Relativistic Viscous Hydrodynamics for Multi-Component Systems in Heavy Ion Collisions

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 June 25th2010, La Londe-Les-Maures, France

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 A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Introduction Quark-Gluon Plasma (QGP) at Relativistic Heavy Ion Collisions RHIC experiments (2000-) Well-described in relativistic ideal hydrodynamic models Small discrepancies; non-equilibrium effects? LHC experiments (2009-) Asymptotic freedom -> Less strongly-coupled QGP? Relativistic viscous hydrodynamic models are the key A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Introduction Hydrodynamic modeling of heavy ion collisions for RHIC particles t Freezeout surface Hadronic cascade picture Hydro to particles Hydrodynamic picture QGP phase hadronic phase Pre- equilibrium Initial condition CGC/glasma picture? z 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 A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Introduction Elliptic flow coefficients from RHIC data Hirano et al. ( 09) Viscosity Ideal hydro Glauber 1storder Initial condition Eq. of state theoretical prediction experimental data Ideal hydro works well A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Introduction Elliptic flow coefficients from RHIC data Hirano et al. ( 09) Viscosity Ideal hydro Glauber 1storder Lattice-based Initial condition Eq. of state *EoS based on lattice QCD results theoretical prediction experimental data Ideal hydro works maybe A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Introduction Elliptic flow coefficients from RHIC data Hirano et al. ( 09) Viscosity Ideal hydro Glauber CGC 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 A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Introduction Formalism of viscous hydro is not settled yet: Form of viscous hydro equations Song & Heinz ( 08) 1. Fixing the equations is essential in fine- tuning viscosity from experimental data Treatment of conserved currents Low-energy ion collisions are planned at FAIR (GSI) & NICA (JINR) 2. Multiple conserved currents? 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 A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Introduction Categorization of relativistic systems 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 (-) QGP/hadronic gas at heavy ion collisions Cf. etc. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Overview START Energy-momentum conservation Charge conservations Law of increasing entropy Moment equations , Generalized Grad s moment method GOAL (constitutive eqs.) , , , Onsager reciprocal relations: satisfied A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

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 A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Relativistic Hydrodynamics Ideal hydrodynamics Unknowns 5+N Conservation laws 4+N + EoS(1) , , , , , Viscous hydrodynamics Additional unknowns 10+4N , , , , , Constitutive equations !? We derive the equations from the law of increasing entropy perturbation from equilibrium 0thorder theory 1storder theory 2ndorder theory ideal; no entropy production linear response; acausal relaxation effects; causal A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Second Order Theory Kinetic expressions with distribution function : : degeneracy : conserved charge number Conventional formalism 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 A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Extended Second Order Theory Moment equations New eqs. introduced Unknowns (10+4N) Moment eqs. (10+4N) , , All viscous quantities determined in arbitrary frame Expressions of and Determined through the 2ndlaw of thermodynamics where Off-equilibrium distribution is needed A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

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 matrices for dfi Semi-positive definite condition A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Results 2ndorder constitutive equations for systems with multi-components and multi-conserved currents Bulk pressure 1storder terms 2ndorder terms relaxation : relaxation times , : 1st, 2ndorder transport coefficients A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Results (Cont d) Energy current 1storder terms 2ndorder terms relaxation A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Results (Cont d) J-th charge current 1storder terms 2ndorder terms relaxation A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Onsager Cross Effects 1storder terms Dufour effect Vector Soret effect Cool down once for cooking tasty oden (Japanese pot-au-feu) Permeation of ingredients soup potato Thermal gradient Chemical diffusion caused by thermal gradient (Soret effect) It may well be important in our relativistic soup of quarks and gluons A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Results (Cont d) Shear stress tensor 1storder terms 2ndorder terms relaxation Our results in the limit of single component/conserved current Consistent with other results based on Baier et al. ( 08) AdS/CFT approach Renormalization group method Grad s 14-moment method Tsumura and Kunihiro ( 09) Betz et al. ( 09) A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Discussion 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, but vorticity terms need further checking as some are added in their recent revision A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

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 A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Summary and Outlook We formulated generalized 2ndorder 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 Onsager reciprocal relations ( 1storder theory) Justifies the moment expansion 1. 2. 3. Future prospects include applications to Hydrodynamic modeling of Quark-gluon plasma at relativistic heavy ion collisions Cosmological fluid etc A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

The End Thank you for listening! A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

The Law of Increasing Entropy Linear response theory : dissipative current : thermodynamic force : transport coefficient matrix (symmetric; semi-positive definite) Entropy production Theorem: Symmetric matrices can be diagonalized with orthogonal matrix A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Thermodynamic Stability Maximum entropy state condition - Stability condition (1storder) - Stability condition (2ndorder) Preserved for any *Stability conditions are NOT the same as the law of increasing entropy A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

First Order Limits Equilibrium limit 2ndorder constitutive equations thermodynamic forces (Navier-Stokes) transport coefficients (symmetric) thermodynamics forces (2ndorder) Onsager reciprocal relations are satisfied 1storder theory is recovered in the equilibrium limit A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Introduction Hydrodynamic modeling of heavy ion collisions for RHIC particles t Freezeout surface Hadronic cascade picture Hydro to particles Hydrodynamic picture QGP phase hadronic phase Pre- equilibrium Initial condition CGC/glasma picture? z Hydrodynamic model requires Initial condition Equation of state Glauber, color glass condensate, etc. 1storder phase transition, crossover(lattice), etc. Viscosity Ideal hydro, viscous hydro Hadronization JAM, UrQMD, etc. A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

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 A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Distortion of distribution Express in terms of dissipative currents 10+4N (macroscopic) self-consistency conditions , Fix and through 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 A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

Second Order Equations Entropy production Semi-positive definite condition Viscous distortion tensor & vector , Moment equations : symmetric, semi-positive definite matrices Constitutive equations Viscous distortion tensor & vector , Moment equations Dissipative currents , , , , , Matching matrices for dfi Semi-positive definite condition A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

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 A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

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 A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010

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 A power point template created by Akihiko Monnai Akihiko Monnai (The University of Tokyo) , Hot Quarks 2010, La Londe-Les-Maures, France, Jul. 25th2010