Viscous Hydrodynamics

1. AM and T. Hirano, Nucl. Phys. A 847 (2010) 283 Viscous Hydrodynamics Akihiko Monnai Department of Physics, The University of Tokyo Collaborator: Tetsufumi Hirano • Heavy Ion Meeting 2010-12 • December10th 2010, Yonsei University, Korea

2. Outline • Introduction Relativistic hydrodynamics and heavy ion collisions • Relativistic Dissipative Hydrodynamics Extended Israel-Stewart theory from law of increasing entropy • Results and Discussion Constitutive equations in multi-component/conserved current systems • Summary Summary and Outlook Introduction

3. Introduction • Quark-gluon plasma (QGP) at relativistic heavy ion collisions Hadron phase (crossover) QGPphase sQGP (wQGP?) RHICexperiments (2000-) Discovery of QGP as nearly perfect fluid Thermodynamics is at work in QGP at RHIC energies Non-equilibrium effects need investigation for quantitative understandings Introduction

4. Introduction • Hydrodynamic modeling of RHIC particles ｔ t Freezeout Hadronic cascade picture Hydro to particles QGPphase hadronic phase Hydrodynamic picture Pre-equilibrium Initial condition z CGC/glasma picture? • Intermediate stage (~1-10 fm) is described by hydrodynamics • Results are dependent on the inputs: Equation of state, Transport coefficients Initial conditions Hydrodynamic equations Output Introduction

5. 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 Naïve interpretation of dissipative processes Shear viscosity =response to deformation Bulk viscosity = response to expansion Energy dissipation =response to thermal gradient Charge dissipation =response to chemical gradients Introduction

6. Introduction • Elliptic flow coefficients from RHIC data Hirano et al. (‘09) Ideal hydro Viscosity Glauber Initial cond. 1st order Eq. of state theoretical prediction ~ experimental data Ideal hydro works well Introduction

7. Introduction • Elliptic flow coefficients from RHIC data Hirano et al. (‘09) Ideal hydro Viscosity Glauber Initial cond. 1st order Lattice-based Eq. of state theoretical prediction > experimental data Ideal hydro shows slight overshooting Introduction

8. Introduction • Elliptic flow coefficients from RHIC data Hirano et al. (‘09) Ideal hydro Viscosity Color glass condensate Glauber Initial cond. Lattice-based Eq. of state *Gluons in fast moving nuclei are saturated to CGC theoretical prediction > experimental data Viscosityin QGP phase plays important role in reducing v2 Introduction

9. Introduction • Why viscous hydrodynamic models? RHICexperiments (2000-) Success of ideal hydro Necessity of viscous hydro for improved inputs to the hydrodynamic models LHCexperiments (2010-) Viscous hydro? Asymptotic freedom in QCD QGP might become less-strongly coupled 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. Introduction • Viscous hydrodynamics needs improvement • Form of dissipative 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) Only 1 conserved current can be treated # 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 Introduction

11. Introduction • Categorization of relativistic hydrodynamic formalisms Required for QGP/hadron gas at heavy ion collisions Overview

12. Introduction • Categorization of relativistic hydrodynamic formalisms 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. Overview • Formulation of relativistic dissipative hydrodynamics START Energy-momentum conservation Charge conservations Law of increasing entropy Generalized moment method Moment eqs. , GOAL EoM for dissipative currents , , , Onsager reciprocal relationssatisfied Thermodynamics Quantities

14. 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 Thermodynamic Stability

15. Thermodynamic Stability • Maximum entropy state condition - Stability condition (1st order) - Stability condition (2nd order) automatically satisfied in kinetic theory *Stability conditions are NOT the same as the law of increasing entropy Relativistic Hydrodynamics

16. Relativistic Hydrodynamics • Ideal hydrodynamics • Dissipative hydrodynamics (“perturbation” from equilibrium) Unknowns(5+N) Conservation laws (4+N) + EoS(1) , , , , , Additional unknowns (10+4N) Moment equations (10+4N) !? , , , , , , Defined in relativistic kinetic theory as New equations in our work Estimated from the law of increasing entropy Moment Equations

17. Moment Equations • Introduce distortion of distribution *Grad’s moment method extended to multi-conserved current systems 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 Constitutive Equations

18. Constitutive Equations • Bulk Pressure response to expansion cross terms (linear) 2nd order corrections relaxation term Cross termsappear (reciprocal relations) 2nd order termsin full form (multi-conserved currents) Relaxation termappears (causality is preserved) Constitutive Equations

19. Constitutive Equations • Energy current response to temperature gradient cross terms (linear) 2nd order corrections relaxation term Constitutive Equations

20. Constitutive Equations • Charge currents response to chemical gradient (+ cross terms) cross terms (linear) 2nd order corrections relaxation terms Constitutive Equations

21. Constitutive Equations • Shear stress tensor • Discussion - Relaxation term response to deformation 2nd order corrections relaxation term Hiscock & Lindblom (‘85) Linear response Acausal and unstable in relativistic systems Relaxation effect to limit the propagation faster than the light speed Discussion – Cross terms

22. Discussion - Cross terms • Coupling of thermodynamic forces in the dissipative currents ex. Onsager reciprocal relations （　　　　　　　　　　）is satisfied • Cf: “Cooling” process for cooking tasty oden (Japanese soup) ingreadients potato soup thermal gradient Chemical diffusion via thermal gradient Soret effect It should play an important role in our “quark soup” Discussion – 2nd order terms

23. Discussion – 2nd order terms • 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, as vorticity terms are added in their recent revision • Frame-dependent equations Discussion – 2nd order terms

24. Discussion – 2nd order terms • 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 Consistencies suggest we have successfully extended 2nd order theory to multi-component + conserved current systems Summary and Outlook

25. Summary and Outlook • We formulated generalized 2nd order dissipative hydro from the entropy production w/o violating causality • Multi-component systems with multiple conserved currents Inelastic scattering (e.g. pair creation/annihilation) included • 1st order cross terms are present Onsager reciprocal relations are satisfied • 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. The End • Thank you for listening! Appendices