Path Large Deviations for the Kinetic Theory of Weak Turbulence

Abstract

We consider a generic Hamiltonian system of nonlinear interacting waves with 3-wave interactions. In the kinetic regime of wave turbulence, which assumes weak nonlinearity and large system size, the relevant observable associated with the wave amplitude is the empirical spectral density that appears as the natural precursor of the spectral density, or spectrum, for finite system size. Following classical derivations of the Peierls equation for the moment generating function of the wave amplitudes in the kinetic regime, we propose a large deviation estimate for the dynamics of the empirical spectral density, where the number of admissible wavenumbers, which is proportional to the volume of the system, appears as the natural large deviation parameter. The large deviation stochastic Hamiltonian that quantifies the minus of the log-probability of a trajectory is computed within the kinetic regime which assumes the Random Phase approximation for weak nonlinearity. We compare this Hamiltonian with the one for a system of modes interacting in a mean-field way with the empirical spectrum. Its relationship with the Random Phase and Amplitude approximation is discussed. Moreover, for the specific case when no forces and dissipation are present, a few fundamental properties of the large deviation dynamics are investigated. We show that the latter conserves total energy and momentum, as expected for a 3-wave interacting systems. In addition, we compute the equilibrium quasipotential and check that global detailed balance is satisfied at the large deviation level. Finally, we discuss briefly some physical applications of the theory.

Appendices

Appendix A Perturbative Expansion of the Moment Generating Function $Z_{L,\epsilon }$ (21)

Our goal is to compute the moment generating function $Z_{L,\epsilon }$ at an intermediate time $\Delta t$ (which satisfies $1\ll \Delta t\ll \epsilon ^{-2}$ as prescribed by (12)), from time $t=0$, conditioned on ${\hat{n}}(\varvec{\xi },0)=n(\varvec{\xi })$, or equivalently, $\{ \left|b_{\varvec{k}}\right|^{2} \}$. We recall that ${\mathbb {E}}_n$ refers to the average with respect to the uniform measure over the phases (RP).

To do so, we will look for a perturbation expansion in $\epsilon$ of the dynamics (7).

1.1 A.1 Perturbative Expansion of the Modes (7)

In order to anticipate the continuous limit $L\rightarrow \infty$, we will consider a slightly modified but equivalent evolution equation (7). The Kronecker-$\delta$ will be replaced by ${{\left( \tfrac{2\pi }{L}\right) ^{d}}}\chi _{L}^{d}$, with $\chi _{L}^{d}$ the normalized characteristic function of the set $\left[ -\tfrac{\pi }{L},\tfrac{\pi }{L}\right] ^{d}$, defined as

$$\begin{aligned} \chi _{L}^{d}(\varvec{x})={\left\{ \begin{array}{ll} \left( \tfrac{L}{2\pi }\right) ^{d} &{} \text {if }\varvec{x}\in \left[ -\tfrac{\pi }{L},\tfrac{\pi }{L}\right] ^{d}\\ 0 &{} \text {otherwise} \end{array}\right. }. \end{aligned}$$

$\chi _{L}^{d}(\varvec{x})$ is then a precursor of the Dirac-$\delta$ in d-dimension: $\chi _{L}^{d}\underset{L\rightarrow \infty }{\rightarrow }\delta ^{d}$. Therefore, Eq. (7) is replaced by

$$\begin{aligned} i\frac{\text {d}b_{\varvec{k}_{1}}}{\text {d}t} =3\epsilon \left( \frac{2\pi }{L}\right) ^{3d/2}\sum _{\begin{array}{c} {\sigma }_{123}\\ \sigma _{1}=-1 \end{array} }\sum _{\varvec{k}_{2},\varvec{k}_{3}}V_{{\varvec{k}}_{123}}^{{\sigma }_{123}}b_{\varvec{k}_{2}}^{\sigma _{2}}b_{\varvec{k}_{3}}^{\sigma _{3}}\mathrm{e}^{-i\left( {\sigma }_{123}\cdot {\omega }_{123}\right) t}\chi _{L}^{d}({\sigma }_{123}\cdot {\varvec{k}}_{123}) \end{aligned}$$

(A1)

Since a detailed book keeping of the indices will be important in the sequel, we have introduced a new notation to refer to a triad: ${x}_{lmn}=(x_{l},x_{m},x_{n})$. We will note with upper indices the components that take a minus sign. For instance $(x_{1},-x_{2},x_{3})$ will be denoted ${x}_{13}^{2}$. Indices will be always labelled in ascending order to avoid confusion on their position within the triplet $(x_{l},x_{m},x_{n})$.

We are now ready to perform the perturbation expansion for the dynamics (A1). We look for a solution $b_{\varvec{k}}(t)$ for $t\geqslant 0$ as a perturbation expansion in $\epsilon$:

$$\begin{aligned} b_{\varvec{k}}(t)=b_{\varvec{k}}^{(0)}(t)+\epsilon b_{\varvec{k}}^{(1)}(t)+\epsilon ^{2}b_{\varvec{k}}^{(2)}(t)+{\mathcal {O}}\left( \epsilon ^{3}\right) \quad . \end{aligned}$$

(A2)

with $b_{k}(0)=b_{k}^{(0)}(0)$.

Integrating from $t=0$ to $t=\Delta t$, one gets the following hierarchy:

$$\begin{aligned}&b_{\varvec{k}_{1}}^{(0)}(\Delta t) = b_{\varvec{k}}^{(0)}(0) \end{aligned}$$

(A3)

$$\begin{aligned}&b_{\varvec{k}_{1}}^{(1)\sigma _{1}}(\Delta t) \end{aligned}$$

(A4)

$$\begin{aligned}&\; =-3i\left( \frac{2\pi }{L}\right) ^{3d/2} \!\!\! \sigma _{1}\sum _{\sigma _{2},\sigma _{3}}\sum _{\varvec{k}_{2},\varvec{k}_{3}}V_{{\varvec{k}}_{123}}^{{\sigma }_{23}^{1}}b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{3}}^{(0)\sigma _{3}}\Delta _{\Delta t}\left( -{\sigma }_{23}^{1}\cdot {\omega }_{23}^{1}\right) \chi _{L}^{d}({\sigma }_{23}^{1}\cdot {\varvec{k}}_{23}^{1}) \nonumber \\&b_{\varvec{k}_{1}}^{(2)}(\Delta t) \end{aligned}$$

(A5)

$$\begin{aligned}&\; =-6i\left( \frac{2\pi }{L}\right) ^{3d/2} \!\! \sum _{\begin{array}{c} {\sigma } \\ \sigma _{1}=-1 \end{array}} \sum _{\varvec{k}_{2},\varvec{k}_{3}}V_{{\varvec{k}}_{123}}^{{\sigma }_{123}}b_{\varvec{k}_{2}}^{(0)\sigma _{2}} \nonumber \\& \times \left( \int _{0}^{\Delta t}b_{\varvec{k}_{3}}^{(1)\sigma _{3}}(t)\mathrm{e}^{-i({\sigma }_{123}\cdot {\omega }_{123})t}\mathrm{d}t\right) \chi _{L}^{d}({\sigma }_{123}\cdot {\varvec{k}}_{123}) \end{aligned}$$

(A6)

with

$$\begin{aligned} \Delta _{T}(x)=\int _{0}^{T} \mathrm {e}^{ixt}\mathrm {d}t \end{aligned}$$

Injecting (A4) into (A5) yields

$$\begin{aligned}&b_{\varvec{k}_{1}}^{(2)}(\Delta t) =18\left( \frac{2\pi }{L}\right) ^{3d} \end{aligned}$$

(A7)

$$\begin{aligned}&\qquad \times \!\!\sum _{\begin{array}{c} \sigma _{2},\sigma _{3},\sigma _{4},\sigma _{5} \\ \sigma _{1}=-1 \end{array} } \!\!\!\sigma _{1}\sigma _{3}\!\!\!\sum _{\varvec{k}_{2},\varvec{k}_{3},\varvec{k}_{4},\varvec{k}_{5}}\left\{ V_{{\varvec{k}}_{123}}^{{\sigma }_{123}}V_{{\varvec{k}}_{345}}^{{\sigma }_{45}^{3}}\chi _{L}^{d}\left( {\sigma }_{123}\cdot {\varvec{k}}_{123}\right) \chi _{L}^{d}\left( {\sigma }_{45}^{3}\cdot {\varvec{k}}_{345}\right) \right. \nonumber \\&\qquad \qquad \qquad \left. \times \left( b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{4}}^{(0)\sigma _{4}}b_{\varvec{k}_{5}}^{(0)\sigma _{5}}\right) {\tilde{E}}_{\Delta t}\left( -{\sigma }_{123}\cdot {\omega }_{123};-{\sigma }_{45}^{3}\cdot {\omega }_{345}\right) \right\} \end{aligned}$$

(A8)

$$\begin{aligned} {\tilde{E}}_{T}(x,y) =\int _{0}^{T}\mathrm {e}^{ixt}\Delta _{t}(y) \mathrm {d}t =-i\frac{\Delta _{T}(y+x)-\Delta _{T}(x)}{y}. \end{aligned}$$

(A9)

1.2 A.2 Computation of the Moment Generating Function $Z_{L,\epsilon }$ (21)

From the definition of the moment generating function $Z_{L,\epsilon }$ (21), using (8), one obtains

$$\begin{aligned} Z_{L,\epsilon } ={\mathbb {E}}_{n}\left[ \mathrm{e}^{\sum _{\varvec{k}}\lambda (\varvec{k})\left( \left|b_{\varvec{k}}(\Delta t)\right|^{2}-\left|b_{\varvec{k}}(0)\right|^{2}\right) }\right] \, . \end{aligned}$$

(A10)

From Eq. (A2), one obtains

$$\begin{aligned} \left|b_{\varvec{k}}(\Delta t)\right|^{2}&=\left|b_{\varvec{k}}^{(0)}\right|^{2}+\epsilon \left( b_{\varvec{k}}^{(1)}b_{\varvec{k}}^{(0)*}+b_{\varvec{k}}^{(1)*}b_{\varvec{k}}^{(0)}\right) \\& +\epsilon ^{2}\left( b_{\varvec{k}}^{(2)}b_{\varvec{k}}^{(0)*}+b_{\varvec{k}}^{(2)*}b_{\varvec{k}}^{(0)}+b_{\varvec{k}}^{(1)}b_{\varvec{k}}^{(1)*}\right) +{\mathcal {O}}\left( \epsilon ^{3}\right) \,. \end{aligned}$$

Hence, the expansion of $Z_{L,\epsilon }$ with respect to $\epsilon$ reads as

$$\begin{aligned} Z_{L,\epsilon }&={\mathbb {E}}_{n}\left[ 1+\epsilon G_{1}+\epsilon ^{2}\left( G_{2}+\frac{1}{2}G_{1}^{2}\right) +{\mathcal {O}}\left( \epsilon ^{3}\right) \right] \end{aligned}$$

(A11)

with

$$\begin{aligned} G_{1}&=\sum _{\varvec{k}}\lambda (\varvec{k})\left( b_{\varvec{k}}^{(1)}b_{\varvec{k}}^{(0)*}+b_{\varvec{k}}^{(1)*}b_{\varvec{k}}^{(0)}\right) \\ G_{2}&=\sum _{\varvec{k}}\lambda (\varvec{k})\left( b_{\varvec{k}}^{(2)}b_{\varvec{k}}^{(0)*}+b_{\varvec{k}}^{(2)*}b_{\varvec{k}}^{(0)}+b_{\varvec{k}}^{(1)}b_{\varvec{k}}^{(1)*}\right) \,. \end{aligned}$$

1.2.1 A.2.1 Computation of ${\mathcal {O}}\left( \epsilon ^{1}\right)$ Term

From Eqs. (A3) and (A4), $G_{1}$ reads as

$$\begin{aligned} G_{1}&= 3i\left( \frac{2\pi }{L}\right) ^{3d/2} \!\! \sum _{{\sigma }_{123}}\sum _{{\varvec{k}}_{123}}\lambda (\varvec{k}_{1})\sigma _{1} \nonumber \\&\quad \times \left[ V_{{\varvec{k}}_{123}}^{{\sigma }_{123}}b_{\varvec{k}_{1}}^{(0)\sigma _{1}}b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{3}}^{(0)\sigma _{3}}\Delta _{\Delta t}\left( -{\sigma }_{123}\cdot {\omega }_{123}\right) \chi _{L}^{d}({\sigma }_{123}\cdot {\varvec{k}}_{123})\right] \;. \end{aligned}$$

(A12)

However, since

$$\begin{aligned} {\mathbb {E}}_{n}\left[ b_{\varvec{k}_{1}}^{(0)\sigma _{1}}b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{3}}^{(0)\sigma _{3}}\right] =0 \end{aligned}$$

for any ${\sigma }=(\sigma _{1},\sigma _{2},\sigma _{3})$, we deduce ${\mathbb {E}}_n\left[ G_1 \right] =0$, namely that the ${\mathcal {O}}\left( \epsilon ^{1}\right)$ term of $Z_{L,\epsilon }$ vanishes.

As a remark, one may notice that ${\mathbb {E}}_{n}\left[ \prod _{i=1}^{p}b_{\varvec{k}_{i}}^{(0)\sigma _{i}}\right] =0$ for any odd integer p. We thus deduce that there is not any correction of order $\epsilon ^{p}$ (p odd) in the moment generating function $Z_{L,\epsilon }$.

1.2.2 A.2.2 Computation of ${\mathcal {O}}\left( \epsilon ^{2}\right)$ Term

Expressions of $G_{2}$ and $G_{1}^{2}$

From Eqs. (A3), (A4) and (A5), $G_{2}$ reads as

$$\begin{aligned} G_{2}&=\left( \frac{2\pi }{L}\right) ^{3d}\sum _{\varvec{k}_{1}}\lambda (\varvec{k}_{1}) \\&\quad \times \Bigg \{ 18 \!\!\!\! \sum _{\sigma _{1},\sigma _{2},\sigma _{3},\sigma _{4},\sigma _{5}} \!\!\! \sigma _{1}\sigma _{3} \!\!\! \sum _{\varvec{k}_{2},\varvec{k}_{3},\varvec{k}_{4},\varvec{k}_{5}} \!\!\! V_{{\varvec{k}}_{123}}^{{\sigma }_{123}}V_{{\varvec{k}}_{345}}^{{\sigma }_{45}^{3}}\chi _{L}^{d}({\sigma }_{123}\cdot {\varvec{k}}_{123})\chi _{L}^{d}({\sigma }_{45}^{3}\cdot {\varvec{k}}_{345}) \\& \times \left( b_{\varvec{k}_{1}}^{(0)\sigma _{1}}b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{4}}^{(0)\sigma _{4}}b_{\varvec{k}_{5}}^{(0)\sigma _{5}}\right) {\tilde{E}}_{\Delta t}\left( -{\sigma }_{123}\cdot {\omega }_{123};-{\sigma }_{45}^{3}\cdot {\omega }_{345}\right) \\& +\frac{9}{2}\sum _{\sigma _{1},\sigma _{2},\sigma _{3},\sigma _{4},\sigma _{5}} \!\!\! \sigma _{1}^{2} \!\! \sum _{\varvec{k}_{2},\varvec{k}_{3},\varvec{k}_{4},\varvec{k}_{5}}V_{{\varvec{k}}_{123}}^{{\sigma }_{23}^{1}}V_{{\varvec{k}}_{145}}^{{\sigma }_{1}^{45}}\left( b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{3}}^{(0)\sigma _{3}}b_{\varvec{k}_{4}}^{(0)-\sigma _{4}}b_{\varvec{k}_{5}}^{(0)-\sigma _{5}}\right) \\& \times \Delta _{\Delta t}\left( -{\sigma }_{23}^{1}\cdot {\omega }_{123}\right) \Delta _{\Delta t}^{*}\left( {\sigma }_{1}^{45}\cdot {\omega }_{145}\right) \chi _{L}^{d}({\sigma }_{23}^{1}\cdot {\varvec{k}}_{123})\chi _{L}^{d}({\sigma }_{1}^{45}\cdot {\varvec{k}}_{145}) \Bigg \} \end{aligned}$$

Furthermore, one gets from Eq. (A12):

$$\begin{aligned} G_{1}^{2}&=-9\left( \frac{2\pi }{L}\right) ^{3d} \!\! \sum _{{\sigma }_{123},{\sigma }_{456}}\sum _{{\varvec{k}}_{123},{\varvec{k}}_{456}}\sigma _{1}\lambda (\varvec{k}_{1})\sigma _{4}\lambda (\varvec{k}_{4}) \\& \times \left[ V_{{\varvec{k}}_{123}}^{{\sigma }_{123}}b_{\varvec{k}_{1}}^{(0)\sigma _{1}}b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{3}}^{(0)\sigma _{3}}\Delta _{\Delta t}\left( {\sigma }_{123}\cdot {\omega }_{123}\right) \chi _{L}^{d}({\sigma }_{123}\cdot {\varvec{k}}_{123})\right] \\& \times \left[ V_{{\varvec{k}}_{456}}^{{\sigma }_{456}}b_{\varvec{k}_{4}}^{(0)\sigma _{4}}b_{\varvec{k}_{5}}^{(0)\sigma _{5}}b_{\varvec{k}_{6}}^{(0)\sigma _{6}}\Delta _{\Delta t}\left( {\sigma }_{456}\cdot {\omega }_{456}\right) \chi _{L}^{d}({\sigma }_{456}\cdot {\varvec{k}}_{456})\right] \end{aligned}$$

Random phase averages ${\mathbb {E}}_{n}\left[ G_{2}\right]$ and ${\mathbb {E}}_{n}\left[ G_{1}^{2}\right]$ The contributions $G_{2}$ and $G_{1}^{2}$ contain respectively terms of order 4 and 6 in $b_{\varvec{k}}$, whose average over the uniform phase distribution (RP) yields non zero contributions. ${\mathbb {E}}_{n}\left[ b_{\varvec{k}_{1}}^{(0)\sigma _{1}}b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{3}}^{(0)\sigma _{3}}b_{\varvec{k}_{4}}^{(0)\sigma _{4}}\right]$ and ${\mathbb {E}}_{n}\left[ b_{\varvec{k}_{1}}^{(0)\sigma _{1}}b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{3}}^{(0)\sigma _{3}}b_{\varvec{k}_{4}}^{(0)\sigma _{4}}b_{\varvec{k}_{5}}^{(0)\sigma _{5}}b_{\varvec{k}_{6}}^{(0)\sigma _{6}}\right]$ will contain non zero contributions as long as the number of b matches the number of $b^{*}$ in order to form non oscillating terms.

• The first term in $G_{2}$ is ${\mathbb {E}}_{n}\left[ b_{\varvec{k}_{1}}^{(0)\sigma _{1}}b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{4}}^{(0)\sigma _{4}}b_{\varvec{k}_{5}}^{(0)\sigma _{5}}\right]$. The latter can be explicitly computed under the RP distribution. It contains three non-vanishing terms only:

  $$\begin{aligned}&{\mathbb {E}}_{n}\left[ b_{\varvec{k}_{1}}^{(0)\sigma _{1}}b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{4}}^{(0)\sigma _{4}}b_{\varvec{k}_{5}}^{(0)\sigma _{5}}\right] \\&\quad =\delta _{\sigma _{1},-\sigma _{2}}\delta _{\sigma _{4},-\sigma _{5}}\delta _{\varvec{k}_{1},\varvec{k}_{2}}\delta _{\varvec{k}_{4},\varvec{k}_{5}}\left|b_{\varvec{k}_{1}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{4}}^{(0)}\right|^{2}\\&\quad \quad +\left( \delta _{\sigma _{1},-\sigma _{4}}\delta _{\sigma _{2},-\sigma _{5}}\delta _{\varvec{k}_{1},\varvec{k}_{4}}\delta _{\varvec{k}_{2},\varvec{k}_{5}}+\delta _{\sigma _{1},-\sigma _{5}}\delta _{\sigma _{2},-\sigma _{4}}\delta _{\varvec{k}_{1},\varvec{k}_{5}}\delta _{\varvec{k}_{2},\varvec{k}_{4}}\right) \left|b_{\varvec{k}_{1}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2} \end{aligned}$$

• The second term in $G_{2}$ is ${\mathbb {E}}_{n}\left[ b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{3}}^{(0)\sigma _{3}}b_{\varvec{k}_{4}}^{(0)-\sigma _{4}}b_{\varvec{k}_{5}}^{(0)-\sigma _{5}}\right]$ and reads as

  $$\begin{aligned}&{\mathbb {E}}_{n}\left[ b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{3}}^{(0)\sigma _{3}}b_{\varvec{k}_{4}}^{(0)-\sigma _{4}}b_{\varvec{k}_{5}}^{(0)-\sigma _{5}}\right] \\&\quad =\delta _{\sigma _{2},-\sigma _{3}}\delta _{\sigma _{4},-\sigma _{5}}\delta _{\varvec{k}_{2},\varvec{k}_{3}}\delta _{\varvec{k}_{4},\varvec{k}_{5}}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{4}}^{(0)}\right|^{2}\\&\quad \quad +\left( \delta _{\sigma _{2},\sigma _{4}}\delta _{\sigma _{3},\sigma _{5}}\delta _{\varvec{k}_{2},\varvec{k}_{4}}\delta _{\varvec{k}_{3},\varvec{k}_{5}}+\delta _{\sigma _{2},\sigma _{5}}\delta _{\sigma _{3},\sigma _{4}}\delta _{\varvec{k}_{2},\varvec{k}_{5}}\delta _{\varvec{k}_{3},\varvec{k}_{4}}\right) \left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2} \end{aligned}$$

• Finally, the only term in $G_{1}^{2}$ is ${\mathbb {E}}_{n}\left[ b_{\varvec{k}_{1}}^{(0)\sigma _{1}}b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{3}}^{(0)\sigma _{3}}b_{\varvec{k}_{4}}^{(0)\sigma _{4}}b_{\varvec{k}_{5}}^{(0)\sigma _{5}}b_{\varvec{k}_{6}}^{(0)\sigma _{6}}\right]$. It contains 15 non-vanishing terms:

  $$\begin{aligned}&{\mathbb {E}}_{n}\left[ b_{\varvec{k}_{1}}^{(0)\sigma _{1}}b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{3}}^{(0)\sigma _{3}}b_{\varvec{k}_{4}}^{(0)\sigma _{4}}b_{\varvec{k}_{5}}^{(0)\sigma _{5}}b_{\varvec{k}_{6}}^{(0)\sigma _{6}}\right] \\&\quad = \delta _{\sigma _{1},-\sigma _{2}}\delta _{\varvec{k}_{1},\varvec{k}_{2}}\left|b_{\varvec{k}_{1}}^{(0)}\right|^{2} \left( \delta _{\sigma _{3},-\sigma _{4}}\delta _{\sigma _{5},-\sigma _{6}}\delta _{\varvec{k}_{3},\varvec{k}_{4}}\delta _{\varvec{k}_{5},\varvec{k}_{6}}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{5}}^{(0)}\right|^{2} \right. \\& + \delta _{\sigma _{3},-\sigma _{5}}\delta _{\sigma _{4},-\sigma _{6}}\delta _{\varvec{k}_{3},\varvec{k}_{5}}\delta _{\varvec{k}_{4},\varvec{k}_{6}}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{4}}^{(0)}\right|^{2} \\& \left. + \, \delta _{\sigma _{3},-\sigma _{6}}\delta _{\sigma _{4},-\sigma _{5}}\delta _{\varvec{k}_{3},\varvec{k}_{6}}\delta _{\varvec{k}_{4},\varvec{k}_{5}}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{4}}^{(0)}\right|^{2} \right) \\&\qquad +\delta _{\sigma _{1},-\sigma _{3}}\delta _{\varvec{k}_{1},\varvec{k}_{3}}\left|b_{\varvec{k}_{1}}^{(0)}\right|^{2} \left( \delta _{\sigma _{2},-\sigma _{4}}\delta _{\sigma _{5},-\sigma _{6}}\delta _{\varvec{k}_{2},\varvec{k}_{4}}\delta _{\varvec{k}_{5},\varvec{k}_{6}}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{5}}^{(0)}\right|^{2} \right. \\& + \delta _{\sigma _{2},-\sigma _{5}}\delta _{\sigma _{4},-\sigma _{6}}\delta _{\varvec{k}_{2},\varvec{k}_{5}}\delta _{\varvec{k}_{4},\varvec{k}_{6}}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{4}}^{(0)}\right|^{2} \\& \left. + \delta _{\sigma _{2},-\sigma _{6}}\delta _{\sigma _{4},-\sigma _{5}}\delta _{\varvec{k}_{2},\varvec{k}_{6}}\delta _{\varvec{k}_{4},\varvec{k}_{5}}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{4}}^{(0)}\right|^{2}\right) \\&\qquad +\delta _{\sigma _{1},-\sigma _{4}}\delta _{\varvec{k}_{1},\varvec{k}_{4}}\left|b_{\varvec{k}_{1}}^{(0)}\right|^{2} \left( \delta _{\sigma _{2},-\sigma _{3}}\delta _{\sigma _{5},-\sigma _{6}}\delta _{\varvec{k}_{2},\varvec{k}_{3}}\delta _{\varvec{k}_{5},\varvec{k}_{6}}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{5}}^{(0)}\right|^{2} \right. \\& + \delta _{\sigma _{2},-\sigma _{5}}\delta _{\sigma _{3},-\sigma _{6}}\delta _{\varvec{k}_{2},\varvec{k}_{5}}\delta _{\varvec{k}_{3},\varvec{k}_{6}}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2} \\& \left. +\delta _{\sigma _{2},-\sigma _{6}}\delta _{\sigma _{3},-\sigma _{5}}\delta _{\varvec{k}_{2},\varvec{k}_{6}}\delta _{\varvec{k}_{3},\varvec{k}_{5}}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2}\right) \\&\qquad + \delta _{\sigma _{1},-\sigma _{5}}\delta _{\varvec{k}_{1},\varvec{k}_{5}}\left|b_{\varvec{k}_{1}}^{(0)}\right|^{2} \left( \delta _{\sigma _{2},-\sigma _{3}}\delta _{\sigma _{4},-\sigma _{6}}\delta _{\varvec{k}_{2},\varvec{k}_{3}}\delta _{\varvec{k}_{4},\varvec{k}_{6}}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{4}}^{(0)}\right|^{2} \right. \\& + \delta _{\sigma _{2},-\sigma _{4}}\delta _{\sigma _{3},-\sigma _{6}}\delta _{\varvec{k}_{2},\varvec{k}_{4}}\delta _{\varvec{k}_{3},\varvec{k}_{6}}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2} \\& \left. + \delta _{\sigma _{2},-\sigma _{6}}\delta _{\sigma _{3},-\sigma _{4}}\delta _{\varvec{k}_{2},\varvec{k}_{6}}\delta _{\varvec{k}_{3},\varvec{k}_{4}}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2}\right) \\&\qquad + \delta _{\sigma _{1},-\sigma _{6}}\delta _{\varvec{k}_{1},\varvec{k}_{6}}\left|b_{\varvec{k}_{1}}^{(0)}\right|^{2} \left( \delta _{\sigma _{2},-\sigma _{3}}\delta _{\sigma _{4},-\sigma _{5}}\delta _{\varvec{k}_{2},\varvec{k}_{3}}\delta _{\varvec{k}_{4},\varvec{k}_{5}}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{4}}^{(0)}\right|^{2} \right. \\& + \delta _{\sigma _{2},-\sigma _{4}}\delta _{\sigma _{3},-\sigma _{5}}\delta _{\varvec{k}_{2},\varvec{k}_{4}}\delta _{\varvec{k}_{3},\varvec{k}_{5}}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2} \\& \left. +\delta _{\sigma _{2},-\sigma _{5}}\delta _{\sigma _{3},-\sigma _{4}}\delta _{\varvec{k}_{2},\varvec{k}_{5}}\delta _{\varvec{k}_{3},\varvec{k}_{4}}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2}\right) \end{aligned}$$

Gathering all the contributions of $G_2$, we obtain

$$\begin{aligned}&{\mathbb {E}}_{n}\left[ G_{2}\right] =\left( \frac{2\pi }{L}\right) ^{2d}\sum _{\varvec{k}_{1}}\lambda (\varvec{k}_{1})\\& \times \left\{ 36\sum _{{\sigma }_{123}}\sigma _{1}\sigma _{3}\sum _{\varvec{k}_{2},\varvec{k}_{3}}\left|V_{{\varvec{k}}_{123}}^{{\sigma }_{123}}\right|^{2}\left|b_{\varvec{k}_{1}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\chi _{L}^{d}({\sigma }_{123}\cdot {\varvec{k}}_{123}) \right. \\& \times {\tilde{E}}_{\Delta t}\left( -{\sigma }_{123}\cdot {\omega }_{123};{\sigma }_{123}\cdot {\omega }_{123}\right) \\& \left. + 9 \, \sum _{{\sigma }_{123}}\sigma _{1}^{2}\sum _{\varvec{k}_{2},\varvec{k}_{3}}\left|V_{{\varvec{k}}_{123}}^{{\sigma }_{123}}\right|^{2}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2}\left|\Delta _{\Delta t}\left( {\sigma }_{123}\cdot {\omega }_{123}\right) \right|^{2}\chi _{L}^{d}({\sigma }_{123}\cdot {\varvec{k}}_{123})\right\} \\& +R(G_{2}) \end{aligned}$$

with $R(G_{2})$ the remaining term coming from the internal pairing of indices within the triads:

$$\begin{aligned} R(G_{2})&=\left( \frac{2\pi }{L}\right) ^{2d}18\sum _{\sigma _{1},\sigma _{3},\sigma _{4}}\sum _{\varvec{k}_{1},\varvec{k}_{3},\varvec{k}_{4}}\sigma _{1}\sigma _{3}\lambda (\varvec{k}_{1})V_{\varvec{k}_{1},\varvec{k}_{1},\varvec{k}_{3}}^{\sigma _{1},-\sigma _{1},\sigma _{3}}V_{\varvec{k}_{3},\varvec{k}_{4},\varvec{k}_{4}}^{-\sigma _{3},\sigma _{4},-\sigma _{4}} \\& \times \left|b_{\varvec{k}_{1}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{4}}^{(0)}\right|^{2} {\tilde{E}}_{\Delta t}\left( -\sigma _{3}\omega (\varvec{k}_{3});\sigma _{3}\omega (\varvec{k}_{3})\right) \chi _{L}^{d}(\sigma _{3}\varvec{k}_{3})\\&\; +\left( \frac{2\pi }{L}\right) ^{2d}\frac{9}{2}\sum _{\sigma _{1},\sigma _{2},\sigma _{4}}\sum _{\varvec{k}_{1},\varvec{k}_{2},\varvec{k}_{4}}\sigma _{1}^{2}\lambda (\varvec{k}_{1})V_{\varvec{k}_{1}, \varvec{k}_2, \varvec{k}_2}^{-\sigma _1,\sigma _2,-\sigma _2} V_{\varvec{k}_{1}, \varvec{k}_4, \varvec{k}_4}^{\sigma _1,\sigma _4,-\sigma _4} \\& \times \left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{4}}^{(0)}\right|^{2} \Delta _{\Delta t}\left( -\sigma _{1}\omega (\varvec{k}_{1})\right) \Delta _{\Delta t}^{*}\left( \sigma _{1}\omega (\varvec{k}_{1})\right) \chi _{L}^{d}(\sigma _{1}\varvec{k}_{1}) \end{aligned}$$

Note that $R(G_{2})$ vanishes because $\lim _{\varvec{k}_{1}\rightarrow 0}V_{\varvec{k}_{1},\varvec{k}_{2},\varvec{k}_{3}}^{{\sigma }}=0$ for any ${\sigma }$ and any $\varvec{k}_{2},\varvec{k}_{3}$. We have used the equality $\left[ \chi _{L}^{d}(\varvec{x})\right] ^{2}=\left( \tfrac{L}{2\pi }\right) ^{d}\chi _{L}^{d}(\varvec{x})$ to get the proper scaling in L.

Similar calculations yields, for ${\mathbb {E}}\left[ G_{1}^{2}\right]$:

$$\begin{aligned} {\mathbb {E}}_{n}\left[ G_{1}^{2}\right]&=18\left( \frac{2\pi }{L}\right) ^{2d}\sum _{{\sigma }_{123}}\sum _{{\varvec{k}}_{123}}\left( \sigma _{1}\lambda (\varvec{k}_{1})\right) ^{2}\left|V_{{\varvec{k}}_{123}}^{{\sigma }_{123}}\right|^{2}\left|b_{\varvec{k}_{1}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2} \\& \times \left|\Delta _{\Delta t}\left( {\sigma }_{123}\cdot {\omega }_{123}\right) \right|^{2}\chi _{L}^{d}({\sigma }_{123}\cdot {\varvec{k}}_{123})\\&\; +36\left( \frac{2\pi }{L}\right) ^{2d}\sum _{{\sigma }_{123}}\sum _{{\varvec{k}}_{123}}\left( \sigma _{1}\lambda (\varvec{k}_{1})\right) \left( \sigma _{2}\lambda (\varvec{k}_{2})\right) \left|V_{{\varvec{k}}_{123}}^{{\sigma }_{123}}\right|^{2}\left|b_{\varvec{k}_{1}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2} \\& \times \left|\Delta _{\Delta t}\left( {\sigma }_{123}\cdot {\omega }_{123}\right) \right|^{2}\chi _{L}^{d}({\sigma }_{123}\cdot {\varvec{k}}_{123})\\&\; +R(G_{1}^{2})\,. \end{aligned}$$

We do not write explicitly the contribution $R(G_{1}^{2})$ (that is slightly lengthy) which comes from the internal pairing of indices within triads (i.e. all the terms in ${\mathbb {E}}_{n}\left[ b_{\varvec{k}_{1}}^{(0)\sigma _{1}}b_{\varvec{k}_{2}}^{(0)\sigma _{2}}b_{\varvec{k}_{3}}^{(0)\sigma _{3}}b_{\varvec{k}_{4}}^{(0)\sigma _{4}}\right. \left. b_{\varvec{k}_{5}}^{(0)\sigma _{5}}b_{\varvec{k}_{6}}^{(0)\sigma _{6}}\right]$ except those proportional to $\left|b_{\varvec{k}_{1}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2}$ ). The remainder $R(G_{1}^{2})$ is also vanishing because of the property $\lim _{\varvec{k}_{1}\rightarrow 0}V_{\varvec{k}_{1},\varvec{k}_{2},\varvec{k}_{3}}^{{\sigma }}=0$ for any ${\sigma }$ and any $\varvec{k}_{2},\varvec{k}_{3}$.

Finally, one gets by permutation symmetry the more compact expressions

$$\begin{aligned} {\mathbb {E}}_{n}\left[ G_{1}^{2}\right]&=6\left( \frac{2\pi }{L}\right) ^{2d}\sum _{{\sigma }_{123}}\sum _{{\varvec{k}}_{123}}\left( {\sigma }_{123}\cdot {\lambda }_{123}\right) ^{2}\left|V_{{\varvec{k}}_{123}}^{{\sigma }_{123}}\right|^{2} \nonumber \\& \times \left|b_{\varvec{k}_{1}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2}\left|\Delta _{\Delta t}\left( {\sigma }_{123}\cdot {\omega }_{123}\right) \right|^{2}\chi _{L}^{d}({\sigma }_{123}\cdot {\varvec{k}}_{123})\nonumber \\ \end{aligned}$$

(A13)

$$\begin{aligned} {\mathbb {E}}_{n}\left[ G_{2}\right]&= 6 \left( \frac{2\pi }{L}\right) ^{2d}\sum _{{\sigma }_{123}}\sum _{{\varvec{k}}_{123}}\left|V_{{\varvec{k}}_{123}}^{{\sigma }_{123}}\right|^{2}\chi _{L}^{d}({\sigma }_{123}\cdot {\varvec{k}}_{123}) \nonumber \\& \times \left\{ \frac{1}{2}\left|\Delta _{\Delta t}\left( {\sigma }_{123}\cdot {\omega }_{123}\right) \right|^{2} \left( \sigma _{1}^{2}\lambda (\varvec{k}_{1})\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2} \right. \right. \nonumber \\& +\sigma _{2}^{2}\lambda (\varvec{k}_{2})\left|b_{\varvec{k}_{1}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{3}}^{(0)}\right|^{2} \nonumber \\& \left. +\sigma _{3}^{2}\lambda (\varvec{k}_{3})\left|b_{\varvec{k}_{1}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\right) \nonumber \\& +{\tilde{E}}_{\Delta t}\left( -{\sigma }_{123}\cdot {\omega }_{123};{\sigma }_{123}\cdot {\omega }_{123}\right) \nonumber \\& \left. \times \left( \sigma _{1}\lambda (\varvec{k}_{1})\left|b_{\varvec{k}_{1}}^{(0)}\right|^{2}\left|b_{\varvec{k}_{2}}^{(0)}\right|^{2}\sigma _{3}+\text {cyclic. perm.}\right) \right\} \end{aligned}$$

(A14)

Expressions in terms of the empirical spectral density n Since the average is conditioned on $n(\varvec{\xi })=\left( \tfrac{2\pi }{L}\right) ^{d}\sum _{\varvec{k}}\left|b_{\varvec{k}}^{(0)}\right|^{2}\delta (\varvec{\xi }-\varvec{k})$, one can replace all the discrete sums with respect to the wavevectors $\varvec{k}$ in (A13, A14) by continuous integrals:

$$\begin{aligned} {\mathbb {E}}_{n}\left[ G_{2}\right]&=6\left( \frac{L}{2\pi }\right) ^{d}\sum _{{\sigma }_{123}}\int \mathrm{d}^{d}\xi _{1}\int \mathrm{d}^{d}\xi _{2}\int \mathrm{d}^{d}\xi _{3}\left|V_{{\varvec{\xi }}_{123}}^{{\sigma }_{123}}\right|^{2}\chi _{L}^{d}\left( {\sigma }_{123}\cdot {\varvec{\xi }}_{123}\right) \\& \times \left\{ {\tilde{E}}_{\Delta t}\left( -{\sigma }_{123}\cdot {\omega }_{123};{\sigma }_{123}\cdot {\omega }_{123}\right) \left( \sigma _{1}\sigma _{3}\lambda (\varvec{\xi }_{1})n(\varvec{\xi }_{1})n(\varvec{\xi }_{2})+\text {cyclic. perm.}\right) \right. \\& \left. +\frac{1}{2}\left|\Delta _{\Delta t}\left( {\sigma }_{123}\cdot {\omega }_{123}\right) \right|^{2}{{\left( \sigma _{1}^{2}\lambda (\varvec{\xi }_{1})n(\varvec{\xi }_{2})n(\varvec{\xi }_{3})+ \text {cyclic. perm.} \right) }}\right\} \\ {\mathbb {E}}_{n}\left[ G_{1}^{2}\right]&=6\left( \frac{L}{2\pi }\right) ^{d}\sum _{{\sigma }_{123}}\int \mathrm{d}^{d}\xi _{1}\int \mathrm{d}^{d}\xi _{2}\int \mathrm{d}^{d}\xi _{3}\left|V_{{\varvec{\xi }}_{123}}^{{\sigma }_{123}}\right|^{2} \chi _{L}^{d}\left( {\sigma }_{123}\cdot {\varvec{\xi }}_{123}\right) \\&\qquad \times \left( {\sigma }_{123}\cdot {\lambda }_{123}\right) ^{2}n(\varvec{\xi }_{1})n(\varvec{\xi }_{2})n(\varvec{\xi }_{3})\left|\Delta _{\Delta t}\left( {\sigma }_{123}\cdot {\omega }_{123}\right) \right|^{2}\,. \end{aligned}$$

Asymptotic expressions of ${\mathbb {E}}_{n}\left[ G_{1}^{2}\right]$, ${\mathbb {E}}_{n}\left[ G_{2}\right]$ and $Z_{L,\epsilon }$ in the kinetic limit As explained in Sect. 3.2.1, the kinetic limit refers to the joint limit $L\rightarrow \infty$ and $\epsilon \rightarrow 0$ such that $L\epsilon ^{2}$ is infinite or at least bounded from below in order to get a large number of quasiresonances [2].

Here, the time $\Delta t$ is chosen such that the inequality (12) is fulfilled. Fixing $\Delta \tau ={\mathcal {O}}(1)$, we have to evaluate ${\mathbb {E}}_{n}[G_{1}^{2}]$ and ${\mathbb {E}}_{n}[G_{2}]$ in the kinetic limit. The quantities ${\mathbb {E}}_{n}[G_{1}^{2}]$ and ${\mathbb {E}}_{n}[G_{2}]$ are of the kind

$$\begin{aligned} \left( \frac{L}{2\pi }\right) ^{d}\sum _{{\sigma }_{123}}\int \mathrm{d}^{d}\xi _{1}\int \mathrm{d}^{d}\xi _{2}\int \mathrm{d}^{d}\xi _{3}X(\varvec{\xi }_{1},\varvec{\xi }_{2},\varvec{\xi }_{3}){\tilde{\chi }}_{\Delta t}\left( {\sigma }_{123}\cdot {\omega }_{123}\right) \chi _{L}^{d}\left( {\sigma }_{123}\cdot {\varvec{\xi }}_{123}\right) \end{aligned}$$

with ${\tilde{\chi }}_{\Delta t}(x)=\left|\Delta _{\Delta t}(x)\right|^{2}$ or ${\tilde{\chi }}_{\Delta t}(x)={\tilde{E}}_{\Delta t}(-x,x)$, and $X(\varvec{\xi }_1, \varvec{\xi }_2 ,\varvec{\xi }_3 )$ the remaining factors.

Proceeding formally, one can use the following asymptotic limits (see [2, Eqs. (6.41, 6.42)])

$$\begin{aligned} \lim _{\Delta t\rightarrow \infty }\frac{{\tilde{E}}_{\Delta t}\left( -{\sigma }\cdot {\omega };{\sigma }\cdot {\omega }\right) }{\Delta t}&=\pi \delta \left( {\sigma }\cdot {\omega }\right) \\ \lim _{\Delta t\rightarrow \infty }\frac{\left|\Delta _{\Delta t}\left( {\sigma }\cdot {\omega }\right) \right|^{2}}{\Delta t}&=2\pi \delta \left( {\sigma }\cdot {\omega }\right) \\ \lim _{L\rightarrow \infty }\chi _{L}^{d}({\sigma }\cdot {\varvec{\xi }})&=\delta \left( {\sigma }\cdot {\varvec{\xi }}\right) \end{aligned}$$

to conclude that

$$\begin{aligned}&\left( \frac{L}{2\pi }\right) ^{d}\sum _{{\sigma }_{123}}\int \!\! \mathrm{d}^{d}\xi _{1} \!\! \int \!\! \mathrm{d}^{d}\xi _{2} \!\! \int \!\! \mathrm{d}^{d}\xi _{3} X(\varvec{\xi }_{1},\varvec{\xi }_{2},\varvec{\xi }_{3}){\tilde{\chi }}_{\Delta t}\left( {\sigma }_{123}\cdot {\omega }_{123}\right) \chi _{L}^{d}\left( {\sigma }_{123}\cdot {\varvec{\xi }}_{123}\right) \\& \underset{\text {kin.}}{\sim }c\pi \left( \frac{L}{2\pi }\right) ^{d} \!\! \frac{\Delta \tau }{\epsilon ^{2}}\sum _{{\sigma }_{123}}\int \!\! \mathrm{d}^{d}\xi _{1} \!\! \int \!\! \mathrm{d}^{d}\xi _{2} \!\! \int \!\! \mathrm{d}^{d}\xi _{3} X(\varvec{\xi }_{1},\varvec{\xi }_{2},\varvec{\xi }_{3})\delta ({\sigma }_{123}\cdot {\omega }_{123})\delta ({\sigma }_{123}\cdot {\varvec{\xi }}_{123}) \end{aligned}$$

with $c=1$ for ${\tilde{\chi }}_{\Delta t}(x)={\tilde{E}}_{\Delta t}(-x,x)$ and $c=2$ for ${\tilde{\chi }}_{\Delta t}(x)=\left|\Delta _{\Delta t}(x)\right|^{2}$. This can be seen with the choice $L\rightarrow \infty$ first, followed by $\epsilon \rightarrow 0$ as done for instance in [2, Chap. 6].

One finally obtains

$$\begin{aligned}&\lim _{\Delta \tau \rightarrow 0}\text {Kin lim}\left\{ \left( \frac{2\pi }{L}\right) ^{d}\frac{\epsilon ^{2}}{\Delta \tau }{\mathbb {E}}_{n}\left[ G_{2}\right] \right\} \\& =6\pi \int \mathrm{d}^{d}\xi _{1}\lambda (\varvec{\xi }_{1})\sum _{{\sigma }}\int \mathrm{d}^{d}\xi _{2}\int \mathrm{d}^{d}\xi _{3}\left|V_{{\varvec{\xi }}}^{{\sigma }}\right|^{2}\delta \left( {\sigma }\cdot {\omega }\right) \delta \left( {\sigma }\cdot {\varvec{\xi }}\,\right) \\& \times \left( {\sigma }\cdot {\lambda }\right) \left( \sigma _{1}n(\varvec{\xi }_{2})n(\varvec{\xi }_{3})+\sigma _{2}n(\varvec{\xi }_{1})n(\varvec{\xi }_{3})+\sigma _{3}n(\varvec{\xi }_{1})n(\varvec{\xi }_{2})\right) \\&\lim _{\Delta \tau \rightarrow 0} \text {Kin lim}\left\{ \left( \frac{2\pi }{L}\right) ^{d}\frac{\epsilon ^{2}}{\Delta \tau }{\mathbb {E}}_{n}\left[ G_{1}^{2}\right] \right\} \\& =12\pi \sum _{{\sigma }}\int \mathrm{d}^{d}\xi _{1}\int \mathrm{d}^{d}\xi _{2}\int \mathrm{d}^{d}\xi _{3}\left|V_{{\varvec{\xi }}}^{{\sigma }}\right|^{2} \delta \left( {\sigma }\cdot {\omega }\right) \delta \left( {\sigma }\cdot {\varvec{\xi }} \,\right) \\& \times \left( {\sigma }\cdot {\lambda }\right) ^{2} n(\varvec{\xi }_{1})n(\varvec{\xi }_{2})n(\varvec{\xi }_{3})\;. \end{aligned}$$

Coming back to the expression of $Z_{L,\epsilon }$ (A11), the previous calculations allows one to get

$$\begin{aligned}&\lim _{\Delta \tau \rightarrow 0} \text {Kin lim}\left\{ \left( \frac{2\pi }{L}\right) ^{d}\frac{\epsilon ^{2}}{\Delta \tau }\left( {\mathbb {E}}_{n}\left[ G_{2}\right] +\frac{1}{2}{\mathbb {E}}_{n}\left[ G_{1}^{2}\right] \right) \right\} \\& = 6\pi \left\{ \sum _{{\sigma }}\int \mathrm{d}^{d}\xi _{1}\int \mathrm{d}^{d}\xi _{2}\int \mathrm{d}^{d}\xi _{3}\left|V_{{\varvec{\xi }}}^{{\sigma }}\right|^{2}\delta \left( {\sigma }\cdot {\omega }\right) \delta \left( {\sigma }\cdot {\varvec{\xi }}\,\right) \right. \\& \qquad \qquad +\left( {\sigma }\cdot {\lambda }\right) \left( \sigma _{1}n(\varvec{\xi }_{2})n(\varvec{\xi }_{3})+\sigma _{2}n(\varvec{\xi }_{1})n(\varvec{\xi }_{3})+\sigma _{3}n(\varvec{\xi }_{1})n(\varvec{\xi }_{2})\right) \\& \qquad \qquad +\left. \left( {\sigma }\cdot {\lambda }\right) ^{2}\times n(\varvec{\xi }_{1})n(\varvec{\xi }_{2})n(\varvec{\xi }_{3})\right\} \;. \end{aligned}$$

Therefore, defining

$$\begin{aligned} {\mathcal {R}}_{1}(L,\epsilon )&=\left( \frac{2\pi }{L}\right) ^{d}\frac{\epsilon ^{2}}{\Delta \tau }\left( {\mathbb {E}}_{n}\left[ G_{2}\right] +\frac{1}{2}{\mathbb {E}}_{n}\left[ G_{1}^{2}\right] \right) \\&\qquad -\lim _{\Delta \tau \rightarrow 0}\text {Kin lim}\left\{ \left( \frac{2\pi }{L}\right) ^{d}\frac{\epsilon ^{2}}{\Delta \tau }\left( {\mathbb {E}}_{n}\left[ G_{2}\right] +\frac{1}{2}{\mathbb {E}}_{n}\left[ G_{1}^{2}\right] \right) \right\} \, , \end{aligned}$$

one finally obtains the asymptotic estimate (22) of $Z_{L,\epsilon }$:

$$\begin{aligned} Z_{L,\epsilon }&=1+\Delta \tau \left( \frac{L}{2\pi }\right) ^{d}\left\{ 6\pi \sum _{{\sigma }}\int \mathrm{d}^{d}\xi _{1}\int \mathrm{d}^{d}\xi _{2}\int \mathrm{d}^{d}\xi _{3}\left|V_{{\varvec{\xi }}}^{{\sigma }}\right|^{2}\delta \left( {\sigma }\cdot {\omega }\right) \delta \left( {\sigma }\cdot {\varvec{\xi }}\,\right) \right. \\&\qquad \qquad \qquad +\left( {\sigma }\cdot {\lambda }\right) \left( \sigma _{1}n(\varvec{\xi }_{2})n(\varvec{\xi }_{3})+\sigma _{2}n(\varvec{\xi }_{1})n(\varvec{\xi }_{3})+\sigma _{3}n(\varvec{\xi }_{1})n(\varvec{\xi }_{2})\right) \\&\qquad \qquad \qquad +\left. \left( {\sigma }\cdot {\lambda }\right) ^{2} n(\varvec{\xi }_{1})n(\varvec{\xi }_{2})n(\varvec{\xi }_{3})+{\mathcal {R}}_{1}(L,\epsilon )+\epsilon ^{2}{\mathcal {R}}_{2}(L,\epsilon )\right\} \;. \end{aligned}$$

where ${\mathcal {R}}_{2}(L,\epsilon )$ corresponds to the contribution of higher order (${\mathcal {O}}(\epsilon ^{4})$ and beyond) terms that have not been considered in the perturbative expansion (A2). As explained in Sect. 3.3, we will assume here that $\text {Kin lim }\epsilon ^{2}{\mathcal {R}}_{2}(L,\epsilon )=0$.

Appendix B Quasipotential at Equilibrium and Entropy

In this appendix, we compute the equilibrium quasipotential for the empirical spectral density. Its relation with the entropy associated with the microcanonical measure (at fixed energy E and momentum $\varvec{K}$) is discussed.

In order to regularize the ultraviolet divergence that occurs for the Rayleigh-Jeans spectrum at equilibrium as well as to define properly the microcanonical distribution, we assume that the set of wavenumbers is restricted to ${\mathbb {K}}_{L}^{d}=\left\{ \varvec{k}=\frac{2\pi }{L}{\mathbb {Z}}^{d}|| \varvec{k}| \leqslant k_{\mathrm{max}}\right\}$. Therefore, we consider only a finite number of modes ${\mathcal {N}}_{L}\sim \left( \tfrac{L}{2\pi }k_{\mathrm{max}}\right) ^{d}$ in this appendix. For the sake of simplicity, we keep the notation $\sum _{\varvec{k}}$ to refer to $\sum _{\varvec{k}\in {\mathbb {K}}_{L}^{d}}$.

We define the microcanonical distribution at fixed energy E and momentum $\varvec{K}$ on the space of the re-scaled amplitudes $a_{\varvec{k}}$:

$$\begin{aligned} \text {d}\mu _{E,\varvec{K},L,\epsilon }=\frac{1}{\Gamma _{E,\varvec{K},L,\epsilon }}\delta \left( E-\tilde{\mathcal {{H}}}\right) \delta \left( \varvec{K}-\left( \tfrac{2\pi }{L}\right) ^{d}\sum _{\varvec{k}}\varvec{k}\left|a_{\varvec{k}}\right|^{2}\right) \prod _{\varvec{k}}\mathrm{d}a_{\varvec{k}}\mathrm{d}a_{\varvec{k}}^{*}, \end{aligned}$$

(B1)

with $\Gamma _{E,\varvec{K},L,\epsilon }(E,\varvec{K})=\prod _{\varvec{k}}\int \mathrm{d}a_{\varvec{k}}\mathrm{d}a_{\varvec{k}}^{*}\delta \left( E-\tilde{{\mathcal {H}}}\right) \delta \left( \varvec{K}-\left( \tfrac{2\pi }{L}\right) ^{d}\sum _{\varvec{k}}\varvec{k}\left|a_{\varvec{k}}\right|^{2}\right)$ the volume of the phase space associated with the macrostate $(E,\varvec{K})$. We keep the indices $(L,\epsilon )$ in order to remember that the energy $\tilde{{\mathcal {H}}}$ depends on $\epsilon$ and that the system is of linear size L. The energy

$$\begin{aligned} \tilde{\mathcal {{H}}}\equiv \epsilon ^{-2}{\mathcal {H}}=\tilde{{\mathcal {H}}}_{2}+\epsilon \tilde{{\mathcal {H}}}_{3} \end{aligned}$$

is the microscopic energy ${\mathcal {H}}$ of the microscopic modes (2) rescaled by $\epsilon$. In terms of the rescaled amplitudes $\left|a_{\varvec{k}}\right|^{2}$ and the phases $\left\{ \varphi _{\varvec{k}}\right\} ,$ we get

$$\begin{aligned} \tilde{\mathcal {{H}}}_{2}=\left( \frac{2\pi }{L}\right) ^{d}\sum _{\varvec{k}}\omega _{\varvec{k}} \vert a_{\varvec{k}}\vert ^{2} \end{aligned}$$

and

$$\begin{aligned} \tilde{\mathcal {{H}}}_{3}=\left( \frac{2\pi }{L}\right) ^{3d/2}\sum _{{\sigma }}\sum _{{\varvec{k}}}V_{{\varvec{k}}}^{{\sigma }}\left|a_{\varvec{k}_{1}}\right|\left|a_{\varvec{k}_{2}}\right|\left|a_{\varvec{k}_{3}}\right|\mathrm{e}^{i\left( \varphi _{\varvec{k}_{1}}+\varphi _{\varvec{k}_{2}}+\varphi _{\varvec{k}_{3}}\right) }\delta _{{\sigma }\cdot {\varvec{k}},0}. \end{aligned}$$

Our goal is to estimate the probability distribution of the empirical density ${\hat{n}}$, that is a macroscopic state of the system, from the microcanonical measure (B1) with fixed energy E and fixed momentum $\varvec{K}$. This distribution of the empirical spectral density ${\hat{n}}$ is denoted $P_{E,\varvec{K},L,\epsilon }\left[ n\right] ={\mathbb {E}}_{E,\varvec{K}L,\epsilon }\left[ \delta \left( {\hat{n}}-n\right) \right]$, with ${\mathbb {E}}_{E,\varvec{K},L,\epsilon }$ the average with respect to the microcanonical distribution (B1). One obtains

$$\begin{aligned} P_{E,\varvec{K},L,\epsilon }[n]=\frac{\Gamma _{E,\varvec{K},L,\epsilon }[n]}{\Gamma _{E,\varvec{K},L,\epsilon }}\,, \end{aligned}$$

(B2)

where $\Gamma _{E,\varvec{K},L,\epsilon }[n]$ is the volume of the phase space associated with the macrostate $(E,\varvec{K},n)$:

$$\begin{aligned} \Gamma _{E,\varvec{K},L,\epsilon }[n]=\prod _{\varvec{k}}\int \mathrm{d}a_{\varvec{k}}\mathrm{d}a_{\varvec{k}}^{*}\delta \left( E-\tilde{{\mathcal {H}}}\right) \delta \left( \varvec{K}-\left( \tfrac{2\pi }{L}\right) ^{d}\sum _{\varvec{k}}\varvec{k}\left|a_{\varvec{k}}\right|^{2}\right) \delta \left( n-{\hat{n}}\right) . \end{aligned}$$

In the limit $L\rightarrow \infty$, it is natural to expect a large deviation principle

$$\begin{aligned} Q_{E,\varvec{K},\epsilon }[n]=-\lim _{L\rightarrow \infty }\left( \frac{2\pi }{L}\right) ^{d}\log P_{E,\varvec{K},L,\epsilon }[n]\,. \end{aligned}$$

(B3)

Although $Q_{E,\varvec{K},\epsilon }$ can be defined for any value of $\epsilon$, we will compute $Q_{E,\varvec{K}}=\lim _{\epsilon \rightarrow 0}Q_{E,\varvec{K},\epsilon }$ since it is the relevant contribution for the large deviations of the empirical spectral density in the kinetic limit (13). At leading order ($\epsilon \rightarrow 0$), the cubic correction to the energy $\tilde{{\mathcal {H}}}=\tilde{{\mathcal {H}}}_{2}+\epsilon \tilde{{\mathcal {H}}}_{3}$ is vanishing and one gets

$$\begin{aligned} \Gamma _{E,\varvec{K},L}[n]&=\lim _{\epsilon \rightarrow 0}\Gamma _{E,\varvec{K},L,\epsilon } \\&=\prod _{\varvec{k}}\int \mathrm{d}a_{\varvec{k}}\mathrm{d}a_{\varvec{k}}^{*}\delta \left( E-\tilde{{\mathcal {H}}}_{2}\right) \delta \left( \varvec{K}-\left( \tfrac{2\pi }{L}\right) ^{d}\sum _{\varvec{k}}\varvec{k}\left|a_{\varvec{k}}\right|^{2}\right) \delta \left( n-{\hat{n}}\right) . \end{aligned}$$

Using Eq. (B2), the quasipotential is naturally expressed in terms of the entropy $s_{E,\varvec{K}}[n]=\lim _{L\rightarrow \infty }\left( \frac{2\pi }{L}\right) ^{d}\log \Gamma _{E,\varvec{K},L}[n]$ of the macrostate $(n,E,\varvec{K})$ and the entropy $s_{E,\varvec{K}}=\lim _{L\rightarrow \infty }\left( \frac{2\pi }{L}\right) ^{d}\log \Gamma _{E,\varvec{K},L}$ of the macrostate $(E,\varvec{K})$ as

$$\begin{aligned} Q_{E,\varvec{K}}[n]=s_{E,\varvec{K}}-s_{E,\varvec{K}}[n]. \end{aligned}$$

Since $Q_{E,\varvec{K}}[n]\geqslant 0$ and $\inf _{n}Q_{E,\varvec{K}}=0$ by definition, one obtains by contraction $s_{E,\varvec{K}}=\max _{n}\left\{ s_{E,\varvec{K}}[n]\right\}$. The entropy $s_{E,\varvec{K}}[n]$ can be computed by using the inverse Laplace transform of the Dirac-$\delta$, or equivalently going to the canonical ensemble. We thus considers the free energy

$$\begin{aligned} f_{\beta ,\varvec{\mu }}\left[ \lambda \right]&=-\lim _{L\rightarrow \infty }\left( \frac{2\pi }{L}\right) ^{d}\log \left[ \prod _{\varvec{k}}\int \mathrm{d}a_{\varvec{k}}\mathrm{d}a_{\varvec{k}}^{*}\mathrm{e}^{-\left[ \beta \sum _{\varvec{k}}\omega _{\varvec{k}}\left|a_{\varvec{k}}\right|^{2}+\varvec{\mu }\cdot \sum _{\varvec{k}}\varvec{k}\left|a_{\varvec{k}}\right|^{2}+\sum _{\varvec{k}}\lambda (\varvec{k})\left|a_{\varvec{k}}\right|^{2}\right] }\right] \nonumber \\&=\int \mathrm{d}^{d}\xi \,\log \left( \frac{\beta \omega _{\varvec{\xi }}+\varvec{\mu }\cdot \varvec{\xi }+\lambda (\varvec{\xi })}{2\pi }\right) \;. \end{aligned}$$

(B4)

The entropy $s_{E,\varvec{K}}[n]$ is obtained as the Legendre-Fenchel transform of the free energy (B4) that is differentiable everywhere on its domain. One gets

$$\begin{aligned} s_{E,\varvec{K}}[n]&=\inf _{\beta ,\varvec{\mu },\lambda }\left\{ \beta E+\varvec{\mu }\cdot \varvec{K}+\int \mathrm{d}^{d}\xi \,\lambda (\varvec{\xi })n(\varvec{\xi })-f_{\beta ,\varvec{\mu }}\left[ \lambda \right] \right\} \nonumber \\&={\left\{ \begin{array}{ll} \int \mathrm{d}^{d}\xi \,\left[ 1+\log \left( 2\pi \right) +\log n(\varvec{\xi })\right] &{} \text {if }E=\int \mathrm{d}^{d}\xi \,\omega _{\varvec{\xi }}n(\varvec{\xi })\,,\;\varvec{K}=\int \mathrm{d}^{d}\xi \,\varvec{\xi }n(\varvec{\xi })\\ -\infty &{} \text {otherwise} \end{array}\right. }. \end{aligned}$$

(B5)

The constant term (bounded for $k_\mathrm{max}<\infty$) within $s_{E,\varvec{K}}[n]$ can be safely discarded because only difference of entropy matters. Looking for the supremum (with respect to n) of the entropy (B5), The entropy of the macrostate $(E,\varvec{K})$ reads

$$\begin{aligned} s_{E,\varvec{K}}=\int \mathrm{d}^{d}\xi \,\log n^{*}(\varvec{\xi })\,, \end{aligned}$$

(B6)

with

$$\begin{aligned} n^{*}(\varvec{\xi })=\left[ \beta \omega _{\varvec{\xi }}+\varvec{\mu }\cdot \varvec{\xi }\right] ^{-1} \end{aligned}$$

the so called Rayleigh-Jeans spectrum [2, Chap. 9], and $(\beta ,\varvec{\mu })$ such that $E=\int \mathrm{d}^{d}\xi \,\omega _{\varvec{\xi }}n^{*}(\varvec{\xi })$, $\varvec{K}=\int \mathrm{d}^{d}\xi \,\varvec{\xi }n^{*}(\varvec{\xi })$.

This explicit expression of the equilibrium Rayleigh-Jeans spectrum clearly shows the appearance of an ultraviolet catastrophe [2, Chap. 9] that prevents the physical existence of such solution in absence of any cut-off $k_{\mathrm{max}}$ on the wavenumbers. Indeed, considering for instance the conserved quantity $\beta E+\varvec{\mu }\cdot \varvec{K}=\int \mathrm{d}^{d}\xi$, one sees that the latter cannot remains finite if the set of allowed wavenumbers is not of finite volume.

Finally, from the expressions of the entropies (B5) and (B6), one obtains

$$\begin{aligned} Q_{E,\varvec{K}}[n]={\left\{ \begin{array}{ll} -\int \mathrm{d}^{d}\xi \,\log \left( \frac{n(\varvec{\xi })}{n^{*}(\varvec{\xi })}\right) &{} \text {if }E=\int \mathrm{d}^{d}\xi \,\omega _{\varvec{\xi }}n(\varvec{\xi })\,,\;\varvec{K}=\int \mathrm{d}^{d}\xi \,\varvec{\xi }n(\varvec{\xi })\\ +\infty &{} \text {otherwise} \end{array}\right. }\,. \end{aligned}$$

(B7)

Note that because $E=\int \mathrm{d}^{d}\xi \,\omega _{\varvec{\xi }}n(\varvec{\xi })$ and $\varvec{K}=\int \mathrm{d}^{d}\xi \,\varvec{\xi }n(\varvec{\xi })$ are conserved and $\int n(\varvec{\xi })/n^{*}(\varvec{\xi }) \mathrm {d}^d\xi = \beta E + \varvec{\mu }\cdot \varvec{K}$ one can rewrite the quasipotential as

$$\begin{aligned} Q_{E,\varvec{K}}[n]={\left\{ \begin{array}{ll} \int \mathrm{d}^{d}\xi \,\left( \frac{n(\varvec{\xi })}{n^{*}(\varvec{\xi })} - 1 - \log \left( \frac{n(\varvec{\xi })}{n^{*}(\varvec{\xi })}\right) \right) &{} \text {if }E=\int \mathrm{d}^{d}\xi \,\omega _{\varvec{\xi }}n(\varvec{\xi })\,,\;\varvec{K}=\int \mathrm{d}^{d}\xi \,\varvec{\xi }n(\varvec{\xi })\\ +\infty &{} \text {otherwise} \end{array}\right. }\, .\nonumber \\ \end{aligned}$$

(B8)

The positivity and convexity of the quasipotential $Q_{E,\varvec{K}}$ can then directly be deduced from the properties of the function $x\mapsto x-1 - \log (x)$ that has a single minimum at $x=1$.