(Nearly) perfect fluidity in cold atomic gases: Recent results. Thomas Schaefer North Carolina State University

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1 (Nearly) perfect fluidity in cold atomic gases: Recent results Thomas Schaefer North Carolina State University

2 Fluids: Gases, Liquids, Plasmas,... Hydrodynamics: Long-wavelength, low-frequency dynamics of conserved or spontaneously broken symmetry variables. τ τ micro τ λ Historically: Water (ρ, ǫ, π)

3 Simple non-relativistic fluid Simple fluid: Conservation laws for mass, energy, momentum ρ t + (ρ v) = 0 ǫ t + j ǫ = 0 t (ρv i) + Π ij = 0 x j Constitutive relations: Energy momentum tensor Π ij = Pδ ij + ρv i v j + η ( i v j + j v i 23 ) δ ij k v k + O( 2 ) reactive dissipative 2nd order Expansion Π 0 ij δπ1 ij δπ2 ij

4 Regime of applicability Expansion parameter Re 1 = η( v) ρv 2 = η ρlv 1 1 Re = η n fluid property mvl flow property Consider mvl : Hydrodynamics requires η/( n) < 1

5 Shear viscosity in kinetic theory Kinetic theory: conserved quantities carried by quasi-particles f p t + v x f p + F p f p = C[f p ] η 1 3 n p l mfp Weakly interacting gas: l mfp 1/(nσ) η 1 3 Strongly interacting gas: η(σ 0) η n pl mfp but: kinetic theory not reliable! p σ

6 Holographic duals: Transport properties Thermal (conformal) field theory AdS 5 black hole CFT entropy shear viscosity Strong coupling limit η s = 4πk B Hawking-Bekenstein entropy area of event horizon Graviton absorption cross section area of event horizon η s Son and Starinets (2001) h 4πk B Strong coupling limit universal? Provides lower bound for all theories? 0 g 2 N c Answer appears to be no; e.g. theories with higher derivative gravity duals.

7 Effective theories for fluids (Unitary Fermi Gas, T > T F ) L = ψ ( i M ) ψ C 0 2 (ψ ψ) 2 f p t + v x f p = C[f p ] ω < T t (ρv i) + x j Π ij = 0 ω < T n η

8 Effective theories (Strong coupling) L = λ(iσ D)λ 1 4 Ga µνg a µν +... S = 1 2κ 2 5 d 5 x gr +... SO(d + 2,2) Schr(d) AdS d+3 X d+3 t (ρv i) + x j Π ij = 0 (ω < T)

9 Kinetics vs No-Kinetics AdS/CFT low viscosity goo gravitational dual η/s 1/(4π) kinetic liquid quasi-particles η/s > 1

10 Outline I. Conformal second order hydrodynamics II. Fluctuations III. Kinetic theory IV. Experiment V. Outlook: QGP vs Cold Atoms

11 I. Scale invariant fluid dynamics Many body system: Effective cross section σ tr n 2/3 (or σ tr λ 2 ) Systems remains hydrodynamic despite expansion

12 Scale and conformal symmetry Gallilean boosts x = x + vt t = t scale trafo x = e s x t = e 2s t conformal trafo x = x/(1 + ct) 1/t = 1/t + c Ideal fluid dynamics Π 0 ij = Pδ ij + ρv i v j, P = 2 3 E First order viscous hydrodynamics ( δ (1) Π ij = ησ ij, σ ij = i v j + j v i 2 ) 3 δ ij( v), ζ = 0

13 Second order conformal hydrodynamics Relaxation of shear stress is a second order hydro term. Complete list [ δ (2) Π ij = ητ π Dσ ij + 2 ] 3 σij ( v) + λ 1 σ i k σj k + λ 2 σ i k Ωj k + λ 3 Ω i k Ωj k + γ 1 i T j T + γ 2 i P j P + γ 3 i j T +... D = 0 + v A ij = 1 2 A ij + A ji 3 2 g ij Ak k Ω ij = i v j j v i New transport coefficients τ π, λ i, γ i Can be written as a relaxation equation for π ij δπ ij π ij = ησ ij τ π [ Dπ ij ( v)πij ] +... Chao, Schaefer (2011)

14 Why second order fluid dynamics? Scaling ( Hubble ) expansion ρ(x i, t) = ρ 0 (b i (t)x i ), v i (x j, t) = α i (t)x i, α i (t) = ḃi(t)/b i (t) Compare ideal and dissipative stresses Π 0 xx δ (1) Π xx Π 0 ii = P + ρα ix 2 i δ (1) Π ii = η(α i 4 3 j α j) x x Ideal stresses propagate with speed c s, dissipative stresses propagate with infinite speed. Hydro always breaks down in the dilute corona. Solved by relaxation time τ π η P.

15 II. Fluctuations If hydrodynamics is an effective (field?) theory then where are the loop corrections?

16 Hydrodynamic variables fluctuate Thermal fluctuations δv i (x, t)δv j (x, t) = T ρ δ ijδ(x x ) Linearized hydrodynamics propagates fluctuations as shear or sound δv T i δv T j ω,k = 2T ρ (δ ij ˆk iˆkj ) δv L i δv L j ω,k = 2T ρ ˆk iˆkj νk 2 ω 2 + (νk 2 ) 2 shear ωk 2 Γ (ω 2 c 2 sk 2 ) 2 + (ωk 2 Γ) 2 sound v = v T + v L : v T = 0, v L = 0 ν = η/ρ, Γ = 4 3 ν +...

17 Hydro Loops: Breakdown of second order hydro Response function G xyxy R = θ(t)[π xy,π xy ] ω,k Π xy = ρv x v y v T ρ ρ v T vl v L v L ρ ρ ρ vt ρ G xyxy R = P + δp + iω[η + δη] + ω 2 [ητ π + δ(ητ π )] δη T ( ) 2 ( ρ P η ρ ) 1/2 δ(ητ π ) 1 ( ) 3/2 ρ ω η

18 Hydro Loops: Breakdown of second order hydro δη T ( ) 2 ( ρ P η ρ ) 1/2 δ(ητ π ) 1 ( ) 3/2 ρ ω η Small shear viscosity enhances fluctuation corrections. Small η leads to large δη: There must be a bound on η/n. Relaxation time diverges: 2nd order hydro without fluctuations inconsistent.

19 Fluctuation induced bound on η/n η/n fluctuations 0.5 η(ω)/n 0.4 fluctuations kinetic theory kinetic theory T/T F ω/t spectral function (η/n) min 0.3 non-analytic ω term see also Kovtun, Moore, Romatschke (2011)

20 III. Linear response and kinetic theory Consider background metric g ij (t,x) = δ ij + h ij (t,x). Linear response δπ ij = 1 2 Gijkl R h kl Kubo relation: η(ω) = 1 ω ImGxyxy R (ω, 0) Kinetic theory: Boltzmann equation ( ( t + pi m x i g il ġ lj p j + Γ i jk p j p k m ) ) p i f(t,x,p) = C[f] C[f] =

21 Kinetic theory linearize f = f 0 + δf, solve for δf, δπ ij, G R, η(ω) η(ω) = η 1 + ω 2 τ 2 π η = π (mt)3/2 τ π = η nt shear channel sound channel xyxy (,k)/( k F a= T T/T F =1, k=0.5 T/T F =1, k=1.0 T/T F =2, k=0.5 T/T F =2, k=1.0 T/T F =3, k=0.5 T/T F =3, k=1.0 zzzz (,k)/( k F a=100 T/T F =1, k=0.5 T/T F =1, k=1.0 T/T F =2, k=0.5 T/T F =2, k=1.0 T/T F =3, k=0.5 T/T F =3, k= T Braby, Chao, T.S. (2010)

22 Second order hydrodynamics from kinetic theory Boltzmann equation (BGK approximation) [ Dσ ij σij ( v) δ (2) Π ij = η2 P + η2 P [ σ i k σj k + σ i k Ωj k] + O(κη i j T) ] relaxation time τ π = η P η nt

23 Shear & bulk viscosity: Sum rules Randeria & Taylor proved the sum rules (corrected by Enss & Zwerger) [ ] 1 C dw η(ω) π 15π = E mω 3 C 10πma ( ) 1 1 C dw ζ(ω) = π 72πma 2 a 1 where C is Tan s contact, n k C/k 4. Model spectral function: Kinetic theory for ω < T, OPE for ω > T. η(ω)/n T/T F = 0.5, 1, ω/t F

24 Lattice data: η/s PIMC, N x = 8 PIMC, N x =10 Enss et al. kinetic theory phonons KSS bound 0.6 η/s T/ε F Wlazlowski, Magierski & Drut, arxiv:

25 Lattice data: Spectral functions T/ε F =0.16 SVD MEM initial model final model tail asymptotics T/ε F =0.28 SVD MEM initial model final model tail asymptotics 0.4 η(ω)/n 0.1 η(ω)/n ω/ε F ω/ε F Wlazlowski, Magierski & Drut, arxiv:

26 IV. Experiments: Flow and Collective Modes Hydrodynamic expansion converts σ σ µ coordinate space anisotropy σ σ to momentum space anisotropy O Hara et al. (2002)

27 Elliptic flow: High T limit Quantum viscosity η = η 0 (mt) 3/ Aspect Ratio η = η 0 (mt) 3/2 τπ = η/p Cao et al., Science (2010) Time After Release (µs) fit: η 0 = 0.33 ± 0.04 theory: η 0 = π = 0.26

28 Elliptic flow: Freezeout? switch from hydro to (weakly collisional) kinetics at scale factor b fr = 1,5,10,20 A R t [ms] no freezeout seen in the data

29 Elliptic flow: Shear vs bulk viscosity Dissipative hydro with both η, ζ A R 1.5 ideal 1.0 η ζ t [ms]

30 Elliptic flow: Shear vs bulk viscosity Dissipative hydro with both η, ζ β η,ζ = [η, ζ] n E F E 1 (3λN) 1/ β η 0.2 η ζ β ζ Dusling, Schaefer (2010)

31 Viscosity to entropy density ratio consider both collective modes (low T) and elliptic flow (high T) Cao et al., Science (2010) η/s 0.4

32 V. Outlook: Temperature/density dependence QGP: η/s typically assumed to be constant. Probably o.k. (in QGP phase, at 30% level), but corrections difficult to study. CAG: η/n has significant dependence on T 3/2 n 1. Difficult to unfold. Local s/n known to high accuracy ( 2%). Initial state QGP: Biggest source of uncertainty. Opportunity: Initial state fluctuations generate odd harmonics. CAG: Well characterized. Some ability to control initial state, but this opportiunity has not been exploited.

33 Final state QGP: Kinetic afterburners standard. Flow does not saturate, some puzzling data about energy dependence (v 2 growth due to mean p T increase?) and high p T (non-hydro) flow. CAG: No need for freezeout, flow saturates. But: need kinetics for corona (currently: relaxation time approach interpolates to kinetic limit). Higher harmonics QGP: Important constraint on viscosity, but need initial state models. CAG: Some data on higher multipole collective modes. Need data from single experiment.

34 Bulk viscosity QGP: Hard to measure from flow. Important effects on p T spectra and hadro-chemistry? CAG: Zero at unitarity. Can be measured from either flow or collective mode damping. Leading behavior away from unitarity not known. Fluctuations QGP: Dominated by initial state fluctuations? Possible effects near critical points. Some attempts at stochastic hydro. CAG: Dominated by thermal fluctuations. Possible effects near T c and in two dimensions.

35 Outlook: New ideas (?) QGP: Phenomenology: What is the best observable to constrain η/s? Integrated v 2 in ultra-central collisions? Theory: Anomalous hydrodynamics, new ideas about thermalization (hydro before isotropization and thermalization). CAG: Experiment: Hydrodynamic engineering; can we design flow profiles? Muiple modes, artificial gauge fields (?) Theory: How to do hydro/kinetic interface: 2nd order hydro, lattice Boltzmann, BGK kinetics? Holographic Schr(d): Still no dual of unitary fermions, d = 2 easier?