non-equilibrium effects need investigation for quantitative understandings Introduction

4. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Introduction Hydrodynamic modeling of RHIC particles hadronic phase QGP phase Freezeout Pre- equilibrium Hydrodynamic picture CGC/glasma picture? Hadronic cascade picture Initial condition Hydro to particles ｔ z t Intermediate stage (~1-10 fm) is described by hydrodynamics Results are dependent on the inputs: Hydrodynamic equationsInitial conditionsOutput Equation of state, Transport coefficients Introduction

5. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Introduction Properties of QCD fluid Equation of state: relation among thermodynamic variables sensitive to degrees of freedom in the system Transport coefficients: responses to thermodynamic forces sensitive to interaction in the system Shear viscosity = response to deformation Bulk viscosity = response to expansion Energy dissipation = response to thermal gradient Charge dissipation = response to chemical gradients Naïve interpretation of dissipative processes Introduction

6. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Introduction Elliptic flow coefficients from RHIC data Hirano et al. (‘09) Ideal hydro Glauber 1 st order Viscosity Eq. of state Initial cond. theoretical prediction ~ experimental data Ideal hydro works well Introduction

7. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Introduction Elliptic flow coefficients from RHIC data Hirano et al. (‘09) Ideal hydro Glauber 1 st order Viscosity Eq. of state Initial cond. theoretical prediction > experimental data Lattice-based Ideal hydro shows slight overshooting Introduction

8. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Introduction Elliptic flow coefficients from RHIC data Viscosity in QGP phase plays important role in reducing v 2 Hirano et al. (‘09) Ideal hydro Glauber Viscosity Eq. of state Initial cond. theoretical prediction > experimental data Color glass condensate Lattice-based *Gluons in fast moving nuclei are saturated to CGC Introduction

9. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Introduction Why viscous hydrodynamic models? RHIC experiments (2000-) Success of ideal hydro for improved inputs to the hydrodynamic models Necessity of viscous hydro QGP might become less-strongly coupled LHC experiments (2010-) Asymptotic freedom in QCD Viscous hydro? CERN Press release, November 26, 2010: “… confirms that the much hotter plasma produced at the LHC behaves as a very low viscosity liquid” Viscous hydro is likely to work also at the LHC energies Introduction

10. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Introduction Viscous hydrodynamics needs improvement 1. Form of dissipative hydro equations 2. Treatment of conserved currents 3. Treatment of multi-component systems Fixing the equations is essential 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) Only 1 conserved current can be treated We need to construct a firm framework of viscous hydro Song & Heinz (‘08) Introduction

11. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Introduction Categorization of relativistic hydrodynamic formalisms 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 (-) Required for QGP/hadron gas at heavy ion collisions Overview

12. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Introduction Categorization of relativistic hydrodynamic formalisms 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) Required for QGP/hadron gas at heavy ion collisions In this work we formulate relativistic dissipative hydro for multi-component + multi-conserved current systems Overview

13. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Overview Formulation of relativistic dissipative hydrodynamics Energy-momentum conservation Charge conservations Law of increasing entropy START GOAL Onsager reciprocal relations satisfied Moment eqs., Generalized moment method EoM for dissipative currents,,, Thermodynamics Quantities

14. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: 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: Thermodynamic Stability

15. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Thermodynamic Stability Maximum entropy state condition - Stability condition (1 st order) - Stability condition (2 nd order)automatically satisfied in kinetic theory *Stability conditions are NOT the same as the law of increasing entropy Relativistic Hydrodynamics

16. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Relativistic Hydrodynamics Ideal hydrodynamics Dissipative hydrodynamics (“perturbation” from equilibrium),,, Conservation laws (4+N) + EoS(1)Unknowns (5+N), Defined in relativistic kinetic theory as Additional unknowns (10+4N),,,,, !? Moment equations (10+4N), Estimated from the law of increasing entropy New equations in our work Moment Equations

17. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Moment Equations Introduce distortion of distribution *Grad’s moment method extended to multi-conserved current systems 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 Constitutive Equations

18. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Constitutive Equations Bulk Pressure response to expansioncross terms (linear) relaxation term 2 nd order corrections Cross terms appear (reciprocal relations) Relaxation term appears (causality is preserved) 2 nd order terms in full form (multi-conserved currents) Constitutive Equations

19. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Constitutive Equations Energy current response to temperature gradientcross terms (linear) relaxation term 2 nd order corrections Constitutive Equations

20. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Constitutive Equations Charge currents response to chemical gradient (+ cross terms)cross terms (linear) relaxation terms 2 nd order corrections Constitutive Equations

21. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Constitutive Equations Shear stress tensor Discussion - Relaxation term response to deformation relaxation term 2 nd order corrections Linear responseAcausal and unstable in relativistic systems Hiscock & Lindblom (‘85) Relaxation effect to limit the propagation faster than the light speed Discussion – Cross terms

22. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Discussion - Cross terms Coupling of thermodynamic forces in the dissipative currents potato thermal gradient ingreadients soup Chemical diffusion via thermal gradient Cf: “Cooling” process for cooking tasty oden (Japanese soup) It should play an important role in our “quark soup” Soret effect ex. Onsager reciprocal relations （ ） is satisfied Discussion – 2 nd order terms

23. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Discussion – 2 nd order terms 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, as vorticity terms are added in their recent revision Frame-dependent equations Discussion – 2 nd order terms

24. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Discussion – 2 nd order terms Comparison with Grad’s 14-moment approach The form of their equations are consistent with that of ours Multiple conserved currents are not supported in 14-moment method Betz et al. (‘09) Consistencies suggest we have successfully extended 2 nd order theory to multi-component + conserved current systems Summary and Outlook

25. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: Summary and Outlook We formulated generalized 2 nd order dissipative hydro from the entropy production w/o violating causality 1. Multi-component systems with multiple conserved currents Inelastic scattering (e.g. pair creation/annihilation) included 2. 1 st order cross terms are present Onsager reciprocal relations are satisfied 3. Frame independent Independent equations for energy and charge currents Future prospects include applications to… Numerical estimation of viscous hydrodynamic models for relativistic heavy ion collisions Cosmological fluid, and more AM & T. Hirano, in preparation

26. Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Akihiko Monnai (The University of Tokyo) Viscous Hydrodynamics Next slide: The End Thank you for listening! Appendices