Field Theory of Non-Newtonian Turbulence
Esteban Calzetta























    calzetta@df.uba.arUniversidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Física, Buenos Aires, Argentina,
        
        and CONICET-Universidad de Buenos Aires, Instituto de Física de Buenos Aires (IFIBA), Buenos Aires, Argentina  § ABSTRACT 
        Providing a compelling derivation of Kolmogorov turbulence is a fascinating open challenge in field theory. Here, we pose a more modest question: if we had a field-theoretic description of Kolmogorov turbulence, could we use it to describe deviations caused, for example, by adding a polymer additive or by relativistic corrections? To investigate this issue, we assume a description of developed, homogeneous, and isotropic turbulence along the lines of Martin, Siggia, and Rose, and we work out the first corrections to the equal-time, two-point spectrum caused by adding non-Newtonian terms to the fluid stress tensor. While the results are not conclusive, they show a promising resemblance to turbulent spectra found in both experiments and large-scale numerical simulations.
        Field Theory of Non-Newtonian Turbulence
            Esteban Calzetta
            September 9, 2024
        ========================================
        
        
        
        § INTRODUCTION
        
        
        Almost since the inception of the Kolmogorov theory of turbulence <cit.>, there have been attempts to derive it from the Navier-Stokes equations by treating them as a non-relativistic field theory <cit.>. While substantial contributions have been made <cit.>, it is fair to say that this is still an open challenge <cit.>. But we can also ask whether, if we had a field theoretic description of Kolmogorov turbulence, we could use it to investigate other situations of interest, such as turbulence in strongly relativistic fluids <cit.>, or in viscoelastic fluids <cit.>. 
        
        Actually, there are grounds to expect that such an extension of the theory is feasible. Some time ago Goldenfeld, Gioia, Chakraborthy and others <cit.> proposed a quantitative link between friction in a wall bounded flow and the turbulent spectrum in isotropic turbulence. Subsequently the original authors and others showed that the relevant correlation between friction and spectrum subsisted in more general situations, such as flows in different number of dimensions <cit.>, time dependent flows <cit.>, and polymeric solutions <cit.>. In the present work we similarly ask whether a theory built to describe Kolmogorov turbulence under the Navier-Stokes equations, may still be relevant when the fundamental equations are perturbed.
        
        Concretely, we shall begin by assuming that the Navier-Stokes equations (NSE) provide the “bare” description of the flow and will use the Martin - Siggia and Rose (MSR) formalism <cit.> to build an “effective action” (EA) <cit.> from which the full correlations of the theory may be derived. If the EA were computed in a diagrammatic expansion it would reproduce the formalism by Wyld and Lee, see <cit.>. Actually, we shall never need to write down this effective action, but only assume we know what the full correlations are in homogeneous, isotropic Kolmogorov turbulence.
        
        Then we shall perturb the NSE by the addition of non-Newtonian terms in the fluid stress tensor <cit.>. The particular perturbation we shall consider arises in models of viscoelastic behavior, such as characteristic of polymeric solutions <cit.>. We shall show in Appendix <ref> that this model also describes the nonrelativistic limit of a conformal fluid <cit.>.  We then work out the first order correction to the spectrum from the perturbed EA. We  compare this result with spectra found in viscoelastic  turbulence both in experiment and numerical simulations. 
        
        This paper is organized as follows. Next section <ref> presents the basic notations and the equations of the fluid, both the NES and the perturbed one. To make the work self-contained, we have included a basic introduction to the MSR approach and the EA therefrom. In section <ref> we proceed to compute the kernels which are necessary to write down the perturbed Schwinger-Dyson equations. In section <ref> we solve the Schwinger-Dyson equations and find the energy spectrum.
        We conclude with some brief final remarks in section <ref>. 
        
        We have included three appendices. In Appendix <ref> we show how the model in Section <ref> describes the nonrelativistic limit of a conformal fluid. In Appendix <ref> we discuss how to account for the random Galilean invariance <cit.> of the NSE in the EA formalism, a subject that we left out of the main text for simplicity, but has deep implications for the development of the theory. Appendix (<ref>) fills in the details of one of the derivations in the text.
        
        
        
        § THE MODEL
        
        The model is represented by the equations
        
              Q^i=V^j_,t+V^kV^j_,k+P^jk_,k+1/ρP^,j=0      Q^ij=τ_1[P^jk_,t+V^lP^jk_,l]-τ_2[P^jlV^k_,l+V^j_,lP^lk]+P^jk+νΣ^jk=0
         
        V^j_,j=P^jk_,jk=0
        
        where V^j is the incompresible fluid velocity, P^ij is the stress tensor,  P is the pressure, μ is the constant fluid mass density, σ^ij is the shear tensor
        
        Σ^jk=V^j,k+V^k,j
        and ν is the kinematic viscosity. When τ_1,2→ 0 at fixed ν we get an ordinary Newtonian fluid. When τ_1=τ_2=τ, the derivative terms in the second of eqs. (<ref>) add up to the upper convected derivative of p^ij. When τ_2=0 it reduces to a material derivative, which is the case that describes the nonrelativistic limit of a conformal fluid, see Appendix <ref>. In this note we shall assume τ_1=τ_2=τ. 
        
        We note that the right hand side of equations (<ref>) ought to display stochastic sources necessary to put the fluid in motion. However, since we wish to work in the regime where fluid fluctuations are self-sustained, we shall not consider these sources explicitly.
        
        To be able to derive equations (<ref>) from a variational principle we introduce Lagrange multipliers A_j and B_jk such that A^j_,j=0, and write
        
        
        S=∫ d^3ydt {A_jQ^j+B_jkQ^jk}
        We delete the pressure term from Q^j, since it integrates to zero anyway.
        
        
        
         §.§ The MSR EA and its perturbations
        
        
        We see that the action functional eq. (<ref>) depends on four different fields, the physical fields V^j and P^ij and the auxiliary fields A_j and B_ij. This diversity makes for a rather complex field theory. 
        
        To avoid unnecessary complications, we shall adopt an scheme based on three levels of description. Eqs. (<ref>) and (<ref>) belong to the first level, where we treat both physical and auxiliary fields as distinct. In the second level, however, we drop this distinction and gather together the physical fields into a single string V^a=( V^j,P^jk), and similarly the auxiliary fields into a string A_a=(A_j,B_jk). For higher compression, in the third level of description we regard all variables as components of a single object X^J=( V^a,A_a). In the second and third levels space-time indexes are included into the indexes a,J and we apply Einstein's convention to sums over indexes, both discrete and continuous.
        
        Given an action S[ X] we define a generating functional 
        
         
        e^iW[ J] =∫ DX e^i( S[ X] +J_KX^K)  
        were the J_K are a string of external sources. Differentiation yields the mean fields 
        
        X̅^J=δ W/δ J_J 
        We shall work under conditions where symmetry forces all background fields to zero, namely homogeneous, isotropic turbulence. Further differentiation produces the higher cumulants, in particular the two-point correlations
        
        δ^2 W/δ J_Jδ J_K=i⟨ X^JX^K⟩ 
        where we are already using that the mean fields vanish. It is convenient to choose the mean fields, rather than the sources, as independent variables. To achieve this, we introduce the effective action Γ as the Legendre transform of the generating functional 
        
        Γ[ X̅] =W[ J] -J_KX̅^K
         
        whereby we get the equations of motion for the mean fields
        
        δΓ/δX̅^J=-J_J
          
        Differentiating eq. (<ref>) with respect to the mean fields and using eq. (<ref>) we get 
        
        δ^2 Γ/δX̅^JδX̅^K⟨ X^KX^L⟩ =iδ^L_J
        δ^L_J denotes the identity operator in the corresponding functional space. Similarly, from eq. (<ref>) we get 
        
        ⟨ X^JX^K⟩δ^2 Γ/δX̅^KδX̅^L =iδ^J_L 
         
        These are the Schwinger-Dyson equations of the theory. From either of these equations we can derive the two-point correlations from the effective action. 
        
        
        
        
        
         §.§ Auxiliary and physical fields
        
        
        We will now elaborate on the analysis above by distinguishing physical fields V^a from auxiliary fields A_a. We also distinguish the external sources J_a coupled to physical fields from the sources K^a coupled to auxiliary fields. The action eq. (<ref>) is written as
        
        
        S=A_aQ^a[V]
         
        The equations of motion Q^a are causal and we assume (<cit.>)
        
        Detδ Q^a/δ V^b= constant 
        We may choose the constant to be 1. The generating functional eq. (<ref>) is expanded into
        
        
        e^iW[J,K]=∫ DADV  e^i(A_aQ^a[V]+A_aK^a+J_aV^a) 
        Observe that
        
        
        W[0,K]=0
         
        identically, so all the expectation values of products of auxiliary fields vanish. Eq. (<ref>) becomes
        
        ([ Γ_,A̅_aA̅_b Γ_,A̅_aV̅^b; Γ_,V̅^aA̅_b Γ_,V̅^aV̅^b ])([         0 ⟨ A_bV^c⟩; ⟨ V^bA_c⟩ ⟨ V^bV^c⟩ ])=i([ δ^a_c     0;     0 δ^c_a ])
        
        This implies that Γ_,A̅_aV̅^b and ⟨V̅^bA̅_c⟩ are non singular, since
        
        Γ_,A̅_aV̅^b⟨ V^bA_c⟩=iδ^a_c
        
        and then it must be 
        
        Γ_,V̅^aV̅^b=0
         
        when the mean auxiliary fields vanish. 
        
        
        
        § COMPUTING THE EA WITH THE BACKGROUND FIELD METHOD
        
        
        According to the usual rule <cit.>, the EA is the classical action plus a “quantum” correction
        
        Γ=S+Γ_Q
        
        To compute Γ_Q, we split all fields into a background value plus a fluctuation X^J→X̅^J+ x^J etc., expand the action eq. (<ref>) and discard terms independendent or linear in the fluctuations. Then
        
        Γ_Q=(-i)ln∫ Dx^J e^i(S[x]+X̅^JS̅_J[x]+J_QJx^J)
        where S is just the action eq. (<ref>) evaluated on the fluctuation fields, and S̅ is linear on the background fields. The sources J_Q enforce the constraints that the expectation value of the fluctuations vanish ⟨ x^J⟩=0. For this reason all one particle insertions in the diagrammatic evaluation of the effective action vanish, and it is enough to consider one-particle irreducible graphs only. 
        
        Expanding the exponential, we see that the contribution to the quadratic part of the EA from Γ_Q is 
        
        Γ_Q^(2)=i/2X̅^JX̅^K⟨S̅_J[x]S̅_K[x]⟩ 
        where 
        
        ⟨𝒳⟩=∫ Dx^J e^i(S[x]+J_QAx^A)𝒳
        We are using that ⟨S̅_J[x]⟩=0 at zero background fields, which is easily verified.
        Expanding 
        
         
        S=S_0+τ S_1
        
        then to first order in τ we have 
        
        ⟨𝒳⟩=⟨𝒳⟩_0+iτ⟨𝒳S_1⟩_0
         
        where
        
        ⟨𝒳⟩_0=∫ Dx^J e^i(S_0[x]+J_QAx^A)𝒳
        The S̅_J may be similarly expanded 
        
        S̅_J=S̅_0J+τS̅_1J 
        Therefore, up to first order in τ we get 
        
        Γ_Q^(2)=i/2X̅^JX̅^K{⟨S̅_0J[x]S̅_0K[x](1+iτ S_1)⟩_0+2τ⟨S̅_0J[x]S̅_1K[x]⟩_0} 
        We shall assume viscosity effects are negligible in computing Γ_Q, whereby
        
        
        S_0   =   ∫ d^3yds [a_j(v^j_,t+v^kv^j_,k+p^jk_,k)+b_jkp^jk]
        S_1   =   ∫ d^3yds b_jk(p^jk_,t+v^lp^jk_,l-p^jlv^k_,l-v^j_,lp^lk)
         
        Because S_0 only contains b_jk in the combination b_jkp^jk, we find a Novikov-type formula
        
        ⟨ p_jk𝒳⟩_0=i⟨δ𝒳/δ b_jk⟩_0
        
        To apply this formula, we use
        
        δ b_jk(y,s)/δ b_j'k'(y',s')=δ(s-s')Δ^j'k'_jk(y-y')
         
        where 
        
        Δ^j'k'_jk(y-y')=1/2{δ^j'_jδ^k'_k+δ^j'_kδ^k'_j}δ(y-y')
        
        For our purposes we need the noise kernels, when both X̅^J and X̅^K are auxiliary fields, and the self-energies, when one is an auxiliary field and the other a physical field. The noise kernels are computed with the substitutions
        
        X̅^J=A̅_j⇒S̅_J= S^j_a(x,t)=S^j_0a(x,t)=(v^kv^j_,k)(x,t)
         
        or
        
        X̅^J=B̅_jk⇒S̅_J=  S^jk_b(x,t)=S^jk_1b(x,t)=(v^lp^jk_,l-p^jlv^k_,l-v^j_,lp^lk)(x,t)
         
        We see that expressions containing S^jk_b display more p^jk than b_jk factors, therefore according to the Novikov formula are obviously zero. So
        
        δ^2Γ_Q/δA̅_j(x,t)δB̅_j'k'(x',t')=δ^2Γ_Q/δB̅_jk(x,t)δB̅_j'k'(x',t')=0
        
        The calculation of δ^2Γ_Q/δB̅_jn(x,t)δV̅^k(y,t') involves S^jk_b and 
        
              X̅^J=V̅^j⇒S̅_0J=  S_0vj(x,t)=(a_kv^k_,j-a_k,jv^k)(x,t)      X̅^J=V̅^j⇒S̅_1J=  S_1vj(x,t)=[a_lkp^kl_,j+2(b_jkp^kl)_,l](x,t)
         
        Once again we see that there are more  p^jk than b_jk factors, therefore
        
        δ^2Γ_Q/δB̅_jn(x,t)δV̅^k(y,t')=0
         
        Let's spell out the expectation values involved in computing δ^2Γ_Q/δA̅_j(x,t)δV̅^k(y,t')⟨(v^lv^j_,l)(x,t)(a_mv^m_,k-a_m,kv^m)(y,t')(1+iτ S_1)⟩_0+2τ⟨(v^lv^j_,l)(x,t)[a_mnp^mn_,k+2(a_knp^mn)_,m](y,t')⟩_0
        
        Now
        
        ⟨(v^lv^j_,l)(x,t)(a_mv^m_,k-a_m,kv^m)(y,t')iτ S_1⟩_0=⟨(v^lv^j_,l)(x,t)[a_mnp^mn_,k+2(a_knp^mn)_,m](y,t')⟩_0=0
        
        We conclude that there are no first order corrections to the noise and dissipation kernels, which remain at their Kolmogorov values
        
        δΓ_Q/δA̅_j(x,t)δV̅^k(y,t')=δ(t-t')∫d^3k/(2π)^3e^ik(x-y)Δ^k_jκ_k
        
        where on dimensional grounds
        
        κ_k=ν_0(ϵ k^2)^1/3ν_0 is a dimensionless constant, and ϵ is the constant in Kolmogorov's 4/5 law, which we also assume (<cit.>)
        
        ⟨[r̂_j(v^j(r)-v^j(0))]^3⟩=-4/5ϵ r
        
        We also have
        
        δΓ_Q0/δA̅_j(x,t)δA̅_k(y,t')=iδ(t-t')∫d^3k/(2π)^3e^ik(x-y)Δ^kjN_k
        
        Since A_j has dimensions of L^-4T, N_k has dimensions of L^5T^-3. Because N_k is analytical and isotropic near k≈ 0, there N_k≈ k^2. It peaks at a scale k_c≈ 1/L, where L is the linear dimension of the flow. In the inertial range, N_k may depend only on ϵ and k, so N_k≈ϵ/k^3. We interpolate between these behaviors as
        
         
        N_k=N_0ϵ k^2/( k_c^2+k^2)^5/2N_0 is a dimensionless constant. 
        
        To compute variations with respect to P^km we need 
        
        X̅^J=P̅^jk⇒S̅_J=  S_1pkm=-[v^lb_km,l+b_klv^l_,m+b_mlv^l_,k]
         
        It is clear that δ^2Γ_Q/δB̅_jn(x,t)δP̅^km(y,t')=O(τ^2) at least.
        
        The only remaining kernel we need to compute the Schwinger-Dyson equations to first order in τ is 
        
        δ^2Γ_Q/δA̅_j(x,t)δP̅^km(y,t')=-τ⟨(v^lv^j_,l)(x,t)[v^nb_km,n+b_knv^n_,m+b_mnv^n_,k](y,t')⟩_0
        
        We assume this kernel is local in time to enforce random Galilen invariance, and then on dimensional grounds (P^km has units of L^2T^-2, B_km has units of L^-5T)
        
        δ^2Γ_Q/δA̅_j(x,t)δP̅^km(y,t')=i/2τδ(t-t')∫d^3k/(2π)^3e^ik(x-y)[Δ^k_jk^m+Δ^m_jk^k]Λκ_k
        
        where Λ is dimensionless 
        
        Λ≈α N_0(k/k_c)^2/3
        where
        
        α=1/12π^2Γ[13/6]Γ[1/3]/Γ[5/2]≈ 0.018
        
        See appendix (<ref>).
        
        
        
        § COMPUTING THE ENERGY SPECTRUM
        
        
        Because of eq. (<ref>), the velocity-velocity correlation reduces to
        
        ⟨ v^j(x,t)v^k(x',t')⟩=i∫ d^3yds d^3y'ds' ⟨ v^j(x,t)a_l(y,s)⟩δ^2Γ/δA̅_l(y,s)δA̅_m(y',s')⟨ a_m(y',s')v^k(x',t”)⟩
        To find the causal propagator, we ought to solve the system
        
              ∫ d^3ydt'{δ^2Γ/δA̅_j(x,t)δV̅^k(y,t')⟨ v^k(y,t')a_l(x',t”)⟩
        +δ^2Γ/δA̅_j(x,t)δP̅^km(y,t')⟨ p^km(y,t')a_l(x',t”)⟩}=iΔ^j_l      ∫ d^3ydt'{δ^2Γ/δB̅_jn(x,t)δV̅^k(y,t')⟨ v^k(y,t')a_l(x',t”)⟩
        +δ^2Γ/δB̅_jn(x,t)δP̅^km(y,t')⟨ p^km(y,t')a_l(x',t”)⟩}=0
        
        The relevant kernels are computed in eqs. (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), so we may finally write down the Schwinger-Dyson equations. Write
        
              ⟨ v^j(x,t)a_k(x',t')⟩=∫d^3k/(2π)^3dω/(2π)e^i[k(x-x')-ω(t-t')]Δ^j_kG[k,ω]      ⟨ p^jm(x,t)a_k(x',t')⟩=i∫d^3k/(2π)^3dω/(2π)e^i[k(x-x')-ω(t-t')][Δ^j_kk^m+Δ^m_kk^j]G'[k,ω]
         
        
        Then
        
        [-iω+κ_k]G-k^2[1+Λκ_kτ]G'   =   iν G+[-iωτ+1]G'   =   0
        
        Elliminating G'[-iω+κ_k+ν k^2[1+Λκ_kτ]/[1-iωτ]]G=i
        
        The correction is only meaningful when iω≈κ, so we may write
        
        [-iω+κ_k+ν k^2[1+Λκ_kτ]/[1-κτ]]G=i
        
        To first order in τ[-iω+κ_k+ν k^2+ν k^2κτ[1+Λ]]G=i
        
        In the inertial range ν k^2≤κ, so we may write further
        
        {-iω+κ_k[1+ν k^2τ(1+Λ)]}G=i
        
        Once G is known, we may compute
        
        ⟨ v^j(x,t)v^k(x',t')⟩=∫d^3k/(2π)^3dω/(2π)e^i[k(x-x')-ω(t-t')]Δ^jkG_1[k,ω]
        
        where
        
        
        G_1[k,ω]=|G[k,ω]|^2N_k
        
        The energy spectrum is computed from the coincidence limit of the velocity-self correlation
        
        
        E[k]=k^2∫dω/(2π)G_1[k,ω]=k^2N_k/κ_k[1+ν k^2τ(1+Λ)] 
        We plot a typical spectrum in fig. (<ref>)
        
        
        
        
        
        § FINAL REMARKS
        
        
        Eq. (<ref>) shows that non-Newtonian effects become relevant above the scale
        
        
        k_NN=√(1/ντ) 
        It is important to compare this scale with the scale
        
         
        k_D=(ϵ/ν^3)^1/4 
        which marks the upper limit of the inertial range. Observe that we may write
        
        ϵ/ν=Re/T^2 
        where T≈ L/V is the revolving time of the largest eddies. Then
        
        k_NN/k_D=1/(Re)^1/4√(T/τ) 
        which shows that non-Newtonian effects could be relevant well within the inertial range for turbulent enough flows.
        
        Our goal in this paper has been to explore whether the well known difficulties of field theoretic approaches to turbulence were an intrinsic limitation of field theory, or just confined to Kolmogorov turbulence. We have attempted to show that if one had a field theory capable to account for the Kolmogorov spectrum  and 4/5 Law, then this theory would retain some predictive power even in more general situations.
        
        A more close to Earth goal was to establish a bridge between the theory of turbulence in non-Newtonian and in relativistic turbulence, with the hope that our growing understanding of the former could guide us in the exploration of the latter.
        
        
        
        
        I thank P. Mininni for multiple talks.
        
        E. C. acknowledges financial support from Universidad de Buenos Aires through Grant No. UBACYT
        20020170100129BA, CONICET Grant No. PIP2017/19:11220170100817CO and ANPCyT Grant No. PICT 2018: 03684. 
        
         
        
        § NON-NEWTONIAN FLUID AS THE NON-RELATIVISTIC LIMIT OF A CONFORMAL FLUID
        
        
        We consider a relativistic fluid of massless particles. 
        
        At the macroscopic level, the theory is described by the energy-momentum tensor (EMT) T^μν. Adopting the Landau prescription for the four velocity u^μ and the energy density ρ
        T^μ_νu^ν=-ρ u^μ
        and observing that T^μν is traceless, we are led to write
        
        
        T^μν=ρ[u^μu^ν+1/3Δ^μν+Π^μν]
        
        where
        
        Δ^μν=η^μν+u^μu^ν 
        and
        
        Π^μ_νu^ν=Π^μ_μ=0
        
        We must also provide an entropy flux. For an ideal fluid, namely when Π^μ_ν=0, the entropy density is 
        
        
        s=s_0=1/T(ρ+P)
        
        where T is the temperature and P=ρ/3 is the pressure. From the thermodynamic relation
        
        
        s_0=∂ P/∂ T 
        we conclude that
        
        ρ=σ_SB T^4
        
        for some constant σ_SB. The entropy flux is then
        
        
        S_0^μ=s_0u^μ
        When we consider the real fluid, Π^μ_ν≠0, we observe that because of (<ref>) we cannot make a vector out of u^μ and Π^μν. Therefore it makes sense to write
        
        
        S^μ=su^μ
        The entropy density ought to be maximum when the fluid is in equilibrium, namely when Π^μν vanishes. So at least close to equilibrium we should have
        
        
        s=4/3σ_SBT^3e^-3/2λΠ^μνΠ_μν 
        for some dimensionless constant λ. If we further write 
        
        
        T=T_0e^δ
        Then the conservation laws are 
        
        
        0   =   δ_,νu^ν+1/3u^ν_,ν+1/4Π^μνu_μ,ν
        0   =   δ_,ν[Δ^μν+3Π^μν]+u^μ_,νu^ν+3/4Δ^μ_ρΠ^ρν_,ν 
        On the other part, positive entropy creation yields
        
        
        0≤1/3S^μ_μ=u^ν[δ_,ν-λΠ^ρσΠ_ρσ,ν]+1/3u^ν_,ν
        which using the conservation laws and the transversality of Π^ρσ may be written as
        
        Π^ρσ[λ u^νΠ_ρσ,ν+1/8σ_ρσ]≤ 0
        
        where
        
        σ^ρσ=[Δ^ρμΔ^σν+Δ^ρνΔ^σμ-2/3Δ^ρσΔ^μν]u_μ,ν
        is the covariant form of the shear tensor eq. (<ref>).
        Therefore, positive entropy creation is achieved by adopting the Cattaneo-Maxwell equation
        
        λ u^νΠ^ρσ_,ν+1/t_RΠ^ρσ+1/8σ^ρσ=0
        
        We shall now consider the nonrelativistic limit. We write explicitly x^0=ct and
        
        u^μ   =   ( 1,u^k/c) /√(1-u^2/c^2)Π^μν   =   ( [ Π_lmu^lu^m/c^2      Π_klu^l/c;      Π_jmu^m/c           Π_jk ]) +Π_lmu^lu^m/c^2/3-u^2/c^2( [ u^2/c^2   u^k/c;   u^j/c    δ_jk ])
         
        where Π^j_j=0.  Observe that 
        
        
        Δ^μ_ν=1/1-u^2/c^2( [                  -u^2/c^2                     u^k/c;                    -v_j/c δ^jk+(u^ju^k-u^2δ^jk)/c^2 ])
        
        The first nontrivial terms in the energy conservation equation are of order 1/c and read
        
        
        0=δ_,t+u^jδ_,j+1/3u^j_,j+1/4Π^jkv_j,k 
        From the momentum conservation equation we get
        
        
        0=δ_,k[δ^jk+3Π^jk]+3/4Π^jk_,k+1/c^2[u^j_,t+u^ku^j_,k]
        
        The Cattaneo-Maxwell equation (<ref>) yields
        
        λ[Π^jk_,t+u^lΠ^jk_,l]+1/t_RΠ^jk+1/8(v_j,k+v_k,j)=0
        
        A consistent nonrelativistic limit requires δ,Π^jk∝ 1/c^2. Then from energy conservation we get u^j_,j=0 to lowest order. Let us write
        
         
        u_j   =   v_j+1/c^2ϕ_,jΠ^jk   =   4/3c^2p^jkδ   =   1/c^2ϵλ   =   3c^2/32ντ
        t_R   =   32ν/3c^2
        where v^j_,j=0.
        
        Collecting again the leading terms we get 
        
        
        0=ϵ_,t+v^jϵ_,j+1/3Δϕ+1/3v^j,kp_jk 
        0=ϵ_,j+p^jk_,k+[v^j_,t+v^kv^j_,k]
        
        Taking the divergence of this equation we get
        
        
        0=Δϵ+v^k,jv^j_,k+p^jk_,jk
        so we may write a scalar-free equation of motion
        
        
        Q^l=Δ^l_j[v^j_,t+v^kv^j_,k+p^jk_,k]=0
        
        where
        
        Δ_jk=δ_jk-∂_jΔ^-1∂_k
        
        Finally
        
        
        Q^jk=τ[p^jk_,t+v^lp^jk_,l]+p^jk+ν[v_j,k+v_k,j]=0
        
        We may define a mass density
        
        μ=ρ/c^2 
        Then μ is constant to order 1/c^2. We see that ϵ=P/μ, where P is the non-constant part of the pressure. P is not a dynamical variable but it is determined from the constraint
        
        
        0=1/μΔ P+ p^jk_,jk+v^k_,jv^j_,k
        We see that we recover equations (<ref>) in the particular case τ_2=0. 
        
        
        
        § RANDOM GALILEAN INVARIANCE
        
        
        Let us go back to the action functional eq. (<ref>) and the corresponding generating functional eq. (<ref>), whose Legendre transform yields the 1PI effective action Γ, eq. (<ref>). 
        
        This construction misses the fact that the equations of motion (<ref>) are random galilean invariant, that is, they are invariant under the transformation
        
              v^j(x^j,t)→ v^j(x^j-ϵ^j(t),t)+ϵ̇^j(t)       p^jk(x^j,t)→ p^jk(x^j-ϵ^j(t),t)      A_j(x^j,t)→ A_j(x^j-ϵ^j(t),t)      A_jk(x^j,t)→ A_jk(x^j-ϵ^j(t),t)
        
        where ϵ^j(t) is an arbitrary time dependent field. Of course we are using that
        
        ∫ d^3x A_jϵ̈^j=∫ d^3x A_j∂^j(ϵ̈_kx^k)=0
        
        For this reason the path integral defining the generating functional, eq. (<ref>), is redundant. To eliminate the overcounting, we consider the non-invariant function
        
        
        P^j(t)=∫ d^3x μ v^j
         
        Assuming that μ transforms as μ(x^j,t)→μ(x^j-ϵ^j(t),t) we see that
        
        
        P^j→ P^j+Mϵ̇^j(t)
        
        where M is the total mass of the fluid. We now observe that
        
         
        1=∫ Dϵ^j detδ P^j[ϵ]/δϵ^kδ(P^j[ϵ]-C^j)
         
        Introducing this identity into the path integral, we can take the ϵ integral out as a constant factor (for this we make a change of variables within the integral, with unit Jacobian), integrate over the C^j with a Gaussian weight and exponentiate the determinant introducing Grassmann variables ζ_j and η^j, where now 
        
        
        e^iW[Z_a,H^a,z_a,h^a]=∫ DX^aDA_a e^i(S_RGI+Z_aX^a+H^aA_a+z_aη^a+h^aζ_a) 
        where
        
         
        S_RGI=∫ dtd^3x {A_jQ^j+A_jkQ^jk}+1/2α∫ dt P_jP^j+i∫ dt ζ_jMη̇^j
         
        Note that the ghost fields are decoupled. This action is still invariant under a BRST transformation defined as follows: the matter and auxiliary fields transform as in a random galilean transformation with parameter ϵ^j=θη^j, where θ is a Grassmann constant, ζ_j transforms into ζ_j+iθ P_j/α, and η^j is invariant. We thus obtain the Zinn-Justin equation
        
              ∫ d^dxdt {δΓ/δ v^j(η^l(t)v^j_,l(x^l,t)-η̇^j(t))+δΓ/δ p^jkη^l(t) p^jk_,l(x^l,t)+δΓ/δ A_jη^l(t)A_j,l(x^l,t).   +   .δΓ/δ A_jkη^l(t)A_jk,l(x^l,t)}-i/α∫ dt δΓ/δζ_jP_j(t)=0
        
        Since the integral over ghost fields is just a decoupled Gaussian integral, we have
        
        δΓ/δζ_j=iMη̇^j
        
        Moreover
        
        ∫ d^dxdt η^l(t)v^j_,l(x^l,t)δ/δ v^j∫ dt P_k(t)P^k(t)=0
        
        So eq. (<ref>) is consistent with 
        
        Γ=Γ_0+1/2α∫ dt P_j(t)P^j(t)
         
        where Γ_0 is independent of α. Then Γ_0=lim_α→∞Γ, namely, it is an effective action without the Fadeev-Popov procedure. This implies that Γ_0 is identically zero when the auxiliary fields vanish, independently of the physical fields.
        
        Taking a derivative of eq. (<ref>) with respect to η^j we see that the Zinn-Justin equation for Γ_0 is local in time, and so it must be Γ_0 itself.
        
        In the presence of the gauge-fixing term  ⟨ vA⟩ and ⟨ vv⟩ are unchanged, and now
        
        ⟨ AA⟩=μ^2/αδ(k)/ω^2+ν^2[0]
        Because the velocity-velocity correlation vanishes at zero momentum, this does not affect the perturbation theory at non-vanishing momenta. This means there is no loss of generality if we take α→∞.
        
        
        
        § DERIVATION OF EQ. (<REF>)
        
        
        We may estimate Λ as follows. First note that at τ=0p^jk becomes a lagrange multiplier enforcing the constraint
        
        
        b_jk=1/2[a_j,k+a_k,j]
        
        Now use a quasi-Gaussian approximation to get
        
              δ^2Γ_Q/δA̅_j(x,t)δP̅^km(y,t')=-τ/2⟨(v^lv^j_,l)(x,t)[v^n[a_k,mn+a_m,kn]+[a_k,n+a_n,k]v^n_,m+[a_m,n+a_n,m]v^n_,k](y,t')⟩_0   =   -τ/2⟨(v^lv^j_,l)(x,t)[∂_m(v^na_k,n)+∂_k(v^na_m,n)+a_n,kv^n_,m+a_n,mv^n_,k](y,t')⟩_0   =   -τ/2{∂^2/∂ y^m∂ y^n[⟨ v^l(x,t)v^n(y,t')⟩⟨ v^j_,l(x,t)a_k(y,t')⟩+⟨ v^l(x,t)a_k(y,t')⟩⟨ v^j_,l(x,t)v^n(y,t')⟩].   +   .⟨ v^l(x,t)a_n,k(y,t')⟩⟨ v^j_,l(x,t)v^n_,m(y,t')⟩+⟨ v^l(x,t)v^n_,m(y,t')⟩⟨ v^j_,l(x,t)a_n,k(y,t')⟩+(k↔ m)}
        We use the Kolmogorov values 
        
              ⟨ v^j(x,t)a_k(x',t')⟩=∫d^3k/(2π)^3Δ^j_ke^[ik(x-x')-κ_k(t-t')]θ(t-t')      ⟨ v^j(x,t)v^k(x',t')⟩=∫d^3k/(2π)^3Δ^j_ke^[ik(x-x')-κ_k|t-t'|]N_k/2κ_k
        Where N_k has been defined in eq. (<ref>). Then
        
              δ^2Γ_Q/δA̅_j(x,t)δP̅^km(y,t')=iτ/2θ(t-t')∫d^3k/(2π)^3e^ik(x-y)      [ (k_mk_n)∫d^3k'/(2π)^3Δ_k'^lne^-[κ_k'+κ_(k-k')](t-t')(k-k')_lΔ^j_(k-k')kN_k'/2κ_k'.   +   (k_mk_n)∫d^3k'/(2π)^3Δ_k'^jne^-[κ_k'+κ_(k-k')](t-t')k'_lΔ^l_(k-k')kN_k'/2κ_k'   +   ∫d^3k'/(2π)^3Δ_k'^jne^-[κ_k'+κ_(k-k')](t-t')k'_lk'_mΔ^l_(k-k')n(k-k')_kN_k'/2κ_k'   +   .∫d^3k'/(2π)^3Δ_k'^lne^-[κ_k'+κ_(k-k')](t-t')k'_mΔ^j_(k-k')n(k-k')_l(k-k')_kN_k'/2κ_k']+(k↔ m)
        
        Since the k' integral is dominated by the infrared band, we approximate k'≪ k      δ^2Γ_Q/δA̅_j(x,t)δP̅^km(y,t')=iτ/2θ(t-t')∫d^3k/(2π)^3e^ik(x-y)      [ (k_mk_n)∫d^3k'/(2π)^3Δ_k'^lne^-κ_k(t-t')k_lΔ^j_(k)kN_k'/2κ_k'.   +   (k_mk_n)∫d^3k'/(2π)^3Δ_k'^jne^-κ_k(t-t')k'_lΔ^l_(k)kN_k'/2κ_k'   +   ∫d^3k'/(2π)^3Δ_k'^jne^-κ_(k)(t-t')k'_lk'_mΔ^l_(k)nk_kN_k'/2κ_k'   +   .∫d^3k'/(2π)^3Δ_k'^lne^-κ_(k)(t-t')k'_mΔ^j_(k)nk_lk_kN_k'/2κ_k']+(k↔ m)
        
        The k' integral now has spherical symmetry, so it simplifies to
        
              δ^2Γ_Q/δA̅_j(x,t)δP̅^km(y,t')=iτ/2θ(t-t')∫d^3k/(2π)^3e^ik(x-y)      [ (k_mk_n)∫d^3k'/(2π)^3Δ_k'^lne^κ_k(t-t')k_lΔ^j_(k)kN_k'/2κ_k'.   +   .∫d^3k'/(2π)^3Δ_k'^jne^-κ_(k)(t-t')k'_lk'_mΔ^l_(k)nk_kN_k'/2κ_k']+(k↔ m)
        
        Using the symmetry again this becomes (we also filter out a term proportional to k^j)
        
              δ^2Γ_Q/δA̅_j(x,t)δP̅^km(y,t')=iτ/2θ(t-t')∫d^3k/(2π)^3e^ik(x-y)      [ 2/3k_mk^2Δ^j_(k)k∫d^3k'/(2π)^3 e^-κ_k(t-t')N_k'/2κ_k'.   +   .1/15Δ^j_mk_k∫d^3k'/(2π)^3e^-κ_(k)(t-t')N_k'k'^2/2κ_k']+(k↔ m)
        
        The dominant integral is the first. We also approximate
        
        
        e^-κ_k(t-t')≈δ(t-t')/κ_k
        The final step is a rescaling k'=rk whereby
        
              δ^2Γ_Q/δA̅_j(x,t)δP̅^km(y,t')=iτ/2δ(t-t')∫d^3k/(2π)^3e^ik(x-y)κ_k[ k_mΔ^j_(k)k+ k_kΔ^j_(k)m]      ∫d^3r/(2π)^3 N_0 r^4/3/3[ r_c^2+r^2] ^5/2
        Where r_c=k_c/k. Computing the integral we find eq. (<ref>)
        
        
        
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