Introduction to Relativistic Hydrodynamicscrunch.ikp.physik.tu-darmstadt.de/nhc/pages/lectures/rhiseminar12-13/... · Relativistic hydrodynamics Ideal hydrodynamics and dissipative

Introduction to Relativistic HydrodynamicsHeavy 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 NowakowskiTU Darmstadt, Institut für Kernphysik

Seminar “Relativistische Schwerionenphysik”, WS 12/13

November 22nd, 2012 | TUD - IKP | D. Nowakowski | 1

MotivationHeavy ion collisions

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K. Heckmann, TU Darmstadt, Nov. 2011


Heavy ion collisionsA very naive picture(?)

Vapour

Pressure

Temperature

Solid Liquid


MotivationHeavy ion collisions

Question:

Do we discover deconfined matter in these collisions; can we extract informationabout the properties of quarks and gluons?

Possible answer:

Hydrodynamics

I extract local temperatures, energy densitiesI describe collective effectsI no / little detailed knowledge of microscopic physics needed, if relevant input

provided externallyI analyse experimental data in this framework

Applicable to detect signatures of deconfined matter with a hydrodynamical model?


Relativistic hydrodynamicsIntroduction

I matter produced in heavy ion collisions has varying degrees of freedom; applicability for hydrodynamical description? Limited! No clear startingpoint

I 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 thermodynamicsA short glance

HydrodynamicsI continuous media with collective behaviorI pressure and temperature slowly varying

ThermodynamicsI change from extensive to intensive variables

ε =EV

, s =SN

, n =NV

ε + P = Ts + µn , dε = Tds + µdn , c2s =

∂P∂ε


Relativistic hydrodynamicsBasic equations

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

uµ =(γ, γ~v

), uµuµ = 1 (flat spacetime)

I local thermal equilibrium is requiredI energy-momentum conservation yields

∂µTµν = 0 ⇒ 4 equations

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

∂µNµi = 0 ⇒ 1 equation


Relativistic hydrodynamicsParametrization of hydrodynamical quantities

I energy-momentum tensor Tµν (10 independent components)

Tµν = εuµuν − P∆µν + Wµuν + W νuµ + πµν

uµ,∆µν = gµν − uµuν “projection vector, tensor”ε energy densityP = PS + Π hydrostatic + bulk pressureWµ energy / heat currentπµν shear stress tensor

I conserved current Nµi (4 · k independent components)

Nµi = niuµ + Vµ

i

ni = uµNµi charge density

Vµi = ∆µ

νNνi charge current

gµν = diag(1,−13×3)


Relativistic hydrodynamicsBasic equations

I projection vector and tensor are orthogonal to each other

uµ∆µν = 0

I each term of Tµν and Nµi can be obtained by contraction of these quantities

with uµ and ∆µν or combinations of them, like

P = −13∆µνTµν

ε = uµTµνuν

I need initial / boundary conditions and equation of state


Relativistic hydrodynamicsWhat is flow?

Two definitions of flow

1. Flow of energy, Wµ = 0 (Landau)Landau, Lifshitz, Fluid mechanics, Pergamon Press (1959)

uµL =Tµν uνL√

uαL TβαTβγuγL

=1ε

Tµν uνL

2. Flow of conserved charge, Vµ = 0 (Eckart)Eckart, Phys. Rev. 58, 919 (1940)

uµE =Nµ

√NνNν

Vμ

uLμ

Wμ

uEμ

Both definitions are related to each other by Lorentz transformations

uµE = ΛµνuνL , uµL ≈ uµE + Wµ

ε+Ps, uµE ≈ uµL + Vµ

n


Relativistic hydrodynamicsOverview

∂µTµν = 0, ∂µNµi = 0

Continuity equation: ∂∂t (γn) +∇

(γn~v

)= 0

Conservation of energy: ∂∂t T

00 +∇iT i0 = 0

Euler equation: ∂∂t (ε + p) γ2v i +∇j (ε + p) γ2v iv j = −∇iP

Problem11 + 4k unknown variables, only 5 equations.

Solution1. choose suited frame→ Landau: Wµ = 0, but now uµ dynamical variable

2. only ideal fluids: 5 + 1k unknowns

3. additional input needed!


Relativistic hydrodynamicsIdeal hydrodynamics and dissipative effects

I separate ideal and dissipative parts (Landau frame)

Tµν = Tµν0 + δTµν

Nµ = Nµ0 + δNµ

Ideal DissipativePressure P = Ps + Π Π = 0 Π 6= 0Energy / heat current Wµ Wµ = 0 Wµ = 0Shear stress tensor πµν πµν = 0 πµν 6= 0Charge current Vµ

i Vµi = 0 Vµ

i 6= 0

I consider only ideal hydrodynamics for now1. local thermal equilibrium

2. isotropy for the pressure

3. unique flow uµL = uµ

E

4. no viscous corrections


Relativistic ideal hydrodynamicsLocal rest frame

Tµν0 = (ε + Ps) uµuν − Ps gµν , Nµ0 = nuµ

in the local rest frame (LRF) with uµ = (1, 0, 0, 0)I uµuν (∆µν ) project time (space)-like quantitiesI the energy-momentum tensor is given by

T LRF = ε(u · u) + P∆ =(ε

Ps13×3

)

I 4 + 1 + 1 equations for 7 free variables (ε, P, n, uµ)

∂µTµν0 = 0, ∂µNµ0 = 0, uµuµ = 1⇒ another equation needed!

I equation of state P = P(ε, n) gives constraints


Relativistic ideal hydrodynamicsSummary

Energy-momentum tensor and conserved current

Tµν0 = (ε + Ps) uµuν − Ps gµν = εuµuν − Ps∆µν ,

Nµ0 = nuµ

Basic equations

∂µTµν0 = 0, ∂µNµ0 = 0

I local rest frame simplifies equations, e. g.

∂µTµνLRF = 0 ⇒ ∂0ε + ∂iP i = 0

I entropy density s resp. entropy current Sµ = suµ conservation

uν∂µTµνLRF = 0 ⇒ ∂µSµ = 0


Relativistic viscous hydrodynamicsGradient expansion

Energy-momentum tensor

Tµν = Tµν0 + δTµν

I δTµν can contain first, second,... spatial gradientsI hierarchy of orders

1. Zeroth order: Ideal Hydrodynamics2. First order: Viscous Hydrodynamics (“Navier-Stokes”)3. Second order: Viscous Hydrodynamics (“Israel-Müller-Stewart theory”)

I corresponds to modifying the entropy current according to

Sµ = suµ +O (δTµν ) +O(

(δTµν )2)

+ ...

I. Müller, Zeitschrift f. Physik 198, 329-344 (1967)

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


Relativistic viscous hydrodynamicsBasic equations

I allow dissipative terms, but no charge in the systemI assume entropy current has additional linear dissipative terms (first-order

theory)

Sµ = suµ + αVµ

I phenomenological definitions resp. so called constitutive equations for theshear stress tensor πµν and for the bulk pressure Π

πµν = 2η(

12

(∆µα∆

νβ + ∆µ

β∆να

)− 1

3∆µν∆αβ

)∆ακ∂κuβ

Π = −ξ∂µuµ

I transport coefficients: η shear viscosity, ξ bulk viscosity, ...I characterize deviation from thermal equilibrium


Relativistic viscous hydrodynamicsTransport coefficients

I Shear viscosity: fluid’s resistance to shear forces

I Bulk viscosity: fluid’s resistance to compression


Relativistic viscous hydrodynamicsSummary

I entropy current conserved or increasing for viscous hydrodynamics

T∂µSµ = T(uµ∂µs + s∂µuµ

), ∂µNµ = 0, n = 0

= uν∂µTµν0 , Ts = ε + P − µn

= −uν∂µ (δTµν ) , ∂µTµν = 0,

inserting definitions from constitutive equations yields

∂µSµ =πµνπ

µν

2η+Π2

ξ⇒ ∂µSµ ≥ 0


Relativistic hydrodynamicsEquations of motion

Describing dynamics of the system

1. From uν∂µTµν = 0

ε = −(ε + Ps + Π)θ + πµν

(12

(∆µα∆

νβ + ∆µ

β∆να

)− 1

3∆µν∆αβ

)∆ακ∂κuβ

2. From ∆µα∂βTαβ = 0

(ε + Ps + Π)uµ = ∆µν∂ν (Ps + Π)−∆µα∆βγ∂γπαβ + πµαuα

where

θ = ∂µuµ expansion scalar (u V/V )a = uµ∂µa substantial (co-moving) time derivative


Relativistic hydrodynamicsEquations of motion

First equation of motion

ε = −(ε + Ps + Π)θ + πµν

(12

(∆µα∆

νβ + ∆µ

β∆να

)− 1

3∆µν∆αβ

)∆ακ∂κuβ

= −(ε + Ps)θ︸︷︷︸a.

+ ξθ2 + 2η((

12

(∆µα∆

νβ + ∆µ

β∆να

)− 1

3∆µν∆αβ

)∆ακ∂κuβ

)2

︸︷︷︸b.

Time evolution of the energy density in the co-moving system

a. change of energy density and hydrostatic pressure due to expansion / dilutionresp. changing volume

b. production of entropy due to dissipative effects ; heating of the system


Heavy ion collisions

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

Idea: study heavy ion collisions with hydrodynamicsE. Fermi, Prog. Theor. Phys. 5 (1950) 570; Phys. Rev. 81 (1951) 683.


Heavy ion collisionsSchematic time line

z

t

I proper time

τ =√

t2 − z2

I space-time rapidity

ηs = 1/2 · ln (t + z)(t − z)

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

red: τ = const, green: ηs = const



z

t

I heavy ions collideI quark gluon plasmaI freeze out

pre-equilibrium | hydrodynamics | free-streaming



z

t 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


Heavy ion collisionsInitial conditions

I dynamics of particle production cannot be described in hydrodynamicsI specify thermodynamical state of matter and initial velocityI 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 initialconditions

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


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


Heavy ion collisionsBjorken model

I early thermalizationI vanishing Baryon number for the fluidI one-dimensional expansion

z

t

I boost symmetry of initial conditionsI no initial dependence on rapidity y because no

dependence on Lorentz boost angleI fluid rapidity is the same as spacetime rapidity

(E large)

ηs = y


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

I valid for times of the order from τ ≈ 1 fm/c to τ ≈ 5− 10 fm/cI specify to expansion along z directionI introduce a boost-invariant four-velocity

uµ =xµ

τ=

tτ

(1, 0, 0,

zt

)=

1√t2 − z2

(t , 0, 0, z)

I due to Lorentz-symmetry and initial conditions

ε = ε(τ , y )→ ε(τ )

P = P(τ , y )→ P(τ )

T = T (τ , y )→ T (τ ) = β−1(τ )



I first equation of motion simplifies to

dεdτ

= −ε + Ps

τ+

43

1τ 2

(ε + Ps) ηTs

+ε + Ps

Tsξ

τ 2

= −ε + Ps

τ

(1− 4

3τTη

s− 1τT

ξ

s

)

a. time-evolution of energy density is governed by the sum e + Ps per proper time τfor ideal hydrodynamics

b. last two terms on the RHS are viscous corrections with appearing dimensionlessquantities

η/s and ξ/s

which characterize intrinsic properties of the fluid



I first equation of motion simplifies to

dεdτ

= −ε + Ps

τ

I description of the time evolution of the system in a simple (solvable) wayI good approximation, but for detailed calculations viscous effects need to be

considered


Heavy ion collisionsBjorken model and an equation of state

I simple equation of state

P = γε, γideal gas =13

I solutions for equations of motion

ε(τ ) = ε0

(τ0

τ

)1+γ

T (τ ) = T0

(τ0

τ

)γand

s0τ0 = sτ


Heavy ion collisionsBjorken model and observables

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

dEdy

=d3Vdy

ε(τf ) = πR2︸︷︷︸d2x

τf ε0

(τ0

τf

)1+γ

= πR2ε0τ0

(τ0

τf

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

ε0 =1

πR2τ0

dEdy

=〈mt〉πR2

dNdz|z=0 =

〈mt〉πR2

dydz|z=0︸︷︷︸

1/τ0

dNdy

I allows to estimate the initial energy density


Heavy ion collisionsBjorken model: Does a QGP occur in HICs?

I energy density @ RHIC

0 100 200 300

2

4

200 GeV130 GeV19.6 GeV

pN

[GeV

fm-2c-

1 ]τ

Bj

∈

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


Heavy ion collisionsElliptic flow

bx

y

I initial geometric anisotropy gets transformed to anisotropies in particlemomenta spectrum

I expanding system develops flow patternI azimuthal distribution of emitted particles with respect to reaction plane

dNdpT dφdy

=∑

n

vn(pT ) cos nφ

I elliptic flow v2 sensitive to viscous effects


Heavy ion collisionsBjorken model: Elliptic flow

I Euler equation for vx

∂tvx = − 1e + P

∂P∂x

= −c2s∂ ln s∂x

I assume gaussian entropy profile from the collision

s(x , y ) = s0 exp

(−1

2

(σ2

y x2 + σ2x y2

σ2xσ

2y

))I solution of Euler equation

vx (t) =c2

s

σ2x

tx + vx ,0, vy (t) =c2

s

σ2y

ty + vy ,0

I non-central collision (σx < σy ) implies |vx | > |vy |I anisotropy in particle spectrum→ details next week


Bjorken modelViolation of causality in first order theory

I linearized Euler equation for small perturbations of vy → vy + δvy

∂t(δvy)− η

ε + P∂2

x

(δvy) = 0

I allow sinusoidal perturbation of the form

δvy (t , x) ∝ exp (ωt − ikx)

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

ω =η

ε + Pk2

I estimate speed of mode with wave-number k

v (k ) =dωdk

=2ηε + P

k →∞ for k →∞

⇒ perturbations with k →∞ propagate with infinite speed


Heavy ion collision and the Bjorken modelShort summary

Bjorken modelI assumes Boost-invariance of initial conditionsI describes heavy ion collisions within hydrodynamical frameworkI one-dimensional expansion along z for τ ≈ 1− 10 fm/c

Problems in first order theory:I violation of causalityI 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)

I introduce relaxation time in equations of motionI no acausality


Summary

HydrodynamicsI offers simple formalism to describe heavy ion collisionsI strong assumption required: local thermal equilibriumI relies on initial conditions, equation of state and freeze-out descriptionI Bjorken modelI 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 producedin those violent collisions satisfy the criteria justifying a dynamical treatment interms of a macroscopic theory which follows idealized laws.”

U. Heinz, arXiv:nucl-th/0901.4355

I systematical improvements of hydrodynamical descriptionI many numerical simulations availableI other (effective) description of heavy ion collisionsI extend to include anisotropies, turbulence, non-equilibrium...


Literature

[1] T. Hirano, N. van der Kolk, A. BilandzicHydrodynamic and FlowarXiv:nucl-th/0809.2684 (2008)

[2] P. Huovinen and P. V. RuuskanenHydrodynamic Models for Heavy Ion CollisionsAnnu. Rev. Nucl. Particle Science 56 (2006), arXiv:nucl-th/0605008

[3] U. HeinzEarly collective expansion: Relativistic hydrodynamics and the transportproperties of QCD matterarXiv:nucl-th/0901.4355 (2009)


Documents

Introduction to Relativistic Hydrodynamicscrunch.ikp.physik.tu-darmstadt.de/nhc/pages/lectures/rhiseminar12-13/... · Relativistic hydrodynamics Ideal hydrodynamics and dissipative