Physics | Hydrodynamics » Paul Romatschke - Hydrodynamics

Hydrodynamics Paul Romatschke FIAS, Frankfurt, Germany Quark Matter, May 2011 Paul Romatschke Hydro/HIC Asymptotic Freedom of QCD Coupling [Particle Data Group]: αs(µ) 0.3 0.2 0.1 0 1 10 µ GeV 10 2 Nobel Prize 2004: Gross, Politzer, Wilczek Paul Romatschke Hydro/HIC QCD Phase Transition QCD Energy-density from lattice QCD: Rapid Rise of  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 > Tc ∼ 200 p MeV, equation of state (P = P()), speed of sound cs = dP/d Experimental setup: collide (large) nuclei at high speeds to reach T > Tc Paul Romatschke Hydro/HIC Hydrodynamics Paul Romatschke Hydro/HIC Fluid Dynamics = Conservation of Energy+Momentum for long wavelength modes1 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:  (General Relativity): space-time metric gµν 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 , p, vector u µ , tensors of rank two: u µ u ν , g µν T µν must be of form T µν = A(, p)u µ u ν + B(, p)g µν Local rest frame (vanishing fluid velocity, u µ = (1, 0, 0, 0)):    0 0 0  0 −p 0 0   T µν =   0 0 −p 0  0 0 0 −p Can use to determine A, B ! Paul Romatschke Hydro/HIC Tidµν = u µ u ν − p(g µν − u µ u ν ) (Fluid EMT, no gradients) + ∂µ T µν = 0 (“EMT Conservation”) = Ideal Fluid Dynamics Paul Romatschke Hydro/HIC Proof

Take ∂µ T µi = 0, take non-relativistic limit (neglect u 2 /c 2  1, p  mc 2 ): 1 ∂t u i + u m ∂m u i = − ∂j δ ij p  “Euler Equation” [L. Euler, 1755] Euler Equation: non-linear, non-dissipative: “ideal fluid dynamics” Take ∂µ T µ0 = 0, take non-relativistic limit (neglect u/c  1, p  mc 2 ): ∂i u i + ∂t  + u i ∂i  = 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 = Tidµν (Fluid EMT, no gradients) + ∂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

= Tidµν is approximation! Lift approximation: T µν = Tidµν + Πµν Build Πµν : e.g first order gradients on , u µ , g µν Πµν = η∇<µ u ν> + ζ∆µν ∇ · u Paul Romatschke Hydro/HIC Relativistic Viscous Fluid Dynamics T µν = Tidµν + Πµν (Fluid EMT, 1st o. gradients) + ∂µ T µν = 0 (“EMT Conservation”) = Relativistic Navier-Stokes Equation Paul Romatschke Hydro/HIC Relativistic Viscous Fluid Dynamics L. Euler, 1755: 1 ∂t u i + u m ∂m u i = − ∂j δ ij p  C. Navier, 1822; G Stokes 1845:  1  ∂t u i + u m ∂m u i = − ∂j δ ij p + Πij ,   i  l ∂u ∂u j 2 ij ∂u l ij ij ∂u Π = −η + − δ − ζδ , ∂x j ∂x i 3 ∂x l ∂x l η, ζ. transport coefficients (“viscosities”) Paul Romatschke Hydro/HIC Gradients and Hydro T µν : fluid dofs (, p, u µ , g µν ), no gradients gives Ideal Hydrodynamics T µν : fluid dofs (, p, u µ , g µν ) up to 1st order gradients gives

Navier-Stokes equation T µν : fluid dofs (, p, u µ , g µν ) up to 2nd , 3rd , . 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 ∂ 2 δu y = 0 +p x ∂t δu y − Diffusion speed of wavemode k : vT (k ) = 2k η ∞ (k  1) +p Know how to regulate: “second-order” theories: τπ ∂t2 δu y + ∂t δu y − η ∂ 2 δu y = 0 +p x [Maxwell (1867), Cattaneo (1948)] Paul Romatschke Hydro/HIC Second Order Fluid Dynamics Limiting speed is finite s lim vL (k ) = k ∞ cs2 + ζ 4η + 3τπ ( + p) τΠ ( + p) [Romatschke, 2009] τπ , τΠ . : dynamics “2nd order” regulators for

“1st order” fluid Regulators acts in UV, low momentum (fluid dynamics) regime is still Navier-Stokes Paul Romatschke Hydro/HIC Second Order Fluid Dynamics T µν = Tidµν + Πµν (Fluid EMT, 2nd o. gradients) + ∂µ T µν = 0 (“EMT Conservation”) = “Causal” Relativistic Viscous Fluid Dynamics First complete 2nd theory for shear only in 2007 ! [Baier et al. 2007; Bhattacharyya et al 2007] Paul Romatschke Hydro/HIC Second Order Fluid Dynamics T µν = Tidµν + Πµν (Fluid EMT, 2nd o. gradients) + ∂µ T µν = 0 (“EMT Conservation”) = “Causal” Relativistic Viscous Fluid Dynamics First complete 2nd 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 = Tidµν + Πµν Πµν is given by small gradient expansion Πµν = η∇<µ u ν> + . Hydrodynamics breaks down if gradient expansion breaks down: Π ∼ Tidµν 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: , u µ at τ = τ0 Need equation of state p = p(), which gives cs2 = dp d 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 proper

time τ = t 2 − z 2 and transverse space x⊥ . Models for energy density distribution: Glauber/Color-Glass-Condensate Starting time τ0 : 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 Tc [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 Z µν µν Thydro = (+p)u µ u ν −Pg µν +Πµν = Tparticles = f (x, p)pµ pν p No dissipation (ideal hydro) = equilibrium: f (p, x) = e−p·u/T Only shear dissipation: Quadratic ansatz   pα pβ Παβ −p·u/T 3 f (p, x) = e 1+ + O(p ) 2( + p)T 2

Paul Romatschke Hydro/HIC Experimental Observables dN/dp/dφ  Paul Romatschke Hydro/HIC Experimental Observables For ultrarelativistic heavy-ion collisions, dN dN =h iφ (1 + 2v2 (p⊥ ) cos(2φ) + . ) dp⊥ dφdy dp⊥ dφdy Radial flow: h dpdN iφ ⊥ dy Elliptic flow: v2 (p⊥ ) Paul Romatschke Hydro/HIC Putting things together Source:model https://doksi.net Hydro 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 < v3 > and < v2 >2 6=< v2 >2 Paul Romatschke Hydro/HIC LHC

results v2 0.12 0.1 0.08 0.06 v2{2} v2{2} (same charge) v2{4} v2{4} (same charge) v2{q-dist} v2{LYZ} v2{EP} STAR v2{LYZ} STAR 0.04 0.02 0 0 10 20 30 40 50 60 70 80 90 centrality percentile [ALICE Collaboration, 2010] If < v2 >2 =< v2 >2 then v2 {2} = v2 {4}. Paul Romatschke Hydro/HIC 3D Evolution h+/- v3 [%] h+/- v2 [%] 8 6 4 2 PHOBOS 15-25% central ideal, avg 0 2 1 ideal, e-b-e η/s=0.08, e-b-e η/s=0.16, e-b-e 0 -4 -2 0 ηp 2 4 [B. Schenke et al, 2010] Non-Bjorken flow in longitudinal direction. Paul Romatschke Hydro/HIC Freeze-Out χ(p/T) / [4π η/(sT)] 20 Quadratic 15 Coll. 10 LO 5 Linear 0 0 2 4 6 8 p/T 10 12 14 16 [K. Dusling et al, 2009] Quadratic ansatz may be inaccurate Paul Romatschke Hydro/HIC Thermalization How does system get T µν that is close to hydro? Far from equilibration dynamics: non-perturbative, real-time: hard! Attempts to thermalization: pQCD inspired (’plasma instabilities’);

AdS/CFT inspired (’collision of shock waves’) Paul Romatschke Hydro/HIC Things to keep in mind for this week Dynamics: anything less than 2+1D is not realistic (’Bjorken hydro’) Ideal hydro does not indicate its own breakdown. Does not mean results are accurate! Keep in mind that ideal hydro only exists with numerical viscosity (value?) All working 2+1D viscous hydro codes are ’second order hydro’ Different names (’Israel-Stewart’, ’full IS’, ’BRSSS’) correspond to different choices for values of τπ , . Paul Romatschke Hydro/HIC A personal appeal: theory/data comparisons A new theorists calculation/model should first be rigorously studied before ’fitting’ data Example: hydrodynamic calculations on a grid; physical results correspond to limit of vanishing grid spacing Theorists: Please first check your model/calculation before you compare to experimental data Experimentalists: Please don’t blindly trust (or promote!) a model just because it

fits data Paul Romatschke Hydro/HIC .and all the rest There are many topics/details I couldn’t cover today! Some lecture notes: “New Developments in Relativistic Viscous Hydrodynamics”, PR, arXiv:0902.3663 “Nearly Perfect Fluidity: From Cold Atomic Gases to Hot Quark Gluon Plasmas”, T. Schäfer and D Teaney, arXiv:0904.3107 “Early collective expansion: Relativistic hydrodynamics and the transport properties of QCD matter”, U.W Heinz, arXiv:0901.4355 Paul Romatschke Hydro/HIC