Introduction to Relativistic Hydrodynamics

Transcription

1 Introduction to Relativistic Hydrodynamics Heavy Ion Collisions and Hydrodynamics modified from B. Schenke, S. Jeon, C. Gale, Phys. Rev. Lett. 106, (2011), schenke/, Nov 17, 2012 Daniel Nowakowski TU Darmstadt, Institut für Kernphysik Seminar Relativistische Schwerionenphysik, WS 12/13 November 22nd, 2012 TUD - IKP D. Nowakowski 1

2 Motivation Heavy ion collisions #, % -/. * *!!+ )(& "# $ % $ % * 1 # &'( 23!' -. 0 November 22nd, 2012 TUD - IKP D. Nowakowski 3 K. Heckmann, TU Darmstadt, Nov. 2011

3 Temperature Heavy ion collisions A very naive picture(?) Solid Liquid Pressure Vapour November 22nd, 2012 TUD - IKP D. Nowakowski 4

4 Motivation Heavy ion collisions Question: Do we discover deconfined matter in these collisions; can we extract information about the properties of quarks and gluons? Possible answer: Hydrodynamics extract local temperatures, energy densities describe collective effects no / little detailed knowledge of microscopic physics needed, if relevant input provided externally analyse experimental data in this framework Applicable to detect signatures of deconfined matter with a hydrodynamical model? November 22nd, 2012 TUD - IKP D. Nowakowski 5

5 Relativistic hydrodynamics Introduction matter produced in heavy ion collisions has varying degrees of freedom applicability for hydrodynamical description? Limited! No clear starting point strong correlations in quark matter in vicinity of the phase transition boundary ideal-fluid like behavior of the Quark Gluon Plasma taken from U. Heinz, arxiv: nucl-th/ ; Ref. therein: J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 92, (2004) November 22nd, 2012 TUD - IKP D. Nowakowski 6

6 Classical hydrodynamics and thermodynamics A short glance Hydrodynamics continuous media with collective behavior pressure and temperature slowly varying Thermodynamics change from extensive to intensive variables ɛ = E V, s = S N, n = N V ɛ + P = Ts + µn, dɛ = Tds + µdn, c 2 s = P ɛ November 22nd, 2012 TUD - IKP D. Nowakowski 7

7 Relativistic hydrodynamics Basic equations system characterized by 4-velocity ( = c = k B = 1) u µ = ( γ, γ v ), u µ u µ = 1 (flat spacetime) local thermal equilibrium is required energy-momentum conservation yields µ T µν = 0 4 equations current conservation (Baryon number,...) requires µ N µ i = 0 1 equation November 22nd, 2012 TUD - IKP D. Nowakowski 8

8 Relativistic hydrodynamics Parametrization of hydrodynamical quantities energy-momentum tensor T µν (10 independent components) T µν = ɛu µ u ν P µν + W µ u ν + W ν u µ + π µν u µ, µν = g µν u µ u ν projection vector, tensor ɛ energy density P = P S + Π hydrostatic + bulk pressure W µ energy / heat current π µν shear stress tensor conserved current N µ i (4 k independent components) N µ i = n i u µ + V µ i n i = u µn µ i V µ i = µ ν N ν i charge density charge current g µν = diag(1, 1 3 3) November 22nd, 2012 TUD - IKP D. Nowakowski 9

9 Relativistic hydrodynamics Basic equations projection vector and tensor are orthogonal to each other u µ µν = 0 each term of T µν and N µ i can be obtained by contraction of these quantities with u µ and µν or combinations of them, like P = 1 3 µνt µν ɛ = u µ T µν u ν need initial / boundary conditions and equation of state November 22nd, 2012 TUD - IKP D. Nowakowski 10

10 Relativistic hydrodynamics What is flow? Two definitions of flow 1. Flow of energy, W µ = 0 (Landau) Landau, Lifshitz, Fluid mechanics, Pergamon Press (1959) u µ L = T µ ν uν L u α L T β α T βγ u γ L = 1 ɛ T µ ν uν L 2. Flow of conserved charge, V µ = 0 (Eckart) Eckart, Phys. Rev. 58, 919 (1940) u µ E = N µ Nν N ν ul μ u E μ V μ W μ Both definitions are related to each other by Lorentz transformations November 22nd, 2012 TUD - IKP D. Nowakowski 11 u µ E = Λµ ν uν L, uµ L uµ E + W µ ɛ+p s, u µ E uµ L + V µ n

11 Relativistic hydrodynamics Overview µ T µν = 0, µ N µ i = 0 Continuity equation: Conservation of energy: Euler equation: t (γn) + ( γn v ) = 0 t T 00 + i T i0 = 0 t (ɛ + p) γ2 v i + j (ɛ + p) γ 2 v i v j = i P Problem k unknown variables, only 5 equations. Solution 1. choose suited frame Landau: W µ = 0, but now u µ dynamical variable 2. only ideal fluids: 5 + 1k unknowns 3. additional input needed! November 22nd, 2012 TUD - IKP D. Nowakowski 12

12 Relativistic hydrodynamics Ideal hydrodynamics and dissipative effects separate ideal and dissipative parts (Landau frame) T µν = T µν 0 + δt µν N µ = N µ 0 + δnµ Ideal Dissipative Pressure P = P s + Π Π = 0 Π 0 Energy / heat current W µ W µ = 0 W µ = 0 Shear stress tensor π µν π µν = 0 π µν 0 Charge current V µ i V µ i = 0 V µ i 0 consider only ideal hydrodynamics for now 1. local thermal equilibrium 3. unique flow u µ L = uµ E 2. isotropy for the pressure 4. no viscous corrections November 22nd, 2012 TUD - IKP D. Nowakowski 13

13 Relativistic ideal hydrodynamics Local rest frame T µν 0 = (ɛ + P s ) u µ u ν P s g µν, N µ 0 = nuµ in the local rest frame (LRF) with u µ = (1, 0, 0, 0) u µ u ν ( µν ) project time (space)-like quantities the energy-momentum tensor is given by ( ɛ T LRF = ɛ(u u) + P = P s ) equations for 7 free variables (ɛ, P, n, u µ ) µ T µν 0 = 0, µ N µ 0 = 0, u µu µ = 1 another equation needed! equation of state P = P(ɛ, n) gives constraints November 22nd, 2012 TUD - IKP D. Nowakowski 14

14 Relativistic ideal hydrodynamics Summary Energy-momentum tensor and conserved current Basic equations T µν 0 = (ɛ + P s ) u µ u ν P s g µν = ɛu µ u ν P s µν, = nuµ N µ 0 µ T µν 0 = 0, µ N µ 0 = 0 local rest frame simplifies equations, e. g. µ T µν LRF = 0 0ɛ + i P i = 0 entropy density s resp. entropy current S µ = su µ conservation u ν µ T µν LRF = 0 µs µ = 0 November 22nd, 2012 TUD - IKP D. Nowakowski 15

15 Relativistic viscous hydrodynamics Gradient expansion Energy-momentum tensor T µν = T µν 0 + δt µν δt µν can contain first, second,... spatial gradients hierarchy of orders 1. Zeroth order: Ideal Hydrodynamics 2. First order: Viscous Hydrodynamics ( Navier-Stokes ) 3. Second order: Viscous Hydrodynamics ( Israel-Müller-Stewart theory ) corresponds to modifying the entropy current according to ( S µ = su µ + O (δt µν ) + O (δt µν ) 2) +... I. Müller, Zeitschrift f. Physik 198, (1967) W. Israel, J. M. Stewart, Phys. Lett. A 58, 4 (1976), November 22nd, 2012 TUD - IKP D. Nowakowski 16

16 Relativistic viscous hydrodynamics Basic equations allow dissipative terms, but no charge in the system assume entropy current has additional linear dissipative terms (first-order theory) S µ = su µ + αv µ phenomenological definitions resp. so called constitutive equations for the shear stress tensor π µν and for the bulk pressure Π ( 1 ( ) π µν = 2η µ 2 α ν β + µ β ν α 1 ) 3 µν αβ ακ κ u β Π = ξ µ u µ transport coefficients: η shear viscosity, ξ bulk viscosity,... characterize deviation from thermal equilibrium November 22nd, 2012 TUD - IKP D. Nowakowski 17

17 Relativistic viscous hydrodynamics Transport coefficients Shear viscosity: fluid s resistance to shear forces Bulk viscosity: fluid s resistance to compression November 22nd, 2012 TUD - IKP D. Nowakowski 18

18 Relativistic viscous hydrodynamics Summary entropy current conserved or increasing for viscous hydrodynamics T µ S µ = T ( u µ µ s + s µ u µ), µ N µ = 0, n = 0 = u ν µ T µν 0, Ts = ɛ + P µn = u ν µ (δt µν ), µ T µν = 0, inserting definitions from constitutive equations yields µ S µ = π µνπ µν 2η + Π2 ξ µ S µ 0 November 22nd, 2012 TUD - IKP D. Nowakowski 19

19 Relativistic hydrodynamics Equations of motion Describing dynamics of the system 1. From u ν µ T µν = 0 ɛ = (ɛ + P s + Π)θ + π µν ( 1 2 ( ) µ α ν β + µ β ν α 1 ) 3 µν αβ ακ κ u β 2. From µα β T αβ = 0 where (ɛ + P s + Π) u µ = µν ν (P s + Π) µα βγ γ π αβ + π µα u α θ = µ u µ expansion scalar ( V /V ) ȧ = u µ µ a substantial (co-moving) time derivative November 22nd, 2012 TUD - IKP D. Nowakowski 20

20 Relativistic hydrodynamics Equations of motion First equation of motion ( 1 ɛ = (ɛ + P s + Π)θ + π µν 2 (( 1 + ξθ 2 + 2η 2 = (ɛ + P s )θ }{{} a. ( ) µ α ν β + µ β ν α 1 ) 3 µν αβ ακ κ u β ( ) µ α ν β + µ β ν α 1 ) ) 2 3 µν αβ ακ κ u β }{{} b. Time evolution of the energy density in the co-moving system a. change of energy density and hydrostatic pressure due to expansion / dilution resp. changing volume b. production of entropy due to dissipative effects heating of the system November 22nd, 2012 TUD - IKP D. Nowakowski 21

21 Heavy ion collisions Oct 30, 2012 Idea: study heavy ion collisions with hydrodynamics E. Fermi, Prog. Theor. Phys. 5 (1950) 570; Phys. Rev. 81 (1951) 683. November 22nd, 2012 TUD - IKP D. Nowakowski 22

22 Heavy ion collisions Schematic time line t proper time τ = t 2 z 2 z space-time rapidity η s = 1/2 ln (t + z) (t z) Coordinates: t = τ cosh η s and z = τ sinh η s, red: τ = const, green: η s = const v z = z t November 22nd, 2012 TUD - IKP D. Nowakowski 23

23 Heavy ion collisions Schematic time line t z heavy ions collide quark gluon plasma freeze out pre-equilibrium hydrodynamics free-streaming November 22nd, 2012 TUD - IKP D. Nowakowski 23

24 Heavy ion collisions Schematic time line t z Different stages of a HIC 1. initial stage initial conditions 2. intermediate stage equation of state 3. final stage decoupling hydrodynamical description? pre-equilibrium hydrodynamics free-streaming November 22nd, 2012 TUD - IKP D. Nowakowski 23

25 Heavy ion collisions Initial conditions dynamics of particle production cannot be described in hydrodynamics specify thermodynamical state of matter and initial velocity two sets of initial conditions: 1. Landau: nuclei stopped by collision, no initial dependence Landau, Izv. Akad. Nauk Ser. Fiz Bjorken: particle production is frame-independent, boost invariance of initial conditions Aad, G. and Gray, H. M. and Marshall, Z. and Mateos, D. Lopez and Perez, K. et al., ATLAS collaboration, Phys. Lett. B710 (2012) November 22nd, 2012 TUD - IKP D. Nowakowski 24

26 Heavy ion collisions Bjorken model (Bjorken, Phys. Rev. D 27, (1983)) November 22nd, 2012 TUD - IKP D. Nowakowski 25

27 Heavy ion collisions Bjorken model t early thermalization vanishing Baryon number for the fluid one-dimensional expansion z boost symmetry of initial conditions no initial dependence on rapidity y because no dependence on Lorentz boost angle fluid rapidity is the same as spacetime rapidity (E large) η s = y November 22nd, 2012 TUD - IKP D. Nowakowski 26

28 Heavy ion collisions Bjorken model (Bjorken, Phys. Rev. D 27, (1983)) valid for times of the order from τ 1 fm/c to τ 5 10 fm/c specify to expansion along z direction introduce a boost-invariant four-velocity u µ = x µ τ = t τ ( 1, 0, 0, z ) = t due to Lorentz-symmetry and initial conditions ɛ = ɛ(τ, y) ɛ(τ) P = P(τ, y) P(τ) 1 (t, 0, 0, z) t 2 z2 T = T (τ, y) T (τ) = β 1 (τ) November 22nd, 2012 TUD - IKP D. Nowakowski 27

29 Heavy ion collisions Bjorken model first equation of motion simplifies to dɛ dτ = ɛ + P s τ = ɛ + P s τ 1 (ɛ + P s ) η τ 2 Ts + 4 ( τT η s 1 τt + ɛ + P s ) Ts ξ s ξ τ 2 a. time-evolution of energy density is governed by the sum e + P s per proper time τ for ideal hydrodynamics b. last two terms on the RHS are viscous corrections with appearing dimensionless quantities η/s and ξ/s which characterize intrinsic properties of the fluid November 22nd, 2012 TUD - IKP D. Nowakowski 28

30 Heavy ion collisions Bjorken model first equation of motion simplifies to dɛ dτ = ɛ + P s τ description of the time evolution of the system in a simple (solvable) way good approximation, but for detailed calculations viscous effects need to be considered November 22nd, 2012 TUD - IKP D. Nowakowski 28

31 Heavy ion collisions Bjorken model and an equation of state simple equation of state solutions for equations of motion P = γɛ, γ ideal gas = 1 3 ( τ0 ) 1+γ ɛ(τ) = ɛ 0 ( τ τ0 ) γ T (τ) = T 0 and τ s 0 τ 0 = sτ November 22nd, 2012 TUD - IKP D. Nowakowski 29

32 Heavy ion collisions Bjorken model and observables energy change per unit of rapidity can be measured and calculated de dy = d 3 V dy ɛ(τ f ) = }{{} πr 2 d 2 x τ f ɛ 0 ( τ0 τ f ) 1+γ = πr 2 ɛ 0 τ 0 ( τ0 assume no hydrodynamical expansion any more (τ τ f ) ɛ 0 = 1 de πr 2 τ 0 dy = m t dn πr 2 dz z=0 = m t dy πr 2 dz dn z=0 }{{} dy 1/τ 0 allows to estimate the initial energy density τ f ) γ November 22nd, 2012 TUD - IKP D. Nowakowski 30

33 Heavy ion collisions Bjorken model: Does a QGP occur in HICs? energy RHIC [GeV fm -2 c -1 ] Bj τ GeV 130 GeV 19.6 GeV N p S. S. Adler et al. (PHENIX collaboration), Phys. Rev. C 71, (2005) November 22nd, 2012 TUD - IKP D. Nowakowski 31

34 Heavy ion collisions Elliptic flow y b x initial geometric anisotropy gets transformed to anisotropies in particle momenta spectrum expanding system develops flow pattern azimuthal distribution of emitted particles with respect to reaction plane dn dp T dφdy = v n (p T ) cos nφ elliptic flow v 2 sensitive to viscous effects n November 22nd, 2012 TUD - IKP D. Nowakowski 32

35 Heavy ion collisions Bjorken model: Elliptic flow Euler equation for v x t v x = 1 P e + P x = ln s c2 s x assume gaussian entropy profile from the collision ( ( )) s(x, y) = s 0 exp 1 σ 2 y x 2 + σx 2y 2 2 σxσ 2 y 2 solution of Euler equation v x (t) = c2 s tx + v σx 2 x,0, v y (t) = c2 s ty + v σy 2 y,0 non-central collision (σ x < σ y ) implies v x > v y anisotropy in particle spectrum details next week November 22nd, 2012 TUD - IKP D. Nowakowski 33

36 Bjorken model Violation of causality in first order theory linearized Euler equation for small perturbations of v y v y + δv y ( ) t δv y η ( ) ɛ + P 2 x δv y = 0 allow sinusoidal perturbation of the form δv y (t, x) exp (ωt ikx) dispersion relation with wave-number k is given by η ω = ɛ + P k 2 estimate speed of mode with wave-number k v(k) = dω dk = 2η ɛ + P k for k perturbations with k propagate with infinite speed November 22nd, 2012 TUD - IKP D. Nowakowski 34

37 Heavy ion collision and the Bjorken model Short summary Bjorken model assumes Boost-invariance of initial conditions describes heavy ion collisions within hydrodynamical framework one-dimensional expansion along z for τ 1 10 fm/c Problems in first order theory: violation of causality solutions show instabilities W. A. Hiscock, L. Lindblom, Phys. Rev. D 31, (1985) Solution: Use second-order theory W. Israel and J. M. Stewart, Annals Phys. 118, 341 (1979) introduce relaxation time in equations of motion no acausality November 22nd, 2012 TUD - IKP D. Nowakowski 35

38 Summary Hydrodynamics offers simple formalism to describe heavy ion collisions strong assumption required: local thermal equilibrium relies on initial conditions, equation of state and freeze-out description Bjorken model experimental data (might) agree well with predictions in a certain range see talk next week by J. Onderwaater November 22nd, 2012 TUD - IKP D. Nowakowski 36

39 Outlook... it is by no means clear that the highly excited, but still small systems produced in those violent collisions satisfy the criteria justifying a dynamical treatment in terms of a macroscopic theory which follows idealized laws. U. Heinz, arxiv:nucl-th/ systematical improvements of hydrodynamical description many numerical simulations available other (effective) description of heavy ion collisions extend to include anisotropies, turbulence, non-equilibrium... November 22nd, 2012 TUD - IKP D. Nowakowski 37

40 Literature [1] T. Hirano, N. van der Kolk, A. Bilandzic Hydrodynamic and Flow arxiv:nucl-th/ (2008) [2] P. Huovinen and P. V. Ruuskanen Hydrodynamic Models for Heavy Ion Collisions Annu. Rev. Nucl. Particle Science 56 (2006), arxiv:nucl-th/ [3] U. Heinz Early collective expansion: Relativistic hydrodynamics and the transport properties of QCD matter arxiv:nucl-th/ (2009) November 22nd, 2012 TUD - IKP D. Nowakowski 38