Introduction to Relativistic Hydrodynamics

Introduction to Relativistic Hydrodynamics 

Heavy Ion Collisions and Hydrodynamics 

modified from B. Schenke, S. Jeon, C. Gale, Phys. Rev. Lett. 106, 042301 (2011), http://www.physics.mcgill.ca/˜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

Motivation 

Heavy ion collisions 

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November 22nd, 2012 | TUD - IKP | D. Nowakowski | 3 

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K. Heckmann, TU Darmstadt, Nov. 2011


A very naive picture(?) 


Pressure 

Solid Liquid 

Vapour 

Temperature

Motivation 


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? 


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/0901.4355; Ref. therein: J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 92, 052302 (2004) 


Classical hydrodynamics and thermodynamics 

A short glance 


◮ 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 ∂P 

s = 

∂ɛ


Basic equations 

◮ system characterized by 4-velocity ( = c = kB = 1) 

u µ = γ, γv , u µ uµ = 1 (flat spacetime) 

◮ local thermal equilibrium is required 

◮ energy-momentum conservation yields 

∂µT µν = 0 ⇒ 4 equations 

◮ current conservation (Baryon number, ...) requires 


∂µN µ 

i 

= 0 ⇒ 1 equation


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 = PS + Π hydrostatic + bulk pressure 

W µ 

energy / heat current 

π µν 

shear stress tensor 

◮ conserved current N µ 

i (4 · k independent components) 

ni = uµN µ 

i 

V µ 

i = ∆µ ν N ν 

i 


N µ 

i = niu µ + V µ 

i 

charge density 

charge current 

g µν = diag(1, −13×3)



◮ projection vector and tensor are orthogonal to each other 

◮ each term of T µν and N µ 

i 

uµ∆ µν = 0 

can be obtained by contraction of these quantities 

with u µ and ∆ µν or combinations of them, like 

P = − 1 

3 

ɛ = uµT µν uν 

∆µνT µν 

◮ need initial / boundary conditions and equation of state 



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 Tβγu 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 

√ 

NνN ν 

Both definitions are related to each other by Lorentz transformations 


u µ 

E = Λµ ν uν µ 

µ 

W V 

L , uµ 

L ≈ uµ 

E + , uµ 

ɛ+Ps E ≈ uµ 

L + n 

uL μ 

u E μ 

V μ 

W μ


Overview 

Continuity equation: 

Conservation of energy: 

Euler equation: 

∂µT µν = 0, ∂µN µ 

i 

∂ 

∂t (γn) + ∇ γnv = 0 

∂ 

∂t T 00 + ∇iT i0 = 0 

Problem 

11 + 4k unknown variables, only 5 equations. 

Solution 

= 0 

∂ 

∂t (ɛ + p) γ2 v i + ∇j (ɛ + p) γ 2 v i v j = −∇ i P 

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! 



Ideal hydrodynamics and dissipative effects 

◮ separate ideal and dissipative parts (Landau frame) 

T µν = T µν 

0 

N µ = N µ 

0 

+ δT µν 

+ δNµ 

Ideal Dissipative 

Pressure P = Ps + 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 

2. isotropy for the pressure 

µ 

L = uµ 

E 

4. no viscous corrections 


Relativistic ideal hydrodynamics 

Local rest frame 

T µν 

0 = (ɛ + Ps) u µ u ν − Ps 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∆ = 

◮ 4 + 1 + 1 equations for 7 free variables (ɛ, P, n, u µ ) 

∂µT µν 

0 

Ps3×3 

= 0, ∂µN µ 

0 = 0, uµu µ = 1 ⇒ another equation needed! 

◮ equation of state P = P(ɛ, n) gives constraints 


Relativistic ideal hydrodynamics 

Summary 

Energy-momentum tensor and conserved current 


T µν 

0 

N µ 

0 

= (ɛ + Ps) u µ u ν − Ps g µν = ɛu µ u ν − Ps∆ µν , 

= nuµ 

∂µT µν 

0 

= 0, ∂µN µ 

0 

◮ local rest frame simplifies equations, e. g. 

= 0 

∂µT µν 

LRF = 0 ⇒ ∂0ɛ + ∂iP i = 0 

◮ entropy density s resp. entropy current S µ = su µ conservation 


uν∂µT µν 

LRF = 0 ⇒ ∂µS µ = 0

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, 329-344 (1967) 

W. Israel, J. M. Stewart, Phys. Lett. A 58, 4 (1976), 213-215 




◮ 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 

π µν 

1 

 

= 2η ∆ 

2 

µ α∆ ν β + ∆µ 

β∆ν 

α − 1 

3 ∆µν 

∆αβ ∆ ακ ∂κu β 

Π = −ξ∂µu µ 

◮ transport coefficients: η shear viscosity, ξ bulk viscosity, ... 

◮ characterize deviation from thermal equilibrium 



Transport coefficients 

◮ Shear viscosity: fluid’s resistance to shear forces 

◮ Bulk viscosity: fluid’s resistance to compression 



Summary 

◮ entropy current conserved or increasing for viscous hydrodynamics 

T ∂µS µ = T u µ ∂µs + s∂µu µ , ∂µN µ = 0, n = 0 

= uν∂µT µν 

0 , Ts = ɛ + P − µn 

= −uν∂µ (δT µν ) , ∂µT µν = 0, 

inserting definitions from constitutive equations yields 

∂µS µ = 


πµνπ µν 

2η 

+ Π2 

ξ ⇒ ∂µS µ ≥ 0


Equations of motion 

Describing dynamics of the system 

1. From uν∂µT µν = 0 

˙ɛ = −(ɛ + Ps + Π)θ + πµν 

2. From ∆µα∂βT αβ = 0 

where 


 

1 

 

∆ 

2 

µ α∆ ν β + ∆µ 

β∆ν 

α − 1 

3 ∆µν 


(ɛ + Ps + Π) ˙u µ = ∆ µν ∂ν(Ps + Π) − ∆ µα ∆ βγ ∂γπαβ + π µα ˙uα 

θ = ∂µu µ expansion scalar (≅ ˙V /V ) 

˙a = uµ∂ µ a substantial (co-moving) time derivative


Equations of motion 

First equation of motion 

 

1 

 

˙ɛ = −(ɛ + Ps + Π)θ + πµν ∆ 

2 

µ α∆ ν β + ∆µ 

β∆ν 

α − 1 

3 ∆µν 


= −(ɛ + Ps)θ + ξθ 

 

a. 

2 

1 

 

+ 2η ∆ 

2 

µ α∆ ν β + ∆µ 

β∆ν 

α − 1 

3 ∆µν 


2 

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 




http://cdsweb.cern.ch/record/1305398, Oct 30, 2012 

Idea: study heavy ion collisions with hydrodynamics 

E. Fermi, Prog. Theor. Phys. 5 (1950) 570; Phys. Rev. 81 (1951) 683.


Schematic time line 

t 

z 

◮ proper time 

 

τ = t 2 − z2 ◮ space-time rapidity 

ηs = 1/2 · ln 

Coordinates: t = τ cosh ηs and z = τ sinh ηs, vz = z 

t 


red: τ = const, green: ηs = const 

(t + z) 

(t − z)




t 

z 

◮ heavy ions collide 

◮ quark gluon plasma 

◮ freeze out 

pre-equilibrium | hydrodynamics | free-streaming




t 

z 

Different stages of a HIC 

1. initial stage 

→ initial conditions 

2. intermediate stage 

→ equation of state 

3. final stage 

→ decoupling 

pre-equilibrium | hydrodynamics | free-streaming 

hydrodynamical description?


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. 17 51 

2. 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) 363-382 



Bjorken model (Bjorken, Phys. Rev. D 27, 140-151 (1983)) 



Bjorken model 

◮ early thermalization 

◮ vanishing Baryon number for the fluid 

◮ one-dimensional expansion 


◮ 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) 

ηs = y 

t 

z


Bjorken model (Bjorken, Phys. Rev. D 27, 140-151 (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 

 

= 1, 0, 0, 

τ 

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)


Bjorken model 

◮ first equation of motion simplifies to 

dɛ 

dτ 

1 

τ 2 

(ɛ + Ps) η 

Ts 

+ Ps 

= −ɛ + 

τ 

4 

+ 

3 

ɛ + Ps 

 

Ts 

ɛ + Ps 

= − 1 − 

τ 

4 

 

η 1 ξ 

− 

3τT s τT s 

a. time-evolution of energy density is governed by the sum e + Ps 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 


ξ 

τ 2


Bjorken model 

◮ first equation of motion simplifies to 

dɛ 

dτ 

= −ɛ + Ps 

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 



Bjorken model and an equation of state 

◮ simple equation of state 

◮ solutions for equations of motion 


P = γɛ, γideal gas = 1 

3 

 

τ0 

1+γ 

ɛ(τ) = ɛ0 

 

τ 

τ0 

T (τ) = T0 

τ 

s0τ0 = sτ 

γ 

and


Bjorken model and observables 

◮ energy change per unit of rapidity can be measured and calculated 

dE 

dy = d 3V dy ɛ(τf ) = πR 2 

 

d 2 x 

1+γ τ0 

τf ɛ0 = πR 

τf 

2 γ τ0 

ɛ0τ0 

τf 

◮ assume no hydrodynamical expansion any more (τ ≥ τf ) 

ɛ0 = 

1 

πR 2 τ0 

dE 

dy 

◮ allows to estimate the initial energy density 


〈mt〉 

= 

πR2 dN 

dz |z=0 = 〈mt〉 

πR2 dy 

dz |z=0 

dN 

 

dy 

1/τ0


Bjorken model: Does a QGP occur in HICs? 

◮ energy density @ RHIC 

[GeV fm -2 c -1 ] 

τ 

Bj 

∈ 


4 

2 

0 100 200 300 

200 GeV 

130 GeV 

19.6 GeV 

Np 

S. S. Adler et al. (PHENIX collaboration), Phys. Rev. C 71, 034908 (2005)


Elliptic flow 

b 

◮ 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 

dpT dφdy = 

◮ elliptic flow v2 sensitive to viscous effects 


 

vn(pT ) cos nφ 

n 

y 

x


Bjorken model: Elliptic flow 

◮ Euler equation for vx 

∂tvx = − 1 ∂P 

e + P ∂x 

∂ ln s 

= −c2 s 

∂x 

◮ assume gaussian entropy profile from the collision 

 

s(x, y) = s0 exp − 1 

2 

◮ solution of Euler equation 

 

2 σy x 2 + σ2 2 

x y 

σ2 xσ 2 y 

 

vx(t) = c2 s 

σ2 tx + vx,0, vy(t) = 

x 

c2 s 

σ2 ty + vy,0 

y 

◮ non-central collision (σx < σy) implies |vx| > |vy| 

◮ anisotropy in particle spectrum → details next week 


Bjorken model 

Violation of causality in first order theory 

◮ linearized Euler equation for small perturbations of v y → v y + δv y 

y 

∂t δv − η 

ɛ + P ∂2 

y 

x δv = 0 

◮ allow sinusoidal perturbation of the form 

δv y (t, x) ∝ exp (ωt − ikx) 

◮ “dispersion relation” with wave-number k is given by 

η 2 

ω = k 

ɛ + P 

◮ estimate speed of mode with wave-number k 

v(k) = dω 2η 

= k → ∞ for k → ∞ 

dk ɛ + P 

⇒ perturbations with k → ∞ propagate with infinite speed 


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, 725-733 (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 


Summary 


◮ 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 


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.” 

◮ systematical improvements of hydrodynamical description 

◮ many numerical simulations available 

◮ other (effective) description of heavy ion collisions 

◮ extend to include anisotropies, turbulence, non-equilibrium... 


U. Heinz, arXiv:nucl-th/0901.4355

Literature 

[1] T. Hirano, N. van der Kolk, A. Bilandzic 

Hydrodynamic and Flow 

arXiv:nucl-th/0809.2684 (2008) 

[2] P. Huovinen and P. V. Ruuskanen 

Hydrodynamic Models for Heavy Ion Collisions 

Annu. Rev. Nucl. Particle Science 56 (2006), arXiv:nucl-th/0605008 

[3] U. Heinz 

Early collective expansion: Relativistic hydrodynamics and the transport 

properties of QCD matter 

arXiv:nucl-th/0901.4355 (2009)

Introduction to Relativistic Hydrodynamics

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