Linear transport and hydrodynamic fluctuations
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1 Linear transport and hydrodynamic fluctuations Pavel Kovtun University of Victoria Vienna, August 25, 2010 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
2 Basic idea Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
3 Basic idea Hydrodynamics as a quantum field theory, and what it may be good for Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
4 Basic idea Hydrodynamics as a quantum field theory, and what it may be good for Old concepts, going back to studies of turbulence and dynamic critical phenomena Well known to some people in this room, perhaps not so well known to others Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
5 Some literature L.Kadanoff and P.Martin, Hydrodynamic equations and correlation functions, Ann. Phys P. Hohenberg and B.Halperin, Theory of dynamical critical phenomena, RMP 1977 J.Zinn-Justin, Quantum field theory and critical phenomena V.Lebedev, Macroscopic fluctuation effects in condensed matter V.L vov and I.Procaccia, chao-dyn/ Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
6 Outline 1. Prologue: hydro fluctuations 2. Brownian motion 3. Dynamic critical phenomena 4. Effective action for hydrodynamics 5. Renormalization of transport coefficients 6. What I would like to understand Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
7 Outline Prologue: hydro fluctuations 1. Prologue: hydro fluctuations 2. Brownian motion 3. Dynamic critical phenomena 4. Effective action for hydrodynamics 5. Renormalization of transport coefficients 6. What I would like to understand Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
8 Prologue: hydro fluctuations Simple non-relativistic ideal fluids Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
9 Prologue: hydro fluctuations Simple non-relativistic ideal fluids Hydro equations = conservation laws Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
10 Prologue: hydro fluctuations Simple non-relativistic ideal fluids Hydro equations = conservation laws mass : t ρ + i (ρv i ) = 0, momentum : t (ρv i ) + j Π ij = 0, Π ij = P δ ij + ρv i v j, ( ) ) energy : t ɛ + ρv2 + i ((w+ ρv2 2 2 )v i = 0. Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
11 Prologue: hydro fluctuations Simple non-relativistic ideal fluids Hydro equations = conservation laws mass : t ρ + i (ρv i ) = 0, momentum : t (ρv i ) + j Π ij = 0, Π ij = P δ ij + ρv i v j, ( ) ) energy : t ɛ + ρv2 + i ((w+ ρv2 2 2 )v i = 0. Here w ɛ + P, and EoS for example is P = P (ρ, ɛ) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
12 Prologue: hydro fluctuations Simple non-relativistic viscous fluids Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
13 Prologue: hydro fluctuations Simple non-relativistic viscous fluids Hydro equations = conservation laws Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
14 Prologue: hydro fluctuations Simple non-relativistic viscous fluids Hydro equations = conservation laws mass : t ρ + i (ρv i ) = 0, momentum : t (ρv i ) + j Π ij = 0, Π ij = P δ ij + ρv i v j σ ij, ( ) ) energy : t ɛ + ρv2 + i ((w+ ρv2 2 2 )v i σ ij v j κ i T = 0. Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
15 Prologue: hydro fluctuations Simple non-relativistic viscous fluids Hydro equations = conservation laws mass : t ρ + i (ρv i ) = 0, momentum : t (ρv i ) + j Π ij = 0, Π ij = P δ ij + ρv i v j σ ij, ( ) ) energy : t ɛ + ρv2 + i ((w+ ρv2 2 2 )v i σ ij v j κ i T = 0. σ ij = η( i v j + j v i 2 3 δ ij k v k ) + ζδ ij k v k η = shear viscosity, ζ = bulk viscosity, κ = thermal conductivity Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
16 Prologue: hydro fluctuations Linearized hydrodynamics Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
17 Prologue: hydro fluctuations Linearized hydrodynamics Small fluctuations around the static equilibrium P = P 0, ɛ = ɛ 0, ρ = ρ 0, v = 0 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
18 Prologue: hydro fluctuations Linearized hydrodynamics Small fluctuations around the static equilibrium P = P 0, ɛ = ɛ 0, ρ = ρ 0, v = 0 Linearize, find eigenmodes, eigenfrequencies: Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
19 Prologue: hydro fluctuations Linearized hydrodynamics Small fluctuations around the static equilibrium P = P 0, ɛ = ɛ 0, ρ = ρ 0, v = 0 Linearize, find eigenmodes, eigenfrequencies: Fluctuations of v : ω = iγ η k 2, γ η = η ρ, Fluctuations of v, T, ρ : ω = idk 2, D = κ ρ c P, Fluctuations of v, T, ρ : ω = ±v s k i Γ 2 k2, Γ = ζ+ 4η 3 + κ(γ 1). ρ ρ c P Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
20 Prologue: hydro fluctuations Linearized hydrodynamics Small fluctuations around the static equilibrium P = P 0, ɛ = ɛ 0, ρ = ρ 0, v = 0 Linearize, find eigenmodes, eigenfrequencies: Fluctuations of v : ω = iγ η k 2, γ η = η ρ, Fluctuations of v, T, ρ : ω = idk 2, D = κ ρ c P, Fluctuations of v, T, ρ : ω = ±v s k i Γ 2 k2, ( Here vs 2 = P ρ ) S,N = (speed of sound) 2, γ c P /c V. Γ = ζ+ 4η 3 + κ(γ 1). ρ ρ c P Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
21 Prologue: hydro fluctuations Hydro fluctuations and correlation functions Linear response: Kadanoff+Martin, 1963 Solutions to the classical hydro equations dictate the form of correlation functions at ω 0, k 0 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
22 Prologue: hydro fluctuations Hydro fluctuations and correlation functions Linear response: Kadanoff+Martin, 1963 Solutions to the classical hydro equations dictate the form of correlation functions at ω 0, k 0 Frequencies of hydro modes poles of the correlation functions In particular, S ρρ (ω, k) = (something) ω 2 +(Dk 2 ) 2 + (something) (ω 2 v 2 sk 2 ) 2 + (ωk 2 Γ) 2 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
23 Prologue: hydro fluctuations Dynamic structure factor in the hydro regime Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
24 Prologue: hydro fluctuations Dynamic structure factor in the hydro regime Can measure spectral function = dynamic structure factor S(ω, k) = FT of ρ(x, t)ρ(x, t ) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
25 Prologue: hydro fluctuations Dynamic structure factor in the hydro regime Can measure spectral function = dynamic structure factor S(ω, k) = FT of ρ(x, t)ρ(x, t ) S(ω, k) diffusive peak at ω=0 S(ω, k) sonic peak at ω=v s k Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
26 Prologue: hydro fluctuations Dynamic structure factor in the hydro regime Can measure spectral function = dynamic structure factor S(ω, k) = FT of ρ(x, t)ρ(x, t ) S(ω, k) diffusive peak at ω=0 S(ω, k) sonic peak at ω=v s k Figure: Bodensteiner et al, Phys. Rev. A 45, 5709 (1992) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
27 Prologue: hydro fluctuations Dynamic structure factor in the hydro regime Let s have a deeper look at the fuctuations Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
28 Outline Brownian motion 1. Prologue: hydro fluctuations 2. Brownian motion 3. Dynamic critical phenomena 4. Effective action for hydrodynamics 5. Renormalization of transport coefficients 6. What I would like to understand Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
29 Langevin equation Brownian motion Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
30 Brownian motion Langevin equation Brownian particle: m d2 x dt 2 = (6πηa)dx dt + f(t), (6πηa) = friction coefficient (Stokes law) f(t) = random force Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
31 Brownian motion Langevin equation Brownian particle: m d2 x dt 2 = (6πηa)dx dt + f(t), (6πηa) = friction coefficient (Stokes law) f(t) = random force Take q dx, Langevin equation: dt q(t) + γq(t) = ξ(t) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
32 Brownian motion Langevin equation Brownian particle: m d2 x dt 2 = (6πηa)dx dt + f(t), (6πηa) = friction coefficient (Stokes law) f(t) = random force Take q dx, Langevin equation: dt q(t) + γq(t) = ξ(t) Noise properties: ξ(t) = 0, ξ(t)ξ(t ) = Cδ(t t ). C determines the strength of the noise Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
33 Brownian motion Correlation function of q(t) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
34 Brownian motion Correlation function of q(t) Take the Langevin equation q(t) + γq(t) = ξ(t) Solve for q(t) in terms of ξ(t) Find q(t)q(t ) by averaging over ξ(t) When γt, γt 1, find q(t)q(t ) = C 2γ e γ t t Fourier transform: S(ω) = C ω 2 + γ 2 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
35 Noise strength Brownian motion Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
36 Noise strength Brownian motion Recall ξ(t)ξ(t ) = Cδ(t t ) What determines the noise strength C? Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
37 Noise strength Brownian motion Recall ξ(t)ξ(t ) = Cδ(t t ) What determines the noise strength C? Assume thermal equilibrium Demand that the correlation functions satisfy the FDT: Im G R (ω) = ω 2T S(ω) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
38 Noise strength Brownian motion Recall ξ(t)ξ(t ) = Cδ(t t ) What determines the noise strength C? Assume thermal equilibrium Demand that the correlation functions satisfy the FDT: Im G R (ω) = ω 2T S(ω) To find G R, introduce source (external force) δq(t) = dt G R (t t ) δf(t ) Langevin equation gives G R (ω) = i ω+iγ Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
39 Noise strength Brownian motion Recall ξ(t)ξ(t ) = Cδ(t t ) What determines the noise strength C? Assume thermal equilibrium Demand that the correlation functions satisfy the FDT: Im G R (ω) = ω 2T S(ω) To find G R, introduce source (external force) δq(t) = dt G R (t t ) δf(t ) Langevin equation gives G R (ω) = Demand FDT: C = 2T i ω+iγ Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
40 Brownian motion Path integral for Brownian particle Let us now represent the Brownian motion as Quantum Mechanics (0+1 dimensional quantum field theory) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
41 Brownian motion Path integral for Brownian particle Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
42 Brownian motion Path integral for Brownian particle Step 1 Write Langevin equation as EoM ( q + F q ξ) = 0 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
43 Brownian motion Path integral for Brownian particle Step 1 Step 2 Write Langevin equation as EoM ( q + F ξ) = 0 q Gaussian noise:... = Dξ e W [ξ] (...), where W [ξ] = 1 dt ξ(t ) 2. 2C Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
44 Brownian motion Path integral for Brownian particle Step 1 Step 2 Write Langevin equation as EoM ( q + F ξ) = 0 q Gaussian noise:... = Dξ e W [ξ] (...), where W [ξ] = 1 dt ξ(t ) 2. 2C Step 3 Recall δ(f(x)) δ(x x 0 ), where x 0 solves f(x 0 ) = 0. So Dq J δ(eom) q(t 1 ) q(t 2 )... = q ξ (t 1 ) }{{} q ξ(t 2 ) }{{}... satisfy EoM(q, ξ) = 0 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
45 Brownian motion Path integral for Brownian particle Step 1 Step 2 Write Langevin equation as EoM ( q + F ξ) = 0 q Gaussian noise:... = Dξ e W [ξ] (...), where W [ξ] = 1 dt ξ(t ) 2. 2C Step 3 Recall δ(f(x)) δ(x x 0 ), where x 0 solves f(x 0 ) = 0. So Dq J δ(eom) q(t 1 ) q(t 2 )... = q ξ (t 1 ) }{{} q ξ(t 2 ) }{{}... satisfy EoM(q, ξ) = 0 Step 4 Write δ(eom) = Dp e i p EoM, do the integral over ξ(t). Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
46 Brownian motion Path integral for Brownian particle (2) When the dust settles: q(t 1 )... q(t n ) = Dq Dp J e is[q,p] q(t 1 )... q(t n ) where S[q, p] = dt ( p q + p F q + ic ) 2 p2. Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
47 Brownian motion Path integral for Brownian particle (2) When the dust settles: q(t 1 )... q(t n ) = Dq Dp J e is[q,p] q(t 1 )... q(t n ) where S[q, p] = dt ( p q + p F q + ic ) 2 p2. For the simple Langevin equation F (q) = 1 2 γq2, as expected. S(ω) = FT of q(t)q(t ) = C ω 2 + γ 2, Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
48 Bottomline: Brownian motion In the stochastic model q(t) + F (q) = ξ(t) q }{{}}{{} relaxation term noise term correlation functions can be derived from field theory with ( S[q, p] = dt p q + p F q + ic ) 2 p2 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
49 Outline Dynamic critical phenomena 1. Prologue: hydro fluctuations 2. Brownian motion 3. Dynamic critical phenomena 4. Effective action for hydrodynamics 5. Renormalization of transport coefficients 6. What I would like to understand Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
50 Fields Dynamic critical phenomena Many variables: q i (t) φ(x, t) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
51 Fields Dynamic critical phenomena Many variables: q i (t) φ(x, t) Langevin equation: F (q) q(t) = + ξ(t) [φ] φ(x, t) = ΓδF + ξ(x, t) q t δφ Functional F [φ] depends on the problem Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
52 Fields Dynamic critical phenomena Many variables: q i (t) φ(x, t) Langevin equation: F (q) q(t) = + ξ(t) [φ] φ(x, t) = ΓδF + ξ(x, t) q t δφ Functional F [φ] depends on the problem e.g. ( a F [φ] = d d x 2 φ2 + b 2 ( φ)2 + λ ) 24 φ4 is model A in the classification of dynamic critical phenomena by Hohenberg and Halperin, RMP, 1977 Also called time-dependent Landau-Ginzburg theory Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
53 Dynamic critical phenomena Effective action Gaussian noise: ξ(x 1, t 1 )ξ(x 2, t 2 ) = C δ(x 1 x 2 )δ(t 1 t 2 ) Correlation functions: φ(x 1, t 1 )...φ(x n, t n ) = Dφ Dχ Je is[φ,χ] φ(x 1, t 1 )...φ(x n, t n ), where S[φ, χ] = dt d d x ( χ t φ + χγ δf ) δφ + ic 2 χ2. Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
54 Dynamic critical phenomena Effective action Gaussian noise: ξ(x 1, t 1 )ξ(x 2, t 2 ) = C δ(x 1 x 2 )δ(t 1 t 2 ) Correlation functions: φ(x 1, t 1 )...φ(x n, t n ) = Dφ Dχ Je is[φ,χ] φ(x 1, t 1 )...φ(x n, t n ), where S[φ, χ] = dt d d x ( χ t φ + χγ δf ) δφ + ic 2 χ2. In model A (λ = 0) : S φφ (ω, k) = ( ) FT of φ(x 1, t 1 )φ(x 2, t 2 ) = C ω 2 + Γ 2 (a + bk 2 ) 2 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
55 Retarded function Dynamic critical phenomena Effective action for model A (Langevin eqn for fields) : ( S[φ, χ] = dt d d x χ t φ + χγ δf ) δφ + ic 2 χ2. Add source as F [φ] F [φ] dt d d x h φ Response of the field: δ φ(x, t) = iγ dt d d x G(t t, x x )δh(x, t ) where G(t t, x x ) φ(x, t)χ(x, t ). Can identify G R (t, x) = iγ φ(x, t)χ(0), G A (t, x) = iγ φ(0)χ(x, t). Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
56 Dynamic critical phenomena Fluctuation-dissipation theorem Note: S φφ (x, t) φ(x, t)φ(0) and G(x, t) φ(x, t)χ(0) are not independent. Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
57 Dynamic critical phenomena Fluctuation-dissipation theorem Note: S φφ (x, t) φ(x, t)φ(0) and G(x, t) φ(x, t)χ(0) are not independent. Integrate out χ: S φφ (ω, k) = C Re G(ω, k) ω Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
58 Dynamic critical phenomena Fluctuation-dissipation theorem Note: S φφ (x, t) φ(x, t)φ(0) and G(x, t) φ(x, t)χ(0) are not independent. Integrate out χ: S φφ (ω, k) = C Re G(ω, k) ω This is FDT in the effective field theory for φ Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
59 Dynamic critical phenomena Fluctuation-dissipation theorem Note: S φφ (x, t) φ(x, t)φ(0) and G(x, t) φ(x, t)χ(0) are not independent. Integrate out χ: S φφ (ω, k) = C Re G(ω, k) ω This is FDT in the effective field theory for φ G R (ω, k) and S φφ (ω, k) are related by FDT provided the noise strength is C = 2T Γ Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
60 Dynamic critical phenomena Fluctuation-dissipation theorem Note: S φφ (x, t) φ(x, t)φ(0) and G(x, t) φ(x, t)χ(0) are not independent. Integrate out χ: S φφ (ω, k) = C Re G(ω, k) ω This is FDT in the effective field theory for φ G R (ω, k) and S φφ (ω, k) are related by FDT provided the noise strength is C = 2T Γ In model A (λ = 0) G R (ω, k) = Γ iω Γ(a+bk 2 ), S φφ(ω, k) = C ω 2 + Γ 2 (a+bk 2 ) 2 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
61 Model A Dynamic critical phenomena Nice singulatities of correlation functions, but still not quite hydrodynamics Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
62 Dynamic critical phenomena Diffusion Note that model A (Langevin eqn for fields) does not describe diffusion of a conserved density Field φ is referred to as a non-conserved order parameter Diffusion equation t n(t, x) = D 2 n(t, x) predicts G R (ω, k) = Dχk2 iω Dk 2, S nn(ω, k) = where χ n / µ is static susceptibility 2DT χk2 ω 2 + (Dk 2 ) 2 Guess: take model A, with Γ Dχk 2. This is model B in the classification of Hohenberg and Halperin, RMP, 1977 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
63 Model B Dynamic critical phenomena Stochastic equation with the free energy F [n] = and Gaussian noise δf [n] n(x, t) = γ 2 + ξ(x, t) t δn ( a d d x 2 n2 + b ) 2 ( n) ξ(x, t)ξ(x, t ) = 2T γ 2 δ(x x )δ(t t ) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
64 Bottomline Dynamic critical phenomena Correlation functions for the simple diffusion equation: n(x, t)n(x, t )... = Dn Dψ e is[n,ψ] n(x, t)n(x, t )... S[n, ψ] = t,x ( ψ n t ψd 2 n + idχt ( ψ) 2) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
65 Bottomline Dynamic critical phenomena Correlation functions for the simple diffusion equation: n(x, t)n(x, t )... = Dn Dψ e is[n,ψ] n(x, t)n(x, t )... S[n, ψ] = t,x ( ψ n t ψd 2 n + idχt ( ψ) 2) Can integrate out ψ, get a non-local effective action for n only S eff [n] = 1 E(x, t)d(x, x )E(x, t) 2 t,x,x where E(x, t) ( n t D 2 n), and 2 D(x, x ) = 1 2DχT δ(x x ). Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
66 Bottomline Dynamic critical phenomena Correlation functions for the simple diffusion equation: n(x, t)n(x, t )... = Dn Dψ e is[n,ψ] n(x, t)n(x, t )... S[n, ψ] = t,x ( ψ n t ψd 2 n + idχt ( ψ) 2) Can integrate out ψ, get a non-local effective action for n only S eff [n] = 1 E(x, t)d(x, x )E(x, t) 2 t,x,x where E(x, t) ( n t D 2 n), and 2 D(x, x ) = 1 2DχT δ(x x ). This effective action produces the correct hydro response functions Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
67 Bottomline Dynamic critical phenomena Now that we know how to construct the effective action for diffusion, can do the same for hydrodynamics Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
68 Outline Effective action for hydrodynamics 1. Prologue: hydro fluctuations 2. Brownian motion 3. Dynamic critical phenomena 4. Effective action for hydrodynamics 5. Renormalization of transport coefficients 6. What I would like to understand Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
69 Simple hydro Effective action for hydrodynamics Relativistic hydro with µ = 0: ɛ t + π = 0, π i t + jt ij = 0. T ij = P δ ij γ η ( i π j + j π i 2 ) d δ ij π γ ζ δ ij π +... γ η η/ w, γ ζ ζ/ w, and w = ɛ+ P. Fluctuations of π : ω = iγ η k 2, Fluctuations of π, ɛ : ω = ±v s k i γ s 2 k2, γ s γ ζ + 2d 2 d γ η. Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
70 Effective action for hydrodynamics Stochastic model for linearized hydro ɛ t = π, π i t = v2 s i ɛ + M ij π j + ξ i (x, t). Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
71 Effective action for hydrodynamics Stochastic model for linearized hydro Dissipative terms: ɛ t = π, π i t = v2 s i ɛ + M ij π j + ξ i (x, t). Noise correlations: M ij γ η ( 2 δ ij i j ) + γ s i j ξ i (x, t)ξ j (x, t ) = 2 wt M ij δ(x x )δ(t t ) Note the same M ij must appear both in the hydro equations, and in the noise correlations Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
72 Effective action for hydrodynamics Functional integral for hydro Correlation functions in linearized hydro: ɛ(x, t)π k (x, t )... = Dɛ Dπ Dη Dλ e is ɛ(x, t)π k (x, t )... S = t,x ( η ( ɛ t + π) + λ i ( πi t +v2 s i ɛ M ij π j ) i wt λ i M ij λ j ) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
73 Effective action for hydrodynamics Functional integral for hydro Correlation functions in linearized hydro: ɛ(x, t)π k (x, t )... = Dɛ Dπ Dη Dλ e is ɛ(x, t)π k (x, t )... S = t,x ( η ( ɛ t + π) + λ i ( πi t +v2 s i ɛ M ij π j ) i wt λ i M ij λ j ) Can integrate out the auxiliary field λ: S eff [ɛ, π] = 1 E i (t, x)d ij (x, x )E j (t, x ) 2 t,x,x where E i ( π i t +v2 s i ɛ M ij π j ), and M ij D jk = 1 2 wt δ(x x )δ ik Note the action S eff [ɛ, π] is time-reversal invariant, as it should be Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
74 Effective action for hydrodynamics Functional integral for hydro Correlation functions in linearized hydro: ɛ(x, t)π k (x, t )... = Dɛ Dπ Dη Dλ e is ɛ(x, t)π k (x, t )... S = t,x ( η ( ɛ t + π) + λ i ( πi t +v2 s i ɛ M ij π j ) i wt λ i M ij λ j ) Can integrate out the auxiliary field λ: S eff [ɛ, π] = 1 E i (t, x)d ij (x, x )E j (t, x ) 2 t,x,x where E i ( π i t +v2 s i ɛ M ij π j ), and M ij D jk = 1 2 wt δ(x x )δ ik Note the action S eff [ɛ, π] is time-reversal invariant, as it should be This effective action produces the correct hydro response functions Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
75 Effective action for hydrodynamics Correlation functions Once know S πi π j (ω, k), the others follow from energy conservation: ωs ɛπi (ω, k) = k l S πl π i (ω, k), ωs ɛɛ (ω, k) = k l S πl ɛ(ω, k). Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
76 Effective action for hydrodynamics Correlation functions Once know S πi π j (ω, k), the others follow from energy conservation: ωs ɛπi (ω, k) = k l S πl π i (ω, k), ωs ɛɛ (ω, k) = k l S πl ɛ(ω, k). Can read off correlation functions from the effective action S eff [ɛ, π]: ( S πi π j (ω, k) = δ ij k ) ik j 2γη wt k 2 k 2 ω 2 +(γ η k 2 ) +k ik j 2γ s wt k 2 ω 2 2 k 2 (ω 2 vsk 2 2 ) 2 + (γ s k 2 ω) 2 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
77 Effective action for hydrodynamics Correlation functions Once know S πi π j (ω, k), the others follow from energy conservation: ωs ɛπi (ω, k) = k l S πl π i (ω, k), ωs ɛɛ (ω, k) = k l S πl ɛ(ω, k). Can read off correlation functions from the effective action S eff [ɛ, π]: ( S πi π j (ω, k) = δ ij k ) ik j 2γη wt k 2 k 2 ω 2 +(γ η k 2 ) +k ik j 2γ s wt k 2 ω 2 2 k 2 (ω 2 vsk 2 2 ) 2 + (γ s k 2 ω) }{{}}{{} 2 shear mode sound mode Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
78 Outline Renormalization of transport coefficients 1. Prologue: hydro fluctuations 2. Brownian motion 3. Dynamic critical phenomena 4. Effective action for hydrodynamics 5. Renormalization of transport coefficients 6. What I would like to understand Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
79 Renormalization of transport coefficients Interaction of low-energy modes E.g. look at model A. Effective action: ( S[φ, χ] = dt d d x χ t φ + χγ δf ) δφ + ic 2 χ2. where F [φ] = ( 1 d d x 2 ( φ)2 + a 2 φ2 + λ ) 24 φ4 The φ 4 term will couple the low-energy modes As a result, a, λ, and Γ will be renormalized Scale dependence is similar to the usual φ 4 theory Many similar models have been studied extensively since 1970 s Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
80 Renormalization of transport coefficients Interaction of hydro modes In hydro, there are no arbitrary coupling constants like λ Coefficients of non-linear terms are fixed by symmetry (Galilean or Lorentz) E.g. J µ = nu µ + ν µ, T µν = (ɛ+p )u µ u ν + P η µν + τ µν. All transport coefs η, ζ, κ are present already in linearized hydro Interaction of modes will change hydro correlation functions Was known since late 1960 s mode-mode coupling Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
81 Long-time tails Renormalization of transport coefficients Start with J = D n + nv, take k = 0. Schematically: J(t)J(0) d d x n(t, x)v(t, x)n(0)v(0) = d d x n(t, x)n(0) v(t, x)v(0) d d k e Dk2t e γηk2 t [ 1 (D+γ η )t ] d/2 See e.g. Arnold+Yaffe, PRD 1997 (known since late 1960 s) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
82 Long-time tails Renormalization of transport coefficients Start with J = D n + nv, take k = 0. Schematically: J(t)J(0) d d x n(t, x)v(t, x)n(0)v(0) = d d x n(t, x)n(0) v(t, x)v(0) d d k e Dk2t e γηk2 t [ 1 (D+γ η )t ] d/2 When FT, the convective contribution to S(ω) is S(ω) ω 1/2, d = 3 S(ω) ln(ω), d = 2 See e.g. Arnold+Yaffe, PRD 1997 (known since late 1960 s) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
83 Renormalization of transport coefficients Correction to Kubo formulas Recall Kubo formula for the diffusion constant: 1 DχT = lim ω 0 2d S ii(ω, k=0) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
84 Renormalization of transport coefficients Correction to Kubo formulas Recall Kubo formula for the diffusion constant: 1 DχT = lim ω 0 2d S ii(ω, k=0) This was derived in linear response. With the non-linear temrs: ( D full = lim D + const ω 1/2, ω 0 d = 3 D full = lim (D + const ln(ω)), ω 0 d = 2 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
85 Renormalization of transport coefficients Correction to Kubo formulas Recall Kubo formula for the diffusion constant: 1 DχT = lim ω 0 2d S ii(ω, k=0) This was derived in linear response. With the non-linear temrs: ( D full = lim D + const ω 1/2, ω 0 d = 3 D full = lim (D + const ln(ω)), ω 0 d = 2 Same applies to shear viscosity: ( η full = lim η + const ω 1/2, ω 0 d = 3 η full = lim (η + const ln(ω)), ω 0 d = 2 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
86 Renormalization of transport coefficients Correction to Kubo formulas Recall Kubo formula for the diffusion constant: 1 DχT = lim ω 0 2d S ii(ω, k=0) This was derived in linear response. With the non-linear temrs: ( D full = lim D + const ω 1/2, ω 0 d = 3 D full = lim (D + const ln(ω)), ω 0 d = 2 Same applies to shear viscosity: ( η full = lim η + const ω 1/2, ω 0 d = 3 η full = lim (η + const ln(ω)), ω 0 d = 2 In 2+1 dimensional hydro, transport coefficients blow up Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
87 Renormalization of transport coefficients Comment In AdS/CFT, the ln(ω) correction is 1/N 3/2 suppressed Transport coefficients come out finite in dimensional classical gravity Long-time tails come from quantum corrections to classical gravity Kovtun+Yaffe, 2003 Caron-Huot + Saremi, 2009 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
88 Renormalization of transport coefficients Higher derivative corrections to hydro Take diffusion equation, add higher-derivative terms n t = D 2 n + D 2 4 n +... Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
89 Renormalization of transport coefficients Higher derivative corrections to hydro Take diffusion equation, add higher-derivative terms n t = D 2 n + D 2 4 n +... Long-time tails imply: D blows up in 2+1 dim, but is finite in 3+1 dim Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
90 Renormalization of transport coefficients Higher derivative corrections to hydro Take diffusion equation, add higher-derivative terms n t = D 2 n + D 2 4 n +... Long-time tails imply: D blows up in 2+1 dim, but is finite in 3+1 dim D 2 blows up even in 3+1 dim, D 2 = lim ω 0 const ω 3/2 DeSchepper + Van Beyeren + Ernst, 1974 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
91 Renormalization of transport coefficients Higher derivative corrections to hydro Take diffusion equation, add higher-derivative terms n t = D 2 n + D 2 4 n +... Long-time tails imply: D blows up in 2+1 dim, but is finite in 3+1 dim D 2 blows up even in 3+1 dim, D 2 = lim ω 0 const ω 3/2 DeSchepper + Van Beyeren + Ernst, 1974 Alternatively, the dispersion of hydro modes has no analytic expansion in powers of k, i.e. ω c 1 k + c 2 k 2 + c 4 k Interaction of hydro modes produces many fractional powers ω = c 1 k + c 2 k 2 + a 1 k 5/2 + a 2 k 11/ Ernst + Dorfman, 1975 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
92 Renormalization of transport coefficients Higher derivative corrections to hydro Take diffusion equation, add higher-derivative terms n t = D 2 n + D 2 4 n +... Long-time tails imply: D blows up in 2+1 dim, but is finite in 3+1 dim D 2 blows up even in 3+1 dim, D 2 = lim ω 0 const ω 3/2 DeSchepper + Van Beyeren + Ernst, 1974 Alternatively, the dispersion of hydro modes has no analytic expansion in powers of k, i.e. ω c 1 k + c 2 k 2 + c 4 k Interaction of hydro modes produces many fractional powers ω = c 1 k + c 2 k 2 + a 1 k 5/2 + a 2 k 11/ So is 2+1 hydro meaningless? Is 3+1 hydro meaningless beyond first derivatives? Ernst + Dorfman, 1975 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
93 Hope for hydro Renormalization of transport coefficients Hydro is not meaningless. Rather, viscosity, conductivity etc become scale-dependent running couplings in the low-energy effective hydro theory Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
94 Renormalization of transport coefficients Toy model in 2+1 dim Incompressible fluid: impose π = 0 Forster, Nelson, Stephen, PRA 1977 (throw away the sound mode, keep shear mode only) Momentum conservation: t π i = j T ij + ξ i, T ij = P δ ij γ η ( i π j + j π i ) + π iπ j w Stochastic model: t π i = i P + γ η 2 π i + (π )π i w + ξ i, ξ i (x, t)ξ j (x, t ) = 2γ η T ( i j 2 δ ij ) δ(x x )δ(t t ) Can write down the action S eff [π, λ], pressure drops out Can do perturbation theory with the non-linear term Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
95 Renormalization of transport coefficients Running viscosity in 2+1 dim Start with the effective action S eff [π, λ] Integrate out fields with momentum Λ < p < Λ Derive RG equations Shear viscosity η const ln(rλ) at large length scales R Forster, Nelson, Stephen, PRA 1977 Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
96 Renormalization of transport coefficients Running viscosity in 2+1 dim Start with the effective action S eff [π, λ] Integrate out fields with momentum Λ < p < Λ Derive RG equations Shear viscosity η const ln(rλ) at large length scales R Forster, Nelson, Stephen, PRA 1977 Viscosity grows at long distances (good news for the viscosity bound) Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
97 Outline What I would like to understand 1. Prologue: hydro fluctuations 2. Brownian motion 3. Dynamic critical phenomena 4. Effective action for hydrodynamics 5. Renormalization of transport coefficients 6. What I would like to understand Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
98 What I would like to understand A comment on AdS/CFT Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
99 What I would like to understand A comment on AdS/CFT So far, there was no AdS/CFT. Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
100 What I would like to understand A comment on AdS/CFT So far, there was no AdS/CFT. In gravity, hydro is usually discussed either at the level of correlation functions (e.g. dispersion relations of QNM), or at the level of the equations of motion (fluid-gravity correspondence). Effective action for hydrodynamics from gravity? Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
101 What I would like to understand A comment on AdS/CFT So far, there was no AdS/CFT. In gravity, hydro is usually discussed either at the level of correlation functions (e.g. dispersion relations of QNM), or at the level of the equations of motion (fluid-gravity correspondence). Effective action for hydrodynamics from gravity? This was already done for the Brownian motion by Son+Teaney, The effective action S[q, p] is the ra version of the Schwinger- Keldysh formalism. The auxiliary field responsible for the noise arises because of the second half of the Kruskal extension of the black hole. Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
102 What I would like to understand What I would like to understand I only showed the effective action for linearized hydro. Can one make it covariant and extend it to the full non-linear relativistic hydro? Effective action for relativistic superfluids? How do transport coefficients in 2+1 dim flow at non-zero density? How do transport coefficients in 2+1 dim flow in external magnetic field? Scale dependence of 2-nd order transport coefficients in relativistic hydro? Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
103 What I would like to understand THE END! Pavel Kovtun (University of Victoria) Linear transport and hydro fluctuations Vienna, August 25, / 48
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