Perfect Fluidity in Cold Atomic Gases?

Transcription

1 Perfect Fluidity in Cold Atomic Gases? Thomas Schaefer North Carolina State University 1

2 Hydrodynamics Long-wavelength, low-frequency dynamics of conserved or spontaneoulsy broken symmetry variables τ τ micro τ λ 1 Historically: Water (ρ, ǫ, π) 2

3 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

4 Questions Existence of solutions (1M$) Structure of solutions (Turbulence, Shocks,...) How to find constitutive relations from micro physics Determine transport coefficients from micro physics 4

5 BNL and RHIC 5

6 Heavy Ion Collision Star TPC 6

7 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) 7

8 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)? 8

9 Viscosity Bound: Rough Argument Kinetic theory estimate of shear viscosity Uncertainty relation implies η 1 3 n vml = 2 3 n ( 1 2 m v2 ) l v = 2 3 nǫτ mft η s ǫτ mft k B n E avτ mft k B k B where we have used s k B n. Danielewicz & Gyulassy (1984) Validity of kinetic theory as Eτ? Why η/s? Why not η/n? 9

10 Holographic Duals at Finite Temperature Thermal (superconformal) field theory CFT temperature CFT entropy AdS 5 black hole Hawking temperature of black hole Hawking-Bekenstein entropy area of event horizon Strong coupling limit s = π2 2 N2 c T 3 = 3 4 s 0 s weak coupling strong coupling Gubser and Klebanov λ = g 2 N 10

11 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) 11

12 Holographic Duals: Transport Properties Thermal (superconformal) field theory CFT entropy shear viscosity Strong coupling limit AdS 5 black hole 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 12

13 Viscosity Bound: Common Fluids 4π η h s Helium 0.1MPa Nitrogen 10MPa Water 100MPa Viscosity bound T, K 13

14 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 14 σ = 4π k 2

15 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 15

16 Elliptic Flow Hydrodynamic expansion converts σ σ µ coordinate space anisotropy σ σ to momentum space anisotropy 16

17 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 17

18 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) 18

19 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 2 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. 19

20 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 20

21 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? 21

22 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? 22