Fluid dynamics as an effective theory. Thomas Schaefer, North Carolina State University

Transcription

1 Fluid dynamics as an effective theory Thomas Schaefer, North Carolina State University

2 Hydroynamics Hydrodynamics (undergraduate version): Newton s law for continuous, deformable media.

3 Fluids: Gases, liquids, plasmas,... Hydrodynamics (postmodern): Effective theory of non-equilibrium long-wavelength, low-frequency dynamics of any many-body system. non conserved density conserved density τ τ micro τ λ τ τ micro : Dynamics of conserved charges. Water: (ρ,ǫ, π)

4 παντ α ρει (everything flows) Heraclitus The mountains flowed before the Lord. Prophet Deborah, Judges, 5:5

5 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 s η

6 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 2 d AdS d+3 Schr 2 d t (ρv i)+ x j Π ij = 0 (ω < T)

7 Outline I. Unitary Fermi gas II. Gradient expansion III. Symmetries: Galilean and conformal IV. Fluctuations V. Effective field theory? VI. Kinetic theory VII. Quantum field theory VIII. Holography

8 I. Non-relativistic fermions in unitarity limit Consider simple square well potential a < 0 a =, ǫ B = 0 a > 0, ǫ B > 0

9 Non-relativistic fermions in unitarity limit Now take the range to zero, keeping ǫ B 0 Universal relations T = 1 ik +1/a ǫ B = 1 2ma 2 ψ B 1 ar exp( r/a)

10 Fermi gas at unitarity: Field Theory Non-relativistic fermions at low momentum L eff = ψ ( i M ) ψ C 0 2 (ψ ψ) 2 Unitary limit: a (DR: C 0 ) This limit is smooth (HS-trafo, Ψ = (ψ,ψ ) [ ] L = Ψ 2 i 0 +σ 3 Ψ+ ( Ψ σ + Ψφ+h.c. ) 1 φ φ, 2m C 0

11 II. Gradient expansion (simple non-relativistic fluid) Simple fluid: Conservation laws for mass, energy, momentum ρ t + j ρ = 0 ǫ t + j ǫ = 0 π i t + Π ij = 0 x j Ward identity: mass current = momentum density j ρ ρ v = π Constitutive relations: Gradient expansion for currents 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 )

12 Brief remark: Regimes of fluid dynamics Ideal stress tensor Π ij = Pδ ij +ρv i v j. Relative importance of the two terms governed by Mach number Ma = v c s sound speed c 2 s = P ρ Ma 1: Incompressible flow, gas dynamics Ma 1: Compressible flow

13 Fluid dynamic expansion Gradient expansion for currents, e.g. Π ij = Π ij (ρ,v,t) Π ij = Pδ ij +ρv i v j +η ( i v j + j v i 23 ) δ ij k v k +O( 2 ) 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 Nearly Perfect Fluid

14 Additional remarks 1. Incompressible flows: Expansion parameter Ma 2 Re 1. Re 1 : Turbulent flow Re 1 : Highly viscous flow 2. Shocks: Breakdown of gradient expansion η v ρv 2 Shock profile unreliable, but jump conditions insensitive to gradient corrections 3. Navier-Stokes problem (finite time blow-up). Relevant to physics? Not clear (to me). Blow-up could be irrelevant to coarse-grained observables.

15 III. Scale invariant fluid dynamics Consider a many body system with σ tr n 2/3 Systems remains hydrodynamic despite expansion

16 Conformal fluid dynamics: Symmetries Symmetries of a conformal non-relativistic fluid Galilean boost 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 This is known as the Schrödinger algebra (= the symmetries of the free Schrödinger equation) Generators: Mass, momentum, angular momentum M = dxρ P i = dxj i J ij = dxǫ ijk x j j k Boost, dilations, special conformal K i = dxx i ρ D = dxx j C = dxx 2 ρ/2

17 Spurion method: Local symmetries Diffeomorphism invariance δx i = ξ i (x,t) δg ij = L ξ g ij = ξ k k g ij +... Gauge invariance δψ = iα(x, t)ψ δa 0 = α ξ k k A 0 A k ξk δa i = i α ξ k k A i A k i ξ k +mg ik ξk Conformal transformations δt = β(t) δo = βȯ 1 2 βo O More recent work: Newton-Cartan geometry

18 Example: Stress tensor Determine transformation properties of fluid dynamic variables δρ = L ξ ρ δs = L ξ s δv = L ξ v + ξ Stress tensor: Ideal fluid dynamics Π 0 ij = Pg ij +ρv i v j, P = 2 3 E First order viscous hydrodynamics δ (1) Π ij = ησ ij ζg ij σ ζ = 0 σ ij = ( i v j + j v i 2 ) 3 g ij σ σ = v + ġ 2g Son (2007)

19 Simple application: Kubo formula Consider background metric g ij (t,x) = δ ij +h ij (t,x). Linear response δπ xy = 1 2 Gxyxy R h xy Harmonic perturbation h xy = h 0 e iωt Kubo relation: G xyxy R = P iηω +... η = lim ω 0 [ 1 ω ImGxyxy R (ω, 0) ]

20 Second order conformal hydrodynamics Second order gradient corrections to stress tensor [ 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 +O( 2 T) D = 0 + v A ij = 1 2 ( A ij + A ji 2 ) 3 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)

21 Second order fluid dynamics: Causality Speed of diffusive wave in Navier-Stokes theory v D = ω k = 2η ρ k May encounter v D c s Not a fundamental problem (should impose k < Λ), but a nuisance in simulations. Second order fluid dynamics, relaxation type iω = νk2 1 iωτ π ( resummed hydro ) Limiting speed v D η/(ρτ π ) Find v D c s for τ π = η/p.

22 IV. Beyond gradients: Hydrodynamic fluctuations Hydrodynamic variables fluctuate 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 +...

23 Hydro Loops: Breakdown of second order hydro Correlation function in hydrodynamics G xyxy S = {Π xy,π xy } ω,k ρ 2 0 {v x v y,v x v y } ω,k v T ρ ρ v T vl v L v L ρ ρ ρ vt Match to response function in ω 0 (Kubo) limit ρ G xyxy R = P +δp iω[η +δη]+ω 2 [ητ π +δ(ητ π )] with δp TΛ 3 δη TρΛ η δ(ητ π ) 1 ω Tρ 3/2 η 3/2

24 Hydro Loops: RG and breakdown of 2nd order hydro Cutoff dependence can be absorbed into bare parameters. Non-analytic terms are cutoff independent. Fluid dynamics is a renormalizable effective theory. ( ) 2 ( ρ P Small η enhances fluctuation corrections: δη T η ρ ) 1/2 Small η leads to large δη: There must be a bound on η/n. Relaxation time diverges: δ(ητ π ) 1 ω ( ρ η ) 3/2 2nd order hydro without fluctuations inconsistent.

25 Fluctuation induced bound on η/s η/s η(ω)/s 0.20 fluctuations kinetic theory (η/s) min 0.2 T/T F ω/t F spectral function non-analytic ω term Schaefer, Chafin (2012), see also Kovtun, Moore, Romatschke (2011)

26 V. Effective field theory? Consider T = 0 systems with broken U(1) (superfluids). Goldstone mode ϕ. Impose exact gauge & Galilei invariance L = P(X)+O( X) Standard EFT: Expand in gradients X = µ 0 ϕ ( ϕ)2 2m L = f 2[ ( 0 ϕ) 2 c 2 s( ϕ) 2] +g( 0 ϕ)( ϕ) Define hydrodynamic variables n = P (X) v s = 1 m ϕ Use Euler equation for ϕ & Gibbs-Duhem P = n µ 0 n+ 1 m (n ϕ) = 0 0v s v2 s = 1 m µ Superfluid hydrodynamics

27 Remarks (Quantum) loop corections? Calculable, but highly suppressed. Can this be extended to T 0? Not directly, fluid dynamics is irreversible. Can construct generating functional for linearized hydro. Simple example: Diffusion 0 n = D 2 n (model B) Z = DnDψe is, S = dxdt [ ψ 0 n ψd 2 n+idψχt( ψ) 2] Analytic structure? Higher order terms?

28 VI. Kinetic theory Microscopic picture: Quasi-particle distribution function f p (x,t) ρ(x,t) = dγ p gmfp (x,t) π i (x,t) = dγ p gpi f p (x,t) Π ij (x,t) = dγ p gpi v j f p (x,t) Boltzmann equation ( t + pi m x i ( g il ġ lj p j +Γ i jk p j p k m ) ) p i f p (t,x,) = C[f] C[f] = Solve order-by-order in Knudsen number Kn = l mfp /L

29 Kinetic theory: Knudsen expansion Chapman-Enskog expansion f = f 0 +δf 1 +δf Gradient exp. δf n = O( n ) Knudsen exp. δf n = O(Kn n ) First order result Bruun, Smith (2005) δ (1) Π ij = ησ ij η = π (mt)3/2 Second order result Chao, Schaefer (2012), Schaefer (2014) [ δ (2) Π ij = η2 Dσ ij + 2 ] P 3 σij ( v) + η2 P [ ] σ i k σj k σ i k Ωj k relaxation time τ π = η/p +O(κη i j T)

30 Knudsen expansion Knudsen vs fugacity expansion δf (1) η ρt 2 pi p j σ ij δf (2) O(η 2 σ 2 ) Fugacity expansion η = (nλ 3 ) { η 0 +η 1 (nλ 3 )+... } Analog of virial expansion for equilibrium properties. Dense gas (nλ 3 ) 1: Kinetic theory not applicable. Relativistic theories (nλ 3 ) = 1: Coupling constant expansion.

31 Frequency dependence, breakdown of kinetic theory Consider harmonic perturbation h xy e iωt+ikx. Use schematic collision term C[f 0 p +δf p ] = δf p /τ. δf p (ω,k) = 1 2T iωp x v y iω +i v k +τ 1 0 Leads to Lorentzian line shape of transport peak f 0 p h xy. η(ω) = η(0) 1+ω 2 τ 2 0 Pole at ω = iτ 1 0 (τ 0 = η/(st)) controls range of convergence of gradient expansion. High frequency behavior misses short range correlations for ω > T.

32 Bulk viscosity and conformal symmetry breaking Conformal symmetry breaking (thermodynamics) 1 2E 3P = O C 12πmaP 1 6π nλ3 λ a How does this translate into ζ 0? Momentum dependent m (p). ImΣ(k) zt Bulk viscosity ζ = π λ 3 ( zλ a ReΣ(k) zt λ a T ǫ k Erf T ǫ k F D ) 2 ζ ( 1 2E 3P ( ǫk T ) ( ǫk ) 2 η T T )

33 Can be used to extrapolate Boltzmann result to T T F = VII. Quantum Field Theory The diagrammatic content of the Boltzmann equation is known: Kubo formula with Maki-Thompson + Azlamov-Larkin + Self-energy viscosity η [ h n] classical 2.77 T 3/2 viscosity η(ω=0) η(ω) [ h n] T=10 T= 5 T= 2 T= 1 T=0.5 T= T [T F ] ω [E F / h] Enss, Zwerger (2011)

34 Operator product expansion (OPE) Short time behavior: OPE η(ω) = n O n ω ( n d)/2 O n (λ 2 t,λx) = λ n O n (t,x) Leading operator: Contact density (Tan) O C = C 2 0ψψψ ψ = ΦΦ C = 4 η(ω) O C / ω. Asymptotic behavior + analyticity gives sum rule [ 1 dw η(ω) O ] C π 15π = E mω 3 Randeria, Taylor (2010), Enss, Zwerger (2011), Hoffman (2013)

35 VIII. Holography DLCQ idea: Light cone compactification of relativistic theory in d+2 p µ p µ = 2p + p p 2 = 0 p = p2 p + = 2n+1 2p + L Galilean invariant theory in d+1 dimensions. String theory embedding: Null Melvin Twist AdS d+3 NMT Schr 2 d Iso(AdS d+3 ) = SO(d+2,2) Schr(d) Son (2008), Balasubramanian et al. (2008) Other ideas: Horava-Lifshitz (Karch, 2013)

36 Schrödinger Metric Coordinates (u,v, x,r), periodic in v, x = (x,y) {[ ] ds 2 = r2 1 f(r) k(r) 2/3 4β 2 r 2 f(r) du 2 + β2 r+ 4 } r 4 dv 2 [1+f(r)] dudv +k(r) 1/3 { r 2 d x 2 + dr2 r 2 f(r) Fluctuations δg y x = e iωu χ(ω,r) satisfy (u = (r + /r) 2 ) χ (ω,u) 1+u2 f(u)u χ (ω,u)+ u f(u) 2w2 χ(ω,u) = 0 } Retarded correlation function G R (ω) = βr3 + v 4πG 5 f(u)χ (ω,u) uχ(ω, u). u 0 Adams et al. (2008), Herzog et al. (2008)

37 Spectral function η(ω)/s η(0)/s = 1/(4π) η(ω ) ω 1/ ω/(πt) Kubo relation (incl. τ π ): G R (ω) = P iηω +τ π ηω 2 +κ R k 2 τ π T = log(2) 2π AdS 5 : τ π T = 2 log(2) 2π Range of validity of fluid dynamics: ω < T Sch 2 : Cannot be matched to relaxation type hydro? Schaefer (2014), BRSSS (2008)

38 Quasi-normal modes Re w Re w Im w 4 Im w Sch 2 2 AdS 5 QNM s are stable, Imλ < 0. Pole at ω it limits convergence of fluid dynamics. Also: Gradient expansion only asymptotic. Schaefer (2014), Starinets (2002), Heller (2012)

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

40 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

41 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

42 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

43 Viscosity to entropy density ratio (recent update) (η/n) drops to zero in superfluid phase (η/s) has a minimum near T c Joseph et al. (2014)

44 Outlook Fluid dynamics as an E(F)T: Many interesting questions remain. Experiment: Main issue is temperature, density dependence of η/s. How to unfold? Need hydro codes that exit gracefully (LBE, anisotropic hydro, hydro+cascade) Quasi-particles vs quasi-normal modes (kinetics vs holography) unresolved. Need better holographic models, improved lattice calculations.