Perfect Fluidity in Cold Atomic Gases?
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1 Perfect Fluidity in Cold Atomic Gases? Thomas Schaefer North Carolina State University 1
2 2
3 Hydrodynamics Long-wavelength, low-frequency dynamics of conserved or spontaneoulsy broken symmetry variables. τ τ micro τ λ 1 Historically: Water (ρ, ǫ, π) 3
4 Example: Simple Fluid Continuity equation ρ t + (ρ v) = 0 Euler (Navier-Stokes) equation t (ρv i) + Π ij = 0 x j Energy momentum tensor Π ij = Pδ ij + ρv i v j + η reactive ( i v j + j v i 23 δ ij k v k ) dissipative
5 Elliptic Flow Hydrodynamic expansion converts coordinate space anisotropy to momentum space anisotropy Anisotropy Parameter v Hydro model π K p Λ PHENIX Data π + +π K +K p+p STAR Data 0 K S Λ+Λ b Transverse Momentum p T (GeV/c) source: U. Heinz (2005) 5
6 Elliptic Flow II v 2 /ε 0.25 HYDRO limits E lab /A=11.8 GeV, E877 E lab /A=40 GeV, NA E lab /A=158 GeV, NA49 s NN =130 GeV, STAR s NN =200 GeV, STAR Prelim (1/S) dn ch /dy Requires perfect fluidity (η/s < 0.1?) (s)qgp saturates (conjectured) universal bound η/s = 1/(4π)? 6
7 Viscosity Bound: Rough Argument Kinetic theory estimate of shear viscosity η 1 3 n vml = 2 3 n ( 1 2 m v2 ) l v = 2 3 neτ mft Entropy density: s k B n. Uncertainty relation implies η s neτ mft k B n Eτ mft k B k B Validity of kinetic theory as Eτ? Why η/s? Why not η/n? 7
8 Holographic Duals at Finite Temperature Thermal (conformal) field theory AdS 5 black hole CFT temperature CFT entropy Strong coupling limit s = π2 2 N2 c T 3 = 3 4 s 0 Gubser and Klebanov Hawking temperature of black hole Hawking-Bekenstein entropy area of event horizon s weak coupling strong coupling λ = g 2 N Extended to transport properties by Policastro, Son and Starinets 8
9 Quark Gluon Plasma Equation of State (Lattice) 7 6 p/t ε/t 4 ε SB /T 4 15% n f =(2+1),N τ =4 N τ =6 n f =2, N τ =4 N τ =6 p SB /T p4, N τ =4, n f =3 asqtad, N τ =6, n f =2+1 0 T/T T/T c Compilation by F. Karsch (SciDAC) 9
10 Holographic Duals: Transport Properties Thermal (conformal) field theory AdS 5 black hole CFT entropy shear viscosity Strong coupling limit Hawking-Bekenstein entropy area of event horizon Graviton absorption cross section area of event horizon η s η s = 4πk B Son and Starinets h 4πk B Strong coupling limit universal? Provides lower bound for all theories? 0 g 2 N c 10
11 Viscosity Bound: Common Fluids 4π η h s Helium 0.1MPa Nitrogen 10MPa Water 100MPa Viscosity bound T, K 11
12 Viscosity Bound: Counter Examples? non relativistic systems: can make S/N large η s = 1 log(n s ) c mt a 2 n modified conjecture: applies to systems that can be embedded in a relativistic (gauge?) theory T. Cohen: Consider heavy-light mesons in QCD with N F = N c m Q = m 0 QN F n = η s 1 log(n s ) n 0 log(n F ), T = T 0 N F log(n F ) 1/2 stable fluid? s well defined? 12
13 Designer Fluids Atomic gas with two spin states: and V closed open r Feshbach resonance a(b) = a 0 ( 1 + B B 0 ) Unitarity limit a 13 σ = 4π k 2
14 Why are these systems interesting? System is intrinsically quantum mechanical cross section saturates unitarity bound System is scale invariant at unitarity. Universal thermodynamics E ( E ) A = ξ = ξ 3 ( k 2 F A 0 5 2M System is strongly coupled but dilute (k F a) (k F r) 0 ) Strong elliptic flow observed experimentally 14
15 Elliptic Flow Hydrodynamic expansion converts σ σ µ coordinate space anisotropy σ σ to momentum space anisotropy 15
16 Collective Modes Radial breathing mode Kinast et al. (2005) 2.00 B (Gauss) Frequency (ω/ω ) /k F a Ideal fluid hydrodynamics, equation of state P n 5/3 n t + (n v) = 0 v ( t + v ) v = 1 P mn 1 V m ω = 10 3 ω 16
17 Damping of Collective Excitations 60 (a) (<x 2 >) (1/2) ( µm) (b) (c) Damping rate (1/τω ) T ~ t hold (ms) 6 8 T/T F = (0.5,0.33,0.17) τω: decay time trap frequency Kinast et al. (2005) 17
18 Viscous Hydrodynamics Energy dissipation (η, ζ, κ: shear, bulk viscosity, heat conductivity) Ė = η d 3 x ( i v j + j v i 23 ) 2 2 δ ij k v k ζ d 3 x ( i v i ) 2 κ d 3 x ( i T) 2 T Shear viscosity to entropy ratio (assuming ζ =κ=0) η/s η s = 3 4 ξ 1 2 (3N) 1 3 Γ ω ω N ω S see also Bruun, Smith, Gelman et T/T F al. 18
19 Damping dominated by shear viscosity? Study dependence on flow pattern Study particle number scaling viscous hydro: Γ N 1/3 Boltzmann: Γ N 1/3 Role of thermal conductivity? suppressed for scaling flows: δt T(δn/n) const 19
20 Kinetic Theory Quasi-Particles: Kinetic Theory T ij = d 3 p p ip j E p f p, Boltzmann equation f p t + v x f p + F p f p = C[f p ] Linearized theory (Chapman-Enskog): f p = f 0 p(1 + χ p /T) η χ X 2 χ C χ χ X = d 3 p f 0 p χ p p ij v ij v ij = v 2 δ ij 3v i v j 20
21 Low T: Phonons High T: Atoms Η s T T F η ( T ) 1 ) 3/2 log ( T ( TF ) 8 η + + s T s T F T F 21
22 Free scaling expansion n(r, r z ) = 1 b 2 b z b = Viscous damping n 0 ( r b, r z b z ) ω 2 Elliptic Flow R i /R 0 z b (b 2 b z) γ Ė = 4 E/(µN) (ḃ ḃz ) 2 d x η(x) b b z E kin E int E(3 KSS) E = 0.05 E(KSS) dt Ė converges quickly R R z ω t ω t 22
23 Elliptic Flow (cont) Can define v 2 = cos(2φ) as in HI collisions 1 v ǫ = ǫ = 2z2 x 2 + y 2 z 2 + x 2 + y v/v max Can also sweep to BEC regime and simulate recombination models 23
24 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 T F 24
25 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. No quasi-particles in sqgp? 25
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