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

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

PART 1 Hydrodynamics in Heavy Ion Collisions

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

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

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

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

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.

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

PART 2 Collective Flow

Sufficient Energy Density? Bjorken energy density : proper time y: rapidity R: effective transverse radius m T : transverse mass Bjorken(’83) total energy (observables)

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

Centrality Dependence of Energy Density PHENIX(’05) STAR(’08)  c from lattice Well above  c from lattice in central collision at RHIC, if assuming =1fm/c.

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)

Matter in (Chemical) Equilibrium? Two fitting parameters: T ch,  B direct Resonance decay

Amazing Fit! T=177MeV,  B = 29 MeV Close to T c from lattice

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 T c ?  No chemical eq. in hadron phase!?  Essentially dynamical problem! Expansion rate  Scattering rate see, e.g., U.Heinz, nucl-th/

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

Matter in (Kinetic) Equilibrium? uu Kinetically equilibrated matter at rest Kinetically equilibrated matter at finite velocity pxpx pypy pxpx pypy Isotropic distribution Lorentz-boosted distribution

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

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

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

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

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

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

Ollitrault (’92) Hydro behavior Spatial Anisotropy Momentum Anisotropy INPUT OUTPUT Interaction among produced particles dN/d  No secondary interaction 0 22 dN/d  0 22 2v22v2 x y  What is Elliptic Flow? --How does the system respond to spatial anisotropy?--

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

Eccentricity Fluctuation 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 Adopted from D.Hofman(PHOBOS), talk at QM2006 A sample event from Monte Carlo Glauber model

Elliptic Flow in Hydro Saturate in first several femto-meters v 2 signal is sensitive to initial stage. Response of the system (= v 2 /) is almost constant. Pocket formula:v 2 ~0.2 Kolb and Heinz (’03)

Elliptic Flow in Kinetic Theory b = 7.5fm generated through secondary collisions saturated in the early stage sensitive to cross section (~1/m.f.p.~1/viscosity) v 2 is Zhang et al.(’99)ideal hydro limit t(fm/c) v2v2 : Ideal hydro : strongly interacting system

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

Several Remarks on the Discovery 1.Chemical Composition 2.Differential Elliptic Flow 3.Smallness of Transport Coefficients 4.Importance of Dynamics 5.Applicability of Boltzmann Eq. 6.Applicability of Blast Wave Model 7.Dependence of Initial Conditions

Inputs for Hydrodynamic Simulations for perfect fluids 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?  Instead, initial conditions for hydro simulations 0 z t

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

Interface 1: Initial Condition Initial conditions (tuned to reproduce dN ch /d): initial time, energy density, flow velocity Transverse plane Reaction plane Energy density distribution (Lorentz-contracted) nuclei

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

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.

Utilization of Hadron Transport Model for Freezeout Process (1) Sudden freezeout: QGP+hadron fluids (2) Gradual freezeout: QGP fluid + hadron gas Automatically describe chemical and thermal freezeouts 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

Several Remarks on the Discovery 1.Chemical Composition 2.Differential Elliptic Flow 3.Smallness of Transport Coefficients 4.Importance of Dynamics 5.Applicability of Boltzmann Eq. 6.Applicability of Blast Wave Model 7.Dependence of Initial Conditions

1. Data Properly Reproduced? T.H. and K.Tsuda (’02) MeV 100MeV transverse momentum (GeV/c) CE: chemical equilibrium (not consistent with exp. yield) PCE: partial chemical equilibrium Final differential v 2 depends on hadronic chemical compositions.

Cancel between v 2 and Cancel between v 2 and pTpT v 2 (p T ) v2v2 pTpT v 2 (p T ) v2v2 pTpT v 2 (p T ) v2v2 Chemical Eq. Chemical F.O. At hadronization CE: increase CFO: decrease freezeout

Intuitive Picture Chemical Freezeout Chemical Equilibrium Mean E T decreases due to pdV work MASS energy KINETIC energy

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

3. Is viscosity really small in QGP? 1+1D Bjorken flow Bjorken(’83) Baym(’84)Hosoya,Kajantie(’85)Danielewicz,Gyulassy(’85)Gavin(’85)Akase et al.(’89)Kouno et al.(’90)… (Ideal) (Viscous)  : shear viscosity (MeV/fm 2 ), s : entropy density (1/fm 3 ) /s is a good dimensionless measure (in the natural unit) to see viscous effects. Shear viscosity is small in comparison with entropy density!

Quiz: Which has larger viscosity at room temperature, water or air? (Dynamical) Viscosity : ~1.0x10 -3 [Pa s] (Water 20 ℃ ) ~1.8x10 -5 [Pa s] (Air 20 ℃ ) Kinetic Viscosity =/: ~1.0x10 -6 [m 2 /s] (Water 20 ℃ ) ~1.5x10 -5 [m 2 /s] (Air 20 ℃ ) [Pa] = [N/m 2 ] Non-relativistic Navier-Stokes eq. (a simple form) Neglecting external force and assuming incompressibility.  water >  air BUT water < air

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

5. Boltzmann at work?  ~ 15 *  pert ! Caveat 1: Where is the “dilute” approximation in Boltzmann simulation? Is ~0.1fm o.k. for the Boltzmann description? Caveat 2: Differential v 2 is tricky. dv 2 /dp T ~v 2 /. Difference of v 2 is amplified by the difference of. Caveat 3: Hadronization/Freezeout are different %reduction Molnar&Gyulassy(’00)Molnar&Huovinen(’04) gluonicfluid

6. Does v 2 (p T ) really tell us smallness of  /s in the QGP phase? Not a result from dynamical calculation, but a “fitting” to data. No QGP in the model  0 is not a initial time, but a freeze-out time.  s /t 0 is not equal to h/s, but to 3/4sT 0  0 (in 1+1D). Being smaller T 0 from p T dist., t0 should be larger (~10fm/c). D.Teaney(’03)

7. Initial condition is a unique? Novel initial conditions from Color Glass Condensate lead to large eccentricity. Need viscosity and/or softer EoS in the QGP! Hirano and Nara(’04), Hirano et al.(’06) Kuhlman et al.(’06), Drescher et al.(’06)

Summary QM2004

Summary So Far Interpretation of RHIC results involves many subtle issues in hydrodynamic modeling of reactions Three pillars: Glauber initial condition + Ideal QGP + dissipative hadron gas Need to check each modeling to get conclusive interpretation Next task: viscosity  Lecture on the 2 nd day