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On the Energy Spectrum of Non-Newtonian Turbulence

Esteban Calzetta calzetta@df.uba.ar Universidad 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

The goal of this paper is to propose a theoretical framework to study homogeneous and isotropic turbulence in a viscoelastic fluid, regarded as a perturbation of a Newtonian incompressible fluid, where the fluid relaxation time, or else the Weissenberg number, plays the role of small parameter. We use a Martin-Siggia-Rose framework to obtain a formal expression for the velocity correlation function of the non-Newtonian flow, and we expand this formal expression to linear order in the relaxation time. The coefficients in this expansion are correlation functions of the base Newtonian flow. We do not derive these correlations, instead we replace them by their values according to K41 theory, which could be regarded as an extreme form of renormalization. While substantial work will be necessary to validate the model against numerical and experimental data, preliminary results are encouraging.

I Introduction

The goal of this paper is to propose a theoretical framework for studying turbulent flows in situations beyond K41 theory [1, 2, 3] yet connected to it. The example we discuss is the spectrum of velocity fluctuations in homogeneous, isotropic turbulence of viscoelastic fluids [4, 5, 6, 7, 8, 9, 10, 11, 12], which reduce to Newtonian incompressible fluids when the relaxation time is taken to 0. Another example, closely connected to the first, see Appendix A, is the turbulent flow of a relativistic viscous fluid [13, 14], which becomes a Non-Newtonian fluid when the speed of light cc\to\infty.

The basic idea is to formulate the problem within a Martin-Siggia-Rose (MSR) framework [15, 16, 17, 18, 19, 20]. In this approach, the Schwinger-Dyson equations for the causal response and the velocity correlations are derived from an effective action (EA) [21, 22, 23, 24, 25, 26, 27], which has a formal expression as a path integral over velocity and stress tensor fluctuations (details are provided in Section II). We then expand this formal expression in powers of the relaxation time (or its dimensionless equivalent, the Weissenberg number). The coefficients in this expansion can be written in terms of correlation functions in the base Newtonian flow. We do not derive these correlations from the MSR EA itself; instead, we substitute them with convenient ansätze suggested by K41 theory. This will allow us to sidestep some known criticisms of the MSR formalism in the literature [28, 29, 30, 31, 32, 33]. We discuss this strategy in the Final Remarks Section V; for the time being, we simply propose it as a matter of expediency.

Concretely, we begin by assuming that the Navier-Stokes equations (NSE) provide the “bare” description of homogeneous, isotropic turbulence in a Newtonian incompressible fluid. Then we perturb the NSE by adding non-Newtonian terms in the fluid stress tensor [34, 35, 36, 37]. The perturbation we consider arises in models of polymeric solutions [38, 39, 40, 41, 42, 43]. In Appendix A we show that this model also describes the non-relativistic limit of a conformal fluid [44, 45, 46]. We then work out the first order correction to the spectrum from the perturbed EA. We thus find a concrete analytical expression for the velocity correlation function of the viscoelastic flow, see eq. (56) below. Validating this expression against numerical and experimental data is a massive undertaking; preliminary comparisons at moderate Reynolds numbers and low Weissenberg numbers, such as the data presented in [10], are encouraging.

This paper is organized as follows. Next section II presents the basic notations and the equations of the fluid, both the NSE and the perturbed one. To make the work self-contained, we include an introduction to the MSR approach and the EA therefrom. We show how to derive the correlation functions from the EA and develop their perturbative expansion. In this Section we have adopted a highly compressed notation that best displays the structure of the problem, but we return to a more natural notation in the remaining Sections.

In section III we proceed to compute the kernels necessary to write down the perturbed Schwinger-Dyson equations, solve for the causal response function and finally derive the analytical expression for the spectrum of velocity correlations eq. (56), which is the main result of this paper. We begin with a brief review of the correlations in K41 theory. While this contains no new material, it is convenient to have all the basic formulae in one place.

In section IV we introduce the overall velocity scale u1u_{1} (see eq. (57)) and the Taylor microscale λ0\lambda_{0} (see eq. (58)) [47]. This allows us to define two dimensionless numbers, Reynolds number and Weissenberg number (see eqs. (59) and (60)) [37], which make it much easier to compare our results with the literature. We conclude with some brief final remarks in section V.

The paper includes four appendices. In Appendix A we show how the model in Section II describes the nonrelativistic limit of a conformal fluid. In Appendix B we discuss how to account for the random Galilean invariance [48, 49, 50, 51, 52] of the NSE in the EA formalism. This subject, which we left out of the main text for simplicity, has deep implications for the development of the theory. Appendixes Cand D contain some technical details.

II The model

The model is represented by the equations

Qj=V,tj+VkV,kj+P,kjk+1μP,j=0\displaystyle Q^{j}=V^{j}_{,t}+V^{k}V^{j}_{,k}+P^{jk}_{,k}+\frac{1}{\mu}P^{,j}=0
Qjk=τ1[P,tjk+VlP,ljk]τ2[PjlV,lk+V,ljPlk]+Pjk+νΣjk=0\displaystyle Q^{jk}=\tau_{1}\left[P^{jk}_{,t}+V^{l}P^{jk}_{,l}\right]-\tau_{2}\left[P^{jl}V^{k}_{,l}+V^{j}_{,l}P^{lk}\right]+P^{jk}+\nu\Sigma^{jk}=0 (1)
V,jj=0V^{j}_{,j}=0 (2)

where VjV^{j} is the incompresible fluid velocity, PijP^{ij} is the stress tensor, PP is the pressure, μ\mu is the constant fluid mass density, Σij\Sigma^{ij} is the shear tensor

Σjk=Vj,k+Vk,j\Sigma^{jk}=V^{j,k}+V^{k,j} (3)

and ν\nu is the kinematic viscosity. When τ1,20\tau_{1,2}\to 0 at fixed ν\nu we get an ordinary Newtonian fluid. When τ1=τ2=τ\tau_{1}=\tau_{2}=\tau, the derivative terms in the second of eqs. (1) add up to the upper convected derivative of PijP^{ij} [37]. When τ2=0\tau_{2}=0 it reduces to a material derivative, which is the case that describes the nonrelativistic limit of a conformal fluid, see Appendix A. In this note we shall assume τ1=τ2=τ\tau_{1}=\tau_{2}=\tau.

We note that the right hand side of equations (1) 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 (1) from a variational principle we introduce Lagrange multipliers AjA_{j} and BjkB_{jk} such that A,jj=0A^{j}_{,j}=0, and write

S=d3y𝑑t{AjQj+BjkQjk}S=\int d^{3}ydt\;\left\{A_{j}Q^{j}+B_{jk}Q^{jk}\right\} (4)

We delete the pressure term from QjQ^{j}, since it integrates to zero anyway.

II.1 The MSR EA

We see that the action functional eq. (4) depends on four different fields, the physical fields VjV^{j} and PijP^{ij} and the auxiliary fields AjA_{j} and BijB_{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. (1) and (4) 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 Va=(Vj,Pjk)V^{a}=\left(V^{j},P^{jk}\right), and similarly the auxiliary fields into a string Aa=(Aj,Bjk)A_{a}=\left(A_{j},B_{jk}\right). For higher compression, in the third level of description we regard all variables as components of a single object XJ=(Va,Aa)X^{J}=\left(V^{a},A_{a}\right). In the second and third levels space-time indexes are included into the indexes a,Ja,J and we apply Einstein’s convention to sums over indexes, both discrete and continuous.

Given an action S[X]S\left[X\right] we define a generating functional

eiW[J]=DXei(S[X]+JKXK)e^{iW\left[J\right]}=\int DX\;e^{i\left(S\left[X\right]+J_{K}X^{K}\right)} (5)

were the JKJ_{K} are a string of external sources. Differentiation yields the mean fields

X¯J=δWδJJ\bar{X}^{J}=\frac{\delta W}{\delta J_{J}} (6)

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

δ2WδJJδJK=iXJXK\frac{\delta^{2}W}{\delta J_{J}\delta J_{K}}=i\left\langle X^{J}X^{K}\right\rangle (7)

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\Gamma as the Legendre transform of the generating functional

Γ[X¯]=W[J]JKX¯K\Gamma\left[\bar{X}\right]=W\left[J\right]-J_{K}\bar{X}^{K} (8)

whereby we get the equations of motion for the mean fields

δΓδX¯J=JJ\frac{\delta\Gamma}{\delta\bar{X}^{J}}=-J_{J} (9)

Differentiating eq. (6) with respect to the mean fields and using eq. (7) we get

δ2ΓδX¯JδX¯KXKXL=iδJL\frac{\delta^{2}\Gamma}{\delta\bar{X}^{J}\delta\bar{X}^{K}}\left\langle X^{K}X^{L}\right\rangle=i\delta^{L}_{J} (10)

δJL\delta^{L}_{J} denotes the identity operator in the corresponding functional space. Similarly, from eq. (9) we get

XJXKδ2ΓδX¯KδX¯L=iδLJ\left\langle X^{J}X^{K}\right\rangle\frac{\delta^{2}\Gamma}{\delta\bar{X}^{K}\delta\bar{X}^{L}}=i\delta^{J}_{L} (11)

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.

II.2 Auxiliary and physical fields

We will now elaborate on the analysis above by distinguishing physical fields VaV^{a} from auxiliary fields AaA_{a}. We also distinguish the external sources JaJ_{a} coupled to physical fields from the sources KaK^{a} coupled to auxiliary fields. The action eq. (4) is written as

S=AaQa[V]S=A_{a}Q^{a}\left[V\right] (12)

The equations of motion QaQ^{a} are causal and we assume ([81])

DetδQaδVb=constant{\rm{Det}}\frac{\delta Q^{a}}{\delta V^{b}}=\;{\rm{constant}} (13)

We may choose the constant to be 11. The generating functional eq. (5) is expanded into

eiW[J,K]=DADVei(AaQa[V]+AaKa+JaVa)e^{iW\left[J,K\right]}=\int DADV\;e^{i\left(A_{a}Q^{a}\left[V\right]+A_{a}K^{a}+J_{a}V^{a}\right)} (14)

Observe that

W[0,K]=0W\left[0,K\right]=0 (15)

identically, so all the expectation values of products of auxiliary fields vanish. Eq. (10) becomes

(Γ,A¯aA¯bΓ,A¯aV¯bΓ,V¯aA¯bΓ,V¯aV¯b)(0AbVcVbAcVbVc)=i(δca00δac)\left(\begin{array}[]{cc}\Gamma_{,\bar{A}_{a}\bar{A}_{b}}&\Gamma_{,\bar{A}_{a}\bar{V}^{b}}\\ \Gamma_{,\bar{V}^{a}\bar{A}_{b}}&\Gamma_{,\bar{V}^{a}\bar{V}^{b}}\end{array}\right)\left(\begin{array}[]{cc}0&\left\langle A_{b}V^{c}\right\rangle\\ \left\langle V^{b}A_{c}\right\rangle&\left\langle V^{b}V^{c}\right\rangle\end{array}\right)=i\left(\begin{array}[]{cc}\delta^{a}_{c}&0\\ 0&\delta^{c}_{a}\end{array}\right) (16)

This implies that Γ,A¯aV¯b\Gamma_{,\bar{A}_{a}\bar{V}^{b}} and V¯bA¯c\left\langle\bar{V}^{b}\bar{A}_{c}\right\rangle are non singular, since

Γ,A¯aV¯bVbAc=iδca\Gamma_{,\bar{A}_{a}\bar{V}^{b}}\left\langle V^{b}A_{c}\right\rangle=i\delta^{a}_{c} (17)

and then it must be

Γ,V¯aV¯b=0\Gamma_{,\bar{V}^{a}\bar{V}^{b}}=0 (18)

when the mean auxiliary fields vanish.

The correlations of type VbAc\left\langle V^{b}A_{c}\right\rangle are the response functions of the theory. Once they are found, the physical correlations VaVb\left\langle V^{a}V^{b}\right\rangle follow from

VaVb=iVaAcVbAdΓ,A¯cA¯d\left\langle V^{a}V^{b}\right\rangle=i\left\langle V^{a}A_{c}\right\rangle\left\langle V^{b}A_{d}\right\rangle\Gamma_{,\bar{A}_{c}\bar{A}_{d}} (19)

The second derivatives Γ,A¯cA¯d\Gamma_{,\bar{A}_{c}\bar{A}_{d}} are the so-called “noise kernels” [22]

II.3 Computing the EA with the background field method

According to the usual rule [22], the EA is the classical action plus a “quantum correction”

Γ=S+ΓQ\Gamma=S+\Gamma_{Q} (20)

To compute ΓQ\Gamma_{Q}, we split all fields into a background value plus a fluctuation XJX¯J+xJX^{J}\to\bar{X}^{J}+x^{J} etc., expand the action eq. (4) and discard terms independent or linear in the fluctuations. Then

eiΓQ=Dxei(S[x]+S¯bg[X¯,x]+JQJxJ)e^{i\Gamma_{Q}}=\int Dx\;e^{i\left(S\left[x\right]+\bar{S}_{bg}\left[\bar{X},x\right]+J_{QJ}x^{J}\right)} (21)

where SS is just the action eq. (4) evaluated on the fluctuation fields. S¯bg\bar{S}_{bg} is linear on the background fields and quadratic on the fluctuation fields. The sources JQJ_{Q} enforce the constraints that the expectation value of the fluctuations vanish. For this reason, all one-particle insertions in the diagrammatic evaluation of the effective action cancel out, and it is enough to consider one-particle irreducible graphs only. We shall no longer write the sources explicitly, they are assumed to be included into S[x]S[x].

We normalize the integration measure so that

DxeiS[x]=1\int Dx\;e^{iS\left[x\right]}=1 (22)

So that ΓQ[X¯J=0]=0\Gamma_{Q}[\bar{X}^{J}=0]=0. Then we find

δΓQδX¯J|X¯J=0=DxeiS[x]δSbgδX¯J=0\frac{\delta\Gamma_{Q}}{\delta\bar{X}^{J}}|_{\bar{X}^{J}=0}=\int Dx\;e^{iS\left[x\right]}\frac{\delta S_{bg}}{\delta\bar{X}^{J}}=0 (23)

Taking one more derivative

δ2ΓQδX¯JδX¯K|X¯J=0=iDxeiS[x]δSbgδX¯JδSbgδX¯Ki<δSbgδX¯JδSbgδX¯K>\frac{\delta^{2}\Gamma_{Q}}{\delta\bar{X}^{J}\delta\bar{X}^{K}}|_{\bar{X}^{J}=0}=i\int Dx\;e^{iS\left[x\right]}\frac{\delta S_{bg}}{\delta\bar{X}^{J}}\frac{\delta S_{bg}}{\delta\bar{X}^{K}}\equiv i<\frac{\delta S_{bg}}{\delta\bar{X}^{J}}\frac{\delta S_{bg}}{\delta\bar{X}^{K}}> (24)

Expanding

S\displaystyle S =\displaystyle= S0+τS1\displaystyle S_{0}+\tau S_{1}
Sbg\displaystyle S_{bg} =\displaystyle= Sbg0+τSbg1\displaystyle S_{bg0}+\tau S_{bg1} (25)

then to first order in τ\tau we have

δ2ΓQδX¯JδX¯K|X¯J=0=i<δSbg0δX¯JδSbg0δX¯K>0\displaystyle\frac{\delta^{2}\Gamma_{Q}}{\delta\bar{X}^{J}\delta\bar{X}^{K}}|_{\bar{X}^{J}=0}=i<\frac{\delta S_{bg0}}{\delta\bar{X}^{J}}\frac{\delta S_{bg0}}{\delta\bar{X}^{K}}>_{0} (26)
+\displaystyle+ iτ<δSbg0δX¯JδSbg1δX¯K>0+iτ<δSbg1δX¯JδSbg0δX¯K>0τ<δSbg0δX¯JδSbg0δX¯KS1>0\displaystyle i\tau<\frac{\delta S_{bg0}}{\delta\bar{X}^{J}}\frac{\delta S_{bg1}}{\delta\bar{X}^{K}}>_{0}+i\tau<\frac{\delta S_{bg1}}{\delta\bar{X}^{J}}\frac{\delta S_{bg0}}{\delta\bar{X}^{K}}>_{0}-\tau<\frac{\delta S_{bg0}}{\delta\bar{X}^{J}}\frac{\delta S_{bg0}}{\delta\bar{X}^{K}}S_{1}>_{0}

where

X0=DxeiS0[x]X\left\langle{X}\right\rangle_{0}=\int Dx\;e^{iS_{0}\left[x\right]}{X} (27)

In computing the path integral only one-particle irreducible graphs should be considered.

The point of this analysis is that the expectation values in eq. (26) are computed at τ=0\tau=0, that is, for a Newtonian theory. We shall not attempt to derive them from the path integral representation of ΓQ\Gamma_{Q}, but rather assume that they take values that are consistent with K41 theory. We shall come back to discussing the validity of this procedure in the final remarks.

III Perturbative energy spectrum

In this Section we shall derive the energy spectrum to first order in τ\tau, which will be evaluated in next Section IV. It is convenient to first review the response function and noise kernel in the K41 theory, which will be used later to build the corresponding kernels for viscoelastic flow.

III.1 Response and noise kernels in K41 theory

The K41 theory is built on the observation that turbulent fluctuations are non-Gaussian, with skewness

[r^j(vj(𝒓)vj(𝟎))]3=45ϵr\left\langle\left[\hat{r}_{j}\left(v^{j}\left(\bm{r}\right)-v^{j}\left(\bm{0}\right)\right)\right]^{3}\right\rangle=-\frac{4}{5}\epsilon r (28)

This is the so-called Kolmogorov’s 4/54/5 law ([3]). The dynamically generated scale ϵ\epsilon has dimensions of L2T3L^{2}T^{-3}. It measures the transport of energy accross the turbulent cascade because of nonlinear interactions.

The K41 theory recognizes three flow regimes. There is a scale Lkc1L\approx k_{c}^{-1}, basically the linear dimension of the flow, where energy is being injected. This scale and larger wavelengths form the “creation range”. Out of ϵ\epsilon and the molecular viscosity ν\nu we can form the Kolmogorov scale

kK=(ϵν3)1/4k_{K}=(\frac{\epsilon}{\nu^{3}})^{1/4} (29)

At smaller length scales the flow is dominated by viscosity. This is the dissipation range.

From kck_{c} to kKk_{K} we are in the inertial range, where ϵ\epsilon is the only relevant dimensionful parameter. We may use ϵ\epsilon to build the second derivative of the quantum action

δΓQ0δA¯j(x,t)δV¯k(y,t)=d3k(2π)3dω(2π)ei[k(xy)ω(tt)]Δjkζ(ϵk2)1/3,\frac{\delta\Gamma_{Q0}}{\delta\bar{A}_{j}(x,t)\delta\bar{V}^{k}(y,t^{\prime})}=\int\frac{d^{3}k}{(2\pi)^{3}}\frac{d\omega}{(2\pi)}e^{i[k(x-y)-\omega(t-t^{\prime})]}\Delta^{k}_{j}\zeta\left(\epsilon k^{2}\right)^{1/3}, (30)

The 0 subscript denotes that this result holds for τ=0\tau=0, ζ\zeta is a dimensionless constant, and

Δkj=δkjkjkkk2\Delta^{j}_{k}=\delta^{j}_{k}-\frac{k^{j}k_{k}}{k^{2}} (31)

is a projector that enforces incompressibility. Adding this to the linearized NSE we obtain the response function [53, 54]

vj(x,t)ak(x,t)0\displaystyle\left\langle v^{j}\left(x,t\right)a_{k}\left(x^{\prime},t^{\prime}\right)\right\rangle_{0} =\displaystyle= (1)d3k(2π)3dω(2π)ei[k(xy)ω(tt)][ω+iκk]Δjk\displaystyle(-1)\int\frac{d^{3}k}{(2\pi)^{3}}\frac{d\omega}{(2\pi)}\frac{e^{i[k(x-y)-\omega(t-t^{\prime})]}}{[\omega+i\kappa_{k}]}\Delta^{k}_{j} (32)
=\displaystyle= id3k(2π)3Δkje[i𝒌(𝒙𝒙)κk(tt)]θ(tt)\displaystyle i\int\frac{d^{3}k}{(2\pi)^{3}}\Delta^{j}_{k}e^{\left[i\bm{k(x-x^{\prime})}-\kappa_{k}\left(t-t^{\prime}\right)\right]}\theta\left(t-t^{\prime}\right)

where

κk=ζ(ϵk2)1/3+νk2\kappa_{k}=\zeta\left(\epsilon k^{2}\right)^{1/3}+\nu k^{2} (33)

For the noise kernel, we assume

δΓ0δA¯j(x,t)δA¯k(y,t)=iδ(tt)d3k(2π)3ei𝒌(𝒙𝒚)ΔkjNk\frac{\delta\Gamma_{0}}{\delta\bar{A}_{j}(x,t)\delta\bar{A}_{k}(y,t^{\prime})}=i\delta(t-t^{\prime})\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\bm{k(x-y)}}\Delta^{kj}N_{k} (34)

and then we may compute the correlation function from eq. (19)

vj(x,t)vk(x,t)0=d3k(2π)3Δkje[i𝒌(𝒙𝒙)κk|tt|]Nk2κk\left\langle v^{j}\left(x,t\right)v^{k}\left(x^{\prime},t^{\prime}\right)\right\rangle_{0}=\int\frac{d^{3}k}{(2\pi)^{3}}\Delta^{j}_{k}e^{\left[i\bm{k(x-x^{\prime})}-\kappa_{k}\left|t-t^{\prime}\right|\right]}\frac{N_{k}}{2\kappa_{k}} (35)

Taking the coincidence limit yields the energy spectrum

E0\displaystyle E_{0} =\displaystyle= 𝑑kE0[k]\displaystyle\int\;dk\;E_{0}[k] (36)
=\displaystyle= 12<v2>=1(2π)2𝑑kk2Nkκk\displaystyle\frac{1}{2}<v^{2}>=\frac{1}{(2\pi)^{2}}\int\;dk\;\frac{k^{2}N_{k}}{\kappa_{k}}

AjA_{j} has dimensions of L4TL^{-4}T and NkN_{k} has dimensions of L5T3L^{5}T^{-3}. Since we are working in a frame where the fluid is globally at rest, see Appendix (B), we expect Nk0N_{k}\to 0 when k0k\to 0. NkN_{k} peaks at the scale kck_{c}. In the inertial range, NkN_{k} may depend only on ϵ\epsilon and kk, so Nkϵ/k3N_{k}\approx\epsilon/k^{3}.

E0[k]k4E_{0}[k]\approx k^{4} in the creation range [55] and falls off exponentially in the dissipation range [56]. We interpolate between these regimes adopting [57, 58]

E0[k]=CKϵ2/3k4(kc2+k2)17/6eβ(k/kK)E_{0}[k]=\frac{C_{K}\epsilon^{2/3}k^{4}}{\left(k_{c}^{2}+k^{2}\right)^{17/6}}e^{-\beta(k/k_{K})} (37)

where CK0.5C_{K}\approx 0.5 is the Kolmogorov constant [59]. See fig. (1).

Refer to caption
Figure 1: (Color online) A typical K41 energy spectrum as given by eq. (37). For this particular plot we chose Re0=435{\rm{Re}}_{0}=435. The dotted line is the K41 spectrum; we have included two more lines, dashed corresponding to Ek5/3E\approx k^{-5/3} and dashed-dotted corresponding Ek2.3E\approx k^{-2.3} for comparison. We see that for moderate Reynolds number, the full spectrum can be fitted to power laws other than 5/35/3.

Then

Nk=(2π)2CKϵ2/3k2κk(kc2+k2)17/6eβ(k/kK)N_{k}=\frac{(2\pi)^{2}C_{K}\epsilon^{2/3}k^{2}\kappa_{k}}{\left(k_{c}^{2}+k^{2}\right)^{17/6}}e^{-\beta(k/k_{K})} (38)

where β1/2\beta\approx 1/2 is a dimensionless constant. Note that the value of ζ\zeta remains undetermined.

In this paper we shall only consider two-point correlations. For the constraints that K41 theory puts on higher correlations see ([60, 61, 62]).

III.2 Noise kernels in viscoelastic flow

After reviewing the K41 theory, we return to the viscoelastic flow.

Let us begin by writing in full the expression for the velocity-velocity correlation, eq. (19). Since the “classical” action eq. (4) is linear on the auxiliary fields, the noise kernels come entirely from ΓQ\Gamma_{Q}, whereby

<vj(x,t)vk(x,t)=d3y𝑑sd3y𝑑s\displaystyle<v^{j}(x,t)v^{k}(x^{\prime},t^{\prime})=\int d^{3}yds\;d^{3}y^{\prime}ds^{\prime} (39)
[<vj(x,t)al(y,s)><vk(x,t)am(y,s)>δ2ΓQA¯l(y,s)A¯m(y,s)\displaystyle[<v^{j}(x,t)a_{l}(y,s)><v^{k}(x^{\prime},t^{\prime})a_{m}(y^{\prime},s^{\prime})>\frac{\delta^{2}\Gamma_{Q}}{\partial\bar{A}_{l}(y,s)\partial\bar{A}_{m}(y^{\prime},s^{\prime})}
+\displaystyle+ <vj(x,t)al(y,s)><vk(x,t)bmn(y,s)>δ2ΓQA¯l(y,s)B¯mn(y,s)\displaystyle<v^{j}(x,t)a_{l}(y,s)><v^{k}(x^{\prime},t^{\prime})b_{mn}(y^{\prime},s^{\prime})>\frac{\delta^{2}\Gamma_{Q}}{\partial\bar{A}_{l}(y,s)\partial\bar{B}_{mn}(y^{\prime},s^{\prime})}
+\displaystyle+ <vj(x,t)blp(y,s)><vk(x,t)am(y,s)>δ2ΓQB¯lp(y,s)A¯m(y,s)\displaystyle<v^{j}(x,t)b_{lp}(y,s)><v^{k}(x^{\prime},t^{\prime})a_{m}(y^{\prime},s^{\prime})>\frac{\delta^{2}\Gamma_{Q}}{\partial\bar{B}_{lp}(y,s)\partial\bar{A}_{m}(y^{\prime},s^{\prime})}
+\displaystyle+ <vj(x,t)blp(y,s)><vk(x,t)bmn(y,s)>δ2ΓQB¯lp(y,s)B¯mn(y,s)]\displaystyle<v^{j}(x,t)b_{lp}(y,s)><v^{k}(x^{\prime},t^{\prime})b_{mn}(y^{\prime},s^{\prime})>\frac{\delta^{2}\Gamma_{Q}}{\partial\bar{B}_{lp}(y,s)\partial\bar{B}_{mn}(y^{\prime},s^{\prime})}]

Our first concern will be to show that, of the four noise kernels listed in eq. (39), only the first survives in the high Reynolds number limit, where viscosity effects are negligible. In other words, when ν0\nu\to 0

δ2ΓQA¯l(y,s)B¯mn(y,s)δ2ΓQB¯lp(y,s)B¯mn(y,s)0\frac{\delta^{2}\Gamma_{Q}}{\partial\bar{A}_{l}(y,s)\partial\bar{B}_{mn}(y^{\prime},s^{\prime})}\approx\frac{\delta^{2}\Gamma_{Q}}{\partial\bar{B}_{lp}(y,s)\partial\bar{B}_{mn}(y^{\prime},s^{\prime})}\approx 0 (40)

Let us begin by listing S0S_{0}, S1S_{1}, Sbg0S_{bg0} and Sbg1S_{bg1}, cfr. eqs (25), in the ν=0\nu=0 limit.

S0\displaystyle S_{0} =\displaystyle= d3y𝑑s[aj(v,tj+vkv,kj+p,kjk)+bjkpjk]\displaystyle\int d^{3}yds\;\left[a_{j}\left(v^{j}_{,t}+v^{k}v^{j}_{,k}+p^{jk}_{,k}\right)+b_{jk}p^{jk}\right]
S1\displaystyle S_{1} =\displaystyle= d3y𝑑sbjk(p,tjk+vlp,ljkpjlv,lkv,ljplk)\displaystyle\int d^{3}yds\;b_{jk}\left(p^{jk}_{,t}+v^{l}p^{jk}_{,l}-p^{jl}v^{k}_{,l}-v^{j}_{,l}p^{lk}\right) (41)
Sbg0\displaystyle S_{bg0} =\displaystyle= d3y𝑑s[A¯jvkv,kj+V¯j(akv,jkaj,kvk)]\displaystyle\int d^{3}yds\;\left[\bar{A}_{j}v^{k}v^{j}_{,k}+\bar{V}^{j}(a_{k}v^{k}_{,j}-a_{j,k}v^{k})\right]
Sbg1\displaystyle S_{bg1} =\displaystyle= d3yds[B¯jk(vlp,ljkpjlv,lkv,ljplk)\displaystyle\int d^{3}yds\;[\bar{B}_{jk}\left(v^{l}p^{jk}_{,l}-p^{jl}v^{k}_{,l}-v^{j}_{,l}p^{lk}\right) (42)
+\displaystyle+ V¯j(bklp,jkl+2l(bjkpkl)P¯jk(vlbjk,l+2bjlv,kl)]\displaystyle\bar{V}^{j}(b_{kl}p^{kl}_{,j}+2\partial_{l}(b_{jk}p^{kl})-\bar{P}^{jk}(v^{l}b_{jk,l}+2b_{jl}v^{l}_{,k})]

Let us return to verifying eq. (40). Since δSbg0/δB¯jk=0\delta S_{bg0}/\delta\bar{B}_{jk}=0, there is simply nothing to compute when applying eq. (26) to δ2ΓQ/δB¯jkδB¯lm\delta^{2}\Gamma_{Q}/\delta\bar{B}_{jk}\delta\bar{B}_{lm}, which must be at least 0f O(τ2)O(\tau^{2}).

With respect to δ2ΓQ/δA¯jδB¯kl\delta^{2}\Gamma_{Q}/\delta\bar{A}_{j}\delta\bar{B}_{kl}, from eq. (26) we find

δ2ΓQδA¯j(y,s)δB¯kl(y,s)=iτ<δSbg0δA¯j(y,s)δSbg1δB¯kl(y,s)>0\displaystyle\frac{\delta^{2}\Gamma_{Q}}{\delta\bar{A}_{j}(y,s)\delta\bar{B}_{kl}(y^{\prime},s^{\prime})}=i\tau<\frac{\delta S_{bg0}}{\delta\bar{A}_{j}(y,s)}\frac{\delta S_{bg1}}{\delta\bar{B}_{kl}(y^{\prime},s^{\prime})}>_{0} (43)
=\displaystyle= <(vmv,mj)(y,s)(vnp,nklpknv,nlv,nkpnl)(y,s)>0\displaystyle<(v^{m}v^{j}_{,m})(y,s)\left(v^{n}p^{kl}_{,n}-p^{kn}v^{l}_{,n}-v^{k}_{,n}p^{nl}\right)(y^{\prime},s^{\prime})>_{0}

It is not necessary to filter out the longitudinal modes, since the whole expression vanishes. Because S0S_{0} only contains bjkb_{jk} in the combination bjkpjkb_{jk}p^{jk}, we find a Novikov-type formula [63].

pjkX0=iδXδbjk0\left\langle p_{jk}{X}\right\rangle_{0}=i\left\langle\frac{\delta{X}}{\delta b_{jk}}\right\rangle_{0} (44)

From this formula it is obvious that eq. (43) vanishes.

It is an important point that we have been able to show the validity of eq. (40) directly from the properties of the path integral, without relying on any particular turbulent flow model. However, in the remaining, we shall need to rely on the K41 theory to move forward.

Since only the first line of eq. (39) survives, we only need one response function, namely <vj(x,t)al(y,s)><v^{j}(x,t)a_{l}(y,s)>, and one noise kernel, namely δ2ΓQ/δA¯jδA¯k\delta^{2}\Gamma_{Q}/\delta\bar{A}_{j}\delta\bar{A}_{k}. We shall consider the latter here, and the former in next section.

At τ=0\tau=0 the noise kernel is given by eqs. (34) and (38). We shall now show that the linear order corrections from eq. (26) vanish. Indeed since δSbg1/δA¯j=0\delta S_{bg1}/\delta\bar{A}_{j}=0, the only possible correction is

<δSbg0δA¯j(x,t)δSbg0δA¯k(x,tS1>0=d3y𝑑s<(vlv,lj)(x,t)(vmv,mk)(x,t)(bnp(p,tnp+vqp,qnppnqv,qpv,qppqn))(y,s)><\frac{\delta S_{bg0}}{\delta\bar{A}_{j}(x,t)}\frac{\delta S_{bg0}}{\delta\bar{A}_{k}(x^{\prime},t^{\prime}}S_{1}>_{0}=\int d^{3}yds<(v^{l}v^{j}_{,l})(x,t)(v^{m}v^{k}_{,m})(x^{\prime},t^{\prime})(b_{np}\left(p^{np}_{,t}+v^{q}p^{np}_{,q}-p^{nq}v^{p}_{,q}-v^{p}_{,q}p^{qn}\right))(y,s)> (45)

which is seen to vanish from eq. (44). Once again, we do not filter our the longitudinal modes.

III.3 The response functions

The remaining step to compute the viscoelastic spectrum is to find the response function <vj(x,t)ak(x,t)><v^{j}(x,t)a_{k}(x^{\prime},t^{\prime})>. To do this we must solve the system

d3y𝑑t{δ2ΓδA¯j(x,t)δV¯k(y,t)vk(y,t)al(x,t′′)+δ2ΓδA¯j(x,t)δP¯km(y,t)pkm(y,t)al(x,t′′)}=iΔlj\displaystyle\int d^{3}ydt^{\prime}\left\{\frac{\delta^{2}\Gamma}{\delta\bar{A}_{j}\left(x,t\right)\delta\bar{V}^{k}\left(y,t^{\prime}\right)}\left\langle v^{k}\left(y,t^{\prime}\right)a_{l}\left(x^{\prime},t^{\prime\prime}\right)\right\rangle+\frac{\delta^{2}\Gamma}{\delta\bar{A}_{j}\left(x,t\right)\delta\bar{P}^{km}\left(y,t^{\prime}\right)}\left\langle p^{km}\left(y,t^{\prime}\right)a_{l}\left(x^{\prime},t^{\prime\prime}\right)\right\rangle\right\}=i\Delta^{j}_{l}
d3y𝑑t{δ2ΓδB¯jn(x,t)δV¯k(y,t)vk(y,t)al(x,t′′)+δ2ΓδB¯jn(x,t)δP¯km(y,t)pkm(y,t)al(x,t′′)}=0\displaystyle\int d^{3}ydt^{\prime}\left\{\frac{\delta^{2}\Gamma}{\delta\bar{B}_{jn}\left(x,t\right)\delta\bar{V}^{k}\left(y,t^{\prime}\right)}\left\langle v^{k}\left(y,t^{\prime}\right)a_{l}\left(x^{\prime},t^{\prime\prime}\right)\right\rangle+\frac{\delta^{2}\Gamma}{\delta\bar{B}_{jn}\left(x,t\right)\delta\bar{P}^{km}\left(y,t^{\prime}\right)}\left\langle p^{km}\left(y,t^{\prime}\right)a_{l}\left(x^{\prime},t^{\prime\prime}\right)\right\rangle\right\}=0 (46)

It is easy to see that the first order corrections to δ2Γ/δA¯j(x,t)δV¯k(y,t)\delta^{2}\Gamma/\delta\bar{A}_{j}\left(x,t\right)\delta\bar{V}^{k}\left(y,t^{\prime}\right), δ2Γ/δB¯jn(x,t)δV¯k(y,t)\delta^{2}\Gamma/\delta\bar{B}_{jn}\left(x,t\right)\delta\bar{V}^{k}\left(y,t^{\prime}\right) and δ2Γ/δB¯jn(x,t)δP¯km(y,t)\delta^{2}\Gamma/\delta\bar{B}_{jn}\left(x,t\right)\delta\bar{P}^{km}\left(y,t^{\prime}\right) vanish because of eq. (44).

The only remaining kernel we need to compute the Schwinger-Dyson equations to first order in τ\tau is

δ2ΓQδA¯j(x,t)δP¯km(y,t)=iτ(vlv,lj)(x,t)[vnbkm,n+bknv,mn+bmnv,kn](y,t)0\frac{\delta^{2}\Gamma_{Q}}{\delta\bar{A}_{j}\left(x,t\right)\delta\bar{P}^{km}\left(y,t^{\prime}\right)}=-i\tau\left\langle\left(v^{l}v^{j}_{,l}\right)\left(x,t\right)\left[v^{n}b_{km,n}+b_{kn}v^{n}_{,m}+b_{mn}v^{n}_{,k}\right]\left(y,t^{\prime}\right)\right\rangle_{0} (47)

Where only transverse modes contribute to the variational derivative with respect to A¯j(x,t)\bar{A}_{j}\left(x,t\right). We assume this kernel is local in time, and then on dimensional grounds (PkmP^{km} has units of L2T2L^{2}T^{-2}, BkmB_{km} has units of L5TL^{-5}T)

δ2ΓQδA¯j(x,t)δP¯km(y,t)=i2τδ(tt)d3k(2π)3ei𝒌(𝒙𝒚)[Δjkkm+Δjmkk]Λκk\frac{\delta^{2}\Gamma_{Q}}{\delta\bar{A}_{j}\left(x,t\right)\delta\bar{P}^{km}\left(y,t^{\prime}\right)}=-\frac{i}{2}\tau\delta(t-t^{\prime})\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\bm{k(x-y)}}\left[\Delta^{k}_{j}k^{m}+\Delta^{m}_{j}k^{k}\right]\Lambda\kappa_{k} (48)

where Λ\Lambda is dimensionless

Λ2CK3k2ϵ2/3κk2kc2/3\Lambda\approx\frac{2C_{K}}{3}\frac{k^{2}\epsilon^{2/3}}{\kappa_{k}^{2}k_{c}^{2/3}} (49)

See appendix (C).

We may now solve the system eq. (46). Let us parametrize

vj(x,t)ak(x,t)=d3k(2π)3dω(2π)ei[𝒌(𝒙𝒙)ω(tt)]ΔkjG[k,ω]\displaystyle\left\langle v^{j}\left(x,t\right)a_{k}\left(x^{\prime},t^{\prime}\right)\right\rangle=\int\frac{d^{3}k}{(2\pi)^{3}}\frac{d\omega}{(2\pi)}e^{i\left[\bm{k(x-x^{\prime})}-\omega\left(t-t^{\prime}\right)\right]}\Delta^{j}_{k}G\left[k,\omega\right]
pjm(x,t)ak(x,t)=id3k(2π)3dω(2π)ei[𝒌(𝒙𝒙)ω(tt)][Δkjkm+Δkmkj]G[k,ω]\displaystyle\left\langle p^{jm}\left(x,t\right)a_{k}\left(x^{\prime},t^{\prime}\right)\right\rangle=i\int\frac{d^{3}k}{(2\pi)^{3}}\frac{d\omega}{(2\pi)}e^{i\left[\bm{k(x-x^{\prime})}-\omega\left(t-t^{\prime}\right)\right]}\left[\Delta^{j}_{k}k^{m}+\Delta^{m}_{k}k^{j}\right]G^{\prime}\left[k,\omega\right] (50)

Then

[iω+ζ(ϵk2)1/3]Gk2[1Λκkτ]G\displaystyle\left[-i\omega+\zeta(\epsilon k^{2})^{1/3}\right]G-k^{2}\left[1-\Lambda\kappa_{k}\tau\right]G^{\prime} =\displaystyle= i\displaystyle i
νG+[iωτ+1]G\displaystyle\nu G+\left[-i\omega\tau+1\right]G^{\prime} =\displaystyle= 0\displaystyle 0 (51)

Solving for GG

G[k,ω]=(i)[1iωτ]P[ω]G[k,\omega]=(-i)\frac{[1-i\omega\tau]}{P[\omega]} (52)

where

P[ω]=τω2+iω[1+τζ(ϵk2)1/3]κk[1Λτνk2]P[\omega]=\tau\omega^{2}+i\omega[1+\tau\zeta(\epsilon k^{2})^{1/3}]-\kappa_{k}[1-\Lambda\tau\nu k^{2}] (53)

Once GG is known, we may compute

vj(x,t)vk(x,t)=d3k(2π)3dω(2π)ei[𝒌(𝒙𝒙)ω(tt)]ΔjkG1[k,ω]\left\langle v^{j}\left(x,t\right)v^{k}\left(x^{\prime},t^{\prime}\right)\right\rangle=\int\frac{d^{3}k}{(2\pi)^{3}}\frac{d\omega}{(2\pi)}e^{i\left[\bm{k(x-x^{\prime})}-\omega\left(t-t^{\prime}\right)\right]}\Delta^{jk}G_{1}\left[k,\omega\right] (54)

where

G1[k,ω]=|G[k,ω]|2NkG_{1}\left[k,\omega\right]=\left|G\left[k,\omega\right]\right|^{2}N_{k} (55)

NkN_{k} as in eq. (38). The energy spectrum is computed from the coincidence limit of the velocity-self correlation

E[k]=1(2π)2k2Nkdω(2π)|G[k,ω]|2E0[k][1+κkτ+Λτνk2][1+ζτ(ϵk2)1/3]E\left[k\right]=\frac{1}{\left(2\pi\right)^{2}}k^{2}N_{k}\int\frac{d\omega}{(2\pi)}\left|G\left[k,\omega\right]\right|^{2}\equiv E_{0}[k]\frac{[1+\kappa_{k}\tau+\Lambda\tau\nu k^{2}]}{\left[1+\zeta\tau(\epsilon k^{2})^{1/3}\right]} (56)

E0E_{0} is defined in eq. (37), Λ\Lambda in eq. (49). See the details in Appendix D

IV Results

To be able to compare the spectrum found in eq. (56) with results in the literature we need to introduce several relevant scales.

The spectrum eq. (56) defines a velocity scale

u12=23𝑑kE[k]23CK(ϵkc)2/3u_{1}^{2}=\frac{2}{3}\int\;dk\;E\left[k\right]\approx\frac{2}{3}C_{K}(\frac{\epsilon}{k_{c}})^{2/3} (57)

We also introduce the Taylor microscale λ\lambda [47]

1λ2=10u12𝑑kk2E[k]10kc2/3kK4/3\frac{1}{\lambda^{2}}=\frac{10}{u_{1}^{2}}\int\;dk\;k^{2}\;E\left[k\right]\approx 10k_{c}^{2/3}k_{K}^{4/3} (58)

We next introduce two dimensionless parameters, the Newtonian Reynolds number

Re=u1λ0νCK15(kKkc)2/3{\rm{Re}}=\frac{u_{1}\lambda_{0}}{\nu}\approx\sqrt{\frac{C_{K}}{15}}(\frac{k_{K}}{k_{c}})^{2/3} (59)

and the Weissenberg number [37]

Wi=τνλ02{\rm{Wi}}=\frac{\tau\nu}{\lambda^{2}_{0}} (60)

From eqs. (58) and (59) we obtain

λ0kcCK1501Re\displaystyle\lambda_{0}k_{c}\approx\sqrt{\frac{C_{K}}{150}}\frac{1}{{\rm{Re}}}
λ0kK(320CK)1/4Re1/2\displaystyle\lambda_{0}k_{K}\approx(\frac{3}{20C_{K}})^{1/4}{\rm{Re}}^{1/2} (61)

We may now write

νk2τ\displaystyle\nu k^{2}\tau =\displaystyle= Wi(λ0k)2\displaystyle{\rm{Wi}}\;(\lambda_{0}k)^{2}
(ϵk2)(1/3)\displaystyle(\epsilon k^{2})^{(}1/3) =\displaystyle= (320CK)1/3Re2/3Wi(λ0k)2/3\displaystyle(\frac{3}{20C_{K}})^{1/3}{\rm{Re}}^{2/3}{\rm{Wi}}(\lambda_{0}k)^{2/3} (62)

and

Λ329CK2/3Re2/3(λ0k)2/3[ζ+(20CK3Re2)1/3(λ0k)4/3]2\Lambda\approx\frac{32}{9}C_{K}^{2/3}\frac{{\rm{Re}}^{2/3}(\lambda_{0}k)^{2/3}}{[\zeta+(\frac{20C_{K}}{3{\rm{Re}}^{2}})^{1/3}(\lambda_{0}k)^{4/3}]^{2}} (63)

Observe that in the terms of the viscoelastic model presented in ref. [10] we are working in the limit where the Bingham number goes to infinity.

We plot a typical spectrum in fig. (2)

Refer to caption
Figure 2: (Color online) A typical energy spectrum as given by eq. (56). For this particular plot we chose Re0=435{\rm{Re}}_{0}=435, Wi=103{\rm{Wi}}=10^{-3} and the constant ζ\zeta in eq. (33) as ζ=25\zeta=25. The full line is the actual spectrum; we have included two more lines, dotted corresponding to the Kolmogorov spectrum eq. (37) and dashed-dotted corresponding Ek2.3E\approx k^{-2.3} for comparison. We have normalized the spectrum so that it converges to the universal Kolmogorov spectrum for kλ01k\approx\lambda_{0}^{-1}. Compare with fig. 2 of [10].

V Final remarks

Our goal in this paper is to develop a practical tool to explore problems that can be regarded as perturbations of homogeneous, isotropic turbulence in Newtonian incompressible fluids.

To achieve this goal we have deployed functional methods, such as the EA approach to the MSR formalism, to obtain formal expressions that display the parametric dependence of the correlation functions in full. These formal expressions can be used to construct perturbative series in terms of a suitable small parameter, the Weissenberg number in the case of viscoelastic flow, or the inverse speed of light for a relativistic fluid.

Of course, to obtain results which may be matched against numerical and experimental data requires computing the expansion coefficients to a high accuracy. Achieving this accuracy may be regarded as a unsolved problem in itself. We sidestep this issue by relying on our understanding of fully developed turbulence. Thus we replace the formal path integrals by simple ansätze known to work in Kolmogorov turbulence. We regard this procedure as an extreme, but valid, kind of renormalization [64].

As a final remark on this subject, we note that the application of field theory methods to turbulence is almost as old as the K41 theory itself [65, 66, 67, 68, 69, 70, 71, 72, 73, 74]. The MSR approach is typically seen as the most systematic way to translate a classical stochastic problem into a field theory. It has been used extensively in turbulence (to cite a few examples, see [75, 76, 77, 78, 79]). However, there remain lingering doubts on whether these methods can provide a convincing derivation of the K41 theory, let alone a means to improve it, such as providing a better prediction of the scaling exponents for the higher structure functions [28, 29, 30, 31, 32, 33]. Replacing the unknown correlations by their K41 form is a way to mimic the yet undiscovered non-perturbative methods necessary to compute them from first principles.

We would like to conclude with a few comments on our results.

It is customary to fit turbulent spectra to power laws and in this sense a 7/37/3 exponent for velocity correlations in Non Newtonian fluids has been reported, see eg. [8, 10]. This scaling does not seem to emerge from our analysis. However, as fig. (1) shows, for moderate Reynolds numbers the spectrum spans a range of scaling laws in the transition from the inertial to the dissipative range. For Re=435{\mathrm{Re}}=435 (as shown in fig. (1)) it actually displays a 7/37/3 scaling around the Kolmogorov scale. The Non-Newtonian spectrum in fig. (2) shows a similar behavior.

This could be a case where two completely different analytic expressions nevertheless yield similar results. This is not uncommon, the agreement between the Colebrook formula and the Blasius law for the friction factor in pipe turbulence comes to mind [80].

A remarkable property of the spectrum eq. (56), clearly shown in fig. (2), is the enhancement of velocity fluctuations for very small scales, deep in the dissipative range, relative to the already strongly suppressed Newtonian spectrum of eq. (37). To the best of our understanding, this is not ruled out by available data, such as those in refs. ([6]), ([8]) and ([10]), but it may well be a make or break issue for the validation of the model.

Regarding this last point, we believe that the most relevant feature of our results is the way they display the functional dependence of the non-Newtonian spectrum on the different parameters of the theory. Validating this theoretical prediction requires comparing it against numerical and experimental data accross a full range of Reynolds and Weissenberg numbers.

We stress that we have discussed flows which are strictly homogeneous and isotropic. Data obtained from different configurations, such as wall-bounded flow ([8]), while relevant, cannot be directly compared to our theory.

In conclusion, our goal is to establish the cogency of our proposal, with its validation to be discussed in future communications.

Acknowledgements.
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.

Appendix A 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) TμνT^{\mu\nu}. Adopting the Landau prescription for the four velocity uμu^{\mu} and the energy density ρ\rho

Tνμuν=ρuμT^{\mu}_{\nu}u^{\nu}=-\rho u^{\mu} (64)

and observing that TμνT^{\mu\nu} is traceless, we are led to write

Tμν=ρ[uμuν+13Δμν+Πμν]T^{\mu\nu}=\rho\left[u^{\mu}u^{\nu}+\frac{1}{3}\Delta^{\mu\nu}+\Pi^{\mu\nu}\right] (65)

where

Δμν=ημν+uμuν\Delta^{\mu\nu}=\eta^{\mu\nu}+u^{\mu}u^{\nu} (66)

and

Πνμuν=Πμμ=0\Pi^{\mu}_{\nu}u^{\nu}=\Pi^{\mu}_{\mu}=0 (67)

We must also provide an entropy flux. For an ideal fluid, namely when Πνμ=0\Pi^{\mu}_{\nu}=0, the entropy density is

s=s0=1T(ρ+P)s=s_{0}=\frac{1}{T}\left(\rho+P\right) (68)

where TT is the temperature and P=ρ/3P=\rho/3 is the pressure. From the thermodynamic relation

s0=PTs_{0}=\frac{\partial P}{\partial T} (69)

we conclude that

ρ=σSBT4\rho=\sigma_{SB}T^{4} (70)

for some constant σSB\sigma_{SB}. The entropy flux is then

S0μ=s0uμS_{0}^{\mu}=s_{0}u^{\mu} (71)

When we consider the real fluid, Πνμ0\Pi^{\mu}_{\nu}\not=0, we observe that because of (67) we cannot make a vector out of uμu^{\mu} and Πμν\Pi^{\mu\nu}. Therefore it makes sense to write

Sμ=suμS^{\mu}=su^{\mu} (72)

The entropy density ought to be maximum when the fluid is in equilibrium, namely when Πμν\Pi^{\mu\nu} vanishes. So at least close to equilibrium we should have

s=43σSBT3e32λΠμνΠμνs=\frac{4}{3}\sigma_{SB}T^{3}e^{-\frac{3}{2}\lambda\Pi^{\mu\nu}\Pi_{\mu\nu}} (73)

for some dimensionless constant λ\lambda. If we further write

T=T0eδT=T_{0}e^{\delta} (74)

Then the conservation laws are

0\displaystyle 0 =\displaystyle= δ,νuν+13u,νν+14Πμνuμ,ν\displaystyle\delta_{,\nu}u^{\nu}+\frac{1}{3}u^{\nu}_{,\nu}+\frac{1}{4}\Pi^{\mu\nu}u_{\mu,\nu}
0\displaystyle 0 =\displaystyle= δ,ν[Δμν+3Πμν]+u,νμuν+34ΔρμΠ,νρν\displaystyle\delta_{,\nu}\left[\Delta^{\mu\nu}+3\Pi^{\mu\nu}\right]+u^{\mu}_{,\nu}u^{\nu}+\frac{3}{4}\Delta^{\mu}_{\rho}\Pi^{\rho\nu}_{,\nu} (75)

On the other part, positive entropy creation yields

013Sμμ=uν[δ,νλΠρσΠρσ,ν]+13u,νν0\leq\frac{1}{3}S^{\mu}_{\mu}=u^{\nu}\left[\delta_{,\nu}-\lambda\Pi^{\rho\sigma}\Pi_{\rho\sigma,\nu}\right]+\frac{1}{3}u^{\nu}_{,\nu} (76)

which using the conservation laws and the transversality of Πρσ\Pi^{\rho\sigma} may be written as

Πρσ[λuνΠρσ,ν+18σρσ]0\Pi^{\rho\sigma}\left[\lambda u^{\nu}\Pi_{\rho\sigma,\nu}+\frac{1}{8}\sigma_{\rho\sigma}\right]\leq 0 (77)

where

σρσ=[ΔρμΔσν+ΔρνΔσμ23ΔρσΔμν]uμ,ν\sigma^{\rho\sigma}=\left[\Delta^{\rho\mu}\Delta^{\sigma\nu}+\Delta^{\rho\nu}\Delta^{\sigma\mu}-\frac{2}{3}\Delta^{\rho\sigma}\Delta^{\mu\nu}\right]u_{\mu,\nu} (78)

is the covariant form of the shear tensor eq. (3). Therefore, positive entropy creation is achieved by adopting the Cattaneo-Maxwell equation

λuνΠ,νρσ+1tRΠρσ+18σρσ=0\lambda u^{\nu}\Pi^{\rho\sigma}_{,\nu}+\frac{1}{t_{R}}\Pi^{\rho\sigma}+\frac{1}{8}\sigma^{\rho\sigma}=0 (79)

We shall now consider the nonrelativistic limit. We write explicitly x0=ctx^{0}=ct and

uμ\displaystyle u^{\mu} =\displaystyle= (1,uk/c)1u2/c2\displaystyle\frac{\left(1,u^{k}/c\right)}{\sqrt{1-u^{2}/c^{2}}}
Πμν\displaystyle\Pi^{\mu\nu} =\displaystyle= (Πlmulum/c2Πklul/cΠjmum/cΠjk)+Πlmulum/c23u2/c2(u2/c2uk/cuj/cδjk)\displaystyle\left(\begin{array}[]{cc}\Pi_{lm}u^{l}u^{m}/c^{2}&\Pi_{kl}u^{l}/c\\ \Pi_{jm}u^{m}/c&\Pi_{jk}\end{array}\right)+\frac{\Pi_{lm}u^{l}u^{m}/c^{2}}{3-u^{2}/c^{2}}\left(\begin{array}[]{cc}u^{2}/c^{2}&u^{k}/c\\ u^{j}/c&\delta_{jk}\end{array}\right) (84)

where Πjj=0\Pi^{j}_{j}=0. Observe that

Δνμ=11u2/c2(u2/c2uk/cvj/cδjk+(ujuku2δjk)/c2)\Delta^{\mu}_{\nu}=\frac{1}{1-u^{2}/c^{2}}\left(\begin{array}[]{cc}-u^{2}/c^{2}&u^{k}/c\\ -v_{j}/c&\delta^{jk}+\left(u^{j}u^{k}-u^{2}\delta^{jk}\right)/c^{2}\end{array}\right) (85)

The first nontrivial terms in the energy conservation equation are of order 1/c1/c and read

0=δ,t+ujδ,j+13u,jj+14Πjkvj,k0=\delta_{,t}+u^{j}\delta_{,j}+\frac{1}{3}u^{j}_{,j}+\frac{1}{4}\Pi^{jk}v_{j,k} (86)

From the momentum conservation equation we get

0=δ,k[δjk+3Πjk]+34Π,kjk+1c2[u,tj+uku,kj]0=\delta_{,k}\left[\delta^{jk}+3\Pi^{jk}\right]+\frac{3}{4}\Pi^{jk}_{,k}+\frac{1}{c^{2}}\left[u^{j}_{,t}+u^{k}u^{j}_{,k}\right] (87)

The Cattaneo-Maxwell equation (79) yields

λ[Π,tjk+ulΠ,ljk]+1tRΠjk+18(vj,k+vk,j)=0{\lambda}\left[\Pi^{jk}_{,t}+u^{l}\Pi^{jk}_{,l}\right]+\frac{1}{t_{R}}\Pi^{jk}+\frac{1}{8}\left(v_{j,k}+v_{k,j}\right)=0 (88)

A consistent nonrelativistic limit requires δ,Πjk1/c2\delta,\Pi^{jk}\propto 1/c^{2}. Then from energy conservation we get u,jj=0u^{j}_{,j}=0 to lowest order. Let us write

uj\displaystyle u_{j} =\displaystyle= vj+1c2ϕ,j\displaystyle v_{j}+\frac{1}{c^{2}}\phi_{,j}
Πjk\displaystyle\Pi^{jk} =\displaystyle= 43c2pjk\displaystyle\frac{4}{3c^{2}}p^{jk}
δ\displaystyle\delta =\displaystyle= 1c2ϵ\displaystyle\frac{1}{c^{2}}\epsilon
λ\displaystyle\lambda =\displaystyle= 3c232ντ\displaystyle\frac{3c^{2}}{32\nu}\tau
tR\displaystyle t_{R} =\displaystyle= 32ν3c2\displaystyle\frac{32\nu}{3c^{2}} (89)

where v,jj=0v^{j}_{,j}=0.

Collecting again the leading terms we get

0=ϵ,t+vjϵ,j+13𝚫ϕ+13vj,kpjk0=\epsilon_{,t}+v^{j}\epsilon_{,j}+\frac{1}{3}\bm{\Delta}\phi+\frac{1}{3}v^{j,k}p_{jk} (90)
0=ϵ,j+p,kjk+[v,tj+vkv,kj]0=\epsilon_{,j}+p^{jk}_{,k}+\left[v^{j}_{,t}+v^{k}v^{j}_{,k}\right] (91)

Taking the divergence of this equation we get

0=𝚫ϵ+vk,jv,kj+p,jkjk0=\bm{\Delta}\epsilon+v^{k,j}v^{j}_{,k}+p^{jk}_{,jk} (92)

so we may write a scalar-free equation of motion

Ql=Δjl[v,tj+vkv,kj+p,kjk]=0Q^{l}=\Delta^{l}_{j}\left[v^{j}_{,t}+v^{k}v^{j}_{,k}+p^{jk}_{,k}\right]=0 (93)

where

Δjk=δjkj𝚫1k\Delta_{jk}=\delta_{jk}-\partial_{j}\bm{\Delta}^{-1}\partial_{k} (94)

Finally

Qjk=τ[p,tjk+vlp,ljk]+pjk+ν[vj,k+vk,j]=0Q^{jk}=\tau\left[p^{jk}_{,t}+v^{l}p^{jk}_{,l}\right]+p^{jk}+\nu\left[v_{j,k}+v_{k,j}\right]=0 (95)

We may define a mass density

μ=ρc2\mu=\frac{\rho}{c^{2}} (96)

Then μ\mu is constant to order 1/c21/c^{2}. We see that ϵ=P/μ\epsilon=P/\mu, where PP is the non-constant part of the pressure. PP is not a dynamical variable but it is determined from the constraint

0=1μ𝚫P+p,jkjk+v,jkv,kj0=\frac{1}{\mu}{\bm{\Delta}}P+p^{jk}_{,jk}+v^{k}_{,j}v^{j}_{,k} (97)

We see that we recover equations (1) in the particular case τ2=0\tau_{2}=0.

Appendix B Random Galilean invariance

Let us go back to the action functional eq. (4) and the corresponding generating functional eq. (5), whose Legendre transform yields the 1PI effective action Γ\Gamma, eq. (8).

This construction misses the fact that the equations of motion (1) are random galilean invariant, that is, they are invariant under the transformation

vj(xj,t)vj(xjϵj(t),t)+ϵ˙j(t)\displaystyle v^{j}\left(x^{j},t\right)\to v^{j}\left(x^{j}-\epsilon^{j}\left(t\right),t\right)+\dot{\epsilon}^{j}\left(t\right)
pjk(xj,t)pjk(xjϵj(t),t)\displaystyle p^{jk}\left(x^{j},t\right)\to p^{jk}\left(x^{j}-\epsilon^{j}\left(t\right),t\right)
Aj(xj,t)Aj(xjϵj(t),t)\displaystyle A_{j}\left(x^{j},t\right)\to A_{j}\left(x^{j}-\epsilon^{j}\left(t\right),t\right)
Ajk(xj,t)Ajk(xjϵj(t),t)\displaystyle A_{jk}\left(x^{j},t\right)\to A_{jk}\left(x^{j}-\epsilon^{j}\left(t\right),t\right) (98)

where ϵj(t)\epsilon^{j}\left(t\right) is an arbitrary time dependent field. Of course we are using that

d3xAjϵ¨j=d3xAjj(ϵ¨kxk)=0\int d^{3}x\;A_{j}\ddot{\epsilon}^{j}=\int d^{3}x\;A_{j}\partial^{j}\left(\ddot{\epsilon}_{k}x^{k}\right)=0 (99)

For this reason the path integral defining the generating functional, eq. (5), is redundant. To eliminate the overcounting, we consider the non-invariant function

Pj(t)=d3xμvjP^{j}\left(t\right)=\int d^{3}x\;\mu v^{j} (100)

Assuming that μ\mu transforms as μ(xj,t)μ(xjϵj(t),t)\mu\left(x^{j},t\right)\to\mu\left(x^{j}-\epsilon^{j}\left(t\right),t\right) we see that

PjPj+Mϵ˙j(t)P^{j}\to P^{j}+M\dot{\epsilon}^{j}\left(t\right) (101)

where MM is the total mass of the fluid. We now observe that

1=DϵjdetδPj[ϵ]δϵkδ(Pj[ϵ]Cj)1=\int\;D\epsilon^{j}\;{\rm{det}}\frac{\delta P^{j}\left[\epsilon\right]}{\delta\epsilon^{k}}\delta\left(P^{j}\left[\epsilon\right]-C^{j}\right) (102)

Introducing this identity into the path integral, we can take the ϵ\epsilon integral out as a constant factor (for this we make a change of variables within the integral, with unit Jacobian), integrate over the CjC^{j} with a Gaussian weight and exponentiate the determinant introducing Grassmann variables ζj\zeta_{j} and ηj\eta^{j}, where now

eiW[Za,Ha,za,ha]=DXaDAaei(SRGI+ZaXa+HaAa+zaηa+haζa)e^{iW\left[Z_{a},H^{a},z_{a},h^{a}\right]}=\int\;DX^{a}DA_{a}\;e^{i\left(S_{RGI}+Z_{a}X^{a}+H^{a}A_{a}+z_{a}\eta^{a}+h^{a}\zeta_{a}\right)} (103)

where

SRGI=𝑑td3x{AjQj+BjkQjk}+12α𝑑tPjPj+i𝑑tζjMη˙jS_{RGI}=\int dtd^{3}x\;\left\{A_{j}Q^{j}+B_{jk}Q^{jk}\right\}+\frac{1}{2\alpha}\int\;dt\;P_{j}P^{j}+i\int\;dt\;\zeta_{j}M\dot{\eta}^{j} (104)

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 ϵj=θηj\epsilon^{j}=\theta\eta^{j}, where θ\theta is a Grassmann constant, ζj\zeta_{j} transforms into ζj+iθPj/α\zeta_{j}+i\theta P_{j}/\alpha, and ηj\eta^{j} is invariant. We thus obtain the Zinn-Justin equation [81]

ddxdt{δΓδV¯j(ηl(t)v,lj(xl,t)η¯˙j(t))+δΓδP¯jkηl(t)p,ljk(xl,t)+δΓδA¯jηl(t)aj,l(xl,t)\displaystyle\int\;d^{d}xdt\;\left\{\frac{\delta\Gamma}{\delta\bar{V}^{j}}\left(\left\langle\eta^{l}\left(t\right)v^{j}_{,l}\left(x^{l},t\right)\right\rangle-\dot{\bar{\eta}}^{j}\left(t\right)\right)+\frac{\delta\Gamma}{\delta\bar{P}^{jk}}\left\langle\eta^{l}\left(t\right)p^{jk}_{,l}\left(x^{l},t\right)\right\rangle+\frac{\delta\Gamma}{\delta\bar{A}_{j}}\left\langle\eta^{l}\left(t\right)a_{j,l}\left(x^{l},t\right)\right\rangle\right. (105)
+\displaystyle+ δΓδB¯jkηl(t)bjk,l(xl,t)}iαdtδΓδζ¯jPj(t)=0\displaystyle\left.\frac{\delta\Gamma}{\delta\bar{B}_{jk}}\left\langle\eta^{l}\left(t\right)b_{jk,l}\left(x^{l},t\right)\right\rangle\right\}-\frac{i}{\alpha}\int\;dt\;\frac{\delta\Gamma}{\delta\bar{\zeta}_{j}}\left\langle P_{j}\left(t\right)\right\rangle=0

Since the integral over ghost fields is just a decoupled Gaussian integral, the binary products decouple, namely

ηl(t)v,lj(xl,t)=η¯l(t)V¯,lj(xl,t)\left\langle\eta^{l}\left(t\right)v^{j}_{,l}\left(x^{l},t\right)\right\rangle=\bar{\eta}^{l}\left(t\right)\bar{V}^{j}_{,l}\left(x^{l},t\right) (106)

etc., and

δΓδζ¯j=iMη¯˙j\frac{\delta\Gamma}{\delta\bar{\zeta}_{j}}=iM\dot{\bar{\eta}}^{j} (107)

Moreover

ddx𝑑tη¯l(t)V¯,lj(xl,t)δδvj𝑑tPk(t)Pk(t)=0\int\;d^{d}xdt\;\bar{\eta}^{l}\left(t\right)\bar{V}^{j}_{,l}\left(x^{l},t\right)\frac{\delta}{\delta v^{j}}\int\;dt\;P_{k}\left(t\right)P^{k}\left(t\right)=0 (108)

So eq. (105) is consistent with

Γ=Γ0+12α𝑑tPj(t)Pj(t)\Gamma=\Gamma_{0}+\frac{1}{2\alpha}\int\;dt\;P_{j}\left(t\right)P^{j}\left(t\right) (109)

where Γ0\Gamma_{0} is independent of α\alpha. Γ0\Gamma_{0} is an effective action without the Fadeev-Popov procedure. This implies that Γ0\Gamma_{0} is identically zero when the auxiliary fields vanish, independently of the physical fields.

In the presence of the gauge-fixing term vA\left\langle vA\right\rangle and vv\left\langle vv\right\rangle are unchanged, and now

AA=μ2αδ(k)ω2+ν2[0]\left\langle AA\right\rangle=\frac{\mu^{2}}{\alpha}\frac{\delta\left(k\right)}{\omega^{2}+\nu^{2}\left[0\right]} (110)

When α0\alpha\to 0 this forces the noise kernels and the self energies to vanish at zero momentum, as we have assumed in the text.

Appendix C Derivation of eq. (49)

We may estimate Λ\Lambda as follows. First note that at τ=0\tau=0 pjkp^{jk} becomes a Lagrange multiplier enforcing the constraint

bjk=12[aj,k+ak,j]b_{jk}=\frac{1}{2}\left[a_{j,k}+a_{k,j}\right] (111)

Now use a quasi-Gaussian approximation to get

δ2ΓQδA¯j(x,t)δP¯km(y,t)=iτ2(vlv,lj)(x,t)[vn[ak,mn+am,kn]+[ak,n+an,k]v,mn+[am,n+an,m]v,kn](y,t)0\displaystyle\frac{\delta^{2}\Gamma_{Q}}{\delta\bar{A}_{j}\left(x,t\right)\delta\bar{P}^{km}\left(y,t^{\prime}\right)}=-\frac{i\tau}{2}\left\langle\left(v^{l}v^{j}_{,l}\right)\left(x,t\right)\left[v^{n}\left[a_{k,mn}+a_{m,kn}\right]+\left[a_{k,n}+a_{n,k}\right]v^{n}_{,m}+\left[a_{m,n}+a_{n,m}\right]v^{n}_{,k}\right]\left(y,t^{\prime}\right)\right\rangle_{0} (112)
=\displaystyle= iτ2(vlv,lj)(x,t)[mn2(vnak)+kn2(vnam)+an,kv,mn+an,mv,kn](y,t)0\displaystyle-\frac{i\tau}{2}\left\langle\left(v^{l}v^{j}_{,l}\right)\left(x,t\right)\left[\partial^{2}_{mn}\left(v^{n}a_{k}\right)+\partial^{2}_{kn}\left(v^{n}a_{m}\right)+a_{n,k}v^{n}_{,m}+a_{n,m}v^{n}_{,k}\right]\left(y,t^{\prime}\right)\right\rangle_{0}
=\displaystyle= iτ2{2ymyn[vl(x,t)vn(y,t)v,lj(x,t)ak(y,t)+vl(x,t)ak(y,t)v,lj(x,t)vn(y,t)]\displaystyle-\frac{i\tau}{2}\left\{\frac{\partial^{2}}{\partial y^{m}\partial y^{n}}\left[\left\langle v^{l}\left(x,t\right)v^{n}\left(y,t^{\prime}\right)\right\rangle\left\langle v^{j}_{,l}\left(x,t\right)a_{k}\left(y,t^{\prime}\right)\right\rangle+\left\langle v^{l}\left(x,t\right)a_{k}\left(y,t^{\prime}\right)\right\rangle\left\langle v^{j}_{,l}\left(x,t\right)v^{n}\left(y,t^{\prime}\right)\right\rangle\right]\right.
+\displaystyle+ vl(x,t)an,k(y,t)v,lj(x,t)v,mn(y,t)+vl(x,t)v,mn(y,t)v,lj(x,t)an,k(y,t)+(km)}\displaystyle\left.\left\langle v^{l}\left(x,t\right)a_{n,k}\left(y,t^{\prime}\right)\right\rangle\left\langle v^{j}_{,l}\left(x,t\right)v^{n}_{,m}\left(y,t^{\prime}\right)\right\rangle+\left\langle v^{l}\left(x,t\right)v^{n}_{,m}\left(y,t^{\prime}\right)\right\rangle\left\langle v^{j}_{,l}\left(x,t\right)a_{n,k}\left(y,t^{\prime}\right)\right\rangle+\left(k\leftrightarrow m\right)\right\}

We replace eqs. (32) and (35)

δ2ΓQδA¯j(x,t)δP¯km(y,t)=iτ2θ(tt)d3k(2π)3ei𝒌(𝒙𝒚)\displaystyle\frac{\delta^{2}\Gamma_{Q}}{\delta\bar{A}_{j}\left(x,t\right)\delta\bar{P}^{km}\left(y,t^{\prime}\right)}=-i\frac{\tau}{2}\theta\left(t-t^{\prime}\right)\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\bm{k(x-y)}} (113)
[(kmkn)d3k(2π)3Δklne[κk+κ(kk)](tt)(kk)lΔ(kk)kjNk2κk\displaystyle\left[\left(k_{m}k_{n}\right)\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\Delta_{k^{\prime}}^{ln}e^{-\left[\kappa_{k^{\prime}}+\kappa_{\left(k-k^{\prime}\right)}\right]\left(t-t^{\prime}\right)}\left(k-k^{\prime}\right)_{l}\Delta^{j}_{\left(k-k^{\prime}\right)k}\frac{N_{k^{\prime}}}{2\kappa_{k^{\prime}}}\right.
+\displaystyle+ (kmkn)d3k(2π)3Δkjne[κk+κ(kk)](tt)klΔ(kk)klNk2κk\displaystyle\left(k_{m}k_{n}\right)\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\Delta_{k^{\prime}}^{jn}e^{-\left[\kappa_{k^{\prime}}+\kappa_{\left(k-k^{\prime}\right)}\right]\left(t-t^{\prime}\right)}k^{\prime}_{l}\Delta^{l}_{\left(k-k^{\prime}\right)k}\frac{N_{k^{\prime}}}{2\kappa_{k^{\prime}}}
+\displaystyle+ d3k(2π)3Δkjne[κk+κ(kk)](tt)klkmΔ(kk)nl(kk)kNk2κk\displaystyle\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\Delta_{k^{\prime}}^{jn}e^{-\left[\kappa_{k^{\prime}}+\kappa_{\left(k-k^{\prime}\right)}\right]\left(t-t^{\prime}\right)}k^{\prime}_{l}k^{\prime}_{m}\Delta^{l}_{\left(k-k^{\prime}\right)n}\left(k-k^{\prime}\right)_{k}\frac{N_{k^{\prime}}}{2\kappa_{k^{\prime}}}
+\displaystyle+ d3k(2π)3Δklne[κk+κ(kk)](tt)kmΔ(kk)nj(kk)l(kk)kNk2κk]+(km)\displaystyle\left.\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\Delta_{k^{\prime}}^{ln}e^{-\left[\kappa_{k^{\prime}}+\kappa_{\left(k-k^{\prime}\right)}\right]\left(t-t^{\prime}\right)}k^{\prime}_{m}\Delta^{j}_{\left(k-k^{\prime}\right)n}\left(k-k^{\prime}\right)_{l}\left(k-k^{\prime}\right)_{k}\frac{N_{k^{\prime}}}{2\kappa_{k^{\prime}}}\right]+\left(k\leftrightarrow m\right)

Since the kk^{\prime} integral is dominated by the infrared band, we approximate kkk^{\prime}\ll k. Observe that two of the integrals vanish from symmetry, so

δ2ΓQδA¯j(x,t)δP¯km(y,t)=iτ2θ(tt)d3k(2π)3ei𝒌(𝒙𝒚)eκk(tt)\displaystyle\frac{\delta^{2}\Gamma_{Q}}{\delta\bar{A}_{j}\left(x,t\right)\delta\bar{P}^{km}\left(y,t^{\prime}\right)}=-i\frac{\tau}{2}\theta\left(t-t^{\prime}\right)\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\bm{k(x-y)}}e^{-\kappa_{k}\left(t-t^{\prime}\right)}
[(kmkn)klΔ(k)kjd3k(2π)3ΔklnNk2κk+Δ(k)nlkkd3k(2π)3ΔkjnklkmNk2κk]+(km)\displaystyle\left[\left(k_{m}k_{n}\right)k_{l}\Delta^{j}_{\left(k\right)k}\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\Delta_{k^{\prime}}^{ln}\frac{N_{k^{\prime}}}{2\kappa_{k^{\prime}}}+\Delta^{l}_{\left(k\right)n}k_{k}\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\Delta_{k^{\prime}}^{jn}k^{\prime}_{l}k^{\prime}_{m}\frac{N_{k^{\prime}}}{2\kappa_{k^{\prime}}}\right]+\left(k\leftrightarrow m\right) (114)

We filter out a longitudinal term and use again the spherical symmetry to get

δ2ΓQδA¯j(x,t)δP¯km(y,t)=iτ2θ(tt)d3k(2π)3ei𝒌(𝒙𝒚)eκk(tt)(Δkjkm+Δmjkk)\displaystyle\frac{\delta^{2}\Gamma_{Q}}{\delta\bar{A}_{j}\left(x,t\right)\delta\bar{P}^{km}\left(y,t^{\prime}\right)}=-i\frac{\tau}{2}\theta\left(t-t^{\prime}\right)\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\bm{k(x-y)}}e^{-\kappa_{k}\left(t-t^{\prime}\right)}(\Delta^{j}_{k}k_{m}+\Delta^{j}_{m}k_{k})
[23k2k𝑑kE[k]+115k𝑑kk2E[k]]\displaystyle\left[\frac{2}{3}k^{2}\int^{k}dk^{\prime}\;E[k^{\prime}]+\frac{1}{15}\int^{k}dk^{\prime}\;k^{\prime 2}E[k^{\prime}]\right] (115)

E[k]E[k^{\prime}] is given in eq. (37). Since the kk^{\prime} integrals are cut off at kkk^{\prime}\leq k, the first integral is the dominant one. We observe that the integral diverges as kc0k_{c}\to 0, keeping only the most divergent term we get

k𝑑kE[k]CKϵ2/3kc2/3\int^{k}dk^{\prime}\;E[k^{\prime}]\approx\frac{C_{K}\epsilon^{2/3}}{k_{c}^{2/3}} (116)

We finally approximate

θ(tt)eκk(tt)δ(tt)κk\theta\left(t-t^{\prime}\right)e^{-\kappa_{k}\left(t-t^{\prime}\right)}\approx\frac{\delta\left(t-t^{\prime}\right)}{\kappa_{k}} (117)

whereby we find eqs. (48) and (49).

Appendix D Computation of the Non-Newtonian spectrum

In this Appendix we give details of the derivation of eq. (56). We begin by writing P[ω]P[\omega] from eq. (53) as

P[ω]=τ(ω+iκkν+)(ω+iκkν)P[\omega]=\tau(\omega+i\kappa_{k}\nu_{+})(\omega+i\kappa_{k}\nu_{-}) (118)

Both ν±\nu_{\pm} are real and positive; when τ0\tau\to 0, ν+1\nu_{+}\to 1 and ν\nu_{-}\to\infty; this shows we are dealing with a singular limit.

The integral over ω\omega in (56) is computed by contour methods and yields

dω(2π)|G[k,ω]|2=[1+κk2τ2ν+ν]2κk3τ2ν+ν(ν++ν)\int\frac{d\omega}{(2\pi)}\left|G\left[k,\omega\right]\right|^{2}=\frac{[1+\kappa_{k}^{2}\tau^{2}\nu_{+}\nu_{-}]}{2\kappa_{k}^{3}\tau^{2}\nu_{+}\nu_{-}(\nu_{+}+\nu_{-})} (119)

Comparing eqs. (53) and (118) we find

ν++ν\displaystyle\nu_{+}+\nu_{-} =\displaystyle= 1τκk[1+τζ(ϵk2)1/3]\displaystyle\frac{1}{\tau\kappa_{k}}[1+\tau\zeta(\epsilon k^{2})^{1/3}]
ν+ν\displaystyle\nu_{+}\nu_{-} =\displaystyle= 1τκk[1Λτνk2]\displaystyle\frac{1}{\tau\kappa_{k}}[1-\Lambda\tau\nu k^{2}] (120)

To obtain eq. (56) we substitute these expressions and develop the result to first order in Λ\Lambda.

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