Relativistic Viscous Hydrodynamics for Multi-Component Systems with Multiple Conserved Currents

Transcription

1 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 25 th 2010, La Londe-Les-Maures, France

2 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 Introduction Quark-Gluon Plasma (QGP) at Relativistic Heavy Ion Collisions Hadron phase QGP phase Tc ~0.2 T (GeV) 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

4 Introduction Hydrodynamic modeling of heavy ion collisions for RHIC particles t Freezeout surface Σ Hadronic cascade picture t Hydro to particles hadronic phase QGP phase Preequilibrium z Hydrodynamic picture Initial condition CGC/glasma picture? 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

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

6 Introduction Elliptic flow coefficients from RHIC data Hirano et al. ( 09) Viscosity Initial condition Eq. of state Ideal hydro Glauber Lattice-based 1 st order *EoS based on lattice QCD results theoretical prediction experimental data Ideal hydro works maybe

7 Introduction Elliptic flow coefficients from RHIC data Hirano et al. ( 09) Viscosity Initial condition Eq. of state Ideal hydro Glauber CGC Lattice-based *Gluons in fast nuclei may form color glass condensate (CGC) theoretical prediction experimental data Viscous hydro in QGP plays important role in reducing v 2

8 Introduction Formalism of viscous hydro is not settled yet: 1. Form of viscous hydro equations Fixing the equations is essential in finetuning viscosity from experimental data 2. Treatment of conserved currents Low-energy ion collisions are planned at FAIR (GSI) & NICA (JINR) Multiple conserved currents? 3. Treatment of multi-component systems Song & Heinz ( 08) # 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

9 Types of interactions Introduction Categorization of relativistic systems Number of components Single component with binary collisions Israel & Stewart ( 79), etc Single component with inelastic scatterings (-) Multi-components with binary collisions Prakash et al. ( 91) Multi-components with inelastic scatterings Monnai & Hirano ( 10) QGP/hadronic gas at heavy ion collisions Cf. etc.

10 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

11 Thermodynamic Quantities Tensor decompositions by flow where 2+N equilibrium quantities Energy density: Hydrostatic pressure: J-th charge density: *Stability conditions should be considered afterward is the projection operator 10+4N dissipative currents Energy density deviation: Bulk pressure: Energy current: Shear stress tensor: J-th charge density dev.: J-th charge current:

12 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 0 th order theory 1 st order theory 2 nd order theory ideal; no entropy production linear response; acausal relaxation effects; causal

13 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),,,,, frame fixing, stability conditions one-component, elastic scattering Not extendable for multi-component/conserved current systems

14 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 2 nd law of thermodynamics where Off-equilibrium distribution is needed

15 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 Dissipative currents,,,,, Viscous distortion tensor & vector, Moment equations Matching matrices for dfi Semi-positive definite condition

16 Results 2 nd order constitutive equations for systems with multi-components and multi-conserved currents Bulk pressure 1 st order terms 2 nd order terms relaxation : relaxation times, : 1 st, 2 nd order transport coefficients

17 Results (Cont d) Energy current 1 st order terms 2 nd order terms relaxation

18 Results (Cont d) J-th charge current 1 st order terms 2 nd order terms relaxation

19 Onsager Cross Effects 1 st order terms Vector Dufour effect Soret effect Cool down once for cooking tasty oden (Japanese pot-au-feu) Permeation of ingredients potato soup Thermal gradient Chemical diffusion caused by thermal gradient (Soret effect) It may well be important in our relativistic soup of quarks and gluons

20 Results (Cont d) Shear stress tensor 1 st order terms 2 nd order terms relaxation Our results in the limit of single component/conserved current Consistent with other results based on AdS/CFT approach Baier et al. ( 08) Renormalization group method Tsumura and Kunihiro ( 09) Grad s 14-moment method Betz et al. ( 09)

21 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

22 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 multicomponent/conserved current formalism is a natural extension

23 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 Quark-gluon plasma at relativistic heavy ion collisions Cosmological fluid etc

24 Thank you for listening! The End

25 Relativistic hydrodynamics Introduction 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

26 The Law of Increasing Entropy Linear response theory : dissipative current : thermodynamic force : transport coefficient matrix (symmetric; semi-positive definite) Entropy production Theorem: Symmetric matrices orthogonal matrix can be diagonalized with

27 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

28 First Order Limits 2 nd order constitutive equations Equilibrium limit transport coefficients (symmetric) thermodynamics forces (2 nd order) thermodynamic forces (Navier-Stokes) Onsager reciprocal relations are satisfied 1 st order theory is recovered in the equilibrium limit