Frankfurt Institute for Advanced Studies June 27, 2011 with G. Denicol, E. Molnar, P. Huovinen, D. H. Rischke
1 Fluid dynamics (Navier-Stokes equations) Conservation laws momentum conservation Thermal equilibrium Navier-Stokes equations 2 Relativistic fluid dynamics Relativistic conservation laws Relativistic Navier-Stokes equations Transient fluid dynamics 3 Heavy-ion collisions Strongly interacting matter Stages Stages Equation of State Initial conditions Freeze-out η/s 4 RHIC 5 LHC 2760 GeV 6 LHC 5500 GeV 7 Summary
Part I: Fluid dynamics
Conservation laws In fluid dynamics dynamics a state of continuous matter is described in terms of few macroscopic fields like: mass density ρ(t, x) fluid velocity v(t, x) t = time, x = (x, y, z) Basic laws coverning the dynamics of the system are conservation of energy and momentum. e.g. conservation of mass tρ + (vρ) = 0 or tρ = v ρ ρ v v ρ = change of density by matter carried away by the velocity field ρ v = change of density by expansion of the matter
momentum conservation momentum density M = ρ(t, x)v(t, x) Conservation of momentum tm + v M + M v = P + π P = isotropic pressure π = π ij = shear stress tensor i, j = x, y, z (non-isotropic viscous pressure ) In general P and π ij depend on the microscopic state of the matter equations exact, but not yet closed We need to express P and π ij in terms of ρ and v
Thermal equilibrium Equilibrium Put N particles with total energy E into a box with volume V and wait... If we wait sufficiently long, the system thermalizes i.e. it forgets its history and the state of the matter depends only on E, N and V. Especially pressure is given by the Equation of State P = P eq(e, N, V ) Local Equilibrium At each point (t, x) state of the matter is determined by n = N/V and e = E/V P = P eq(e, n) Locally isotropic state (no preferred direction) π ij = 0
Euler equations If pressure determined by P = P eq(ρ) (can be good approximation when temperature T constant): Euler equations tρ + (vρ) = 0 tm + v M + M v = P eq(ρ) Non-viscous or perfect fluid dynamics Generally one needs one additional equation for the energy conservation + additional field e(t, x) = energy density
Navier-Stokes equations Gradient expansion: system still close to local equilibrium: P = P eq ζ v π ij = 2η <i v j> ζ = bulk viscosity coefficient ( v = expansion rate) η = shear viscosity coefficient ( <i v j> = shear rate) Bulk and shear viscosity tell how fluid resists deformations Navier-Stokes equations tρ + (vρ) = 0 tm i + v M i + M i v = i P + j π ij
Relativistic conservation laws Usually mass in not conserved in the relativistic systems More straightforward to consider energy conservation Define energy density T 00, momentum density T 0i Flux of x-momentum to y-direction T xy, etc. energy-momentum tensor T µν (µ, ν = t, x, y, z) Energy and momentum conservation µt µν = 0 T µν = (e + P)u µ u ν Pg µν + π µν 4-velocity u µ = γ(1, v), where γ = (1 v 2 ) 1
Relativistic Navier-Stokes equations Gradient expansion: (state of the matter determined by e and u µ ) P = P eq(e) ζ µu µ π µν = 2η <µ u ν> Both relativistic and non-relativistic Navier-Stokes equations are parabolic unlimited signal propagation speed (ACAUSAL: not good for relativistic fluids!) Acausality leads to unstable relativistic theory
Transient fluid dynamics (Israel-Stewart theory) π µν not directly proportional to the shear rate <µ u ν> Instead π µν relaxes to the Navier-Stokes value within relaxation time τ π relaxation equation for π µν d dt π µν = 1 ( π µν 2η µ u ν ) τ π Stable and causal for range of the transport coefficients η and τ π In the limit τ π 0 reduces back to Navier-Stokes theory State of the matter: e, u µ, π µν
Part II: Ultrarelativistic heavy-ion collisions
Strongly interacting matter Strongly interacting matter = matter where the fundamental building blocks are quarks and gluons. Theory = Quantum ChromoDynamics (QCD) At low temperature/density quarks and gluons are bound to hadrons (protons, neutrons, pions etc.) QCD calculations Sufficiently high temperature or density: transition from hadronic matter to Quark-Gluon Plasma (QGP) Question: Where can we find this matter? Neutrons stars (too far...) Early universe (too long ago...) Can we produce it in a laboratory?
Yes! Smash heavy ions together!
Stages of the heavy-ion collisions Initial particle production τ 1 fm/c: Two (Lorentz-contracted) nuclei go through each other, leaving highly exited matter between Non-equilibrium evolution of the produced matter (aka thermalization) τ 1 fm/c (Fluid dynamical) evolution of the QGP τ 5 fm/c
Stages of the heavy-ion collisions Transition back to hadronic matter (QCD phase transition) Hadronic evolution (interacting hadron gas) Transition to free particles (Freeze-out)
Israel-Stewart hydrodynamics Neglect net-baryon number, bulk viscosity & heat flow µt µν = 0 Dπ µν = 1 τ π ( π µν 2η µ u ν ) 4 3 πµν ( λ u λ) Longitudinal expansion is treated using boost invariance: p = 0, v η z = z s t To solve this set of equations we need at τ = τ 0 Equation of state p = p(e) and T = T (e) Initial condition T µν (τ 0, x, y) Shear viscous coefficient η(t ) and relaxation time τ π(t )
Equation of State 1 s95p-v1 s95p-pce150-v1 Bag Model 0.3 0.25 0.2 p [GeV/fm 3 ] 0.1 0.01 T [GeV] 0.15 0.1 0.05 0.001 0.01 0.1 1 10 e [GeV/fm 3 ] 0 0.01 0.1 1 10 e [GeV/fm 3 ] Lattice parametrization by Petreczky/Huovinen: Nucl. Phys. A837, 26-53 (2010), [arxiv:0912.2541 [hep-ph]]. Chemical equilibrium (s95p-v1) (partial) chemical freeze-out at T chem = 150 MeV (s95p-pce150-v1) for comparison bag-model EoS Hadron Resonance Gas (HRG) includes all hadronic states up to m 2 GeV
Initial profiles Initial energy density proportional to the density of binary nucleon-nucleon collisions (optical Glauber) Centrality selection according to Glauber Initial shear viscosity π µν = 0 τ 0 = 1.0 fm (RHIC) τ 0 = 0.6 fm (LHC) Initial velocity v x = v y = 0 snn [GeV] τ 0 [fm] ε 0 [GeV/fm 3 ] T max [MeV] 200 1.0 24.0 335 2760 0.6 187.0 506 5500 0.6 240.0 594
Freeze-out where Converting fluid to particles e, u µ, π µν E dn d 3 p Standard Cooper-Frye freeze-out for particle i E dn d 3 p = g i (2π) 3 dσ µ p µf i (p, x), [ ] f i (p, x) = f i,eq (T, {µ i }) 1 + πµν p µp ν 2T 2 (e + p) Integral over constant temperature hypersurface 2- and 3-body decays of unstable hadrons included Here T dec = 100 MeV dn dydp 2 T dφ = v 2 (p T ) = elliptic flow coefficient dn dydpt 2 (1 + 2v 2 (p T ) cos(2φ) + )
Temperature dependent η/s 1.0 0.8 LH-LQ LH-HQ HH-LQ HH-HQ η/s 0.6 0.4 0.2 0.0 0.10 0.20 0.30 0.40 0.50 T [GeV] Can we separate hadronic viscosity from the QGP viscosity? Try 4 different parametrization of η/s(t ).
HRG vs. QGP viscosity at RHIC Au+Au s NN = 200 GeV dn/dydp T 2 [1/GeV 2 ] 10 3 10 2 10 1 10 0 10-1 10-2 10-3 10-4 0.25 0.20 p/100 K + /10 0-5 % RHIC 200 AGeV π + LH-LQ LH-HQ HH-LQ HH-HQ PHENIX η/s STAR v 2 {4} 0.0 0.10 0.20 0.30 0.40 0.50 1.0 0.8 0.6 0.4 0.2 LH-LQ LH-HQ HH-LQ HH-HQ T [GeV] v 2 (p T ) 0.15 0.10 0.05 0.00 0 p T [GeV] 20-30 % charged hadrons Elliptic flow insensitive to the QGP viscosity Weak sensitivity of p T -slopes on the QGP viscosity Behavior of v 2 (p T ) dominated by HRG viscosity
HRG vs. QGP viscosity at RHIC: Change the minimum η/s dn/dydp T 2 [1/GeV 2 ] 10 3 10 2 10 1 10 0 10-1 10-2 10-3 10-4 0.25 0-5 % K + /10 p/100 RHIC 200 AGeV STAR v 2 [4] π + η/s min = 0.08 η/s min = 0.16 PHENIX η/s 1.0 0.8 0.6 0.4 0.2 HH-HQ η/s min = 0.16 0.20 0.0 0.10 0.20 0.30 0.40 0.50 T [GeV] v 2 (p T ) 0.15 0.10 20-30 % charged hadrons Increase minimum η/s by a factor 2 0.05 v 2 (p T ) sensitive to the minimum value 0.00 0 1 2 3 p T [GeV]
HRG vs. QGP viscosity at LHC Pb+Pb 2760 AGeV dn/dydp T 2 [1/GeV 2 ] 10 3 10 2 10 1 10 0 10-1 10-2 p/100 K + /10 0-5 % π + LH-LQ LH-HQ HH-LQ HH-HQ η/s 1.0 0.8 0.6 LH-LQ LH-HQ HH-LQ HH-HQ 10-3 10-4 0.25 0.20 LHC 2760 AGeV ALICE v 2 [4] 0.4 0.2 0.0 0.10 0.20 0.30 0.40 0.50 T [GeV] v 2 (p T ) 0.15 0.10 20-30 % charged hadrons LHC s = 2760 AGeV Both QGP and HRG η/s change v 2 (p T ) 0.05 Stronger effect of QGP η/s to p T -slopes 0.00 0 1 2 3 p T [GeV]
HRG vs. QGP viscosity at LHC Pb+Pb 5500 AGeV dn/dydp T 2 [1/GeV 2 ] 10 3 10 2 10 1 10 0 10-1 10-2 p/100 K + /10 0-5 % π + LH-LQ LH-HQ HH-LQ HH-HQ η/s 1.0 0.8 0.6 0.4 LH-LQ LH-HQ HH-LQ HH-HQ v 2 (p T ) 10-3 10-4 LHC 5500 AGeV 0.25 0.20 0.15 20-30 % 0.10 charged hadrons 0.05 0.00 0 1 2 3 p T [GeV] 0.2 0.0 0.10 0.20 0.30 0.40 0.50 T [GeV] LHC s = 5500 AGeV multiplicity from minijet+saturation model (prediction) Eskola et al., Phys. Rev. C 72, 044904 (2005). Note the difference: v 2 (p T ) curves group according to the QGP viscosity!!
Summary RHIC Au+Au s NN = 200 GeV v 2 (p T ) is almost independent of high-temperature η/s, but very sensitive to the hadronic η/s Still some sensitivity to minimum value of η/s LHC Pb+Pb s NN = 5.5 TeV (prediction) v 2 (p T ) depends on the high-temperature η/s v 2 (p T ) almost independent of the hadronic viscosity LHC Pb+Pb s NN = 2.76 TeV Somewhere between: v 2 (p T ) sensitive on the QGP and hadronic viscosity