Non-universality of hydrodynamics

We investigate the effects of stochastic interactions on hydrodynamic correlation functions using the Schwinger-Keldysh effective field theory. We identify new"stochastic transport coefficients"that are invisible in the classical constitutive relations, but nonetheless affect the late-time behaviour of hydrodynamic correlation functions through loop corrections. These results indicate that classical transport coefficients do not provide a universal characterisation of long-distance, late-time correlations even within the framework of fluctuating hydrodynamics.

Hydrodynamics is often referred to as the "universal" low-energy effective description of many-body systems near thermal equilibrium. It is argued that if one waits long enough for all the high-energy "fast" modes to thermalise, the spectrum of a system can effectively be captured by the remaining "slow" modes associated with conserved operators (such as energy, momentum, and particle number). Fluctuations of conserved operators persist over long scales as they need to be transported out to infinity to thermalise. A hydrodynamic system is characterised by its fluxes expressed in terms of densities (or chemical potentials) and their derivatives, known as constitutive relations, with dynamics governed by the associated conservation equations.
It is known that this "classical" picture of hydrodynamics is incomplete. Hydrodynamic equations can be used to obtain the physically observable retarded correlation functions; see [1]. But these results can potentially be contaminated by interactions between the slow hydrodynamic modes and a background of fast modes [2,3]. A more complete picture is offered by the formalism of stochastic hydrodynamics, wherein the collective excitations of fast modes are modelled by random small-scale noise in the hydrodynamic equations [2][3][4][5][6]. The shortranged stochastic interactions are fine-tuned to reproduce the classical hydrodynamic results at tree level. However, consistently including loop corrections one finds that, for instance, the 2-point correlation function of fluid velocity has non-analytic behaviour in ω, referred to as "longtime tails", that is not predicted by classical hydrodynamics [7]. This formalism, however, is not exhaustive as the requirement to reproduce classical hydrodynamics does not uniquely fix the structure of stochastic interactions. Importantly, assuming these random interactions to be Gaussian, as is typically done, still leaves room for ambiguities. Physically, the ambiguities correspond to some high-energy "fast physics", that has been ignored at the classical level, leaking into the low-energy correlation functions via interactions. This would mean that, contrary to what is typically believed, the hydrodynamic transport coefficients do not universally characterise the low-energy spectrum of thermal systems.
The stochastic contamination in hydrodynamics can also be motivated from general considerations in thermal field theory. Fluctuation-dissipation theorems (FDT) imply that all the information in 2-and 3-point thermal correlation functions in a system can be captured by the respective retarded functions. However, for 4or higher-point correlations, retarded functions are no longer enough [8]. Classical hydrodynamics is only sensitive to tree-level retarded correlations of conserved operators, and is consequently blind to any information that might be contained in non-retarded higher-point correlation functions. These higher-point correlations can nonetheless affect the classically observable retarded functions of conserved densities and fluxes through loop corrections. The point of this letter is to make the above qualitative arguments precise and to explore the limits of hydrodynamics.
To probe these questions effectively, one needs a systematic prescription to include stochastic noise into the hydrodynamic framework. While classically, hydrodynamics is posed as a system of conservation equations, there now exists a complete effective field theory (EFT) for hydrodynamics based on the Schwinger-Keldysh (SK) formalism of thermal field theories [9][10][11]; see [12] for a review. The effective Kubo-Martin-Schwinger (KMS) condition in this formalism ensures that the FDT requirements are built into the EFT, and can be used to investigate stochastic signatures in hydrodynamic response functions. This formalism has recently been used to revisit long-time tails due to diffusive fluctuations at oneloop order in [13].
We argue that the effective action for hydrodynamics can be naturally organised into what we call "KMS blocks." The first KMS block contains all the information about fully retarded tree-level correlation functions, i.e. classical hydrodynamics, plus a minimal set of higherpoint interactions enforced by KMS conditions. Aligning with the expectations from thermal field theory, the second KMS block starts at the level of four-point interactions and contains all the residual information about correlation functions that are retarded in all but one momenta, and so on for higher KMS blocks. Interactions in the nth KMS block are typically suppressed with at least (2n − 1) derivatives compared to ideal hydrodynamics, therefore the first stochastic signatures creep into hydrodynamics at third derivative order. This signals a nonuniversality of higher-derivative corrections in hydrodynamics.
Stochastic interactions in simple diffusion.-For a concrete realisation of these ideas, we consider a simplified model with a single conserved density J t = n(µ), where µ is the associated chemical potential. Classical evolution of n is governed by its conservation (diffusion) equation ∂ µ J µ = 0 with J i = −σ(µ)∂ i µ, with diffusion constant given by D = σ(µ)/n ′ (µ). The conductivity σ(µ) is a non-negative classical "transport coefficient".
The EFT for diffusion is described by a phase field ϕ r and an associated stochastic noise field ϕ a [10]. We introduce background gauge fields φ r,a = (A r,aµ ) coupled to the noise and physical current operators O a,r = (J µ a,r ) respectively. The effective action S of the theory is constructed out of the background gauge invariant building blocks Φ r,a = (B r,aµ = A r,aµ + ∂ µ ϕ r,a ). Connected correlation functions of O r,a are computed via a path integral over the dynamical fields ψ = (ϕ r,a ), i.e. [14] where α = r, a andᾱ = a, r is its conjugate, while n a is the number of a type fields on the left. G raa computes the retarded function, while G rrr computes the symmetric one, with all the remaining combinations in between [15]. The theory is required to satisfy a set of SK constraints Here £ β denotes a Lie derivative along the thermal vector β µ = 1/T 0 δ µ t , with T 0 being the constant global temperature, and Θ = PT represents a discrete spacetime parity transformation. In particular, (2b) implements the inequality in the second law of thermodynamics, while the KMS symmetry (2c) implements FDT [16]. The theory also has a local phase shift symmetry Given these requirements, at leading order in derivatives, we find the effective Lagrangian [10] Here µ = B rt = ∂ t ϕ r + A rt . Given that σ is nonnegative, conditions (2a) and (2b) are trivially satisfied. The second term maps to itself under (2c), while the first term merely generates an additional total derivative term ∂ t p(µ) such that p ′ (µ) = n(µ). The classical diffusion equation can be recovered upon varying the action with respect to ϕ a , restricting to configurations with ϕ a = 0, and setting the background to A rµ = µ 0 δ t µ , A aµ = 0. While the action (4a) is sufficient to reproduce classical evolution, the formalism does allow for extra terms that are at least quadratic in noise fields and hence leave the classical dynamics untouched. For instance where ϑ 1,2 (µ) are arbitrary "stochastic coefficients". Each term here involves at least four fields, so the stochastic coefficients only contribute to 4-and higher-point nonfully-retarded correlation function at tree level, as argued in the introduction. For example, denoting "r" type fields by solid and "a" type by wavy lines, the partially-retarded function G rraa of n receives a tree-level stochastic contribution due to interactions in (4b) (see appendix) for p 1 = (ω, k, 0, 0) and p 2 = (ω, k cos θ, k sin θ, 0). Here D = σ/χ is the diffusion constant and χ = ∂n/∂µ is the susceptibility. Ellipsis denote further non-stochastic corrections due to terms in eq. (4a). One can use the retarded functions G raaa , G raa , G ra to cancel these terms, and obtain a Kubo formula for ϑ 1 , ϑ 2 using (5a).
Hereλ = ∂D/∂n + D/χ ∂χ/∂n. The middle ellipsis in (5) denote subleading terms coming from (4a), while the final ellipsis denote terms higher order in k 2 . Detailed calculations for finite k 2 are given in the appendix. These results illustrate that the hydrodynamic correlation functions start to receive higher-derivative corrections that are not fixed by the constitutive relations. We note that the ϑ 1 -contribution to the effective action (4b) is quadratic in the noise field. Gaussian noise can be captured by the conventional stochastic model, wherein one introduces a random microscopic term r i in the flux J i = −σ(µ)∂ i µ + r i . Correlation functions are obtained by path integrating over r i weighted by a Gaussian factor exp(−1/4 d 4 x r i r j λ ij [µ]) [17]. Imposing FDT constrains the form of the coefficient λ ij in terms of hydrodynamic transport coefficients. At leading order in derivatives, FDT uniquely fixes λ ij = δ ij /(T 0 σ(µ)). However, this uniqueness is violated by higher derivative corrections pertaining to stochastic coefficients, such as ϑ 1,2 , that are not fixed by FDT. For example, the ϑ 1 -term from (4b) appears as Stochastic interactions in hydrodynamics.-The EFT for full relativistic hydrodynamics is considerably more involved, but the discussion of stochastic interactions follows a similar flow. In addition to the phase pair ϕ r , ϕ a associated with density fluctuations, the theory also contains the Lagrangian coordinates σ A=0,...,3 of the fluid elements and respective noise X µ a as fundamental fields associated with energy-momentum fluctuations [10] [18]. We take σ 0 to define the local rest frame associated with the global thermal state. The thermal vector β µ is no longer a constant, but is given by β µ = 1/T 0 ∂x µ (σ(x))/∂σ 0 (x). Introducing background fields φ r,a = (g r,aµν , A r,aµ ) coupled to noise and physical energy-momentum tensor/charge current operators O a,r = (T µν a,r , J µ a,r ) respectively, the correlation functions can be computed by (1), with the path integral over ψ = (ϕ r,a , σ A , X µ a ). The building blocks for the respective effective action S, besides β µ , are (see [12]) Denoting Φ r,a = (G r,aµν , B r,aµ ), the SK constraints and phase shift symmetry are still given by (2), (3). Expressing S = d 4 x √ −g r L, up to leading order in derivatives, the effective action for relativistic hydrodynamics satisfying these requirements is given as where ∆ µν = g µν r + u µ u ν . Velocity u µ (with u µ u µ = −1), temperature T , and chemical potential µ are defined via u µ /T = β µ , µ/T = β µ B rµ . Energy density ǫ, pressure p, number density n, viscosities η, ζ, and conductivity σ are functions of T , µ. They satisfy dp = sdT + ndµ, ǫ + p = T s + µn for entropy density s. Condition (2b) requires η, ζ, σ to be non-negative. Classical conservation equations of hydrodynamics are obtained by varying the action with respect to X µ a , ϕ a in a configuration with X µ a = ϕ a = 0, and setting the background to g rµν = η µν , A rµ = µ 0 δ t µ , g aµν = A aµ = 0.
Similar to (4b), the full hydrodynamic action can also be modified with arbitrary stochastic terms based on the symmetries of the theory. For instance we have with ϑ i being a few stochastic coefficients; we do not perform the exhaustive counting exercise here.
Contributions from stochastic interactions in (7b) to hydrodynamic response functions can be computed similar to (5). We leave this analysis for future work. We note, however, that non-stochastic interactions in the simplified diffusion model only set in at one-derivative order as opposed to full non-linear hydrodynamics where momentum/velocity fluctuations in (7a) lead to ideal-order interactions; see [17]. Since part of the derivative suppression of stochastic signatures in (5b), (5c) arises from non-stochastic vertices, we expect this suppression to be relaxed in full hydrodynamics.
The stochastic coefficients ϑ i also arise in the context of non-relativistic (Galilean) hydrodynamics, in complete analogy with its relativistic counterpart. The quantitative details can be worked out along the lines of [19].
KMS blocks.-In our discussion so far, we introduced stochastic terms in the effective action by hand. In this section, we discuss a general procedure to construct such terms and, in doing so, classify the generic structure of stochastic interactions admissible by the hydrodynamic EFT. For a generic thermal system, the effective Lagrangian can be organised as where the nth "KMS block" L n contains the most generic terms involving n number of "a" fields allowed by symmetries, plus a set of terms with higher number of "a" fields required in order to satisfy KMS/FDT requirements. By definition, classical dynamics of the system, and treelevel fully retarded correlation functions G ra...a , are completely characterised by L 1 . Higher KMS blocks L n for n > 1 contain stochastic interactions that contribute to tree-level non-fully-retarded correlators G r...ra...a involving at least n instances of "r" type operators. The decomposition (8) is not unique; we can always redefine a KMS block with terms from higher KMS blocks. Such ambiguity in L 1 is precisely the non-universality of classical hydrodynamics. Nonetheless, we provide a particular characterisation of KMS blocks for the hydrodynamic effective Lagrangian satisfying the requirements (2). Condition (2a) implies that the L can be arranged in a power series in Φ a starting from the linear term. We start with a parametrisation (see appendix) . (9) Here D m are a set of totally-symmetric real multi-linear maps, allowing m arguments, made out of Φ r and β µ .
Here ×n denotes n identical arguments. For instance, the diffusive Lagrangian (4) corresponds to the choice for arbitrary vectors W µ , X µ , Y µ , Z µ . Recall that µ = B rt and Φ r,a = (B r,aµ ). This form is particularly convenient because each term in the summation is closed under KMS: the n instances of Φ a , Φ a + i£ β Φ r map to each other up to Θ, and Φ a + i/2 £ β Φ r maps to itself. Requiring (9) to respect (2b) and (2c) (up to a total derivative), we find for some vector N µ 0 . Note that changing any argument of D m from Φ a to £ β Φ r flips its Θ-parity, therefore its contribution to L is generically not Θ-even. For (10), these are satisfied with N µ 0 = p(µ)β µ (such that p ′ (µ) = n(µ)) and σ(µ) ≥ 0. Note that only D 1,2,3 from (9) can contribute to the classical constitutive relations. These generically satisfy an emergent second law of thermodynamics for some S µ ; see appendix. (11) guarantees the positivity of entropy production, within the derivative expansion. Generically, D n contain all structures allowed by symmetries at a given derivative order. We refer to the contribution of each such structure in the effective action (9) as a "KMS group". Each KMS group is independently invariant under the KMS transformation. The "nth KMS block" can be defined as the set of all KMS groups wherein the least nonzero power of Φ a fields is n. Inspecting (9), it follows that each group in D n falls at least in the ⌊n/2⌋-th KMS block. Here ⌈n/2⌉ and ⌊n/2⌋ denote ceiling and floor functions. We say "at least" because there can be groups in D n that identically vanish (up to a total derivative) when one or more of their arguments are £ β Φ r , e.g. ϑ 1,2 contribution in D 2 in (10). As seen from (9), such groups are pushed to higher KMS blocks. To account for these subtleties, we can decompose D n = n m=0 D n,m , where D n,m can be thought of as the component of D n with m of its arguments projected transverse to £ β Φ r . More precisely, it contains groups from D n that do not vanish for up to n − m instances of £ β Φ r in their arguments, but vanish for n − m + 1 instances. In particular, D n,0 does not vanish for any number of £ β Φ r , while D n,n vanishes for even one. Note that D 1,0 = 0 due to (11).
Plugging this decomposition into (9), we can work out the first KMS block explicitly as which completely characterises classical hydrodynamics. The first two terms correspond to "adiabatic" transport, as they do not contribute to entropy production in (12). Their respective contribution to the constitutive relations is Θ-even and Θ-odd respectively. The last two terms correspond to Θ-odd and Θ-even "dissipative" transport leading to entropy production in (12). From our examples in (4b) and (7b), ǫ, p, n ∈ D 1,1 and η, ζ, σ ∈ D 2,0 ; other contributions show up at higher order in derivatives. Technically, D 3,1 also appears in (13), but can be pushed to L 2 by redefining D 1,1 ; see appendix.
The first non-trivial stochastic corrections to classical hydrodynamics come from the 2nd KMS block. Up to leading order in derivatives, this is given as At this point, we are unable to ascertain any physical distinction between various contributions. In our examples, (ϑ 1 + 2 3 ϑ 2 ), ϑ 3 ∈ D 2,2 , while ϑ 2 , ϑ 4 ∈ D 4,0 . Higher KMS blocks can be worked out in a similar manner.
The count derivative ordering in hydrodynamics, we use the canonical scheme from [10], where Φ r ∼ O(∂ 0 ) and Φ a , £ β Φ r ∼ O(∂ 1 ). Projecting m arguments against £ β Φ r in D n,m requires the introduction of m copies of £ β Φ r . Hence, the contribution of D n,m to L, and to the hydrodynamic observables, is typically suppressed with O(∂ n+m−1 ) compared to the ideal order thermodynamic terms in D 1,1 . Consequently, effects of stochastic KMS blocks L n for n > 1 are suppressed with O(∂ 2n−1 ) compared to L 1 , in addition to any loop-suppression.
Outlook.-Hydrodynamics is an effective theory and an immensely successful one at that. However, like any effective theory, it has a limited scope of applicability. It posits that the low-energy dynamics of a many-body thermal system can be effectively captured by the long-range transport properties of its conserved charges. While it is certainly true to a leading approximation, short-range stochastic interactions must be included into the framework to consistently describe interactions between hydrodynamic modes. In this letter, we took this fine-print a step further and investigated the sensitivity of hydrodynamics to the choice of stochastic interactions.
We used the EFT framework of hydrodynamics developed recently [10] and identified new "stochastic transport coefficients" in (4b) and (7b) characterising shortrange information. The stochastic coefficients do not enter the classical constitutive relations, but nonetheless affect retarded correlation functions in the hydrodynamic regime at subleading orders in derivatives through loop interactions. We explicitly derived the stochastic signatures in 2-and 3-point retarded functions for diffusive fluctuations in (3 + 1) dimensions in (5). In particular, we found the stochastic correction to 3-point function to be non-analytic in frequency at one-loop order. It is worth noting that these results are different from the usual "long-time tails" as the effects we are describing are characterised by entirely new transport coefficients which are invisible in classical constitutive relations. Finally, we classified the general structure of stochastic interactions through KMS blocks. Classical physics is completely characterised by the first KMS block, while the higher KMS blocks characterise a plethora of stochastic coefficients.
We conclude that the sensitivity of hydrodynamic observables to short-range stochastic information signifies a breakdown of the celebrated universality of hydrodynamics in describing long-range correlations. It would be interesting to find physical systems where the signatures of stochastic coefficients are enhanced enough to overcome the loop suppression. As we already discussed in the letter, part of this could be achieved by revisiting the computation in the presence of momentum modes in Galilean or relativistic hydrodynamics. The stochastic signatures are also enhanced in lower spatial dimensions. We plan to return to these questions in the future.
This work was supported in part by the NSERC Dis-

Linearised fluctuations in diffusion EFT
In this appendix we give details of loop calculations in diffusion EFT. Analysis for full hydrodynamics proceeds in a similar manner.
Linearised action.-To compute various correlation functions, we need to expand the effective action orderby-order in self interactions. It is convenient to work with the density n as a fundamental degree of freedom instead of µ. We can expand the Lagrangian (4a) up to forth order in the fields δn = n − n(µ 0 ) and ϕ a , in the absence of background fields, to obtain Here χ = ∂n/∂µ is the susceptibility and D = σ/χ is the diffusion constant, along with In a typical diffusive model, ω ∼ k 2 . Taking D, χ, T 0 ∼ 1, and noting that L ∼ k d ω ∼ k d+2 , we can infer that ϕ a , δn ∼ k d/2 . Therefore, higher order interactions in δn and ϕ a are successively more irrelevant in k and can be consistently dropped within the derivative expansion. This form of the diffusive action was recently derived in [13]. The coefficients λ 4 andλ 4 are denoted as λ ′ andλ ′ in [13]; we reserve primes to denote derivatives with respect to µ. For the stochastic Lagrangian (4b), we get the first non-trivial contribution as in (A1) is the free Lagrangian and leads to the tree propagators where p = (ω, k) and F (p) = ω + iDk 2 . We denote δn by solid and ϕ a by wavy lines. The remaining terms in (A1) and (A3) lead to various interaction vertices Stochastic vertices from (A3) are denoted in bold. The respective Feynman rules can be read off from directly from (A1) and (A3). Energy-momentum conservation at each vertex is understood. One final piece of information that we need is the coupling to sources. As we shall only be interested in correlation functions of density, we only keep the scalar sources A r,a t , and truncate to forth order in A r,a t , δn, ϕ a , We do not get any contribution from L 2 . The first two terms in L 2pt 1s are the usual linear couplings between fundamental fields and sources, while the remaining nonlinear couplings can be represented by the vertices We have denoted A rt by dotted and A at by dashed lines. Vertices in the last line only couple to sources and lead to contact terms in the correlation functions. The associated Feynmann rules can be obtained from (A6). We will not need the four point interactions in (A7) in the following calculation, but we have enlisted them anyway for completeness. We can now utilise (1) to compute various correlation functions order-by-order in loops. We are working with the conventions of [10] for the definition of correlation functions. These are related to the conventions of [8] as G WH α... = i/2 (−1) na (−2i) nr G α... and those of [17] as G K α... = (−1) na G α... . We note that the free propagators δn(p)ϕ a (−p) 0 and δn(p)ϕ a (−p) 0 in [13] have incorrect overall signs compared to our (A4). To account for this, their one-loop results should be modified with λ → −λ and λ ′ → −λ ′ ; these are reviewed below. Note that the "ra" type propagator δn(p)ϕ a (−p) 0 in (A4) is purely retarded while the "ar" type propagator ϕ a (p)δn(−p) 0 is purely advanced, which is a generic feature of these EFTs. This allows us to ignore any diagrams that contain a loop made entirely of "ra" or entirely of "ar" propagators, as they trivially drop out upon performing the frequency integral with a KMS consistent renormalisation scheme [20]. Another fact to note is that the "rr" propagator δn(p)δn(−p) 0 can be decomposed into a retarded and advanced piece as seen in (A4), which is essentially the statement of FDT [8]. This allows us to drop any diagrams with a single "rr" propagator closed in a loop, as the loop integral splits into a purely retarded and a purely advanced piece and trivially drops out. We shall not enlist such diagrams in our discussion below.
One-loop 2-point function.-Let us start the discussion with one-loop corrections to the retarded 2-point function. At this order there are no possible diagrams involving a stochastic vertex. However, it is still helpful to revisit the contribution coming from hydrodynamic diagrams to set up some ground work (see [13]). Let us parametrise the loop corrections to δn(p)ϕ a (−p) 0 in (A4) as Σ(p) can be understood as a momentum-dependent correction to the diffusion constant. We have two diagrams that can possibly contribute to this process It is straightforward to compute these and obtain the one-loop correction In obtaining the second equality, we have expanded the second term in the brackets into a retarded and advanced piece. The term purely retarded in p ′ drops out of the integral. The integration has been be performed with a hard momentum cutoff and cutoff-dependent terms have been ignored; see (A34). We can also compute the respective contribution to the retarded two-point correlation function G ra . Using (1) and (A6) we can parametrise it as Here δσ(p) is seen as correction to conductivity in the language of [13], while Γ(p) is the physically measurable correction to the two-point function. The . . . in the second line represents contributions involving the source couplings from (A7), given by diagrams The first diagram contributes a contact term, while the remaining two diagrams follows along (A10) leading to In total, we find the correction to the correlation function One can also read out δσ using (A11) We see that the physical observables exhibit non-analytic behaviour due to factors of (k 2 − 2iω/D) 1/2 . These are the so called "long-time tails" in diffusion model. These results at k = 0 were originally derived in [13] (the long-time tails at k = 0 is a simpler exercise, see e.g. [17]). Instead of using the generating functional, [13] computed the symmetric function G rr first and used Kramers-Kronig relations and FDT to obtain G ra .
One-loop 3-point function.-Moving to the case at point, we want to probe the signatures of stochastic vertices in hydrodynamic correlation functions. The simplest place to look at turns out to be the three-point "raa" correlator. At tree level, this correlator is controlled by the hydrodynamic interaction coupling λ through where p 2 + p 3 = p 1 and δλ(p 1 ; p 2 , p 3 ) parametrises possible loop corrections. At one loop order, there is only one diagram involving a stochastic vertex that contributes to this process Introducing a hard momentum cutoff, we can compute its contribution to δλ as Refer (A34) for the explicit integral. Again, the cutoff dependent terms have been ignored. We find a non-analytic correction coming from the stochastic coefficients ϑ 1 , ϑ 2 . Note that there can still be other corrections at this loop order coming from hydrodynamic vertices that we have not taken into account here. We can probe the correction δλ using the three-point retarded function G raa . Using (1) in conjunction with (A6), and keeping track of all the contact terms, we can find that Similar to (A11), the . . . contributions in the second line arise due to non-linear background couplings in (A7). These generate tree-level terms in the last step due to (A21) There are, however, no analogous one-loop diagrams involving a stochastic vertex and background fields.
as reported in (5c). Ellipsis in the middle denote higher derivative non-stochastic contributions that we have not computed here, while ellipsis at the end denote terms further suppressed in k 2 .
Two-loop 2-point function.-The quest for stochastic signatures in 2-point functions is considerably more involved. As there are no stochastic contributions at oneloop order, we need to go to two-loops to get a non-trivial effect. Focusing on the "ra" propagator (A8), we find 7 independent qualifying diagrams, but only 2 actually they combine to give an analytic correction. Accordingly, it is expected that non-analytic stochastic contributions will kick in the two-point function at three-loop order.
We can combine these expressions with the one-loop results and expand in k 2 ≪ ω/D limit. Through a straightforward computation, we find reproducing (5b). Middle ellipsis denote further nonstochastic corrections to the correlator that we have not considered here.
Tree-level 4-point functions.-Let us finally look at the stochastic contributions to tree-level 4-point functions. It is clear that stochastic vertices do not contribute to retarded functions at tree-level. However they do contribute to non-retarded ones. For instance, we can compute the contribution to the partially-retarded 4-point function G rraa through the diagram This evaluates to Note that there are no possible contact terms due to (A7) in this computation. Ellipsis denote that we have ignored any contribution coming from non-stochastic interactions. Let us take p 1 = p 4 = (ω, k 1 ) and p 2 = p 3 = (ω, k 2 ) with |k 1 | = |k 2 | = k and k 1 · k 2 = k 2 cos θ, corresponding to two propagating density packets intersecting each other at angle θ. This reduces to G rraa = . . . + 2ω 2 k 4 (ω + iDk 2 ) 4 2ϑ 2 cos 2 θ − ϑ 1 sin 2 θ .
There can be other contributions coming from nonstochastic hydrodynamic vertices as well.
Integrals.-The following loop integrals were used in the calculation above. The integrals have been performed with a hard momentum cutoff and cutoff-dependent terms have been ignored. We have Here p = (ω, k), p ′ = (ω ′ , k ′ ), and F (p) = ω + iDk 2 . Round brackets denote symmetrisation over all the enclosing indices, divided by the number of permutations.

Details of KMS block manipulations
In this appendix we elaborate on some calculational details regarding the derivation of KMS blocks.
We note that the most general Lagrangian consistent with (2a) can be arranged in a power series in Φ a starting from the linear term with appropriate powers of i to account for the imaginary part Here F m are a set of totally symmetric real multi-linear maps, with allowing m arguments, made out of Φ r and β µ . The "under-brace" notation represents m identical arguments. From here, one can check that (9) follows by a resummation of (A35) with where c mn is an invertible matrix given by c mn = (−1) n/2+1 n/2 n−m for n even, (−1) (n+1)/2 m+1 n+1 (n+1)/2 n−m for n odd.
To illustrate the second law of thermodynamics, we note that we can use integration by parts to define This equation essentially says that if D n acts on Φ a as a derivative operator, we can always extract out Φ a by adding total derivatives. This allows us to derive the classical constitutive relations performing Euler-Lagrange derivatives of (9) and setting "a" type fields to zero Note that operators higher than n = 3 do not appear in the constitutive relations. In terms of these, the entropy current is defined as We can reuse eq. (A38), with Φ a replaced by £ β Φ r , and the classical conservation equations ∇ µ T µν = F νρ J ρ , ∇ µ J µ = 0 to derive the second law statement in (12).
To derive the KMS blocks in (13) and (14), we expand D n into components (A41) Here "•" is a placeholder for an arbitrary argument of the operator. As mentioned in the main text, D n,m can be seen as the component of D n which has m of its arguments projected transverse to £ β Φ r . We can insert this into (9) The summation maxes out at s = 2n − m as D 2n,m vanishes for higher number of £ β Φ r in its arguments. For m ≤ n the summation never reaches this limit and D 2n,m for m ≤ n falls in the nth KMS block. D 2n,m for m > n, however, has the least power of Φ a to be m and falls in the mth KMS block. Similarly for D 2n+1,m (n = 0), we find D 2n+1,m (Φ a + i 2 £ β Φ r , Φ a , . . .
For m ≤ n, we never hit this limit and hence D 2n+1,m for m ≤ n falls in the nth KMS block. For m > n, however, we find that D 2n+1,m falls in the mth KMS block. The same result also applies to D 1,1 which falls in the 1st KMS block. We can now rearrange (A42) according to the contributions to KMS blocks and obtain L = (iD 2n,m +D 2n+1,m ) .
This is still a Θ-even linear operator that vanishes upon replacing its argument with £ β Φ r , as required by the KMS structure. Note that we cannot push D 2,1 to L 2 in a similar manner because that would require shifting D 1,1 with a Θ-odd term. The remaining terms in (A46) make up the first KMS block given in (13). Similarly, we can work out the second term in the series in (A45), including D 3,1 shifted from the previous term, and obtain iD 2,2 (Φ a , Φ a ) + D 3,2 (Φ a , Φ a , Φ a + 3i 2 £ β Φ r ) + 2 m=0 iD 4,m (Φ a , Φ a , Φ a +i£ β Φ r , Φ a +i£ β Φ r ) We can again shift to push D 4,2 , D 5,1 , and D 5,2 to L 3 and obtain the full expression for L 2 as The expression for L 2 given in (14) in the main text has been truncated to the leading derivative order. This procedure can be in principle be iterated to obtain higher KMS blocks as well.