Nonlinear Fluctuations in Relativistic Causal Fluids

In the Second Order Theories (SOT) of real relativistic fluids, the non-ideal properties of the flows are described by a new set of dynamical tensor variables. In this work we explore the non-linear dynamics of those variables in a conformal fluid. Among all possible SOTs, we choose to work with the Divergence Type Theories (DTT) formalism, which ensures that the second law of thermodynamics is fulfilled non-perturbatively. The tensor modes include two divergence-free modes which have no analog in theories based on covariant generalizations of the Navier-Stokes equation, and that are particularly relevant because they couple linearly to a gravitational field. To study the dynamics of this irreducible tensor sector, we observe that in causal theories such as DTTs, thermal fluctuations induce a stochastic stirring force, which excites the tensor modes while preserving energy momentum conservation. From fluctuation-dissipation considerations it follows that the random force is Gaussian with a white spectrum. The irreducible tensor modes in turn excite vector modes, which back-react on the tensor sector, thus producing a consistent non-linear, second order description of the divergence-free tensor dynamics. Using the Martin-Siggia-Rose (MSR) formalism plus the Two-Particle Irreducible Effective Action (2PIEA) formalism, we obtain the equations for the relevant two-point correlation functions of the model: the retarded propagator and the Hadamard function. The overall result of the self-consistent dynamics of the irreducible tensor modes at this order is a depletion of the relaxation time for the hard modes, a fact that suggests that tensor modes could sustain an inverse cascade of entropy.


Introduction
Fluid description of relativistic matter proved to be a powerful tool for a clearer understanding of high energy phenomena [1,2].Examples are the thermalization [3] and isotropization [4] of the quark-gluon plasma created in the Relativistic Heavy Ion Collider (RHIC) facilities; the behaviour of matter in the inner cores of Neutron Stars (NS) [5][6][7]; the state of the plasma around the cosmological phase transitions [8]; etc. where, in general, the features of the processes observed in those systems cannot be explained using ideal relativistic fluids.
Unlike non-relativistic hydrodynamics, where there is a successful theory to describe non-ideal fluids, namely the Navier-Stokes equation, there is no definite mathematical model to study real relativistic fluids.The history of the development of such theory begins with the recognition of the parabolic character of Navier-Stokes and Fourier equations1 [9], which implies that they cannot be naively extended to relativistic regimes.In fact, the first attempts by Eckart and Landau [10,11] to build a relativistic theory of dissipative fluids starting from the non-relativistic formulation, also encountered this pathology.
The paradox about the non-causal structure of Navier-Stokes and Fourier equations, known as First Order Theories (FOTs), was resolved phenomenologically in 1967 by I. Müller [12].He showed that by including second order terms in the heat flow and the stresses in the conventional expression for the entropy, it was possible to obtain a system of phenomenological equations which was consistent with the linearized form of Grad kinetic equations [13], i.e., equations that describe transient effects that propagate with finite velocities.These equations, constitute the so-called Second Order Theories (SOTs), whose main difference with respect to FOTs is that the stresses are upgraded to dynamical variables that satisfy a set of Maxwell-Cattaneo equations [14][15][16][17].Latter on, Müller's phenomenological theory was extended to the relativistic regime by W. Israel and others [18][19][20][21][22][23][24][25][26][27][28][29][30][31].
An improved, more systematic description of relativistic thermodynamics was introduced in 1986 by Liu, Müller and Ruggieri [32], who developed a field-like description of particle density, particle flux and energy-momentum components.The resulting field equations were the conservation of particle number, energy momentum and balance of fluxes, and were strongly constrained by the relativity principle, the requirement of hyperbolicity and the entropy principle.The only unknown functions of the formalism were the shear and bulk viscosities and the heat conductivity, and all propagation speeds were finite.Several years latter, Geroch and Lindblom extended the analysis of Liu et al. and wrote down a general theory were all the dynamical equations can be written as totaldivergence equations [33,34], see also Refs.[35][36][37][38][39][40].This theory, known as Divergence Type Theory (DTT) is causal in an open set of states around equilibrium states.Moreover, all the dynamics is determined by a single scalar generating functional of the dynamical variables, a fact that allows to cast the theory in a simple mathematical form.Besides the dynamical equations an extra four-vector current is introduced, the entropy four-current, whose divergence is non-negative and, by the sole virtue of the dynamical equations, is a function of the basic fields and not of any of their derivatives.This fact guarantees that the second law is automatically enforced at all orders in a perturbative development.In contrast, as Israel-Stewart-like theories must be built order by order, the Second Law of thermodynamics must be enforced in each step of the construction [41].In other words, DTTs are exact hydrodynamic theories that do not rely on velocity gradient expansions and therefore go beyond Israel-Stewart-like second-order theories.
The novelty of SOTs, is the introduction of tensor dynamical variables to account for non-ideal features of the flow.This means that besides the scalar (spin 0) and vector (spin 1) modes already present in Landau-Lifshitz or Eckart theories, it is possible to excite tensor (spin 2) perturbations.This fact enlarges the set of hydrodynamic effects that a real relativistic fluid can sustain.As is familiar for the gravitational field, the tensor sector can be further decomposed into scalar, vector, and divergence-free components.If present in the Early Universe plasma, the latter could excite primordial gravitational waves [42], or seed primordial electromagnetic fluctuations [43].Another scenario where tensor modes could play a relevant role are high energy astrophysical compact objects as, e.g., Neutron Stars [6].It is well known that rotational tensor normal modes of those stars can source gravitational waves, however at present there is no compelling hydrodynamical model of those objects that satisfactory models those modes.
Second Order Theories have been successfully applied to describe the thermalization and isotropization of the out-of-equilibrium flow created in RHICs.Besides these phenomenological applications, some steps were given toward the construction of a complete hydrodynamic theory of real relativistic fluids [3].Among all the possible lines to pursue this task, there is the study of the stochastic flows induced by the fluid's own thermal fluctuations.
The development of fluctuating hydrodynamics was pioneered by Landau and Lifshitz [11,[44][45][46][47], who applied the fluctuation dissipation theorem [48] to the Navier-Stokes equation.The extension of fluctuating hydrodynamics to the Eckart formulation of relativistic theory of fluids was done by Zimdahl [49,50] and the result applied to cosmology.Due to the above mentioned problems of FOTs, to which Eckart theory belongs, it was necessary to write down a fluctuating hydrodynamics for the Second Order Theories, where the pathologies of FOTs are absent.A first step toward that goal was done by Calzetta in 1998 [39], who characterized hydrodynamic fluctuations in DTTs theories.Forcing by thermal noise is relevant to the calculation of the transport coefficients of the fluid [51][52][53] as well as the phenomenon of long time tails [54][55][56][57][58]. Noise, whether thermal or not, can also play an important role in early Universe phenomena such as primordial magnetic field induction [59,60] and phase transitions [61,62] to cite a few.
In the literature on relativistic fluctuating hydrodynamics, no distinctions were made between scalar and vector fluctuations, not even tensor fluctuations were mentioned.So, as the distinctive feature of SOTs is that they sustain tensor modes, in this work we shall concentrate on the study of fluctuations in the pure tensor sector.Among all possible SOTs, we choose to work with DTTs.The reasons were quickly mentioned above: they are thermodynamically and relativistically consistent in arbitrary flows and independent of any approximations.Consequently in a perturbative development we do not have to worry about the fulfillment of the Second Law.
To address the subjet of study, we use effective field theory methods, which long ago began to be used in the study of turbulence [63][64][65], and continue to be a powerful tool to study random flows [66][67][68][69].Among all methods, the Effective Action formalism allows to express the different N-point correlation functions of the theory in terms of loop diagrams, which adds a new source of intuition in the intepretation of the correlations.In this manuscript we use the Two-Particle Irreducible Effective Action (2PIEA) formalism, through which we write down the evolution equations for the relevant two-point functions of the problem under study, namely the Retarded and the Hadamard propagators.This field method allows to build the propagators as the contribution of all closed interacting diagrams that cannot be separated by cutting two of their internal lines.
The paper is organized as follows.In section 2 we begin by quickly reviewing the Landau-Lifshitz hydrodynamics, after which we build the minimal conformal Divergence-Type-Theory beyond LL.We end this section by setting the criterion for incompressibility.In section 3 we give an abridged presentation of the fluctuation-dissipation theorem in DTTs consistent with a causal theory [39] and outline the Martin-Siggia-Rose formalism for the Two-Particle-Irreducible Effective Action.We briefly show how this formalism allows to write down evolution equations for the main propagators of the theory, namely the retarded (or causal) propagator and the Hadamard two-point function.In Section 4 we study the effect of linear fluctuations around an equilibrium state and find the lowest order causal propagators as well as the vector and tensor Hadamard propagators.In Section 5 we extended the analysis of Section 4 to nonlinear fluctuations of tensor modes in order to find the corrections that they induce in the propagators found within the linear approximation.In the free-streaming regime under consideration (τ → ∞) we find that the equation for the causal tensor propagator acquires a new term which renormalize the relaxation time introducing a scale dependence proportional to the fourth power of the momentum.In consequence the Hadamard correlation function is also modified in a way that shows a transition from a flat spectrum, for low values of p, to a power law spectrum (∝ p −4 ) with increasing p.We end this section by briefly discussing the effect of the tensor fluctuations on the mean value of the entropy, which is also depleted in the "large-p range" of the spectrum, arguing that this may be an indication of an inverse cascade of entropy.Finally in Section 6 we discuss the results we obtained, draw our main conclusions and suggest possible lines to pursue the study developed in this manuscript.We left for the Appendix A the discussion of the scaling law of the main diagrams.

The model
To make this manuscript self-contained, we begin this section with a condensed review of Landau-Lifshitz hydrodynamics for a conformal neutral fluid, in order to show that a FOT does not guarantee fulfillment of the Second Law of thermodynamics.Among all possible SOTs, we choose a DTT to build what is arguably the minimal extension of Landau-Lifshitz hydrodynamics which enforces the Second Law of thermodynamics non-perturbatively, and where the dynamics of the neutral fluid is given by the conservation laws of the energy-momentum tensor (EMT) T µν and of a third order tensor A µνρ that encodes the non-ideal properties of the flow.The theory is completed by considering an entropy current S µ whose conservation equation enforces the Second Law of thermodynamics.T µν is symmetric and traceless, and A µνρ is totally symmetric and traceless on any two indices [88].We linearize the evolution equations and find the propagation speed for the scalar, vector and tensor modes, from which we write down a criterion to define incompressibility.

Landau-Lifshitz hydrodynamics in a nut-shell
Let us consider the simplest model for a conformal fluid, for which there is no particle number current and the energy-momentum tensor is traceless.The energy density ρ is defined by the Landau prescription with normalization u 2 = −1.Observe that eq.(2.1) is also the definition of u µ as an eigenvector of T µν with eigenvalue −ρ.For an ideal fluid the energy momentum tensor must be isotropic in the rest frame, so where is the projector onto hypersurfaces orthogonal to u µ .Tracelessness implies the equation of state p = ρ 3 . (2.4) From the entropy density s = (ρ + p) /T LL we build the entropy flux with β µ LL = u µ /T LL and where subindex LL refers to Landau-Lifshitz frame.The differential form for the first law, ds = dρ/T LL , implies which gives that an ideal fluid flows with no entropy production, i.e., Besides from p = ρ/3 we have s = 4ρ/3T LL and ds/dρ = 1/T LL and we get ρ = σ SB T 4 LL , where σ SB is the Stefan-Boltzmann constant.
A real fluid departs from an ideal one in that now where Π µν encodes the non-ideal properties of the flow and satisfies Π µν u ν = 0.If we still consider S µ 0 to be the entropy flux, we now have Positive entropy production is satisfied if where σ µν is the shear tensor and η ∝ T 3 LL is the fluid viscosity.This constitutive relation leads to Landau-Lifshitz hydrodynamics, namely a convariant Navier-Stokes equation, which violates causality [23].
We may intend to solve the problem by upgrading Π µν to a dynamical variable and adopting a Maxwell-Cattaneo equation for it, having eq.(2.10) as an asymptotic limit.We then write Π µν = − ησ µν + τ Πµν . (2.12) This would follow from demanding positive entropy production with an entropy production term and identifying later on τ = ςη.There arises the problem of what is S µ .A natural choice would be which is thermodynamically satisfactory, but leads to The extra term may be expected to be small, as it is of third order in deviations from equilibrium, but it is not nonnegative definite, and so we cannot be certain that the Second Law is properly enforced.To guarantee that it is, we should go to higher order in eq.(2.12), a step that would stem from including a new higher order term in the expression (2.13), and then impose a condition equivalent to (2.14), and so on.In other words, in order to have a thermodynamically consistent hydrodynamics we should enforce the Second Law order by order in deviations from equilibrium.

Minimal conformal DTT beyond Landau-Lifshitz hydrodynamics
Instead of patching the theory one step at a time, DTTs attempt to formulate a consistent theory in its own right by postulating new currents, besides T µν , which together determine the entropy flux.In its simplest form there is only one further current, A µνρ , satisfying a divergence-type equation [88] where I µν is a tensor source of irreversibility.A simple count of degrees of freedom tells us that we need 5 independent equations to complement the 4 equations from the energy momentum conservation.We impose A µνρ to be totally symmetric and traceless on any two indices and take the transverse, traceless part of eq.(2.16) as providing the required equations.
The big assumption of DTTs is that we have a local First Law of the form with ζ µν a new tensor variable that encodes the non-ideal properties of the flow.In particular, this leads to So the Second Law is enforced as long as Another consequence of eq. ( 2.17) is that if we consider the Massieu function density [33,34] and further, the symmetry of T µν allows us to write Thus the theory is defined by specifying the scalar Φ and the tensor I µν as local functions of the vector β µ and the tensor ζ µν , subjected to eq. (2.19).Although a DTT may be derived from an underlying microscopic description, such as kinetic or field theory, when such is available, one of its appealing features is that one can go a long way into finding the right DTT from purely macroscopic arguments.In the case at hand of a conformal theory, and if we consider up to second order deviations from equilibrium, then the DTT is essentially unique, as we shall show below.We start by writing the generating function Φ in terms of 0, 1st and 2nd order in deviations from equilibrium In equilibrium ζ µν = 0, so where Thus the only choice is (2.30) For the first order terms we have whereby To obtain the right number of degrees of freedom for a conformal fluid we ask that for physically meaningful fields ζ λ λ = ζ λσ β σ = 0.When these conditions hold we shall say we are "on shell".As A µνρ must be symmetric on any pair of indices we have that φ 2 = 2φ 1 , and by demanding A µνρ to be traceless on any pair of indices Xφ 2 + 3φ 2 = 0.So with a a constant that sets the intensity of the first order deviations around equilibrium, whose precise value will depend on the specific system under study, and In writing second order terms, we leave out terms that do not contribute to the currents "on shell".This leaves and From symmetry of A µνρ we get φ 4 = 4φ 3 .So we can write with b a constant that sets the amplitude of the second order deviations around equilibrium.
Its precise value will depend on the specific system under study.The energy density then is The entropy current reads We now define where α is to be defined below, to get These constitutive relations define the theory.By comparing to the Landau-Lifshitz theory above we see that on dimensional grounds we may write X −1 = −T 2 , where T has dimensions of temperature, while Z µν is dimensionless.Writing β µ = u µ /T the constitutive relations take the form To fix the remaining constants we ask that the theory reproduces Landau-Lifshitz hydrodynamics to first order in deviations from equilibrium.This requires In the DTT framework, eq.(2.57) ought to follow from the transverse, traceless part of the first-order conservation law for A µνρ where is the complete transverse and traceless spatial projector.Therefore we must have It is convenient to introduce the relaxation time τ through the Anderson-Witting prescription [82] We may estimate η from the AdS/CFT bound [83], η ≥ (4/3) σ SB T 3 0 /4π, T 0 being a fiducial equilibrium temperature.In the next subsection we show that causality requires α 2 ≤ 2/3 and so T τ ≥ 3η/2σ SB T 3 0 ≥ 1/2π.In what follows we shall be interested in the "free streaming" limit τ → ∞, thus α → 0.
As we shall show in next section, when thermal fluctuations are considered I µν acquires a stochastic component, I µν → I µν + F µν .F µν is a stochastic source which may be derived from fluctuation-dissipation considerations and will be described in more detail below.Observe that this force sources entropy, and not energy, as there is no stirring in the equations for the conservation of T µν .

Propagation speeds and "incompressibility"
Before we proceed, we shall derive the propagation speeds for different types of linearized fluctuations around equilibrium [84], and show that in the α → 0 limit scalar modes may be regarded as frozen, thus the fluid behaves as an incompressible one.
We are looking at a situation where a discontinuity is propagating along the surface z = ct, c being the desired propagation speed.Above the front the fluid is in equilibrium, so T = T 0 = constant, u µ = U µ = (1, 0, 0, 0) and Z µν = 0. Hydrodynamic variables are continuous across the front.Observe that any variable X which remains constant at the front must obey Ẋ = −cX , where X = X ,3 , where subindex 3 refers to the z coordinate.Moreover the conditions u 2 = −1 and Z µν u ν = 0 show that u 0 ,ρ = Z µ0 ,ρ = 0, and the condition Z ρ ρ = 0 shows that Z a a = −Z 3 3 (indices a, b run from 1 to 2, which denote x and y coordinates).
The theory decomposes into tensor modes Z ab + (1/2) δ ab Z 3 3 , which do not propagate, vector modes u a and Z a3 and scalar modes T , u 3 and Z 33 .Writing the conservation equations T µν ,ν = 0 and Λ µν ρλ A ρλσ ,σ = 0 (since I µν = 0 at the front), and eliminating time derivatives, we get two sets of equations.For the vector modes which shows that vector modes propagate with speed For scalar modes we get which admits a non-propagating mode with u 3 = 0, T /T 0 = − (3/4) αZ 33 , and two propagating modes with speed So causality demands α 2 ≤ 2/3, and c S c V when α 2 2/3.This means that the only interaction of interest is between tensor and vector modes.

The Martin-Siggia-Rose effective action
As stated above, we shall be interested in fluids stirred by their own thermal fluctuations.Therefore in this section we shall review the fluctuation dissipation theorem appropriate to causal relativistic real fluids [39] (see also [85]), and then use the MSR formulation [70][71][72][73][74][75] to develop an effective action from where we can get the Dyson equations for the stochastic correlations of vector and tensor hydrodynamic variables.

Fluctuation-dissipation theorem in a DTT framework
In this subsection we summarize the derivation of the fluctuation-dissipation theorem as applied to causal relativistic fluid theories.Following Ref. [39] we define the following shorthand notation for the variables described above with Φ a the vector generating functional.The entropy production is given by To include thermal fluctuations we add random sources F B in the equations of motion which then become Langevin-type equations, namely A satisfactory theory must predict vanishing mean entropy production in equilibrium, so However, because the coincidence limit may not be well defined, we impose a stronger condition due to elementary causality considerations, which is for every space-like pair (x, x ).In the following we shall assume that we have defined time in such a way that x and x belong to the same equal time surface, namely t = t .In the linear approximation I B is a linear function of X C , then with Only the symmetrized derivative occurs in (3.11) due to the symmetry of the stochastic average.Assuming Gaussian white noise we have that the correlations between fluxes and noise is where t = t .The fluctuation dissipation theorem follows from whenever t = t , which implies or equivalently We use this version of the fluctuation-dissipation theorem in order to set the correlation function of the noise source.Of course, as we show in the main text, in the limit in which our DTT converges to the Landau-Lifshitz hydrodynamics, the correlation of the stochastic energy-momentum tensor converges to the well-known Landau-Lifshitz noise [44,45].
To verify Eq. (3.15), let us multiply both sides by the non-singular matrix where n a is the unit normal field to the equal time surface containing both x and x .In the linear approximation Φ a is quadratic on X A and In equilibrium we may apply the Einstein's formula, relating the thermodynamic potentials to the distribution function of fluctuations, to conclude that where δ (3) (x, x ) is the three-dimensional covariant delta function on the Cauchy surface.This is a generalized version of the equipartition theorem.On the other hand It is possible to write the equations of motion (3.8) as where L A involves the field variables on the surface, but not their normal derivatives, and The factor 1/2 takes into account the average of the derivative evaluated in x = x − and x = x + .Therefore (3.21) and (3.22) are equal.Due to the non-singularity of M AB , equation (3.15) holds.
In the case at hand, these results imply that only the equation for A µνρ acquires a random source, and then where

MSR and the 2PI Effective action
We now proceed with the analysis of the correlations in the theory.From the analysis of the propagation velocities we know that in the free streaming regime the scalar modes propagate faster than the vector ones and therefore can be considered as frozen.In other words, in the considered limit the flow may be regarded as "incompressible" (cfr.subsection (2.3)).The MSR formalism will allow to convert the problem of classical fluctuations into a quantum field theory one, for which we shall derive the Two-Particle-Irreducible Effective Action (2PIEA).The advantage of this formalism is that variations of this Action yields directly the Schwinger-Dyson equations for the propagators, instead of directly the propagators as would be the case in the One-Particle-Irreducible Effective Action formulation.Before going on an important remark is in order.In hydrodynamics there is no explicit single 'small' parameter, such as in quantum field theory, which could be used to organize the perturbative expansion.For this reason it has been propossed that the loop expansion should be understood as an expansion in 'the complexity of the interaction' [76], since due to the randomness of the stirring, the sum of higher order terms will tend to cancel.On the other hand, it is possible to identify the relevant small parameters in the theory through the scaling behavior of restricted sets of graphs.In the case at hand, this analysis suggests that the loop expansion is an expansion in powers of p 3 /(c 2 V σ SB T 3 ) (see Appendix A).In consequence the loop expasion is consistent while Let us return to the construction of the 2PIEA.We continue to use the abridged notation from eqs. (3.1)-(3.6).The equations of motion have the form P A = F A , where the P 's are the left hand sides of the EMT T µν and the nonequilibrium current A µνρ conservation equations.If the sources F A are given, we call X A [F ] the solution to the equations.Under thermal noise, all X A = 0. Therefore we can write a generating functional for the correlation functions where P F A is the Gaussian probability density for the sources, K AB are the currents introduced in the formalism to couple the variables of the theory and the integration is performed over all the noise realizations.Observe that where the determinant can be proved to be a constant [86] and will be consequently disregarded.We exponentiate the delta function by adding auxiliary fields Introducing the source correlations we finally obtain where In fact, we have mapped the stochastic hydrodynamic problem into a nonequilibrium field theory one, where S from eq. (3.33) plays the role of "classical" action.We may formally add new sources coupled to the auxiliary fields and consider the whole string X K = X A , Y A as degrees of freedom of the theory.The Legendre transform of the generating function is the 2PIEA Γ G JK , where the G JK = X J X K are the thermal correlations we seek.Once the 2PIEA is known, the actual correlations are obtained as extrema The 2PIEA has the structure [87] Γ = 1 2 where Γ 2Q is the sum of all two-particle irreducible Feynman graphs for a theory whose interactions are the terms cubic or higher in S, and carrying propagators G JK .It so happens that Y A Y B = 0, and also so actually we get two sets of equations, one for the retarded propagators and another for the actual thermal correlations (3.39)

Induced dynamics of tensor modes
We now begin to investigate the induced dynamic of tensor modes in the presence of thermal fluctuations.As we have seen, it is given by eqs.(3.37) for the causal, or retarded correlators and (3.38) for the symmetric, or Hadamard two-point functions.In equilibrium we have X K = 0.The MSR "classical" action (3.33) may be written as where S Q is quadratic and S I contains the interaction terms; in our case we only keep terms cubic in the fields in S C .The 2PIEA is given by eq.(3.35), with and where the G KL are the propagators and N ∝ DetG KL −1/2 [87].We use the notation The normalization is set up so that 1 = 1.If, as in our case, S C = 0, then the lowest order contribution to Γ 2Q is and then the self energies read The expectation value on the r.h.s. is developed in terms of Feynman graphs with propagators G KL in the internal legs, and where only 2PI graphs are considered [87].Again to lowest order, we may replace the full propagators by their lowest order approximations which describe the correlations of linearized fluctuations around equilibrium.The equations for nonlinear fluctuations (3.37) and (3.38) can then be written in compact form as

Linear fluctuations around equilibrium
From the discussion above, to formulate a MSR effective action for the minimal DTT, we consider a "classical" action of the form (cfr. eq.(3.33)) Since we are disregarding scalar modes, the auxiliary fields Y µ and Y µν contain only vector and tensor degrees of freedom.This means that the independent variables are a three vector Y j and a three tensor Y jk obeying Y j ;j = Y jk ;j = Y jk ;k = 0.The time components are then constrained through Observe that Y µ has units of T −1 , while Y µν has units of T −2 .Explicitly expr.(4.1) can be decomposed as S = S N + S T + S A + S I with 2)

Identifying the physical degrees of freedom
We now apply the above formalism to study thermal fluctuations around a fiducial equilibrium configuration with velocity U µ = (1, 0, 0, 0) and temperature T 0 .We shall keep only terms which are quadratic (S q ) or cubic (S c , not to be confused with the 'classical Action' referred above) in deviations from equilibrium.We shall derive the quadratic terms in the next subsection, and come back to the cubic terms later on, in subsection (5.1).We write This suggests taking V k = ∆ k ν V ν as independent variables, whereby Any transverse tensor admits a similar decomposition, namely where h.o.means 'higher orders'.Moreover Y k k = Z k k = 0. We can thus identify the quadratic and cubic terms in the "classical" action.The quadratic terms are and recalling that Our next step is to Fourier transform all degrees of freedom.We adopt the convention to get and We may simplify these expressions by using the linear equations of motion derived from the quadratic terms, namely where O ;0 refers to time derivative, to get Finally, we discriminate between vector and proper tensor modes by writing where p j Z j T = p j Y j T = p j Z jk TT = p j Y jk TT = 0.

The quadratic action
After separating vector and tensor proper modes, the quadratic terms decouple.For the vectors we get and for the tensors

The lowest order propagators
We now return to the conformal fluid case, where the string of fields X K may be split into physical vector fields V j and Z j T , auxiliary vector fields Y j and Y j T , the physical tensor proper field Z TT ij and the auxiliary tensor proper field Y TT ij .The correlations between a physical and an auxiliary field yield the causal propagators; if the physical field is to the left, then it is a retarded propagator.The correlations between physical fields are the symmetric correlations in the theory; the correlations between auxiliary fields vanish identically.To lowest order, we obtain decoupled equations for correlations involving only vector fields and those involving only tensor fields.

Causal vector correlations
The causal vector correlations are V j Y k and Z T j Y k on one hand, and V j Y T k and Z T j Y T k on the other.These two pairs are decoupled from each other.They all have the structure where The equations of motion for the whose solution is where c V = √ 3α/2 is the propagation speed for vector modes (cfr.subsection (2.3)), which in the limit we are interested satisfies c V 1.For the second pair V j Y T k and Z T j Y T k we obtain with solution In summary, the causal correlations of vector fields all have the structure 50) The different correlations are summarized in Table 1.

Causal tensor correlations
The only causal tensor correlation is where It obeys the equation Observe that the tensor modes do not propagate, and their momentum dependence is trivial.

Hadamard vector correlations
The vector correlations are They have the structure They may be derived from eq. (3.39).For example Explicitly, Similarly Now, from the equations for the propagators Since the integrand vanishes at the upper limit, we find The correlation G 1Z T Z T is even in t − t , and so there is no loss of generality in assuming t > t .In this case, we get In summary, the structure of vector correlations is where the projector S TT is defined in eq.(4.52), and We may check that in the limit τ → 0 we recover Landau-Lifshitz theory.Indeed in this limit we may approximate In this limit, the self correlation for the tensor proper part of the viscous EMT (cfr.eq.(2.54)) is where we have used eq.(2.62).We thus recover the Landau-Lifshitz result [44,45] in this limit.
5 Nonlinear fluctuations around equilibrium

Cubic terms in the "classical" action
We continue the discussion of the cubic terms in the "classical" action eq.(4.1).We shall not need the cubic terms which do not contain tensor modes.For those which contain tensor fields, we distinguish a) Terms that only contain Y TTjk (−p, t): These terms naturally split into two D jkrs (p, q) = δ rk δ js E jkrs (p, q) = q 2 + p l q l δ sj δ kr + q k (δ js p r + p s δ jr ) ( b) Terms that only contain Z TTlm (q, t): These terms also split as We shall not need the explicit form of the remaining terms: c) terms that contain both Y TTjk (−p, t) and Z TTlm (q, t), d) terms quadratic in Z TTlm (q, t).

Tensor self-energy
We now turn to the derivation of the self energy for tensor modes We may split the self energy into two contributions, one with only vector propagators in internal lines, and the other with one vector and one tensor modes.Our goal is to derive the momentum dependence of the self-energy and the noise kernel.Now, because the lowest order tensor propagators are momentum independent, the Feynman graphs containing them are momentum independent too.For this reason we shall not compute them.The self-energy, considering only the Feynman graphs with vector propagators in internal lines, is It is understood that after computing the functional derivatives one must project back onto tensor proper modes.The terms involving S Y TT V V are suppressed by one power of τ and will not be computed.Let us begin with Observe that Σ jklm vanishes unless t ≥ t , which we assume.Moreover, the graph in the second line of the right hand side vanishes when t = t and we shall not compute it.We then have where jklm = d 3 q E jkrs p, q F uvlm q , p P rv p − q P su q |p − q | q ω q ω p−q cos ω p−q t − t − ϕ p−q cos ω q t − t + ϕ q (5.11) -25 -On dimensional grounds, the tensor self energy has units of T 4 as it should.σ jklm has units of p 5 .
To obtain the true self energy we must project back on the transverse components, symmetric and traceless in jk and lm.If p lies in the z direction, this means we only need σ abcd , where the indices run from 1 to 2. Since there are no preferred directions, we will obtain σ (V 1) abcd = Aδ ac δ bd + Bδ ad δ bc + Cδ ab δ cd (5.12) Symmetrization on ab yields and removing the trace on ab we get where S TTabcd is the restriction to the case where p is on the third direction of the projector eq.(4.52).This means that the physical self energy takes the form where phys = 1 2 d 3 q W p, q |p − q | ω p−q cos ω p−q t − t − ϕ p−q cos ω q t − t + ϕ q (5.16) W p, q = 1 q ω q S TT ijkl [p] E jkrs p, q F uvlm q , p P rv p − q P su q (5.17) observe that W has units of p 2 .
To proceed, we shall make an important simplification.As we have seen, the "classical" equations for the correlations eqs.(4.43, 4.47, 4.53) are local in time.We assume that the main loop corrections to these equations are those that are local in time too.Therefore we shall seek only the singular terms in the self energy and the noise kernel, namely the terms which are proportional to δ (t − t ).Since the propagators themselves are regular functions, any such singular term can only result from the asymptotic, large |q | region of the integration domain.In this region ω q and ω p−q are both real.
To compute σ phys we keep only the terms which are slow and not zero in the coincidence limit cos ω p−q t − t − ϕ p−q cos ω q t − t + ϕ q 1 2 cos ω p−q − ω q t − t cos ϕ p−q cos ϕ q (5.18 phys ≈ 1 4 d 3 q W p, q |p − q | ω p−q cos ω p−q − ω q t − t cos ϕ p−q cos ϕ q (5.19) Assume again the p j = (0, 0, p).Then ω p−q − ω q vanishes when q = (q T , p/2), with q T in the x, y plane.We write q = (q T , p/2 + δq 3 ) to get (where explicit, q = (q T , p/2)) Evaluating the prefactors at δq 3 = 0 we may integrate over δq 3 to obtain the singular part of the self-energy where, in the free streaming limit τ → ∞, The remaining task is to compute (we neglect terms proportional to α 2 ) (5.23) We have integrals of the form (5.24) We compute them in the scheme of dimensional regularization in D = 2 − dimensions.Following [89,90] we get (5.25) As we see the limit → 0 is well defined for any integer n, so we take = 0 (D = 2) straightforwardly.Further in the free-streaming limit τ → ∞, we get We now consider the other graph Repeating the same steps as in the previous case we get phys we keep only the terms which are slow and not zero in the coincidence limit Finally, in the free-streaming limit, we get The total physical self-energy induced by vector fluctuations reads, in the free-streaming limit, Σ where γ Σ = 187 384 π 2 (2π) 3  (5.34)

Nonlinear tensor correlations
Adding the singular self-energy term to the classical equation (4.53) we get the one-loop corrected equation for the causal tensor propagator with p L in (3.27), being the largest value of p that makes the loop expansion consistent.Since c V τ ∼ τ /T , in the free-streaming limit (τ → ∞) we get p T p L .Therefore the expression (5.47) clearly shows a transition from a flat spectrum for p p T to a power law spectrum p −4 for p T p < p L .For the latter range we find (5.49)

Conclusions and Discussion
In this paper we began the study of the non-linear hydrodynamics of a real relativistic conformal fluid within the framework of Divergence Type Theories, which have the advantage that the Second Law of Thermodynamics is satisfied non-perturbatively.
In Second Order Theories such as DTTs, the fact that non-ideal effects are described by a new independent tensor variable permits to enlarge the set of hydrodynamic effects, as now quadrupolar oscillations represented by purely tensor modes are allowed in the flow, besides the scalar and vector ones already present in First Order theories.
This fact was previously exploited in [42] to investigate the induction of primordial gravitational waves by the presence of these modes in the Early Universe plasma, and also in [43] in the context of Early Universe magnetogenesis.
In this manuscript we began to develop the nonlinear hydrodynamics of real relativistic fluids by studying the effect that thermally induced tensor fluctuations have on vector fluctuations.
We consider a simple situation where tensor modes are excited by a Gaussian noise with a white spectrum.As was just said, this noise is due to the fluid own thermal fluctuations and the spectrum can be computed from the fluctuation-dissipation theorem.
Moreover, we show that the energy and momentum contents of the system are not affected by the noise and consequently, due to the structure of the theory, it can be thought as if entropy is added to the system, while keeping constant its energy content.This rather academic model allows to simplify the mathematics while retaining the most important features of the dynamics as e.g., the scaling properties of two point functions.
Using techniques borrowed from Quantum Field Theory to study non-linear hydrodynamics, such as the Two-Particle-Irreducible Effective Action and the Martin-Siggia-Rose formalism, we wrote down the evolution equations for the retarded and Hadamard propagators for both the vector and tensor sectors.We first found the lowest linear order expression for the two-point functions and latter non-linear fluctuations around equilibrium were considered.It was found that the non-linearities renormalized the relaxation time of the theory in a way that induces a depletion of the tensor correlations in the range p T p ≤ p L , with p L the biggest value of p for which the loop expansion is consistent.Stated otherwise, we found that tensor fluctuations have a flat spectrum for the largest scales, which turns to a power law p −4 spectrum in the small length-scale sector.
To lowest order, the correction to the entropy density is quadratic in the fluctuations and consequently it is also diminished in the large p range.This result suggests that tensor modes could sustain a turbulent inverse cascade of entropy [91], and we intend to study this issue in a forthcoming work.
Besides the studies mentioned just above, other systems where fluid tensor modes can play an important role are Neutron Stars [5][6][7] and Early Universe plasmas [8], to mention a few.In both systems, the fluids are non-ideal relativistic plasma.Therefore it is important to have a solid hydrodynamic theory in order to understand the features of those systems.This work is a small step toward that goal and sets the basis for more complete studies of tensor turbulence where energy injection can also be taken into account.