Perfect Fluidity
in Cold Atomic Gases?
Thomas Schaefer
Hydrodynamics
Long-wavelength, low-frequency dynamics of
conserved or spontaneoulsy broken symmetry
variables.
Example: Simple Fluid
Continuity equation
∂ρ
∂t + ∇~ (ρ~v) = 0
Euler (Navier-Stokes) equation
∂
∂t(ρvi) +
∂
∂xj Πij = 0
Energy momentum tensor
Πij = P δij + ρvivj + η ∂ivj + ∂jvi − 2 3δij∂kvk + . . . reactive dissipative
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 ΛElliptic 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 3nEτmft
Entropy density: s ∼ kBn. Uncertainty relation implies
η s ∼ nEτmft kBn ∼ Eτmft kB ≥ ~ kB
Validity of kinetic theory as Eτ ∼ ~?
Holographic Duals at Finite Temperature
Thermal (conformal) 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
strong 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 pSB/T4 nf=(2+1),Nτ=4 Nτ=6 nf=2, Nτ=4 Nτ=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/Tc ε/T4 εSB/T 4 p4, Nτ=4, nf=3 asqtad, Nτ=6, nf=2+1 15%Holographic Duals: Transport Properties
Thermal (conformal) 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 =
~ 4πkB
Son and Starinets
0 ¯ h 4πkB η s g2Nc
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 hViscosity Bound: Counter Examples?
non relativistic systems: can make S/N large
η s = 1 log(Ns) c√mT a2 n
modified conjecture: applies to systems that can be embedded in a relativistic (gauge?) theory
T. Cohen: Consider heavy-light mesons in QCD with NF = Nc → ∞
mQ = m0QNF n = n0 log(NF ), T = T0 NF log(NF )1/2 η s ∼ 1
Designer Fluids
Atomic gas with two spin states: “↑” and “↓”
open closed V r Feshbach resonance a(B) = a0 1 + ∆ B − B0
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
(kF a) → ∞ (kF 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 B) Hydrodynamic expansion converts coordinate space anisotropy to momentum space anisotropyCollective 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/kFa 680 730 834 1114 B (Gauss)
Ideal fluid hydrodynamics, equation of state P ∼ n5/3
∂n ∂t + ∇ ·~ (n~v) = 0 ∂~v ∂t + ~v · ∇~ ~v = − 1 mn∇~ P − 1 m∇~ 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) thold(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~Viscous Hydrodynamics
Energy dissipation (η, ζ, κ: shear, bulk viscosity, heat conductivity)
˙ E = −η 2 Z d3x ∂ivj + ∂jvi − 2 3δij∂kvk 2 − ζ Z d3x (∂ivi)2 − κ T 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 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 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?
Kinetic Theory
Quasi-Particles: Kinetic Theory
Tij = Z d3p pipj Ep fp, Boltzmann equation ∂fp ∂t + ~v · ∇~xfp + F~ · ∇~pfp = C[fp]
Linearized theory (Chapman-Enskog): fp = fp0(1 + χp/T )
η ≥ hχ|Xi 2 hχ|C|χi hχ|Xi = Z d3p fp0 χp pijvij vij = v2δij − 3vivj
Low T: Phonons High T: Atoms 0.2 0.4 0.6 0.8 1 1.2 TTF 0.1 0.2 0.5 1 2 5 Η s + + η ∼ TF 8 η ∼ T 3/2 log T −1
Elliptic Flow
Free scaling expansion
n(r⊥, rz) = 1 b2 ⊥bz n0 r ⊥ b⊥ , rz bz ¨b ⊥ = ω⊥2 b⊥(b2⊥bz)γ 50 100 150 200 2 4 6 8 10 Ri/R0z ω⊥t R⊥ Rz Viscous damping ˙ E = −4 3 ˙ b⊥ b⊥ − ˙ bz bz 2 Z d3x η(x) ∆E = Z dt E˙ converges quickly 5 10 15 20 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 E/(µN) ω⊥t Ekin Eint ∆E(KSS) ∆E(3·KSS)
Elliptic Flow (cont)
Can define v2 = hcos(2φ)i as in HI collisions
ǫ = h2z 2 − x2 + y2i hz2 + x2 + y2 i 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 v2 v/vmax ǫ = 0.045 0.25 0.50 0.75
Final Thoughts
Cold atomic gases provide interesting, strongly coupled, model system in which to study sources of dissipation.
η/s ∼ 1/2
Smaller than any other known liquid (except for QGP?). Since other sources of dissipation exist, this is really an upper bound.
There are reliable calculations of η/s at high T (Bruun, Smith, . . . ) and low T (Rupak and T.S). Extrapolate to T ∼ TF
Conjectured bound has a smooth non-relativistic limit. Note that the
a → ∞ limit can also be realized in QCD (by tuning µ, µe and mq).
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.