Hydrodynamic Fluctuations in relativistic heavy ion collisions

Transcription

1 Hydrodynamic Fluctuations in relativistic heavy ion collisions J.I. Kapusta, BM & M. Stephanov, PRC 85, (2012) Berndt Müller INT Workshop on the Ridge 5-11 May 2012

2 Sources of fluctuations Initial-state fluctuations Hydrodynamic fluctuations Fluctuations during hadronization Jet-medium interactions Fluctuations are generated during the entire evolution Initial-state, hadronization, and jet-induced fluctuations are controlled by other physics. Hydrodynamic fluctuations are uniquely predictable. 2

3 Molecular Dynamics Lubrication Equation Stochastic Lubrication Equation

4 Particle number fluctuations (ΔN) 2 ~ N ~ ρv? 4

5 Particle number fluctuations λf (ΔN) 2 ~ N ~ ρv? (ΔN) 2 ~ ρλf 4

6 Particle number fluctuations λf (ΔN) 2 ~ N ~ ρv? (ΔN) 2 ~ ρλf (Δp) 2 ~ ρλf p ~ η 4

7 Relativistic Dissipative Fluid Dynamics In the Landau-Lifshitz approach u μ is the velocity of energy transport. ΔT µν = η( Δ µ u ν + Δ ν u µ ) + ( 2 3η ζ )H µν ρ u ρ ΔJ µ B = χ ( n B T / w) 2 Δ µ ( µ B / T ) s µ = su µ µ B T ΔJ µ B µ s µ = η ( 2T iu j + j u i 2 3δ ij k u k ) 2 + ζ ( T ku k ) 2 + χ T 2 ( k T + T u k ) 2 > 0

8 Landau s theory of hydrodynamic fluctuations Stochastic source Fluctuation - dissipation theorem: S µν is related to ΔT µν Text µν S vis ( x)s vis αβ ( ) + ζ 2 3η ( y) = 2T η H µα H νβ + H µβ H να ( )H µν H αβ δ 4 ( x y) µν S heat αβ ( x)s heat ( y) = 2χT 2 H µα u ν u β + H νβ u µ u α + H µβ u ν u α + H να u µ u β δ 4 ( x y) µν S vis αβ ( x)s heat ( y) = 0 N. Salie, R. Wuffert, and W. Zimdahl, J. Phys. A 16, 3533 (1983). E. Calzetta, Class. Quant. Grav. 15, 653 (1998).

9 Solution procedure µ T µν ideal + µ ΔT µν vis = µ S µν Choose initial conditions Solve hydro equations for arbitrary sources S μν Calculate correlations / fluctuations of observables Average over stochastic sources Apply thermal freeze-out smearing (Cooper-Frye)

10 Bjorken scaling hydrodynamics τ = t 2 z 2, ξ = tanh 1 (z / t), t = τ coshξ, z = τ sinh u 0 = cosh( ξ + ω ), u 3 = sinh( ξ + ω ) T = T 0 (τ ) + δt (ξ,τ ) P = P 0 (τ ) + δ P(ξ,τ ) ε = ε 0 (τ ) + δε(ξ,τ ) δε = c V (T )δt w = ε + P δ P = s(t )δt δ s = δε / T ρ = δ s / s ΔT µν vis = ( 4 3 η + ς )( u)h µν S µν = w(τ ) f (ξ,τ )h µν f (ξ,τ ) f (ξ ',τ ') = 2T (τ ) Aτw(τ ) η(τ ) + ς(τ ) [ ]δ (τ τ ')δ (ξ ξ ')

11 Equations of motion Simplifying assumptions: v 2 s = 1 3 = const., γ = 1/ 1 v 2 s s, ν= ( 4 3 η + ς ) / s = 1 3π Fourier transform: 9

12 Correlators Set 10

13 Inviscid case Analytic solution for Green functions: with 11

14 Sound horizon Singular, when there is noise source propagating to ξ at τf. ξ1 source ξ Gaussian smeared Gsing source ξ1 =ξ2 Grr sing x 12

15 Regular part of G Grr reg x Greg = G - Gsing Sound horizon: 13

16 Viscous case / thermal noise permits approximate analytic solution: Physical noise is not delta-function in ξ. Thermal noise correlator: with 14

17 Freeze-out smearing Cooper-Frye formula: Rapidity distribution of emitted particles: Rapidity fluctuation of emitted particles: 15

18 Rapidity fluctuations Relative rapidity fluctuations, with Inviscid case with thermal freeze-out smearing only 16

19 Rapidity fluctuations Relative rapidity fluctuations, with 3.0 KHDhL Inviscid case with thermal freeze-out smearing only Dh 16

20 Rapidity fluctuations Relative rapidity fluctuations, with 3.0 KHDhL Inviscid case with thermal freeze-out smearing only τ0 = 0.16 fm/c T0 = 600 MeV τf = 10 fm/c 0.0 Tf = 150 MeV Dh 16

21 Initial time dependence 6 KHDhL T0 = 600 MeV T0 = 400 MeV T0 = 300 MeV τ0 = 0.16 fm/c τ0 = 0.52 fm/c τ0 = 1.25 fm/c Dh 17

22 Various scenarios (all include freeze-out smearing) KHDhL Viscosity only Dh 18

23 Various scenarios (all include freeze-out smearing) KHDhL Viscosity only Dh KHDhL Thermal noise Dh 18

24 Various scenarios (all include freeze-out smearing) KHDhL Viscosity only Dh KHDhL Thermal noise Dh KHDhL Viscosity and thermal noise Dh 18

25 Various scenarios (all include freeze-out smearing) KHDhL KHDhL Viscosity only Dh Viscosity and thermal noise Dh KHDhL K ini Thermal noise Dh Initial fluctuation and viscosity, no hydro fluctuations Dh 18

26 Summary Hydrodynamic fluctuations can be model independently predicted given a hydro scenario. Hydrodynamic fluctuations encode important information on transport coefficients and speed of sound. Hydrodynamic fluctuations can generate rather long range (Δη ~ 4) rapidity correlations in e-by-e observables. There is plenty of work to do: Numerical implementation of fluctuating hydrodynamics Interplay between initial-state and hydrodynamic fluctuations Theory of thermal broadening of hydrodynamical fluctuations Incorporation of critical fluctuations Fluctuations of locally conserved quantities