Dynamical Large Deviations for an Inhomogeneous Wave Kinetic Theory: Linear Wave Scattering by a Random Medium

Abstract

The wave kinetic equation predicts the averaged temporal evolution of a continuous spectral density of waves either randomly interacting or scattered by the fine structure of a medium. In a wide range of systems, the wave kinetic equation is derived from a fundamental equation of wave motion, which is symmetric through time reversal. By contrast, the corresponding wave kinetic equations is time irreversible: Its solutions monotonically increase an entropy-like quantity. A similar paradox appears whenever one makes a mesoscopic description of the evolution of a very large number of microscopic degrees of freedom, the paradigmatic example being the kinetic theory of dilute gas molecules leading to the Boltzmann equation. Since Boltzmann, it has been understood that a probabilistic understanding solves the apparent paradox. More recently, it has been understood that the kinetic description itself, at a mesoscopic level, should not break time reversal symmetry (Bouchet in J Stat Phys 181(2):515-550, 2020). The time reversal symmetry remains a fundamental property of the mesoscopic stochastic process: Without external forcing, the path probabilities obey a detailed balance relation with respect to an equilibrium quasipotential. The proper theoretical or mathematical tool to derive fully this mesoscopic time reversal stochastic process is large deviation theory: A large deviation principle uncovers a time reversible field theory, characterized by a large deviation Hamiltonian, for which the deterministic wave kinetic equation appears as the most probable evolution. Its irreversibility appears as a consequence of an incomplete description, rather than as a consequence of the kinetic limit itself, or some related chaotic hypothesis. This paper follows Bouchet (J Stat Phys 181(2):515-550, 2020) and a series of other works that derive the large deviation Hamiltonians of the main classical kinetic theories, for instance, Guioth et al. (J Stat Phys 189(20):20, 2022) for homogeneous wave kinetics. We propose here a derivation of the large deviation principle in an inhomogeneous situation, for the linear scattering of waves by a weak random potential. This problem involves microscopic scales corresponding to the typical wavelengths and periods of the waves and mesoscopic ones which are the scales of spatial inhomogeneities in the spectral density of both the random scatterers and the wave spectrum, and the time needed for the random scatterers to alter the wave spectrum. The main assumption of the kinetic regime is a large separation of these microscopic and mesoscopic scales. For the sake of simplicity, we consider a generic model of wave scattering by weak disorder: the Schrödinger equation with a random potential. We derive the path large deviation principle for the local spectral density and discuss its main properties. We show that the mesoscopic process obeys a time reversal symmetry at the level of large deviations.

Appendices

A Properties of the Stochastic System Specified by a Large Deviation Hamiltonian

1.1 A.1 Path Large Deviation

This appendix presents some general properties of a stochastic process $X^\epsilon (t)$ whose probability conditioned on an initial value, $X^\epsilon (t_i) = X(t_i)$, is specified at the large deviation level via a formula,

$$\begin{aligned} {\mathbb {P}} \left[ \left\{ X^\epsilon (t) = X(t) \right\} _{t_i \leqslant t \leqslant t_f} \right] \underset{\epsilon \rightarrow 0}{\asymp } \exp \left( - \frac{{\mathcal {S}}[X]}{\epsilon } \right) \end{aligned}$$

(60a)

$$\begin{aligned} {\mathcal {S}}[X] = \int _{t_i}^{t_f} dt {\mathcal {L}}(X, \dot{X}) \equiv \int _{t_i}^{t_f} dt \sup _P \left\{ P \cdot \dot{X} - {\mathcal {H}}(X, P) \right\} . \end{aligned}$$

(60b)

Here, $X^\epsilon (t)$ can be a scalar, vector, or continuous function defined on some space. Basic requirements are that an inner product is properly defined, and the dynamical property of the system is controlled by a single nonnegative parameter $\epsilon$. For the simplicity, we regard X as a scalar but the following consideration can be immediately extended to general cases. The large deviation Hamiltonian ${\mathcal {H}}(X, P)$ is a convex function of P and satisfies ${\mathcal {H}}(X, 0) = 0$ for any X. From the definition, the Lagrangian ${\mathcal {L}}$ satisfies ${\mathcal {L}}(X, \dot{X}) \ge P \cdot \dot{X} - {\mathcal {H}}(X, P)$ for any X, $\dot{X}$ and P. Therefore, inserting $P=0$, we know ${\mathcal {L}} \ge 0$.

1.1.1 A.1.1 Relaxation Path

Clearly, in the limit of $\epsilon \rightarrow 0$, the system becomes deterministic with a single path that minimizes the action ${\mathcal {S}}[X]$ for a prescribed initial condition $X(t_i)$—named the relaxation path. Since ${\mathcal {L}} \ge 0$, if there exists a function R(X) that satisfies ${\mathcal {L}}(X, R(X)) = 0$, a path solving $\dot{X} = R(X)$ minimizes the action and yields $\min _{X} {\mathcal {S}}[X] = 0$. From the facts that ${\mathcal {L}}(X, \dot{X}) = P \cdot \dot{X} - {\mathcal {H}}(X, P)$ with P solving $\dot{X} - \partial {\mathcal {H}} / \partial P = 0$ and ${\mathcal {H}}(X, 0) = 0$ for any X, we understand that a function $R(X) = \left. \partial {\mathcal {H}} / \partial P \right| _{P=0}$ fulfills ${\mathcal {L}}(X, R(X)) = 0$. We thus assert that an equation

$$\begin{aligned} \dot{X}^r = R(X^r) \equiv \frac{\partial {\mathcal {H}}}{\partial P} (X^r, 0) \end{aligned}$$

(61)

determines the relaxation path $X^r(t)$.

1.1.2 A.1.2 Optimal Path

Slightly changing the situation, if we fix both the initial and final states, $X(t_i) = x_i$ and $X(t_f) = x_f$, respectively, the most probable path from $x_i$ to $x_f$, namely the optimal path, or the instanton, is obtained by again minimizing ${\mathcal {S}}[X]$. This problem is equivalent to the principle of least action in analytical mechanics. In this context, P is called the generalized momentum and represented by $P = \partial {\mathcal {L}} / \partial \dot{X}$ which is no longer 0. The optimal path in phase space is governed by a set of canonical equations,

$$\begin{aligned} \dot{X}&= \frac{\partial {\mathcal {H}}}{\partial P} \end{aligned}$$

(62a)

$$\begin{aligned} \dot{P}&= - \frac{\partial {\mathcal {H}}}{\partial X} , \end{aligned}$$

(62b)

and we shall write their solutions as $X^o[x_f, t_f; x_i, t_i]$ and $P^o[x_f, t_f; x_i, t_i]$. For the simplicity, we fix the initial conditions, $t_i$ and $x_i$, and rewrite the final state as x and t. We then introduce the Hamilton’s principal function ${\mathcal {Q}}$ as an integration of the action following the optimal path as

$$\begin{aligned} {\mathcal {Q}}(x, t) = \int _{t_i}^t d\tau {\mathcal {L}} (X^o(\tau ), \dot{X}^o(\tau )) . \end{aligned}$$

(63)

It is known in analytical mechanics that in this case the generalized momentum at the final time is represented as $P^o(t) = \left. \partial {\mathcal {L}} / \partial \dot{X} \right| _{t} = \partial {\mathcal {Q}} / \partial x$, and ${\mathcal {Q}}$ solves the Hamilton–Jacobi equation,

$$\begin{aligned} \frac{\partial {\mathcal {Q}}}{\partial t} + {\mathcal {H}}\left( x, \frac{\partial {\mathcal {Q}}}{\partial x} \right) = 0 . \end{aligned}$$

(64)

In the definition of ${\mathcal {Q}}(x, t)$, a set of arguments, (x, t), is arbitrary chosen. When we pick up an optimal path $\left\{ X^o(\tau ), P^o(\tau ) \right\}$, at any point on this path, the generalized momentum P and the Hamilton’s principle function ${\mathcal {Q}}$ are related via

$$\begin{aligned} P^o(\tau ) = \frac{\partial {\mathcal {Q}}}{\partial x} (X^o(\tau ), \tau ) . \end{aligned}$$

(65)

Therefore, combining (65) and the first line of (62), we formally obtain a single equation determining the optimal path,

$$\begin{aligned} \frac{d X^o}{d \tau } = \frac{\partial {\mathcal {H}}}{\partial P} \left( X^o, \frac{\partial {\mathcal {Q}}}{\partial x} (X^o, \tau ) \right) . \end{aligned}$$

(66)

This equation is, however, not generally useful because ${\mathcal {Q}}(x, t)$ is inaccessible in most cases.

1.1.3 A.1.3 Quasipotential and Fluctuation Path

Going back to the original stochastic model, the meaning of ${\mathcal {S}}$ is understood as the rate function for the probability that $X^\epsilon$ reaches $x_f$ at $t = t_f$. Indeed, we may derive from (60) an expression

$$\begin{aligned} {\mathbb {P}} \left[ X^\epsilon (t) = x \vert X^\epsilon (t_i) = x_i \right] \underset{\epsilon \rightarrow 0}{\asymp }\ \exp \left( - \frac{{\mathcal {Q}}(x, t)}{\epsilon } \right) \end{aligned}$$

(67)

based on the contraction principle. Now, we shall consider the stationary distribution of the probability density of $X^\epsilon$. This can be done simply setting $t_i = -\infty$ in (67). To make the discussion more specific, let us assume that the relaxation dynamics (61) has a unique global attractor $x_0$, where $R(x_0) = 0$. Then, we set $x_i = x_0$ and also $t = 0$, to write a large deviation formula for the stationary distribution

$$\begin{aligned} {\mathbb {P}}_s (x) \underset{\epsilon \rightarrow 0}{\asymp }\ \exp \left( - \frac{{\mathcal {U}}(x)}{\epsilon } \right) \end{aligned}$$

(68)

with

$$\begin{aligned} {\mathcal {U}}(x) = \inf _{X(t) \vert X(-\infty ) = x_0 \text{ and } X(0) = x} \int _{-\infty }^0 dt {\mathcal {L}} (X(t), \dot{X}(t)) . \end{aligned}$$

(69)

The rate function ${\mathcal {U}}$ is called the quasipotential. Since ${\mathcal {U}}$ is the special case of ${\mathcal {Q}}$ but independent of t, it solves the stationary version of the Hamilton–Jacobi Eq. (64),

$$\begin{aligned} {\mathcal {H}} \left( x, \frac{\partial {\mathcal {U}}}{\partial x} \right) = 0 . \end{aligned}$$

(70)

For the present case, the optical path $X^o(\tau )$ represents the most probable route from an attractor $x_0$ to a specific point x. This route is called the fluctuation path and is denoted by $X^f(t)$. Once we obtain the quasipotential ${\mathcal {U}}(x)$, (66) provides an equation determining the fluctuation path as

$$\begin{aligned} \dot{X}^f = F(X^f) \equiv \frac{\partial {\mathcal {H}}}{\partial P} \left( X^f, \frac{\partial {\mathcal {U}}}{\partial x} (X^f) \right) . \end{aligned}$$

(71)

Since the vector field F(x) does not depend on t, this equation is more useful than the original one (66). On the fluctuation path, the generalized momentum is computed based on (65) as $P^f = \partial {\mathcal {U}} / \partial x (X^f)$. Combining (70) with the fact that ${\mathcal {H}}$ is constant along the optical path, we understand that ${\mathcal {H}}(X^f, P^f) = 0$ always holds.

1.1.4 A.1.4 Quasipotential as a Lyapunov Function

A relaxation path and a fluctuation path have distinct properties for the variations in ${\mathcal {U}}$. For a relaxation path, we have

$$\begin{aligned} \frac{d {\mathcal {U}}}{d t}(X^r)&= \dot{X}^r \frac{\partial {\mathcal {U}}}{\partial x}(X^r) = \frac{\partial {\mathcal {H}}}{\partial P} (X^r, 0) \frac{\partial {\mathcal {U}}}{\partial x}(X^r) \\&= \underbrace{{\mathcal {H}}(X^r, 0)}_{= 0} - \underbrace{{\mathcal {H}} \left( X^r, \frac{\partial {\mathcal {U}}}{\partial x} (X^r)) \right) }_{= 0} + \frac{\partial {\mathcal {H}}}{\partial P} (X^r, 0) \frac{\partial {\mathcal {U}}}{\partial x}(X^r) \le 0 , \end{aligned}$$

where we have used the general expressions, ${\mathcal {H}}(X, 0) = {\mathcal {H}}(X, \partial {\mathcal {U}} / \partial x (X)) = 0$, and the convexity of ${\mathcal {H}}(X, P)$ for P. For a fluctuation path, we have

$$\begin{aligned}&\frac{d {\mathcal {U}}}{d t}(X^f) = \dot{X}^f \frac{\partial {\mathcal {U}}}{\partial x}(X^f) = \frac{\partial {\mathcal {H}}}{\partial P} \left( X^f, \frac{\partial {\mathcal {U}}}{\partial x}(X^f) \right) \frac{\partial {\mathcal {U}}}{\partial x}(X^f) \\&\quad = \underbrace{{\mathcal {H}}(X^f, 0)}_{= 0} - \underbrace{{\mathcal {H}} \left( X^f, \frac{\partial {\mathcal {U}}}{\partial x} (X^f)) \right) }_{= 0} + \frac{\partial {\mathcal {H}}}{\partial P} \left( X^f, \frac{\partial {\mathcal {U}}}{\partial x}(X^f) \right) \frac{\partial {\mathcal {U}}}{\partial x}(X^f) \ge 0 , \end{aligned}$$

again from the convexity of ${\mathcal {H}}$. We have thus learned that the quasipotential is a Lyapunov function because it monotonically decreases in a relaxation path, while increases in a fluctuation path. These results are natural consequence from a basic property that ${\mathcal {U}}(x)$ is minimum at the attractor $x_0$ and the relaxation and fluctuation paths represent routes to and from the attractor.

1.2 A.2 Properties of Large Deviation Hamiltonian

1.2.1 A.2.1 Conservation Law

From now on, we shall regard X and P as vectors so that there are multiple directions in X space. For this case, when the Hamiltonian ${\mathcal {H}}$ possesses a kind of symmetry, it is related to the conservation law of the system. More specifically, let us suppose that we find a function ${\mathcal {C}}(X)$ that satisfies

$$\begin{aligned} {\mathcal {H}} \left( X, P + \alpha \frac{\partial {\mathcal {C}}}{\partial X} \right) = {\mathcal {H}} (X, P) \end{aligned}$$

(72)

for any X, P, and $\alpha$. This condition is equivalent to

$$\begin{aligned} \frac{\partial {\mathcal {H}}}{\partial P}(X, P) \cdot \frac{\partial {\mathcal {C}}}{\partial X} (X) = 0 \end{aligned}$$

for any X and P. Now that ${\mathcal {H}}(X, \cdot )$ is flat in the direction of $\partial {\mathcal {C}} / \partial X$, from the property of the Legendre–Fenchel transform [67], the corresponding Lagrangian has a property,

$$\begin{aligned} {\mathcal {L}} (X, \dot{X}) = + \infty \quad \text{ if } \quad \dot{X} \cdot \frac{\partial {\mathcal {C}}}{\partial X} (X) \ne 0 . \end{aligned}$$

(73)

This expression indicates that the probability for a path crossing a contour of ${\mathcal {C}}$ is strictly 0. This constraint applies not only to the optimal path but also to any path with random fluctuations. We thus understand that (72) serves as a condition for ${\mathcal {C}}$ being an invariant of the system.

1.2.2 A.2.2 Detailed Balance

The detailed balance is a property of equilibrium states which asserts time reversibility of the process, meaning that the probabilities of any trajectory and its reversed counterpart are equal. A basic expression of detailed balance for a stationary stochastic process is ${\mathbb {P}}_{\Delta t}(y; x) {\mathbb {P}}_S(x) = {\mathbb {P}}_{\Delta t}(x; y) {\mathbb {P}}_S(y)$, where ${\mathbb {P}}_{\Delta t}(y; x)$ is the transition probability from a state x to another state y during a time interval $\Delta t$, and ${\mathbb {P}}_S(x)$ is the stationary probability distribution.

Since we are now considering a continuous Markov process, it is enough to regard $\Delta t$ as arbitrary small. For the limit of $\Delta t \rightarrow 0$, we may write $y \sim x + \dot{x} \Delta t$ and redefine the transition probability as ${\mathbb {P}}_{\Delta t}(x, \dot{x}) \sim {\mathbb {P}}_{\Delta t}(x + \dot{x} \Delta t; x)$. Assuming the continuity of ${\mathbb {P}}$ and ${\mathbb {P}}_S$, the detailed balance condition is rewritten as

$$\begin{aligned} {\mathbb {P}}_{\Delta t}(x, \dot{x}) {\mathbb {P}}_S(x) \sim {\mathbb {P}}_{\Delta t}(x + \dot{x} \Delta t, - \dot{x}) {\mathbb {P}}_S(x + \dot{x} \Delta t) . \end{aligned}$$

(74)

For the present problem, the probability distribution is specified as ${\mathbb {P}}_{\Delta t}(x, \dot{x}) \asymp \exp (- \Delta t {\mathcal {L}}(x, \dot{x}) / \epsilon )$ and ${\mathbb {P}}_S (x) \asymp \exp (- {\mathcal {U}}(x) / \epsilon )$. Therefore, the detailed balance condition (74) is rewritten as

$$\begin{aligned} {\mathcal {L}}(x, \dot{x}) - {\mathcal {L}}(x, - \dot{x}) = \dot{x} \cdot \frac{\partial {\mathcal {U}}}{\partial x} . \end{aligned}$$

(75)

This condition is modified in terms of the Hamiltonian via the Legendre–Fenchel transform as

$$\begin{aligned} {\mathcal {H}}(x, -p) = {\mathcal {H}} \left( x, p + \frac{\partial {\mathcal {U}}}{\partial x} \right) . \end{aligned}$$

(76)

Because ${\mathcal {H}}(x, 0) = 0$ in the current problem, the stationary Hamilton–Jacobi Eq. (70) is a necessary (but not sufficient) condition for ${\mathcal {U}}$ being the quasipotential.

Once the detailed balance condition (76) is verified, we understand that the probabilities of a path and its reverse are related at the large deviation level via the expression,

$$\begin{aligned} \frac{{\mathbb {P}} \left[ \left\{ X^\epsilon (t) = X(t) \right\} _{t_i \leqslant t \leqslant t_f} \right] }{{\mathbb {P}}\left[ \left\{ X^\epsilon (t) = X(t_f + t_i - t) \right\} _{t_i \leqslant t \leqslant t_f}\right] } \underset{\epsilon \rightarrow 0}{\asymp }\ \exp \left( - \frac{{\mathcal {U}}(x_f) - {\mathcal {U}}(x_i)}{\epsilon } \right) , \end{aligned}$$

(77)

an equivalent form of the Crooks fluctuation theorem. Another outcome of the detailed balance is that the fluctuation path is the time reverse of the relaxation path. This property is derived from

$$\begin{aligned} R(X) = \frac{\partial {\mathcal {H}}}{\partial P}(X, 0) = - \frac{\partial {\mathcal {H}}}{\partial P} \left( X, \frac{\partial {\mathcal {U}}}{\partial X} \right) = - F(X) . \end{aligned}$$

B Microcanonical Ensemble and Quasipotential for the Schrödinger Equation

In this appendix, we consider the microcanonical ensemble of the dynamics governed by the Schrödinger Eq. (7). The aim is to compute the quasipotential of the local empirical spectral density, i.e., the large deviation rate function of $n^\mu$, in the small $\mu$ limit. We will prove that

$$\begin{aligned} {\mathbb {P}}^\mu _{A,m}[n^\mu = n] \underset{\mu \rightarrow 0}{\asymp }\ \exp \left( -\frac{{\mathcal {U}}_A[n]}{(2 \pi \mu )^d} \right) , \end{aligned}$$

(78)

with ${\mathbb {P}}^\mu _{A,m}$ being the probabilities with respect to the microcanonical measure with the constraints,

$$\begin{aligned} {\mathcal {A}}_\omega [n^\mu ]&\equiv \int h \left( \omega - \frac{\vert \varvec{p}\vert ^2}{2} \right) n^\mu (\varvec{x}, \varvec{p}) d\varvec{x} d\varvec{p} = A(\omega ), \end{aligned}$$

(79)

where h is the Heaviside function, and $A: {\mathbb {R}}^+ \rightarrow {\mathbb {R}}^+$ is a prescribed function. The expected form of the rate function, namely, the quasipotential, is

$$\begin{aligned} {\mathcal {U}}_A[n] = {\left\{ \begin{array}{ll} - \int d\varvec{x} d\varvec{p} \log \left( \dfrac{n(\varvec{x}, \varvec{p})}{{n^A_h}(\varvec{p})} \right) &{} \text{ if } \quad {\mathcal {A}}_\omega [n] = A(\omega ) \\ + \infty &{} \text{ otherwise } \end{array}\right. } , \end{aligned}$$

(80)

where

$$\begin{aligned} n^A_{h}(\varvec{p}) = \dfrac{A'(|\varvec{p} |^2 / 2)}{\int d\varvec{x} d\varvec{\eta } \delta (|\varvec{p} |^2 / 2 - |\varvec{\eta } |^2 / 2)} . \end{aligned}$$

(81)

In the following proof, we first define the microcanonical measure and then derive the rate function (80).

If we consider an infinite domain with a finite amount of total wave action, $\int _{{\mathbb {R}}^d} |\psi ^\mu |^2 d\varvec{x} < \infty$, since the wave action density will be diluted to absolutely 0 everywhere, an equilibrium state of the microcanonical ensemble does not make sense. Here, we instead assume a spatial periodicity of the scalar field $\psi ^\mu (\varvec{x})$ and concentrate our attention on a d-dimensional cubic domain, $[0, 2\pi )^d \equiv \Gamma \subset {\mathbb {R}}^d$. In this setting, the empirical local spectral density (10) consists of delta functions in wave-vector space like

$$\begin{aligned} n^\mu (\varvec{x}, \varvec{p}) = \sum _{\varvec{k} \in (1/2) {\mathbb {Z}}^d} n^\mu _{\varvec{k}} (\varvec{x}) \delta (\varvec{p} - \mu \varvec{k}) , \end{aligned}$$

(82)

where $n^\mu _k (\varvec{x})$ is a discrete form of Wigner distribution adapted to periodic domains, defined as

$$\begin{aligned} n^\mu _{\varvec{k}} (\varvec{x})&= \frac{1}{(2 \pi )^d} \int _\Gamma d\varvec{y} \textrm{e}^{- 2 \textrm{i} \varvec{k} \cdot \varvec{y}} \psi ^\mu \left( \varvec{x} + \varvec{y} \right) \psi ^{\mu \dag } \left( \varvec{x} - \varvec{y} \right) . \end{aligned}$$

(83)

We also introduce the Fourier coefficients of $\psi ^\mu$ as

$$\begin{aligned} \hat{\psi }^\mu _{\varvec{k}}&= \frac{1}{(2\pi )^d} \int _\Gamma \textrm{e}^{- \text {i} \varvec{k} \cdot \varvec{x}} \psi ^\mu (\varvec{x}) d\varvec{x} \end{aligned}$$

(84a)

$$\begin{aligned} \psi ^\mu (\varvec{x})&= \sum _{\varvec{k} \in {\mathbb {Z}}^d} \textrm{e}^{\text {i} \varvec{k} \cdot \varvec{x}} \hat{\psi }^\mu _{\varvec{k}} . \end{aligned}$$

(84b)

Perceval’s theorem allows us to write the wave action density per unit volume in three forms,

$$\begin{aligned} \frac{1}{(2\pi )^d} \int _{\Gamma \times {\mathbb {R}}^d} n^\mu (\varvec{x}, \varvec{p}) d\varvec{x} d\varvec{p} = \frac{1}{(2\pi )^d} \int _{\Gamma } |\psi ^\mu (\varvec{x}) |^2 d\varvec{x} = \sum _{\varvec{k}} |\hat{\psi }^\mu _{\varvec{k}} |^2 . \end{aligned}$$

(85)

We shall set $\mu \rightarrow 0$ while keeping this action density finite. When we fix a volume element in $\varvec{p}$ space, the number of $\varvec{k}$ vectors which are involved there increases as $\sim \mu ^{-d}$. Therefore, the typical amplitude of the Fourier coefficients depends on $\mu$ as $\hat{\psi }^\mu _{\varvec{k}} \sim {\mathcal {O}}(\mu ^{d/2})$. Now, we scale $\hat{\psi }^\mu$ by introducing a new coefficient, $a^\mu _{\varvec{p}}$ with $\varvec{p} \in \mu {\mathbb {Z}}^d$, as $\hat{\psi }^\mu _{\varvec{k}} = \mu ^{d/2} a^\mu _{\mu \varvec{k}}$, i.e.,

$$\begin{aligned} a^\mu _{\varvec{p}} = \frac{1}{(2 \pi \mu ^{1/2})^d} \int _\Gamma \textrm{e}^{- \text {i} \varvec{p} \cdot \varvec{x} / \mu } \psi ^\mu (\varvec{x}) d\varvec{x} . \end{aligned}$$

(86)

Note that $a^\mu _{\varvec{p}}$ remains finite for $\mu \rightarrow 0$, but its norm, $\sum _{\varvec{p} \in \mu {\mathbb {Z}}^d} |a^\mu _{\varvec{p}} |^2 = (2\pi \mu )^{-d} \int _{\Gamma \times {\mathbb {R}}^d} n^\mu (\varvec{x}, \varvec{p}) d\varvec{x} d\varvec{p}$, diverges in the same limit.

To apply equilibrium statistical mechanics, we consider the phase space spanned by the scaled coefficients, $\{ a^\mu _{\varvec{p}} \}_{\varvec{p} \in \mu {\mathbb {Z}}^d}$. In this space, the Lebesgue measure $\mathfrak {m}$ is represented as

$$\begin{aligned} d \mathfrak {m} = \prod _{\varvec{p}} d a^\mu _{\varvec{p}} d a^{\mu \dag }_{\varvec{p}} \equiv \prod _{\varvec{p}} d a^r_{\varvec{p}} d a^i_{\varvec{p}} , \end{aligned}$$

(87)

where $a^\mu _{\varvec{p}} = (a^r_{\varvec{p}} + \textrm{i} a^i_{\varvec{p}}) / \sqrt{2}$ is understood. This measure makes sense only when an upper limit of the wave vector is set to truncate the infinite product. As a result, the number of degrees of freedom of $\psi ^\mu$ in physical space is also restricted. Let us define the bounded set of $\varvec{k}$ as ${\mathbb {K}}_\Delta \equiv \left\{ - 1 / (2\Delta ) + 1, \ldots , 1 / (2\Delta ) \right\} ^d$, and accordingly that of $\varvec{p}$ as $\mu {\mathbb {K}}_\Delta$. The number of elements in ${\mathbb {K}}_\Delta$ is ${\mathcal {N}} \equiv 1 / \Delta ^d$. Then, $\psi ^\mu$ is specified by the values at ${\mathcal {N}}$ points, $\Gamma _\Delta \equiv \left\{ 0, \Delta , \ldots , 2\pi - \Delta \right\} ^d$, and the values in $\Gamma \setminus \Gamma _\Delta$ are determined by interpolation. The Lebesgue measure is now represented in either wave vector or position space as

$$\begin{aligned} d\mathfrak {m} = \prod _{\varvec{p} \in \mu {\mathbb {K}}_\Delta } d a^\mu _{\varvec{p}} d a^{\mu \dag }_{\varvec{p}} = J^\mu _{\Delta } \prod _{\varvec{x} \in \Gamma _\Delta } d \psi ^\mu (\varvec{x}) d \psi ^{\mu \dag }(\varvec{x}) , \end{aligned}$$

(88)

where $J^\mu _\Delta$ is the Jacobian of the function that maps $a^\mu _{\varvec{p}}$ to $\psi ^\mu$. To compute this Jacobian, we consider the integral,

$$\begin{aligned} \int d\mathfrak {m} \exp \left[ - \frac{1}{(2\pi \mu )^d} \int _{\Gamma \times {\mathbb {R}}^d} n^\mu (\varvec{x}, \varvec{p}) d\varvec{x} d\varvec{p} \right] \end{aligned}$$

(89)

that we express in two different ways:

$$\begin{aligned}&\int \left( \prod _{\varvec{p} \in \mu {\mathbb {K}}_\Delta } d a^\mu _{\varvec{p}} d a^{\mu \dag }_{\varvec{p}} \right) \exp \left[ - \sum _{\varvec{p} \in \mu {\mathbb {K}}_\Delta } |a^\mu _{\varvec{p}} |^2 \right] \nonumber \\&\qquad = J^\mu _\Delta \int \left( \prod _{\varvec{x} \in \Gamma _\Delta } d \psi ^\mu (\varvec{x}) d \psi ^{\mu \dag }(\varvec{x}) \right) \exp \left[ - \left( \frac{\Delta }{2 \pi \mu } \right) ^d \sum _{\varvec{x} \in \Gamma _\Delta } |\psi ^\mu (\varvec{x}) |^2 \right] . \end{aligned}$$

(90)

The left-hand side turns out to be $(2 \pi )^{\mathcal {N}}$, and the right-hand side is $J^\mu _\Delta (2 \pi )^{\mathcal {N}} (2 \pi \mu )^{d{\mathcal {N}}} {\mathcal {N}}^{\mathcal {N}}$. We therefore obtain $J^\mu _\Delta$ and, accordingly,

$$\begin{aligned} d \mathfrak {m} = \prod _{\varvec{x} \in \Gamma _\Delta } \frac{d \psi ^\mu (\varvec{x}) d \psi ^{\mu \dag }(\varvec{x})}{(2\pi \mu )^d {\mathcal {N}}} . \end{aligned}$$

(91)

We denote by ${\mathbb {E}}$ integrals over the Lebesgue measure (91). The microcanonical measure with constraints on A is then defined as

$$\begin{aligned} d \mathfrak {m}_A = \frac{\Pi _\omega \delta \left( {\mathcal {A}}_\omega \left[ n^\mu \right] - A(\omega ) \right) }{{\mathbb {E}}\left[ \Pi _\omega \delta \left( {\mathcal {A}}_\omega \left[ n^\mu \right] - A(\omega ) \right) \right] } d \mathfrak {m}, \end{aligned}$$

(92)

where $\Pi _\omega \delta \left( {\mathcal {A}}_\omega \left[ n^\mu \right] - A(\omega ) \right)$ means that we constrain the values of all the invariants ${\mathcal {A}}_\omega$ for any $\omega$.

Our goal is to compute the probability distribution of $n^\mu$ for a microcanonical measure constrained by A, i.e.,

$$\begin{aligned} {\mathbb {P}}^\mu _{A,m}[n^\mu = n] = \frac{{\mathbb {E}}\left[ \delta \left( n^\mu - n \right) \Pi _\omega \delta \left( {\mathcal {A}}_\omega \left[ n^\mu \right] - A(\omega ) \right) \right] }{{\mathbb {E}}\left[ \Pi _\omega \delta \left( {\mathcal {A}}_\omega \left[ n^\mu \right] - A(\omega ) \right) \right] }. \end{aligned}$$

(93)

It will be mathematically convenient, for intermediate computations, to use in the following a normalizable Gaussian measure $d \mathfrak {m}_G$,

$$\begin{aligned} d \mathfrak {m}_G&= \prod _{\varvec{p} \in \mu {\mathbb {K}}_\Delta } \textrm{e}^{- |a^\mu _{\varvec{p}} |^2 } \frac{d a^\mu _{\varvec{p}} d a^{\mu \dag }_{\varvec{p}}}{2 \pi } \nonumber \\ {}&= \exp \left[ - \frac{1}{(2\pi \mu )^d} \int _{\Gamma \times {\mathbb {R}}^d} n^\mu (\varvec{x}, \varvec{p}) d\varvec{x} d\varvec{p} - {\mathcal {N}} \log 2\pi \right] d \mathfrak {m} , \end{aligned}$$

(94)

which satisfies $\int d\mathfrak {m}_G = 1$. We note that the ${\mathcal {N}}$-dependent term diverges in the $\Delta \rightarrow 0$ limit, but this divergence will be compensated in ${\mathbb {P}}_{A, m}^\mu$ below. We denote ${\mathbb {E}}_G$ averages with respect to this Gaussian measure.

Then, (93) can be rewritten as

$$\begin{aligned}&{\mathbb {P}}^\mu _{A,m}[n^\mu = n] \nonumber \\&\qquad = \dfrac{\exp \left[ (2\pi \mu )^{-d} \int _{\Gamma \times {\mathbb {R}}^d} n (\varvec{x}, \varvec{p}) d\varvec{x} d\varvec{p} \right] {\mathbb {E}}_G\left[ \delta \left( n^\mu - n \right) \Pi _\omega \delta \left( {\mathcal {A}}_\omega \left[ n^\mu \right] - A(\omega ) \right) \right] }{{\mathbb {E}}_G \left\{ \exp \left[ (2\pi \mu )^{-d} \int _{\Gamma \times {\mathbb {R}}^d} n^\mu (\varvec{x}, \varvec{p}) d\varvec{x} d\varvec{p} \right] \Pi _\omega \delta \left( {\mathcal {A}}_\omega \left[ n^\mu \right] - A(\omega ) \right) \right\} }. \end{aligned}$$

(95)

In the following, when we consider the Gaussian measure $\mathfrak {m}_G$, the continuous limit $\Delta \rightarrow 0$ is always understood.

We look for a large deviation principle

$$\begin{aligned} {\mathbb {E}}_G\left[ \delta \left( n^\mu - n \right) \Pi _\omega \delta \left( {\mathcal {A}}_\omega \left[ n^\mu \right] - A(\omega ) \right) \right] \underset{\mu \rightarrow 0}{\asymp }\ \exp \left( -\frac{{\mathcal {I}}_G[n, A]}{(2 \pi \mu )^d} \right) . \end{aligned}$$

(96)

Our strategy is to compute its rescaled cumulant-generating function and to apply the Gärtner–Ellis theorem. We shall then define a free energy as

$$\begin{aligned} f_G[\lambda , \beta ]&\equiv - \lim _{\mu \rightarrow 0} \left( 2 \pi \mu \right) ^d \log {\mathbb {E}}_G \Biggl \{ \exp \Biggl [ - \frac{1}{\left( 2 \pi \mu \right) ^d} \int _{\Gamma \times {\mathbb {R}}^d} d\varvec{x} d\varvec{p} \nonumber \\&\quad \times \biggl ( \int _{{\mathbb {R}}^+} \beta (\omega ) h \left( \omega - \frac{|\varvec{p} |^2}{2} \right) d\omega + \lambda (\varvec{x}, \varvec{p}) \biggr ) n^\mu (\varvec{x}, \varvec{p}) \Biggr ] \Biggr \}, \end{aligned}$$

(97)

where $\beta : {\mathbb {R}}^+ \rightarrow {\mathbb {R}}$ is a real continuous function representing the chemical potential and $\lambda : \Gamma \times {\mathbb {R}}^d \rightarrow {\mathbb {R}}$ is also a real continuous function. Because $n^\mu$ is quadratic in $\psi ^\mu$, the expectation is a Gaussian integral. To make it explicit, the expression in the square brackets is rewritten as

$$\begin{aligned} - \frac{1}{\left( 2 \pi \mu \right) ^d} \int \psi ^{\mu \dag } (\varvec{x}) {\mathcal {L}}_{\tilde{\lambda }} \psi ^\mu (\varvec{x}) d\varvec{x} , \end{aligned}$$

(98)

where ${\mathcal {L}}_{\tilde{\lambda }}$ is a pseudo-differential operator defined as

$$\begin{aligned} {\mathcal {L}}_{\tilde{\lambda }} \psi ^\mu (\varvec{x})&\equiv \int L_{\tilde{\lambda }} (\varvec{x}, \varvec{x}') \psi ^\mu (\varvec{x}')d \varvec{x}' \end{aligned}$$

(99a)

$$\begin{aligned} L_{\tilde{\lambda }} (\varvec{x}, \varvec{x}')&= \frac{1}{(2 \pi \mu )^d} \int _{{\mathbb {R}}^d} \tilde{\lambda } \left( \frac{\varvec{x} + \varvec{x}'}{2}, \varvec{p} \right) \textrm{e}^{\text {i} \varvec{p} \cdot (\varvec{x} - \varvec{x}') / \mu } d \varvec{p} \end{aligned}$$

(99b)

$$\begin{aligned} {\tilde{\lambda }} (\varvec{x}, \varvec{p})&\equiv {\left\{ \begin{array}{ll} \lambda (\varvec{x}, \varvec{p}) + \int _{{\mathbb {R}}^+} \beta (\omega ) h(\omega - |\varvec{p} |^2 / 2) d\omega \quad (\varvec{x} \in \Gamma ) \\ 0 \quad (\varvec{x} \notin \Gamma ) . \end{array}\right. } \end{aligned}$$

(99c)

In (98) and (99a), the integration range is unbounded. However, Riemann–Lebesgue lemma applied to (99b) assures that the kernel function $L_{\tilde{\lambda }}$ vanishes in the limit of $\mu \rightarrow 0$ except in the vicinity of the points of $\varvec{x} = \varvec{x}'$. Consequently, further taking into account (99c), the range of the integration of (98) and (99a) can be reduced from ${\mathbb {R}}^d$ to $\Gamma$.

To obtain a simpler form of $f_G[\lambda , \beta ]$, we need to compute the functional determinant of ${\mathcal {L}}_{\tilde{\lambda }}$. This is done here straightforwardly:

$$\begin{aligned}&f_G[\lambda , \beta ] = - \lim _{\mu \rightarrow 0} \left( 2 \pi \mu \right) ^d \log {\mathbb {E}}_G \left\{ \exp \left[ - \frac{1}{\left( 2 \pi \mu \right) ^d} \int _{{\mathbb {R}}^d} \psi ^{\mu \dag } (\varvec{x}) {\mathcal {L}}_{\tilde{\lambda }} \psi ^\mu (\varvec{x}) d\varvec{x} \right] \right\} \nonumber \\&\quad \; = - \lim _{\mu \rightarrow 0} \lim _{\Delta \rightarrow 0} \left( 2 \pi \mu \right) ^d \log \int \prod _{\varvec{x} \in \Gamma _\Delta } \frac{d \psi ^\mu (\varvec{x}) d \psi ^{\mu \dag }(\varvec{x})}{2 \pi (2\pi \mu )^d {\mathcal {N}}} \nonumber \\&\quad \; \times \exp \left[ - \frac{\Delta ^{2d}}{\left( 2 \pi \mu \right) ^d} \sum _{\varvec{x}, \varvec{x}' \in \Gamma _\Delta } \psi ^{\mu \dag } (\varvec{x}) L_{\tilde{\lambda }} (\varvec{x}, \varvec{x}') \psi ^\mu (\varvec{x}') - \frac{\Delta ^{d}}{\left( 2 \pi \mu \right) ^d} \sum _{\varvec{x} \in \Gamma _\Delta } |\psi ^\mu (\varvec{x}) |^2 \right] \nonumber \\&\quad \; = \lim _{\mu \rightarrow 0} \lim _{\Delta \rightarrow 0} \left( 2 \pi \mu \right) ^d \log \textrm{det} \left( \varvec{I} + \varvec{L}^\Delta _{\tilde{\lambda }} \right) . \end{aligned}$$

(100)

We have carried out the Gaussian integration. The problem thus reduces to computing the determinant of an ${\mathcal {N}} \times {\mathcal {N}}$ matrix, $\varvec{I} + \varvec{L}^\Delta _{\tilde{\lambda }}$, where $\varvec{I}$ is a unit matrix and $\varvec{L}^\Delta _{\tilde{\lambda }}$ consists of $\left\{ \Delta ^d L_{\tilde{\lambda }} (\varvec{x}, \varvec{x}') \ \vert \ \varvec{x},\varvec{x}' \in \Gamma _\Delta \right\}$. Let us use the following expression,

$$\begin{aligned} \log \textrm{det} \left( \varvec{I} + \varvec{L}^\Delta _{\tilde{\lambda }} \right)&= \textrm{tr} \log \left( \varvec{I} + \varvec{L}^\Delta _{\tilde{\lambda }} \right) \nonumber \\&= - \sum _{j=1}^\infty \frac{(-1)^j}{j} \textrm{tr} \varvec{L}^{\Delta j}_{\tilde{\lambda }} , \end{aligned}$$

(101)

which holds for sufficiently small $\tilde{\lambda }$. We know that $\varvec{L}_{\tilde{\lambda }}^{\Delta j}$ consists of $\left\{ \Delta ^d L^j_{\tilde{\lambda }} (\varvec{x}, \varvec{x}') \ \vert \ \varvec{x},\varvec{x}' \in \Gamma _\Delta \right\}$ where $L^j_{\tilde{\lambda }} (\varvec{x}, \varvec{x}')$ corresponds to the kernel function of an operator,

$$\begin{aligned} {\mathcal {L}}^j_{\tilde{\lambda }} \equiv \underbrace{{\mathcal {L}}_{\tilde{\lambda }} \ldots {\mathcal {L}}_{\tilde{\lambda }}}_j , \end{aligned}$$

(102)

in the small $\Delta$ limit. In general, a product between pseudo-differential operators corresponds to a star product, or a Moyal product, between symbols [68, 69]. The star product is expanded in terms of $\mu$ with the leading term equivalent to the ordinary product. Accordingly, ${\mathcal {L}}^j_{\tilde{\lambda }} = {\mathcal {L}}_{\tilde{\lambda }^j} + {\mathcal {O}}(\mu )$ holds, where $\tilde{\lambda }^j$ is the jth power of $\tilde{\lambda }$. We thus derive

$$\begin{aligned} \lim _{\Delta \rightarrow 0} L^j_{\tilde{\lambda }} (\varvec{x}, \varvec{x})&= \frac{1}{(2 \pi \mu )^d} \int _{{\mathbb {R}}^d} \tilde{\lambda }^j (\varvec{x}, \varvec{p}) d\varvec{p} + {\mathcal {O}}(\mu ) \nonumber \\ \therefore \lim _{\Delta \rightarrow 0} \textrm{tr} \varvec{L}^{\Delta j}_{\tilde{\lambda }}&= \frac{1}{(2 \pi \mu )^d} \int _{\Gamma \times {\mathbb {R}}^d} \tilde{\lambda }^j (\varvec{x}, \varvec{p}) d\varvec{x} d\varvec{p} + {\mathcal {O}}(\mu ) , \end{aligned}$$

(103)

and hence

$$\begin{aligned} f_G[\lambda , \beta ] = \int _{\Gamma \times {\mathbb {R}}^d} \log \left( \lambda (\varvec{x}, \varvec{p}) + \int _{{\mathbb {R}}^+} \beta (\omega ) h \left( \omega - \frac{|\varvec{p} |^2}{2} \right) d\omega + 1 \right) d\varvec{x} d\varvec{p}.\qquad \end{aligned}$$

(104)

From this formula, the Gärtner–Ellis theorem yields the rate function ${\mathcal {I}}_G$ (96) as

$$\begin{aligned} {\mathcal {I}}_G [n, A]=- \inf _{\lambda , \beta } \left\{ \int d\varvec{x} d\varvec{p} \lambda (\varvec{x}, \varvec{p}) n(\varvec{x}, \varvec{p}) + \int _{{\mathbb {R}}^+} d \omega \beta (\omega ) A(\omega ) - f[\lambda , \beta ] \right\} , \end{aligned}$$

(105)

which is computed as

$$\begin{aligned} {\mathcal {I}}_G [n, A] = {\left\{ \begin{array}{ll} \int d\varvec{x} d\varvec{p} \left( n - 1 - \log n \right) &{} \text{ if } \quad {\mathcal {A}}_\omega [n] = A(\omega ) \\ + \infty &{} \text{ otherwise } \end{array}\right. } . \end{aligned}$$

(106)

As should have been expected, the minimum value of ${\mathcal {I}}_G[n, A]$ is 0, which is realized when and only when $n = 1$ and $A(\omega ) = {\mathcal {A}}_\omega [1]$.

Based on the large deviation result (96), the numerator in (95) turns out to be

$$\begin{aligned} {}&\exp \left[ \frac{1}{(2\pi \mu )^d} \int _{\Gamma \times {\mathbb {R}}^d} n (\varvec{x}, \varvec{p}) d\varvec{x} d\varvec{p} \right] {\mathbb {E}}_G\left[ \delta \left( n^\mu - n \right) \Pi _\omega \delta \left( {\mathcal {A}}_\omega \left[ n^\mu \right] - A(\omega ) \right) \right] \nonumber \\ {}&\underset{\mu \rightarrow 0}{\asymp }\ \exp \left( \frac{S_A[n]}{(2 \pi \mu )^d} \right) \end{aligned}$$

(107)

with

$$\begin{aligned} S_A[n] = {\left\{ \begin{array}{ll} \int d\varvec{x} d\varvec{p} \left( 1 + \log n \right) &{} \text{ if } \quad {\mathcal {A}}_\omega [n] = A(\omega ) \\ - \infty &{} \text{ otherwise } \end{array}\right. } . \end{aligned}$$

(108)

The finite part of this function defines the entropy for a mesoscopic state specified by n, and it coincides with the Lyapunov function (25) for the wave kinetic equation. Because the denominator of (95) is the integration of the numerator over all n, the Laplace’s principle enables us to compute it as

$$\begin{aligned} {}&{\mathbb {E}}_G \left\{ \exp \left[ \frac{1}{(2\pi \mu )^d} \int _{\Gamma \times {\mathbb {R}}^d} n^\mu (\varvec{x}, \varvec{p}) d\varvec{x} d\varvec{p} \right] \Pi _\omega \delta \left( {\mathcal {A}}_\omega \left[ n^\mu \right] - A(\omega ) \right) \right\} \nonumber \\ {}&\underset{\mu \rightarrow 0}{\asymp }\ \exp \left( \frac{s[A]}{(2 \pi \mu )^d} \right) , \end{aligned}$$

(109)

with $s[A] = \sup _n \left\{ S_A[n] \right\}$. The supremum is achieved when ${\mathcal {A}}_\omega [n] = A(\omega )$, and

$$\begin{aligned} n(\varvec{x}, \varvec{p}) = n_h^A(\varvec{p}) = N \left( \frac{|\varvec{p}|^2}{2} \right) ; \end{aligned}$$

(110)

that is, n is homogeneous in space and depends only on the magnitude of its wave vector. The function $N(\omega )$ is related to $A(\omega )$ by the condition ${\mathcal {A}}_\omega [n^A_h] = A(\omega )$. This gives formula (81) for $n_h^A$. We also have

$$\begin{aligned} s[A] = \int d\varvec{x} d\varvec{p} \left( 1 + \log n_h^A \right) . \end{aligned}$$

(111)

Finally, starting from (95), and using the two asymptotic relations (107) and (109), as well as (108) and (111), we obtain

$$\begin{aligned} {\mathbb {P}}^\mu _{A,m}[n^\mu = n] \underset{\mu \rightarrow 0}{\asymp }\ \exp \left( -\frac{{\mathcal {U}}_A[n]}{(2 \pi \mu )^d} \right) \end{aligned}$$

(112)

where the quasipotential ${\mathcal {U}}_A[n]$ is given by Eq. (80). We have established the announced results.

C Computations of Scattering Terms

This appendix describes somewhat intricate derivation of the scattering terms that appear in the wave kinetic equation and the large deviation Hamiltonian. For this purpose, we prepare some useful formulae,

$$\begin{aligned} \int _0^t d\tau _1 \textrm{e}^{\textrm{i} \omega \tau _1 / \mu } \int _0^{\tau _1} d\tau _2 \textrm{e}^{-\textrm{i} \omega \tau _2 / \mu }&= \pi \mu t \delta (\omega ) + o(\mu ) \end{aligned}$$

(113a)

$$\begin{aligned} \int _0^t d\tau _1 \textrm{e}^{\textrm{i} \omega \tau _1 / \mu } \int _0^t d\tau _2 \textrm{e}^{-\textrm{i} \omega \tau _2 / \mu }&= 2 \pi \mu t \delta (\omega ) + o(\mu ) \end{aligned}$$

(113b)

$$\begin{aligned} \frac{1}{(2 \pi \mu )^d} \int _{{\mathbb {R}}^d} d \varvec{\xi } \textrm{e}^{\text {i} \varvec{p} \cdot (\varvec{\xi } - \varvec{x}) / \mu } f(\varvec{\xi })&= \sum _{|\alpha |\ge 0} \frac{(- \text {i} \mu )^{|\alpha |}}{\alpha !} \nabla _x^\alpha f(\varvec{x}) \nabla _p^\alpha \delta (\varvec{p}) . \end{aligned}$$

(113c)

Equations (113a) and (113b) are often used in the literature of weak turbulence [2]. The residual terms denoted by $o(\mu )$ make negligible contributions in the limit of $\mu \rightarrow 0$ compared to the leading-order terms when integrated with respect to $\omega$. In (113c), a multi-index notation is used. In the following computation, integration is always carried out over ${\mathbb {R}}^d$, except for the basic positional coordinates represented by $\varvec{x}$ whose integration range is $\Gamma$.

1.1 C.1 Terms Appearing in the Classical Wave Kinetic Equation

We first compute the scattering terms in the wave kinetic equation, specifically ${\mathbb {E}} \left[ w^\mu (\psi ^\mu _2, \psi ^\mu _0) \right]$, ${\mathbb {E}} \left[ w^\mu (\psi ^\mu _0, \psi ^\mu _2) \right]$, and ${\mathbb {E}} \left[ w^\mu (\psi ^\mu _1, \psi ^\mu _1) \right]$. These terms are common with those linear to $\lambda$ in the scattering part of the large deviation Hamiltonian, ${\mathcal {H}}_S$. The computations are slightly involved but mostly straightforward. A detailed procedure is presented only for the ${\mathbb {E}} \left[ w^\mu (\psi ^\mu _2, \psi ^\mu _0) \right]$ case.

From (16) and (18), we have

$$\begin{aligned} w^\mu (\psi ^\mu _2, \psi ^\mu _0)&= \frac{1}{- \mu ^2 (2 \pi \mu )^d} \int _0^t d\tau _1 \int _0^{\tau _1} d\tau _2 \int d\varvec{y} d\varvec{\xi }_{1234} \textrm{e}^{- \textrm{i} \varvec{p} \cdot \varvec{y} / \mu } \\ {}&\quad \times G^\mu \left( \varvec{x} + \frac{\varvec{y}}{2} - \varvec{\xi }_1, t - \tau _1 \right) V^\mu (\varvec{\xi }_1) G^\mu (\varvec{\xi }_1 - \varvec{\xi }_2, \tau _1 - \tau _2) V^\mu (\varvec{\xi }_2)\\ {}&\quad \times G^\mu (\varvec{\xi }_2 - \varvec{\xi }_3, \tau _2) \psi ^\mu (\varvec{\xi }_3, 0) \\ {}&\quad \times G^{\mu \dag } \left( \varvec{x} - \frac{\varvec{y}}{2} - \varvec{\xi }_4, t \right) \psi ^{\mu \dag } (\varvec{\xi }_4, 0) . \end{aligned}$$

Taking ensemble average, writing the propagators $G^\mu$ as Fourier integrals (17) with wave vectors $\varvec{\eta }_1, \varvec{\eta }_2, \varvec{\eta }_3$ and $\varvec{\eta }_4$ in this order, and setting

$$\begin{aligned} {\mathbb {E}} \left[ V^\mu (\varvec{\xi }_1) V^\mu (\varvec{\xi }_2) \right]&= \int d \varvec{\eta }_5 \textrm{e}^{\textrm{i} \varvec{\eta }_5 \cdot (\varvec{\xi }_1 - \varvec{\xi }_2) / \mu } \Pi (\varvec{\eta }_5) \\ \psi ^\mu (\varvec{\xi }_3, 0) \psi ^{\mu \dag } (\varvec{\xi }_4, 0)&= \int d \varvec{\eta }_6 \textrm{e}^{\textrm{i} \varvec{\eta }_6 \cdot (\varvec{\xi }_3 - \varvec{\xi }_4) / \mu } n ((\varvec{\xi }_3 + \varvec{\xi }_4)/2, \varvec{\eta }_6) , \end{aligned}$$

we derive

$$\begin{aligned} {}&{\mathbb {E}} \left[ w^\mu (\psi ^\mu _2, \psi ^\mu _0) \right] \\ {}&= \frac{1}{- \mu ^2 (2 \pi \mu )^{5d}} \int d\varvec{y} d\varvec{\xi }_{1234} d\varvec{\eta }_{123456} \textrm{e}^{- \textrm{i} \varvec{p} \cdot \varvec{y} / \mu } \\ {}&\qquad \times \textrm{e}^{-\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_4\vert ^2)t / 2 \mu } \int _0^t d\tau _1 \textrm{e}^{\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_2\vert ^2) \tau _1 / 2 \mu } \int _0^{\tau _1} d\tau _2 \textrm{e}^{\textrm{i} (\vert \varvec{\eta }_2\vert ^2 - \vert \varvec{\eta }_3\vert ^2) \tau _2 / 2 \mu } \\ {}&\qquad \times \textrm{e}^{\textrm{i} \varvec{\eta }_1 \cdot (\varvec{x} + \varvec{y} / 2 - \varvec{\xi }_1) / \mu } \textrm{e}^{\textrm{i} \varvec{\eta }_2 \cdot (\varvec{\xi }_1 - \varvec{\xi }_2) / \mu } \textrm{e}^{\textrm{i} \varvec{\eta }_3 \cdot (\varvec{\xi }_2 - \varvec{\xi }_3) / \mu } \textrm{e}^{- \textrm{i} \varvec{\eta }_4 \cdot (\varvec{x} - \varvec{y} / 2 - \varvec{\xi }_4) / \mu } \\ {}&\qquad \times \Pi (\varvec{\eta }_5) \textrm{e}^{\textrm{i} \varvec{\eta }_5 \cdot (\varvec{\xi }_1 - \varvec{\xi }_2)} n((\varvec{\xi }_3 + \varvec{\xi }_4) / 2, \varvec{\eta }_6) \textrm{e}^{\textrm{i} \varvec{\eta }_6 \cdot (\varvec{\xi }_3 - \varvec{\xi }_4) / \mu } . \end{aligned}$$

Integration of this expression with respect to $\varvec{y}$, $\varvec{\xi }_1$, and $\varvec{\xi }_2$ yields

$$\begin{aligned} {}&{\mathbb {E}} \left[ w^\mu (\psi ^\mu _2, \psi ^\mu _0) \right] \\ {}&\qquad = \frac{1}{- \mu ^2 (2 \pi \mu )^{2d}} \int d\varvec{\xi }_{34} d\varvec{\eta }_{123456} \\ {}&\qquad \times \textrm{e}^{-\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_4\vert ^2)t / 2 \mu } \int _0^t d\tau _1 \textrm{e}^{\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_2\vert ^2) \tau _1 / 2 \mu } \int _0^{\tau _1} d\tau _2 \textrm{e}^{\textrm{i} (\vert \varvec{\eta }_2\vert ^2 - \vert \varvec{\eta }_3\vert ^2) \tau _2 / 2 \mu } \\ {}&\qquad \times \textrm{e}^{\textrm{i} (\varvec{\eta }_1 \cdot \varvec{x} - \varvec{\eta }_3 \cdot \varvec{\xi }_3 - \varvec{\eta }_4 \cdot \varvec{x} + \varvec{\eta }_4 \cdot \varvec{\xi }_4 + \varvec{\eta }_6 \cdot \varvec{\xi }_3 - \varvec{\eta }_6 \cdot \varvec{\xi }_4) / \mu } \\ {}&\qquad \times \delta ((\varvec{\eta }_1 + \varvec{\eta }_4) / 2 - \varvec{p}) \delta (\varvec{\eta }_1 - \varvec{\eta }_2 - \varvec{\eta }_5) \delta (\varvec{\eta }_3 - \varvec{\eta }_2 - \varvec{\eta }_5) \\ {}&\qquad \times \Pi (\varvec{\eta }_5) n((\varvec{\xi }_3 + \varvec{\xi }_4) / 2, \varvec{\eta }_6) . \end{aligned}$$

We change the variables as

$$\begin{aligned} \varvec{X} = \frac{\varvec{\xi }_3 + \varvec{\xi }_4}{2}, \quad \varvec{Y} = \varvec{\xi }_3 - \varvec{\xi }_4 \end{aligned}$$

and carry out the integration with respect to $\varvec{Y}$ to get

$$\begin{aligned} {}&{\mathbb {E}} \left[ w^\mu (\psi ^\mu _2, \psi ^\mu _0) \right] \\ {}&\qquad = \frac{1}{- \mu ^2 (2 \pi \mu )^d} \int d\varvec{X} d\varvec{\eta }_{123456} \\ {}&\qquad \times \textrm{e}^{-\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_4\vert ^2)t / 2 \mu } \int _0^t d\tau _1 \textrm{e}^{\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_2\vert ^2) \tau _1 / 2 \mu } \int _0^{\tau _1} d\tau _2 \textrm{e}^{\textrm{i} (\vert \varvec{\eta }_2\vert ^2 - \vert \varvec{\eta }_3\vert ^2) \tau _2 / 2 \mu } \\ {}&\qquad \times \textrm{e}^{\textrm{i} (\varvec{\eta }_1 - \varvec{\eta }_4) \cdot (\varvec{x} - \varvec{X}) / \mu } \delta ((\varvec{\eta }_3 + \varvec{\eta }_4) / 2 - \varvec{\eta }_6) \\ {}&\qquad \times \delta ((\varvec{\eta }_1 + \varvec{\eta }_4) / 2 - \varvec{p}) \delta (\varvec{\eta }_1 - \varvec{\eta }_2 - \varvec{\eta }_5) \delta (\varvec{\eta }_3 - \varvec{\eta }_2 - \varvec{\eta }_5) \\ {}&\qquad \times \Pi (\varvec{\eta }_5) n(\varvec{X}, \varvec{\eta }_6) . \end{aligned}$$

We understand that $\varvec{\eta }_1 = \varvec{\eta }_3$ and $\varvec{\eta }_6 = \varvec{p}$ hold in the integrand. Integration with respect to $\varvec{\eta }_3$ and $\varvec{\eta }_6$ yields

$$\begin{aligned}&{\mathbb {E}} \left[ w^\mu (\psi ^\mu _2, \psi ^\mu _0) \right] \\ {}&\qquad = \frac{1}{- \mu ^2 (2 \pi \mu )^{d}} \int d\varvec{X} d\varvec{\eta }_{1245} \\ {}&\qquad \times \textrm{e}^{-\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_4\vert ^2)t / 2 \mu } \int _0^t d\tau _1 \textrm{e}^{\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_2\vert ^2) \tau _1 / 2 \mu } \int _0^{\tau _1} d\tau _2 \textrm{e}^{\textrm{i} (\vert \varvec{\eta }_2\vert ^2 - \vert \varvec{\eta }_1\vert ^2) \tau _2 / 2 \mu } \\ {}&\qquad \times \textrm{e}^{\textrm{i} (\varvec{\eta }_1 - \varvec{\eta }_4) \cdot (\varvec{x} - \varvec{X}) / \mu } \delta ((\varvec{\eta }_1 + \varvec{\eta }_4) / 2 - \varvec{p}) \delta (\varvec{\eta }_1 - \varvec{\eta }_2 - \varvec{\eta }_5) \\ {}&\qquad \times \Pi (\varvec{\eta }_5) n(\varvec{X}, \varvec{p}) . \end{aligned}$$

We use (113a) to derive

$$\begin{aligned} {}&\int _0^t d\tau _1 \textrm{e}^{\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_2\vert ^2) \tau _1 / 2 \mu } \int _0^{\tau _1} d\tau _2 \textrm{e}^{\textrm{i} (\vert \varvec{\eta }_2\vert ^2 - \vert \varvec{\eta }_1\vert ^2) \tau _2 / 2 \mu } \\ {}&= \pi \mu t \delta \left( \frac{\vert \varvec{\eta }_1\vert ^2}{2} - \frac{\vert \varvec{\eta }_2\vert ^2}{2} \right) + o(\mu ) . \end{aligned}$$

We also use (113c) to derive

$$\begin{aligned} {}&\frac{1}{(2 \pi \mu )^{d}} \int d\varvec{X} d\varvec{\eta }_{4} \textrm{e}^{-\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_4\vert ^2)t / 2 \mu } \textrm{e}^{\textrm{i} (\varvec{\eta }_1 - \varvec{\eta }_4) \cdot (\varvec{x} - \varvec{X}) / \mu } \delta ((\varvec{\eta }_1 + \varvec{\eta }_4) / 2 - \varvec{p}) n (\varvec{X}, \varvec{p}) \\ {}&\quad = \delta (\varvec{\eta }_1 - \varvec{p}) n(\varvec{x}, \varvec{p}) + {\mathcal {O}} (\mu , t) . \end{aligned}$$

Consequently, we obtain

$$\begin{aligned} {\mathbb {E}} \left[ w^\mu (\psi ^\mu _2, \psi ^\mu _0) \right]&= - \frac{t}{2 \mu } \int d\varvec{\eta } \sigma (\varvec{p}, \varvec{\eta }) n (\varvec{x}, \varvec{p}) + o \left( \frac{t}{\mu } \right) . \end{aligned}$$

(114)

Because $w^\mu (\psi ^\mu _0, \psi ^\mu _2) = [w^\mu (\psi ^\mu _2, \psi ^\mu _0)]^\dag$, and $\sigma$ and n are real functions, we also have

$$\begin{aligned} {\mathbb {E}} \left[ w^\mu (\psi ^\mu _0, \psi ^\mu _2) \right]&= - \frac{t}{2 \mu } \int d\varvec{\eta } \sigma (\varvec{p}, \varvec{\eta }) n (\varvec{x}, \varvec{p}) + o \left( \frac{t}{\mu } \right) . \end{aligned}$$

(115)

Finally, we consider ${\mathbb {E}}[w^\mu (\psi ^\mu _1, \psi ^\mu _1)]$. From (16) and (18), we have

$$\begin{aligned} w^\mu (\psi ^\mu _1, \psi ^\mu _1)&= \frac{1}{\mu ^2 (2 \pi \mu )^d} \int _0^t d\tau _1 \int _0^t d\tau _2 \int d\varvec{y} d\varvec{\xi }_{1234} \textrm{e}^{- \textrm{i} \varvec{p} \cdot \varvec{y} / \mu }\\ {}&\qquad \times G^\mu \left( \varvec{x} + \frac{\varvec{y}}{2} - \varvec{\xi }_1, t - \tau _1 \right) V^\mu (\varvec{\xi }_1) G^\mu (\varvec{\xi }_1 - \varvec{\xi }_2, \tau _1) \psi ^\mu (\varvec{\xi }_2, 0) \\ {}&\qquad \times G^{\mu \dag } \left( \varvec{x} - \frac{\varvec{y}}{2} - \varvec{\xi }_3, t - \tau _2 \right) V^\mu (\varvec{\xi }_3) G^{\mu \dag } (\varvec{\xi }_3 - \varvec{\xi }_4, \tau _2) \psi ^{\mu \dag } (\varvec{\xi }_4, 0) . \end{aligned}$$

Taking the ensemble average, introducing the Fourier integrals, and integrating some variables in the same manner as the previous case, we derive

$$\begin{aligned} {}&{\mathbb {E}} \left[ w^\mu (\psi ^\mu _1, \psi ^\mu _1) \right] \\ {}&= \frac{1}{\mu ^2 (2 \pi \mu )^d} \int d\varvec{X} d\varvec{\eta }_{123456} \\ {}&\qquad \times \textrm{e}^{-\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_3\vert ^2)t / 2 \mu } \int _0^t d\tau _1 \textrm{e}^{\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_2\vert ^2) \tau _1 / 2 \mu } \int _0^t d\tau _2 \textrm{e}^{- \textrm{i} (\vert \varvec{\eta }_3\vert ^3 - \vert \varvec{\eta }_4\vert ^2) \tau _2 / 2 \mu } \\ {}&\qquad \times \textrm{e}^{\textrm{i} (\varvec{\eta }_1 - \varvec{\eta }_3) \cdot (\varvec{x} - \varvec{X}) / \mu } \delta ( (\varvec{\eta }_2 + \varvec{\eta }_4) / 2 - \varvec{\eta }_6) \\ {}&\qquad \times \delta ((\varvec{\eta }_1 + \varvec{\eta }_3) / 2 - \varvec{p}) \delta (\varvec{\eta }_1 - \varvec{\eta }_2 - \varvec{\eta }_5) \delta (\varvec{\eta }_3 - \varvec{\eta }_4 - \varvec{\eta }_5) \\ {}&\qquad \times \Pi (\varvec{\eta }_5) n(\varvec{X}, \varvec{\eta }_6) . \end{aligned}$$

Applying (113c), understanding $\varvec{\eta }_1 = \varvec{\eta }_3$ and $\varvec{\eta }_2 = \varvec{\eta }_4$ in the integrand, using (113b), and integrating all the possible variables, we finally obtain

$$\begin{aligned} {\mathbb {E}} \left[ w^\mu (\psi ^\mu _1, \psi ^\mu _1) \right] = \frac{t}{\mu } \int d\varvec{\eta } \sigma (\varvec{p}, \varvec{\eta }) n(\varvec{x}, \varvec{\eta }) + o \left( \frac{t}{\mu } \right) . \end{aligned}$$

(116)

1.2 C.2 Quadratic Terms in the Hamiltonian

We compute the terms in ${\mathcal {H}}_S$ quadratic in $\lambda$, originating from four expressions,

$$\begin{aligned} \frac{1}{(2\pi \mu )^{2d}} \int d\varvec{x}_{12} d\varvec{p}_{12} \lambda (\varvec{x}_1, \varvec{p}_1) \lambda (\varvec{x}_2, \varvec{p}_2) {\mathbb {E}}[w^\mu (\psi ^\mu _1, \psi ^\mu _0)(\varvec{x}_1, \varvec{p}_1) w^\mu (\psi ^\mu _0, \psi ^\mu _1)(\varvec{x}_2, \varvec{p}_2)] \end{aligned}$$

(117a)

$$\begin{aligned} \frac{1}{(2\pi \mu )^{2d}} \int d\varvec{x}_{12} d\varvec{p}_{12} \lambda (\varvec{x}_1, \varvec{p}_1) \lambda (\varvec{x}_2, \varvec{p}_2) {\mathbb {E}}[w^\mu (\psi ^\mu _0, \psi ^\mu _1)(\varvec{x}_1, \varvec{p}_1) w^\mu (\psi ^\mu _1, \psi ^\mu _0)(\varvec{x}_2, \varvec{p}_2)] \end{aligned}$$

(117b)

$$\begin{aligned} \frac{1}{(2\pi \mu )^{2d}} \int d\varvec{x}_{12} d\varvec{p}_{12} \lambda (\varvec{x}_1, \varvec{p}_1) \lambda (\varvec{x}_2, \varvec{p}_2) {\mathbb {E}}[w^\mu (\psi ^\mu _1, \psi ^\mu _0)(\varvec{x}_1, \varvec{p}_1) w^\mu (\psi ^\mu _1, \psi ^\mu _0)(\varvec{x}_2, \varvec{p}_2)] \end{aligned}$$

(117c)

$$\begin{aligned} \frac{1}{(2\pi \mu )^{2d}} \int d\varvec{x}_{12} d\varvec{p}_{12} \lambda (\varvec{x}_1, \varvec{p}_1) \lambda (\varvec{x}_2, \varvec{p}_2) {\mathbb {E}}[w^\mu (\psi ^\mu _0, \psi ^\mu _1)(\varvec{x}_1, \varvec{p}_1) w^\mu (\psi ^\mu _0, \psi ^\mu _1)(\varvec{x}_2, \varvec{p}_2)] . \end{aligned}$$

(117d)

Among these, the two pairs, (117a)–(117b) and (117c)–(117d), are complex conjugate, respectively. Therefore, we need to compute only two expressions. Although the number of factors involved in the integration is greater than those in the linear terms, the computation procedures are largely the same. For the sake of conciseness, we denote time t instead of $\Delta t$.

From (16) and (18), we write the Wigner transforms in the integrand of (117a) as

$$\begin{aligned} w^\mu (\psi ^\mu _1, \psi ^\mu _0) (\varvec{x}_1, \varvec{p}_1) =&\frac{1}{\textrm{i} \mu (2 \pi \mu )^d} \int _0^t d\tau d \varvec{y} d\varvec{\xi }_{123} \textrm{e}^{- \textrm{i} \varvec{p}_1 \cdot \varvec{y} / \mu } \\ {}&\times G^\mu \left( \varvec{x}_1 + \frac{\varvec{y}}{2} - \varvec{\xi }_1, t - \tau \right) V^\mu (\varvec{\xi }_1) G^\mu (\varvec{\xi }_1 - \varvec{\xi }_2, \tau ) \psi ^\mu (\varvec{\xi }_2, 0) \\ {}&\times G^{\mu \dag } \left( \varvec{x}_1 - \frac{\varvec{y}}{2} - \varvec{\xi }_3, t \right) \psi ^{\mu \dag } (\varvec{\xi }_3, 0) \\ w^\mu (\psi ^\mu _0, \psi ^\mu _1) (\varvec{x}_2, \varvec{p}_2) =&\frac{1}{- \textrm{i} \mu (2 \pi \mu )^d} \int _0^t d\tau d \varvec{y} d\varvec{\xi }_{123} \textrm{e}^{- \textrm{i} \varvec{p}_2 \cdot \varvec{y} / \mu } \\ {}&\times G^\mu \left( \varvec{x}_2 + \frac{\varvec{y}}{2} - \varvec{\xi }_1, t \right) \psi ^\mu (\varvec{\xi }_1, 0) \\ {}&\times G^{\mu \dag } \left( \varvec{x}_2 - \frac{\varvec{y}}{2} - \varvec{\xi }_2, t - \tau \right) V^\mu (\varvec{\xi }_2) G^{\mu \dag } (\varvec{\xi }_2 - \varvec{\xi }_3, \tau ) \psi ^{\mu \dag } (\varvec{\xi }_3, 0) . \end{aligned}$$

Taking the ensemble average of the product of these expressions, it would be straightforward to have

$$\begin{aligned} {}&{\mathbb {E}} \left[ w^\mu (\psi ^\mu _1, \psi ^\mu _1) \right] \\ {}&= \frac{1}{\mu ^2 (2 \pi \mu )^d} \int d\varvec{X} d\varvec{\eta }_{123456} \\ {}&\qquad \times \textrm{e}^{-\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_3\vert ^2)t / 2 \mu } \int _0^t d\tau _1 \textrm{e}^{\textrm{i} (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_2\vert ^2) \tau _1 / 2 \mu } \int _0^t d\tau _2 \textrm{e}^{- \textrm{i} (\vert \varvec{\eta }_3\vert ^3 - \vert \varvec{\eta }_4\vert ^2) \tau _2 / 2 \mu } \\ {}&\qquad \times \textrm{e}^{\textrm{i} (\varvec{\eta }_1 - \varvec{\eta }_3) \cdot (\varvec{x} - \varvec{X}) / \mu } \delta ( (\varvec{\eta }_2 + \varvec{\eta }_4) / 2 - \varvec{\eta }_6) \\ {}&\qquad \times \delta ((\varvec{\eta }_1 + \varvec{\eta }_3) / 2 - \varvec{p}) \delta (\varvec{\eta }_1 - \varvec{\eta }_2 - \varvec{\eta }_5) \delta (\varvec{\eta }_3 - \varvec{\eta }_4 - \varvec{\eta }_5) \\ {}&\qquad \times \Pi (\varvec{\eta }_5) n(\varvec{X}, \varvec{\eta }_6) . \end{aligned}$$

Applying (113c) and (113a) and integrating all the possible variables yield

$$\begin{aligned}&\frac{1}{(2\pi \mu )^{2d}} \int d\varvec{x}_{12} d\varvec{p}_{12} \lambda (\varvec{x}_1, \varvec{p}_1) \lambda (\varvec{x}_2, \varvec{p}_2) {\mathbb {E}}[w^\mu (\psi ^\mu _1, \psi ^\mu _0)(\varvec{x}_1, \varvec{p}_1) w^\mu (\psi ^\mu _0, \psi ^\mu _1)(\varvec{x}_2, \varvec{p}_2)] \nonumber \\&= \frac{t}{\mu (2\pi \mu )^d} \int d\varvec{x}_1 d\varvec{p}_1 d\varvec{\eta }_2 \lambda (\varvec{x}_1, \varvec{p}_1)^2 \sigma (\varvec{p}_1, \varvec{\eta }_2) n(\varvec{x}_1, \varvec{p}_1) n(\varvec{x}_1, \varvec{\eta }_2) + o \left( \frac{t}{\mu (2\pi \mu )^d} \right) . \end{aligned}$$

(118)

From the condition of complex conjugate, we also have

$$\begin{aligned}&\frac{1}{(2\pi \mu )^{2d}} \int d\varvec{x}_{12} d\varvec{p}_{12} \lambda (\varvec{x}_1, \varvec{p}_1) \lambda (\varvec{x}_2, \varvec{p}_2) {\mathbb {E}}[w^\mu (\psi ^\mu _0, \psi ^\mu _1)(\varvec{x}_1, \varvec{p}_1) w^\mu (\psi ^\mu _1, \psi ^\mu _0)(\varvec{x}_2, \varvec{p}_2)] \nonumber \\&= \frac{t}{\mu (2\pi \mu )^d} \int d\varvec{x}_1 d\varvec{p}_1 d\varvec{\eta }_2 \lambda (\varvec{x}_1, \varvec{p}_1)^2 \sigma (\varvec{p}_1, \varvec{\eta }_2) n(\varvec{x}_1, \varvec{p}_1) n(\varvec{x}_1, \varvec{\eta }_2) + o \left( \frac{t}{\mu (2\pi \mu )^d} \right) . \end{aligned}$$

(119)

For the computation of (117c), we write

$$\begin{aligned} w^\mu (\psi ^\mu _1, \psi ^\mu _0) (\varvec{x}_1, \varvec{p}_1) =&\frac{1}{i \mu (2 \pi \mu )^d} \int _0^t d\tau d \varvec{y} d\varvec{\xi }_{123} e^{- i \varvec{p}_1 \cdot \varvec{y} / \mu } \\&\quad \times G^\mu \left( \varvec{x}_1 + \frac{\varvec{y}}{2} - \varvec{\xi }_1, t - \tau \right) V^\mu (\varvec{\xi }_1) G^\mu (\varvec{\xi }_1 - \varvec{\xi }_2, \tau ) \psi ^\mu (\varvec{\xi }_2, 0) \\&\quad \times G^{\mu \dag } \left( \varvec{x}_1 - \frac{\varvec{y}}{2} - \varvec{\xi }_3, t \right) \psi ^{\mu \dag } (\varvec{\xi }_3, 0) \end{aligned}$$

Taking the ensemble average of the product of this expression, it would be straightforward to have

$$\begin{aligned}&\frac{1}{(2\pi \mu )^{2d}} \int d\varvec{x}_{12} d\varvec{p}_{12} \lambda (\varvec{x}_1, \varvec{p}_1) \lambda (\varvec{x}_2, \varvec{p}_2) {\mathbb {E}}[w^\mu (\psi ^\mu _1, \psi ^\mu _0)(\varvec{x}_1, \varvec{p}_1) w^\mu (\psi ^\mu _1, \psi ^\mu _0)(\varvec{x}_2, \varvec{p}_2)] \\&= \frac{1}{- \mu ^2 (2 \pi \mu )^{4d}} \int d\varvec{x}_{12} d\varvec{p}_{12} d\varvec{X}_{12} d\varvec{\eta }_{123456789} \lambda (\varvec{x}_1, \varvec{p}_1) \lambda (\varvec{x}_2, \varvec{p}_2) \\&\quad \times e^{-i (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_3\vert ^2)t / 2 \mu } \int _0^t d\tau _1 e^{i (\vert \varvec{\eta }_1\vert ^2 - \vert \varvec{\eta }_2\vert ^2)\tau _1 / 2 \mu } \\&\quad \times e^{-i (\vert \varvec{\eta }_4\vert ^2 - \vert \varvec{\eta }_6\vert ^2)t / 2 \mu } \int _0^t d\tau _2 e^{i (\vert \varvec{\eta }_4\vert ^2 - \vert \varvec{\eta }_5\vert ^2)\tau _2 / 2 \mu } \\&\quad \times \delta (\varvec{\eta }_1 / 2 + \varvec{\eta }_3 / 2 - \varvec{p}_1) \delta (\varvec{\eta }_4 / 2 + \varvec{\eta }_6 / 2 - \varvec{p}_2) \delta (\varvec{\eta }_1 - \varvec{\eta }_2 - \varvec{\eta }_7) \delta (\varvec{\eta }_4 - \varvec{\eta }_5 + \varvec{\eta }_7) \\&\quad \times \delta (\varvec{\eta }_2 / 2 + \varvec{\eta }_6 / 2 - \varvec{\eta }_8) \delta (\varvec{\eta }_5 / 2 + \varvec{\eta }_3 / 2 - \varvec{\eta }_9) \\&\quad \times e^{i (\varvec{\eta }_1 - \varvec{\eta }_3) \cdot \varvec{x}_1 / \mu } e^{i (\varvec{\eta }_4 - \varvec{\eta }_6) \cdot \varvec{x}_2 / \mu } e^{- i (\varvec{\eta }_2 - \varvec{\eta }_6) \cdot \varvec{X}_1 / \mu } e^{- i (\varvec{\eta }_5 - \varvec{\eta }_3) \cdot \varvec{X}_2 / \mu } \\&\quad \times \Pi (\varvec{\eta }_7) n(\varvec{X}_1, \varvec{\eta }_8) n(\varvec{X}_2, \varvec{\eta }_9) . \end{aligned}$$

Applying (113c) and (113a) and integrating all the possible variables yield

$$\begin{aligned} {}&\frac{1}{(2\pi \mu )^{2d}} \int d\varvec{x}_{12} d\varvec{p}_{12} \lambda (\varvec{x}_1, \varvec{p}_1) \lambda (\varvec{x}_2, \varvec{p}_2) {\mathbb {E}}[w^\mu (\psi ^\mu _1, \psi ^\mu _0)(\varvec{x}_1, \varvec{p}_1) w^\mu (\psi ^\mu _1, \psi ^\mu _0)(\varvec{x}_2, \varvec{p}_2)] \nonumber \\ {}&= - \frac{t}{\mu (2\pi \mu )^d} \int d\varvec{x}_1 d\varvec{p}_{12} \lambda (\varvec{x}_1, \varvec{p}_1) \lambda (\varvec{x}_1, \varvec{p}_2) \sigma (\varvec{p}_1, \varvec{p}_2) n(\varvec{x}_1, \varvec{p}_1) n(\varvec{x}_1, \varvec{p}_2) \nonumber \\ {}&\quad + o \left( \frac{t}{\mu (2\pi \mu )^d} \right) . \end{aligned}$$

(120)

Finally, from the condition of complex conjugate, we have

$$\begin{aligned} {}&\frac{1}{(2\pi \mu )^{2d}} \int d\varvec{x}_{12} d\varvec{p}_{12} \lambda (\varvec{x}_1, \varvec{p}_1) \lambda (\varvec{x}_2, \varvec{p}_2) {\mathbb {E}}[w^\mu (\psi ^\mu _0, \psi ^\mu _1)(\varvec{x}_1, \varvec{p}_1) w^\mu (\psi ^\mu _0, \psi ^\mu _1)(\varvec{x}_2, \varvec{p}_2)] \nonumber \\ {}&= - \frac{t}{\mu (2\pi \mu )^d} \int d\varvec{x}_1 d\varvec{p}_{12} \lambda (\varvec{x}_1, \varvec{p}_1) \lambda (\varvec{x}_1, \varvec{p}_2) \sigma (\varvec{p}_1, \varvec{p}_2) n(\varvec{x}_1, \varvec{p}_1) n(\varvec{x}_1, \varvec{p}_2) \nonumber \\ {}&\quad + o \left( \frac{t}{\mu (2\pi \mu )^d} \right) . \end{aligned}$$

(121)

D Derivation of the Hamiltonian in the Diffusive Limit

In this appendix, we show how to obtain the large deviation Hamiltonian (58) from  (42). Since we are dealing with the scattering term that is local in position space, one can omit the dependence on $\varvec{x}$. As explained in the main text, the important point is to show how the diffusion kernel $\Sigma [n]$ transforms when only infinitesimal deviation from incoming wave vector is allowed by the cross section. For this specific purpose, we introduce a parameter $\nu$ and write the typical correlation length of V as $\mu / \nu$. Our strategy is to expand the Hamiltonian in terms of $\nu$.

Because the potential spectrum $\Pi (\varvec{p})$ is supposed to have a finite support with the extent of ${\mathcal {O}}(\nu )$, it is convenient to rewrite the cross section as $\sigma (\varvec{p}_1,\varvec{p}_2) = \nu ^{-d}\tilde{\sigma }((\varvec{p}_1-\varvec{p}_2) / \nu ; \varvec{p}_1)$. For any test functions, f and g, the diffusive kernel (39) satisfies

$$\begin{aligned}&\int d \varvec{p}_1 d \varvec{p}_2 f(\varvec{p}_1)\Sigma (\varvec{p}_1,\varvec{p}_2)g(\varvec{p}_2) \nonumber \\&\quad = c \int d \varvec{p}d \varvec{q}f(\varvec{p})n(\varvec{p})\tilde{\sigma }(\varvec{q};\varvec{p})n(\varvec{p}- \nu \varvec{q})\left[ g(\varvec{p}) - g(\varvec{p}- \nu \varvec{q})\right] . \end{aligned}$$

(122)

Taylor expanding the integrand with respect to $\nu$, one obtains

$$\begin{aligned}&\int d \varvec{p}_1 d \varvec{p}_2 f(\varvec{p}_1)\Sigma (\varvec{p}_1, \varvec{p}_2)g(\varvec{p}_2) \nonumber \\&\quad = c \int d \varvec{p}f(\varvec{p})n(\varvec{p}) \Biggl \{\nu n(\varvec{p}) \sum _{n=1}^{d} \left( \int dq_n \tilde{\sigma }(\varvec{q}; \varvec{p}) q_n\right) \partial _{p_n}g(\varvec{p}) \nonumber \\&\quad - \nu ^2 \sum _{n=1}^{d}\sum _{m=1}^{d} \left( \int dq_n dq_m \tilde{\sigma }(\varvec{q}; \varvec{p}) q_n q_m \right) \partial _{p_n}n(\varvec{p}) \partial _{p_{m}}g(\varvec{p}) \nonumber \\&\quad - \frac{\nu ^2}{2} n(\varvec{p}) \sum _{n=1}^{d}\sum _{m=1}^{d} \left( \int dq_n dq_m \tilde{\sigma }(\varvec{q}; \varvec{p}) q_n q_m \right) \partial _{p_n}\partial _{p_m} g(\varvec{p}) + {\mathcal {O}} \left( \nu ^3 \right) \Biggr \} . \end{aligned}$$

(123)

To evaluate the integrations with respect to $\varvec{q}$, we shall exponentiate the Dirac-$\delta$ in the definition of the cross section (23) and use the inverse Fourier transform of $\Pi$ in (57). The resulting expression is

$$\begin{aligned}&\tilde{\sigma }(\varvec{q}; \varvec{p}) = \frac{1}{\nu (2\pi )^{d}} \int d\varvec{y}\int _{{\mathbb {R}}} ds \textrm{e}^{- i\varvec{q}\cdot \varvec{y}} R^\nu \left( s\varvec{p}+ \varvec{y}-\frac{\nu }{2}s\varvec{q}\right) \nonumber \\&\text {and} \quad R^\nu (\varvec{y}) = \int d \varvec{p}\Pi (\varvec{p}) \textrm{e}^{i\varvec{p}\cdot \varvec{y}/ \nu } , \end{aligned}$$

(124)

where $R^\nu (\varvec{y}) = R(\varvec{y} / \nu )$ (see Eq. (57)) is a scaled correlation function of the random potential with the correlation length of ${\mathcal {O}}(1)$. Again Taylor expanding Eq. (124) with respect to $\nu$, one gets

$$\begin{aligned} c \nu \int d \varvec{q}\tilde{\sigma }(\varvec{q}; \varvec{p}) q_n&= - \nu \sum _{m=1}^{d} \partial _{p_m} D^\nu _{nm}(\varvec{p}) + {\mathcal {O}}(\nu ^2) \\ c \nu ^2 \int d \varvec{q}\tilde{\sigma }(\varvec{q}; \varvec{p}) q_n q_m&= 2 \nu D^\nu _{nm}(\varvec{p}) + {\mathcal {O}}(\nu ^2) \end{aligned}$$

with $\varvec{D}^\nu (\varvec{p}) = - (c/2) \int _{{\mathbb {R}}} \nabla \otimes \nabla R^\nu (s \varvec{p}) ds$. Here, $\int _{{\mathbb {R}}} \nabla R^\nu (s \varvec{p}) ds$ vanishes because of the point symmetry in $R^\nu$. Inserting these expressions into (123), one obtains

$$\begin{aligned}&\int d \varvec{p}_1 d \varvec{p}_2 f(\varvec{p}_1)\Sigma (\varvec{p}_1,\varvec{p}_2)g(\varvec{p}_2) \nonumber \\&\quad = - c \nu \int d \varvec{p}fn \sum _{n=1}^d \sum _{m=1}^d \left\{ n\partial _{p_{n}} D^\nu _{nm} \partial _{p_{m}}g + 2D^\nu _{nm} \partial _{p_{n}} n \partial _{p_{m}} g + n D^\nu _{nm} \partial _{p_{n}} \partial _{p_{m}} g \right\} \nonumber \\&\quad \quad + {\mathcal {O}}(\nu ^2) . \end{aligned}$$

(125)

Evidently from this result, the dominant term is of order ${\mathcal {O}}(\nu )$ in the present scaling. Therefore, in order for diffusion in wave-vector space to be comparable to the free propagation in position space, one needs to rescale the position coordinate. However, for the sake of simplicity here, we shall set $\nu = 1$ while ignoring ${\mathcal {O}}(\nu ^2)$ terms. Consequently, after reorganizing derivatives of products and performing integration by parts, we confirm that (125) transforms into  (56).