Relativistic hydrodynamics for heavy-ion physics

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1 heavy-ion physics Universität Heidelberg June 27, / 26

2 Collision time line 2 / 26

3 3 / 26

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5 Space-time diagram proper time: τ = t 2 z 2 space-time rapidity η s : t = τ cosh(η s ) z = τ sinh(η s ) η s = 1 t+z 2 ln t z fluid rapidity Y : v z = tanh (Y ) 5 / 26

6 Why hydrodynamics? inviscid hydrodynamics describe experimental data Kn 1 with Kn λ/r λ mean free path; R characteristic dimension of system one (strong) assumption: µ p and µ T small local thermodynamic equilibrium 6 / 26

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8 Equations of hydrodynamics 4-velocity u µ : u 0 = 1 1 v 2 ; ui = v i 1 v 2 Baryon number conservation: µ n µ = µ (nu µ ) = 0 energy-momentum tensor: T µν = (ɛ + p)u µ u ν pg µν with T 00 energy density, T 0j momentum density, T i0 energy flux and T ij momentum flux. In rest frame: ɛ T = 0 p p p conservation of energy and momentum: µ T µν = 0 8 / 26

9 (Hydrodynamic Basics) We have five equations: µ T µν = 0 and µ (nu µ ) = 0 6 Variables: ɛ(x), p(x), n B (x), u(x) equation of state needed 9 / 26

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11 Equations of thermodynamics differential of internal energy : du = pdv + TdS + µdn V, S and N extensive variables: U = pv + TS + µn Gibbs-Duhem: Vdp = SdT + Ndµ need intensive quantities ɛ U/V, s S/V, n N/V : ɛ = p + Ts + µn dp = sdt + ndµ dɛ = Tds + µdn 11 / 26

12 Velocity of sound ( ) 1 velocity of sound c s is defined by: c s p 2 ɛ with a small disturbance δɛ and δp, so that ɛ = ɛ 0 + δɛ and p = p 0 + δp delivers: t (δɛ) + (ɛ 0 + p 0 ) v = 0 (ɛ 0 + p 0 ) v t + δp = 0 using definition of c s as δp = c s 2 δɛ: 2 (δɛ) t 2 c s 2 (δɛ) = 0 12 / 26

13 Equations of state for massless particles let µ = 0 ɛ = g 3π2 90 T 4 [ Fermions ɛ = g 3π2 90 T ] p = ɛ 3 n = g π2 90 T 3 ζ(3) π2 ζ(4) [ Fermions n = g 90 T 3 7ζ(3) 6ζ(4) ] g 40 number of degrees of freedom g = 37 [u,d] ; g = 47.5 [u,d,s] With ɛ + p = Ts: s = 3nT +nt T Estimation of c s : c 2 s = p ɛ = 1 3 = 4n 13 / 26

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15 Bjorken Model I: Simplified model for massless particles Expansion only in z-direction u µ = 1 τ 0 (t, 0, 0, z) ɛ(τ, y) = ɛ(τ), p(τ, y) = p(τ), β(τ, y) = β(τ) µ T µν = 0 simplifies to dɛ dτ = ɛ+p τ = 4 3 ɛ(τ) = ɛ(τ 0 ) ( τ τ 0 ) 4 3 ɛ τ µ s µ = 0 ds dτ = s τ s(τ) = s(τ 0) τ 0 τ hence: s T = ɛ + p = 4 3 ɛ T (τ) = T (τ 0 )( τ τ 0 ) 1 3 Estimation of the lifetime of the QGP: τ c τ 0 = τ 0 (( T 0 T c ) 3 1) 15 / 26

16 Bjorken Model II [ more general with mass ] Tr T 0 ɛ 3p With ɛ τ = ɛ+p τ ( τ τ 0 ) 4 3 ɛ(τ) ɛ(τ 0 ) ( τ τ 0 ) 1 n(τ, y) = n(τ) leads to constant rapidity density = const dn dy initial condition ɛ(τ 0 )τ 0 =? 16 / 26

17 Estimation of the energy density de = dn < m T cosh(y) > y=0 ɛ 0 = <m T > dn A dz = <m T > A dn dy dy ; [ dy dz z=0 = d dz atanh( z τ ) z=0 = 1 τ ] dz y=0 = <m T > Aτ 0 dn dy ɛ 0 τ 0 = 1 A de T dy 17 / 26

18 Estimation of the energy density A πr 2 Pb 140fm2 dn dy y= ± 92 from (Pb+Pb@ s NN = 2.76TeV) ɛ 0 τ 0 (11.7 ± 0.43) GeV fm 2 for an initial time of τ 0 1fm this leads to ɛ 0 10 GeV fm 3 from ɛ 0 = g π2 30 T 0 4 we get initial temperature T 0 280MeV and with T c 170MeV the livetime τ c τ fm c 18 / 26

19 Bjorken energy density from data 19 / 26

20 Longitudinal acceleration more general distribution of s(τ, x, y, η s ) acceleration x,y,z-direction Assumption: s(x, y, η s ) exp( 1 2 (( x ) 2 + ( y ) 2 + ( η ) 2 ) σ x σ y σ η from Euler-equation we get: v z t = 1 p ɛ + p z = ln(s) c2 s z Y (τ) = (1 + c2 s ln τ τ 0 )η ση 2 s σ η leads to the Bjorkencondition 20 / 26

21 Transversal acceleration vx t = 1 p ɛ+p x = c2 ln(s) s x v x = cs 2 x t σx 2 v y = cs 2 y t σy 2 σ x < σ y non-central collision dn dφ = 1 + 2v 2cos(2Φ), v 2 elliptic flow v 2 ɛ σ2 y σ2 x σ 2 y +σ2 x 21 / 26

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23 radial flow u = u 1 u 1 + u 2 u 2 ; m0 2 = m2 t pt 2 d dm t log u 0+um t/p t T dn 2πp tdp tdp z 23 / 26

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25 hydrodynamics describe experimental data simplifications enable one to solve hydrodynamic equations we can estimate the order of magnitude of ɛ, p and T as a function of time 25 / 26

26 References J-Y Ollitraut, heavy-ion collision, Eur.J.Phys.29: ,2008, arxiv: v2 J.D. Bjorken, Highly relativistic nucleus-nucleus collision: The central rapidity region, Phys. Rev. D27 1 (1983) 26 / 26