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Hydrodynamical Simulation of Relativistic Heavy Ion Collisions

Strongly Coupled Plasmas: Electromagnetic, Nuclear and Atomic. Hydrodynamical Simulation of Relativistic Heavy Ion Collisions. Tetsufumi Hirano. Introduction. Features of heavy ion collision at RHIC System of strongly interacting particles Quantum ChromoDynamics Quarks & Gluons / Hadrons

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Hydrodynamical Simulation of Relativistic Heavy Ion Collisions

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

  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 t Thermal freezeout Chemical freezeout QCD phase transtion (1st or crossover?) t = -z/c Thermalized matter QGP? t = z/c Chemical equilibration (Quark Gluon Plasma) Time scale 10 fm/c~10-23sec Temperature scale 100MeV/kB~1010K Local thermalization (Gluon Plasma) z:collision axis 0 Parton distribution function in colliding nuclei Gold nucleus Gold nucleus v~0.99c

  5. Dynamical Modeling Based on Hydrodynamics

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

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

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

  9. Main Ingredient: Equation of State One can test many kinds of EoS in hydrodynamics. Typical EoS in hydro model Lattice QCD simulations H: resonance gas(RG) Q: QGP+RG P.Kolb and U.Heinz(’03) F.Karsch et al. (’00) p=e/3 Latent heat Lattice QCD predicts cross over phase transition. Nevertheless, energy density explosively increases in the vicinity of Tc.  Looks like 1st order.

  10. Interface 1: Initial Condition • Need initial conditions (energy density, flow velocity,…) Initial time t0 ~ thermalization time Energy density distribution Rapidity distribution of produced charged hadrons Perpendicular to the collision axis Reaction plane (Note: Vertical axis represents expanding coordinate hs) T.H. and Y.Nara(’04) mean energy density ~5.5-6.0GeV/fm3 (Lorentz-contracted) nucleus

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

  12. Observable: Elliptic Flow

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

  14. y f x Elliptic Flow Ollitrault (’92) Response of the system to initial spatial anisotropy No secondary interaction Hydrodynamic behavior Input Spatial anisotropy e Interaction among produced particles 2v2 Output Momentum anisotropy v2 dN/df dN/df 0 f 2p 0 f 2p

  15. Elliptic Flow from a Parton Cascade Model Time evolution of v2 hydro limit View from collision axis b = 7.5fm • Gluons uniformly distributed • in the overlap region • dN/dy ~ 300 for b = 0 fm • Thermal distribution with • T = 500 MeV/kB Zhang et al.(’99) generated through secondary collisions saturated in the early stage sensitive to cross section (~viscosity) v2 is

  16. Comparison of Hydro Results with Experimental Data

  17. Particle Density Dependence of Elliptic Flow NA49(’03) Kolb, Sollfrank, Heinz (’00) • Dimension • 2D+boost inv. • EoS • QGP + hadrons (chem. eq.) • Decoupling • Sudden freezeout (response)=(output)/(input) • Hydrodynamic response is • const. v2/e ~ 0.2 @ RHIC • Exp. data reach hydrodynamic • limit at RHIC for the first time. Number density per unit transverse area Dawn of the hydro age?

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

  19. Particle Density Dependence of Elliptic Flow (contd.) Teaney, Lauret, Shuryak(’01) • Dimension • 2D+boost inv. • EoS • Parametrized by latent heat • (LH8, LH16, LH-infinity) • Hadrons • QGP+hadrons (chem. eq.) • Decoupling • Hybrid (Boltzmann eq.) • Deviation at lower energies can be filled by “viscosity” in hadron gases • Latent heat ~0.8 GeV/fm3 is favored.

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

  21. “Fine Structure” of v2: Transverse Momentum Dependence PHENIX(’03) STAR(’03) • Correct pT dependence • up to pT=1-1.5 GeV/c • Mass ordering • Deviation in small wave lengthregions •  Effects other than hydro Huovinen et al.(’01) • Dimension • 2D+boost inv. • EoS • QGP + RG (chem. eq.) • Decoupling • Sudden freezeout

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

  23. Summary, Discussion and Outlook • Large magnitude of v2, observed at RHIC, is consistent with hydrodynamic prediction. • Long wave length components obey hydrodynamics scaling. • Hybrid approach gives a good description (v2 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 Almost Perfect Fluid of quark-gluon matter Gas of Hadrons pre-thermalization? proper time t Thermalization time ~0.5-1.0fm/c Mean energy density ~5.5-6 GeV/fm3 @1fm/c Colliding nuclei “Latent heat” ~0.8 GeV/fm3

  25. BACKUP SLIDES

  26. “Coupling Parameter” S.Ichimaru et al.(’87) Plasma Physics G = (Average Coulomb Energy)/(Average Kinetic Energy) G =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 Tc M.H.Thoma (’04) C: Casimir (4/3 for quark or 3 for gluon) g: strong coupling constant T: Temperature d: Distance between partons

  27. Hydro or Boltzmann ? Molnar and Huovinen (’04) • Comparison between hydro and Boltzmann • Pure gluon system • Elastic scattering • (gggg) • Number conservation in hydro • Need to check more realistic model elastic cross section At the initial stage, interaction among gluons are so strong that many body correlation could be important. Almost perfect fluid? Knudsen number =(mean free path)/(typical size) ~10-4 @ t = 0.1 fm/c (~initial time) ~10-1 @ t = 10 fm/c (~final time)

  28. Discussion and Outlook

  29. Hydrodynamic Simulations for Viscous Fluids Non-relativistic case (Based on discussion by Cattaneo (1948)) Balance eq.: Constitutive eq.: Fourier’s law t 0 t : “relaxation time” Parabolic equation (heat equation) ACAUSAL!! (Similar difficulty is known in relativistic hydrodynamic equations.) finite t Hyperbolic equation (telegraph equation) No full 3D calculation yet. (D.Teaney, A.Muronga…)

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

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

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