Fluctuations in Fluid Dynamics Thomas Sch afer North Carolina - PowerPoint PPT Presentation

Fluctuations in Fluid Dynamics Thomas Sch¨ afer North Carolina State University

Why consider fluctuations? For consistency: Satisfy fluctuation-dissipation relations. Fluid dynamics as an EFT: Fluctuations determine non- analyticities in ( ω, l ) , and encode the resolution depen- dence of low energy parameters (such as transport coef- ficients). Role of fluctuations enhanced in nearly perfect fluids ( η/s < ∼ 1 ). Fluctuations are dominant near critical points.

Beyond gradients: Hydrodynamic fluctuations Hydrodynamic variables fluctuate � δv i ( x, t ) δv j ( x ′ , t ) � = T ρ δ ij δ ( x − x ′ ) Linearized hydrodynamics propagates fluctuations as shear or sound νk 2 j � ω,k = 2 T ρ ( δ ij − ˆ k i ˆ � δv T i δv T k j ) shear ω 2 + ( νk 2 ) 2 ωk 2 Γ j � ω,k = 2 T k i ˆ ˆ � δv L i δv L k j sound ( ω 2 − c 2 s k 2 ) 2 + ( ωk 2 Γ) 2 ρ Γ = 4 v = vT + vL : ∇ · vT = 0 , ∇ × vL = 0 ν = η/ρ , 3 ν + . . .

Hydro Loops: “Breakdown” of second order hydro Correlation function in hydrodynamics G xyxy = �{ Π xy , Π xy }� ω,k ≃ ρ 2 0 �{ v x v y , v x v y }� ω,k S v T v L v L ρ ρ ρ ρ ρ ρ � � � v T v L v T Match to response function in ω → 0 (Kubo) limit = P + δP − iω [ η + δη ] + ω 2 [ ητ π + δ ( ητ π )] G xyxy R with Tρ 3 / 2 δη ∼ Tρ Λ 1 δP ∼ T Λ 3 δ ( ητ π ) ∼ √ ω η 3 / 2 η

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. � 1 / 2 � 2 � P � ρ Small η enhances fluctuation corrections: δη ∼ T η ρ Small η leads to large δη : There must be a bound on η/n . � 3 / 2 � ρ 1 √ ω Relaxation time diverges: δ ( ητ π ) ∼ η 2nd order hydro without fluctuations inconsistent.

Fluctuation induced bound on η/s 1.0 0.25 η /s η ( ω ) /s fluctuations 0.8 0.20 0.6 0.15 0.4 0.10 0.2 kinetic theory 0.05 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 ω/T F T/T F spectral function ( η/s ) min ≃ 0 . 2 non-analytic √ ω term Schaefer, Chafin (2012), see also Kovtun, Moore, Romatschke (2011)

Fluctuation induced bulk stresses Kubo relation for bulk viscosity ω → 0 Im 1 � dtd 3 x e − iωt � [Π ii ( t, x ) , Π jj (0)]Θ( t ) � ζ = − lim 9 ω Scale invariance not manifest  ǫ = 0 to rewrite Kubo formula May use conservation of energy ∂ t E + � ∇ · � ω → 0 Im 1 O = 1 3Π ii − 2 ζ = − lim ω � [ O ( t, x ) , O (0)] � ωk 3 E and consider coupling to fluctuations of ρ and T O = O 0 + a ρρ (∆ ρ ) 2 + a ρT ∆ ρ ∆ T + a T T (∆ T ) 2 + . . . .

Fluctuation induced bulk stresses Fluctuation contribution to bulk spectral function ( A i ∼ ( P − 2 3 E ) 2 ): √ ω � (2 D T ) 3 / 2 + A Γ A T � √ ζ ( ω ) = ζ (0) − 2 π . Γ 3 / 2 36 Fluctuation bound √ � � � A T 5 A Γ T √ ζ min = + m . 2 D 2 3Γ 2 T Consider λ/a ∼ 1 . Get ζ/s > ∼ 0 . 1

Digression: Diffusion Consider a Brownian particle � ζ ( t ) ζ ( t ′ ) � = κδ ( t − t ′ ) p ( t ) = − γ D p ( t ) + ζ ( t ) ˙ drag (dissipation) white noise (fluctuations) For the particle to eventually thermalize � p 2 � = 2 mT drag and noise must be related κ = mT γ D Einstein (Fluctuation-Dissipation)

Hydrodynamic equation for critical mode Equation of motion for critical mode φ (“model H”) ∂φ ∂t = D ∇ 2 δ F ∇ φ · δ F T δφ − g� π + ζ φ δ� Diffusive Reactive White Noise Free energy functional: Order parameter φ , momentum density � π = ρ� v � 1 � � ∇ φ ) 2 + r 2 φ 2 + λφ 4 + 1 2( � d d x π 2 F = 2 � Fluctuation-Dissipation relation � ζ φ ( x, t ) ζ φ ( x ′ , t ′ ) � = 2 DTδ ( x − x ′ ) δ ( t − t ′ ) ensures P [ φ ] ∼ exp( −F [ φ ] /T )

Linearized analysis (non-critical fluid) v + ν ∇ 2 � Navier-Stokes equation: ∂ 0 � v = mode couplings + noise − νk 2 P T j � ω,k = 1 ν = η ij � δv T i δv T Linearized propagator: ρ − iω + νk 2 ρ Fluctuation correction: Renormalized viscosity: √ ω Tρ 3 / 2 Tρ Λ η = η 0 + c η − c τ η 3 / 2 η 0 0 Hydro is a renormalizable stochastic field theory

Linearized analysis (critical fluid) Consider order parameter mode ∂ 0 φ = − D ∇ 2 δ F δφ + mode couplings + ζ φ � 1 2( ∇ φ ) 2 + r 2 φ 2 + λφ 4 + 1 � � d 3 x π 2 F = 2 � Dispersion relation iω = Dq 2 ( r + q 2 ) + . . . Use r ∼ ξ − 2 . Relaxation time for modes q ∼ ξ − 1 : τ ∼ ξ z ′′ Critical slowing down ′′ ( z = 4) A more sophisticated analysis gives z ≃ 3 and η ∼ ξ 0 . 05 κ ∼ ξ 0 . 9 ζ ∼ ξ 2 . 8

Numerical Simulation: Stochastic Diffusion Stochastic diffusion equation � δ F � ∂ t n B ( x, t ) = Γ ∇ 2 + ∇ · J ( x, t ) δn B √ � 2 T Γ � � ζ i ( x, t ) ζ j ( x ′ , t ′ ) � = δ ( x − x ′ ) δ ( t − t ′ ) δ ij J ( x, t ) = ζ ( x, t ) Free energy functional � m 2 (∆ n B ) 2 + K � d 3 x ( ∇ n B ) 2 F [ n B ] = T 2 n 2 2 n 2 c c � + λ 3 (∆ n B ) 3 + λ 4 (∆ n B ) 4 + λ 6 (∆ n B ) 6 3 n 3 4 n 4 6 n 6 c c c Scale m 2 ∼ ξ − 2 , λ 3 ∼ ξ − 3 / 2 etc., parameterize ξ ( t ) with t = T − T c T c .

Numerical results (diffusion in expanding critical fluid) 1 0.1 τ * [fm/c] 0.01 0.5 1 1.5 2 2.5 3 3.5 T/T c Dynamical scaling: Consider correlation function C 2 ( t ) = � ∆ n B ( k, 0)∆ n B ( − k, t ) � for k = k ∗ ∼ ξ − 1 Determine decay rate C 2 ( t ) ∼ exp( − t/τ ∗ ) . Blue line: Expectation for z = 4 . M. Nahrgang et al. (2018)

0.08 Numerical results 0.06 (diffusion in expanding critical fluid) V σ 2 0.04 0.02 Variance 0 0 −0.001 (S σ ) V Skewness −0.002 0 −0.01 ( κσ 2 ) V Kurtosis −0.02 −0.03 0 1 2 3 4 5 6 7 8 τ − τ 0 [fm/c] M. Nahrgang et al. (2018)

Outlook Obtain higher order cumulants from Gaussian noise and mode couplings. Find significant finite size effects in correlation length and higher order cumulants. Full 3d simulations in progress.

Outlook Parotto et al. (2018)