1. Hydrodynamical Simulation of Relativistic Heavy Ion Collisions Tetsufumi Hirano Strongly Coupled Plasmas: Electromagnetic, Nuclear and Atomic

2. Introduction Features of heavy ion collision at RHIC –System of strongly interacting particles Quantum ChromoDynamics Quarks & Gluons / Hadrons –“Phase transition” from Quark Gluon Plasma to hadrons –Dynamically evolving system –Transient state (life time ~ 10 fm/c ~ 10 -23 sec) –No heat bath. Control parameters: collision energy and the size of nucleus. –The number of observed hadrons ~ <5000 –“Impact parameter” can be used to categorize events through the number of observed hadrons.

3. Introduction (contd.) Need dynamical modeling of heavy ion collisions  How? –Local thermal equilibrium? Non-equilibrium? –Fluids (hydrodynamics)? Gases (Boltzmann)? –Perfect? Viscous? Lots of “stages” in collision (next slide) –Ultimate purpose: Dynamical description of the whole stage –Current status: Description of “intermediate stage” based on hydrodynamics

4. Space-Time Evolution of Relativistic Heavy Ion Collisions Parton distribution function in colliding nuclei Local thermalization (Gluon Plasma) Chemical equilibration (Quark Gluon Plasma) QCD phase transtion (1 st or crossover?) Chemical freezeout Thermal freezeout Gold nucleus Gold nucleus v~0.99c 0 z:collision axis t Time scale 10 fm/c~10 -23 sec Temperature scale 100MeV/k B ~10 10 K t = z/c t = -z/c Thermalized matter QGP?

5. Dynamical Modeling Based on Hydrodynamics

6. Rapidity and Boost Invariant Ansatz z t 0 midrapidity:y=0 forward rapidity y>0 y=infinity Rapidity as a “relativistic velocity” Boost invariant ansatz Bjorken (’83)  Dynamics depends on , not on  s.  =const.  s =const. t, z

7. Hydrodynamic Equations for a Perfect Fluid Baryon number Energy Momentum e : energy density,P : pressure, : four velocity

8. Inputs for Hydrodynamic Simulations Final stage: Free streaming particles  Need decoupling prescription Intermediate stage: Hydrodynamics can be valid as far as local thermalization is achieved.  Need EoS P(e,n) Initial stage: Particle production, pre-thermalization, instability?  Instead, initial conditions for hydro simulations Need modeling (1) EoS, (2) Initial cond., and (3) Decoupling 0 z t

9. Main Ingredient: Equation of State Latent heat One can test many kinds of EoS in hydrodynamics. Lattice QCD predicts cross over phase transition. Nevertheless, energy density explosively increases in the vicinity of T c.  Looks like 1 st order. Lattice QCD simulations Typical EoS in hydro model H: resonance gas(RG) p=e/3 Q: QGP+RG F.Karsch et al. (’00) P.Kolb and U.Heinz(’03)

10. Interface 1: Initial Condition Need initial conditions (energy density, flow velocity,…) Initial time  0 ~ thermalization time Perpendicular to the collision axis Reaction plane (Note: Vertical axis represents expanding coordinate  s ) Energy density distribution Rapidity distribution of produced charged hadrons (Lorentz-contracted) nucleus T.H. and Y.Nara(’04) mean energy density ~5.5-6.0GeV/fm 3

11. Interface 2: Freezeout (1) Sudden freezeout(2) Transport of hadrons via Boltzman eq. (hybrid) Continuum approximation no longer valid at the late stage  Molecular dynamic approach for hadrons ( ,K,p,…) 0 z t 0 z t At T=T f, =0 (ideal fluid)  =infinity (free stream) T=TfT=Tf QGP fluid Hadron fluid QGP fluid

12. Observable: Elliptic Flow

13. Anisotropic Flow in Atomic Physics Fermionic 6 Li atoms in an optical trap Interaction strength controlled via Feshbach resonance Releasing the “cloud” from the trap Superfluid? Or collisional hydrodynamics? How can we “see” anisotropic flow in heavy ion collisions? K.M.O’Hara et al., Science298(2002)2179

14. Elliptic Flow Response of the system to initial spatial anisotropy Ollitrault (’92) Hydrodynamic behavior Spatial anisotropy  Momentum anisotropy v 2 Input Output Interaction among produced particles dN/d   No secondary interaction 0 22 dN/d   0 22 2v22v2 x y 

15. Elliptic Flow from a Parton Cascade Model b = 7.5fm Time evolution of v 2 generated through secondary collisions saturated in the early stage sensitive to cross section (~viscosity) Gluons uniformly distributed in the overlap region dN/dy ~ 300 for b = 0 fm Thermal distribution with T = 500 MeV/k B v 2 is Zhang et al.(’99) View from collision axis hydro limit

16. Comparison of Hydro Results with Experimental Data

17. Particle Density Dependence of Elliptic Flow Hydrodynamic response is const. v 2 /  ~ 0.2 @ RHIC Exp. data reach hydrodynamic limit at RHIC for the first time. (response)=(output)/(input) Number density per unit transverse area Dimension 2D+boost inv. EoS QGP + hadrons (chem. eq.) Decoupling Sudden freezeout NA49(’03) Kolb, Sollfrank, Heinz (’00) Dawn of the hydro age?

18. “Wave Length” Dependence Short wave length Long wave length Dimension Full 3D (  s coordinate) EoS QGP + hadrons (chem. frozen) Decoupling Sudden freezeout T.H.(’04) particle density low high spatial anisotropy large small Long wave length components (small transverse momentum) obey “hydrodynamics scaling” Short wave length components (large transverse momentum) deviate from hydro scaling. (response)=(output)/(input)

19. Particle Density Dependence of Elliptic Flow (contd.) Dimension 2D+boost inv. EoS Parametrized by latent heat (LH8, LH16, LH-infinity) Hadrons QGP+hadrons (chem. eq.) Decoupling Hybrid (Boltzmann eq.) Teaney, Lauret, Shuryak(’01) Deviation at lower energies can be filled by “viscosity” in hadron gases Latent heat ~0.8 GeV/fm 3 is favored.

20. Rapidity Dependence of Elliptic Flow Dimension Full 3D (  s coordinate) EoS 1.QGP + hadrons (chem. eq.) 2.QGP + hadrons (chem. frozen) Decoupling Sudden freezeout Density  low  Deviation from hydro Forward rapidity at RHIC ~ Midrapidity at SPS? Heinz and Kolb (’04) T.H. and K.Tsuda(’02)

21. “Fine Structure” of v 2 : Transverse Momentum Dependence Dimension 2D+boost inv. EoS QGP + RG (chem. eq.) Decoupling Sudden freezeout PHENIX(’03) Correct p T dependence up to p T =1-1.5 GeV/c Mass ordering Deviation in small wave length regions  Effects other than hydro Huovinen et al.(’01) STAR(’03)

22. Viscous Effect on Distribution Parametrization of hydro field + dist. fn. with viscous correction 1 st order correction to dist. fn.: : Sound attenuation length : Tensor part of thermodynamic force Reynolds number in boost invariant scaling flow Nearly perfect fluid !? D.Teaney(’03) G.Baym(’84)

23. Summary, Discussion and Outlook Large magnitude of v 2, observed at RHIC, is consistent with hydrodynamic prediction. Long wave length components obey hydrodynamics scaling. Hybrid approach gives a good description (v 2 at midrapidity, mass splitting, density dependence). –Ideal hydro for the QGP “liquid” –Molecular dynamics for the hadron “gas” No full 3D {hybrid, viscous} hydro model yet.

24. Summary: A Probable Scenario Colliding nuclei proper time t Almost Perfect Fluid of quark-gluon matter pre-thermalization? Thermalization time ~0.5-1.0fm/c Mean energy density ~5.5-6 GeV/fm 3 @1fm/c “Latent heat” ~0.8 GeV/fm 3 Gas of Hadrons

25. BACKUP SLIDES

26. “Coupling Parameter” S.Ichimaru et al.(’87)  (Average Coulomb Energy)/(Average Kinetic Energy) Plasma Physics  =O(10 -4 ) for laser plasma O(0.1) for interior of Sun O(50) for interior of Jupiter O(100) for white dwarf Quark Gluon Plasma near T c C: Casimir (4/3 for quark or 3 for gluon) g: strong coupling constant T: Temperature d: Distance between partons M.H.Thoma (’04)

27. Hydro or Boltzmann ? Molnar and Huovinen (’04) elastic cross section At the initial stage, interaction among gluons are so strong that many body correlation could be important.  Almost perfect fluid? Comparison between hydro and Boltzmann Pure gluon system Elastic scattering (gg  gg) Number conservation in hydro Need to check more realistic model Knudsen number =(mean free path)/(typical size) ~10 -4 @  = 0.1 fm/c (~initial time) ~10 -1 @  = 10 fm/c (~final time)

28. DiscussionandOutlook

29. Hydrodynamic Simulations for Viscous Fluids Non-relativistic case (Based on discussion by Cattaneo (1948)) Fourier’s law  : “relaxation time” Parabolic equation (heat equation)  ACAUSAL!! (Similar difficulty is known in relativistic hydrodynamic equations.) finite  Hyperbolic equation (telegraph equation) No full 3D calculation yet. (D.Teaney, A.Muronga…) Balance eq.: Constitutive eq.:   0

30. Hydro + Rate Eq. in the QGP phase Including gg  qqbar and gg  ggg Collision term: T.S.Biro et al.,Phys.Rev.C48(’93)1275. Assuming “multiplicative” fugacity, EoS is unchanged.

31. 2 nd order formula… 14 equations… 1 st order2 nd order How obtain additional equations? In order to ensure the second law of thermodynamics, one can choose Balance eqs. Constitutive eqs.