Fluid dynamics: theory and applications in heavy-ion physics

Transcription

1 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Fluid dynamics: theory and applications in heavy-ion physics Dirk H. Rischke Institut für Theoretische Physik and Frankfurt Institute for Advanced Studies thanks to: Barbara Betz, Ioannis Bouras, Gabriel S. Denicol, Carsten Greiner, Pasi Huovinen, Etele Molnár, Harri Niemi, Jorge Noronha, Zhe Xu

2 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Heavy-ion collisions: the experimentalist s view Pb+Pb collision at s NN = 17.3 GeV NA49 CERN-SPS Au+Au collision at s NN = 2 GeV STAR BNL-RHIC Pb+Pb collision at s NN = 2.76 TeV ALICE CERN-LHC

3 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Heavy-ion accelerators Accelerator Location Operation Period s NN [GeV] Systems Experiments BEVALAC LBNL Ne+Ne, Ar+KCl, Au+Au Plastic Ball, EOS SIS-18 GSI Ni+Ni, U+U FOPI, KAOS, HADES AGS BNL O+Au, Si+Au, E866, E877, E891, Au+Au E895, E91, E917 SPS CERN NA34, NA35, NA36, NA38, NA39, NA44, p+p, C+C, S+S, NA45, NA49, NA5, O+Pb, In+In, NA52, NA6, WA8, Pb+Pb WA97/NA57, WA98, NA61 RHIC BNL p+p, d+au, Cu+Cu, PHENIX, STAR, Au+Au, U+U BRAHMS, PHOBOS LHC CERN p+p, Pb+Pb ALICE, ATLAS, CMS SIS-1/3 FAIR 21? p+p,..., U+U CBM SIS: SchwerIonenSynchrotron AGS: Alternating Gradient Synchrotron SPS: Super Proton Synchrotron RHIC: Relativistic Heavy Ion Collider LHC: Large Hadron Collider LBNL: Lawrence Berkeley National Laboratory GSI: Gesellschaft für SchwerIonenforschung (Darmstadt) BNL: Brookhaven National Laboratory CERN: Conseil Européen pour la Recherche Nucléaire FAIR: Facility for Antiproton and Ion Research (Darmstadt)

4 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Heavy-ion collisions: the theorist s view Bjorken s picture: J.D. Bjorken, PRD 27 (1983) 14 Below certain temperature T f hadrons freeze-out Below certain temperature T c QGP hadronizes... quark-gluon plasma (QGP) Leave in their wake highly excited quark-gluon matter which thermalizes and becomes a... Highly Lorentz-contracted nuclei pass through each other Note: Bjorken assumed that evolution of QGP and hadronic phase prior to freeze-out can be described by ideal fluid dynamics (1+1-d boost-invariant scaling solution) = physics constant on proper-time hypersurfaces τ = t 2 z 2 = const.

5 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Fluid dynamics implies collective flow y (fm) 1 non-central heavy-ion collision in transverse (x y) plane (z = ): x (fm) If particles do not interact with each other, they stream freely towards the detector = information on initial geometry is lost = single-inclusive particle spectrum: E dn d 3 p E E dn, p dp z d 2 = p 2 x p + p2 y transverse momentum dn dn dp z p dp dϕ tanh y p z E, dy p dp dϕ, y : longitudinal rapidity is independent of azimuthal angle ϕ = information on initial geometry is lost But: If particles interact strongly (like in a fluid), collective flow develops = initial spatial asymmetry is, by difference in pressure gradients, converted to final momentum anisotropy

6 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Characterization of collective flow Event-averaged single-inclusive particle spectrum at y = as function of ϕ: 2 dn/(dy p T dp T d )/dn/(dy p T dp T ) ϕ π v 2 = preferential emission of particles in the reaction (x z) plane ϕ = Fourier decomposition of single-inclusive particle spectrum: E dn d 3 p dn dy p dp dϕ 1 2π dn dy p dp n=1 v n (y, p ) cos(nϕ) v 1 : directed flow, v 2 : elliptic flow v 3 : triangular flow, etc.

7 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Data confronts theory v 2 /ε.25 HYDRO limits E lab /A=11.8 GeV, E877 E lab /A=4 GeV, NA49 E lab /A=158 GeV, NA49 s NN =13 GeV, STAR s NN =2 GeV, STAR Prelim (1/S) dn ch /dy Anisotropy Parameter v Hydro model π K p Λ PHENIX Data π + +π K +K p+p STAR Data K S Λ+Λ Transverse Momentum p T (GeV/c) = approach to fluid-dynamical limit with increasing centrality and beam energy = quantitative description of elliptic flow at RHIC within ideal fluid dynamics = no dissipative effects! = RHIC physicists serve up the perfect fluid

8 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Two problems (I) 1. There is no real ideal fluid! shear viscosity η T average scattering cross section σ σ minimal value for shear viscosity to entropy density ratio: η (i) from uncertainty principle ( quantum limit ): s 1 12 P. Danielewicz, M. Gyulassy, PRD 31 (1985) 53 (ii) from AdS/CFT correspondence: conjectured lower bound P. Kovtun, D.T. Son, A. Starinets, PRL 94 (25) = What is If η s η s of hot and dense hadronic matter? 1 = matter is strongly interacting! η s = 1 4π = strongly coupled quark-gluon plasma (sqgp)

9 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Two problems (II) 2. Fluid-dynamical equations of motion: µ T µν = = partial differential equations = require initial conditions on a space-time hypersurface energy-momentum tensor T µν (τ, x) on initial space-time hypersurface τ t 2 z 2 τ = const. = continuum of parameters to fit to experimental data = experimental data may allow for non-zero viscosity! = need calculations within dissipative fluid dynamics and with realistic initial conditions

10 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Dissipation vs. initial conditions M. Luzum, P. Romatschke, PRC78 (28) 34915, Erratum C79 (29) 3993 Glauber Glauber v η/s= η/s=.8 PHOBOS v 2 (percent) STAR non-flow corrected (est.) STAR event-plane η/s=1-4 η/s=.8 η/s=.16 Glauber initial conditions: η/s.8.2 η/s=.16 5 v η/s=.16.4 η/s= η/s=1-4 η/s=.8 N Part CGC PHOBOS v 2 (percent) 1 2 p T [GeV] 3 4 CGC 25 η/s= STAR non-flow corrected (est). STAR event-plane η/s=.8 η/s=.16 η/s=.24 Color-Glass Condensate (CGC) initial conditions: η/s N Part p T [GeV]

11 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Event-by-event fluctuations (I) event by event: fluctuations of initial geometry = rotate participant plane vs. reaction plane ψ 2 = induce higher flow harmonics v n, n = 3, 4,... = additional constraint on η/s and initial condition! B. Schenke, S. Jeon, C. Gale, PRC85 (212) 2491 Glauber initial conditions v n v 2 2-3% v 3 2-3% v 4 2-3% v 5 2-3% PHENIX v 2 PHENIX v 3 PHENIX v 4 η/s= v 2 2-3% v 3 2-3% v 4 2-3% v 5 2-3% PHENIX v 2 PHENIX v 3 PHENIX v 4 η/s= v 2 2-3% v 3 2-3% v 4 2-3% v 5 2-3% PHENIX v 2 PHENIX v 3 PHENIX v 4 η/s= p T [GeV] p T [GeV] p T [GeV]

12 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Event-by-event fluctuations (II) state-of-the-art initial conditions: IP-Glasma (vs. MC-Glauber and MC-KLN) B. Schenke, presentation at QM 212, Washington, D.C v 2 v 3 v 4 v 5 ATLAS 1-2%, EP.2.15 v 2 v 3 v 4 v 5 ATLAS 2-3%, EP v 2 v 3 v 4 v 5 ATLAS 4-5%, EP v n 2 1/2.1 η/s =.2 v n 2 1/2.1 η/s =.2 v n 2 1/2.15 η/s = p T [GeV] p T [GeV] p T [GeV]

13 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Fluid dynamics: degrees of freedom 1. Net charge current: N µ = n u µ + n µ u µ fluid 4-velocity, u µ u µ = u µ g µν u ν = 1 g µν diag(+,,, ) (West coast!!) metric tensor n u µ N µ net charge density in fluid rest frame n µ µν N ν N <µ> diffusion current (flow of net charge relative to u µ ), n µ u µ = µν = g µν u µ u ν projector onto 3-space orthogonal to u µ, µν u ν = 2. Energy-momentum tensor: T µν = ǫ u µ u ν (p + Π) µν + 2 q (µ u ν) + π µν ǫ u µ T µν u ν p energy density in fluid rest frame pressure in fluid rest frame Π bulk viscous pressure, p + Π 1 3 µν T µν q µ µν T νλ u λ heat flux current (flow of energy relative to u µ ), q µ u µ = π µν T <µν> shear stress tensor, π µν u µ = π µν u ν =, π µ µ = a (µν) 1 2 (aµν + a νµ ) symmetrized tensor ( a <µν> α (µ ν) β 1 ) 3 µν αβ a αβ symmetrized, traceless spatial projection

14 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Fluid dynamics: equations of motion 1. Net charge (e.g., strangeness) conservation: µ N µ = ṅ + n θ + n = ȧ u µ µ a convective (comoving) derivative (time derivative in fluid rest frame, ȧ RF t a) θ µ u µ expansion scalar 2. Energy-momentum conservation: µ T µν = energy conservation: u ν µ T µν = ǫ + (ǫ + p + Π) θ + q q u π µν µ u ν = acceleration equation: µν λ T νλ = (ǫ+p) u µ = µ (p+π) Π u µ µν q ν q µ θ q u µ µν λ π νλ µ µν ν 3-gradient, (spatial gradient in fluid rest frame, u µ RF (1,,, ))

15 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Solvability Problem: 5 equations, but 15 unknowns (for given u µ ): ǫ, p, n, Π, n µ (3), q µ (3), π µν (5) Solution: 1. clever choice of frame (Eckart, Landau,...): eliminate n µ or q µ = does not help! Promotes u µ to dynamical variable! 2. ideal fluid limit: all dissipative terms vanish, Π = n µ = q µ = π µν = = 6 unknowns: ǫ, p, n, u µ (3) (not quite there yet...) = fluid is in local thermodynamical equilibrium = provide equation of state (EOS) p(ǫ, n) to close system of equations 3. provide additional equations for dissipative quantities = dissipative relativistic fluid dynamics (a) First-order theories: e.g. generalization of Navier-Stokes (NS) equations to the relativistic case (Landau, Lifshitz) (b) Second-order theories: e.g. Israel-Stewart (IS) equations

16 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Navier-Stokes equations Navier-Stokes (NS) equations: first-order, dissipative relativistic fluid dynamics 1. bulk viscous pressure: Π NS = ζ θ ζ bulk viscosity 2. diffusion current: n µ NS = κ n µ α β 1/T inverse temperature, α β µ, µ chemical potential, κ n 3. shear stress tensor: π µν NS = 2 η σ µν η net-charge diffusion coefficient shear viscosity, σ µν = <µ u ν> shear tensor = algebraic expressions in terms of thermodynamic and fluid variables = simple... but: unstable and acausal equations of motion!! W.A. Hiscock, L. Lindblom, PRD 31 (1985) 725

17 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Israel-Stewart equations Israel-Stewart (IS) equations: second-order, dissipative relativistic fluid dynamics W. Israel, J.M. Stewart, Ann. Phys. 118 (1979) 341 Simplified version: τ Π Π + Π = Π NS τ n ṅ <µ> + n µ = n µ NS τ π π <µν> + π µν = π µν NS = dynamical (instead of algebraic) equations for dissipative terms! solution: e.g. bulk viscous pressure Π(t) = Π NS ( 1 e t/τ Π ) + Π() e t/τ Π = dissipative quantities Π, n µ, π µν relax to their respective NS values Π NS, n µ NS, π µν NS on time scales τ Π, τ n, τ π = stable and causal fluid dynamical equations of motion!

18 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Power counting (I) 3 length scales: 2 microscopic, 1 macroscopic thermal wavelength λ th β 1/T mean free path l mfp ( σ n) 1 σ averaged cross section, n T 3 = β 3 λ 3 th length scale over which macroscopic fluid fields vary L hydro, µ L 1 hydro Note: since η ( σ λ th ) 1 = l mfp λ th 1 σ n 1 λ th λ3 th σ λ th λ3 th σ λ th η s s entropy density, s n T 3 = β 3 λ 3 th = η s solely determined by the 2 microscopic length scales! Note: similar argument holds for ζ s, κ n β s

19 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Power counting (II) 3 regimes: dilute gas limit l mfp λ th η s 1 σ λ2 th = weak-coupling limit viscous fluids l mfp λ th η s 1 σ λ2 th interactions happen on the scale λ th = moderate coupling ideal fluid limit l mfp λ th η s 1 σ λ2 th = strong-coupling limit gradient (derivative) expansion: l mfp µ l mfp L hydro K δ 1 K Knudsen number = equivalent to k l mfp 1, k typical momentum scale R. Baier, P. Romatschke, D.T. Son, A.O. Starinets, M.A. Stephanov, JHEP 84 (28) 1 = separation of macroscopic fluid dynamics (large scale L hydro ) from microscopic particle dynamics (small scale l mfp )

20 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Power counting (III) Primary quantities: ǫ, p, n, s Dissipative quantities: Π, n µ, π µν If K l mfp µ δ 1, then Π ǫ nµ s πµν ǫ δ 1 Dissipative quantities are small compared to primary quantities = small deviations from local thermodynamical equilibrium! Note: statement independent of value of ζ s, κ n β s, η s! Proof: Gibbs relation: ǫ + p = Ts + µn = β ǫ s! Estimate dissipative terms by their Navier-Stokes values: Π Π NS = ζ θ, n µ n µ NS = κ n µ α, π µν π µν NS = 2 η σµν = Π ǫ ζ β ǫ β θ ζ β λ th l mfp θ l mfp µ u µ δ, s λ th l mfp n µ s κ n s µ α κ n β λ th l mfp µ α l mfp µ α δ, β s λ th l mfp π µν ǫ 2 η β ǫ β σµν 2 η s β λ th λ th l mfp l mfp σ µν l mfp <µ u ν> δ, q.e.d.

21 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Israel-Stewart equations revisited (I) additional relaxation term in IS equation is of second order in K: 1 ǫ τ Π Π 1 ǫ uµ l mfp µ Π K Π ǫ O(K2 ) = to be consistent, have to include other second-order terms as well! τ Π Π + Π = Π NS + K τ n ṅ <µ> + n µ = n µ NS + K µ τ π π <µν> + π µν = π µν NS + K µν K = ζ 1 ω µν ω µν + ζ 2 σ µν σ µν + ζ 3 θ 2 + ζ 4 ( α) 2 + ζ 5 ( p) 2 + ζ 6 α p + ζ 7 2 α + ζ 8 2 p, K µ = κ 1 σ µν ν α + κ 2 σ µν ν p + κ 3 θ µ α + κ 4 θ µ p + κ 5 ω µν ν α + κ 6 µλ ν σ λν + κ 7 µ θ, K µν = η 1 ω <µ λ ω ν>λ + η 2 θ σ µν + η 3 σ <µ λ σ ν>λ + η 4 σ <µ λ ω ν>λ + η 5 <µ α ν> α + η 6 <µ p ν> p + η 7 <µ α ν> p + η 8 <µ ν> α + η 9 <µ ν> p where ω µν <µ u ν> vorticity = second-order gradient expansion! cf. R. Baier, P. Romatschke, D.T. Son, A.O. Starinets, M.A. Stephanov, JHEP 84 (28) 1 P. Romatschke, Class. Quant. Grav. 27 (21) 256

22 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Israel-Stewart equations revisited (II) unfortunately, including second-order gradient terms renders eqs. of motion parabolic = acausal, unstable = in general, K, K µ, K µν have to be omitted!... but there is more: in principle, Π, n µ, π µν are quantities independent from = form new Lorentz-covariants: θ, µ α, µ p, σ µν, ω µν τ Π Π + Π = Π NS + K + J + R τ n ṅ <µ> + n µ = n µ NS + K µ + J µ + R µ τ π π <µν> + π µν = π µν NS + K µν + J µν + R µν J = l Πn n τ Πn n p δ ΠΠ θ Π λ Πn n α + λ Ππ π µν σ µν J µ = ω µν n ν δ nn θ n µ l nπ µ Π+l nπ µν λ π νλ +τ nπ Π µ p τ nπ π µν ν p λ nn σ µν n ν +λ nπ Π µ α λ nπ π µν ν α J µν = 2 π <µ λ ω ν>λ δ ππ θ π µν τ ππ π <µ λ σ ν>λ +λ ππ Π σ µν τ πn n <µ ν> p+l πn <µ n ν> +λ πn n <µ ν> α R = ϕ 1 Π 2 + ϕ 2 n n + ϕ 3 π µν π µν R µ = ϕ 4 π µν n ν + ϕ 5 Π n µ R µν = ϕ 6 Π π µν + ϕ 7 π <µ λ π ν>λ + ϕ 8 n <µ n ν>

23 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Matching to kinetic theory Fluid dynamics is an effective (macroscopic) theory for the long-wavelength, small-frequency limit of a given (microscopic) theory = coefficients in equations of motion can be determined by matching to the underlying theory, e.g. kinetic theory G.S. Denicol, H. Niemi, E. Molnár, DHR, PRD85 (212) 11447

24 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Details (I) 1. Boltzmann eq. k f k = C[f] for single-particle distribution function f k 2. introduce irreducible tensors of rank l: k <µ1 k µl > ν 1 ν l µ 1 µ l k ν1 k νl ν 1 ν l µ 1 µ l are projectors onto subspaces orthogonal to u µ, symmetric in µ i, ν i, and traceless 3. f k can be expanded in terms of k <µ1 k µl > where f k = f k + f k fk N l l= n= H (l) kn ρ µ 1 µ l n k <µ1 k µl > (a) f k = [exp(β u k α )+a] 1 single-particle distribution function in local equilibrium, a = ±1/ for Fermi/Bose/Boltzmann statistics (b) f k = 1 a f k (c) H (l) kn = W (l) l! P (l) kn = n r= N l m=n a(l) mn P (l) a (l) nr Er k km, where are orthogonal polynomials of order n in energy E k u k = H (l) kn are polynomials of order N l in energy E k (d) irreducible moments of δf k f k f k : ρ µ 1 µ l n = dk δf k E n k k<µ1 k µ l>

25 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Details (II) 4. rewrite Boltzmann equation in the form δ f k = f k 1 E k {k (f k + δf k ) C[f]} 5. derive equations of motion for irreducible moments, e.g. up to l = 2: ρ r = C r 1 + α () r θ G 2r θ Π + G 2r σ µν π D 2 D µν + G 3r n + (r 1)σ 2 D µν ρ µν 2 r 2 + r u µ ρ µ r 1 µ ρ µ r 1 1 [ 3 (r + 2)ρr (r 1)m 2 ] ρ r 2 θ r = C r 1 <µ> + α (1) r µ α + ω µ ν ρν r 1 [ 3 (r + 3)ρ µ r (r 1)m 2 ρ µ r 2] θ µ λ ν ρ λν r 1 1 [ 5 (2r + 3)ρ ν r 2(r 1)m 2 ρ ν ] r 2 σ µ ν 1 [ 3 (r + 3)ρr+1 rm 2 ] ρ r 1 u µ ρ <µ> ρ <µν> r + β J r+2,1 ǫ+p ( Π u µ µ Π + µν λ π λν ) µ ( ρ r+1 m 2 ρ r 1 ) + (r 1)ρ µνλ σ λν + r u ν ρ µν r 1 = C r 1 <µν> + 2α (2) r σµν 2 [ 7 (2r + 5)ρ λ<µ r 2(r 1)m 2 ρ λ<µ r 2 [ (r + 4)ρr+2 (2r + 3)m 2 ρ r + (r 1)m 4 ] ρ r 2 σ µν ) 2 ] u ν> ] σ ν> λ + 2 ( 5 <µ ρ ν> r+1 m2 ρ ν> [ r 2 5 (r + 5)ρ <µ r+1 rm 2 ρ <µ r 1 [ (r + 4)ρ µν r (r 1)m 2 ρ µν ] r 2 θ + (r 1)ρ µνλρ r 2 σ λρ µν 1 3, G nm, D nq, J nq thermodynamic functions α (l) r C <µ 1 µ l > r + 2ρ λ<µ r ωλ ν> αβ λ ρ αβλ r 1 + rρ µνλ = dk E r k k<µ1 k µ l> C[f] irreducible moment of collision integral r 1 u λ

26 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Details (III) Remarks: (a) system of infinitely many coupled equations for irreducible moments ρ µ 1 µ l r (b) system completely equivalent to Boltzmann equation (c) by definition ρ = 3 m 2 Π, ρµ = nµ, ρ µν = π µν (d) matching conditions in Landau frame imply ρ 1 = ρ 2 = ρ µ 1 = 6. fluid dynamics only involves tensors up to rank 2 = neglect irreducible moments with l > 2 7. linearize collision integral: C <µ 1 µ l > r 1 = N l n= A (l) rn ρµ 1 µ l n + O(δf 2 k ) 8. diagonalize collision matrix: (Ω 1 ) (l) A (l) Ω (l) = diag(χ (l),..., χ (l) j,...) 9. eigenmodes of linearized equations of motion: X µ 1 µ l i = equations of motion for eigenmodes decouple: Ẋ i + χ () i X i = β () i θ +... Ẋ <µ> i + χ (1) i X µ i = β (1) i µ α +... Ẋ <µν> i + χ (2) i X µν i = β (2) i σ µν +... = N l j= (Ω 1 ) (l) ij ρ µ 1 µ l j

27 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Details (IV) 1. slowest eigenmodes (w/o r.o.g. i = ) remain dynamical, all faster ones (i ) are replaced by their asymptotic (NS) values: X i β() i χ () i θ, X µ i β(1) i χ (1) i µ α, X µν i β(2) i χ (2) i σ µν 11. Since ρ µ 1 µ l i = N l j= Ω (l) ij X µ 1 µ l j : = express X, X µ, X µν in terms of Π, n µ, π µν as well as θ, µ α, σ µν = express ρ i, ρ µ i, ρ µν i in terms of Π, n µ, π µν as well as θ, µ α, σ µν = equations of motion for irreducible moments becomes identical with equations of motion for dissipative quantities Π, n µ, π µν = identify transport coefficients

28 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Discussion (I) 1. Basis of expansion for δf k is orthogonal in irreducible subspaces = truncation at any order in l and N l possible! moment approximation corresponds to choice N = 2, N 1 = 1, N 2 = and leads to IS equations 3. approximation can be systematically improved by increasing N l 4. transport coefficients approach Chapman-Enskog values already for N = 5, N 1 = 4, N 2 = 3 (41-moment approximation) Example: classical massless gas with constant cross section σ, l mfp = (σn) 1 number of moments η τ π [l mfp ] τ ππ [τ π ] λ πn [τ π ] δ ππ [τ π ] l πn [τ π ] τ πn [τ π ] 14 4/(3σβ) 5/3 1/7 4/ /(11σβ) 2 134/77.344/β 4/3.689/β.689/n /(σβ) /β 4/3.687/β.687/n /(σβ) /β 4/3.685/β.685/n number of moments κ τ n [l mfp ] δ nn [τ n ] λ nn [τ n ] λ nπ [τ n ] l nπ [τ n ] τ nπ [τ n ] 14 3/ (16σ) 9/4 1 3/5 β/2 β/ / (128σ) β.118β.295β/p /σ β.119β.297β/p /σ β.119β.297β/p

29 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Discussion (II) 5. approach can be further systematically improved: (a) consider also faster eigenmodes X i, X µ i, X µν i, i >, to be dynamical (b) take into account irreducible moments of tensor rank l > 2 (c) take into account second-order corrections in the collision integral (compute coefficients ϕ 1,..., ϕ 8 )

30 Final Colloquium, GRK 881, University Bielefeld, Germany, September 12 14, Conclusions 1. Fluid dynamics can quantitatively describe collective flow phenomena in heavyion collisions at RHIC and LHC energies 2. Determining transport coefficients by comparison to experimental data requires thorough understanding of initial conditions 3. Second-order fluid dynamics has been systematically derived as long-wavelength, small-frequency limit of kinetic theory 4. Transport coefficients agree with values from Chapman-Enskog expansion 5. Further systematic improvements are possible and should be explored