In Search of the Perfect Fluid

Thomas Schaefer, North Carolina State University

See T. Sch¨afer, D. Teaney, “Perfect Fluidity” [arXiv:0904.3107] Measures of Perfection

Viscosity determines shear stress (“friction”) in ﬂuid ﬂow

∂v F = Aη x ∂y

Dimensionless measure of shear stress: Reynolds number

n Re = mvr η × ﬂuid ﬂow property property

[η/n] = ~ • s Relativistic systems Re = τT • η × Other sources of dissipation (thermal conductivity, bulk viscosity, ...) vanish for certain ﬂuids, but shear viscosity is always non-zero.

There are reasons to believe that η is bounded from

below by a constant times ~s/kB. In a large class of theories η/s ~/(4πk ). ≥ B

A ﬂuid that saturates the bound is a “perfect ﬂuid”. Fluids: Gases, Liquids, Plasmas, ...

Hydrodynamics: Long-wavelength, low-frequency dynamics of conserved or spontaneously broken sym- metry variables.

−1 τ τmicro τ λ ∼ ∼

Historically: Water (ρ, ǫ, ~π) Example: Simple Fluid

Conservation laws: mass, energy, momentum ∂ρ + ~ (ρ~v) = 0 ∂t ∇ ∂ǫ + ~ ~ ǫ = 0 ∂t ∇ ∂ ∂ (ρvi) + Πij = 0 ∂t ∂xj [Euler/Navier-Stokes equation]

Constitutive relations: Energy momentum tensor 2 Π = P δ + ρv v + η ∂ v + ∂ v δ ∂ v + O(∂2) ij ij i j i j j i − 3 ij k k reactive dissipative 2nd order Kinetic Theory

Kinetic theory: conserved quantities carried by quasi-particles ∂f p + ~v ~ f + F~ ~ f = C[f ] ∂t · ∇x p · ∇p p p 1 η n pl¯ mfp ∼ 3 Normalize to density. Uncertainty relation suggests η pl¯ mfp ~ n ∼ ≥ Also: s k n and η/s ~/k ∼ B ≥ B

Validity of kinetic theory as pl¯ mfp ~? ∼ Eﬀective Theories for Fluids (Here: Weak Coupling QCD)

1 =q ¯ (iD/ m )q Ga Ga L f − f f − 4 µν µν

∂f p + ~v ~ f = C[f ] (ω < T ) ∂t · ∇x p p

∂ ∂ 4 (ρvi) + Πij = 0 (ω

Thermal (conformal) ﬁeld theory AdS5 black hole ≡ Hawking-Bekenstein entropy CFT entropy ⇔ area of event horizon ∼ Graviton absorption cross section shear viscosity ⇔ area of event horizon ∼

δS 0 Tµν = gµν = gµν + γµν δgµν Holographic Duals: Transport Properties

h¯

Son and Starinets (2001) 4πkB

2 0 g Nc Strong coupling limit universal? Provides lower bound for all theories? Eﬀective Theories (Strong coupling)

1 1 = λ¯(iσ D)λ Ga Ga + ... S = d5x√ g + ... L · − 4 µν µν ⇔ 2κ2 − R 5 Z

∂ ∂ (ρvi) + Πij = 0 (ω < T ) ∂t ∂xj Kinetics vs No-Kinetics

AdS/CFT low viscosity goo pQCD kinetic plasma Kinetics vs No-Kinetics

Spectral function ρ(ω)=ImGR(ω, 0) associated with Txy

0.6 1 ρxyxy(ω) s 2ω 1/g4 (ω/T )3 0.5 AdS/CFT ∼ ∼ π(ω/2πT)3

ω 0.4 )/2 ω ( 0.3 xyxy ρ 1/s 1 1 0.2 ∼ ∼ η/s=1/4π 0.1 g2 ∼

4 2 0 g T g T gT T ω 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ω/2πT weak coupling QCD strong coupling AdS/CFT transport peak vs no transport peak Perfect Fluids: How to be a contender?

Bound is quantum mechanical need quantum ﬂuids Bound is incompatible with weak coupling and kinetic theory strong interactions, no quasi-particles Model system has conformal invariance (essential?) (Almost) scale invariant systems Perfect Fluids: The contenders

Liquid Helium (T=0.1 meV) QGP (T=180 MeV)

Trapped Atoms (T=0.1 neV) Perfect Fluids: The contenders

Liquid Helium η = 1.7 10−6Pa s QGP η = 5 1011Pa s · · · · Consider ratios Trapped Atoms η/s η = 1.7 10−15Pa s · · Kinetic Theory: Quasiparticles

low temperature high temperature unitary gas phonons atoms

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¡ helium phonons, rotons atoms

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QCD pions quarks, gluons

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¡ Low T: Phonons Goldstone boson ψψ = e2iϕ ψψ h i 5 2 ~ 2 / 3/2 ( ϕ) = c0m µ ϕ˙ ∇ + ... L − − 2m ! Viscosity dominated by ϕ + ϕ ϕ + ϕ → 5 8 ξ TF η = A 3 5 + + cs T

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¡ T.S., G.R. (2007) High T: Atoms Cross section regularized by thermal momentum

15 η = (mT )3/2 32√2

¡ Bruun (2005) Low T: Phonons and Rotons Eﬀective lagrangian

∗ 2 2 2 = ϕ (∂0 v )ϕ + iλϕ˙(~ ϕ) + ... L − ∇ ∗ ∗ 2 + ϕ (i∂0 ∆)ϕ + c0(ϕ ϕ ) + ... R,v − R,v R,v R,v Shear viscosity

cRph ∆/T cph η = ηR + e + 5 + ... √T T

Landau & Khalatnikov High T: Atoms Viscosity governed by hard core (V 1/r12) ∼ 2/3 η = η0(T/T0) Low T: Pions Chiral perturbation theory f 2 = π Tr[∂ U∂µU †] + (BTr[MU] + h.c.) + ... L 4 µ Viscosity dominated by ππ scattering f 4 η = A π T

High T: Quasi-Particles HTL theory (screening, damping, ...) quasi-particle width vαv = dΩ Ga β Ga,µβ 2 LHTL µα (v D)2 γ g T Z · ∼ Viscosity dominated by t-channel gluon exchange

27.13T 3 η = g4 log(2.7/g)

AMY (2003) ¡ ¢ £ ¤ Theory Summary

1.6 1.4 unitary gas 4 η/s η/s He 1.2 6

0.8 4

0.6

0.4 2

0.2

0.2 0.4 0.6 0.8 2 4 6 8 10 12 14

T/TF T [K]

5 QCD η/s 4

100 200 300 400 500 T [MeV ] I. Experiment (Liquid Helium)

200

100

P] 50 µ [ η

10 1 1.5 2 T [K]

Kapitza (1938) Hollis-Hallett (1955) viscosity vanishes below Tc roton minimum, phonon rise capillary ﬂow viscometer rotation viscometer

η/s 0.8 ~/k ≃ B II. Scaling Flows (Cold Gases)

transverse expansion expansion (rotating trap) collective modes

∂n + ~ (n~v) = 0 ∂t ∇ · ∂~v mn + mn ~v ~ ~v = ~ P n~ V ∂t · ∇ −∇ − ∇ Scaling Flows

n5/3 mT Universal equation of state P = f m n2/3 Equilibrium density proﬁle x2 y2 z2 n0(x) = n(µ(x), T ) µ(x) = µ0 1 − R2 − R2 − R2 x y z Scaling Flow: Stretch and rotate proﬁle

µ0 µ0(t), T T0(µ0(t)/µ0), R R (t),... → → x → x Linear velocity proﬁle

~v(x, t) = (α x + (α ω)y,α y + (α + ω)y,α z) x − y z “Hubble ﬂow” Dissipation (Scaling Flows)

Energy dissipation (η,ζ,κ: shear, bulk viscosity, heat conductivity)

2 1 2 E˙ = d3xη(x) ∂ v + ∂ v δ ∂ v −2 i j j i − 3 ij k k Z 2 1 2 d3xζ(x)(∂ v ) d3xκ(x)(∂ T ) − i i − T i Z Z

Have ζ = 0 and T (x) = const. Universality implies T η(x) = s(x) α s µ(x)

d3xη(x) = S α h si Z Navier-Stokes equation

Option 1: Moment method

d3x x (ρv˙ + ...) = d3x x ( P δΠ ) k i k −∇i − ∇j ij Z Z Only involves η /E0. h i Option 2: Scaling ansatz for η(µ, T )

3/2 P (n, T ) η(n, T ) = η0(mT ) + η1 T

Option 3: Numerical solutions. Dissipation

80 Ri θ [ ◦] [µm] 300 60

R E/EF =0.56 ⊥ 200 40 E/EF =2.1

100 20 Rz

0.5 1 1.5 2 0.5 1 1.5 2 2.5 3 3.5 t[ms] t [ms]

R R θ ⊥ ⊥

O’Hara et al (2002), Kinast et al (2005), Clancy et al (2007) Dissipation

80 Ri θ [ ◦] [µm] 300 60

R E/EF =0.56 ⊥ 200 40 E/EF =2.1

100 20 Rz

0.5 1 1.5 2 0.5 1 1.5 2 2.5 3 3.5 t[ms] t [ms]

5 1/3 (δt0)/t0 0.008 α 2 10 S/N 0.85 = h si · (δa)/a   0.024  1/(4π) N 2.3 E0/EF       ◦ t0: “Crossing time” (b⊥ = bz, θ = 45 ) a: amplitude Time Scales

dissipative hydro/free streaming ballistic Ri [µm] 300 R ⊥ 200

100 Rz

0.5 1 1.5 2 t[ms] Collective modes: Small viscous correction exponentiates

a(t) = a0 cos(ωt) exp( Γ t) −

Γ E0 N η/s = (3Nλ)1/3 h i ω E S ⊥ F

1.5 η/s h i

1.0

0.5

0.0 0.2 0.4 0.6 0.8

T/TF

Kinast et al. (2006), Schaefer (2007) Limitations of Scaling Flows

Simple model for η(n, T )

3/2 P (n, T ) η(n, T ) = η0(mT ) + η1 f T Find exact scaling solutions of the Navier Stokes equation

But: η0 completely unconstrained by data 3/2 [η0(mT ) ( v + ...)] = 0 ∇j ∇i j 1.0

η/n 0.8

0.6

0.4

0.2

0.0 0.5 1.0 1.5 2.0 2.5 y Relaxation Time Model

In real systems stress tensor does not relax to Navier-Stokes form instantaneously. Consider ∂ 2 τ + ~v ~ δΠ = δΠ η ∂ v + ∂ v δ ∂ v R ∂t · ∇ ij ij − i j j i − 3 ij · In kinetic theory τ (η/n) T −1 R ≃

0.08

3/2 Γ ω disspiation from η (mT ) : 0.06 • ∼ ⊥

corona excerts drag force. 0.04

modiﬁed T dependence 0.02 •

modifed N scaling 0.2 0.4 0.6 0.8 1.0 1.2 • T/TF Where are we?

high temperature (T > 2.5T ) dominated by corona • c low temperature (T T ): evidence for low viscosity • ∼ c (η/s < 0.4) core also seen in “irrotational ﬂow” data • full (2nd order hydro or hydro+kin) analysis needed • III. Elliptic Flow (QGP)

Hydrodynamic Hydro model PHENIX Data STAR Data + - 0

2 π π expansion converts π + KS + - 0.3 K K +K Λ+Λ coordinate space p p+p Λ anisotropy 0.2 to momentum space anisotropy 0.1

Anisotropy Parameter v Anisotropy Parameter 0

0 2 4 6

b Transverse Momentum p T (GeV/c) source: U. Heinz (2005) Viscosity and Elliptic Flow

25 Consistency condition Tµν δTµν ideal ≫ η/s=0.03 20 (applicability of Navier-Stokes) η/s=0.08 η/s=0.16 4 η + ζ 3 15 STAR 3 (τT ) (percent)

s ≪ 4 2 10 v Danielewicz, Gyulassy (1985) 5 Very restrictive for τ < 1 fm 0 0 1 2 3 4 pT [GeV]

Romatschke (2007), Teaney (2003)

Many questions: Dependence on initial conditions, freeze out, etc. Outlook

Too early to declare a winner.

η/s 0.8 (He), η/s 0.5 (CA), η/s 0.5 (QGP) ≃ ≤ ≤

Other experimental constraints, more analysis needed.

Kinetic theory: o.k. in He (all T ), o.k. close to Tc in CA, QGP?

New theory tools: AdS/Cold Atom correspondence? Field theory approaches in cross over regime (large N, epsilon expansions, ...)