Hydrodynamics and Flow Tetsufumi Hirano Department of Physics The University of Tokyo The University of Tokyo QGP Winter School 2008 February 1-3, 2008, Jaipur, India QGP Winter School 2008 February 1-3, 2008, Jaipur, India My Charge and Disclaimer To lead students/young post-docs to frontier of hydrodynamic description in H.I.C. in order for them to understand talks/posters to be presented in the upcoming QM2008 conference. Not to cover recent studies to be presented at QM2008. Plan of Lecture PART 1 (INTRODUCTION) Introduction to hydrodynamics in H.I.C. PART 2 (FUNDAMENTALS) Formalism of relativistic ideal/viscous hydrodynamics PART 3 (APPLICATIONS) Basic Checks Elliptic flow Ideal hydrodynamic model Application of ideal hydrodynamic model to H.I.C. and comparison with data PART 1 Introduction to hydrodynamics in relativistic 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) Bottom-Up Approach to Heavy Ion Physics The first principle (QuantumChromo Dynamics) Experimental Relativistic Heavy Ion Collider ~200 papers from 4 collaborations since 2000 Phenomenology (hydrodynamics) Connection among Lectures Hydrodynamics QGP fluids Color Glass Condensate (Venugopalan) Jet quenching (Kharzeev,dEnterria) J/psi suppression (Satz, Lourenco) EM probe (Srivastava) Particle production (Stock) Collective flow (Hirano) PART 2 (FUNDAMENTALS) Formalism of relativistic ideal/viscous hydrodynamics Relativistic Hydrodynamics Energy-momentum conservation Current conservation Energy-Momentum tensor The i-th conserved current In H.I.C., N i = N B (net baryon current) Equations of motion in relativistic hydrodynamics Tensor/Vector Decomposition Tensor decomposition with a given time-like and normalized four-vector u where, Projection Tensor/Vector u is perpendicular to . u is local four flow velocity. More precise meaning will be given later. Naively speaking, u ( ) picks up time- (space-)like component(s). Local rest frame (LRF): Decomposition of T :Energy density :(Hydrostatic+bulk) pressure P = P s + :Energy (Heat) current :Shear stress tensor : Symmetric, traceless and transverse to u & u Decomposition of N :charge density :charge current Q. Count the number of unknowns in the above decomposition and confirm that it is 10(T )+4k(N i ). Here k is the number of independent currents. Note: If you consider u as independent variables, you need additional constraint for them. If you also consider P s as an independent variable, you need the equation of state P s =P s (e,n). Ideal and Dissipative Parts Energy Momentum tensor Charge current Ideal part Dissipative part Meaning of u u is four-velocity of flow. What kind of flow? Two major definitions of flow are 2. Flow of conserved charge (Eckart) 1. Flow of energy (Landau) Meaning of u (contd.) Landau (W =0, u L V =0) Eckart (V =0,u E W =0) uLuL VV uEuE WW Just a choice of local reference frame. Landau frame might be relevant in H.I.C. Entropy Conservation in Ideal Hydrodynamics Neglect dissipative part of energy momentum tensor to obtain ideal hydrodynamics. Q. Derive the above equation. Therefore, Entropy Current Assumption (1 st order theory): Non-equilibrium entropy current vector has linear dissipative term(s) constructed from (V , , , u ). We have assumed N = 0, so the second term does not appear in this case. In what follows, we consider the Landau frame only and omit subscript L. For simplicity, we further assume there is no charge in the system. The 2 nd Law of Thermodynamics The 2 nd thermodynamic law tells us Q. Check the above calculation. Constitutive Equations Thermodynamic force Transport coefficient Current tensorshear scalarbulk Equation of Motion : Expansion scalar (Divergence) : Lagrange derivative Equation of Motion Q. Derive the above equations of motion from energy-momentum conservation. Intuitive Interpretation of EoM Work done by pressure Production of entropy Change of volume Dilution Compression Lessons from (Non-Relativistic) Navier-Stokes Equation Assuming incompressible fluids such that, Navier-Stokes eq. becomes Final flow velocity comes from interplay between these two effects. Source of flow (pressure gradient) Diffusion of flow (Kinematic viscosity, /, plays a role of diffusion constant.) Generation of Flow x P Expand Pressure gradient Source of flow Flow phenomena are important in H.I.C to understand EOS Diffusion of Flow Heat equation (: heat conductivity ~diffusion constant) For illustrative purpose, one discretizes the equation in (2+1)D space: Diffusion ~ Smoothing R.H.S. of descretized heat/diffusion eq. i j i j x x y y subtract Suppose i,j is larger (smaller) than an average value around the site, R.H.S. becomes negative (positive). 2 nd derivative w.r.t. coordinates Smoothing Shear Viscosity Reduces Flow Difference Shear flow (gradient of flow) Smoothing of flow Next time step Bjorken s Equation in the 1 st Order Theory (Bjorkens solution) = (1D Hubble flow) Q. Derive the above equation. Viscous Correction Correction from shear viscosity (in compressible fluids) Correction from bulk viscosity If these corrections vanish, the above equation reduces to the famous Bjorken equation. Expansion scalar = theta = 1/tau in scaling solution 1. and are dimensionless quantities in natural units and intrinsic properties of fluids. 2. One often says, viscosity is small. However, one has to say this in the context of heavy ion collisions in a more precise way: Viscous coefficients are small in comparison with entropy density. A Few Remarks is obtained from SUSY Yang-Mills theory. is obtained from lattice. Bulk viscosity has a prominent peak around T c. Recent Topics on Transport Coefficients Need microscopic theory (e.g., Boltzmann eq.) to obtain transport coefficients. Kovtun,Son,Starinet, Nakamura,Sakai, Karzeev,Tuchin,Karsch, Necessity of Relaxation Time Non-relativistic case (Cattaneo(1948)) Fouriers law : relaxation time Parabolic equation (heat equation) ACAUSAL! Finite Hyperbolic equation (telegraph equation) Balance eq.: Constitutive eq.: Entropy Current (2 nd ) Assumption (2 nd order theory): Non-equilibrium entropy current vector has linear + quadratic dissipative term(s) constructed from (V , , , u ). Constitutive Equations Relaxation terms appear ( and are relaxation time). No longer algebraic equations! Dissipative currents become dynamical quantities like thermodynamic variables. Employed in recent viscous fluid simulations. Bjorken s Equation in the 2 nd Order Theory New terms (written in red) appear in the 2 nd order theory. Coupled equations where Summary (part 1&2) Hydrodynamics provides a dynamical framework in heavy ion collisions towards understanding bulk and transport properties of the QGP. Na ve extension of Navier-Stokes equation to its relativistic version has a problem on causality. Need the 2 nd order corrections in entropy current. Contents PART 1 (INTRODUCTION) Introduction to hydrodynamics in H.I.C. PART 2 (FUNDAMENTALS) Formalism of relativistic ideal/viscous hydrodynamics PART 3 (APPLICATIONS) Basic Checks Elliptic flow Ideal hydrodynamic model Application of ideal hydrodynamic model to H.I.C. and comparison with data PART 3 (APPLICATIONS) Basic Checks 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) 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/ Matter in (Kinetic) Equilibrium? uu 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 PART 3 (APPLICATIONS) Elliptic flow 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 22 dN/d 0 22 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 PART 3 (APPLICATIONS) Ideal hydrodynamic model 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 (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 PART 3 (APPLICATIONS) Application of ideal hydrodynamic model to H.I.C. and comparison with data Discovery of Perfect Fluidity!? Response=(output)/(input) Figures taken from STAR white paper(05) Fine structure of elliptic flow Data reaches hydro limit curve QGP+hadron fluids with Glauber I.C. Mug at RHIC/AGS Users Meeting 2005 RHIC serves the perfect liquid A present from Raju when I gave a seminar talk at BNL. Centrality Dependence of v 2 v 2 data are comparable with hydro results. Hadronic cascade cannot reproduce data. Note that, in v 2 data, there exists eccentricity fluctuation which is not considered in model calculations. hadronic cascade result (Courtesy of M.Isse) TH et al. (06) QGP+hadron fluids with Glauber I.C. Pseudorapidity Dependence of v 2 =0 >0 v 2 (data) 20-30% Proof: Assuming, Utilization of Hydro Results Jet quenching J/psi suppression Meson Recombination Coalescence Thermal radiation (photon/dilepton) Information along a path Information on surface Information inside medium Baryon J/psi c c bar Jet Propagation through a QGP Fluid Color: parton density Plot: mini-jets Au+Au 200AGeV, b=8 fm transverse Fragmentation switched off hydro+jet model x y Full 3D ideal hydrodynamics + PYTHIA Parton distribution fn. pQCD 2 2 processes Fragmentation Gyulassy-Levai-Vitev formula Inelastic energy loss TH and Y.Nara (02-) Hydro+J/psi Model Gunji,Hamagaki,Hatsuda,TH(07)