Anisotropic fluid dynamics. Thomas Schaefer, North Carolina State University
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1 Anisotropic fluid dynamics Thomas Schaefer, North Carolina State University
2 Outline We wish to extract the properties of nearly perfect (low viscosity) fluids from experiments with trapped gases, colliding nuclei, etc. The natural tool for these studies is the Navier-Stokes equation, which describes the macroscopic motion of a fluid in which viscous corrections are small. The problem is that this is not the case flor the entire system. There is a dilute corona in which fluid dynamics is not applicable.
3 Hydroynamics Hydrodynamics (undergraduate version): Newton s law for continuous, deformable media.
4 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)
5 Not your grandfathers fluid Consider a many body system (unitary Fermi gas) with σ 1/k 2 Can be made using Feshbach resonances in dilute atomic gases. Systems remains hydrodynamic despite expansion
6 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
7 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)
8 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 )
9 Gradient expansion, 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) Gradient expansion: ω P η s η T. ]
10 Fluid dynamics from kinetic theory Microscopic picture: Quasi-particle distribution function f p (x,t) ρ(x,t) = π i (x,t) = Π ij (x,t) = dγ p mf p (x,t) dγ p p i f p (x,t) dγ p p i v j f p (x,t) Boltzmann equation ( ) t + v x + F p f p (t,x,) = C[f p ] Collision term C[f 1 ] = dγ 234 (f 1 f 2 f 3 f 4 )w(12;34) p - p
11 Fluid dynamics from kinetic theory Conservation laws (collision term) dγ p M p C[f p ] = 0 M p = {1,p,E p } Moments of Boltzmann equation imply fluid dynamic conservation laws ρ t + j ρ = 0 ǫ t + j ǫ = 0 π i t + Π ij = 0 x j Need constitutive equations (and equation of state) j ρ =? j ǫ =? Π ij =?
12 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 ) Zeroth order result: f 0 = exp( β(e p p u µ)) β = 1/T j ρ = π = ρ u j ǫ = (E +P) u P = 2 3 E Π ij = ρu i u j +Pδ ij First order result: δf 1 = f 0 η PT vi v j σ ij +... δ (1) Π ij = ησ ij δ (1) j ǫ i = ηu j σ ij κ i T
13 Kinetic theory: Knudsen expansion For given w(12;34) also obtain prediction for η,κ η = π (mt)3/2 κ = πm (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)
14 Experiments: Elliptic flow σ σ µ σ σ Hydrodynamic expansion converts coordinate space anisotropy to momentum space anisotropy O Hara et al. (2002)
15 Determination of η(n, T) Measurement of A R (t,e 0 ) determines η(n,t). But: gas n(x) fluid The whole cloud is not a fluid. Can we ignore this issue? v(x) transition regime heat current viscous stress δπ( x) No. Hubble flow & low density viscosity η T 3/2 lead to paradoxical fluid dynamics. Q = σ δπ =
16 Possible Solutions Combine hydrodynamics & Boltzmann equation. Not straightforward. Hydrodynamics + non-hydro degrees of freedom (E a ; a = x,y,z) E a t + j a ǫ = P a 2τ E t + j ǫ = 0 P a = P a P E = a E a τ small: Fast relaxation to Navier-Stokes with τ = η/p τ large: Additional conservation laws. Ballistic expansion.
17 Anisotropic hydro from kinetic theory Consider modified expansion f = f A +δf 1 +δf Anisotropic distribution function f A = exp ( (p a mu a ) 2 2mT a µ T ) T = ( T a ) 1/3 f A is an exact solution of the Boltzmann equation in the ballistic limit. The viscous stresses and dissipative corrections to the energy current have the same form as in the Chapman-Enskog theory.
18 Anisotropic Hydrodynamics from kinetic theory Moments of the Boltzmann equation with M p = {1, p,e P }. Navier-Stokes with δπ aa = P a Moments of the Boltzmann equation with p 2 a E a t + j a ǫ = P a 2τ P a = P a P with P a = 2E a (P = 2 3E) and τ = η/p. Solve fluid dynamic equations for small τ δπ aa = P a = ησ aa Ballistic limit τ : Conservation law for E a.
19 Anisotropic Hydrodynamics: Aspect ratio ideal hydrodynamics free streaming Navier-Stokes (α = 0.1) Navier-Stokes (α = 1) A-Hydro (α = 0.1) A-Hydro (α = 1) A-Hydro (α = 1000) A R Consider η = αn and α [0, ) Navier-Stokes: Ideal hydro very viscous hydro. A-hydro: Ideal hydro ballistic expansion. _ t AVH1 hydro code, M. Bluhm & T.S. (2015)
20 Anisotropic Hydrodynamics: Evolution of δπ aa η = α n n η = α T (mt) 3/2 0 0 Π xx (A-Hydro) Π xx (Navier-Stokes) -P x (A-Hydro) Π xx / P Π xx / P Π xx (A-Hydro) Π xx (Navier-Stokes) -P x (A-Hydro) x / x 0 δπ xx (Navier-Stokes) x / x 0 δπ xx (A-Hydro) AVH1 hydro code, M. Bluhm & T.S. (2015)
21 Anisotropic Hydrodynamics: Comparison with Boltzmann T/T F = 0.79, 1.11, 1.54 Dots: Two-body Boltzmann equation with full collision kernel Lines: Anisotropic hydro with η fixed by Chapman-Enskog High temperature (dilute) limit: Perfect agreement! AVH1 hydro code, M. Bluhm & T.S. (2015)
22 Elliptic flow: High T limit Quantum viscosity η = η 0 (mt) 3/2 2 Cao et al., Science (2010) Bluhm et al., PRL (2016) T/T F = 0.79, 1.11, 1.54 fit: η 0 = 0.28±0.02 theory: η 0 = π = 0.269
23 Outlook Fluid dynamics as an E(F)T? Unfold temperature, density dependence of η/s. Applications to other transport problems: Diffusion, superfluid hydrodynamics.
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