Viscous Hydrodynamics in Relativistic Heavy Ion Collisions

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 December 10th2010, Yonsei University, Korea

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

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) Introduction Quark-gluon plasma (QGP) at relativistic heavy ion collisions Hadron phase (crossover) QGP phase sQGP (wQGP?) RHIC experiments (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 Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Introduction

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) Introduction Hydrodynamic modeling of RHIC particles t Freezeout Hadronic cascade picture Hydro to particles Hydrodynamic picture QGP phase hadronic phase Pre- equilibrium Initial condition CGC/glasma picture? z 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 Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Introduction

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) 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 Na ve 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 Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Introduction

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

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

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) Introduction Elliptic flow coefficients from RHIC data Hirano et al. ( 09) Viscosity Ideal hydro Glauber Color glass condensate Initial cond. Lattice-based Eq. of state *Gluons in fast moving nuclei are saturated to CGC theoretical prediction > experimental data Viscosity in QGP phase plays important role in reducing v2 Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Introduction

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) Introduction Why viscous hydrodynamic models? RHIC experiments (2000-) Success of ideal hydro Necessity of viscous hydro for improved inputs to the hydrodynamic models LHC experiments (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 Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Introduction

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) Introduction Viscous hydrodynamics needs improvement Form of dissipative hydro equations Song & Heinz ( 08) 1. Fixing the equations is essential in fine- tuning viscosity from experimental data Treatment of conserved currents 2. Low-energy ion collisions are planned at FAIR (GSI) & NICA (JINR) Only 1 conserved current can be treated Treatment of multi-component systems 3. # 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 Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Introduction

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) Introduction Categorization of relativistic hydrodynamic formalisms Number of components Single component with binary collisions Israel & Stewart ( 79), etc Multi-components with binary collisions Prakash et al. ( 91) Types of interactions Multi-components with inelastic scatterings (-) Single component with inelastic scatterings (-) Required for QGP/hadron gas at heavy ion collisions Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Overview

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) Introduction Categorization of relativistic hydrodynamic formalisms Number of components Single component with binary collisions Israel & Stewart ( 79), etc Multi-components with binary collisions Prakash et al. ( 91) Types of interactions Multi-components with inelastic scatterings Monnai & Hirano ( 10) Single component with inelastic scatterings (-) Required for QGP/hadron gas at heavy ion collisions In this work we formulate relativistic dissipative hydro for multi-component + multi-conserved current systems Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Overview

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

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

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

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) Relativistic Hydrodynamics Ideal hydrodynamics Unknowns (5+N) Conservation laws (4+N) + EoS(1) , , , , , Dissipative hydrodynamics ( perturbation from equilibrium) 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 Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Moment Equations

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) 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 matrices for dfi Semi-positive definite condition Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Constitutive Equations

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

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

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

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

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) 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 soup potato thermal gradient Chemical diffusion via thermal gradient Soret effect It should play an important role in our quark soup Discussion 2ndorder terms Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) Discussion 2ndorder terms Comparison with AdS/CFT+phenomenological approach Baier et al. ( 08) Our approach goes beyond the limit of conformal theory Vorticity-vorticity terms do not appear in kinetic theory Comparison with Renormalization group approach Tsumura & Kunihiro ( 09) Consistent, as vorticity terms are added in their recent revision Frame-dependent equations Discussion 2ndorder terms Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) Discussion 2ndorder 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 2ndorder theory to multi-component + conserved current systems Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea Summary and Outlook

Viscous Hydrodynamics Akihiko Monnai (The University of Tokyo) Summary and Outlook We formulated generalized 2ndorder 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 1storder cross terms are present Onsager reciprocal relations are satisfied Frame independent Independent equations for energy and charge currents 1. 2. 3. Future prospects include applications to Numerical estimation of viscous hydrodynamic models for relativistic heavy ion collisions AM & T. Hirano, in preparation Cosmological fluid, and more Next slide: Heavy Ion Meeting 2010-12, December 10, Yonsei University, Korea

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