\section{Introduction}

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 \emph{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~\cite{1963AnPhy..24..419K}. But
these results can potentially be contaminated by interactions between the slow
hydrodynamic modes and a background of fast
modes~\cite{Martin:1973zz,Pomeau:1974hg}. A more complete picture is offered by
the formalism of \emph{stochastic hydrodynamics}, wherein the collective
excitations of fast modes are modelled by random small-scale noise in the
hydrodynamic equations~\cite{Martin:1973zz, Pomeau:1974hg,
  Hohenberg:1977ym, DeDominicis:1977fw, Khalatnikov:1983ak}. The short-ranged
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 $\omega$, referred to as ``long-time tails'', that is
not predicted by classical hydrodynamics~\cite{Forster:1976zz}.

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 \emph{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
4- or higher-point correlations, retarded functions are no longer
enough~\cite{Wang:1998wg}. 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~\cite{Haehl:2015uoc, Crossley:2015evo,
  Jensen:2017kzi}; see~\cite{Glorioso:2018wxw} 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 one-loop order
in~\cite{Chen-Lin:2018kfl}.

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 higher-point 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 $n$th 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 non-universality of
higher-derivative corrections in hydrodynamics.

\vspace{1em}

\emph{Stochastic interactions in simple diffusion.}---For a concrete realisation
of these ideas, we consider a simplified model with a single conserved density
$J^t = n(\mu)$, where $\mu$ is the associated chemical potential. Classical
evolution of $n$ is governed by its conservation (diffusion) equation
$\dow_\mu J^\mu = 0$ with $J^i = -\sigma(\mu)\dow^i\mu$, with diffusion constant
given by $D = \sigma(\mu)/n'(\mu)$. The conductivity $\sigma(\mu)$ is a
non-negative classical ``transport coefficient''.

The EFT for diffusion is described by a phase field $\varphi_r$ and an
associated stochastic noise field $\varphi_a$~\cite{Crossley:2015evo}. We
introduce background gauge fields $\phi_{r,a} = (A_{r,a\mu})$ coupled to the
noise and physical current operators $\mathcal{O}_{a,r} = (J^\mu_{a,r})$
respectively. The effective action $S$ of the theory is constructed out of the
background gauge invariant building blocks
$\Phi_{r,a} = (B_{r,a\mu} = A_{r,a\mu}+\dow_\mu\varphi_{r,a})$. Connected
correlation functions of $\mathcal{O}_{r,a}$ are computed via a path integral
over the dynamical fields $\psi = (\varphi_{r,a})$, i.e. \footnote{We follow the
  conventions of~\cite{Crossley:2015evo} for correlation functions.}
\begin{align}
  G_{\alpha\ldots}
  = i^{n_a}\lb \frac{-i\delta}{\delta\phi_{\bar \alpha}} \ldots \rb
  \ln
  \int\mathcal{D}\psi\, \exp(i S),
  \label{eq:GreensFunctions}
\end{align}
where $\alpha=r,a$ and $\bar\alpha = a,r$ 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~\cite{Bellac:2011kqa}. The theory is required to satisfy a set of SK
constraints
\begin{subequations}
  \begin{gather}
    \hspace{-0.5em}
    S[\Phi_r,\Phi_a {=} 0;\beta]=0,~
    S[\Phi_r,-\Phi_a;\beta] = -S^*[\Phi_r,\Phi_a;\beta], \hspace{-0.6em}
    \label{eq:SK1} \\
    \Im S[\Phi_r,\Phi_a;\beta] \geq 0, 
    \label{eq:SK2} \\
    S[\Phi_r,\Phi_a;\beta]
    = S[\Theta\Phi_r,\Theta\Phi_a + i\Theta\lie_\beta\Phi_r; \Theta\beta].
    \label{eq:KMS}
  \end{gather}
  \label{eq:SKsymm
\end{subequations}
Here $\lie_\beta$ denotes a Lie derivative along the thermal vector
$\beta^\mu = 1/T_0\,\delta^\mu_t$, with $T_0$ being the constant global
temperature, and $\Theta = \text{PT}$ represents a discrete spacetime parity
transformation. In particular, \eqref{eq:SK2} implements the inequality in the
second law of thermodynamics, while the KMS symmetry \eqref{eq:KMS} implements
FDT \footnote{We have restricted to the KMS symmetry in the statistical (small
  $\hbar$) regime.}.  The theory also has a local phase shift symmetry
\begin{equation}
  \varphi_r(x) \to \varphi_r(x) - \lambda(x)
  ~~\text{such that}~~
  \beta^\mu \dow_\mu \lambda(x) = 0.
  \label{eq:fluid-spacetime-U1}
\end{equation}

Given these requirements, at leading order in derivatives, we find the effective
Lagrangian~\cite{Crossley:2015evo}
\begin{subequations}
  \begin{align}
    \mathcal{L}_{1}
    &= n(\mu) B_{at}
      + i T_0\sigma(\mu) B_{ai} \lb B^i_a + i \lie_\beta B^i_r \rb,
      \label{eq:diff-Lcl} 
  \end{align}
  Here $\mu = B_{rt} = \dow_t\varphi_r + A_{rt}$. Given that $\sigma$ is
  non-negative, conditions \eqref{eq:SK1} and \eqref{eq:SK2} are trivially
  satisfied. The second term maps to itself under \eqref{eq:KMS}, while the
  first term merely generates an additional total derivative term
  $\dow_t p(\mu)$ such that $p'(\mu) = n(\mu)$. The classical diffusion equation
  can be recovered upon varying the action with respect to $\varphi_a$,
  restricting to configurations with $\varphi_a = 0$, and setting the background
  to $A_{r\mu} = \mu_0 \delta^t_\mu$, $A_{a\mu}=0$.
 
 
 
 
 
 
 
 

  While the action \eqref{eq:diff-Lcl} 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
  \begin{align}
    \mathcal{L}_{2}
    &= iT_0^2\vartheta_1(\mu) B_{ai}B_{aj}
      \lb \lie_\beta B_r^i \lie_\beta B_{r}^j
      - \delta^{ij} \lie_\beta B_r^k \lie_\beta B_{rk} \rb \nn\\
    & \hspace{-1em}
      + iT_0^2\vartheta_2(\mu) B^i_{a} B_{ai}
      \lb B^j_a \,{+}\, i\lie_\beta B^j_r \rb 
      \lb B_{aj} \,{+}\, i\lie_\beta B_{rj} \rb ,
      \label{eq:diff-Lst}
  \end{align}
  \label{eq:diff-L
\end{subequations
where $\vartheta_{1,2}(\mu)$ are arbitrary ``stochastic coefficients''. Each
term here involves at least four fields, so the stochastic coefficients only
contribute to 4- and higher-point non-fully-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 \eqref{eq:diff-Lst} (see appendix)
\begin{subequations}
\begin{equation}
  \raisebox{-20pt}{\tikz[thick]{
      \draw [right] (-7mm,7mm)--(-3.5mm,3.5mm);
      \draw [aux] (-3.5mm,3.5mm)--(0,0);
      \node at (-7mm,3mm) {$p_1$};
      \draw [right] (-7mm,-7mm)--(-3.5mm,-3.5mm);
      \draw [aux] (-3.5mm,-3.5mm)--(0,0);
      \node at (-7mm,-3mm) {$p_2$};
      \draw [aux] (7mm,7mm)--(3.5mm,3.5mm);
      \draw [right] (0mm,0)--(3.5mm,3.5mm);
      \node at (7.5mm,3mm) {$p_2$};
      \draw [aux] (7mm,-7mm)--(3.5mm,-3.5mm);
      \draw [right] (0mm,0)--(3.5mm,-3.5mm);
      \node at (7.5mm,-3mm) {$p_1$};
      \draw[fill] (0,0) circle (.4ex);
    }}\quad
  \begin{aligned}
    G_{rraa}
    &= \ldots + \frac{2\omega^2 k^4}{(\omega+iDk^2)^4} \times \\
    & \quad \lb 2\vartheta_2 \cos^2\theta - \vartheta_1 \sin^2\theta \rb.
  \end{aligned}
  \label{eq:kubo}
\end{equation}
for $p_1 = (\omega,k,0,0)$ and $p_2 = (\omega,k\cos\theta,k\sin\theta,0)$. Here
$D = \sigma/\chi$ is the diffusion constant and $\chi = \dow n/\dow\mu$ is the
susceptibility. Ellipsis denote further non-stochastic corrections due to terms
in \cref{eq:diff-Lcl}. One can use the retarded functions $G_{raaa}$, $G_{raa}$,
$G_{ra}$ to cancel these terms, and obtain a Kubo formula for $\vartheta_1$,
$\vartheta_2$ using \eqref{eq:kubo}.

Although stochastic coefficients do not contribute to fully-retarded correlation
functions at tree-level, they do show up in the loop corrections such as
\begin{equation*}
  \raisebox{-10pt}{\tikz[thick]{
      \draw [right] (-10mm,0)--(-5mm,0);
      \draw [aux] (-5mm,0)--(0,0);
      \node at (-5mm,-3mm) {$p$};
     
      \draw [right] (0,0) arc (180:90:6mm);
      \draw [aux] (6mm,6mm) arc (90:0:6mm);
     
     
      \draw [right] (0,0) arc (180:270:6mm);
      \draw [aux] (6mm,-6mm) arc (-90:0:6mm);
     
     
      \draw[fill] (12mm,0) circle (.4ex);
     
      \draw [right] (12mm,0) arc (180:90:6mm);
      \draw [aux] (18mm,6mm) arc (90:0:6mm);
     
     
      \draw [right] (12mm,0) arc (180:270:6mm);
      \draw (18mm,-6mm) arc (-90:0:6mm);
     
     
      \draw [right] (24mm,0)--(29mm,0);
      \draw [aux] (29mm,0)--(34mm,0);
      \node at (29mm,-3mm) {$p$};
    }} \quad
  \raisebox{-20pt}{\tikz[thick]{
    \draw [right] (-10mm,0)--(-5mm,0);
    \draw [aux] (-5mm,0)--(0,0);
    \node at (-5mm,-3mm) {$p$};
   
    \draw [right] (0,0) arc (180:90:6mm);
    \draw [aux] (6mm,6mm) arc (90:0:6mm);
   
   
    \draw [right] (0,0) arc (180:270:6mm);
    \draw [aux] (6mm,-6mm) arc (-90:0:6mm);
   
    \draw[fill] (12mm,0) circle (.4ex);
   
    \draw [right] (12mm,0)--(15.5mm,3.5mm);
    \draw [aux] (15.5mm,3.5mm)--(19mm,7mm);
    \node at (15mm,8mm) {$p/2$};
   
    \draw [right] (12mm,0)--(15.5mm,-3.5mm);
    \draw [aux] (15.5mm,-3.5mm)--(19mm,-7mm);
    \node at (15mm,-8mm) {$p/2$};
  }}
\end{equation*}
Here $p=(\omega,k,0,0)$. Stochastic vertices from \eqref{eq:diff-Lst} are
denoted in bold. We find that the retarded 2-point function of $n$ behaves in
$k^2 \ll \omega/D$ limit as (see appendix)
  \begin{align}
    \frac{\omega}{k^2} \Im G_{ra}
    &= \chi D
      + \frac{\chi^2\lambda^2 T_0}{32\pi D^{3/2}} \omega^{1/2}k^2
      + \ldots \nn\\
    &\qquad
      - \frac{\lambda^2 T_0 (\frac23\vartheta_1+\vartheta_2)}
      {1024\pi^2 D^4} \omega^2 k^4
      + \ldots.
      \label{eq:intro-G2}
  \end{align}
  Here $\lambda = \dow D/\dow n$. The leading correction to $G_{ra}$ dictated by
  the constitutive relations goes as $\omega^{1/2}k^2$~\cite{Chen-Lin:2018kfl},
  while the leading correction due to stochastic coefficients goes as
  $\omega^2k^4$. The retarded 3-point function, on the other hand, receives a
  non-analytic stochastic correction
  \begin{equation}
    -\frac{2\omega^2}{k^4} \Re G_{raa}
    = \chi^2 D \tilde{\lambda}
    + \ldots
    + \frac{\lambda(\frac23\vartheta_1+\vartheta_2)}{32 \pi
      D^{5/2}} \omega^{5/2} + \ldots.
   
   
    \label{eq:intro-G3}
  \end{equation
  \label{eq:intro-results
\end{subequations}
Here $\tilde\lambda = \dow D/\dow n + D/\chi\, \dow\chi/\dow n$. The middle
ellipsis in \eqref{eq:intro-results} denote subleading terms coming from
\eqref{eq:diff-Lcl}, 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 \emph{not fixed by the constitutive
  relations}.

We note that the $\vartheta_1$-contribution to the effective action
\eqref{eq:diff-Lst} 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 $J^i = -\sigma(\mu)\dow^i\mu +
r^i$. Correlation functions are obtained by path integrating over $r^i$ weighted
by a Gaussian factor
$\exp(-1/4 \int \df^4 x\, r^i r^j
\lambda_{ij}[\mu])$~\cite{Kovtun:2012rj}. Imposing FDT constrains the form of
the coefficient $\lambda_{ij}$ in terms of hydrodynamic transport
coefficients. At leading order in derivatives, FDT uniquely fixes
$\lambda_{ij} = \delta_{ij}/(T_0\sigma(\mu))$. However, this uniqueness is
violated by higher derivative corrections pertaining to stochastic coefficients,
such as $\vartheta_{1,2}$, that are \emph{not} fixed by FDT. For example, the
$\vartheta_1$-term from \eqref{eq:diff-Lst} appears as
$\lambda_{ij} \sim -\vartheta_1(\mu)/(T_0\sigma(\mu))^2(\dow_i\mu\dow_j\mu -
\delta_{ij}\dow_k\mu\dow^k\mu)$.

\vspace{1em}

\emph{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 $\varphi_r$,
$\varphi_a$ associated with density fluctuations, the theory also contains the
Lagrangian coordinates $\sigma^{\sA=0,\ldots, 3}$ of the fluid elements and
respective noise $X_a^\mu$ as fundamental fields associated with energy-momentum
fluctuations~\cite{Crossley:2015evo}~\footnote{This is the so-called ``physical
  spacetime'' formulation of hydrodynamic EFT. See~\cite{Glorioso:2018wxw} for
  more details.}. We take $\sigma^0$ to define the local rest frame associated
with the global thermal state. The thermal vector $\beta^\mu$ is no longer a
constant, but is given by
$\beta^\mu = 1/T_0\, \dow x^\mu(\sigma(x))/\dow \sigma^0(x)$. Introducing
background fields $\phi_{r,a} = (g_{r,a\mu\nu},A_{r,a\mu})$ coupled to noise and
physical energy-momentum tensor/charge current operators
$\mathcal{O}_{a,r} = (T^{\mu\nu}_{a,r},J^\mu_{a,r})$ respectively, the
correlation functions can be computed by \eqref{eq:GreensFunctions}, with the
path integral over $\psi = (\varphi_{r,a},\sigma^\sA,X_a^\mu)$. The building
blocks for the respective effective action $S$, besides $\beta^\mu$, are
(see~\cite{Glorioso:2018wxw})
\begin{gather}
  B_{r\mu} = A_{r\mu} + \dow_\mu\varphi_r,\quad
  B_{a\mu} = A_{a\mu} + \dow_\mu\varphi_a + \lie_{X_a}A_{r\mu}, \nn\\
  G_{r\mu\nu} = g_{r\mu\nu}, \quad
  G_{a\mu\nu} = g_{a\mu\nu} + \lie_{X_a} g_{r\mu\nu}.
\end{gather}
Denoting $\Phi_{r,a} = (G_{r,a\mu\nu},B_{r,a\mu})$, the SK constraints and phase
shift symmetry are still given by \eqref{eq:SKsymm},
\eqref{eq:fluid-spacetime-U1}. 

Expressing $S = \int\df^4x\sqrt{-g_r}\,\mathcal{L}$, up to leading order in
derivatives, the effective action for relativistic hydrodynamics satisfying
these requirements is given as
\begin{subequations}
  \begin{align}
    \mathcal{L}_1
    &= \half \lb \epsilon\,u^\mu u^\nu
      + p\,\Delta^{\mu\nu} \rb G_{a\mu\nu}
      + n\, u^\mu B_{a\mu} \nn\\
    &\hspace{-1.5em}
      + \frac{iT}{4}\lb 2\eta\Delta^{\mu\rho}\Delta^{\nu\sigma}
      {+} (\zeta{-}{\textstyle\frac{2}{d}}\eta) \Delta^{\mu\nu}\Delta^{\rho\sigma}\rb
      G_{a\mu\nu} \lb G_{a\mu\nu}{+}i\lie_\beta G_{r\mu\nu} \rb \nn\\
    & + iT\sigma\, \Delta^{\mu\nu} B_{a\mu}
      \lb B_{a\nu} + i\lie_{\beta}B_{r\nu} \rb,
      \label{eq:hydro-Lcl}
  \end{align}
  where $\Delta^{\mu\nu} = g_r^{\mu\nu} + u^\mu u^\nu$.  Velocity $u^\mu$ (with
  $u^\mu u_\mu = -1$), temperature $T$, and chemical potential $\mu$ are defined
  via $u^\mu/T = \beta^\mu$, $\mu/T = \beta^\mu B_{r\mu}$. Energy density
  $\epsilon$, pressure $p$, number density $n$, viscosities $\eta$, $\zeta$, and
  conductivity $\sigma$ are functions of $T$, $\mu$. They satisfy
  $\df p = s\df T + n\df\mu$, $\epsilon+p=Ts+\mu n$ for entropy density
  $s$. Condition \eqref{eq:SK2} requires $\eta$, $\zeta$, $\sigma$ to be
  non-negative. Classical conservation equations of hydrodynamics are obtained
  by varying the action with respect to $X^\mu_a$, $\varphi_a$ in a
  configuration with $X^\mu_a = \varphi_a = 0$, and setting the background to
  $g_{r\mu\nu} = \eta_{\mu\nu}$, $A_{r\mu} = \mu_0 \delta^t_\mu$,
  $g_{a\mu\nu} = A_{a\mu} = 0$.
 
 
 
 
 

  Similar to \eqref{eq:diff-Lst}, the full hydrodynamic action can also be
  modified with arbitrary stochastic terms based on the symmetries of the
  theory. For instance we have
  \begin{align}
    \mathcal{L}_{2}
    &= iT^2\vartheta_1
      \lb \Delta^{\mu\rho}\Delta^{\nu\sigma} -
      \Delta^{\mu\nu}\Delta^{\rho\sigma}\rb
      \lie_\beta B_{r\rho} \lie_\beta B_{r\sigma}\,
      B_{a\mu}B_{a\nu} \nn\\
    & \hspace{-1.5em}
      + iT^2\vartheta_2 \Delta^{\mu\nu} \Delta^{\rho\sigma}
      B_{a\mu} B_{a\nu}
      \lb B_{a\rho} \,{+}\, i\lie_\beta B_{r\rho} \rb 
      \lb B_{a\sigma} \,{+}\, i\lie_\beta B_{r\sigma} \rb  \nn\\
    & \hspace{-1.5em}
      + iT^2\vartheta_3 \lb
      \Delta^{\mu\mu'}\Delta^{\nu\nu'}\Delta^{\rho\rho'}\Delta^{\sigma\sigma'}
      - \Delta^{\mu\rho}\Delta^{\nu\sigma}\Delta^{\mu'\rho'}\Delta^{\nu'\sigma'}
      \rb
      \nn\\
    & G_{a\mu\nu}G_{a\rho\sigma}\, \lie_\beta G_{r\mu'\nu'} \lie_\beta G_{r\rho'\sigma'} \nn\\
    & \hspace{-1.5em}
      + iT^2\vartheta_4
      \Delta^{\mu\nu}\Delta^{\mu'\nu'}\Delta^{\rho\sigma}\Delta^{\rho'\sigma'}
      G_{a\mu\nu}G_{a\mu'\nu'} \nn\\
    & \lb G_{a\rho\sigma}+i\lie_\beta G_{r\rho\sigma} \rb
      \lb G_{a\rho'\sigma'}+i\lie_\beta G_{r\rho'\sigma'} \rb
      + \ldots,
      \label{eq:hydro-Lst}
  \end{align
\end{subequations
with $\vartheta_i$ being a few stochastic coefficients; we do not perform the
exhaustive counting exercise here.

Contributions from stochastic interactions in \eqref{eq:hydro-Lst} to
hydrodynamic response functions can be computed similar to
\eqref{eq:intro-results}. 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 \eqref{eq:hydro-Lcl} lead to ideal-order
interactions; see~\cite{Kovtun:2012rj}. Since part of the derivative suppression
of stochastic signatures in \eqref{eq:intro-G2}, \eqref{eq:intro-G3} arises from
non-stochastic vertices, we expect this suppression to be relaxed in full
hydrodynamics.

The stochastic coefficients $\vartheta_i$ 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~\cite{Jain:2020vgc}.



\vspace{1em}

\emph{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
\begin{equation}
  \mathcal{L} = \sum_{n=1}^\infty \mathcal{L}_n,
  \label{eq:KMS-Block-decomposition}
\end{equation}
where the $n$th ``KMS block'' $\mathcal{L}_n$ 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 tree-level fully
retarded correlation functions $G_{ra\ldots a}$, are completely characterised by
$\mathcal{L}_1$.  Higher KMS blocks $\mathcal{L}_n$ for $n>1$ contain stochastic
interactions that contribute to tree-level non-fully-retarded correlators
$G_{r\ldots ra\ldots a}$ involving at least $n$ instances of ``$r$'' type
operators. The decomposition \eqref{eq:KMS-Block-decomposition} is not unique;
we can always redefine a KMS block with terms from higher KMS blocks. Such
ambiguity in $\mathcal{L}_1$ 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
\eqref{eq:SKsymm}.

Condition~\eqref{eq:SK1} implies that the $\mathcal{L}$ can be arranged in a
power series in $\Phi_a$ starting from the linear term. We start with a
parametrisation (see appendix)
\begin{align}
  \mathcal{L}
  &= \cD_1(\Phi_a)
    + i \sum_{n=1}^\infty
    \cD_{2n}(\underbrace{\Phi_a,\ldots}_{\times n},
    \underbrace{\Phi_a {+} i \lie_\beta\Phi_r,
    \ldots}_{\times n}) \nn\\
  &\hspace{-0.5em}
    + \sum_{n=1}^\infty
    \cD_{2n+1}(\Phi_a{+}{\textstyle\frac{i}{2}}\lie_\beta\Phi_r,
    \underbrace{\Phi_a,\ldots}_{\times n},
    \underbrace{\Phi_a {+} i \lie_\beta\Phi_r,\ldots}_{\times n}).
    \label{eq:L-G}
\end{align}
Here $\cD_m$ are a set of totally-symmetric real multi-linear maps, allowing $m$
arguments, made out of $\Phi_r$ and $\beta^\mu$. Here $\times n$ denotes $n$
identical arguments. For instance, the diffusive Lagrangian \eqref{eq:diff-L}
corresponds to the choice
\begin{align}
  \mathcal{D}_1(X)
  &= n(\mu) X_t, \nn\\
  \mathcal{D}_2(X,Y)
  &= T_0\sigma(\mu) X_i Y^i
    + T_0^2\lb \vartheta_1(\mu) + {\textstyle\frac{2}{3}}\vartheta_2(\mu)\rb  \nn\\
  &\qquad
    (\lie_\beta
B^i_r \lie_\beta B^j_r - \delta^{ij}\lie_\beta B^k_r\lie_\beta B_{rk}) X_iY_j,
    \nn\\
  \mathcal{D}_4(W,X,Y,Z)
  &= T_0^2\vartheta_2(\mu)\,\delta^{(ij}\delta^{kl)}\, W_iX_jY_kZ_l,
    \label{eq:diff-maps}
\end{align}
for arbitrary vectors $W_{\mu}$, $X_{\mu}$, $Y_{\mu}$, $Z_{\mu}$. Recall that
$\mu = B_{rt}$ and $\Phi_{r,a}=(B_{r,a\mu})$. This form is particularly
convenient because each term in the summation is closed under KMS: the $n$
instances of $\Phi_a$, $\Phi_a+i\lie_\beta\Phi_r$ map to each other up to
$\Theta$, and $\Phi_a+i/2\,\lie_\beta\Phi_r$ maps to itself. Requiring
\eqref{eq:L-G} to respect \eqref{eq:SK2} and \eqref{eq:KMS} (up to a total
derivative), we find
\begin{gather}
  \cD_1(\lie_\beta\Phi_r) = \nabla_\mu\mathcal{N}_0^\mu, \qquad
  \cD_m(\Phi_a,\cdots)~\text{are $\Theta$-even}, \nn\\
  \cD_{2}(\Phi_a,\ldots)\big|_{\text{0-derivative}} \geq 0,
  \label{eq:KMS.condition.1}
\end{gather}
for some vector $\mathcal{N}_0^\mu$. Note that changing any argument of
$\mathcal{D}_m$ from $\Phi_a$ to $\lie_\beta\Phi_r$ flips its $\Theta$-parity,
therefore its contribution to $\mathcal{L}$ is generically \emph{not}
$\Theta$-even. For \eqref{eq:diff-maps}, these are satisfied with
$\mathcal{N}_0^\mu = p(\mu)\beta^\mu$ (such that $p'(\mu) = n(\mu)$) and
$\sigma(\mu)\geq0$.  Note that only $\mathcal{D}_{1,2,3}$ from \eqref{eq:L-G} can
contribute to the classical constitutive relations. These generically satisfy an
emergent second law of thermodynamics
\begin{equation}
  \nabla_\mu S^\mu = \cD_2(\lie_\beta\Phi_r,\lie_\beta\Phi_r)
  + \frac{1}{2} \mathcal{D}_3(\lie_\beta\Phi_r,
  \lie_\beta\Phi_r,\lie_\beta\Phi_r),
  \label{eq:EC}
\end{equation}
for some $S^\mu$; see appendix. \eqref{eq:KMS.condition.1} guarantees the
positivity of entropy production, within the derivative expansion.

Generically, $\cD_n$ contain all structures allowed by symmetries at a given
derivative order. We refer to the contribution of each such structure in the
effective action \eqref{eq:L-G} as a ``KMS group''. Each KMS group is
independently invariant under the KMS transformation. The ``$n$th KMS block''
can be defined as the set of all KMS groups wherein the least nonzero power of
$\Phi_a$ fields is $n$. Inspecting \eqref{eq:L-G}, it follows that each group in
$\cD_n$ falls at least in the $\lfloor n/2\rfloor$-th KMS block. Here
$\lceil n/2\rceil$ and $\lfloor n/2\rfloor$ denote ceiling and floor
functions. We say ``at least'' because there can be groups in $\cD_n$ that
identically vanish (up to a total derivative) when one or more of their
arguments are $\lie_\beta\Phi_r$, e.g. $\vartheta_{1,2}$ contribution in
$\mathcal{D}_2$ in \eqref{eq:diff-maps}. As seen from \eqref{eq:L-G}, such
groups are pushed to higher KMS blocks.  To account for these subtleties, we can
decompose $\cD_{n} = \sum_{m=0}^{n} \cD_{n,m}$, where $\cD_{n,m}$ can be thought
of as the component of $\cD_{n}$ with $m$ of its arguments projected transverse
to $\lie_\beta \Phi_r$. More precisely, it contains groups from $\mathcal{D}_n$
that do not vanish for up to $n-m$ instances of $\lie_\beta\Phi_r$ in their
arguments, but vanish for $n-m+1$ instances. In particular, $\cD_{n,0}$ does not
vanish for any number of $\lie_\beta\Phi_r$, while $\cD_{n,n}$ vanishes for even
one. Note that $\cD_{1,0} = 0$ due to \eqref{eq:KMS.condition.1}. 

Plugging this decomposition into \eqref{eq:L-G}, we can work out the first KMS
block explicitly as
\begin{align}
  \mathcal{L}_1
  &= \mathcal{D}_{1,1}(\Phi_a)
    + i\mathcal{D}_{2,1}(\Phi_a,\Phi_a{+}i\lie_\beta\Phi_r) \nn\\
    &\quad
      + i\mathcal{D}_{2,0}(\Phi_a,\Phi_a{+}i\lie_\beta\Phi_r) \nn\\
  &\quad
    + \mathcal{D}_{3,0}(\Phi_a{+}{\textstyle\frac{i}{2}}\lie_\beta\Phi_r,
    \Phi_a,\Phi_a{+}i\lie_\beta\Phi_r),
    \label{eq:L1-general}
\end{align}
which completely characterises classical hydrodynamics. The first two terms
correspond to ``adiabatic'' transport, as they do not contribute to entropy
production in \eqref{eq:EC}. Their respective contribution to the constitutive
relations is $\Theta$-even and $\Theta$-odd respectively. The last two terms
correspond to $\Theta$-odd and $\Theta$-even ``dissipative'' transport leading
to entropy production in \eqref{eq:EC}. From our examples in \eqref{eq:diff-Lst}
and \eqref{eq:hydro-Lst}, $\epsilon,p,n\in\mathcal{D}_{1,1}$ and
$\eta,\zeta,\sigma\in\mathcal{D}_{2,0}$; other contributions show up at higher
order in derivatives. Technically, $\mathcal{D}_{3,1}$ also appears in
\eqref{eq:L1-general}, but can be pushed to $\mathcal{L}_2$ by redefining
$\mathcal{D}_{1,1}$; 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
\begin{align}
  \mathcal{L}_2
  &= i\cD_{22}(\Phi_a,\Phi_a{+}i\lie_\beta\Phi_r)
    + \mathcal{D}_{3,1}(\Phi_a,\Phi_a,
    \Phi_a{+}{\textstyle\frac{3i}{2}}\lie_\beta\Phi_r)
    \nn\\
  &\hspace{-1em}
    + i\cD_{4,0}(\Phi_a,\Phi_a,\Phi_a{+}i\lie_\beta\Phi_r,\Phi_a{+}i\lie_\beta\Phi_r)
    +\mathcal{O}(\dow^5).
    \hspace{-0.1em}
    \label{eq:L2-general}
\end{align}
At this point, we are unable to ascertain any physical distinction between
various contributions. In our examples,
$(\vartheta_1+\frac{2}{3}\vartheta_2),\vartheta_3\in\mathcal{D}_{2,2}$, while
$\vartheta_2,\vartheta_4\in\cD_{4,0}$. Higher KMS blocks can be worked out in a
similar manner.

The count derivative ordering in hydrodynamics, we use the canonical scheme
from~\cite{Crossley:2015evo}, where $\Phi_r\sim\mathcal{O}(\dow^0)$ and
$\Phi_a,\lie_\beta\Phi_r\sim\mathcal{O}(\dow^1)$. Projecting $m$ arguments
against $\lie_\beta\Phi_r$ in $\mathcal{D}_{n,m}$ requires the introduction of
$m$ copies of $\lie_\beta\Phi_r$. Hence, the contribution of $\mathcal{D}_{n,m}$
to $\mathcal{L}$, and to the hydrodynamic observables, is typically suppressed
with $\mathcal{O}(\dow^{n+m-1})$ compared to the ideal order thermodynamic terms
in $\mathcal{D}_{1,1}$. Consequently, effects of stochastic KMS blocks
$\mathcal{L}_n$ for $n>1$ are suppressed with $\mathcal{O}(\dow^{2n-1})$
compared to $\mathcal{L}_1$, in addition to any loop-suppression.
 
\vspace{1em} 

\emph{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~\cite{Crossley:2015evo} and identified new ``stochastic transport
coefficients'' in \eqref{eq:diff-Lst} and \eqref{eq:hydro-Lst} characterising
short-range 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 \eqref{eq:intro-results}. 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.

\vspace{1em}

This work was supported in part by the NSERC Discovery Grant program of Canada.

