Hydrodynamics Paul Romatschke FIAS, Frankfurt, Germany Quark - - PowerPoint PPT Presentation

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Hydrodynamics Paul Romatschke FIAS, Frankfurt, Germany Quark - - PowerPoint PPT Presentation

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Hydrodynamics Paul Romatschke FIAS, Frankfurt, Germany Quark Matter, May 2011 Paul Romatschke Hydro/HIC Asymptotic Freedom of QCD Coupling [Particle Data Group] : 0.3 s ( ) 0.2 0.1 0 2 1 10 10 GeV Nobel Prize 2004: Gross,

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SLIDE 1

Hydrodynamics

Paul Romatschke

FIAS, Frankfurt, Germany

Quark Matter, May 2011

Paul Romatschke Hydro/HIC

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SLIDE 2

Asymptotic Freedom of QCD

Coupling [Particle Data Group]:

0.1 0.2 0.3 1 10 10

2

µ GeV αs(µ)

Nobel Prize 2004: Gross, Politzer, Wilczek

Paul Romatschke Hydro/HIC

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SLIDE 3

QCD Phase Transition

QCD Energy-density from lattice QCD: Rapid Rise of ǫ close to T ∼ 170 MeV.

Paul Romatschke Hydro/HIC

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SLIDE 4

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 for T > Tc ∼ 200 MeV, equation of state (P = P(ǫ)), speed of sound cs =

  • dP/dǫ

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

Paul Romatschke Hydro/HIC

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SLIDE 5

Hydrodynamics

Paul Romatschke Hydro/HIC

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SLIDE 6

Fluid Dynamics = Conservation of Energy+Momentum for long wavelength modes1

1long wavelength modes =

looking at the system for a very long time from very far away

Paul Romatschke Hydro/HIC

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SLIDE 7

Fluid Dynamics: Degrees of Freedom

(Relativistic) Fluids described by: Fluid velocity: uµ Pressure: p (Energy-) Density: ǫ (General Relativity): space-time metric gµν Quantum Field Theory: Energy-Momentum Tensor T µν Conservation of Energy+Momentum: ∂µT µν = 0.

Paul Romatschke Hydro/HIC

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SLIDE 8

Energy Momentum Tensor for Ideal Fluids

T µν symmetric tensor of rank two building blocks for ideal fluids: scalars ǫ, p, vector uµ, tensors of rank two: uµuν, gµν T µν must be of form T µν = A(ǫ, p)uµuν + B(ǫ, p)gµν Local rest frame (vanishing fluid velocity, uµ = (1, 0, 0, 0)): T µν =     ǫ −p −p −p     Can use to determine A, B !

Paul Romatschke Hydro/HIC

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SLIDE 9

T µν

id = ǫuµuν − p(gµν − uµuν) (Fluid EMT, no gradients)

+

∂µT µν = 0

(“EMT Conservation”)

=

Ideal Fluid Dynamics

Paul Romatschke Hydro/HIC

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SLIDE 10

Proof

Take ∂µT µi = 0, take non-relativistic limit (neglect u2/c2 ≪ 1, p ≪ mc2): ∂tui + um∂mui = −1 ǫ ∂jδijp

“Euler Equation” [L. Euler, 1755]

Euler Equation: non-linear, non-dissipative: “ideal fluid dynamics” Take ∂µT µ0 = 0, take non-relativistic limit (neglect u/c ≪ 1, p ≪ mc2): ǫ∂iui + ∂tǫ + ui∂iǫ = 0

“Continuity Equation” [L. Euler, 1755]

Paul Romatschke Hydro/HIC

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SLIDE 11

Non-linear & Non-dissipative: Turbulence

Paul Romatschke Hydro/HIC

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SLIDE 12

Non-linear & Dissipative: Laminar

Paul Romatschke Hydro/HIC

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SLIDE 13

Non-linear & Dissipative: Laminar Viscosity dampens turbulent instability!

Paul Romatschke Hydro/HIC

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SLIDE 14

Relativistic Ideal Fluid Dynamics T µν = T µν

id (Fluid EMT, no gradients)

+

∂µT µν = 0

(“EMT Conservation”)

=

Ideal Fluid Dynamics

Paul Romatschke Hydro/HIC

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SLIDE 15

Relativistic Viscous Fluids

How to include viscous effects? Energy and Momentum Conservation: ∂µT µν = 0 is exact But T µν = T µν

id is approximation!

Lift approximation: T µν = T µν

id + Πµν

Build Πµν: e.g. first order gradients on ǫ, uµ, gµν Πµν = η∇<µuν> + ζ∆µν∇ · u

Paul Romatschke Hydro/HIC

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SLIDE 16

Relativistic Viscous Fluid Dynamics T µν = T µν

id + Πµν (Fluid EMT, 1st o. gradients)

+

∂µT µν = 0

(“EMT Conservation”)

=

Relativistic Navier-Stokes Equation

Paul Romatschke Hydro/HIC

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SLIDE 17

Relativistic Viscous Fluid Dynamics

  • L. Euler, 1755:

∂tui + um∂mui = −1 ǫ ∂jδijp

  • C. Navier, 1822; G. Stokes 1845:

∂tui + um∂mui = −1 ǫ ∂j

  • δijp + Πij

, Πij = −η ∂ui ∂xj + ∂uj ∂xi − 2 3δij ∂ul ∂xl

  • − ζδij ∂ul

∂xl ,

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

Paul Romatschke Hydro/HIC

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SLIDE 18

Gradients and Hydro

T µν: fluid dofs (ǫ, p, uµ, gµν), no gradients gives Ideal Hydrodynamics T µν: fluid dofs (ǫ, p, uµ, gµν) up to 1st order gradients gives Navier-Stokes equation T µν: fluid dofs (ǫ, p, uµ, gµν) up to 2nd, 3rd, ... order: higher

  • rder Hydrodynamics

Paul Romatschke Hydro/HIC

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SLIDE 19

Fluid Dynamics = Effective Theory of Small Gradients

Paul Romatschke Hydro/HIC

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Relativistic Navier-Stokes Equation

Good enough for non-relativistic systems NOT good enough for relativistic systems

Paul Romatschke Hydro/HIC

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Navier-Stokes: Problems with Causality

Consider small perturbations around equilibrium Transverse velocity perturbations obey ∂tδuy − η ǫ + p∂2

xδuy = 0

Diffusion speed of wavemode k: vT(k) = 2k η ǫ + p → ∞ (k ≫ 1) Know how to regulate: “second-order” theories: τπ∂2

t δuy + ∂tδuy −

η ǫ + p∂2

xδuy = 0

[Maxwell (1867), Cattaneo (1948)]

Paul Romatschke Hydro/HIC

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SLIDE 22

Second Order Fluid Dynamics

Limiting speed is finite lim

k→∞ vL(k) =

  • c2

s +

4η 3τπ(ǫ + p) + ζ τΠ(ǫ + p)

[Romatschke, 2009]

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

Paul Romatschke Hydro/HIC

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SLIDE 23

Second Order Fluid Dynamics T µν = T µν

id + Πµν (Fluid EMT, 2nd o. gradients)

+

∂µT µν = 0

(“EMT Conservation”)

=

“Causal” Relativistic Viscous Fluid Dynamics

First complete 2nd theory for shear only in 2007 !

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

Paul Romatschke Hydro/HIC

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SLIDE 24

Second Order Fluid Dynamics T µν = T µν

id + Πµν (Fluid EMT, 2nd o. gradients)

+

∂µT µν = 0

(“EMT Conservation”)

=

“Causal” Relativistic Viscous Fluid Dynamics

First complete 2nd theory for shear only in 2007 !

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

Paul Romatschke Hydro/HIC

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Hydrodynamcis: Limits of Applicability

Remember T µν = T µν

id + Πµν

Πµν is given by small gradient expansion Πµν = η∇<µuν> + . . . Hydrodynamics breaks down if gradient expansion breaks down: Π ∼ T µν

id or

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

Paul Romatschke Hydro/HIC

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Hydro Theory

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

Paul Romatschke Hydro/HIC

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Hydrodynamic Models for Heavy-Ion Collisions

Paul Romatschke Hydro/HIC

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Hydro Models for Heavy-Ion Collisions

Need initial conditions for Hydro: ǫ, uµ at τ = τ0 Need equation of state p = p(ǫ), which gives c2

s = dp dǫ

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

Paul Romatschke Hydro/HIC

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Initial Conditions

IC’s for hydro not known. Here are some popular choices: Fluid velocities are set to zero Boost-invariance: all hydro quantities only depend on proper time τ = √ t2 − z2 and transverse space x⊥. Models for energy density distribution: Glauber/Color-Glass-Condensate Starting time τ0: Should be of order 1 fm, precise value unknown

Paul Romatschke Hydro/HIC

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Equation of State

EoS known (approximately) from lQCD:

Paul Romatschke Hydro/HIC

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Transport Coefficients

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

[Demir and Bass, 2008]

Paul Romatschke Hydro/HIC

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SLIDE 32

Hydro Solvers

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

Paul Romatschke Hydro/HIC

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Freeze-out

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

hydro = (ǫ+p)uµuν−Pgµν+Πµν = T µν 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(ǫ + p)T 2 + O(p3)

  • Paul Romatschke

Hydro/HIC

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Experimental Observables

φ dN/dp/dφ

Paul Romatschke Hydro/HIC

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Experimental Observables

For ultrarelativistic heavy-ion collisions, dN dp⊥dφdy = dN dp⊥dφdy φ (1 + 2v2(p⊥) cos(2φ) + . . .) Radial flow:

dN dp⊥dy φ

Elliptic flow: v2(p⊥)

Paul Romatschke Hydro/HIC

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SLIDE 36

Putting things together

Hydro model simultion of RHIC Au+Au collisions [Luzum & Romatschke, 2008]

Paul Romatschke Hydro/HIC

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Current Research and Open Problems

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

Paul Romatschke Hydro/HIC

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SLIDE 38

Fluctuations

[slide stolen from M. Luzum]

Initial conditions are not smooth! Event average: <> There will be < v3 > and < v2 >2=< v2 >2

Paul Romatschke Hydro/HIC

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SLIDE 39

LHC results

centrality percentile

10 20 30 40 50 60 70 80 90

2

v

0.02 0.04 0.06 0.08 0.1 0.12

{2}

2

v (same charge) {2}

2

v {4}

2

v (same charge) {4}

2

v {q-dist}

2

v {LYZ}

2

v STAR {EP}

2

v STAR {LYZ}

2

v

[ALICE Collaboration, 2010]

If < v2 >2=< v2 >2 then v2{2} = v2{4}.

Paul Romatschke Hydro/HIC

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3D Evolution

1 2

  • 4
  • 2

2 4 h+/- v3 [%] ηp ideal, e-b-e η/s=0.08, e-b-e η/s=0.16, e-b-e 2 4 6 8 h+/- v2 [%] PHOBOS 15-25% central ideal, avg

[B. Schenke et al., 2010]

Non-Bjorken flow in longitudinal direction.

Paul Romatschke Hydro/HIC

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Freeze-Out

5 10 15 20 2 4 6 8 10 12 14 16 χ(p/T) / [4π η/(sT)] p/T Quadratic LO Coll. Linear

[K. Dusling et al., 2009]

Quadratic ansatz may be inaccurate

Paul Romatschke Hydro/HIC

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SLIDE 42

Thermalization

How does system get T µν that is close to hydro? Far from equilibration dynamics: non-perturbative, real-time: hard! Attempts to thermalization: pQCD inspired (’plasma instabilities’); AdS/CFT inspired (’collision of shock waves’)

Paul Romatschke Hydro/HIC

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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 not mean results are accurate! Keep in mind that ideal hydro only exists with numerical viscosity (value?) All working 2+1D viscous hydro codes are ’second order hydro’ Different names (’Israel-Stewart’, ’full IS’, ’BRSSS’) correspond to different choices for values of τπ, . . .

Paul Romatschke Hydro/HIC

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A personal appeal: theory/data comparisons

A new theorists calculation/model should first be rigorously studied before ’fitting’ data Example: hydrodynamic calculations on a grid; physical results correspond to limit of vanishing grid spacing Theorists: Please first check your model/calculation before you compare to experimental data Experimentalists: Please don’t blindly trust (or promote!) a model just because it fits data

Paul Romatschke Hydro/HIC

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...and all the rest...

There are many topics/details I couldn’t cover today! Some lecture notes: “New Developments in Relativistic Viscous Hydrodynamics”, PR, arXiv:0902.3663 “Nearly Perfect Fluidity: From Cold Atomic Gases to Hot Quark Gluon Plasmas”, T. Schäfer and D. Teaney, arXiv:0904.3107 “Early collective expansion: Relativistic hydrodynamics and the transport properties of QCD matter”, U.W. Heinz, arXiv:0901.4355

Paul Romatschke Hydro/HIC