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Relativistic Ideal and Viscous Hydrodynamics

Intensive Lecture YITP, December 10th, 2008. Relativistic Ideal and Viscous Hydrodynamics. Tetsufumi Hirano Department of Physics The University of Tokyo. TH, N. van der Kolk, A. Bilandzic, arXiv:0808.2684[nucl-th]; to be published in “Springer Lecture Note in Physics”.

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Relativistic Ideal and Viscous Hydrodynamics

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  1. Intensive Lecture YITP, December 10th, 2008 Relativistic Ideal and Viscous Hydrodynamics Tetsufumi Hirano Department of Physics The University of Tokyo TH, N. van der Kolk, A. Bilandzic, arXiv:0808.2684[nucl-th]; to be published in “Springer Lecture Note in Physics”.

  2. Plan of this Lecture 1st Day • Hydrodynamics in heavy ion collisions • Collective flow • Dynamical modeling of heavy ion collisions (seminar) 2nd Day • Formalism of relativistic ideal/viscous hydrodynamics • Bjorken’s scaling solution with viscosity • Effect of viscosity on particle spectra (discussion)

  3. PART 1 Hydrodynamics in Heavy Ion Collisions

  4. Why Hydrodynamics? • Static • Quark gluon plasma • under equilibrium • Equation of states • Transport coefficients • etc Energy-momentum: Conserved number: • Dynamics • Expansion, Flow • Space-time evolution of • thermodynamic variables • Local thermalization • Equation of states

  5. Longitudinal Expansion in Heavy Ion Collisions Freezeout “Re-confinement” Expansion, cooling Thermalization First contact (two bunches of gluons)

  6. Strategy: Bottom-Up Approach • The first principle (QuantumChromo Dynamics) • Inputs to phenomenology (lattice QCD) • Complexity • Non-linear interactions of gluons • Strong coupling • Dynamical many body system • Color confinement • Phenomenology (hydrodynamics) • Experimental data • @ Relativistic Heavy Ion Collider • ~200 papers from 4 collaborations • since 2000

  7. Application of Hydro Results Thermal radiation (photon/dilepton) Jet quenching J/psi suppression Heavy quark diffusion Recombination Coalescence Meson J/psi c Baryon c bar Information along a path Information on surface Information inside medium

  8. Why Hydrodynamics? • Goal: To understand the hot QCD matter under equilibrium. • Lattice QCD is not able to describe dynamics in heavy ion collisions. • Analyze heavy ion reaction based on a model with an assumption of local equilibrium, and see what happens and whether it is consistent with data. • If consistent, it would be a starting point of the physics of hot QCD matter under equilibrium.

  9. Plan of this Lecture 1st Day • Hydrodynamics in Heavy Ion Collisions • Collective flow • Dynamical Modeling of heavy ion collisions (seminar) 2nd Day • Formalism of relativistic ideal/viscous hydrodynamics • Bjorken’s scaling solution with viscosity • Effect of viscosity on particle spectra (discussion)

  10. PART 2 Collective Flow

  11. Sufficient Energy Density? Bjorken(’83) Bjorken energy density total energy (observables) t: proper time y: rapidity R: effective transverse radius mT: transverse mass

  12. Critical Energy Density from Lattice Adopted from Karsch(PANIC05) Note that recent results seem to be Tc~190MeV.

  13. Centrality Dependence of Energy Density Well above ec from lattice in central collision at RHIC, if assuming t=1fm/c. ec from lattice PHENIX(’05) STAR(’08)

  14. Caveats (I) • Just a necessary condition in the sense that temperature (or pressure) is not measured. • How to estimate tau? • If the system is thermalized, the actual energy density is larger due to pdV work. • Boost invariant? • Averaged over transverse area. Effect of thickness? How to estimate area? Gyulassy, Matsui(’84) Ruuskanen(’84)

  15. Matter in (Chemical) Equilibrium? direct Resonance decay Two fitting parameters: Tch, mB

  16. Amazing Fit! T=177MeV, mB = 29 MeV Close to Tc from lattice

  17. Caveats (II) • Even e+e- or pp data can be fitted well! See, e.g., Becattini&Heinz(’97) • What is the meaning of fitting parameters? See, e.g., Rischke(’02),Koch(’03) • Why so close to Tc? • No chemical eq. in hadron phase!? • Essentially dynamical problem! Expansion rate  Scattering rate see, e.g., U.Heinz, nucl-th/0407067

  18. Recent example Just a fitting parameter? Where is the region in which we can believe these results as “temperature” and “chemical potential”. STAR, 0808.2041[nucl-ex]

  19. Matter in (Kinetic) Equilibrium? Kinetically equilibrated matter at rest Kinetically equilibrated matter at finite velocity um py py px px Lorentz-boosted distribution Isotropic distribution

  20. Radial Flow Kinetic equilibrium inside matter Blast wave model (thermal+boost) e.g. Sollfrank et al.(’93) Pressure gradient  Driving force of flow  Flow vector points to radial direction

  21. Spectral change is seen in AA! Power law in pp & dAu Adopted from O.Barannikova, (QM05) Convex to Power law in Au+Au • “Consistent” with thermal + boost picture • Large pressure could be built up in AA collisions

  22. Caveats (III) • Flow reaches 50-60% of speed of light!? • Radial flow even in pp? • How does freezeout happen dynamically? STAR, white paper(’05)

  23. Basic Checks  Necessary Conditions to Study the QGP at RHIC • Energy density can be well above ec. • Thermalized? • “Temperature” can be extracted. • Why freezeout happens so close to Tc? • High pressure can be built up. • Completely equilibrated? Importance of systematic study based on dynamical framework

  24. Anisotropic Transverse Flow y f z x x Transverse Plane (perpendicular to collision axis) Reaction Plane Poskanzer & Voloshin (’98)

  25. Directed and Elliptic Flow The 1st mode, v1 directed flow coefficient The 2nd mode, v2 elliptic flow coefficient y z x x • Important in low • energy collisions • Vanish at midrapidity • Important in high energy collisions

  26. Ollitrault (’92) What is Elliptic Flow?--How does the system respond to spatial anisotropy?-- No secondary interaction Hydro behavior y f x INPUT Spatial Anisotropy 2v2 Interaction among produced particles dN/df dN/df OUTPUT Momentum Anisotropy 0 f 2p 0 f 2p

  27. Eccentricity: Spatial Anisotropy y x In hydrodynamics, :Energy density or :Entropy density

  28. Eccentricity Fluctuation Adopted from D.Hofman(PHOBOS), talk at QM2006 A sample event from Monte Carlo Glauber model Interaction points of participants vary event by event. Apparent reaction plane also varies.  The effect is relatively large for smaller system such as Cu+Cu collisions

  29. Kolb and Heinz (’03) Elliptic Flow in Hydro Saturate in first several femto-meters v2 signal is sensitive to initial stage. Response of the system (= v2/e) is almost constant. Pocket formula:v2~0.2e

  30. Elliptic Flow in Kinetic Theory Zhang et al.(’99) ideal hydro limit v2 : Ideal hydro : strongly interacting system b = 7.5fm t(fm/c) generated through secondary collisions saturated in the early stage sensitive to cross section (~1/m.f.p.~1/viscosity) v2 is

  31. Discovery of Perfect Fluidity!? Response=(output)/(input) Fine structure of elliptic flow Data reaches hydro limit curve Figures taken from STAR white paper(’05)

  32. Several Remarks on the Discovery • Chemical Composition • Differential Elliptic Flow • Smallness of Transport Coefficients • Importance of Dynamics • Applicability of Boltzmann Eq. • Applicability of Blast Wave Model • Dependence of Initial Conditions

  33. Inputs for Hydrodynamic Simulations for perfect fluids 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 0 Initial stage: Particle production, pre-thermalization?  Instead, initial conditions for hydro simulations

  34. Main Ingredient: Equation of State Typical EOS in hydro models EOS I Ideal massless free gas EOS H Hadron resonance gas EOS Q QGP: P=(e-4B)/3 Hadron: Resonance gas p=e/3 P.Kolb and U.Heinz(’03) Latent heat Note: Chemically frozen hadronic EOS is needed to reproduce heavy particle yields. (Hirano, Teaney, Kolb, Grassi,…)

  35. Interface 1: Initial Condition Initial conditions (tuned to reproduce dNch/dh): initial time, energy density, flow velocity Energy density distribution Reaction plane Transverse plane (Lorentz-contracted) nuclei

  36. Two Hydro Initial Conditions Which Clear the “First Hurdle” Centrality dependence Rapidity dependence 1.Glauber model Npart:Ncoll = 85%:15% 2. CGC model Matching I.C. via e(x,y,hs) Kharzeev, Levin, and Nardi Implemented in hydro by TH and Nara

  37. Interface 2: Freezeout--How to Convert Bulk to Particles-- Cooper-Frye formula Outputs from hydro in F.O. hypersurface Contribution from resonance decays can be treated with additional decay kinematics. S

  38. t z 0 Utilization of Hadron Transport Model for Freezeout Process (1) Sudden freezeout: QGP+hadron fluids (2) Gradual freezeout: QGP fluid + hadron gas At T=Tf, l=0 (ideal fluid) l=infinity (free stream) Automatically describe chemical and thermal freezeouts T=Tf t Hadron fluid QGP fluid QGP fluid z 0

  39. Several Remarks on the Discovery • Chemical Composition • Differential Elliptic Flow • Smallness of Transport Coefficients • Importance of Dynamics • Applicability of Boltzmann Eq. • Applicability of Blast Wave Model • Dependence of Initial Conditions

  40. 1. Data Properly Reproduced? Final differential v2 depends on hadronic chemical compositions. 100MeV T.H. and K.Tsuda (’02) 140MeV 0.8 1.0 0.2 0.6 0 0.4 0.8 0.2 0.6 0 0.4 transverse momentum (GeV/c) CE: chemical equilibrium (not consistent with exp. yield) PCE: partial chemical equilibrium

  41. Cancel between v2 and <pT> At hadronization Chemical Eq. v2(pT) v2(pT) v2 v2 freezeout pT pT <pT> <pT> Chemical F.O. CE: increase v2(pT) CFO: decrease v2 pT <pT>

  42. Intuitive Picture Chemical Freezeout Mean ET decreases due to pdV work MASS energy KINETIC energy Chemical Equilibrium

  43. 2. Is mass ordering for v2(pT) a signal of the perfect QGP fluid? Pion 20-30% Proton Mass ordering comes from rescattering effect. Interplay btw. radial and elliptic flows Not a direct sign of the perfect QGP fluid Two neglected effects in hydro: chemical freezeout and hadronic dissipation Mass dependence is o.k. from hydro+cascade.

  44. 3. Is viscosity really small in QGP? • 1+1D Bjorken flowBjorken(’83) • Baym(’84)Hosoya,Kajantie(’85)Danielewicz,Gyulassy(’85)Gavin(’85)Akase et al.(’89)Kouno et al.(’90)… (Ideal) (Viscous) h: shear viscosity (MeV/fm2), s : entropy density (1/fm3) h/s is a good dimensionless measure (in the natural unit) to see viscous effects. Shear viscosity is small in comparison with entropy density!

  45. Quiz: Which has larger viscosityat room temperature, water or air? [Pa] = [N/m2] (Dynamical) Viscosity h: ~1.0x10-3 [Pa s] (Water 20℃) ~1.8x10-5 [Pa s] (Air 20℃) Kinetic Viscosity n=h/r: ~1.0x10-6 [m2/s] (Water 20℃) ~1.5x10-5 [m2/s] (Air 20℃) hwater > hair BUT nwater < nair Non-relativistic Navier-Stokes eq. (a simple form) Neglecting external force and assuming incompressibility.

  46. 4. Is h/s enough? • Reynolds number Iso, Mori, Namiki (’59) R>>1 Perfect fluid • (1+1)D Bjorken solution

  47. 5. Boltzmann at work? Molnar&Gyulassy(’00) Molnar&Huovinen(’04) 25-30% reduction gluonic fluid s ~ 15 * spert ! Caveat 1: Where is the “dilute” approximation in Boltzmann simulation? Is l~0.1fm o.k. for the Boltzmann description? Caveat 2: Differential v2 is tricky. dv2/dpT~v2/<pT>. Difference of v2 is amplified by the difference of <pT>. Caveat 3: Hadronization/Freezeout are different.

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