Causal Viscous Hydrodynamics for RelaBvisBc Systems with MulB Components and MulB Conserved Currents

Transcription

1 Reference: AM and T. Hirano, arxiv:1003:3087 Causal Viscous Hydrodynamics for RelaBvisBc Systems with MulB Components and MulB Conserved Currents Akihiko Monnai Department of Physics, The University of Tokyo Collaborator: Tetsufumi Hirano Strong and Electroweak Ma1er 2010 June 29 th 2010, McGill University, Montreal, Canada

2 Outline 1. IntroducBon RelaBvisBc hydrodynamics and Heavy ion collisions 2. FormulaBon of Viscous Hydro Israel Stewart theory for mulb component/conserved current systems 3. Results and Discussion ConsBtuBve equabons and their implicabons 4. Summary Summary and Outlook

3 IntroducBon Quark Gluon Plasma (QGP) at RelaBvisBc Heavy Ion Collisions Hadron phase QGP phase Tc ~0.2 T (GeV) RHIC experiments (2000 ) Well described in relabvisbc ideal hydrodynamic models Small discrepancies; non equilibrium effects? LHC experiments (2009 ) AsymptoBc freedom > Less strongly coupled QGP? RelaBvisBc viscous hydrodynamic models are the key

4 IntroducBon Hydrodynamic modeling of heavy ion collisions parbcles hadronic phase t QGP phase Freezeout surface Σ Preequilibrium z Hadronic cascade picture Hydro to parbcles Hydrodynamic picture IniBal condibon CGC/glasma picture? Hydrodynamics works at the intermediate stage (~1 10 fm/c) Viscous hydro helps us 1. Explain the space Bme evolubon of the QGP 2. Extract viscosibes of the QGP from experimental data

5 IntroducBon Fourier analyses of parbcle spectra from RHIC data Hirano et al. ( 09) Viscosity Ini;al condi;on Eq. of state Ideal hydro Glauber 1 st order theorebcal predicbon experimental data Ideal hydro works well

6 IntroducBon Fourier analyses of parbcle spectra from RHIC data Hirano et al. ( 09) Viscosity Ini;al condi;on Eq. of state Ideal hydro Glauber Lajce based 1 st order *EoS based on lajce QCD results theorebcal predicbon experimental data Ideal hydro works maybe

7 IntroducBon Fourier analyses of parbcle spectra from RHIC data Hirano et al. ( 09) Viscosity Ini;al condi;on Eq. of state Ideal hydro Glauber CGC Lajce based *Gluons in fast nuclei may form color glass condensate (CGC) theorebcal predicbon experimental data Viscous hydro in QGP plays important role in reducing v 2

8 IntroducBon Formalism of viscous hydro is not seuled yet: 1. Form of viscous hydro equabons Important in fine tuning viscosity from experimental data 3. Treatment of conserved currents Low energy ion collisions are planned at FAIR (GSI) & NICA (JINR) MulBple conserved currents? 4. Treatment of mulb component systems # of conserved currents baryon number, strangeness, etc. # of parbcle species pion, proton, quarks, gluons, etc. We need to construct a firm framework of viscous hydro

9 IntroducBon CategorizaBon of relabvisbc systems Number of components Types of interac;ons Single component with binary collisions Israel & Stewart ( 79), etc Single component with inelasbc scauerings ( ) MulB components with binary collisions Prakash et al. ( 91) MulB components with inelasbc scauerings Monnai & Hirano ( 10) Cf. QGP/hadronic gas at relabvisbc heavy ion collisions etc.

10 RelaBvisBc hydrodynamics IntroducBon Macroscopic theory defined on (3+1) D spacebme Flow (vector field) Temperature (scalar field) Chemical potenbals (scalar fields) and are described by the macroscopic fields Gradient in the fields: thermodynamic force Response to the gradients: dissipabve current

11 Overview START Energy momentum conservabon Charge conservabons Law of increasing entropy Moment equabons, Generalized Grad s moment method GOAL (consbtubve eqs.),,, Onsager reciprocal relabons: sa6sfied

12 Thermodynamic QuanBBes Tensor decomposibons by flow where is the projecbon operator 2+N equilibrium quanbbes Energy density: HydrostaBc pressure: J th charge density: *Stability condibons should be considered aterward 10+4N dissipabve currents Energy density deviabon: Bulk pressure: Energy current: Shear stress tensor: J th charge density dev.: J th charge current:

13 RelaBvisBc Hydrodynamics Ideal hydrodynamics Unknowns (5+N),,, ConservaBon laws (4+N) + EoS (1),, Viscous hydrodynamics AddiBonal unknowns (10+4N),,,,, ConsBtuBve equabons!? We derive the equabons from the law of increasing entropy perturbabon from equilibrium 0 th order theory 1 st order theory 2 nd order theory ideal; no entropy producbon linear response; acausal relaxabon effects; causal

14 First Order Theory KineBc expressions with distribubon funcbon : : degeneracy : conserved charge number The law of increasing entropy (1st order)

15 First Order Theory Linear response theory Scalar convenbonal terms cross terms Vector Tensor The cross terms are symmetric due to Onsager reciprocal relabons

16 First Order Theory Linear response theory Dufour effect Soret effect Cool down once for cooking tasty oden (Japanese pot au feu) Vector PermeaBon of ingredients potato soup Thermal gradient Chemical diffusion caused by thermal gradient (Soret effect)

17 Causality Issues Second Order Theory Linear response theory implies instantaneous propagabon RelaxaBon effects are necessary for causality ConvenBonal formalism Israel & Stewart ( 79) DissipaBve currents (14) (9),,,,, frame fixing, stability condibons? Moment equabons (10) (9) one component, elasbc scauering Not extendable for mulb component/conserved current systems

18 Extended Second Order Theory Moment equabons New eqs. introduced Unknowns (10+4N) Moment eqs. (10+4N),, All viscous quanbbes determined in arbitrary frame Expressions of and Determined through the 2 nd law of thermodynamics where Off equilibrium distribu6on is needed

19 Extended Second Order Theory Moment expansion *Grad s 14 moment method extended for mulb conserved current systems so that it is consistent with Onsager reciprocal relabons 10+4N unknowns, are determined in self consistency condibons The entropy producbon is expressed in terms of and DissipaBve currents,,,,, Viscous distorbon tensor & vector, Moment equabons Matching matrices for dfi Semi posibve definite condibon

20 Results 2 nd order consbtubve equabons for systems with mulb components and mulb conserved currents Bulk pressure 1 st order terms 2 nd order terms relaxabon : relaxabon Bmes, : 1 st, 2 nd order transport coefficients

21 Results (Cont d) Energy current Dufour effect 1 st order terms 2 nd order terms relaxabon

22 Results (Cont d) J th charge current Soret effect 1 st order terms 2 nd order terms relaxabon

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

24 Discussion Comparison with AdS/CFT+phenomenological approach Baier et al. ( 08) Our approach goes beyond the limit of conformal theory VorBcity vorbcity terms do not appear in kinebc theory Comparison with RenormalizaBon group approach Tsumura & Kunihiro ( 09) Consistent, but vorbcity terms need further checking as some are added in their recent revision

25 Discussion Comparison with Grad s 14 moment approach Betz et al. ( 09) The form of their equabons are consistent with that of ours Our method can deal with mulbple conserved currents while Grad s 14 moment method does not Consistency with other approaches suggest our mulbcomponent/conserved current formalism is a natural extension

26 Summary and Outlook We formulated generalized 2 nd order theory from the entropy producbon w/o violabng causality 1. MulB component systems with mulbple conserved currents InelasBc scauering (e.g. pair creabon/annihilabon) implied 2. Frame independent Independent equabons for energy and charge currents 3. Onsager reciprocal relabons ( 1 st order theory) JusBfies the moment expansion Future prospects include applicabons to Hydrodynamic modeling of the Quark gluon plasma at heavy ion collisions Cosmological fluid etc

27 Thank you for listening! The End

28 The Law of Increasing Entropy Linear response theory : dissipabve current Entropy producbon : thermodynamic force : transport coefficient matrix (symmetric; semi posibve definite) Theorem: Symmetric matrices can be diagonalized with orthogonal matrix

29 Thermodynamic Stability Maximum entropy state condibon Stability condibon (1 st order) Stability condibon (2 nd order) Preserved for any *Stability condibons are NOT the same as the law of increasing entropy

30 First Order Limits 2 nd order consbtubve equabons Equilibrium limit transport coefficients thermodynamics forces (symmetric) (2 nd order) thermodynamic forces (Navier Stokes) Onsager reciprocal relabons are sabsfied 1 st order theory is recovered in the equilibrium limit

31 DistorBon of distribubon Express in terms of dissipabve currents 10+4N (macroscopic) self consistency condibons, Fix and through matching Moment expansion with 10+4N unknowns, *Grad s 14 moment method extended for mulb conserved current systems (Consistent with Onsager reciprocal relabons) DissipaBve currents,,,,, Viscous distorbon tensor & vector, : Matching matrices

32 Second Order EquaBons Entropy producbon Semi posibve definite condibon Viscous distorbon tensor & vector, Moment equabons ConsBtuBve equabons : symmetric, semi posi6ve definite matrices DissipaBve currents,,,,, Viscous distorbon tensor & vector, Moment equabons Matching matrices for dfi Semi posibve definite condibon

33 Discussion Expanding universe Einstein equabon with Robertson Walker metric in flat space Only ideal hydrodynamic energy momentum tensor is allowed: because the R W metric assumes isotropic and homogeneous expansion. Viscous correcbon to the pressure (= bulk pressure ) is allowed Bulk viscous cosmological model