Hydrodynamics - Indico · 2018. 11. 16. · Title: Hydrodynamics Author: Paul Romatschke Created...

Hydrodynamics

Paul Romatschke

FIAS, Frankfurt, Germany

Quark Matter, May 2011

Paul Romatschke Hydro/HIC

Asymptotic Freedom of QCD

Coupling [Particle Data Group]:

0

0.1

0.2

0.3

1 10 102

µ GeV

α s(µ)

Nobel Prize 2004: Gross, Politzer, Wilczek


QCD Phase Transition

QCD Energy-density from lattice QCD:

Rapid Rise of ε close to T ∼ 170 MeV.


QCD Phase Transition

QCD Phase Transition: Transition from confined matter(neutrons, protons, hadrons) to deconfined matter(quark-gluon plasma )Information from lattice QCD: quark-gluon plasma forT > Tc ∼ 200 MeV, equation of state (P = P(ε)), speed ofsound cs =

√dP/dε

Experimental setup: collide (large) nuclei at high speeds toreach T > Tc


Hydrodynamics


Fluid Dynamics=

Conservation of Energy+Momentum for longwavelength modes1

1long wavelength modes =looking at the system for a very long time from very far away


Fluid Dynamics: Degrees of Freedom

(Relativistic) Fluids described by:Fluid velocity: uµ

Pressure: p(Energy-) Density: ε(General Relativity): space-time metric gµν

Quantum Field Theory:Energy-Momentum Tensor Tµν

Conservation of Energy+Momentum: ∂µTµν = 0.


Energy Momentum Tensor for Ideal Fluids

Tµν symmetric tensor of rank twobuilding blocks for ideal fluids: scalars ε,p, vector uµ,tensors of rank two: uµuν ,gµν

Tµν must be of form

Tµν = A(ε,p)uµuν + B(ε,p)gµν

Local rest frame (vanishing fluid velocity, uµ = (1,0,0,0)):

Tµν =

ε 0 0 00 −p 0 00 0 −p 00 0 0 −p

Can use to determine A,B !


T µνid = εuµuν − p(gµν − uµuν) (Fluid EMT, no gradients)

+∂µT µν = 0 (“EMT Conservation”)

=Ideal Fluid Dynamics


Proof

Take ∂µTµi = 0, take non-relativistic limit (neglectu2/c2 � 1,p � mc2):

∂tui + um∂mui = −1ε∂jδ

ijp

“Euler Equation” [L. Euler, 1755]

Euler Equation: non-linear, non-dissipative: “ideal fluiddynamics”Take ∂µTµ0 = 0, take non-relativistic limit (neglectu/c � 1,p � mc2):

ε∂iui + ∂tε+ ui∂iε = 0

“Continuity Equation” [L. Euler, 1755]


Non-linear & Non-dissipative: Turbulence


Non-linear & Dissipative: Laminar


Non-linear & Dissipative: Laminar

Viscosity dampens turbulent instability!


Relativistic Ideal Fluid Dynamics

T µν = T µνid (Fluid EMT, no gradients)


=Ideal Fluid Dynamics


Relativistic Viscous Fluids

How to include viscous effects?Energy and Momentum Conservation: ∂µTµν = 0 is exactBut Tµν = Tµν

id is approximation!Lift approximation: Tµν = Tµν

id + Πµν

Build Πµν : e.g. first order gradients on ε,uµ,gµν

Πµν = η∇<µuν> + ζ∆µν∇ · u


Relativistic Viscous Fluid Dynamics

T µν = T µνid + Πµν (Fluid EMT, 1st o. gradients)


=Relativistic Navier-Stokes Equation


Relativistic Viscous Fluid Dynamics

L. Euler, 1755:

∂tui + um∂mui = −1ε∂jδ

ijp

C. Navier, 1822; G. Stokes 1845:

∂tui + um∂mui = −1ε∂j[δijp + Πij] ,

Πij = −η(∂ui

∂x j +∂uj

∂x i −23δij ∂ul

∂x l

)− ζδij ∂ul

∂x l ,

η, ζ. . . transport coefficients (“viscosities”)


Gradients and Hydro

Tµν : fluid dofs (ε,p,uµ,gµν), no gradients gives IdealHydrodynamicsTµν : fluid dofs (ε,p,uµ,gµν) up to 1st order gradients givesNavier-Stokes equationTµν : fluid dofs (ε,p,uµ,gµν) up to 2nd,3rd, ... order: higherorder Hydrodynamics


Fluid Dynamics=

Effective Theory of Small Gradients


Relativistic Navier-Stokes Equation

Good enough for non-relativistic systemsNOT good enough for relativistic systems


Navier-Stokes: Problems with Causality

Consider small perturbations around equilibriumTransverse velocity perturbations obey

∂tδuy − η

ε+ p∂2

x δuy = 0

Diffusion speed of wavemode k :

vT (k) = 2kη

ε+ p→∞ (k � 1)

Know how to regulate: “second-order” theories:

τπ∂2t δu

y + ∂tδuy − η

ε+ p∂2

x δuy = 0

[Maxwell (1867), Cattaneo (1948)]


Second Order Fluid Dynamics

Limiting speed is finite

limk→∞

vL(k) =

√c2

s +4η

3τπ(ε+ p)+

ζ

τΠ(ε+ p)

[Romatschke, 2009]

τπ, τΠ. . . ...: “2nd order” regulators for “1st order” fluiddynamicsRegulators acts in UV, low momentum (fluid dynamics)regime is still Navier-Stokes


Second Order Fluid Dynamics

T µν = T µνid + Πµν (Fluid EMT, 2nd o. gradients)


=“Causal” Relativistic Viscous Fluid Dynamics

First complete 2nd theory for shear only in 2007 !

[Baier et al. 2007; Bhattacharyya et al. 2007]


Hydrodynamcis: Limits of Applicability

RememberTµν = Tµν

id + Πµν

Πµν is given by small gradient expansion

Πµν = η∇<µuν> + . . .

Hydrodynamics breaks down if gradient expansion breaksdown: Π ∼ Tµν

id orp ' η∇ · u

Two possible ways: η large (hadron gas!) or ∇ · u large(early times, small systems!)


Hydro Theory

What you should remember:Hydrodynamics is Energy Momentum ConservationHydrodynamics is an Effective Theory for long wavelength(small momenta)Hydrodynamics breaks down for small systems or dilutesystems


Hydrodynamic Models forHeavy-Ion Collisions


Hydro Models for Heavy-Ion Collisions

Need initial conditions for Hydro: ε,uµ at τ = τ0

Need equation of state p = p(ε), which gives c2s = dp

dε

Need functions for transport coefficients η, ζ.Need algorithm to solve (nonlinear!) hydro equationsNeed method to convert hydro information to particles(“freeze-out”)


Initial Conditions

IC’s for hydro not known. Here are some popular choices:Fluid velocities are set to zeroBoost-invariance: all hydro quantities only depend onproper time τ =

√t2 − z2 and transverse space x⊥.

Models for energy density distribution:Glauber/Color-Glass-CondensateStarting time τ0: Should be of order 1 fm, precise valueunknown


Equation of State

EoS known (approximately) from lQCD:


Transport Coefficients

In QCD: known for small and large T, but not for T ' Tc

[Demir and Bass, 2008]


Hydro Solvers

For 3+1D ideal hydro, many groups, well testedFor 2+1D viscous hydro, many groups, well testedFor 3+1D viscous hydro: Schenke, Jeon, Gale 2010


Freeze-out

Partial solution exists: “Cooper-Frye”Idea: Tµν for particles/fluid must be the same

Tµνhydro = (ε+p)uµuν−Pgµν+Πµν = Tµν

particles =

∫p

f (x ,p)pµpν

No dissipation (ideal hydro) = equilibrium:

f (p, x) = e−p·u/T

Only shear dissipation: Quadratic ansatz

f (p, x) = e−p·u/T(

1 +pαpβΠαβ

2(ε+ p)T 2 +O(p3)

)


Experimental Observables

dN/dp/dφ


Experimental Observables

For ultrarelativistic heavy-ion collisions,

dNdp⊥dφdy

= 〈 dNdp⊥dφdy

〉φ (1 + 2v2(p⊥) cos(2φ) + . . .)

Radial flow: 〈 dNdp⊥dy 〉φ

Elliptic flow: v2(p⊥)


Putting things togetherHydro model simultion of RHIC Au+Au collisions

[Luzum & Romatschke, 2008]Paul Romatschke Hydro/HIC

Current Research and Open Problems

Initial Conditions for Hydro: Effect of Fluctuations?3D vs. 2D: quantitative difference for viscoushydrodynamics evolution?Freeze-Out: Consistent coupling of hydro/particledynamics?Thermalization: Can one calculate hydro initial conditions?


Fluctuations

[slide stolen from M. Luzum]

Initial conditions are not smooth! Event average: <>There will be < v3 > and < v2 >

2 6=< v2 >2


LHC results

centrality percentile0 10 20 30 40 50 60 70 80 90

2v

0

0.02

0.04

0.06

0.08

0.1

0.12

{2}2v (same charge){2}2v

{4}2v (same charge){4}2v

{q-dist}2v{LYZ}2v

STAR{EP}2v STAR{LYZ}2v

[ALICE Collaboration, 2010]

If < v2 >2=< v2 >

2 then v2{2} = v2{4}.


3D Evolution

0

1

2

-4 -2 0 2 4

h+

/- v

3 [%

]

ηp

ideal, e-b-e η/s=0.08, e-b-e η/s=0.16, e-b-e

0

2

4

6

8

h+

/- v

2 [%

]

PHOBOS 15-25% central

ideal, avg

[B. Schenke et al., 2010]

Non-Bjorken flow in longitudinal direction.Paul Romatschke Hydro/HIC

Freeze-Out

0

5

10

15

20

0 2 4 6 8 10 12 14 16

χ(p

/T)

/ [4

π η

/(sT

)]

p/T

Quadratic

LO

Coll.

Linear

[K. Dusling et al., 2009]

Quadratic ansatz may be inaccuratePaul Romatschke Hydro/HIC

Thermalization

How does system get Tµν that is close to hydro?Far from equilibration dynamics: non-perturbative,real-time: hard!Attempts to thermalization: pQCD inspired (’plasmainstabilities’); AdS/CFT inspired (’collision of shock waves’)


Things to keep in mind for this week

Dynamics: anything less than 2+1D is not realistic(’Bjorken hydro’)Ideal hydro does not indicate its own breakdown. Does notmean results are accurate!Keep in mind that ideal hydro only exists with numericalviscosity (value?)All working 2+1D viscous hydro codes are ’second orderhydro’Different names (’Israel-Stewart’, ’full IS’, ’BRSSS’)correspond to different choices for values of τπ, . . .


A personal appeal: theory/data comparisons

A new theorists calculation/model should first be rigorouslystudied before ’fitting’ dataExample: hydrodynamic calculations on a grid; physicalresults correspond to limit of vanishing grid spacingTheorists: Please first check your model/calculation beforeyou compare to experimental dataExperimentalists: Please don’t blindly trust (or promote!) amodel just because it fits data


...and all the rest...

There are many topics/details I couldn’t cover today!Some lecture notes:

“New Developments in Relativistic ViscousHydrodynamics”, PR, arXiv:0902.3663“Nearly Perfect Fluidity: From Cold Atomic Gases to HotQuark Gluon Plasmas”, T. Schäfer and D. Teaney,arXiv:0904.3107“Early collective expansion: Relativistic hydrodynamicsand the transport properties of QCD matter”, U.W. Heinz,arXiv:0901.4355


Hydrodynamics - Indico · 2018. 11. 16. · Title: Hydrodynamics Author: Paul Romatschke Created...

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