Hydrodynamics Paul Romatschke FIAS, Frankfurt, Germany Quark - PowerPoint PPT Presentation

Hydrodynamics Paul Romatschke FIAS, Frankfurt, Germany Quark Matter, May 2011 Paul Romatschke Hydro/HIC

Asymptotic Freedom of QCD Coupling [Particle Data Group] : 0.3 α s ( µ ) 0.2 0.1 0 2 1 10 10 µ GeV Nobel Prize 2004: Gross, Politzer, Wilczek Paul Romatschke Hydro/HIC

QCD Phase Transition QCD Energy-density from lattice QCD: Rapid Rise of ǫ close to T ∼ 170 MeV. Paul Romatschke Hydro/HIC

QCD Phase Transition QCD Phase Transition: Transition from confined matter (neutrons, protons, hadrons) to deconfined matter (quark-gluon plasma ) Information from lattice QCD: quark-gluon plasma for T > T c ∼ 200 MeV, equation of state ( P = P ( ǫ ) ), speed of � sound c s = dP / d ǫ Experimental setup: collide (large) nuclei at high speeds to reach T > T c Paul Romatschke Hydro/HIC

Hydrodynamics Paul Romatschke Hydro/HIC

Fluid Dynamics = Conservation of Energy+Momentum for long wavelength modes 1 1 long wavelength modes = looking at the system for a very long time from very far away Paul Romatschke Hydro/HIC

Fluid Dynamics: Degrees of Freedom (Relativistic) Fluids described by: Fluid velocity: u µ Pressure: p (Energy-) Density: ǫ (General Relativity): space-time metric g µν Quantum Field Theory: Energy-Momentum Tensor T µν Conservation of Energy+Momentum: ∂ µ T µν = 0. Paul Romatschke Hydro/HIC

Energy Momentum Tensor for Ideal Fluids T µν symmetric tensor of rank two building blocks for ideal fluids: scalars ǫ, p , vector u µ , tensors of rank two: u µ u ν , g µν T µν must be of form T µν = A ( ǫ, p ) u µ u ν + B ( ǫ, p ) g µν Local rest frame (vanishing fluid velocity, u µ = ( 1 , 0 , 0 , 0 ) ):   ǫ 0 0 0 − p 0 0 0 T µν =     0 0 − p 0   0 0 0 − p Can use to determine A , B ! Paul Romatschke Hydro/HIC

id = ǫ u µ u ν − p ( g µν − u µ u ν ) T µν (Fluid EMT, no gradients) + ∂ µ T µν = 0 (“EMT Conservation”) = Ideal Fluid Dynamics Paul Romatschke Hydro/HIC

Proof Take ∂ µ T µ i = 0, take non-relativistic limit (neglect u 2 / c 2 ≪ 1 , p ≪ mc 2 ): ∂ t u i + u m ∂ m u i = − 1 ǫ ∂ j δ ij p “Euler Equation” [L. Euler, 1755] Euler Equation: non-linear, non-dissipative: “ideal fluid dynamics” Take ∂ µ T µ 0 = 0, take non-relativistic limit (neglect u / c ≪ 1 , p ≪ mc 2 ): ǫ∂ i u i + ∂ t ǫ + u i ∂ i ǫ = 0 “Continuity Equation” [L. Euler, 1755] Paul Romatschke Hydro/HIC

Non-linear & Non-dissipative: Turbulence Paul Romatschke Hydro/HIC

Non-linear & Dissipative: Laminar Paul Romatschke Hydro/HIC

Non-linear & Dissipative: Laminar Viscosity dampens turbulent instability! Paul Romatschke Hydro/HIC

Relativistic Ideal Fluid Dynamics T µν = T µν (Fluid EMT, no gradients) id + ∂ µ T µν = 0 (“EMT Conservation”) = Ideal Fluid Dynamics Paul Romatschke Hydro/HIC

Relativistic Viscous Fluids How to include viscous effects? Energy and Momentum Conservation: ∂ µ T µν = 0 is exact But T µν = T µν id is approximation! Lift approximation: T µν = T µν id + Π µν Build Π µν : e.g. first order gradients on ǫ, u µ , g µν Π µν = η ∇ <µ u ν> + ζ ∆ µν ∇ · u Paul Romatschke Hydro/HIC

Relativistic Viscous Fluid Dynamics T µν = T µν id + Π µν (Fluid EMT, 1 st o. gradients) + ∂ µ T µν = 0 (“EMT Conservation”) = Relativistic Navier-Stokes Equation Paul Romatschke Hydro/HIC

Relativistic Viscous Fluid Dynamics L. Euler, 1755: ∂ t u i + u m ∂ m u i = − 1 ǫ ∂ j δ ij p C. Navier, 1822; G. Stokes 1845: ∂ t u i + u m ∂ m u i = − 1 δ ij p + Π ij � � ǫ ∂ j , � ∂ u i ∂ x j + ∂ u j 3 δ ij ∂ u l − ζδ ij ∂ u l ∂ x i − 2 � Π ij = − η ∂ x l , ∂ x l η, ζ . . . transport coefficients (“viscosities”) Paul Romatschke Hydro/HIC

Gradients and Hydro T µν : fluid dofs ( ǫ, p , u µ , g µν ), no gradients gives Ideal Hydrodynamics T µν : fluid dofs ( ǫ, p , u µ , g µν ) up to 1 st order gradients gives Navier-Stokes equation T µν : fluid dofs ( ǫ, p , u µ , g µν ) up to 2 nd , 3 rd , ... order: higher order Hydrodynamics Paul Romatschke Hydro/HIC

Fluid Dynamics = Effective Theory of Small Gradients Paul Romatschke Hydro/HIC

Relativistic Navier-Stokes Equation Good enough for non-relativistic systems NOT good enough for relativistic systems Paul Romatschke Hydro/HIC

Navier-Stokes: Problems with Causality Consider small perturbations around equilibrium Transverse velocity perturbations obey η ∂ t δ u y − x δ u y = 0 ǫ + p ∂ 2 Diffusion speed of wavemode k : η v T ( k ) = 2 k ǫ + p → ∞ ( k ≫ 1 ) Know how to regulate: “second-order” theories: η t δ u y + ∂ t δ u y − x δ u y = 0 τ π ∂ 2 ǫ + p ∂ 2 [Maxwell (1867), Cattaneo (1948)] Paul Romatschke Hydro/HIC

Second Order Fluid Dynamics Limiting speed is finite � 4 η ζ c 2 k →∞ v L ( k ) = lim s + 3 τ π ( ǫ + p ) + τ Π ( ǫ + p ) [Romatschke, 2009] τ π , τ Π . . . ...: “2 nd order” regulators for “1 st order” fluid dynamics Regulators acts in UV, low momentum (fluid dynamics) regime is still Navier-Stokes Paul Romatschke Hydro/HIC

Second Order Fluid Dynamics T µν = T µν id + Π µν (Fluid EMT, 2 nd o. gradients) + ∂ µ T µν = 0 (“EMT Conservation”) = “Causal” Relativistic Viscous Fluid Dynamics First complete 2 nd theory for shear only in 2007 ! [Baier et al. 2007; Bhattacharyya et al. 2007] Paul Romatschke Hydro/HIC

Second Order Fluid Dynamics T µν = T µν id + Π µν (Fluid EMT, 2 nd o. gradients) + ∂ µ T µν = 0 (“EMT Conservation”) = “Causal” Relativistic Viscous Fluid Dynamics First complete 2 nd theory for shear only in 2007 ! [Baier et al. 2007; Bhattacharyya et al. 2007] Paul Romatschke Hydro/HIC

Hydrodynamcis: Limits of Applicability Remember T µν = T µν id + Π µν Π µν is given by small gradient expansion Π µν = η ∇ <µ u ν> + . . . Hydrodynamics breaks down if gradient expansion breaks down: Π ∼ T µν id or p ≃ η ∇ · u Two possible ways: η large (hadron gas!) or ∇ · u large (early times, small systems!) Paul Romatschke Hydro/HIC

Hydro Theory What you should remember: Hydrodynamics is Energy Momentum Conservation Hydrodynamics is an Effective Theory for long wavelength (small momenta) Hydrodynamics breaks down for small systems or dilute systems Paul Romatschke Hydro/HIC

Hydrodynamic Models for Heavy-Ion Collisions Paul Romatschke Hydro/HIC

Hydro Models for Heavy-Ion Collisions Need initial conditions for Hydro: ǫ, u µ at τ = τ 0 s = dp Need equation of state p = p ( ǫ ) , which gives c 2 d ǫ Need functions for transport coefficients η, ζ . Need algorithm to solve (nonlinear!) hydro equations Need method to convert hydro information to particles (“freeze-out”) Paul Romatschke Hydro/HIC

Initial Conditions IC’s for hydro not known. Here are some popular choices : Fluid velocities are set to zero Boost-invariance: all hydro quantities only depend on √ t 2 − z 2 and transverse space x ⊥ . proper time τ = Models for energy density distribution: Glauber/Color-Glass-Condensate Starting time τ 0 : Should be of order 1 fm, precise value unknown Paul Romatschke Hydro/HIC

Equation of State EoS known (approximately) from lQCD: Paul Romatschke Hydro/HIC

Transport Coefficients In QCD: known for small and large T, but not for T ≃ T c [Demir and Bass, 2008] Paul Romatschke Hydro/HIC

Hydro Solvers For 3+1D ideal hydro, many groups, well tested For 2+1D viscous hydro, many groups, well tested For 3+1D viscous hydro: Schenke, Jeon, Gale 2010 Paul Romatschke Hydro/HIC

Freeze-out Partial solution exists: “Cooper-Frye” Idea: T µν for particles/fluid must be the same � hydro = ( ǫ + p ) u µ u ν − Pg µν +Π µν = T µν T µν f ( x , p ) p µ p ν particles = p No dissipation (ideal hydro) = equilibrium: f ( p , x ) = e − p · u / T Only shear dissipation: Quadratic ansatz 1 + p α p β Π αβ � � f ( p , x ) = e − p · u / T 2 ( ǫ + p ) T 2 + O ( p 3 ) Paul Romatschke Hydro/HIC

Experimental Observables dN/dp/d φ φ Paul Romatschke Hydro/HIC

Experimental Observables For ultrarelativistic heavy-ion collisions, dN dN dp ⊥ d φ dy = � dp ⊥ d φ dy � φ ( 1 + 2 v 2 ( p ⊥ ) cos ( 2 φ ) + . . . ) dN Radial flow: � dp ⊥ dy � φ Elliptic flow: v 2 ( p ⊥ ) Paul Romatschke Hydro/HIC

Putting things together Hydro model simultion of RHIC Au+Au collisions [Luzum & Romatschke, 2008] Paul Romatschke Hydro/HIC

Current Research and Open Problems Initial Conditions for Hydro: Effect of Fluctuations? 3D vs. 2D: quantitative difference for viscous hydrodynamics evolution? Freeze-Out: Consistent coupling of hydro/particle dynamics? Thermalization: Can one calculate hydro initial conditions? Paul Romatschke Hydro/HIC

Fluctuations [slide stolen from M. Luzum] Initial conditions are not smooth! Event average: <> There will be < v 3 > and < v 2 > 2 � = < v 2 > 2 Paul Romatschke Hydro/HIC