# Dynamical large deviations for long-range interacting inhomogeneous systems without collective effects

Ouassim Feliachi, Jean-Baptiste Fouvry

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Ouassim Feliachi, Jean-Baptiste Fouvry. Dynamical large deviations for long-range interacting inhomogeneous systems without collective effects. Physical review. E: Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2024, 110, pp.024108. 10.1103/PhysRevE.110.024108. hal-04249348

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Dynamical large deviations for long-range interacting inhomogeneous systems without collective effects

Ouassim Feliachi¹ and Jean-Baptiste Fouvry²

¹Institut Denis Poisson, Université d'Orléans, CNRS Université de Tours, Orléans, France

²Institut d'Astrophysique de Paris, UMR 7095, 98 bis Boulevard Arago, F-75014 Paris, France

We consider the long-term evolution of an inhomogeneous long-range interacting N-body system. Placing ourselves in the dynamically hot limit, i.e. neglecting collective effects, we derive a large deviation principle for the system's empirical angle-averaged distribution function. This result extends the classical ensemble-averaged kinetic theory given by the so-called inhomogeneous Landau equation, as it specifies the probability of typical and large dynamical fluctuations. We detail the main properties of the associated large deviation Hamiltonian, in particular how it complies with the system's conservation laws and possesses a gradient structure.

# I. INTRODUCTION

As a result of violent relaxation [1], long-range interacting N-body systems generically find themselves to be dynamically frozen on quasi-stationary states. It is then only as a result of finite-N fluctuations that these systems can continue to undergo some statistical relaxation, driving them closer to their thermodynamical equilibrium. Such a long-range dynamics covers quite a wide class of systems like plasmas [2], self-gravitating clusters [3], or even more generic systems [4]. In the present work, we focus our interest on inhomogeneous systems, i.e. systems with a non-trivial integrable mean-field orbital structure, as is for example the case in globular clusters [5].

In order to describe the long-term evolution of these systems, a classical starting point is to consider the ensemble-averaged evolution of the system's distribution function (DF), here averaged over a set of initial conditions. In the context of long-range interacting inhomogeneous systems, this is described by the inhomogeneous Balescu–Lenard (BL) equation [6, 7]. In the limit of a dynamically hot system, i.e. a system which only weakly amplifies perturbations, this kinetic equation reduces to the (simpler) inhomogeneous Landau equation [see, e.g., 8, and references therein]. Both kinetic equations satisfy an H-theorem hence highlighting the irreversibility of the ensemble-averaged dynamics.

Yet, such frameworks, because they solely focus on ensemble-averaged dynamics, cannot predict the detailed probabilities of typical and large dynamical fluctuations away from this mean evolution. Such an extension is the realm of large deviation theory [see, e.g., 9, and the detailed review therein] which describes the statistics of the system's empirical DF. This is the focus of the present paper. In particular, we build upon [10] and derive the large deviation Hamiltonian in the case of a dynamically hot long-range interacting inhomogeneous system. This calculation therefore generalises the inhomogeneous Landau equation, which is immediately recovered from the large deviation theory through an ensemble average. As the upcoming sections will highlight, up to a few additional complications stemming from our accounting of the intricate orbital structure, these calculations share a

lot of similarities with the ones presented in [10].

The paper is organised as follows. In Sec. II, we detail our system, the quasilinear expansion and the inhomogeneous Landau equation. In Sec. III, we derive the system's large deviation Hamiltonian while neglecting collective effects. In Sec. IV, we discuss the main properties of this Hamiltonian. We conclude in Sec. V. Technical details in the main text are kept to a minimum and deferred to Appendices.

# II. DYNAMICS OF LONG-RANGE INTERACTING SYSTEMS

## A. System

We are interested in the long-term evolution of a long-range interacting Hamiltonian system in 2d dimensions. We denote phase space with $\mathbf{w}=(\mathbf{q}, \mathbf{p})$. The system is composed of $N \gg 1$ particles of individual mass, $m=M_{tot}/N$, with $M_{tot}$ the system's fixed total active mass. At any given time, the state of the system can be described by its empirical DF

$$F_d(\mathbf{w}, t) = \sum_{i=1}^{N} m \delta_D[\mathbf{w} - \mathbf{w}_i(t)], \quad (1)$$

with $\mathbf{w}_i(t)$ the location in phase space at time $t$ of particle $i$. We assume that particles are embedded within some given external potential, $U_{ext}(\mathbf{w})$ (e.g., the kinetic energy) and coupled to one another via a long-range pairwise interaction, $U(\mathbf{w}, \mathbf{w}')$. We denote the typical amplitude of $U(\mathbf{w}, \mathbf{w}')$ with $G$. The instantaneous specific empirical Hamiltonian is

$$H_d = U_{\text{ext}}(\mathbf{w}) + \Phi_d(\mathbf{w}, t) \quad (2)$$

with the empirical potential

$$\Phi_d(\mathbf{w}, t) = \int d\mathbf{w}' U(\mathbf{w}, \mathbf{w}') F_d(\mathbf{w}', t). \quad (3)$$

The dynamics of $F_d$ is given by the Klimontovich equation [11]. It reads

$$\frac{\partial F_d}{\partial t} + [F_d, H_d] = 0, \quad (4)$$
with the Poisson bracket

$$[f(\mathbf{w}), h(\mathbf{w})] = \frac{\partial f}{\partial \mathbf{q}} \cdot \frac{\partial h}{\partial \mathbf{p}} - \frac{\partial f}{\partial \mathbf{p}} \cdot \frac{\partial h}{\partial \mathbf{q}}. \quad (5)$$

In the following, we will consider ensemble averages over the initial conditions of the N particles. Denoting the ensemble average as $\langle \cdot \rangle$, we assume then that $\langle F_d(\mathbf{w}, t) \rangle = F(\mathbf{w}, t)$, with $F(\mathbf{w}, t)$ some smooth function, and $H = \langle H_d \rangle$ the associated smooth Hamiltonian. The mean system is assumed to be in an integrable stable quasi-stationnary equilibrium. There exist then canonical angle-action coordinates, $(\theta, \mathbf{J})$ [3], so that

$$F(\mathbf{w}, t) = F(\mathbf{J}, t); \quad H(\mathbf{w}, t) = H(\mathbf{J}, t), \quad (6)$$

hence defining the orbital frequencies, $\Omega(\mathbf{J}) = \partial H / \partial \mathbf{J}$. Such a system is said to be in a quasi-stationary equilibrium since $[F(\mathbf{J}, t), H(\mathbf{J}, t)] = 0$.

## B. Quasilinear expansion

For a given realisation, we define

$$F_N(\mathbf{J}, t) = \int \frac{d\theta}{(2\pi)^d} F_d(\mathbf{w}, t), \quad (7)$$

by averaging over the angles. Following Eq. (3), we assume that $H_N = H_N[F_N]$, only depends on the actions. Since we have $\langle F_N \rangle = F$, it is natural to build the decomposition

$$F_d(\mathbf{w}, t) = F_N(\mathbf{J}, t) + \frac{1}{\sqrt{N}} \delta F(\mathbf{w}, t), \quad (8a)$$

$$H_d(\mathbf{w}, t) = H_N(\mathbf{J}, t) + \frac{1}{\sqrt{N}} \delta\Phi(\mathbf{w}, t), \quad (8b)$$

where the prefactor $1/\sqrt{N}$ ensures that the fluctuations, $\delta F$ and $\delta\Phi$, are of order unity w.r.t. N.

It is crucial to note that Eq. (8) differs from the usual quasilinear decomposition [see, e.g., 7], $F_d = F + \delta F / \sqrt{N}$, performed w.r.t. the mean DF, $F(\mathbf{J}, t)$. Indeed, in the present approach, $F_N$ remains a stochastic quantity that varies from one realisation to another. In this work, our goal is to characterise the statistics of the dynamics of $F_N(\mathbf{J}, t)$ and its deviation away from the mean evolution, i.e. the evolution of $F(\mathbf{J}, t)$.

We can now inject Eqs. (8) into Eq. (4) to obtain evolution equations for $\delta F$ and $F_N$. We get

$$\frac{\partial \delta F}{\partial \tau} + N \left\{ \left[ \delta F, H_N \right] + \left[ F_N, \delta \Phi \right] \right\} = 0, \quad (9a)$$

$$\frac{\partial F_N(\mathbf{J}, \tau)}{\partial \tau} + \int \frac{d\theta}{(2\pi)^d} [\delta F, \delta\Phi] = 0, \quad (9b)$$

where we truncated Eq. (9a) at first order in $1/\sqrt{N}$, performed an angle average in Eq. (9b), and introduced the (slow) time $\tau = t/N$. On the one hand, for $N \gg 1$,

Eq. (9a) for $\delta F$ is a fast process, with a timescale for $\tau$ of order $1/N$: it describes the fast dynamics of fluctuations. On the other hand, Eq. (9b) is associated with a slow process, with a timescale for $\tau$ of order 1: it describes the slow relaxation of orbits.

## C. Kinetic equations

In order to describe the ensemble average of Eq. (9b) for the asymptotic process of $\delta F$ for fixed $F_N$, one typically proceeds as follows: (i) Because the angles $\theta$ are $2\pi$-periodic, fluctuations can be decomposed in Fourier space, hence introducing the associated resonance vectors, $\mathbf{k}, \mathbf{k}' \in \mathbb{Z}^d$; (ii) Assuming a separation between the fast and slow timescales, one solves Eq. (9a) for $\delta F(\mathbf{w}, t)$, assuming a fixed $F_N(\mathbf{J}, \tau)$, see Appendix A; (iii) One finally injects these asymptotic expressions in the r.h.s. of Eq. (9b), to obtain the kinetic collision operator – see [7] for details.

Placing oneself in the dynamically hot limit, i.e. $G \to 0$, one finds that the ensemble-averaged long-term evolution of the system is described by the inhomogeneous Landau equation [8] reading

$$\frac{\partial F(\mathbf{J}, \tau)}{\partial \tau} = \frac{\partial}{\partial \mathbf{J}} \cdot \left[ \sum_{\mathbf{k}, \mathbf{k}'} \mathbf{k} \int d\mathbf{J}' B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') \right. \\ \left. \times \left\{ \mathbf{k} \cdot \frac{\partial F}{\partial \mathbf{J}} F(\mathbf{J}') - \mathbf{k}' \cdot \frac{\partial F}{\partial \mathbf{J}'} F(\mathbf{J}) \right\} \right] + o(G^2), \quad (10)$$

where we wrote $F(\mathbf{J}) = F(\mathbf{J}, \tau)$ to shorten the notations. We also introduced

$$B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') = \pi(2\pi)^d M_{\text{tot}} |\psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')|^2 \quad (11) \\ \times \delta_D[\mathbf{k} \cdot \Omega(\mathbf{J}) - \mathbf{k}' \cdot \Omega(\mathbf{J}')],$$

with $\delta_D$ the Dirac delta. Here, $\psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}', \omega)$ stands for the bare coupling coefficients (see Eq. A3), which account for the system’s inhomogeneity. When accounting for collective effects, i.e. going beyond the limit $G \to 0$, Eq. (10) becomes the BL equation [6, 7]. It is obtained from Eq. (11) by replacing $|\psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')|^2$ with their dressed analogs, $|\psi_{\mathbf{k}\mathbf{k}'}^d(\mathbf{J}, \mathbf{J}', \mathbf{k} \cdot \Omega(\mathbf{J}))|^2$, as defined in Eq. (A10).

## III. LARGE DEVIATION PRINCIPLE

We now want to go beyond the classical computation presented in Eq. (10), by estimating not only the ensemble average of Eq. (9b), but the whole cumulant generating function. This allows one to retrieve not only the average evolution path for the angle-averaged DF, $F_N$, but the whole probability distribution function for any evolution path.

Following the same approach as in [10], we can generically estimate the probability that $F_N(\tau)$ follows a given
time evolution {$F(\tau)$}$_{0\le\tau\le T}$ through<fnref n="1" />

$$
\mathbb{P}(\{F_N(\tau)\}_{0\le\tau\le T} = \{F(\tau)\}_{0\le\tau\le T})_{N\to+\infty} \asymp \quad (12)
$$

$$
\exp\left[-\frac{N(2\pi)^d}{M_{\text{tot}}} \sup_P \int_0^T d\tau \left\{ \left( \int d\mathbf{J} \, \dot{F} P \right) - \mathcal{H}[F, P] \right\} \right],
$$

with $\dot{F} = \partial_{\tau}F$, and the prescription that $F_N(\tau=0)$ con-
verges to $F(\tau=0)$ for $N \to +\infty$. Equation (12) also in-
volves the conjugate field, $\{P(\mathbf{J}, \tau)\}_{0\le\tau\le T}$, over which
the maximisation must be performed. Finally, in
Eq. (12), we introduced the large deviation Hamiltonian
(i.e. the scaled cumulant generating function)

$$
\mathcal{H}[F, P] = \lim_{\Delta \to +\infty} \frac{M_{\text{tot}}}{\Delta(2\pi)^d} \quad (13)
$$

$$
\times \ln \left[ \left\langle \exp \left( \frac{(2\pi)^d}{M_{\text{tot}}} \int_0^\Delta dt \int d\mathbf{J} P(\mathbf{J}) \partial_\tau F_N[\delta F] \right) \right\rangle_F \right],
$$

where both $F(\mathbf{J})$ and $P(\mathbf{J})$ are evaluated at time $\tau$, and $\partial_{\tau}F_N[\delta F]$ follows from Eq. (9b). Here, $\langle \cdot \rangle_F$ denotes an expectation over the fast process $\delta F$ with $F_N=F$ fixed. Equations (12) and (13) are generic results describing the large deviations for the slow evolution of a process driven by a random fast process. We refer to Appendix B for a brief heuristic derivation of Eqs. (12) and (13).

Naturally, here the difficulty lies in the computation of
the average in Eq. (13). Following the same approach
as in [10], we approach this calculation by expanding
Eq. (13) w.r.t. G. We can write

$$
\mathcal{H}[F, P] = \mathcal{H}^{(1)}[F, P] + \mathcal{H}^{(2)}[F, P] + o(G^2). \quad (14)
$$

where $\mathcal{H}^{(1)}$ (resp. $\mathcal{H}^{(2)}$) is the first (resp. second cumu-
lant). Computing these cumulants is a cumbersome cal-
culation. As a first step, this requires the computation
of the asymptotic time evolution of the DF and potential
fluctuations. This is presented in Appendix A. Then,
in Appendix C, we compute explicitly the various first
terms of Eq. (14).

The first cumulant reads (Appendix C 1)

$$
\mathcal{H}^{(1)}[F, P] = - \sum_{\mathbf{k}, \mathbf{k}'} \int d\mathbf{J} d\mathbf{J}' B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') \mathbf{k} \cdot \frac{\partial P}{\partial \mathbf{J}} \quad (15)
$$

$$
\times \left[ \mathbf{k} \cdot \frac{\partial F}{\partial \mathbf{J}} F(\mathbf{J}') - \mathbf{k}' \cdot \frac{\partial F}{\partial \mathbf{J}'} F(\mathbf{J}) \right] + o(G^2),
$$

while the second cumulant reads (Appendix C 2)

$$
\mathcal{H}^{(2)}[F, P] = \sum_{\mathbf{k}, \mathbf{k}'} \int d\mathbf{J} d\mathbf{J}' B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') \mathbf{k} \cdot \frac{\partial P}{\partial \mathbf{J}} \quad (16)
$$

$$
\times \left[ \mathbf{k} \cdot \frac{\partial P}{\partial \mathbf{J}} - \mathbf{k}' \cdot \frac{\partial P}{\partial \mathbf{J}'} \right] F(\mathbf{J}) F(\mathbf{J}') + o(G^2).
$$

Finally, in Appendix C 3, we justify why all cumulants
beyond the two first ones can be neglected in the dynam-
ically hot limit, i.e. they are all of order $o(G^2)$.

Equation (12) in conjunction with Eq. (14) is the main
result of this section. It characterises the dynamical large
deviations of the angle-averaged DF, $F_N(\mathbf{J}, \tau)$. We dis-
cuss their main properties in Sec. IV.

Unfortunately, the expansion from Eq. (14) stops be-
ing effective when accounting for collective effects, i.e. in
the dynamically cold limit. In that case, one must resort
to an explicit computation of the exponential average
from Eq. (13), as all cumulants in Eq. (13) contribute to
the large deviation Hamiltonian. [12] succeeded in per-
forming this calculation in the case of a homogeneous
system. Unfortunately, the theorems and methods used
therein do not lend themselves straightforwardly to the
inhomogeneous case. One reason of these additional dif-
ficulties lies in the fact that in homogeneous systems, the
resonance condition, $\delta_D(\mathbf{k} \cdot [\mathbf{v} - \mathbf{v}'])$, is diagonal w.r.t. the
resonance number $\mathbf{k}$. This is not the case anymore in
inhomogeneous systems where the resonance condition
becomes $\delta_D[\mathbf{k} \cdot \Omega(\mathbf{J}) - \mathbf{k}' \cdot \Omega(\mathbf{J}')]$, as in Eq. (11). This will
be the topic of a future work.

IV. PROPERTIES

In this section, we briefly present the main properties
of the large deviation Hamiltonian from Eq. (14). We
refer to Appendix D for technical details and to sec. 2.1.1
of [10] for thorough discussions.

A. Most probable path

As expected, the most probable path

$$
\frac{\partial F(\mathbf{J}, \tau)}{\partial \tau} = \frac{\delta \mathcal{H}}{\delta P(\mathbf{J})} [F, P=0], \quad (17)
$$

yields exactly the inhomogeneous Landau equation (10).
Equation (17) is the Hamilton equation prescribing the
evolution path that minimizes the large deviation action,
i.e. the most probable evolution path for the empirical
DF [see sec. 7.2.2 in 9].

B. Conservation laws

If $C[F]$ is a conserved quantity for the $N$-body dynam-
ics, the large deviation Hamiltonian must satisfy

$$
\int d\mathbf{J} \frac{\delta C[F]}{\delta F(\mathbf{J})} \frac{\delta \mathcal{H}}{\delta P(\mathbf{J})} = 0. \quad (18)
$$

This symmetry implies that the large deviation ac-
tion (i.e. the exponential argument in the r.h.s. of
Eq. 12) is infinite for evolution paths that do not satisfy

<footnote n="1">¹ In Eq. (12), we introduce the logarithmic equivalence defined via
$a_N \underset{N \to \infty}{\asymp} e^{N a} \iff \lim_{N \to +\infty} \ln(a_N)/N = a.</footnote>
$\int d\mathbf{J} \dot{F} \delta C[F]/\delta F(\mathbf{J})=0$. Phrased differently, any evolution path for the DF violating the conservation laws has zero probability [see sec. 7.2.6 in 9]. In Appendix D 2, we explicitly show that Eq. (14) is consistent with the mass and energy conservation, defined via

$$M[F] = \int d\mathbf{J} F(\mathbf{J}) \quad (\text{mass conservation}), \quad (19a)$$

$$E[F] = \int d\mathbf{J} H(\mathbf{J}) F(\mathbf{J}) \quad (\text{energy conservation}). \quad (19b)$$

### C. Hamilton–Jacobi equation

We define the system's entropy as

$$S[F] = -\int d\mathbf{J} F(\mathbf{J}) \ln[F(\mathbf{J})]. \quad (20)$$

As detailed in Appendix D 3, one can show that it solves the stationary Hamilton–Jacobi equation

$$\mathcal{H}[F, -\delta S/\delta F] = 0, \quad (21)$$

where $\delta S/\delta F$ stands for the function $J \mapsto \delta S[F]/\delta F(\mathbf{J})$. Yet, Eq. (21) is not sufficient to prove that the negative of the entropy is the quasipotential associated with the present large deviation principle [see sec. 7.2.3 in 9]. Indeed, one also has to prove that $S[F]$ has an unique maximum: this is typically not true for self-gravitating systems [see, e.g., 13, 14].

### D. Time-reversal symmetry

The large deviation Hamiltonian complies with the generalised time-reversal symmetry (see Appendix D 4)

$$\mathcal{H}[F, -P] = \mathcal{H}[F, P - \delta S/\delta F]. \quad (22)$$

Such a relation corresponds to a detailed balance condition at the level of large deviations, associated with the symmetry $\tau \to T-\tau$ in Eq. (12) [see sec. 7.3.1 in 9].

### E. Gradient structure

The large deviation Hamiltonian generically induces a gradient flow [15]. Indeed, as detailed in Appendix D 5, the second cumulant from Eq. (16) can be written as

$$\mathcal{H}^{(2)}[F, P] = \int d\mathbf{J} d\mathbf{J}' P(\mathbf{J}) P(\mathbf{J}') Q[F](\mathbf{J}, \mathbf{J}'), \quad (23)$$

where $Q[F]$ reads

$$Q[F](\mathbf{J}, \mathbf{J}') = \sum_{\mathbf{k}, \mathbf{k}'} \mathbf{k} \cdot \frac{\partial}{\partial \mathbf{J}} \left\{ \frac{\partial}{\partial \mathbf{J}'} \left[ -\mathbf{k}' F(\mathbf{J}) F(\mathbf{J}') B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') + \mathbf{k} \delta_{\mathbf{D}}(\mathbf{J} - \mathbf{J}') F(\mathbf{J}) \int d\mathbf{J}'' F(\mathbf{J}'') B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}'') \right] \right\}. \quad (24)$$

The inhomogeneous Landau Eq. (10) can then be rewritten as

$$\frac{\partial F(\mathbf{J}, \tau)}{\partial \tau} = \int d\mathbf{J}' Q[F](\mathbf{J}, \mathbf{J}') \frac{\delta S[F]}{\delta F(\mathbf{J}')}, \quad (25)$$

with the entropy, $S[F]$, defined in Eq. (20). Given that $H[F, P]$ is convex w.r.t. $P$, $Q[F]$ is a positive operator on the space of DFs. Therefore, Eq. (25) illustrates the increase of entropy along solutions of the Landau equation [see sec. 5 in 9].

### F. Stochastic Landau equation

We note that the large deviation Hamiltonian from Eq. (14) is quadratic in $P(\mathbf{J}, t)$. This implies that large deviations are Gaussian [see, e.g., sec. 4.2 in 10]. As a result, as detailed in Appendix D 6, one can setup a stochastic partial differential equation which obeys the large deviation principle from Eq. (12). It reads

$$\frac{\partial F_N(\mathbf{J}, \tau)}{\partial \tau} = \left[ \frac{\partial F_N(\mathbf{J}, \tau)}{\partial \tau} \right]_{\text{Landau}} + \zeta[F_N](\mathbf{J}, \tau), \quad (26)$$

where $[\partial F_N(\mathbf{J})/\partial \tau]_{\text{Landau}}$ is the inhomogeneous Landau collision operator, i.e. the r.h.s. of Eq. (10). We also introduced the Gaussian random field, $\zeta[F](\mathbf{J}, \tau)$. Following sec. 4.2 in [10], it obeys

$$\langle \zeta[F](\mathbf{J}, \tau) \rangle = 0, \quad (27a)$$

$$\langle \zeta[F](\mathbf{J}, \tau) \zeta[F](\mathbf{J}', \tau') \rangle = \frac{2m}{(2\pi)^d} Q[F](\mathbf{J}, \mathbf{J}') \delta_{\mathbf{D}}(\tau - \tau'), \quad (27b)$$

where averages are taken at fixed $F$.

Although intricate, Eq. (26) is an important result as it allows one to “mimic” directly the stochastic evolution of $F_N$ and its large deviations, without ever integrating the N-body equations of motion. Following Sec. IV B, Eq. (26) exactly conserves the total mass and total energy. Finally, in Appendix D 6 (see Eq. D21), we present an alternative writing of $\zeta[F](\mathbf{J}, \tau)$ which (i) eases the effective sampling of the stochastic noise, (ii) highlights explicitly the compliance with the conservation laws.

### G. Homogeneous limit

It is straightforward to recover the results from [10] in the limit of a (multi-periodic) homogeneous system. In that case, (i) the angle-action coordinates become $(\theta, \mathbf{J}) \to (\mathbf{x}, \mathbf{v})$, the volume elements $(2\pi)^d \to L^3$, with $L$ the size of the box, and $N/M_{\text{tot}}$ plays the role of the plasma parameter, $\Lambda$; (ii) the bare coupling coefficients are constrained by symmetry and independent of the orbits, namely $\psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') \to \delta_{\mathbf{k}\mathbf{k}'}/|\mathbf{k}|^2$, up to a prefactor. Equation (14) then falls back on the result from [10].
V. CONCLUSION

In this paper, we investigated dynamical large deviations in systems with long-range interactions. In the view of generalising [10], we focused here on the case of inhomogeneous systems, i.e. systems with a non-trivial orbital structure. Starting from the generic large deviation Hamiltonian of a slow-fast system and placing ourselves in the dynamically hot limit, i.e. neglecting collective effects, we computed the two first cumulants of the associated large deviation Hamiltonian. Equation (14) is the main result of this work, and encodes the likelihood of any given evolution path for the system's angle-averaged DF. In Sec. IV, we highlighted that Eq. (14) complies with all the expected properties, such as (i) recovering the inhomogeneous Landau equation [8] in the ensemble-averaged limit; (ii) ensuring the conservation laws; (iii) and possessing a natural gradient structure. Finally, we emphasised that the quadratic dependence of the large deviation Hamiltonian w.r.t. the conjugate field allows one to construct an effective stochastic partial differential equation with the expected dynamical large deviations.

The present work is only one step toward ever more detailed description of the statistical properties of dynamical large deviations in long-range interacting inhomogeneous systems. We conclude by mentioning a few possible venues for future explorations.

Naturally, it would be useful to generalise Eq. (14) and lift the assumption $G \to 0$, so as to generalise [12] to the inhomogeneous case. Unfortunately, this is no easy undertaking as both the resonance condition and the coupling coefficients get more intricate as one considers an inhomogeneous system. Similarly, approaching these calculations through the BBGKY hierarchy [see, e.g., 2] could also prove insightful.

The generic probability distribution from Eq. (12) is challenging to evaluate in practice. Though, it would be enlightening to compute it numerically, starting with some simple long-range interacting systems. One could begin with the Hamiltonian Mean Field model [16] in an inhomogeneous configuration, for example tracking the statistics of the large deviations of the system's magnetisation. Similarly, one could extend the present work to long-range interacting systems submitted to a stochastic forcing [see, e.g., 17]. These systems exhibit phase transitions whose most likely transition path could be recovered via a large deviation principle – through the resolution of a system of coupled partial differential equations for $F(\mathbf{J}, \tau)$ and $P(\mathbf{J}, \tau)$. Such preliminary explorations are mandatory first steps before applying the present statistical approach to more realistic systems.

Here, the system was always assumed to be (strongly) linearly stable. This allowed us to neglect any possible contributions from the system's damped modes in the linear dynamics of fluctuations (see Appendix A 3). In the case of weakly stable systems, it would be interesting to extend the present calculation to account for the effects of slowly decaying modes and wave-particle inter-

actions [see, e.g., 18], as well as possible dynamical phase transitions toward linear instability [see, e.g., 19]. Additionally, one could investigate the connections between the present dynamical large deviations and the thermal (van-Kampen) fluctuations considered in [20, 21].

Finally, we limited ourselves to diffusion sourced by two-body correlations, i.e. $1/N$ effects. Yet, in the case of one-dimensional systems, such a relaxation can identically vanish. This is a kinetic blocking [see, e.g., 22] and these systems can only relax via the weaker $1/N^2$ effects, i.e. three-body correlations [see, e.g., 23]. It would be informative to investigate the properties of dynamical large deviations in such contrived systems.

Acknowledgments

This work is partially supported by the grant Segal ANR-19-CE31-0017 of the French Agence Nationale de la Recherche and by the Idex Sorbonne Université. We are grateful to J. Barré, C. Pichon, M. Roule for insightful discussions.

Appendix A: Linear dynamics of fluctuations

In this Appendix, we solve for the linear dynamics of the fluctuations, $\delta F$, as driven by Eq. (9a). The calculations below follow closely [7, 24]. In this Appendix, the angle-averaged DF, $F_N$, has simply been denoted with $F$, to shorten the notations.

1. Laplace–Fourier transforms

We define the Fourier transform w.r.t. the angle $\theta$ via

$$f(\mathbf{w}, t) = \sum_{\mathbf{k} \in \mathbb{Z}^d} f_{\mathbf{k}}(\mathbf{J}, t) e^{i\mathbf{k} \cdot \theta}, \quad (A1a)$$

$$f_{\mathbf{k}}(\mathbf{J}, t) = \int \frac{d\theta}{(2\pi)^d} f(\mathbf{w}, t) e^{-i\mathbf{k} \cdot \theta}. \quad (A1b)$$

The self-consistency relation from Eq. (3) when used for the fluctuations then reads

$$\delta\Phi_{\mathbf{k}}(\mathbf{J}, t) = (2\pi)^d \sum_{\mathbf{k}', \mathbf{k}} \int d\mathbf{J}' \psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') \delta F_{\mathbf{k}'}(\mathbf{J}', t). \quad (A2)$$

In that expression, $\psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')$ are the bare coupling coefficients. They follow from the expansion of the pairwise interaction potential as

$$U(\mathbf{w}, \mathbf{w}') = \sum_{\mathbf{k}, \mathbf{k}'} \psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') e^{i(\mathbf{k} \cdot \theta - \mathbf{k}' \cdot \theta')}, \quad (A3)$$

These coefficients satisfy the two symmetries

$$\psi_{\mathbf{k}'\mathbf{k}}(\mathbf{J}', \mathbf{J}) = \psi_{\mathbf{k}\mathbf{k}'}^*(\mathbf{J}, \mathbf{J}'), \quad (A4a)$$

$$\psi_{-\mathbf{k}-\mathbf{k}'}(\mathbf{J}, \mathbf{J}') = \psi_{\mathbf{k}\mathbf{k}'}^*(\mathbf{J}, \mathbf{J}'). \quad (A4b)$$
We define the Laplace transform with the convention

$$ \tilde{f}(\omega) = \int_{0}^{+\infty} dt f(t) e^{i\omega t}; \quad f(t) = \int_{B} \frac{d\omega}{2\pi} \tilde{f}(\omega) e^{-i\omega t}, \quad (A5) $$

where the Bromwich contour, $B$, has to pass above all the poles of the integrand, i.e. $Im[\omega]$ has to be large enough. When expressed in Fourier-Laplace space, Eq. (9a) becomes

$$ \delta \tilde{F}_{\mathbf{k}}(\mathbf{J}, \omega) = - \frac{\mathbf{k} \cdot \partial F / \partial \mathbf{J}}{\omega - \mathbf{k} \cdot \mathbf{\Omega}} \delta \tilde{\Phi}_{\mathbf{k}}(\mathbf{J}, \omega) - \frac{\delta F_{\mathbf{k}}(\mathbf{J}, 0)}{i(\omega - \mathbf{k} \cdot \mathbf{\Omega})} \quad (A6) $$

with $\Omega=\Omega(\mathbf{J})$ and $\delta F_{\mathbf{k}}(\mathbf{J}, 0)$ describing the fluctuations in the DF at the initial time. Once again, we recall that for the sake of shorter notations, we wrote the angle-averaged DF, $F_N$, as $F$.

## 2. Self-consistency

We now act on both sides of Eq. (A6) with the same operator as in the r.h.s. of Eq. (A2). We get

$$ \begin{aligned} \delta \tilde{\Phi}_{\mathbf{k}}(\mathbf{J}, \omega) = &-(2\pi)^d \sum_{\mathbf{k}'} \int d\mathbf{J}' \frac{\psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')}{\omega - \mathbf{k}' \cdot \mathbf{\Omega}'} \mathbf{k}' \cdot \frac{\partial F}{\partial \mathbf{J}'} \delta \tilde{\Phi}_{\mathbf{k}'}(\mathbf{J}', \omega) \\ &-(2\pi)^d \sum_{\mathbf{k}'} \int d\mathbf{J}' \frac{\psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')}{i(\omega - \mathbf{k}' \cdot \mathbf{\Omega}')} \delta F_{\mathbf{k}'}(\mathbf{J}', 0), \end{aligned} \quad (A7) $$

with $\Omega'=\Omega(\mathbf{J}')$.

In the absence of collective effects (i.e. $G \to 0$), the first term in the r.h.s. of Eq. (A7) can be neglected, to get

$$ \delta \tilde{\Phi}_{\mathbf{k}}^{\text{bare}}(\mathbf{J}, \omega) = -(2\pi)^d \sum_{\mathbf{k}'} \int d\mathbf{J}' \frac{\delta F_{\mathbf{k}'}(\mathbf{J}', 0) \psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')}{i(\omega - \mathbf{k}' \cdot \mathbf{\Omega}')} \quad (A8) $$

When collective effects are accounted for, we may assume that the dressed potential perturbations follow the ansatz

$$ \delta \tilde{\Phi}_{\mathbf{k}}^{\text{dress}}(\mathbf{J}, \omega) = -(2\pi)^d \sum_{\mathbf{k}'} \int d\mathbf{J}' \frac{\delta F_{\mathbf{k}'}(\mathbf{J}', 0) \psi_{\mathbf{k}\mathbf{k}'}^{\text{d}}(\mathbf{J}, \mathbf{J}', \omega)}{i(\omega - \mathbf{k}' \cdot \mathbf{\Omega}')} \quad (A9) $$

where the frequency-dependent dressed coupling coefficients, $\psi_{\mathbf{k}\mathbf{k}'}^{\text{d}}(\mathbf{J}, \mathbf{J}', \omega)$, remain to be determined. When injected into Eq. (A7), we find that the dressed coupling coefficients satisfy the self-consistent relation

$$ \begin{aligned} \psi_{\mathbf{k}\mathbf{k}'}^{\text{d}}(\mathbf{J}, \mathbf{J}', \omega) = & \psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') \quad (A10) \\ & - (2\pi)^d \sum_{\mathbf{k}''} \int d\mathbf{J}'' \frac{\psi_{\mathbf{k}\mathbf{k}''}(\mathbf{J}, \mathbf{J}'') \mathbf{k}'' \cdot \partial F / \partial \mathbf{J}''}{\omega - \mathbf{k}'' \cdot \mathbf{\Omega}''} \psi_{\mathbf{k}''\mathbf{k}'}^{\text{d}}(\mathbf{J}'', \mathbf{J}', \omega), \end{aligned} $$

with $\Omega''=\Omega(\mathbf{J}'')$. Since Eq. (A6) was derived assuming $Im[\omega]>0$ large enough, for $\omega_R \in \mathbb{R}$, the resonant denominator in Eq. (A10) has to be interpreted as

$$ \frac{1}{\omega_R - \mathbf{k}'' \cdot \mathbf{\Omega}''} \to \frac{1}{\omega_R - \mathbf{k}'' \cdot \mathbf{\Omega}'' + i\gamma}, \quad (A11) $$

with $\gamma \to 0^+$. We refer to [25] (and references therein) for a discussion of the associated Landau's prescription. We note that an explicit expression for $\psi_{\mathbf{k}\mathbf{k}'}^{\text{d}}$ can be obtained using the basis method [see, e.g., 24, and references therein], but this will not needed here.

Ultimately, we find that the coupling coefficients asymptotically scale w.r.t. $G$, the amplitude of the pairwise interaction, like

$$ \psi_{\mathbf{k}\mathbf{k}'} \propto G, \quad (A12a) $$

$$ \psi_{\mathbf{k}\mathbf{k}'}^{\text{d}} \propto \frac{\psi_{\mathbf{k}\mathbf{k}'}^{\text{d}}}{1 - \psi_{\mathbf{k}\mathbf{k}'}^{\text{d}}} \propto \frac{G}{1 - G}. \quad (A12b) $$

## 3. Time evolution

Taking the inverse Laplace transform of Eq. (A9), the dressed fluctuations generically evolve according to

$$ \begin{aligned} \delta \Phi_{\mathbf{k}}(\mathbf{J}, t) = &-(2\pi)^d \sum_{\mathbf{k}'} \int d\mathbf{J}' \delta F_{\mathbf{k}'}(\mathbf{J}', 0) \\ & \times \int_B \frac{d\omega}{2\pi} \frac{\psi_{\mathbf{k}\mathbf{k}'}^{\text{d}}(\mathbf{J}, \mathbf{J}', \omega)}{i(\omega - \mathbf{k}' \cdot \mathbf{\Omega}')} e^{-i\omega t}, \end{aligned} \quad (A13) $$

where the Bromwich contour, $B$, has to pass above all the poles of the integrand in the complex $\omega$-plane. Because we assumed that the system is linearly stable, the function $\omega \mapsto \psi_{\mathbf{k}\mathbf{k}'}^{\text{d}}(\mathbf{J}, \mathbf{J}', \omega)$ only has poles in the lower half of the complex plane. These are associated with Landau damped modes and correspond to frequencies $\omega_M$ with $Im[\omega_M] < 0$.

As usual, we proceed by distorting the contour $B$ to the lower half of the complex plane so that $|e^{-i\omega t}| \to 0$, snagging onto the poles of the integrand. In Eq. (A13), there is a single pole along the real axis, namely in $\omega = \mathbf{k}' \cdot \mathbf{\Omega}'$. Paying attention to the direction of integration, each pole contributes a $-2i\pi \text{Res}[...]$. Placing ourselves in the limit $t \gg |1/\text{Im}[\omega_M]|$, we neglect the contributions from the damped modes [see, e.g., 18]. Once these have faded away, Eq. (A13) becomes

$$ \begin{aligned} \delta \Phi_{\mathbf{k}}(\mathbf{J}, t) = & (2\pi)^d \sum_{\mathbf{k}'} \int d\mathbf{J}' e^{-i\mathbf{k}' \cdot \mathbf{\Omega}' t} \delta F_{\mathbf{k}'}(\mathbf{J}', 0) \\ & \times \psi_{\mathbf{k}\mathbf{k}'}^{\text{d}}(\mathbf{J}, \mathbf{J}', \mathbf{k}' \cdot \mathbf{\Omega}'). \end{aligned} \quad (A14) $$

Having determined the time-evolution of the potential fluctuations, we now set out to determine the time-evolution of the DF fluctuations themselves. An efficient approach to perform this calculation is to rely on the self-consistency relation from Eq. (A2). We start with Eq. (A14) in which we replace $\psi_{\mathbf{k}\mathbf{k}'}^{\text{d}}$ with the r.h.s. from
The probability of a given time evolution, {$F_N(\tau)$}$_{0\le\tau\le T}$, can then be estimated through the product of conditional probabilities

$$
\mathbb{P}[\{F_N(\tau)\}_{0\le\tau\le T} = \{F(\tau)\}_{0\le\tau\le T}]_{N\to+\infty} \asymp \quad (B9)
$$

$$
\prod_n \mathbb{P}[G_N(n\Delta\tau) = \dot{F}(n\Delta\tau) | F_N(n\Delta\tau) = F(n\Delta\tau)],
$$

with $\dot{F}=\partial_{\tau}F$. Importantly, to obtain this expression, we used a Markovian decomposition of the path of evolution, assuming that increments of $F_N$ over timescales of order $\Delta\tau \gg 1/N$, are independent from one another.

Now, let us compute the probability of a given incre-
ment, $\mathbb{P}(G_N = \dot{F} | F_N = F)$. We use Eq. (B2) with
$N\Delta\tau \gg 1$ as the large deviation rate. We obtain

$$
\mathbb{P}(G_N = \dot{F} | F_N = F) \underset{N \to +\infty}{\asymp} e^{-N \Delta \tau I[F, \dot{F}]}, \quad (B10)
$$

where the dependence w.r.t. *F* emphasises that *F* can
be taken as constant on a time interval of duration Δτ.
In Eq. (B10), the large deviation function follows from
Eq. (B5) and reads

$$
I[F, \dot{F}] = \sup_{P} \{ \dot{F}P - \mathcal{H}[F, P] \}. \quad (B11)
$$

Here, $\mathcal{H}[F, P]$, is the large deviation Hamiltonian It
follows from Eq. (B1), with the large deviation rate,
$\Delta t = N\Delta\tau$, and reads

$$
\mathcal{H}[F, P] = \frac{1}{\Delta t} \ln \left[ \langle e^{\Delta t G_N P} \rangle_F \right], \quad (B12)
$$

where the average is performed over the fast process from
Eq. (B6a) with $F_N = F$. Starting from the definition of
Eq. (B8), we can write

$$
G_N = \frac{1}{\Delta\tau} \int_0^{\Delta\tau} d\tau \partial_\tau F_N = \frac{1}{\Delta\tau} \int_0^{\Delta t} dt \partial_\tau F_N, \quad (B13)
$$

where, for simplicity, we started the time-integral in
$\tau=0$, and $\partial_\tau F_N$ follows from Eq. (B6b). Since $\Delta t \gg 1$
(Eq. B7), we can approximate Eq. (B12) with

$$
\mathcal{H}[F, P] = \lim_{\Delta\tau \to +\infty} \frac{1}{\Delta\tau} \ln \left[ \left\langle \exp \left( P \int_0^\Delta dt \partial_\tau F_N \right) \right\rangle_F \right]. \quad (B14)
$$

The final step of the computation is to use the assump-
tion $\Delta\tau \ll 1$ (Eq. B7) so that the product of probabilities
in Eq. (B9) can be replaced by the exponential of an
integral. More precisely, using Eq. (B10), we obtain

$$
\mathbb{P}[\{F_N(\tau)\}_{0\le\tau\le T} = \{F(\tau)\}_{0\le\tau\le T}]_{N\to+\infty} \asymp \quad (B15)
$$

$$
\exp \left[ -N \sup_{P} \int_{0}^{T} d\tau \left\{ \dot{F}P - \mathcal{H}[F, P] \right\} \right],
$$

where $P=P(\tau)$ is now a field w.r.t. the slow time $\tau$. Nat-
urally, Eqs. (B15) and (B14) bear a lot of similarities with

the generic results from Eqs. (12) and (13). To fully re-
cover these expressions, it only remains to (i) add the
additional dependence of $F_N$ w.r.t. $\mathbf{J}$, which also trans-
mits to $P=P(\mathbf{J}, \tau)$; (ii) account correctly for the various
normalisation prefactors. Let us conclude by pointing out
that the present heuristic proof does not check the differ-
entiability of the large derivation Hamiltonian, $\mathcal{H}[F, P]$,
and hence the possible non-convexity of the large devia-
tion function, $I[F]$.

**Appendix C: Computing the cumulants**

In this Appendix, we compute the first cumulants of
Eq. (13). For a random variable $X$, we recall that

$$
\ln[\langle e^X \rangle] \approx \langle X \rangle + \frac{1}{2} (\langle X^2 \rangle - \langle X \rangle^2) + \dots \quad (C1)
$$

Following the convention from Eq. (A1), we note that
Eq. (9b) can be written as

$$
\frac{\partial F_N(\mathbf{J})}{\partial \tau} = -i \sum_{\mathbf{k}} \mathbf{k} \cdot \frac{\partial}{\partial \mathbf{J}} \left[ \delta F_{\mathbf{k}}(\mathbf{J}) \delta \Phi_{-\mathbf{k}}(\mathbf{J}) \right]. \quad (C2)
$$

Hence, we will apply Eq. (C1) using $X \propto \delta F \delta \Phi$.

1. First cumulant

Owing to Eq. (C2), the first cumulant of Eq. (13) can
be written as

$$
\mathcal{H}^{(1)}[F, P] = \sum_{\mathbf{k}} \int d\mathbf{J} \, \mathbf{k} \cdot \frac{\partial P}{\partial \mathbf{J}} C_{\mathbf{k}}^{(1)}(\mathbf{J}), \quad (C3)
$$

with the correlation function

$$
C_{\mathbf{k}}^{(1)}(\mathbf{J}) = \lim_{\Delta\tau \to +\infty} i \int_0^{\Delta\tau} dt \langle \delta F_{\mathbf{k}}(\mathbf{J}, t) \delta \Phi_{-\mathbf{k}}(\mathbf{J}, t) \rangle_F . \quad (C4)
$$

The next step of the calculation is to inject the time-
dependent expression of the DF and potential fluctua-
tions, from Eqs. (A14) and (A16). These fluctuations
are expressed as a function of the initial conditions,
$\delta F_{\mathbf{k}}(\mathbf{J}, 0)$. We assume that the initial fluctuations stem
from some uncorrelated Poisson shot noise, hence mak-
ing them Gaussian. Following eq. (41) of [7], and paying
attention to the prefactor $1/\sqrt{N}$ in Eq. (8), we have

$$
\langle \delta F_{\mathbf{k}}(\mathbf{J}, 0) \delta F_{\mathbf{k}'}(\mathbf{J}', 0) \rangle_F = M_{\text{tot}} \frac{\delta_{\mathbf{k}, -\mathbf{k}'}}{(2\pi)^d} \delta_{\text{D}}(\mathbf{J} - \mathbf{J}') F(\mathbf{J}). \quad (C5)
$$

Using this statistics, Eq. (C4) becomes

$$
\begin{equation}
\begin{split}
C_{\mathbf{k}}^{(1)}(\mathbf{J}) = {}& i M_{\text{tot}} F(\mathbf{J}) \psi_{\mathbf{k}\mathbf{k}'}^{*}(\mathbf{J}, \mathbf{J}, \mathbf{k} \cdot \mathbf{\Omega}) \\
& + i (2\pi)^d M_{\text{tot}} \mathbf{k} \cdot \frac{\partial F}{\partial \mathbf{J}} \sum_{\mathbf{k}'} \int d\mathbf{J}' F(\mathbf{J}') \frac{|\psi_{\mathbf{k}\mathbf{k}'}^{\text{d}}(\mathbf{J}, \mathbf{J}', \mathbf{k}' \cdot \mathbf{\Omega}')|^2}{\mathbf{k} \cdot \mathbf{\Omega} - \mathbf{k}' \cdot \mathbf{\Omega}' - i\gamma},
\end{split}
\tag{C6}
\end{equation}
$$
where we used the symmetry

$$ \psi_{-\mathbf{k}-\mathbf{k}'}^{d}(\mathbf{J}, \mathbf{J}', -\omega_{\mathrm{R}}) = \psi_{\mathbf{k}\mathbf{k}'}^{d*}(\mathbf{J}, \mathbf{J}', \omega_{\mathrm{R}}), \quad (\mathrm{C}7) $$

for $\omega_R \in \mathbb{R}$.

At this stage, following the truncation from Eq. (10), we must compute Eq. (C6) at order $o(G^2)$. We can generically expand the dressed coupling coefficient as

$$ \psi_{\mathbf{k}\mathbf{k}'}^{d}(\mathbf{J}, \mathbf{J}', \omega) \approx \psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') + \psi_{\mathbf{k}\mathbf{k}'}^{(2)}(\mathbf{J}, \mathbf{J}', \omega) + o(G^2), \quad (\mathrm{C}8) $$

with the scaling $\psi_{\mathbf{k}\mathbf{k}'}^{(2)} \propto G^2$, for $G \to 0$. Relying on the self-consistent relation from Eq. (A10), we get

$$ \psi_{\mathbf{k}\mathbf{k}}^{(2)}(\mathbf{J}, \mathbf{J}, \mathbf{k} \cdot \mathbf{\Omega}) = -(2\pi)^d \sum_{\mathbf{k}'} \int d\mathbf{J}' \frac{|\psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')|^2}{\mathbf{k} \cdot \mathbf{\Omega} - \mathbf{k}' \cdot \mathbf{\Omega}' + i\gamma} \quad (\mathrm{C}9) $$

where we used the symmetry from Eq. (A4a) and the prescription from Eq. (A11).

At order $o(G^2)$, Eq. (C6) then becomes

$$ C_{\mathbf{k}}^{(1)}(\mathbf{J}) = iM_{\text{tot}}F(\mathbf{J})\psi_{\mathbf{k}\mathbf{k}}^{*}(\mathbf{J}, \mathbf{J}) + i(2\pi)^d M_{\text{tot}} \sum_{\mathbf{k}} \int d\mathbf{J}' \frac{|\psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')|^2}{\mathbf{k} \cdot \mathbf{\Omega} - \mathbf{k}' \cdot \mathbf{\Omega}' - i\gamma} \times \left\{ \mathbf{k} \cdot \frac{\partial F}{\partial \mathbf{J}} F(\mathbf{J}') - \mathbf{k}' \cdot \frac{\partial F}{\partial \mathbf{J}'} F(\mathbf{J}) \right\}, \quad (\mathrm{C}10) $$

Performing the symmetrisation $\mathbf{k} \to -\mathbf{k}$ in Eq. (C3), and using the symmetry from Eq. (A4), we find that the term in $\psi_{\mathbf{k}\mathbf{k}}^{*}(\mathbf{J}, \mathbf{J})$ in Eq. (C10) does not contribute to $\mathcal{H}^{(1)}$. The final step of the calculation is to expand the resonant denominator using Plemelj formula

$$ \frac{1}{\omega_R - i\gamma} = \mathcal{P}\left(\frac{1}{\omega_R}\right) + i\pi\delta_D(\omega_R), \quad (\mathrm{C}11) $$

with $\mathcal{P}$ Cauchy principal value. Performing the symmetrisation $(\mathbf{k}, \mathbf{k}') \to (-\mathbf{k}, -\mathbf{k}')$ in Eq. (C3) and (C10), and using once again the symmetries from Eq. (A4), we find that the principal value does not contribute to $\mathcal{H}^{(1)}$. Ultimately, at order $o(G^2)$, we are left with

$$ \mathcal{H}^{(1)}[F, P] = - \sum_{\mathbf{k}, \mathbf{k}'} \int d\mathbf{J} d\mathbf{J}' \mathbf{k} \cdot \frac{\partial P}{\partial \mathbf{J}} B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') \times \left\{ \mathbf{k} \cdot \frac{\partial F}{\partial \mathbf{J}} F(\mathbf{J}') - \mathbf{k}' \cdot \frac{\partial F}{\partial \mathbf{J}'} F(\mathbf{J}) \right\}, \quad (\mathrm{C}12) $$

hence recovering Eq. (15).

## 2. Second cumulant

Let us now compute the second cumulant of Eq. (13). In that case, following Eq. (C1), we must compute averages of order $X^2$, with $X \propto \delta F \delta \Phi$. Since we are computing expressions at order $o(G^2)$, we follow the scalings from Eq. (A17) to simplify the expressions of $\delta \Phi(t)$ and

$\delta F(t)$ to be used in this computation. More precisely, at order $o(G^2)$, we may replace Eq. (A14) with

$$ \delta\Phi_{\mathbf{k}}(\mathbf{J}, t) = (2\pi)^d \sum_{\mathbf{k}'} \int d\mathbf{J}' e^{-i\mathbf{k} \cdot \mathbf{\Omega}'t} \delta F_{\mathbf{k}'}(\mathbf{J}', 0) \times \psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}'), \quad (\mathrm{C}13) $$

and Eq. (A16) with

$$ \delta F_{\mathbf{k}}(\mathbf{J}, t) = e^{-i\mathbf{k} \cdot \mathbf{\Omega} t} \delta F_{\mathbf{k}}(\mathbf{J}, 0). \quad (\mathrm{C}14) $$

This is a key step to simplify the upcoming calculations. The second cumulant of Eq. (13) then reads

$$ \mathcal{H}^{(2)}[F, P] = \sum_{\mathbf{k}, \mathbf{k}'} \int d\mathbf{J} d\mathbf{J}' \mathbf{k} \cdot \frac{\partial P}{\partial \mathbf{J}} \mathbf{k}' \cdot \frac{\partial P}{\partial \mathbf{J}'} C_{\mathbf{k}\mathbf{k}'}^{(2)}(\mathbf{J}, \mathbf{J}'), \quad (\mathrm{C}15) $$

where we introduced the correlation function

$$ C_{\mathbf{k}\mathbf{k}'}^{(2)}(\mathbf{J}, \mathbf{J}') = - \lim_{\Delta \to +\infty} \frac{1}{2} \frac{(2\pi)^d}{\Delta M_{\text{tot}} J_0} \int_{0}^{\Delta} dt \int_{0}^{\Delta} dt' \langle \langle [\delta F_{\mathbf{k}}(\mathbf{J}, t) \delta \Phi_{-\mathbf{k}}(\mathbf{J}, t)] [\delta F_{\mathbf{k}'}(\mathbf{J}', t') \delta \Phi_{-\mathbf{k}'}(\mathbf{J}', t')] \rangle \rangle_F, \quad (\mathrm{C}16) $$

with the notation $\langle\langle XY \rangle\rangle_F = \langle XY \rangle_F - \langle X \rangle_F \langle Y \rangle_F$.

Injecting the dependence from Eqs. (C13) and (C14), we can rewrite Eq. (C16) as

$$ C_{\mathbf{k}\mathbf{k}'}^{(2)}(\mathbf{J}, \mathbf{J}') = - \lim_{\Delta \to +\infty} \frac{1}{2} \frac{(2\pi)^{3d}}{\Delta M_{\text{tot}}} \int_{0}^{\Delta} dt \int_{0}^{\Delta} dt' \sum_{\mathbf{k}_1, \mathbf{k}'_1} \int d\mathbf{J}_1 d\mathbf{J}'_1 \\ \times e^{-i\mathbf{k} \cdot \mathbf{\Omega} t} e^{-i\mathbf{k}_1 \cdot \mathbf{\Omega}_1 t} \psi_{-\mathbf{k}\mathbf{k}_1}(\mathbf{J}, \mathbf{J}_1) \\ \times e^{-i\mathbf{k}' \cdot \mathbf{\Omega}' t} e^{-i\mathbf{k}'_1 \cdot \mathbf{\Omega}'_1 t} \psi_{-\mathbf{k}'\mathbf{k}'_1}(\mathbf{J}', \mathbf{J}'_1) \\ \times \langle\langle [\delta F_{\mathbf{k}}(\mathbf{J}, 0) \delta F_{\mathbf{k}_1}(\mathbf{J}_1, 0)] [\delta F_{\mathbf{k}'}(\mathbf{J}', 0) \delta F_{\mathbf{k}'_1}(\mathbf{J}'_1, 0)] \rangle\rangle_F \quad (\mathrm{C}17) $$

Computing the average from Eq. (C16) requires the computation of the four-point correlation of $\delta F_{\mathbf{k}}(\mathbf{J}, 0)$. We assume that the initial fluctuations are Gaussian so that writing, $\delta F(1) = \delta F_{\mathbf{k}_1}(\mathbf{J}_1, 0)$, we have

$$ \langle\langle [\delta F(1) \delta F(2)] [\delta F(3) \delta F(4)] \rangle\rangle_F = + \langle\langle \delta F(1) \delta F(3) \rangle\rangle_F \langle\langle \delta F(2) \delta F(4) \rangle\rangle_F + \langle\langle \delta F(1) \delta F(4) \rangle\rangle_F \langle\langle \delta F(2) \delta F(3) \rangle\rangle_F. \quad (\mathrm{C}18) $$

Following Eq. (C5), we can then rewrite Eq. (C17) as

$$ C_{\mathbf{k}\mathbf{k}'}^{(2)}(\mathbf{J}, \mathbf{J}') = - \lim_{\Delta \to +\infty} \frac{(2\pi)^d M_{\text{tot}}}{2\Delta} \int_0^\Delta dt \int_0^\Delta dt' \\ \times \left\{ e^{-i(\mathbf{k} \cdot \mathbf{\Omega} - \mathbf{k}' \cdot \mathbf{\Omega}')(t-t')} |\psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')|^2 F(\mathbf{J}) F(\mathbf{J}') \right. \\ + \delta_{\mathbf{k}, -\mathbf{k}'} \delta_{\mathbf{D}}(\mathbf{J} - \mathbf{J}') F(\mathbf{J}) \\ \left. \times \sum_{\mathbf{k}''} \int d\mathbf{J}'' e^{-i(\mathbf{k} \cdot \mathbf{\Omega} - \mathbf{k}'' \cdot \mathbf{\Omega}'')(t-t')} |\psi_{\mathbf{k}\mathbf{k}''}(\mathbf{J}, \mathbf{J}'')|^2 F(\mathbf{J}'') \right\}, \quad (\mathrm{C}19) $$

where we used the symmetries from Eq. (A4).
We now use the identity

$$ \begin{aligned} \lim_{\Delta \to +\infty} \frac{1}{\Delta} \int_0^\Delta dt \int_0^\Delta dt' e^{-i\omega_R(t-t')} &= \int_{-\infty}^{+\infty} dt e^{-i\omega_R t} \\ &= 2\pi\delta_D(\omega_R), \end{aligned} \quad (C20) $$

and Eq. (C19) becomes

$$ \begin{aligned} C_{kk'}^{(2)}(\mathbf{J}, \mathbf{J}') = &-B_{kk'}(\mathbf{J}, \mathbf{J}')F(\mathbf{J})F(\mathbf{J}') \quad (C21) \\ &-\delta_{\mathbf{k}, -\mathbf{k}'}\delta_D(\mathbf{J}-\mathbf{J}')\sum_{\mathbf{k}''} \int d\mathbf{J}'' B_{kk''}(\mathbf{J}, \mathbf{J}'')F(\mathbf{J})F(\mathbf{J}''). \end{aligned} $$

where we used the definition of $B_{kk'}(\mathbf{J}, \mathbf{J}')$ from Eq. (11).

The last step of the calculation is to inject Eq. (C21) into Eq. (C15). We obtain

$$ \mathcal{H}^{(2)}[F, P] = \sum_{\mathbf{k}, \mathbf{k}'} \int d\mathbf{J} d\mathbf{J}' B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') \mathbf{k} \cdot \frac{\partial P}{\partial \mathbf{J}} \times \left[ \mathbf{k} \cdot \frac{\partial P}{\partial \mathbf{J}} - \mathbf{k}' \cdot \frac{\partial P}{\partial \mathbf{J}'} \right] F(\mathbf{J}) F(\mathbf{J}'), \quad (C22) $$

hence recovering Eq. (16).

### 3. High-order cumulants

Following Eq. (C1), cumulants of order higher than two involve averages of order $X^k$, with $k \ge 3$ and $X \propto \delta F \delta \Phi$. Following Eq. (A17), we have $\delta \Phi = \mathcal{O}(G)$ for $G \to 0$. Therefore, one has $X^k = \mathcal{O}(G^k) = o(G^2)$, for $k \ge 3$. As a conclusion, in the dynamically hot limit, cumulants of order higher than two do not contribute to the large deviation Hamiltonian from Eq. (14).

## Appendix D: Properties

In this Appendix, we briefly justify the various properties of the large deviation Hamiltonian from Eq. (14).

### 1. Most probable path

As required by Eq. (17), we need to compute the functional gradient $\delta \mathcal{H} / \delta P(\mathbf{J})$. To do so, we rely on the fundamental identity

$$ \frac{\delta P(\mathbf{J}')}{\delta P(\mathbf{J})} = \delta_D(\mathbf{J} - \mathbf{J}'). \quad (D1) $$

Since $\mathcal{H}^{(2)}$ is quadratic in $P(\mathbf{J})$ (Eq. 16), one has $\partial \mathcal{H}^{(2)} / \partial P(\mathbf{J}) = 0$ in $P=0$. As a consequence, only $\mathcal{H}^{(1)}$ contributes to the most probable path. Integrating by parts w.r.t. $d\mathbf{J}$ the term $\mathbf{k} \cdot \partial P / \partial \mathbf{J}$ in Eq. (15) readily recovers the inhomogeneous Landau equation (10).

### 2. Conservation laws

**Mass conservation.** Following Eq. (19a), we have $\delta M / \delta F(\mathbf{J}) = 1$. We note from Eqs. (15) and (16) that the large deviation Hamiltonian only depends on derivatives of the conjugate field, $P(\mathbf{J})$. To check for mass conservation, we compute terms of the form

$$ \begin{aligned} \int d\mathbf{J} \frac{\delta M}{\delta F(\mathbf{J})} \frac{\delta \mathcal{H}}{\delta P(\mathbf{J})} &= \int d\mathbf{J} d\mathbf{J}' \frac{\delta}{\delta P(\mathbf{J})} \left[ \frac{\partial P}{\partial \mathbf{J}'} \right] \times \dots \\ &= \int d\mathbf{J} d\mathbf{J}' \frac{\partial}{\partial \mathbf{J}'} \left[ \delta_D(\mathbf{J} - \mathbf{J}') \right] \times \dots \\ &= - \int d\mathbf{J} \frac{\partial}{\partial \mathbf{J}} \left[ \dots \right] \\ &= 0, \end{aligned} \quad (D2) $$

This ensures consistency w.r.t. mass conservation.

**Energy conservation.** Following Eq. (19b), we have $\delta E / \delta F(\mathbf{J}) = H(\mathbf{J})$. Following some integration by parts and manipulations, we get from Eq. (15)

$$ \begin{aligned} \int d\mathbf{J} \frac{\delta E}{\delta F(\mathbf{J})} \frac{\delta \mathcal{H}^{(1)}}{\delta P(\mathbf{J})} &= - \sum_{\mathbf{k}, \mathbf{k}'} \int d\mathbf{J} d\mathbf{J}' B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') \mathbf{k} \cdot \mathbf{\Omega} \\ & \times \left[ \mathbf{k} \cdot \frac{\partial F}{\partial \mathbf{J}}(\mathbf{J}') - \mathbf{k}' \cdot \frac{\partial F}{\partial \mathbf{J}'}(\mathbf{J}) \right]. \end{aligned} \quad (D3) $$

Similarly, we get from Eq. (16)

$$ \begin{aligned} \int d\mathbf{J} \frac{\delta E}{\delta F(\mathbf{J})} \frac{\delta \mathcal{H}^{(2)}}{\delta P(\mathbf{J})} &= \sum_{\mathbf{k}, \mathbf{k}'} \int d\mathbf{J} d\mathbf{J}' B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') \mathbf{k} \cdot \mathbf{\Omega} \\ & \times 2F(\mathbf{J}) F(\mathbf{J}') \left[ \mathbf{k} \cdot \frac{\partial P}{\partial \mathbf{J}} - \mathbf{k}' \cdot \frac{\partial P}{\partial \mathbf{J}'} \right]. \end{aligned} \quad (D4) $$

We now perform the symmetrisation $(\mathbf{k}, \mathbf{J}) \leftrightarrow (\mathbf{k}', \mathbf{J}')$ in Eqs. (D3) and (D4) and rely on the symmetry from Eq. (A4). Both equations then involve

$$ B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') [\mathbf{k} \cdot \mathbf{\Omega} - \mathbf{k}' \cdot \mathbf{\Omega}'] = 0, \quad (D5) $$

which vanishes owing to the resonance condition from Eq. (11). As a conclusion, we therefore have

$$ \int d\mathbf{J} \frac{\delta E}{\delta F(\mathbf{J})} \frac{\partial \mathcal{H}}{\partial P(\mathbf{J})} = 0. \quad (D6) $$

This ensures consistency w.r.t. energy conservation.

### 3. Hamilton–Jacobi equation

Following Eq. (20), we have

$$ \frac{\delta S}{\delta F(\mathbf{J})} = - \ln[F(\mathbf{J})] + \text{cst.} \quad (D7) $$

Since $\mathcal{H}[F, P]$ only depends on gradients of $P$ (see Appendix D 2), the constant term in Eq. (D7) does not contribute to $\mathcal{H}[F, -\delta S / \delta F]$. Following Eq. (15), the first
cumulant contributes

$$
\mathcal{H}^{(1)}[F, -\delta S/\delta F] = -\sum_{\mathbf{k},\mathbf{k}'} \int d\mathbf{J} d\mathbf{J}' B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') \mathbf{k} \cdot \frac{\partial \ln[F(\mathbf{J})]}{\partial \mathbf{J}} \\
\times \left[ \mathbf{k} \cdot \frac{\partial F}{\partial \mathbf{J}} F(\mathbf{J}') - \mathbf{k}' \cdot \frac{\partial F}{\partial \mathbf{J}} F(\mathbf{J}) \right]. \quad (\text{D8})
$$

Noting that $B_{\mathbf{k}'\mathbf{k}}(\mathbf{J}', \mathbf{J}) = B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')$ (Eq. 11), we sym-
metrise Eq. (D8) with $(\mathbf{k}, \mathbf{J}) \leftrightarrow (\mathbf{k}', \mathbf{J}')$ to get

$$
\mathcal{H}^{(1)}[F, -\delta S/\delta F] = - \sum_{\mathbf{k}, \mathbf{k}'} \int d\mathbf{J} d\mathbf{J}' \frac{B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')}{F(\mathbf{J})F(\mathbf{J}')} \\
\times \left[ \mathbf{k} \cdot \frac{\partial F}{\partial \mathbf{J}} F(\mathbf{J}') - \mathbf{k}' \cdot \frac{\partial F}{\partial \mathbf{J}'} F(\mathbf{J}) \right]^2 . \quad (\text{D9})
$$

Following Eq. (16), the second cumulant contributes

$$
\mathcal{H}^{(2)}[F, -\delta S/\delta F] = \sum_{\mathbf{k}, \mathbf{k}'} \int d\mathbf{J} d\mathbf{J}' B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') \mathbf{k} \cdot \frac{\partial \ln[F(\mathbf{J})]}{\partial \mathbf{J}} \\
\times \left[ \mathbf{k} \cdot \frac{\partial \ln[F(\mathbf{J})]}{\partial \mathbf{J}} - \mathbf{k}' \cdot \frac{\partial \ln[F(\mathbf{J}')]}{\partial \mathbf{J}'} \right] F(\mathbf{J}) F(\mathbf{J}'). \quad (\text{D10})
$$

Performing the same symmetrisation as in Eq. (D9), we
are left with

$$
\mathcal{H}^{(2)}[F, -\delta S/\delta F] = \sum_{\mathbf{k}, \mathbf{k}'} \int d\mathbf{J} d\mathbf{J}' \frac{B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')}{F(\mathbf{J})F(\mathbf{J}')} \\
\times \left[ \mathbf{k} \cdot \frac{\partial F}{\partial \mathbf{J}} F(\mathbf{J}') - \mathbf{k}' \cdot \frac{\partial F}{\partial \mathbf{J}'} F(\mathbf{J}) \right]^2 . \quad (\text{D11})
$$

Combining Eqs. (D9) and (D11), we ultimately obtain
the expected relation

$$
\begin{align}
\mathcal{H}[F, -\delta S/\delta F] &= \mathcal{H}^{(1)}[F, -\delta S/\delta F] + \mathcal{H}^{(2)}[F, -\delta S/\delta F] \nonumber \\
&= 0. \tag{D12}
\end{align}
$$

4. Time reversal symmetry

Given that $\mathcal{H}^{(1)}[F, P]$ (resp. $\mathcal{H}^{(2)}[F, P]$) is linear (resp. quadratic) w.r.t. $P$, we immediately have

$$
\mathcal{H}[F, -P] = -\mathcal{H}^{(1)}[F, P] + \mathcal{H}^{(2)}[F, P]. \quad (\text{D13})
$$

Similarly, by linearity we have

$$
\mathcal{H}^{(1)}[F, P - \delta S / \delta F] = \mathcal{H}^{(1)}[F, P] + \mathcal{H}^{(2)}[F, -\delta S / \delta F]. \quad (\text{D14})
$$

As for the second cumulant, it reads

$$
\mathcal{H}^{(2)}[F, P - \delta S / \delta F] = \mathcal{H}^{(2)}[F, P] + \mathcal{H}^{(2)}[F, -\delta S / \delta F] \\
\qquad + \tilde{\mathcal{H}}^{(2)}[F, P, -\delta S / \delta F] + \tilde{\mathcal{H}}^{(2)}[F, -\delta S / \delta F, P], \quad (\text{D15})
$$

where, following Eq. (16), we introduced

$$
\tilde{\mathcal{H}}^{(2)}[F, P, Q] = \sum_{\mathbf{k}, \mathbf{k}'} \int d\mathbf{J} d\mathbf{J}' B_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}') \mathbf{k} \cdot \frac{\partial P}{\partial \mathbf{J}} \\
\times \left[ \mathbf{k} \cdot \frac{\partial Q}{\partial \mathbf{J}} - \mathbf{k}' \cdot \frac{\partial Q}{\partial \mathbf{J}'} \right] F(\mathbf{J}) F(\mathbf{J}'). \quad (\text{D16})
$$

With the same symmetrisation as in Eq. (D9), one gets

$$
\tilde{\mathcal{H}}^{(2)}[F, P, -\delta S/\delta F] = -\mathcal{H}^{(1)}[F, P], \quad (\text{D17a})
$$

$$
\tilde{\mathcal{H}}^{(2)}[F, -\delta S/\delta F, P] = -\mathcal{H}^{(1)}[F, P]. \quad (\text{D17b})
$$

Recalling the result from Eq. (D12), all the previous re-
lations lead to the needed result, namely

$$
\mathcal{H}[F, -P] = \mathcal{H}[F, P - \delta S / \delta F]. \quad (\text{D18})
$$

5. Gradient structure

The expression of $Q[F]$ in Eq. (24) follows from integration by parts of Eq. (16). We now compute the r.h.s. of Eq. (25). Following Eq. (D7), we face the term

$$
\int d\mathbf{J}' Q[F](\mathbf{J}, \mathbf{J}') \frac{\partial S[F]}{\partial F(\mathbf{J}')} = \int d\mathbf{J}' \mathbf{k} \cdot \frac{\partial}{\partial \mathbf{J}} \left\{ \frac{\partial}{\partial \mathbf{J}'} \cdot \mathbf{A}(\mathbf{J}, \mathbf{J}') \right\} \\
\times \left\{ -\ln[F(\mathbf{J}')] + \text{cst} \right\}, \quad (\text{D19})
$$

where $\mathbf{A}(\mathbf{J}, \mathbf{J}')$ follows from Eq. (24). Discarding boundary terms, we find that the constant terms do not contribute. Integrating by parts w.r.t. $d\mathbf{J}'$, we get

$$
(D19) = \mathbf{k} \cdot \frac{\partial}{\partial \mathbf{J}} \left\{ \int d\mathbf{J}' \frac{1}{F(\mathbf{J}')} \mathbf{A}(\mathbf{J}, \mathbf{J}') \cdot \frac{\partial F}{\partial \mathbf{J}'} \right\}. \quad (\text{D20})
$$

Injecting $\mathbf{A}(\mathbf{J}, \mathbf{J}')$ from Eq. (24), one recovers Eq. (25).

6. Stochastic Landau equation

As defined in Eq. (27), the correlation function of the
stochastic noise, $\zeta[F](\mathbf{J}, \tau)$, involves Dirac deltas. This
makes the sampling of effective realisations challenging.
In addition, although guaranteed by Eq. (18), it is not
strikingly obvious that Eq. (26), indeed, complies with
the system’s conservation laws. In this Appendix, we
tackle these two issues and devise a “diagonal” rewriting
of $\zeta[F](\mathbf{J}, \tau)$: it is sourced by a normal random Gaussian
field and makes the conservation laws obvious.

For any given resonance vector **k**, we assume that we
can perform the change of variables **J**→(*ω* = **k**·**Ω**, **z**) [see,
e.g., 25, for an explicit example] with a non-degenerate
Jacobian, *J* = |∂(*ω*, **z**) / ∂**J**|. Physically, *ω* is the reso-
nance frequency, and **z** covers the sets of all orbits that
resonate with it. The existence of the present mapping is
mandatory for the resonance condition, δ_D(**k**·**Ω** − **k**'·**Ω**'),
to be generically well-posed in the Landau Eq. (10).

Let us consider the following ansatz for the noise

$$
\zeta[F](\mathbf{J}, \tau) = \frac{\sqrt{\pi} M_{\text{tot}}}{\sqrt{N}} \frac{\partial}{\partial \mathbf{J}} \cdot \left[ \sum_{\mathbf{k}, \mathbf{k}'} \int d\mathbf{J}' \left[ \mathcal{J} \mathcal{J}' \right]^{1/2} \right. \\
\times |\psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')| [F(\mathbf{J}) F(\mathbf{J}')]^{1/2} \delta_D(\omega - \omega') \\
\left. \times \{\eta_{\mathbf{k}\mathbf{k}'}(\mathbf{z}, \mathbf{z}', \omega, \tau) - \eta_{\mathbf{k}'\mathbf{k}}(\mathbf{z}', \mathbf{z}, \omega', \tau)\} \right], \quad (\text{D21})
$$
where we used shortened notations for the change of variables, $\mathbf{J} \leftrightarrow \mathbf{k}(\omega, \mathbf{z})$ and its Jacobian $\mathcal{J}$, and similarly for $\mathbf{J}'$. In Eq. (D21), we also introduced the normal Gaussian random field, $\eta_{\mathbf{k}\mathbf{k}'}(\mathbf{z}, \mathbf{z}', \omega, \tau)$, obeying

$$ \langle \eta_{\mathbf{k}\mathbf{k}'}(\mathbf{z}, \mathbf{z}', \omega, \tau) \rangle = 0, \quad (\text{D22a}) $$

$$ \langle \eta_{\mathbf{k}\mathbf{k}_1}(\mathbf{z}, \mathbf{z}_1, \omega, \tau) \eta_{\mathbf{k}'\mathbf{k}'_1}(\mathbf{z}', \mathbf{z}'_1, \omega', \tau') \rangle = \delta_{\mathbf{k}\mathbf{k}'} \delta_{\mathbf{k}_1\mathbf{k}'_1} \quad (\text{D22b}) \\ \times \delta_{\mathbf{D}}(\mathbf{z}-\mathbf{z}') \delta_{\mathbf{D}}(\mathbf{z}_1-\mathbf{z}'_1) \delta_{\mathbf{D}}(\omega-\omega') \delta_{\mathbf{D}}(\tau-\tau'). $$

Equation (D21) easily complies with the conservation laws from Eqs. (19). Indeed, Eq. (D21) is the divergence of a flux in action space, hence the total mass is conserved. As for the total energy, starting from Eq. (19b), up to prefactors, one must compute

$$ \frac{dE}{dt} \propto \sum_{\mathbf{k}, \mathbf{k}'} \int d\mathbf{J} d\mathbf{J}' \omega \delta_{\mathbf{D}}(\omega - \omega') [\mathcal{J}\mathcal{J}']^{1/2} \\ \times |\psi_{\mathbf{k}\mathbf{k}'}(\mathbf{J}, \mathbf{J}')| [F(\mathbf{J})F(\mathbf{J}')]^{1/2} \\ \times \{\eta_{\mathbf{k}\mathbf{k}'}(\mathbf{z}, \mathbf{z}', \omega, \tau) - \eta_{\mathbf{k}'\mathbf{k}}(\mathbf{z}', \mathbf{z}, \omega', \tau)\}. \quad (\text{D23}) $$

Performing the symmetrisation $(\mathbf{k}, \mathbf{J}) \leftrightarrow (\mathbf{k}', \mathbf{J}')$ leaves us with $(\omega - \omega') \delta_{\mathbf{D}}(\omega - \omega') = 0$, so that $dE/dt = 0$.

Let us now check Eqs. (27). Since $\zeta[F]$ is linear w.r.t. $\eta$, Eq. (D22a) naturally imposes $\langle \zeta[F] \rangle = 0$, as required by Eq. (27a). As for the correlation function, we write

$$ \langle \zeta[F](\mathbf{J}, \tau) \zeta[F](\mathbf{J}', \tau') \rangle = \pi m M_{\text{tot}} \sum_{\substack{\mathbf{k}, \mathbf{k}' \\ \mathbf{k}', \mathbf{k}'_1}} \int d\mathbf{J}_1 d\mathbf{J}'_1 \mathbf{k} \cdot \frac{\partial}{\partial \mathbf{J}} \left\{ \right. \\ \times \mathbf{k}' \cdot \frac{\partial}{\partial \mathbf{J}'} \left[ [\mathcal{J}\mathcal{J}_1 \mathcal{J}'\mathcal{J}'_1]^{1/2} |\psi_{\mathbf{k}\mathbf{k}_1}(\mathbf{J}, \mathbf{J}_1)| |\psi_{\mathbf{k}'\mathbf{k}'_1}(\mathbf{J}', \mathbf{J}'_1)| \right. \\ \times \delta_{\mathbf{D}}(\omega - \omega_1) \delta_{\mathbf{D}}(\omega' - \omega'_1) [F(\mathbf{J})F(\mathbf{J}_1)F(\mathbf{J}')F(\mathbf{J}'_1)]^{1/2} \\ \times \langle \{\eta_{\mathbf{k}\mathbf{k}_1}(\mathbf{z}, \mathbf{z}_1, \omega, \tau) - \eta_{\mathbf{k}_1\mathbf{k}}(\mathbf{z}_1, \mathbf{z}, \omega_1, \tau)\} \\ \times \{\eta_{\mathbf{k}'\mathbf{k}'_1}(\mathbf{z}', \mathbf{z}'_1, \omega', \tau') - \eta_{\mathbf{k}'_1\mathbf{k}}(\mathbf{z}', \mathbf{z}, \omega'_1, \tau')\} \rangle \left. \right\}, \quad (\text{D24}) $$

with $\mathbf{J} \leftrightarrow \mathbf{k}(\omega, \mathbf{z})$ and similarly for $\mathbf{J}', \mathbf{J}_1, \mathbf{J}'_1$.

In Eq. (D24), to compute a given crossed term, say $\langle \eta_{\mathbf{k}\mathbf{k}_1} \eta_{\mathbf{k}'\mathbf{k}'_1} \rangle$, we use Eq. (D22b) and face a product of three resonant Dirac deltas in frequencies. We write it as

$$ \delta_{\mathbf{D}}(\omega - \omega_1) \delta_{\mathbf{D}}(\omega' - \omega'_1) \delta_{\mathbf{D}}(\omega - \omega') = \delta_{\mathbf{D}}(\omega - \omega') \delta_{\mathbf{D}}(\omega_1 - \omega'_1) \delta_{\mathbf{D}}(\omega - \omega_1). \quad (\text{D25}) $$

For the other crossed terms, we pick the appropriate set of differences of frequencies. Finally, we also use

$$ [\mathcal{J}\mathcal{J}']^{1/2} \delta_{\mathbf{k}\mathbf{k}'} \delta_{\mathbf{D}}(\omega - \omega') \delta_{\mathbf{D}}(\mathbf{z} - \mathbf{z}') = \delta_{\mathbf{k}\mathbf{k}'} \delta_{\mathbf{D}}(\mathbf{J} - \mathbf{J}'), \quad (\text{D26}) $$

and similar variations. Following simple manipulations, one ultimately recovers Eq. (27b).

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