Perfect Fluidity in Cold Atomic Gases?

Perfect Fluidity

in Cold Atomic Gases?

Thomas Schaefer

Hydrodynamics

Long-wavelength, low-frequency dynamics of

conserved or spontaneoulsy broken symmetry

variables

τ ∼ τmicro τ ∼ λ−1

Example: Simple Fluid

Continuity equation

∂ρ

∂t + ∇~ (ρ~v) = 0

Euler (Navier-Stokes) equation

∂

∂t(ρvi) +

∂

∂x_j Πij = 0

Energy momentum tensor

Π_ij = P δ_ij + ρv_iv_j + η ∂_iv_j + ∂_jv_i − 2 3δij∂kvk + . . . reactive dissipative

Questions

Existence of solutions (1M$)

Structure of solutions (Turbulence, Shocks, . . .)

How to find constitutive relations from micro physics

Heavy Ion Collision

Elliptic Flow

Hydrodynamic expansion converts coordinate space anisotropy to momentum space anisotropy b 2 Anisotropy Parameter v (GeV/c) T Transverse Momentum p 0 2 4 6 0 0.1 0.2 0.3 -π + + π 0 S K -+K + K p p+ Λ + Λ STAR Data PHENIX Data Hydro model π K p Λ source: U. Heinz (2005)

Elliptic Flow II

0 5 10 15 20 25 30 35 0 0.05 0.1 0.15 0.2 0.25 /dy ch (1/S) dN ε / 2 v HYDRO limits

/A=11.8 GeV, E877

lab

E

/A=40 GeV, NA49

lab

E

/A=158 GeV, NA49

lab

E

=130 GeV, STAR

NN

s

=200 GeV, STAR Prelim.

NN

s

Requires “perfect” fluidity (η/s < 0.1 ?)

Viscosity Bound: Rough Argument

Kinetic theory estimate of shear viscosity

η ∼ 1 3nvml¯ = 2 3n 1 2mv¯ 2 l ¯ v = 2 3nǫτmft

Uncertainty relation implies

η s ∼ ǫτmft k_Bn ∼ E_avτmft k_B ≤ ~ k_B

Danielewicz & Gyulassy (1984)

where we have used s ∼ k_Bn.

Validity of kinetic theory as Eτ ∼ ~?

Holographic Duals at Finite Temperature

Thermal (superconformal)

field theory ⇔ AdS5 black hole

CFT temperature ⇔ Hawking temperature of

black hole

CFT entropy ⇔ Hawking-Bekenstein entropy ∼ area of event horizon

Strong coupling limit

s = π 2 2 N 2 c T 3 = 3 4s0

Gubser and Klebanov

s

λ = g N2

weak coupling

Quark Gluon Plasma Equation of State (Lattice)

0 1 2 3 4 5 6 7 1.0 1.5 2.0 2.5 3.0 3.5 4.0 T/T₀ p/T4 p_SB/T4 n_f=(2+1),N_τ=4 N_τ=6 n_f=2, N_τ=4 N_τ=6 0 2 4 6 8 10 12 14 16 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 T/T_c ε/T4 εSB/T 4 p4, N_τ=4, n_f=3 asqtad, N_τ=6, n_f=2+1 15%

Holographic Duals: Transport Properties

Thermal (superconformal)

field theory ⇔ AdS5 black hole

CFT entropy ⇔ Hawking-Bekenstein entropy ∼ area of event horizon

shear viscosity ⇔ Graviton absorption cross section ∼ area of event horizon

Strong coupling limit

η

s =

~

4πk_B

Son and Starinets

0 ¯ h 4_πkB η s g2N_c

Viscosity Bound: Common Fluids

1 10 100 1000 T, K 0 50 100 150 200 Helium 0.1MPa Nitrogen 10MPa Water 100MPa Viscosity bound 4π η s h

Designer Fluids

Atomic gas with two spin states: “↑” and “↓”

open closed V r Feshbach resonance a(B) = a0 1 + ∆ B − B0 “Unitarity” limit a → ∞ σ = 4π k2

Why are these systems interesting?

System is intrinsically quantum mechanical

cross section saturates unitarity bound

System is scale invariant at unitarity. Universal thermodynamics

E A = ξ E A 0 = ξ 3 5 k2 F 2M

System is strongly coupled but dilute

(k_F a) → ∞ (k_F r) → 0

Elliptic Flow

A) 300 200 100 σ, x σ z ( µ m ) 2.0 1.5 1.0 0.5 0.0 Expansion Time (ms) 2.5 2.0 1.5 1.0 0.5 A s p e c t R a ti o ( σ / x σ ) z 2.0 1.5 1.0 0.5 0.0 Expansion Time (ms) B) Hydrodynamic expansion converts coordinate space anisotropy to momentum space anisotropy

Collective Modes

Radial breathing mode Kinast et al. (2005) 2.00 1.95 1.90 1.85 1.80 1.75 Frequency ( ω/ω ⊥ ) 2 1 0 -1 -2 1/k_Fa 680 730 834 1114 B (Gauss)

Ideal fluid hydrodynamics, equation of state P ∼ n5/3

∂n ∂t + ∇ ·~ (n~v) = 0 ∂~v + ~v · ∇~ ~v = − 1 ∇~ P − 1 ∇~ V ω = r 10 3 ω⊥

Damping of Collective Excitations

50 40 30 20 8 6 4 2 40 30 2.5 2.0 1.5 1.0 0.5 0.0 60 50 40 (<x 2 >) (1/2) ( µ m) 50 40 2.5 2.0 1.5 1.0 0.5 0.0 (c) (b) 50 40 2.5 2.0 1.5 1.0 0.5 0.0 60 50 40 (a) t_hold(ms) 0.10 0.05 0.00 Damping rate (1/ τω ⊥ ) 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 T~

T /TF = (0.5, 0.33, 0.17) τ ω: decay time × trap frequency Kinast et al. (2005)

Viscous Hydrodynamics

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

˙ E = −η 2 Z d3x ∂_iv_j + ∂_jv_i − 2 3δij∂kvk 2 − ζ Z d3x (∂_iv_i)2 − κ 2 Z d3x (∂_iT)2

Shear viscosity to entropy ratio (assuming ζ =κ= 0) η s = 3 4ξ 1 2(3N) 1 3 Γ ω_⊥ ¯ ω ω_⊥ N S

see also Bruun, Smith, Gelman et

0.2 0.4 0.6 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 η/s T /TF

Damping dominated by shear viscosity?

Study dependence on flow pattern

Study particle number scaling

viscous hydro: Γ ∼ N−1/3 Boltzmann: Γ ∼ N1/3

Role of thermal conductivity?

Final Thoughts

Cold atomic gases provide interesting, strongly coupled, model system in which to study sources of dissipation.

η/s ∼ 1/3

Smaller than any other known liquid (except for QGP?). Since other sources of dissipation exist, this is really an upper bound.

Conjectured bound has a smooth non-relativistic limit. Note that the a → ∞ limit can also be realized in QCD (by tuning µ, µ_e and m_q).

But: In non-relativistic systems s ≫ n possible Purely field theoretic proofs?

N = 4 SUSY YM is special because there is no phase transition. In real systems there is a phase transition as the coupling becomes large, and the new phase (confined in QCD, superfluid in the atomic system) has weakly coupled low energy excitations, and a large viscosity.