Hydro fluctuations Outline 1. Why hydro? 2. Hydro fluctuations 3. A simple calculation 4. Fluctuations: Brownian motion 5. Fluctuations: Diffusion equation 6. Fluctuations: Linear hydrodynamics 7. Fluctuations: Non-linear hydrodynamics 8. Conclusions Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 9 / 58
Hydro fluctuations Linearized relativistic hydro Relativistic hydro with µ = 0 : ∂ǫ ∂π i ∂t + ∇ · π = 0 , ∂t + ∂ j T ij = 0 . � � ∂ i π j + ∂ j π i − 2 T ij = Pδ ij − γ η dδ ij ∇ · π − γ ζ δ ij ∇ · π + ... ǫ + ¯ γ η ≡ η/ ¯ w , γ ζ ≡ ζ/ ¯ w , and ¯ w = ¯ P . Fluctuations of π ⊥ : ω = − iγ η k 2 , Fluctuations of π � , ǫ : ω = ± v s | k | − iγ s γ s ≡ γ ζ + 2 d − 2 2 k 2 , γ η . d Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 10 / 58
Hydro fluctuations Simple picture for viscosity Viscosity measures rate of momentum transfer between layers of fluid η = ρv th ℓ mfp Maxwell, 1860 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 11 / 58
Hydro fluctuations Simple picture for viscosity Viscosity measures rate of momentum transfer between layers of fluid η = ρv th ℓ mfp Maxwell, 1860 y x Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 11 / 58
Hydro fluctuations Simple picture for viscosity Viscosity measures rate of momentum transfer between layers of fluid η = ρv th ℓ mfp Maxwell, 1860 y ℓ mfp ∼ 1 nσ ∼ T x λ 2 η 0 ∼ N 2 T 3 λ 2 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 11 / 58
Hydro fluctuations Simple picture for viscosity (2) Elementary excitations are not the only way to transfer momentum. Momentum can also be transfered by collective excitations. Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 12 / 58
Hydro fluctuations Simple picture for viscosity (2) Elementary excitations are not the only way to transfer momentum. Momentum can also be transfered by collective excitations. y x Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 12 / 58
Hydro fluctuations Simple picture for viscosity (2) Elementary excitations are not the only way to transfer momentum. Momentum can also be transfered by collective excitations. 1 y ℓ mfp ∼ η ǫ + P k 2 x � k max ǫ + P k 2 ∼ k max T 2 T d 3 k η 1 ∼ η 0 η 0 /s Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 12 / 58
Hydro fluctuations Simple picture for viscosity (2) Elementary excitations are not the only way to transfer momentum. Momentum can also be transfered by collective excitations. 1 y ℓ mfp ∼ η ǫ + P k 2 x � k max ǫ + P k 2 ∼ k max T 2 T d 3 k η 1 ∼ η 0 η 0 /s Total viscosity η total = η 0 + η 1 is bounded from below This integral IR finite in d = 3+1 , but IR divergent in d = 2+1 Forster+Nelson+Stephen, 1977 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 12 / 58
Hydro fluctuations The rest of the talk will expand on these points Namely How do hydro fluctuations change viscosity in d = 3+1 ? How do hydro fluctuations change second-order hydrodynamics? How do hydro fluctuations change viscosity in d = 2+1 ? Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 13 / 58
A simple calculation Outline 1. Why hydro? 2. Hydro fluctuations 3. A simple calculation 4. Fluctuations: Brownian motion 5. Fluctuations: Diffusion equation 6. Fluctuations: Linear hydrodynamics 7. Fluctuations: Non-linear hydrodynamics 8. Conclusions Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 14 / 58
A simple calculation Interaction of hydro modes In hydro, there are no arbitrary “coupling constants” like g Coefficients of non-linear terms are fixed by symmetry (Galilean or Lorentz) E.g. J µ = nu µ + ν µ , T µν = ( ǫ + P ) u µ u ν + Pη 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) Hydrodynamic fluctuations GGI, May 3, 2011 15 / 58
A simple calculation Long-time tails Start with J = − D ∇ n + n v , take k = 0 . Schematically: � d d x � n ( t, x ) v ( t, x ) n (0) v (0) � � J ( t ) J (0) � ⊃ � d d x � n ( t, x ) n (0) �� v ( t, x ) v (0) � = � d d k e − D k 2 t e − γ η k 2 t ∼ � � d/ 2 1 ∼ ( D + γ η ) t See e.g. Arnold+Yaffe, PRD 1997 (known since late 1960’s) Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 16 / 58
A simple calculation Long-time tails Start with J = − D ∇ n + n v , take k = 0 . Schematically: � d d x � n ( t, x ) v ( t, x ) n (0) v (0) � � J ( t ) J (0) � ⊃ � d d x � n ( t, x ) n (0) �� v ( t, x ) v (0) � = � d d k e − D k 2 t e − γ η k 2 t ∼ � � d/ 2 1 ∼ ( D + γ η ) t See e.g. Arnold+Yaffe, PRD 1997 (known since late 1960’s) When FT, the convective contribution to S ( ω ) is S ( ω ) ∼ ω 1 / 2 , d = 3 S ( ω ) ∼ ln( ω ) , d = 2 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 16 / 58
A simple calculation Correction to Kubo formulas Recall Kubo formula for the diffusion constant: 1 DχT = lim 2 dS ii ( ω, k =0) ω → 0 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 17 / 58
A simple calculation Correction to Kubo formulas Recall Kubo formula for the diffusion constant: 1 DχT = lim 2 dS ii ( ω, k =0) ω → 0 This was derived in linear response. With the non-linear temrs: D full = lim � D + const ω 1 / 2 � , d = 3 ω → 0 D full = lim ω → 0 ( D + const ln( ω )) , d = 2 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 17 / 58
A simple calculation Correction to Kubo formulas Recall Kubo formula for the diffusion constant: 1 DχT = lim 2 dS ii ( ω, k =0) ω → 0 This was derived in linear response. With the non-linear temrs: D full = lim � D + const ω 1 / 2 � , d = 3 ω → 0 D full = lim ω → 0 ( D + const ln( ω )) , d = 2 Same applies to shear viscosity: η full = lim � η + const ω 1 / 2 � , d = 3 ω → 0 η full = lim ω → 0 ( η + const ln( ω )) , d = 2 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 17 / 58
A simple calculation Correction to Kubo formulas Recall Kubo formula for the diffusion constant: 1 DχT = lim 2 dS ii ( ω, k =0) ω → 0 This was derived in linear response. With the non-linear temrs: D full = lim � D + const ω 1 / 2 � , d = 3 ω → 0 D full = lim ω → 0 ( D + const ln( ω )) , d = 2 Same applies to shear viscosity: η full = lim � η + const ω 1 / 2 � , d = 3 ω → 0 η full = lim ω → 0 ( η + const ln( ω )) , d = 2 In 2+1 dimensional hydro, transport coefficients blow up Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 17 / 58
A simple calculation Comment In AdS/CFT, the ln( ω ) correction is 1 /N 3 / 2 suppressed Transport coefficients come out finite in 3 + 1 dimensional classical gravity Long-time tails come from quantum corrections to classical gravity Kovtun+Yaffe, 2003 Caron-Huot + Saremi, 2009 This is an example where long-time limit does not commute with large-N limit Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 18 / 58
A simple calculation Can do the same calculation in momentum space T ij T kl One-loop diagram with sound and/or shear waves in the loop � dω ′ d 3 k S xy,xy ( ω, k =0) = ( ǫ + P ) 2 (2 π ) 3 2 π � � ∆ xx ( ω ′ , k )∆ yy ( ω − ω ′ , − k ) + ∆ xy ( ω ′ , k )∆ yx ( ω − ω ′ , − k ) where ∆ ij = FT of � u i ( x ) u j (0) � Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 19 / 58
A simple calculation When the dust settles... G R xy,xy ( ω ≪ k max , k =0) = − iωη 0 + (1 + i ) ω 3 / 2 (7 + (3 / 2) 3 / 2 ) T − iω 17 Tk max � ( k max γ η 0 ) 2 , ω 2 � + O 120 π 2 γ η 0 240 πγ 3 / 2 η 0 PK+Moore+Romatschke, 2011 The contribution due to hydro fluctuations is suppressed at either small coupling, or large N Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 20 / 58
A simple calculation Implications for the shear viscosity Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 21 / 58
A simple calculation Implications for the shear viscosity The function η + c/η has a minimum, hence viscosity is bounded from below Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 21 / 58
A simple calculation Implications for the shear viscosity The function η + c/η has a minimum, hence viscosity is bounded from below The exact value of the minimum depends on the UV cutoff of the hydro effective theory Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 21 / 58
A simple calculation Implications for the shear viscosity The function η + c/η has a minimum, hence viscosity is bounded from below The exact value of the minimum depends on the UV cutoff of the hydro effective theory Estimate k max γ η 0 ∼ 1 / 2 , then η total /s � 0 . 16 � at T � T c Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 21 / 58
A simple calculation Implications for the shear viscosity The function η + c/η has a minimum, hence viscosity is bounded from below The exact value of the minimum depends on the UV cutoff of the hydro effective theory Estimate k max γ η 0 ∼ 1 / 2 , then η total /s � 0 . 16 � at T � T c Current hydro simulations of QGP are blind to these effects because they simply solve the classical hydro equations and ignore the fluctuations Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 21 / 58
A simple calculation This was for one-derivative hydro Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 22 / 58
A simple calculation This was for one-derivative hydro Take diffusion equation, add higher-derivative terms ∂n ∂t = D ∇ 2 n + D 2 ∇ 4 n + . . . Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 22 / 58
A simple calculation This was for one-derivative hydro Take diffusion equation, add higher-derivative terms ∂n ∂t = D ∇ 2 n + D 2 ∇ 4 n + . . . Hydro loop corrections imply: D blows up in 2+1 dim, but is finite in 3+1 dim Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 22 / 58
A simple calculation This was for one-derivative hydro Take diffusion equation, add higher-derivative terms ∂n ∂t = D ∇ 2 n + D 2 ∇ 4 n + . . . Hydro loop corrections imply: D blows up in 2+1 dim, but is finite in 3+1 dim const D 2 blows up even in 3+1 dim, D 2 = lim ω → 0 ω 1 / 2 DeSchepper + Van Beyeren + Ernst, 1974 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 22 / 58
A simple calculation This was for one-derivative hydro Take diffusion equation, add higher-derivative terms ∂n ∂t = D ∇ 2 n + D 2 ∇ 4 n + . . . Hydro loop corrections imply: D blows up in 2+1 dim, but is finite in 3+1 dim const D 2 blows up even in 3+1 dim, D 2 = lim ω → 0 ω 1 / 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 4 + . . . 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 / 4 + . . . Ernst + Dorfman, 1975 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 22 / 58
A simple calculation Exactly the same happens for second-order relativistic hydro (Israel-Stewart) Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 23 / 58
A simple calculation Exactly the same happens for second-order relativistic hydro (Israel-Stewart) In linearized second order hydro: � ητ Π − κ � ω 2 − κ 2 k 2 + . . . G R xy,xy ( ω, k ) = P − iωη + 2 Baier+Romatschke+Son+Starinets+Stephanov, 2007 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 23 / 58
A simple calculation Exactly the same happens for second-order relativistic hydro (Israel-Stewart) In linearized second order hydro: � ητ Π − κ � ω 2 − κ 2 k 2 + . . . G R xy,xy ( ω, k ) = P − iωη + 2 Baier+Romatschke+Son+Starinets+Stephanov, 2007 But this gets seriously modified by 1-loop hydro fluctuations, G R xy,xy ( ω, k =0) = P − iωη − const | ω | 3 / 2 (1 + i sign( ω )) + . . . Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 23 / 58
A simple calculation Exactly the same happens for second-order relativistic hydro (Israel-Stewart) In linearized second order hydro: � ητ Π − κ � ω 2 − κ 2 k 2 + . . . G R xy,xy ( ω, k ) = P − iωη + 2 Baier+Romatschke+Son+Starinets+Stephanov, 2007 But this gets seriously modified by 1-loop hydro fluctuations, G R xy,xy ( ω, k =0) = P − iωη − const | ω | 3 / 2 (1 + i sign( ω )) + . . . Blindly apply Kubo formula ∂ 2 ητ Π − κ 1 ∂ω 2 Re G R 2 = lim xy,xy ( ω, k =0) → ∞ 2 ω → 0 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 23 / 58
A simple calculation Exactly the same happens for second-order relativistic hydro (Israel-Stewart) In linearized second order hydro: � ητ Π − κ � ω 2 − κ 2 k 2 + . . . G R xy,xy ( ω, k ) = P − iωη + 2 Baier+Romatschke+Son+Starinets+Stephanov, 2007 But this gets seriously modified by 1-loop hydro fluctuations, G R xy,xy ( ω, k =0) = P − iωη − const | ω | 3 / 2 (1 + i sign( ω )) + . . . Blindly apply Kubo formula ∂ 2 ητ Π − κ 1 ∂ω 2 Re G R 2 = lim xy,xy ( ω, k =0) → ∞ 2 ω → 0 This means τ Π does not exist Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 23 / 58
A simple calculation Can we save second-order hydro? Can estimate when ω 3 / 2 term becomes comparable to ω 2 term 2nd-order hydro breaks down below some ω ∗ depends on η 0 /s Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 24 / 58
A simple calculation Can we save second-order hydro? Can estimate when ω 3 / 2 term becomes comparable to ω 2 term 2nd-order hydro breaks down below some ω ∗ depends on η 0 /s If η 0 /s ∼ 0 . 16 , then ω ∗ ∼ T/ 20 , 2nd-order hydro OK for heavy-ion collisions If η 0 /s ∼ 0 . 08 , then ω ∗ ∼ 2 . 5 T , 2nd order hydro makes no sense for heavy-ion collisions PK+Moore+Romatschke, 2011 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 24 / 58
A simple calculation Is there hope for hydrodynamics? Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 25 / 58
A simple calculation Is there hope for hydrodynamics? So is 2+1 hydro meaningless? Is 3+1 hydro meaningless beyond first derivatives? Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 25 / 58
A simple calculation Is there hope for hydrodynamics? So is 2+1 hydro meaningless? Is 3+1 hydro meaningless beyond first derivatives? Hydro is not meaningless. Rather, viscosity, conductivity etc become scale-dependent “running masses” in the low-energy effective hydro theory Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 25 / 58
A simple calculation Is there hope for hydrodynamics? So is 2+1 hydro meaningless? Is 3+1 hydro meaningless beyond first derivatives? Hydro is not meaningless. Rather, viscosity, conductivity etc become scale-dependent “running masses” in the low-energy effective hydro theory To find this low-energy effective hydro theory, need both dissipation (transport coefficients) and fluctuations (thermally excited modes) Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 25 / 58
Fluctuations: Brownian motion Outline 1. Why hydro? 2. Hydro fluctuations 3. A simple calculation 4. Fluctuations: Brownian motion 5. Fluctuations: Diffusion equation 6. Fluctuations: Linear hydrodynamics 7. Fluctuations: Non-linear hydrodynamics 8. Conclusions Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 26 / 58
Fluctuations: Brownian motion Langevin equation Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 27 / 58
Fluctuations: Brownian motion Langevin equation Brownian particle: md 2 x dt 2 = − (6 πηa ) dx dt + f ( t ) , (6 πηa ) = friction coefficient (Stokes law) f ( t ) = random force Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 27 / 58
Fluctuations: Brownian motion Langevin equation Brownian particle: md 2 x dt 2 = − (6 πηa ) dx dt + f ( t ) , (6 πηa ) = friction coefficient (Stokes law) f ( t ) = random force Take q ≡ dx dt , ⇒ Langevin equation: q ( t ) + γq ( t ) = ξ ( t ) ˙ Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 27 / 58
Fluctuations: Brownian motion Langevin equation Brownian particle: md 2 x dt 2 = − (6 πηa ) dx dt + f ( t ) , (6 πηa ) = friction coefficient (Stokes law) f ( t ) = random force Take q ≡ dx dt , ⇒ Langevin equation: q ( t ) + γq ( t ) = ξ ( t ) ˙ Noise properties: � ξ ( t ) ξ ( t ′ ) � = Cδ ( t − t ′ ) . � ξ ( t ) � = 0 , C determines the strength of the noise Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 27 / 58
Fluctuations: Brownian motion Correlation function of q ( t ) Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 28 / 58
Fluctuations: 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 2 γ e − γ | t − t ′ | Fourier transform: C S ( ω ) = ω 2 + γ 2 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 28 / 58
Fluctuations: Brownian motion Noise strength Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 29 / 58
Fluctuations: Brownian motion Noise strength Recall � ξ ( t ) ξ ( t ′ ) � = Cδ ( t − t ′ ) What determines the noise strength C ? Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 29 / 58
Fluctuations: Brownian motion Noise strength 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 2 T S ( ω ) Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 29 / 58
Fluctuations: Brownian motion Noise strength 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 2 T S ( ω ) To find G R , introduce source (external force) � dt ′ G R ( t − t ′ ) δf ( t ′ ) δq ( t ) = i Langevin equation gives G R ( ω ) = ω + iγ Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 29 / 58
Fluctuations: Brownian motion Noise strength 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 2 T S ( ω ) To find G R , introduce source (external force) � dt ′ G R ( t − t ′ ) δf ( t ′ ) δq ( t ) = i Langevin equation gives G R ( ω ) = ω + iγ Demand FDT: C = 2 T Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 29 / 58
Fluctuations: 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) Hydrodynamic fluctuations GGI, May 3, 2011 30 / 58
Fluctuations: Brownian motion Path integral for Brownian particle Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 31 / 58
Fluctuations: Brownian motion Path integral for Brownian particle q + ∂F Step 1 Write Langevin equation as EoM ≡ ( ˙ ∂q − ξ ) = 0 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 31 / 58
Fluctuations: Brownian motion Path integral for Brownian particle q + ∂F Step 1 Write Langevin equation as EoM ≡ ( ˙ ∂q − ξ ) = 0 Step 2 Gaussian noise: � � where W [ ξ ] = 1 dt ′ ξ ( t ′ ) 2 . D ξ e − W [ ξ ] ( ... ) , � ... � = 2 C Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 31 / 58
Fluctuations: Brownian motion Path integral for Brownian particle q + ∂F Step 1 Write Langevin equation as EoM ≡ ( ˙ ∂q − ξ ) = 0 Step 2 Gaussian noise: � � where W [ ξ ] = 1 dt ′ ξ ( t ′ ) 2 . D ξ e − W [ ξ ] ( ... ) , � ... � = 2 C Step 3 Recall δ ( f ( x )) ∼ δ ( x − x 0 ) , where x 0 solves f ( x 0 ) = 0 . So � D q 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) Hydrodynamic fluctuations GGI, May 3, 2011 31 / 58
Fluctuations: Brownian motion Path integral for Brownian particle q + ∂F Step 1 Write Langevin equation as EoM ≡ ( ˙ ∂q − ξ ) = 0 Step 2 Gaussian noise: � � where W [ ξ ] = 1 dt ′ ξ ( t ′ ) 2 . D ξ e − W [ ξ ] ( ... ) , � ... � = 2 C Step 3 Recall δ ( f ( x )) ∼ δ ( x − x 0 ) , where x 0 solves f ( x 0 ) = 0 . So � D q J δ ( EoM ) q ( t 1 ) q ( t 2 ) ... = q ξ ( t 1 ) � �� � q ξ ( t 2 ) � �� � ... satisfy EoM ( q, ξ ) = 0 � � D p e i p EoM , do the integral over ξ ( t ) . Step 4 Write δ ( EoM ) = Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 31 / 58
Fluctuations: Brownian motion Path integral for Brownian particle (2) When the dust settles: � D q D p J e iS [ q,p ] q ( t 1 ) ... q ( t n ) � q ( t 1 ) ... q ( t n ) � = where � � � q + p∂F ∂q + iC 2 p 2 S [ q, p ] = dt p ˙ . Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 32 / 58
Fluctuations: Brownian motion Path integral for Brownian particle (2) When the dust settles: � D q D p J e iS [ q,p ] q ( t 1 ) ... q ( t n ) � q ( t 1 ) ... q ( t n ) � = where � � � q + p∂F ∂q + iC 2 p 2 S [ q, p ] = dt p ˙ . For the simple Langevin equation F ( q ) = 1 2 γq 2 , C S ( ω ) = FT of � q ( t ) q ( t ′ ) � = ω 2 + γ 2 , as expected. Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 32 / 58
Fluctuations: Brownian motion Bottomline: In the stochastic model q ( t ) + ∂F ( q ) ˙ = ξ ( t ) ∂q ���� � �� � relaxation term noise term correlation functions can be derived from field theory with � � � q + p∂F ∂q + iC 2 p 2 S [ q, p ] = dt p ˙ Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 33 / 58
Fluctuations: Diffusion equation Outline 1. Why hydro? 2. Hydro fluctuations 3. A simple calculation 4. Fluctuations: Brownian motion 5. Fluctuations: Diffusion equation 6. Fluctuations: Linear hydrodynamics 7. Fluctuations: Non-linear hydrodynamics 8. Conclusions Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 34 / 58
Fluctuations: Diffusion equation Fields Many variables: q i ( t ) → φ ( x , t ) Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 35 / 58
Fluctuations: Diffusion equation Fields Many variables: q i ( t ) → φ ( x , t ) Langevin equation: q ( t ) = − ∂F ( q ) ∂tφ ( x , t ) = − Γ δF [ φ ] ∂ ˙ + ξ ( t ) → + ξ ( x , t ) ∂q δφ Functional F [ φ ] depends on the problem Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 35 / 58
Fluctuations: Diffusion equation Fields Many variables: q i ( t ) → φ ( x , t ) Langevin equation: q ( t ) = − ∂F ( q ) ∂tφ ( x , t ) = − Γ δF [ φ ] ∂ ˙ + ξ ( t ) → + ξ ( x , t ) ∂q δφ Functional F [ φ ] depends on the problem e.g. � � a � 2 φ 2 + b 2( ∇ φ ) 2 + λ d d x 24 φ 4 F [ φ ] = 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) Hydrodynamic fluctuations GGI, May 3, 2011 35 / 58
Fluctuations: Diffusion equation 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: � D φ D χ Je iS [ φ,χ ] φ ( x 1 , t 1 ) ...φ ( x n , t n ) , � φ ( x 1 , t 1 ) ...φ ( x n , t n ) � = where � � � χ∂ t φ + χ Γ δF δφ + iC dt d d x 2 χ 2 S [ φ, χ ] = . Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 36 / 58
Fluctuations: Diffusion equation 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: � D φ D χ Je iS [ φ,χ ] φ ( x 1 , t 1 ) ...φ ( x n , t n ) , � φ ( x 1 , t 1 ) ...φ ( x n , t n ) � = where � � � χ∂ t φ + χ Γ δF δφ + iC dt d d x 2 χ 2 S [ φ, χ ] = . In model A ( λ = 0 ) : � � C S φφ ( ω, k ) = FT of � φ ( x 1 , t 1 ) φ ( x 2 , t 2 ) � = ω 2 + Γ 2 ( a + b k 2 ) 2 Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 36 / 58
Fluctuations: Diffusion equation Retarded function Effective action for model A (Langevin eqn for fields) : � � � χ∂ t φ + χ Γ δF δφ + iC dt d d x 2 χ 2 S [ φ, χ ] = . � dt d d x h φ Add source as F [ φ ] → F [ φ ] − Response of the field: � dt ′ d d x ′ G ( t − t ′ , x − x ′ ) δh ( x ′ , t ′ ) δ � φ ( x , t ) � = − i Γ where G ( t − t ′ , x − x ′ ) ≡ � φ ( x , t ) χ ( x ′ , t ′ ) � . Can identify G R ( t, x ) = − i Γ � φ ( x , t ) χ (0) � , G A ( t, x ) = − i Γ � φ (0) χ ( x , t ) � . Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 37 / 58
Fluctuations: Diffusion equation 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) Hydrodynamic fluctuations GGI, May 3, 2011 38 / 58
Fluctuations: Diffusion equation 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 ω Re G ( ω, k ) Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 38 / 58
Fluctuations: Diffusion equation 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 ω Re G ( ω, k ) This is FDT in the effective field theory for φ Pavel Kovtun (University of Victoria) Hydrodynamic fluctuations GGI, May 3, 2011 38 / 58
Recommend
More recommend
