#  Superdiffusion transition for a noisy harmonic chain subject to a magnetic
field

Gaëtan Cane  Université Côte d’Azur, CNRS, LJAD
Parc Valrose
06108 NICE Cedex 02, France<EMAIL_ADDRESS>
###### Abstract.

We consider an infinite harmonic chain of charged particles submitted to the
action of a magnetic field of intensity $B$ and subject to the action of a
stochastic noise conserving the energy. In [21, 22] it has been proved that if
$B=0$ the transport of energy is described by a $3/4$-fractional diffusion
while it has been proved in [30] that if $B\neq 0$ it is described by a
$5/6$-fractional diffusion. In this paper we quantify the intensity of the
magnetic field necessary to pass from one regime to the other one. We also
describe the transition mechanism to cross the two different phases.

###### Contents

  1. 1 Introduction
  2. 2 Notations
  3. 3 Microscopic dynamics
  4. 4 Review of previous results
  5. 5 Statement of the results
  6. 6 Proof of Theorem 5.1, Theorem 5.7, Theorem 5.9 and Theorem 5.11
  7. A Proof of the evolution equation of the Wigner’s distribution
  8. B Proof of Proposition 5.4 and Proposition 5.5
  9. C Proof of Proposition 5.6 and Proposition 6.2
  10. D Proof of Lemma 6.3

##  1\. Introduction

Since the seminal numerical experiments of Fermi-Pasta-Ulam-Tsingou in 1953
[15], the understanding of energy transport in very long anharmonic chains of
coupled oscillators attracted a lot of attention but still remains a
fascinating challenging open problem in mathematical physics. During the two
last decades many researchers have been interested in particular to the one
dimensional case for which the energy transport is anomalous (we refer the
reader to the physical reviews [16, 26]). The most elaborated theory to
describe the form of this anomalous transport is probably the recent nonlinear
fluctuating hydrodynamics theory initiated by Spohn [32, 33] which predicts,
for interacting particle systems with several conserved quantities (like
energy, momentum etc.) and local interactions, several different universality
classes containing not only the famous KPZ-fixed point universality class [28]
but also many fractional diffusion classes, apart from the standard Edwards-
Wilkinson class (i.e. normal diffusion class). The theory is macroscopic and
based on formal arguments starting from the hydrodynamic equations associated
to the interacting particle system under investigation. Unfortunately its
validity for systems with more than one conservation law has been proved
rigorously only for few stochastic models [5, 9, 22]. It has also to be
noticed that while the link between the KPZ-fixed point class and the Edwards-
Wilkinson class is provided by the now well understood KPZ equation [19, 20],
almost nothing is known about the potential equations connecting the other
universality classes, even at some heuristic level.

The system we are interested in belongs to the class of systems introduced in
[18], revisited with the heat conduction problem perspective in [2, 3, 10],
and studied since in several subsequent works by various authors (see e.g. [1]
and [25] for some reviews). In this paper we consider the model of [29], i.e.
a harmonic chain of charged particles submitted to a magnetic field and a
stochastic exchange of momentum between neighbor sites. Each particle is
labeled by its rest position $x$ in $\mathbb{Z}$ and lives in the two
dimensional space $\mathbb{R}^{2}$. Its displacement from its rest position is
denoted by $q(x)\in\mathbb{R}^{2}$, its momentum by $p(x)\in\mathbb{R}^{2}$
and its energy by $e(x)$. The total energy $E$, which is conserved by the
dynamics, is thus

$E=\sum_{x\in\mathbb{Z}}e(x)=\frac{1}{2}\left(\sum_{x\in\mathbb{Z}}|p(x)|^{2}+\sum_{x\in\mathbb{Z}}|q(x)-q(x+1)|^{2}\right).$

The magnetic field of intensity $B$ is constant and orthogonal to the plan of
motion of the chain. The dynamics is given by the Hamiltonian dynamics
described above on which is superposed a stochastic noise which exchanges
continuously the velocities of nearest neighbor particles. The noise conserves
the total energy of the chain and the total pseudo-momentum111We recall that
the pseudo-momentum of particle $x$ is $p(x)+B\sigma q(x)$ where
$\sigma=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)$. If $B=0$ the
pseudo-momentum coincides with the momentum. If $B\neq 0$, the momentum is not
conserved by the Hamiltonian dynamics.. The goal of this paper is to
understand the mechanism to cross different universality classes by varying
the intensity of the magnetic field.

### Case without magnetic field, i.e. $B=0$.

In order to study the macroscopic evolution of the energy the authors of [4],
following [31], introduced the Wigner’s distribution
${\mathcal{W}}^{\varepsilon}$ (defined in Sec. 3.4) in the time scale
$t\varepsilon^{-1}$, and in the weak noise limit, i.e. the intensity of the
noise is of order $\varepsilon$. Here $\varepsilon$ is a space scaling
parameter, i.e. the lattice $\mathbb{Z}$ is rescaled in
$\varepsilon\mathbb{Z}$, and going to zero. The Wigner’s distribution is a
kind of localized Fourier transform of the space correlations of the
eigenmodes, also called phonons, of the purely deterministic harmonic chain
(the presence of the stochastic noise couple their time evolutions which
become then non trivial). Each eigenmode is labeled by a $k\in{\mathbb{T}}$,
the continuous torus of length one. For each time $t$,
${\mathcal{W}}^{\varepsilon}(t)$ is a distribution acting on a class of test
functions $J:(u,k)\in(\mathbb{R}\times\mathbb{T})\to J(u,k)\in\mathbb{C}$. To
get some intuition on the relevance of the Wigner’s distribution for the study
of the macroscopic behavior of the energy, we observe that if
$J:(u,k)\in(\mathbb{R}\times\mathbb{T})\to{\tilde{J}}(u)\in\mathbb{R}$ is a
test function not depending on the $k$ variable, we have

$\displaystyle\left\langle{\mathcal{W}}^{\varepsilon}(t),J\right\rangle=\sum_{x\in\mathbb{Z}}\mathbb{E}_{\mu^{\varepsilon}}\left[e\left(t\varepsilon^{-1},x\right)\right]{\tilde{J}}(\varepsilon
x)+\mathcal{O}_{J}(\varepsilon)\ ,$

where $e(t,x)$ is the value of the energy of the particle $x$ at time $t$ and
$\mu^{\varepsilon}$ the initial distribution of the dynamics. From the
previous equation it is clear that if we want to understand the behaviour of
the macroscopic behavior of the energy when $\varepsilon$ goes to zero we have
to understand the behavior of ${\mathcal{W}}^{\varepsilon}$ as $\varepsilon\to
0$.

In [4] it is proved that at kinetic time scale $\varepsilon^{-1}$, the
Wigner’s distribution ${\mathcal{W}}^{\varepsilon}$ converges to the unique
solution $f_{0}:=(f_{0}(1),f_{0}(2))$, of the following phonon linear
Boltzmann’s equation

(1)
$\partial_{t}f_{0}(t,u,k,i)+\frac{\textbf{v}_{0}(k)}{2\pi}\partial_{u}f_{0}(t,u,k,i)=\mathcal{L}_{0}\left[f_{0}\right](t,u,k,i)\
,$

with

(2)
$\mathcal{L}_{0}\left[f_{0}\right](t,u,k,i)=\sum_{j=1}^{2}\int_{\mathbb{T}}R_{0}(k,k^{\prime},i,j)\left[f\left(t,u,k^{\prime},j\right)-f\left(t,u,k,i\right)\right]dk^{\prime}\
.$

Here, $u\in{\mathbb{R}}$ represents the position along the chain after the
kinetic limit, $t\geq 0$ the time and $k\in\mathbb{T}$ the wave number of a
phonon whereas $i$ is the type of phonon. $\mathcal{L}_{0}$ is a collisional
operator completely determined by the noise introduced on the system and
$\textbf{v}_{0}$ is the dispersion relation.

Since $R_{0}(k,k^{\prime},i,j)$ is positive we can interpret the solution of
Eq. (1) as the evolution of the density of a continuous time Markov process
$\left(Z^{0}(\cdot),K^{0}(\cdot),I^{0}(\cdot)\right)$. Here
$\left(K^{0}(\cdot),I^{0}(\cdot)\right)\in{\mathbb{T}}\times\\{1,2\\}$ is the
pure jump Markovian process with generator given by Eq. (2) and $Z^{0}(\cdot)$
is the additive functional defined for any positive time $t$ by

(3) $\displaystyle
Z^{0}(t)=-\int_{0}^{t}\frac{\textbf{v}_{0}\left(K^{0}(s)\right)}{2\pi}ds\ .$

Using this interpretation, the authors of [21] proved first that the finite-
dimensional distributions of $N^{-1}Z^{0}(N^{\frac{3}{2}}\cdot)$ converge to
the finite-dimensional distributions of a Lévy process generated (up to a
constant) by $-(-\Delta)^{\tfrac{3}{4}}$. In a second time, they used the
previous result to show that $f_{0}\left(N^{\frac{3}{2}}t,Nu,\cdot\right)$
converges in $\mathbb{L}^{2}\left(\mathbb{T}\right)$ (see Theorem 4.6) to the
unique solution $\rho_{0}$, of the following fractional diffusion equation

(4) $\forall u\in{\mathbb{R}},\quad\forall
t\in[0,T],\quad\partial_{t}\rho_{0}(t,u)=-D\left(-\Delta\right)^{\tfrac{3}{4}}\left[\rho_{0}\right](t,u),$

where $D$ is a strictly positive constant.

Hence, this two-step argument which consists to first prove the convergence of
the Wigner’s distribution to the unique solution $f_{0}$ of a phonon
Boltzmann’s equation and then the convergence of $f_{0}$ to the solution
$\rho_{0}$ of a fractional Laplacian equation gives the nature of the
superdiffusion of energy proved in [3]. Fractional diffusion equations can
also be derived from some Boltzmann’s equations by purely analysis arguments,
see for example [12, 13, 27] and references therein for more details . In
2015, by investigating further the time evolution of the Wigner’s distribution
in a longer time scale, the authors of [22], proved that we can obtain in one
step Eq. (4) from the microscopic model in a suitable time scale. The results
above are consistent with the predictions of the nonlinear fluctuating
hydrodynamics theory of Spohn.

### Case with a magnetic field $B\neq 0$

By following the strategy initiated in [4, 21], the authors of [30] proved
that at kinetic time scale $\varepsilon^{-1}$ the Wigner’s distribution
converges to the unique solution $f_{B}:=(f_{B}(1),f_{B}(2))$, of a phonon
linear Boltzmann’s equation of the same form as in Eq. (1)

(5)
$\partial_{t}f_{B}(t,u,k,i)+\frac{\textbf{v}_{B}(k)}{2\pi}\partial_{u}f_{B}(t,u,k,i)=\mathcal{L}_{B}\left[f_{B}\right](t,u,k,i)\
,$

with

$\displaystyle\mathcal{L}_{B}\left[f_{B}\right](t,u,k,i)=\sum_{j=1}^{2}\int_{\mathbb{T}}R_{B}(k,k^{\prime},i,j)\left[f_{B}\left(t,u,k^{\prime},j\right)-f_{B}(t,u,k,i)\right]dk^{\prime}\
.$

However, because of the presence of the magnetic field, as it is explained in
the introduction of [30], the relation dispersion $\textbf{v}_{B}$ and the
scattering kernel
$R_{B}(k,i)=\int_{\mathbb{T}}R_{B}\left(k,k^{\prime},i,j\right)dk^{\prime}\delta_{j}(i)$
have different behavior when $k$ goes to zero. Indeed when $B$ is positive we
have for $k$ near to zero

$\displaystyle\textbf{v}_{B}(k)\sim k,\quad R_{B}(k,1)\sim
k^{2}\quad\text{and}\quad R_{B}(k,2)\sim k^{4}\ ,$

whereas in the Boltzmann’s equation (1) obtained in [4]
$R_{0}(k,k^{\prime},i,j)$ does not depend in fact on $i$ and $j$ and satisfies
for $k$ near to zero

$\displaystyle\textbf{v}_{0}(k)\sim 1\quad\text{and}\quad R_{0}(k)\sim
k^{2}\quad\text{where}\quad
R_{0}(k)=\int_{\mathbb{T}}R_{0}(k,k^{\prime},i,j)dk^{\prime}\ .$

This difference has a drastic effect on the energy transport properties of the
chain moving the chain from one universality class to an other one. Indeed,
the energy superdiffusion is still described by a fractional diffusion
equation as in Eq. (4) but with a different exponent. Following [21] it is
proved in [30] that the finite-dimensional distributions of222Here $Z^{B}$ is
defined as $Z^{0}$ in Eq. (3) with $\mathbf{v}_{0}$ replaced by
$\mathbf{v}_{B}$. $N^{-1}Z^{B}\left(N^{\frac{5}{3}}\cdot\right)$ converge
weakly to the finite-dimensional distributions of a Lévy process generated (up
to a constant) by $-(-\Delta)^{\tfrac{5}{6}}$ which implies the convergence of
$f_{B}\left(N^{\frac{5}{3}}t,Nu,\cdot\right)$ in
$\mathbb{L}^{2}\left(\mathbb{T}\right)$ to the unique solution $\rho_{B}$ of
the following fractional diffusion equation

(6) $\forall u\in{\mathbb{R}},\quad\forall
t\in[0,T],\quad\partial_{t}\rho_{B}(t,u)=-D_{B}\left(-\Delta\right)^{\tfrac{5}{6}}\left[\rho_{B}\right](t,u)\
,$

where $\tilde{D}$ is a strictly positive constant. Notice that since the
hydrodynamic limits of this chain are trivial in the Euler time scale, the
nonlinear fluctuating hydrodynamics theory of Spohn does not give any
prediction for this model.

### Contribution

The aim of this paper is to study the transition between the study of [21] and
the one of [30]. As we have seen, the presence of the magnetic field moves the
model from the $3/4$-fractional universality class to the $5/6$-fractional
universality class and it makes sense to ask if we can quantify the intensity
of the magnetic field necessary to cross from one universality class to some
other one, and to understand what is the mechanism occurring at the
transition. As far as we know, while very interesting, these questions did not
receive answer, even at some heuristic level. We notice however that in [6]
and [7, 8] are obtained results describing two transition mechanisms between
standard diffusion universality class and fractional diffusion universality
class in a hamiltonian system stochastically perturbed and having two
conserved quantities. We would like to mention that a family of fractional
diffusion equations have been derived from stochastic harmonic chains with
long-range interaction by Suda in [34].

In a first time we introduce a small magnetic field of intensity
$B\varepsilon^{\delta}$ with $\delta>0$ and $B\neq 0$. We prove first (see
Theorem 5.1) that at the kinetic time scale of order $\varepsilon^{-1}$ the
transition is trivial in the sense that for $\delta=0$ the Wigner’s
distribution converges to the Boltzmann’s equation of [30] and for $\delta>0$
the Wigner’s distribution converges to the one of [4]. We believe however that
the effect of the small magnetic field could be seen in a longer time scale.

Therefore, in a second time, we study the hydrodynamic limit of the solution
$f_{B_{N}}:=(f_{B_{N}}(1),f_{B_{N}}(2))$ of the Boltzmann’s equation (5) when
$B$ is changed to $B_{N}:=BN^{-\delta}$ with $\delta$ in ${\mathbb{R}}^{+}$
and $B\neq 0$. Letting $\alpha_{\delta}=\frac{3}{2}$ if
$\delta\leq\frac{1}{2}$ and $\alpha_{\delta}:=\frac{5-\delta}{3}$ for
$\delta>\frac{1}{2}$, we prove in Theorem 5.7 that the finite-dimensional
distributions of $N^{-1}Z^{B_{N}}\left(N^{\alpha_{\delta}}\cdot\right)$
converge weakly to the finite-dimensional distributions of a Lévy process
generated (up to a constant) by an operator $\mathcal{L}_{\delta}$ whose
action on smooth functions $\phi$ which decay sufficiently fast is defined by

(7) $\forall u\in{\mathbb{R}},\quad\mathfrak{L}_{\delta}\left[\phi\right]\
(u)=\left\\{\begin{array}[]{l}-(-\Delta)^{\frac{3}{4}}\left[\phi\right]\
(u)\quad\text{if}\quad\delta>\frac{1}{2}\ ,\\\ \\\
\mathfrak{L}_{B}\left[\phi\right]\ (u)\quad\text{if}\quad\delta=\frac{1}{2}\
,\\\ \\\ -(-\Delta)^{\frac{5}{6}}\left[\phi\right]\
(u)\quad\text{if}\quad\delta<\frac{1}{2}\ ,\end{array}\right.$

with $\mathfrak{L}_{B}$ defined in Eq. (67). From Theorem 5.7, we prove then
in Theorem 5.9 that $f_{B_{N}}\left(N^{\alpha_{\delta}}t,Nu,\cdot\right)$
converges to the unique solution $\rho_{\delta}$ of the following partial
differential equation

(8)
$\partial_{t}\rho_{\delta}(t,u)=\mathfrak{L}_{\delta}\left[\rho_{\delta}\right](t,u)\
.$

Finally, we prove in Theorem 5.11 that, up to a constant,
$\mathfrak{L}_{B}\rightarrow-(-\Delta)^{\frac{3}{4}}$ when $B\rightarrow 0$
and $\mathfrak{L}_{B}\rightarrow-(-\Delta)^{\frac{5}{6}}$ when
$B\rightarrow\infty$. Hence, $\mathfrak{L}_{B}$ is the infinitesimal generator
of a Lévy process which interpolates between the two fractional universality
classes.

These results are summarized in Fig. 1. On the horizontal axis, $\delta$
represents the intensity of the magnetic field and on the vertical axis
$\alpha_{\delta}$ represents the scaling in space to obtain the hydrodynamic
limit of $f_{B_{N}}$.

### Structure of the paper

In Sec. 2, we precise the notations of the paper. In Sec. 3, we present the
microscopic dynamics and the Wigner’s distribution. In Sec. 4, we recall the
historical results obtained in [4, 21, 30]. In Sec. 5, we state the main
results of this paper which are proved in Sec. 6. In order to make the reading
easier, intermediate results are shown in the Appendices.

Figure 1. Transition graph.

##  2\. Notations

Let $a$ and $b$ be two positive real numbers, we will write $a\lesssim b$ when
there exists a positive constant $C$ such that $a\leq Cb$. The conjugate of a
complex number $z$ will be denoted by $z^{*}$ and $\mathbf{i}$ will denote the
complex number of modulus 1. In order to lighten the notations we will denote
$\mathbb{R}$ \$\\{0\\}$ by $\mathbb{R}^{*}$, the one dimensional torus
$\left[-\tfrac{1}{2},\tfrac{1}{2}\right[$ by $\mathbb{T}$ and the euclidean
norm on $\mathbb{R}^{n}$ by $|\cdot|$.

If $X$ is a topological space we denote the Borelian $\sigma$-field of $X$ by
${\mathcal{B}}(X)$. We denote by ${\mathcal{F}}\left([0,T],X\right)$ the set
of $X$-valued functions on $[0,T]$, by ${\mathcal{C}}\left([0,T],X\right)$ the
subspace of $X$-valued continuous functions on $[0,T]$ and by
${\mathcal{C}}_{\textbf{b}}\left([0,T],X\right)$ the subspace of $X$-valued
bounded continuous functions on $[0,T]$. Let $n$ in
${\mathbb{N}}\cup\\{\infty\\}$, the space of $\mathbb{R}$-valued functions on
$X$ with compact support and $n$ times differentiable is denoted by
${\mathcal{C}}_{\textbf{c}}^{n}\left(X\right)$. The space of $\mathbb{R}$\-
valued càdlàg functions on $[0,T]$ will be denoted by
$\mathcal{D}\left([0,T],{\mathbb{R}}\right).$

For $f$ in $\ell^{1}({\mathbb{Z}})$, we define its (discrete) Fourier’s
transform $\widehat{f}:\mathbb{T}\rightarrow{\mathbb{C}}$ by

$\displaystyle\forall
k\in\mathbb{T},\quad\hat{f}(k)=\sum_{x\in\mathbb{Z}}f(x)\exp(2{\mathbf{i}}k\pi
x)\ .$

As usual we extend this notation for all functions in
$\ell^{2}({\mathbb{Z}})$. In order to study the Wigner’s distribution, defined
in Sec. 3.4, we introduce the set of test functions ${\mathcal{S}}$ given by

${\mathcal{S}}:=\left\\{H\in
C_{c}^{\infty}({\mathbb{R}}\times\mathbb{T})\;\bigg{|}\;\;\forall(n,m,l)\in\mathbb{N}^{3},\;\sup_{k\in\mathbb{T}}\sup_{u\in\mathbb{R}}\left|(1+u)^{l}\partial_{k}^{n}\partial_{u}^{m}H(u,k)\right|<\infty\right\\}\
.$

For any $H$ in ${\mathcal{S}}$, we denote its (continuous) Fourier’s transform
in the first variable by
${\mathcal{F}}\left[H\right]:{\mathbb{R}}\times\mathbb{T}\rightarrow{\mathbb{C}}$
where

$\forall(p,k)\in{\mathbb{R}}\times\mathbb{T},\quad\mathcal{F}\left[H\right](p,k):=\int_{\mathbb{R}}H(u,k)\exp(2{\mathbf{i}}\pi
pu)du\ .$

The set ${\mathcal{S}}$ is stable under the action of ${\mathcal{F}}$.
In the whole paper, for $H$ in ${\mathcal{S}}$, we denote the Laplacian of $H$
in the first variable by $\Delta\left[H\right]$ and for $\alpha$ in $(1,2)$
the fractional Laplacian of $H$ in the first variable by
$-(-\Delta)^{\frac{\alpha}{2}}\left[H\right]$ where we recall that for any $p$
in ${\mathbb{R}}$ and $k$ in $\mathbb{T}$

$\displaystyle\mathcal{F}[\Delta\left[H\right]](p,k)$ $\displaystyle=-(2\pi
p)^{2}\mathcal{F}\left[H\right](p,k)\ .$
$\displaystyle\mathcal{F}\left[-(-\Delta)^{\frac{\alpha}{2}}\left[H\right]\right](p,k)$
$\displaystyle=-|2\pi p|^{\alpha}\mathcal{F}\left[H\right](p,k)\ .$

The space ${\mathcal{S}}\times{\mathcal{S}}$ is equipped with the norm
$\|\cdot\|$ defined by

$\displaystyle\forall
J:=(J_{1},J_{2})\in{\mathcal{S}}\times{\mathcal{S}},\quad\|J\|=\sum_{i=1}^{2}\int_{\mathbb{R}}\sup_{k}|\mathcal{F}\left[J_{i}\right](p,k)|dp\
.$

The space $({\mathcal{S}}\times{\mathcal{S}},\|\cdot\|)$ is then a separable
space. We denote by $({\mathcal{S}}\times{\mathcal{S}})^{\prime}$ the dual of
${\mathcal{S}}\times{\mathcal{S}}$ for the weak-* topology (we refer the
reader to Sec. 3.4 for a precise definition). For ${\mathcal{W}}$ in
$({\mathcal{S}}\times{\mathcal{S}})^{\prime}$ and $J$ in
${\mathcal{S}}\times{\mathcal{S}}$ we denote by
$\left\langle{\mathcal{W}},J\right\rangle$ the duality bracket between
${\mathcal{W}}$ and $J$.

Throughout the article, the random variables are defined on an underlying
probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ and $T$ will
denote a fixed positive time.

##  3\. Microscopic dynamics

In this section, we define the microscopic dynamics studied in this paper.
This dynamic was first introduced in [2, 3, 4] without a magnetic field and
later in [29, 30] with a magnetic field.

###  3.1. Deterministic dynamics

We consider a one dimensional chain of coupled harmonic oscillators each
having two transverse degrees of freedom and subject to the action of a
magnetic field perpendicular to the plane of motion. At rest, the atoms are
aligned according to the lattice $\mathbb{Z}$ and each $x$ in $\mathbb{Z}$
represents the balance position of one atom. We denote the velocity of the
atom with rest position $x$ in $\mathbb{Z}$ by $p(x)=(p_{1}(x),p_{2}(x))$ in
$\mathbb{R}^{2}$ and the displacement from its rest position by
$q(x)=(q_{1}(x),q_{2}(x))$ in $\mathbb{R}^{2}$ . We denote the strength of the
magnetic field by $B$ in $\mathbb{R}$.
The deterministic dynamics is defined at any positive time $t$ and for any $i$
in $\\{1,2\\}$ by

(9) $\begin{array}[]{l}\frac{d}{dt}q_{i}(t,x)=p_{i}(t,x)\ ,\\\ \\\
\frac{d}{dt}p_{i}(t,x)=\Delta_{d}q_{i}(t,x)+\delta_{i,1}Bp_{2}(t,x)-\delta_{i,2}Bp_{1}(t,x)\
,\end{array}$

where $\Delta_{d}$ denotes the discrete Laplacian on $\mathbb{Z}$. We denote a
typical configuration of the system by
$(\mathbf{q},\mathbf{p}):=(q(x),p(x))_{x\in\mathbb{Z}}$ and the configuration
over time by $\\{(\mathbf{q}(t),\mathbf{p}(t);t\geq
0)\\}:=\\{(q(x,t),p(x,t))_{x\in\mathbb{Z}}\;|\;t\geq 0\\}$. The initial
configuration $(q(x,0),p(x,0))_{x\in\mathbb{Z}}$ is denoted by
$(\mathbf{q}^{0},\mathbf{p}^{0})$.
Let $\alpha$ be the function defined on $\mathbb{Z}$ by

$\displaystyle\forall
z\in\mathbb{Z},\quad\alpha(z)=\left\\{\begin{array}[]{l}2\quad\text{if}\quad
z=0\ ,\\\ -1\quad\text{if}\quad|z|=1\ ,\\\ 0\quad\text{otherwise}\
.\end{array}\right.$

Observe that

(10) $\widehat{\alpha}(k)=4\sin^{2}(\pi k)\ ,$

with $\hat{\alpha}(0)=\hat{\alpha}^{\prime}(0)=0$ and
$\hat{\alpha}^{\prime\prime}(0)=8\pi^{2}$.
The formal infinitesimal generator of this Markovian dynamics is given by
$A+BG$ where for every smooth and local complex-valued functions333Here smooth
and local means that for any$i$ in $\\{1,2\\}$, $\phi$ depends only on a
finite number of the sequence $(q_{i}(x),p_{i}(x))_{x\in{\mathbb{Z}}}$ and is
smooth with respect to these coordinates.  $\phi$ we have

$\displaystyle A\left[\phi\right]$
$\displaystyle=\frac{1}{2}\left(\sum_{x\in\mathbb{Z}}\sum_{i=1}^{2}p_{i}(x)\partial_{q_{i}(x)}\left[\phi\right]+\sum_{x,x^{\prime}\in\mathbb{Z}}\sum_{i=1}^{2}\alpha(x-x^{\prime})q_{i}(x^{\prime})\partial_{p_{i}(x)}\left[\phi\right]\right)\
\ .$ $\displaystyle G\left[\phi\right]$
$\displaystyle=\sum_{x\in\mathbb{Z}}\left(p_{2}(x)\partial_{p_{1}(x)}\left[\phi\right]-p_{1}(x)\partial_{p_{2}(x)}\left[\phi\right]\right)\
.$

We denote by $E(\mathbf{q},\mathbf{p})$ the total energy of the configuration
$(\mathbf{q},\mathbf{p})$ where

(11) $\displaystyle E(\mathbf{q},\mathbf{p})$
$\displaystyle=\frac{1}{2}\left(\sum_{x\in\mathbb{Z}}|p(x)|^{2}+\sum_{x\in\mathbb{Z}}|q(x)-q(x+1)|^{2}\right)\
.$

In the whole paper, we will study only configurations
$(\mathbf{q},\mathbf{p})$ with finite total energy. Observe that the energy is
conserved during the time evolution, i.e.

(12) $\displaystyle\forall
t\in[0,T],\quad\frac{d}{dt}E(\mathbf{q}(t),\mathbf{p}(t))$ $\displaystyle=0\
.$

###  3.2. Eigenvalues and eigenvectors of the deterministic dynamics

We define on $\mathbb{T}$ the following real valued functions

(13)
$\omega_{1,B}(k)=\sqrt{\widehat{\alpha}(k)+\frac{|B|^{2}}{4}}+\frac{|B|}{2}\;\;\text{and}\;\;\omega_{2,B}(k)=\sqrt{\widehat{\alpha}(k)+\frac{|B|^{2}}{4}}-\frac{|B|}{2}\
,$ (14) $\forall
i\in\\{1,2\\},\quad\theta_{i,B}(k)=\sqrt{\frac{\omega_{i,B}(k)}{\omega_{1,B}(k)+\omega_{2,B}(k)}}\
,$ (15)
$\textbf{v}_{B}(k):=\frac{d\omega_{1,B}}{dk}(k)=\frac{d\omega_{2,B}}{dk}(k)=\frac{\hat{\alpha}^{\prime}(k)}{2\sqrt{\hat{\alpha}(k)+\frac{|B|^{2}}{4}}}\
,$

where $k$ is in $\mathbb{T}$. Observe that for any $i$ in $\\{1,2\\}$,
$\theta_{i,B}$ is a bounded function.

For every configuration $(\mathbf{q},\mathbf{p})$ we define a couple
$(\widehat{\psi}_{1,B},\widehat{\psi}_{2,B}):\mathbb{T}\rightarrow\mathbb{C}^{2}$
by

(16)
$\widehat{\psi}_{1,B}(\mathbf{q},\mathbf{p})[k]:=\theta_{1,B}(k)\big{[}\widehat{p}_{1}(k)-\mathbf{i}\omega_{2,B}(k)\widehat{q}_{1}(k)+\mathbf{i}\widehat{p}_{2}(k)+\omega_{2,B}(k)\widehat{q}_{2}(k)\big{]}\
.$

(17)
$\widehat{\psi}_{2,B}(\mathbf{q},\mathbf{p})[k]:=\theta_{2,B}(k)\big{[}\widehat{p}_{1}(k)-\mathbf{i}\omega_{1,B}(k)\widehat{q}_{1}(k)-\mathbf{i}\widehat{p}_{2}(k)-\omega_{1,B}(k)\widehat{q}_{2}(k)\big{]}\
.$

######  Lemma 3.1.

We have that for each $k$ in $\mathbb{T}$ and each $i$ in $\\{1,2\\}$,
$\widehat{\psi}_{i,B}(k)$ is an eigenvector of $(A+BG)$ corresponding to the
eigenvalue $-\mathbf{i}\;\omega_{i,B}(k)$, i.e.

$\displaystyle\forall
i\in\\{1,2\\},\quad(A+BG)\left[\widehat{\psi}_{i,B}\right](k)=-\mathbf{i}\;\omega_{i,B}(k)\widehat{\psi}_{i,B}(k)\
.$

###  3.3. Stochastic dynamics

We introduce now the local stochastic perturbation which is defined through
its formal infinitesimal generator $S$ whose action on any smooth and local
complex valued functions $\phi$ is given by

$S\left[\phi\right]=\frac{1}{4}\sum_{x\in\mathbb{Z}}\sum\limits_{\underset{i\neq
j}{i,j=1}}^{2}\left(Y_{x,x+1}^{i,j}\right)^{2}\left[\phi\right]\ ,$

where

$\displaystyle Y_{x,x+1}^{i,j}$ $\displaystyle=$
$\displaystyle(p_{j}(x+1)-p_{j}(x))(\partial_{p_{i}(x+1)}-\partial_{p_{i}(x)})$
$\displaystyle-$
$\displaystyle(p_{i}(x+1)-p_{i}(x))(\partial_{p_{j}(x+1)}-\partial_{p_{j}(x)})\
.$

The operator $S$ conserves the total energy $\left(\text{defined in
Eq.\;\eqref{definition_energy}}\right)$ and the total pseudomomentum defined
by

$\displaystyle\mathcal{P}_{m}$
$\displaystyle=\left(\sum_{x\in\mathbb{Z}}p_{1}(x)-Bq_{2}(x),\sum_{x\in\mathbb{Z}}p_{2}(x)+Bq_{1}(x)\right)\
.$

We introduce a scaling parameter $\varepsilon>0$ and we take $\gamma>0$ which
represents the intensity of the stochastic noise. We define the formal
infinitesimal generator of the dynamics by $L^{\varepsilon}$ by

(18) $L^{\varepsilon}=A+BG+\varepsilon\gamma S\ .$

We denote the Markovian dynamics generated by the formal infinitesimal
generator $L^{\varepsilon}$ by
$(\mathbf{q}^{\varepsilon}(t),\mathbf{p}^{\varepsilon}(t))_{t\geq 0}$ . For a
rigorous definition of the dynamics we refer the reader to [30, Sec 3.4] or to
[14, Chapter 6].

We can check that $L^{\varepsilon}$ conserves the total energy and the total
pseudo-momentum. Since the energy of the dynamics is conserved we will denote
it by $E^{\varepsilon}$. In order to simplify the notations for all $i$ in
$\\{1,2\\}$, $k$ in $\mathbb{T}$ and positive time $t$, we will write
$\widehat{\psi}_{i}^{\varepsilon}(t,k)$ instead of
$\widehat{\psi}_{i,B}\left(\mathbf{q}^{\varepsilon}(t),\mathbf{p}^{\varepsilon}(t)\right)[k]$
where we recall that
$\left(\mathbf{q}^{\varepsilon},\mathbf{p}^{\varepsilon}\right)$ is the
Markovian dynamics generated by the formal operator $L^{\varepsilon}$. Using
the conservation of the energy, we can express the total energy
$E^{\varepsilon}$ in terms of
$\left(\widehat{\psi}^{\varepsilon}_{i}\right)_{i\in\\{1,2\\}}$ in the
following way

$\displaystyle
E^{\varepsilon}=\int_{\mathbb{T}}\left(\left|\widehat{\psi}^{\varepsilon}_{1}(0,k)\right|^{2}+\left|\widehat{\psi}^{\varepsilon}_{2}(0,k)\right|^{2}\right)dk\
.$

We assume that the initial configuration of the system
$\left(\mathbf{q}^{0},\mathbf{p}^{0}\right)$ is distributed according to a
measure $\mu^{\varepsilon}$ which satisfies the following condition

(19)
$K_{0}=\sup_{0<\varepsilon<1}\varepsilon\;\int_{\mathbb{T}}\mathbb{E}_{\mu^{\varepsilon}}\left[\left|\widehat{\psi}^{\varepsilon}_{1}(0,k)\right|^{2}+\left|\widehat{\psi}^{\varepsilon}_{2}(0,k)\right|^{2}\right]dk\quad<\infty\
.$

Since the energy of the system is preserved, this condition is true at any
time $t$, i.e.

(20) $\sup_{0<\varepsilon<1}\sup_{t\geq
0}\;\varepsilon\;\int_{\mathbb{T}}\mathbb{E}_{\mu^{\varepsilon}}\left[|\widehat{\psi}^{\varepsilon}_{1}(t,k)|^{2}+|\widehat{\psi}^{\varepsilon}_{2}(t,k)|^{2}\right]dk=K_{0}\
.$

###  3.4. Wigner’s distribution

The space $({\mathcal{S}}\times{\mathcal{S}})^{\prime}$ is equipped with the
weak-* topology i.e. a sequence
$({\mathcal{W}}^{N})_{N\in\mathbb{N}}=({\mathcal{W}}_{1}^{N},{\mathcal{W}}_{2}^{N})_{N\in\mathbb{N}}$
in $({\mathcal{S}}\times{\mathcal{S}})^{\prime}$ converges to ${\mathcal{W}}$
in $({\mathcal{S}}\times{\mathcal{S}})^{\prime}$ if and only if for any
$J:=(J_{1},J_{2})$ in ${\mathcal{S}}\times{\mathcal{S}}$ and $i$ in
$\\{1,2\\}$

$\displaystyle\lim\limits_{N\rightarrow\infty}\left|\left\langle{\mathcal{W}}_{i}^{N},J_{i}\right\rangle-\left\langle{\mathcal{W}}_{i},J_{i}\right\rangle\right|=0\
.$

We say then that a sequence
$\left({\mathcal{W}}^{N}\right)_{N\in{\mathbb{N}}}$ in
${\mathcal{F}}\left([0,T],({\mathcal{S}}\times{\mathcal{S}})^{\prime}\right)$
converges pointwise to ${\mathcal{W}}$ if and only if for any $t$ in $[0,T]$,
$({\mathcal{W}}^{N}(t))_{N\in{\mathbb{N}}}$ converges to ${\mathcal{W}}(t)$ in
$({\mathcal{S}}\times{\mathcal{S}})^{\prime}$.

For each $t$ in $[0,T]$, we define the Wigner’s distribution, denoted by
${\mathcal{W}}^{\varepsilon}(t)$, as the element of
$({\mathcal{S}}\times{\mathcal{S}})^{\prime}$ defined for any
$J=(J_{1},J_{2})$ in ${\mathcal{S}}\times{\mathcal{S}}$ by

$\langle\mathcal{W}^{\varepsilon}(t),J\rangle=\sum_{i=1}^{2}\langle{\mathcal{W}}_{i}^{\varepsilon}(t),J_{i}\rangle\
,$

where, for $i$ in $\\{1,2\\}$,

(21) $\displaystyle\langle{\mathcal{W}}_{i}^{\varepsilon}(t),J_{i}\rangle$
$\displaystyle=\frac{\varepsilon}{2}\sum_{x,x^{\prime}\in\mathbb{Z}}\mathbb{E}_{\mu^{\varepsilon}}\left[\psi^{\varepsilon}_{i}\left(t\varepsilon^{-1},x^{\prime}\right)^{*}\psi^{\varepsilon}_{i}\left(t\varepsilon^{-1},x\right)\right]\int_{\mathbb{T}}e^{2\mathbf{i}\pi(x^{\prime}-x)k}J_{i}\left(\tfrac{\varepsilon}{2}(x+x^{\prime}),k\right)^{*}dk$
$\displaystyle=\frac{\varepsilon}{2}\int_{\mathbb{R}}\int_{\mathbb{T}}{\mathbb{E}}_{\mu^{\varepsilon}}\left[\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},k-\tfrac{\varepsilon
p}{2}\right)^{*}\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},k+\tfrac{\varepsilon
p}{2}\right)\right]\mathcal{F}\left[J_{i}\right](p,k)^{*}dkdp\ .$

The integrability conditions justifying the existence of these expressions are
proved in Appendix A. The following lemma proves the well posedness of the
Wigner’s distribution as an element of
${\mathcal{C}}\left([0,T],({\mathcal{S}}\times{\mathcal{S}})^{\prime}\right)$
and gives some of its properties.

######  Lemma 3.2.

The Wigner’s distribution satisfies the following properties

  * i)

For all $t$ in $[0,T]$, ${\mathcal{W}}^{\varepsilon}(t)$ belongs to
$({\mathcal{S}}\times{\mathcal{S}})^{\prime}$.

  * ii)

For any $J$ in ${\mathcal{S}}\times{\mathcal{S}}$, the family
$(\left\langle\mathcal{W}^{\varepsilon},J\right\rangle)_{\varepsilon>0}$ is
bounded in
$({\mathcal{C}}([0,T],{\mathbb{C}}),\|\cdot\|_{\infty})$ and

$\displaystyle\|\left\langle\mathcal{W}^{\varepsilon},J\right\rangle\|_{\infty}\lesssim
K_{0}\|J\|\ .$

Furthermore, the application $t\mapsto\mathcal{W}^{\varepsilon}(t)$ belongs to
${\mathcal{C}}([0,T],({\mathcal{S}}\times{\mathcal{S}})^{\prime})$.

###### Proof.

We refer the reader to Appendix A.1. ∎

To study the asymptotic behavior of
$({\mathcal{W}}^{\varepsilon})_{\varepsilon>0}$ we need to introduce two
distributions on ${\mathcal{S}}\times{\mathcal{S}}$ denoted by
$\Gamma^{\varepsilon}$ and $\tilde{\Gamma}^{\varepsilon}$ defined for any
$J:=(J_{1},J_{2})$ in ${\mathcal{S}}\times{\mathcal{S}}$ by

(22)
$\langle\Gamma^{\varepsilon},J\rangle=\sum_{i=1}^{2}\langle\Gamma_{i}^{\varepsilon},J_{i}\rangle\quad\text{and}\quad\langle\tilde{\Gamma}^{\varepsilon},J\rangle=\sum_{i=1}^{2}\langle\tilde{\Gamma}_{i}^{\varepsilon},J_{i}\rangle\
,$

where, for any $i$ in $\\{1,2\\}$, by letting $i^{*}=3-i$, we set

(23) $\displaystyle\langle\Gamma_{i}^{\varepsilon}(t),J_{i}\rangle$
$\displaystyle=\frac{\varepsilon}{2}\int_{\mathbb{R}}\int_{\mathbb{T}}\mathbb{E}_{\mu^{\varepsilon}}\left[\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},\frac{\varepsilon
p}{2}-k\right)\widehat{\psi}^{\varepsilon}_{i^{*}}\left(t\varepsilon^{-1},k+\frac{\varepsilon
p}{2}\right)\right]\mathcal{F}\left[J_{i}\right](p,k)^{*}dkdp\ ,$

and

(24) $\displaystyle\langle\tilde{\Gamma}_{i}^{\varepsilon}(t),J_{i}\rangle$
$\displaystyle=\frac{\varepsilon}{2}\int_{\mathbb{R}}\int_{\mathbb{T}}\mathbb{E}_{\mu^{\varepsilon}}\left[\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},k-\frac{\varepsilon
p}{2}\right)^{*}\widehat{\psi}^{\varepsilon}_{i^{*}}\left(t\varepsilon^{-1},-k-\frac{\varepsilon
p}{2}\right)^{*}\right]\mathcal{F}\left[J_{i}\right](p,k)^{*}dkdp\ .$

##  4\. Review of previous results

In this section, we recall some results on this lattice. We first recall
Theorem 4.2 from [4, 30] which shows the convergence of the Wigner’s
distribution to the solution $f_{B}$ of some Boltzmann’s equation (i)). Then
we recall Theorem 4.5 and Theorem 4.6 from [21, 30] which show the convergence
of $f_{B}$ in some hydrodynamic scaling to the solution of some fractional
heat equations (51).

###  4.1. Kinetic limit of the Wigner’s distribution

Let $J:=(J_{1},J_{2})$ in ${\mathcal{S}}\times{\mathcal{S}}$ and recall Eq.
(13), (14) and (15). We define a collisional operator
$C_{B}:J=\left(J_{1},J_{2}\right)\in{\mathcal{S}}\times{\mathcal{S}}\to\left(\left[C_{B}J\right]_{1},\left[C_{B}J\right]_{2}\right)\in{\mathcal{S}}\times{\mathcal{S}}$
in the following way. For any $u$ in ${\mathbb{R}}$, $k$ in $\mathbb{T}$, $i$
in $\\{1,2\\}$ and $J:=(J_{1},J_{2})$ in ${\mathcal{S}}\times{\mathcal{S}}$,

(25)
$[C_{B}J]_{i}(u,k)=\sum_{j=1}^{2}\int_{\mathbb{T}}\theta_{i,B}^{2}(k)R(k,k^{\prime})\theta_{j,B}^{2}(k^{\prime})\left[J_{j}\left(u,k^{\prime}\right)-J_{i}(u,k)\right]dk^{\prime}\
,$

with

(26) $\forall(k,k^{\prime})\in\mathbb{T}\times\mathbb{T},\quad
R(k,k^{\prime})=16\sin^{2}(\pi k)\sin^{2}(\pi k^{\prime})\ .$

Let ${\mathcal{W}}$ in
$\mathcal{C}\left([0,T],({\mathcal{S}}\times{\mathcal{S}})^{\prime}\right)$
and ${\mathcal{W}}^{0}$ in $({\mathcal{S}}\times{\mathcal{S}})^{\prime}$.

  * i)

We say that ${\mathcal{W}}$ is a weak solution on $[0,T]$ of the linear
Boltzmann’s equation

(27)
$\partial_{t}{\mathcal{W}}+\frac{1}{2\pi}\textbf{v}_{B}\partial_{u}{\mathcal{W}}=\gamma
C_{B}{\mathcal{W}}\ ,$

with ${\mathcal{W}}^{0}$ as initial condition if and only if for any $J$ in
${\mathcal{S}}\times{\mathcal{S}}$ and any $t$ in $[0,T]$

$\displaystyle\left\langle{\mathcal{W}}(t),J\right\rangle-\left\langle{\mathcal{W}}(0),J\right\rangle$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\frac{1}{2\pi}\left\langle{\mathcal{W}}(s),\textbf{v}_{B}\partial_{u}J\right\rangle
ds$ $\displaystyle+$
$\displaystyle\gamma\int_{0}^{t}\left\langle{\mathcal{W}}(s),C_{B}J\right\rangle
ds\ .$

  * ii)

We say that ${\mathcal{W}}$ is a measure valued weak solution on $[0,T]$ of
the linear Boltzmann’s equation (27) if it is a weak solution of (27) such
that for all $t$ in $[0,T]$ and $i$ in $\\{1,2\\}$, ${\mathcal{W}}_{i}(t)$ is
a bounded positive measure on ${\mathbb{R}}\times\mathbb{T}$.

######  Lemma 4.1.

Let $\mu^{0}:=(\mu_{1}^{0},\mu_{2}^{0})$ in
$({\mathcal{S}}\times{\mathcal{S}})^{\prime}$ be a couple of bounded positive
measure on ${\mathbb{R}}\times\mathbb{T}$. Then, there exists a unique measure
valued weak solution $\mu$ on $[0,T]$ to the linear Boltzmann’s equation (27)
with $\mu^{0}$ as initial condition.

###### Proof.

We refer the reader to Appendix A.5. ∎

The following theorem summarizes the results obtained by the authors of [30,
Theorem 1] and by those of [4, Theorem 9] respectively.

######  Theorem 4.2.

[4, 30] Let $T>0$. Assume that the condition (19) holds and that
$(\mathcal{W}^{\varepsilon}(0))_{\varepsilon>0}$ converges in
$({\mathcal{S}}\times{\mathcal{S}})^{\prime}$ to a bounded positive
distribution $\mathcal{W}^{0}$.

  * i)

If $B\neq 0$, then there exists ${\mathcal{W}}$ in
${\mathcal{C}}\left([0,T],({\mathcal{S}}\times{\mathcal{S}})^{\prime}\right)$
such that $({\mathcal{W}}^{\varepsilon})_{\varepsilon>0}$ converges pointwise
to ${\mathcal{W}}$ in
${\mathcal{F}}\left([0,T],({\mathcal{S}}\times{\mathcal{S}})^{\prime}\right)$.
Moreover for each time $t$ in $[0,T]$, the limit $\mathcal{W}(t)$ and
$\mathcal{W}^{0}$ can be extended to a couple of bounded positive measures on
${\mathbb{R}}\times\mathbb{T}$ respectively denoted by
$\mu(t):=(\mu_{1}(t),\mu_{2}(t))$ and $\mu^{0}:=(\mu_{1}^{0},\mu_{2}^{0})$.
Furthermore, $\mu$ belongs to
${\mathcal{C}}\left([0,T],({\mathcal{S}}\times{\mathcal{S}})^{\prime}\right)$
and is the unique measure valued weak solution of the Boltzmann’s equation
(27) with initial condition $\mu^{0}$.

  * ii)

If $B=0$ and furthermore

(29) $\lim\limits_{\rho\rightarrow 0}\limsup\limits_{\varepsilon\rightarrow
0}\frac{\varepsilon}{2}\int_{|k|<\rho}\mathbb{E}_{\mu^{\varepsilon}}\left[\left|\widehat{\psi}^{\varepsilon}_{i}\left(0,k\right)\right|^{2}\right]dk=0\
,$

then the same conclusion as in i) holds with $B=0$ in Eq. (27).

######  Remark 4.3.

When $B=0$, the assumption (29) is required to compensate the lack of
differentiability of the function $\mathbf{v}_{B}$ defined in Eq. (15) at
$k=0$.

###  4.2. Hydrodynamic limit of the Boltzmann’s equation

The aim of this section is to study the hydrodynamic limit of the Boltzmann’s
equation (i)). Let us start by a short reminder about Lévy processes [11].
Given a measure $\nu$ on $\mathbb{R}^{*}$, we say that $\nu$ is a Lévy measure
if and only if

(30) $\int_{\mathbb{R}^{*}}\min\left(1,r^{2}\right)d\nu(r)<\infty\ .$

Let $Y_{u}(\cdot)$ be a real valued stochastic process starting from $u$ in
$\mathbb{R}$. We say that $Y_{u}(\cdot)$ is a Lévy process with (Lévy) measure
$\nu$ if and only if for any positive $t$ and $\theta$ in ${\mathbb{R}}$,

$\displaystyle{\mathbb{E}}\left[\exp\left(\mathbf{i}\theta
Y_{u}(t)\right)\right]=\exp\left(t\Phi_{Y}(\theta)+\mathbf{i}\theta u\right)\
.$

Here, $\Phi_{Y}$ denotes the Lévy exponent associated to the Lévy process
$Y_{u}(\cdot)$ and is given for any $\theta$ in ${\mathbb{R}}$ by

(31)
$\Phi_{Y}(\theta)=a\theta^{2}+\int_{\mathbb{R}^{*}}\left(\exp\left(\mathbf{i}\theta
r\right)-1+\mathbf{i}\theta r\mathds{1}_{\\{|r|<1\\}}\right)d\nu(r)\ ,$

where $a$ is in $\mathbb{R}^{+}$. The action of the infinitesimal generator
$\mathfrak{L}$ of $Y_{u}(\cdot)$ on a smooth function
$\phi:\mathbb{R}\rightarrow\mathbb{R}$ which decays sufficiently fast is given
by

(32) $\forall
p\in{\mathbb{R}},\quad\mathfrak{L}\left[\phi\right](p)=\int_{\mathbb{R}}{\mathcal{F}}\left[\phi\right](\xi)\Phi_{Y}(\xi)\exp\left(2\mathbf{i}\pi
p\xi\right)d\xi\ .$

In this article we will only study pure jumps Lévy processes, i.e. $a=0$. Let
$\alpha$ in $(1,2)$, and assume that $d\nu(r):=|r|^{-\alpha-1}$ then we have
that for any $\theta$ in $\mathbb{R}$

(33)
$\displaystyle\theta\int_{\mathbb{R}^{*}}r\mathds{1}_{\\{|r|<1\\}}d\nu(r):=U<\infty\
.$

Hence, if we change the initial point of the process from $u$ to $u-U$ we have
that (up to a constant)

$\displaystyle\Phi_{Y}(\theta)=-|\theta|^{\alpha}\ .$

For this choice of $\nu$, the infinitesimal generator of $Y_{u}(\cdot)$ is (up
to a constant) the pseudo-differential operator
$-\left(-\Delta\right)^{\tfrac{\alpha}{2}}$. In this case, we say that
$Y_{u}(\cdot)$ is an $\alpha$-stable Lévy process.

For $i$ in $\\{1,2\\}$ and $\left(k,k^{\prime}\right)$ in $\mathbb{T}^{2}$, we
recall that $\theta_{i,B}(k)$ and $R\left(k,k^{\prime}\right)$ are defined in
Eq. (14) and Eq. (26) respectively. Let
$f:\mathbb{T}\times\\{1,2\\}\rightarrow\mathbb{R}$ be a real-valued function
such that for any $i$ in $\\{1,2\\}$
$f(\cdot,i)\in{\mathcal{C}}\left(\mathbb{T},\mathbb{R}\right)$. We define an
operator ${\mathcal{L}}_{B}$ acting on $f$ by

(34) $\displaystyle{\mathcal{L}}_{B}[f](k,i)$
$\displaystyle=\lambda^{-1}_{B}(k,i)\sum_{j=1}^{2}\int_{\mathbb{T}}P_{B}\left(k,i,dk^{\prime},j\right)\left(f\left(k^{\prime},j\right)-f(k,i)\right)dk^{\prime}\
.$

Here for any $\left(k,k^{\prime}\right)$ in $\mathbb{T}^{2}$ and $(i,j)$ in
$\\{1,2\\}^{2}$

(35) $\displaystyle P_{B}\left(k,i,dk^{\prime},j\right)$
$\displaystyle=\gamma\lambda_{B}(k,i)\theta_{i,B}^{2}(k)\theta_{j,B}^{2}\left(k^{\prime}\right)R\left(k,k^{\prime}\right)dk^{\prime}\
,$

where

(36) $\displaystyle\lambda_{B}(k,i)$
$\displaystyle=\left[\gamma\theta_{i,B}^{2}(k)R(k)\right]^{-1}\quad\text{and
}\quad R(k)=\int_{\mathbb{T}}R\left(k,k^{\prime}\right)dk^{\prime}\ .$

Let $\mu$ be the measure valued weak solution of the Boltzmann’s equation (27)
with initial condition $\mu^{0}$. We assume that $\mu=(\mu_{1},\mu_{2})$ and
$\mu^{0}=(\mu_{1}^{0},\mu^{0}_{2})$ have a density denoted by
$f_{B}(\cdot):=(f_{B}(\cdot,1),f_{B}(\cdot,2))$ and
$f^{0}(\cdot):=\left(f^{0}(\cdot,1),f^{0}(\cdot,2)\right)$ respectively with
respect to the measure $dkd\delta$. Then we have for any time $t$ in $[0,T]$,
$k$ in $\mathbb{T}$, $u$ in ${\mathbb{R}}$ and $i$ in $\\{1,2\\}$

(37)
$\partial_{t}f_{B}(t,u,k,i)+\frac{\textbf{v}_{B}(k)}{2\pi}\partial_{u}f_{B}(t,u,k,i)=\mathcal{L}_{B}\left[f_{B}\right](t,u,k,i)\
,$

with $f^{0}(\cdot):=(f^{0}(\cdot,1),f^{0}(\cdot,2))$ as initial condition. To
study the hydrodynamic behavior of $f_{B}$ we interpret the operator
${\mathcal{L}}_{B}$ as the infinitesimal generator of a pure jump continuous
time Markov process on $\mathbb{T}\times\\{1,2\\}$ as follows.

Let
$\left(X_{n}^{B}\right)_{n\in{\mathbb{N}}}:=\left(K_{n}^{B},I_{n}^{B}\right)_{n\in{\mathbb{N}}}$
be the Markov chain on $\mathbb{T}\times\\{1,2\\}$ with transition probability
$P_{B}$ defined in Eq. (35). Let $(\tau_{n})_{n\in{\mathbb{N}}}$ be an i.i.d
sequence of random variables, independent of
$\left(X_{n}^{B}\right)_{n\in{\mathbb{N}}}$ such that
$\tau_{0}\sim\mathcal{E}(1)$. We define the random variable
${\mathcal{T}}_{N}$ by444We decided to not write $\mathcal{T}_{N}^{B}$ in
order to lighten the notations.

(38) $\forall
N\in{\mathbb{N}},\quad{\mathcal{T}}_{N}=\sum_{n=0}^{N}\lambda_{B}\left(X_{n}^{B}\right)\tau_{n}\
.$

Then we can define a pure jump Markovian process
$\left(K^{B}(\cdot),I^{B}(\cdot)\right)$ with values in
$\mathbb{T}\times\\{1,2\\}$ where for any positive time $t$ in $[0,T]$

$\displaystyle K^{B}(t)$ $\displaystyle=K_{n}^{B},\quad\text{and}\quad
I^{B}(t)=I_{n}^{B},\quad\forall
t\in\left[{\mathcal{T}}_{n},{\mathcal{T}}_{n+1}\right[\ .$

The infinitesimal generator of $\left(K^{B}(\cdot),I^{B}(\cdot)\right)$ is
${\mathcal{L}}_{B}$ defined in Eq. (34). From this process, we can define an
additive functional of Markov process $Z_{u}^{B}(\cdot)$ such that for any
time $t$ in $[0,T]$ and $u$ in $\mathbb{R}$

$\displaystyle Z_{u}^{B}(t)$
$\displaystyle=u-\int_{0}^{t}\frac{\textbf{v}_{B}\left(K^{B}(s)\right)}{2\pi}ds\
.$

Then by Dynkin’s formula we get that for any $i$ in $\\{1,2\\}$ and $(u,k)$ in
$\mathbb{R}\times\mathbb{T}$

(39)
$f_{B}(t,u,k,i)=\mathbb{E}_{(k,i)}\left[f^{0}\left(Z_{u}^{B}(t),K^{B}(t),I^{B}(t)\right)\right]\
.$

The hydrodynamic behavior of $f_{B}$ is completely determined by the one of
the process $\left(K^{B}(\cdot),I^{B}(\cdot)\right)$. Hence, in the following
we recall how the authors of [21, 30] studied this process. Let $\pi_{B}$ the
probability measure on $\mathbb{T}\times\\{1,2\\}$ defined as follows

(40)
$\pi_{B}(dk,di)=\sum_{j=1}^{2}\frac{\lambda_{B}(k,j)^{-1}}{\gamma\bar{R}}dk\delta_{j}(di)\quad\text{with}\quad\overline{R}=\int_{\mathbb{T}}R(k)dk\
.$

We can prove that $\pi_{B}$ is a reversible probability measure of the Markov
chain $\left(X_{n}^{B}\right)_{n\in{\mathbb{N}}}$. Observe that

(41) $P_{B}(k,i,dk^{\prime},j)=\pi_{B}(dk^{\prime},j)\ ,$

where $P_{B}$ is defined in Eq. (35).

######  Remark 4.4.

Observe that according to Eq. (41), $\left(X_{n}^{B}\right)_{n\geq 1}$ is an
i.i.d sequence of random variables on $\mathbb{T}\times\\{1,2\\}$ but observe
that $\left(X_{n}^{B}\right)_{n\in\mathbb{N}}$ is not because of $X_{0}^{B}$.
This property will play a crucial role in the proof of our main results stated
in Sec. (6.4).

We define a function $\Psi_{B}$ in the following way

(42) $\forall k\in\mathbb{T},\quad\forall
i\in\\{1,2\\},\quad\Psi_{B}(k,i)=\textbf{v}_{B}(k)\lambda_{B}(k,i)\ .$

The asymptotic behavior of the process $Z_{u}^{B}(\cdot)$ is fully determined
by the tails of the function $\Psi_{B}$. In [21, 30] it is proved that

(43) $\displaystyle\forall
r>0,\quad\lim\limits_{N\rightarrow\infty}N^{\alpha_{B}}\;\pi_{B}\left(\left\\{(k,i)\;\big{|}\quad\Psi_{B}(k,i)>Nr\right\\}\right)$
$\displaystyle=\tilde{\kappa}_{B}|r|^{-\alpha_{B}}\ ,$ (44)
$\displaystyle\forall
r<0,\quad\lim\limits_{N\rightarrow\infty}N^{\alpha_{B}}\;\pi_{B}\left(\left\\{(k,i)\;\big{|}\quad\Psi_{B}(k,i)<Nr\right\\}\right)$
$\displaystyle=\tilde{\kappa}_{B}|r|^{-\alpha_{B}}\ ,$

where

(45) $\displaystyle\alpha_{B}$ $\displaystyle=\frac{3}{2}\quad\text{if}\quad
B=0\quad\text{and}\quad\alpha_{B}=\frac{5}{3},\quad\text{if}\quad B\neq 0\ ,$

and

(49) $\displaystyle\tilde{\kappa}_{B}$
$\displaystyle=\left\\{\begin{array}[]{ll}\tilde{\kappa}_{1}(\widehat{\alpha}^{\prime\prime}(0))^{\tfrac{3}{4}}\gamma^{-\tfrac{3}{2}}\quad\text{if
$B=0$}\ ,\\\ \\\
\tilde{\kappa}_{2}|B|^{-1/3}\widehat{\alpha}^{\prime\prime}(0)\gamma^{-\tfrac{5}{3}}\quad\text{if
$B\neq 0$}\ ,\end{array}\right.$

with $\tilde{\kappa_{1}}$ and $\tilde{\kappa}_{2}$ two positive constants.
Then the following results are proved in [21, Theorem 3.1] and [30, Theorem 3]
respectively.

######  Theorem 4.5 ([21, 30]).

Let $u$ in ${\mathbb{R}}$ and define $u_{N}:=Nu$. We assume that
$X^{B}(0)=\left(K^{B}(0),I^{B}(0)\right)=(k,i)$ with $k\neq 0$ and $i$ in
$\\{1,2\\}$.
Then, under ${\mathbb{P}}_{(k,i)}$ the finite-dimensional distributions of the
scaled process $N^{-1}Z_{u_{N}}^{B}\left(N^{\alpha_{B}}\cdot\right)$ converge
weakly to the finite-dimensional distributions of a Lévy process
$Y_{u}^{B}(\cdot)$ generated by
$-2\gamma\tilde{\kappa}_{B}(-\Delta)^{\frac{\alpha_{B}}{2}}$ where
$\tilde{\kappa}_{B}$ and $\alpha_{B}$ are defined in Eq. (45) and Eq. (49)
respectively.

From these results, the authors of [21, 30] have been able to prove the
following theorem.

######  Theorem 4.6 ([21, 30]).

Let $f_{B}(\cdot):=\left(f_{B}(\cdot,1),f_{B}(\cdot,2)\right)$ be the solution
of Eq. (37) with initial condition $f_{N}^{0}(\cdot)$
$:=(f_{N}^{0}(\cdot,1),f_{N}^{0}(\cdot,2))$ defined as follows

(50) $\displaystyle\forall(u,k)\in\mathbb{R}\times\mathbb{T},\quad
f_{N}^{0}(u,k)=\left(f^{0}\left(\frac{u}{N},k,1\right),f^{0}\left(\frac{u}{N},k,2\right)\right)\
.$

Here
$(f^{0}(\cdot,1),f^{0}(\cdot,2))\in\mathcal{C}^{\infty}_{c}\left(\mathbb{R}\times\mathbb{T}\right)^{2}$.
Let $\overline{f}^{0}:\mathbb{R}\rightarrow\mathbb{R}$ be the real-valued
function defined for any $u$ in $\mathbb{R}$

$\displaystyle\overline{f}^{0}(u)$
$\displaystyle=\sum_{i=1}^{2}\int_{\mathbb{T}}f^{0}(u,k,i)dk\ .$

Then, for any time $t$ in $[0,T]$ and $u$ in $\mathbb{R}$ we have that

$\displaystyle\lim\limits_{N\rightarrow\infty}\
\sum_{i=1}^{2}\int_{\mathbb{T}}\left|f_{B}\left(N^{\alpha_{B}}t,Nu,k,i\right)-\frac{1}{2}\rho_{B}(t,u)\right|^{2}dk=0\
,$

where we recall that $\alpha_{B}$ is defined in Eq. (45) and $\rho_{B}$ is the
solution of

(51) $\forall t\in[0,T],\quad\forall
u\in{\mathbb{R}},\quad\left\\{\begin{array}[]{ll}\partial_{t}\rho_{B}(t,u)=-D_{B}(-\Delta)^{\frac{\alpha_{B}}{2}}\left[\rho_{B}\right](t,u)\
,\\\ \rho_{B}(0,u)=\overline{f}^{0}(u)\ ,\end{array}\right.$

with $D_{B}$ a positive constant.

######  Remark 4.7.

In Eq. (69) we give the explicit values of the constant $D_{B}$ which were not
given in [21, 30].

######  Remark 4.8.

In fact in [21] the result is stated for a unidimensional Boltzmann’s
equation, i.e , $f_{B}$ is a one component function. However, following the
proof given in [30] we can extend the result to a two components function.

##  5\. Statement of the results

In this section, we state our main results. Theorem 5.1 shows that at kinetic
time scale $\varepsilon^{-1}$ under assumption (52) there is no transition in
the convergence of the Wigner’s distribution. Theorem 5.7 and Theorem 5.9 show
that a transition can be observed at some hydrodynamic time scale with the
appearance of an interpolation process $Y^{B}(\cdot)$ for some critical
intensity of the magnetic field. We end this section with Theorem 5.11 which
shows that the interpolation process $Y^{B}(\cdot)$ converges to the
fractional process studied in [21] when $B$ goes to zero and to the one
studied in [30] when $B$ is sent to infinity.

###  5.1. Triviality of the transition in the kinetic time scale

In this section, we assume that the microscopic dynamics $\left(\text{cf.
Eq.}\;\eqref{microscopic_system}\right)$ is submitted to a magnetic field of
intensity $B_{\varepsilon}:=B\varepsilon^{\delta}$ where $\delta>0$ and
$B>0$555In order to lighten the presentation, we choose to only consider the
case $B>0$. However, the case $B<0$ can be treated in a similar way..

######  Theorem 5.1.

Let $\delta>0$, $\eta$ in $]0,1[$ such that $\delta>\eta$ and $\kappa>0$. We
define $\kappa_{\varepsilon}:=\kappa\varepsilon^{\eta}$ and we assume that the
condition (19) holds and that $(\mathcal{W}^{\varepsilon}(0))_{\varepsilon>0}$
converges in $({\mathcal{S}}\times{\mathcal{S}})^{\prime}$ to a bounded
positive distribution $\mathcal{W}^{0}$. We assume furthermore that

(52) $\limsup\limits_{\varepsilon\rightarrow
0}\frac{\varepsilon}{2}\int_{|k|<\kappa_{\varepsilon}}\mathbb{E}_{\mu^{\varepsilon}}\left[\left|\widehat{\psi}^{\varepsilon}_{i}\left(0,k\right)\right|^{2}\right]=0\
.$

Then, there exists ${\mathcal{W}}$ in
${\mathcal{C}}\left([0,T],({\mathcal{S}}\times{\mathcal{S}})^{\prime}\right)$
such that $({\mathcal{W}}^{\varepsilon})_{\varepsilon>0}$ converges pointwise
to ${\mathcal{W}}$ in
${\mathcal{F}}\left([0,T],({\mathcal{S}}\times{\mathcal{S}})^{\prime}\right)$.
Moreover for each time $t$ in $[0,T]$, the limit $\mathcal{W}(t)$ and
$\mathcal{W}^{0}$ can be extended to couples of bounded positive measures on
${\mathbb{R}}\times\mathbb{T}$ denoted respectively by
$\mu(t):=(\mu_{1}(t),\mu_{2}(t))$ and $\mu^{0}:=(\mu_{1}^{0},\mu_{2}^{0})$.
Furthermore, $\mu$ belongs to
$\mathcal{C}\left([0,T],({\mathcal{S}}\times{\mathcal{S}})^{\prime}\right)$
and is the unique measure valued weak solution on $[0,T]$ of the Boltzmann’s
equation (27) with $B=0$ and initial condition
$\mu^{0}:=(\mu^{0}_{1},\mu_{2}^{0})$.

###### Proof.

We refer the reader to Sec. 6.1 for the proof of Theorem 5.1. ∎

From Theorem 5.1, we deduce that under assumption (52) the transition in the
kinetic time scale $\varepsilon^{-1}$ between the case of [4] (zero magnetic
field) and [30] (magnetic field of order one) is trivial in the sense that it
holds for $\delta=0$. We show in Theorem 5.9 that it is not the case in a
longer time scale. To prove Theorem 5.1 we need the following lemma which
shows that the assumption (52) can be extended to times $t\varepsilon^{-1}$.

######  Lemma 5.2.

Let $i$ in $\\{1,2\\}$, then for any time $t$ in $[0,T]$

$\displaystyle\limsup\limits_{\varepsilon\rightarrow
0}\frac{\varepsilon}{2}\int_{|k|<\kappa_{\varepsilon}}\mathbb{E}_{\mu^{\varepsilon}}\left[\left|\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},k\right)\right|^{2}\right]dk=0\
,$

where we recall that $\kappa_{\varepsilon}=\kappa\varepsilon^{\eta}$ with
$\eta$ in $]0,1[$ such that $\delta>\eta$ and $\kappa>0$.

###### Proof.

We refer the reader to Appendix A.4 for the proof of Lemma 5.2. ∎

######  Remark 5.3.

Observe that assumption (52) is weaker than the assumption (29) made in [4] to
compensate the lack of differentiability of the function $\mathbf{v}_{0}$.

###  5.2. Transition in the hydrodynamic time scale

In this section, we study the Markov chain
$\left(X_{n}^{B}\right)_{n\in{\mathbb{N}}}$ introduced in Sec. 4.2 but with a
magnetic field of intensity $B_{N}:=BN^{-\delta}$ where $\delta$ is in
${\mathbb{R}}^{+}$ and $B>0$666As in the previous section, the case $B<0$ can
be studied in a similar way.. Hence, for each $N$ we will denote this chain by
$\left(X_{n}^{B_{N}}\right)_{n\in{\mathbb{N}}}:=\left(K_{n}^{B_{N}},I_{n}^{B_{N}}\right)_{n\in{\mathbb{N}}}$.
In Eq. (13), Eq. (14) and Eq. (15) we replace $B$ by $B_{N}$ and in all the
functions defined in Sec. 4.2. The aim of this section, is to study the
behavior of $f_{B_{N}}(\cdot):=(f_{B_{N}}(\cdot,1),f_{B_{N}}(\cdot,2))$ where
for any positive time $t$, $k$ in $\mathbb{T}$, $u$ in ${\mathbb{R}}$ and $i$
in $\\{1,2\\}$

(53)
$\partial_{t}f_{B_{N}}(t,u,k,i)+\frac{\textbf{v}_{B_{N}}(k)}{2\pi}\partial_{u}f_{B_{N}}(t,u,k,i)={\mathcal{L}}_{B_{N}}\left[f_{B_{N}}\right](t,u,k,i)\
,$

where we recall that $\textbf{v}_{B_{N}}$ and ${\mathcal{L}}_{B_{N}}$ are
defined in Eq. (15) and Eq. (34) respectively. By Dynkin’s formula we get that
for any time $t$ in $[0,T]$, $u$ in ${\mathbb{R}}$, $k$ in $\mathbb{T}$ and
$i$ in $\\{1,2\\}$

(54)
$f_{B_{N}}(t,u,k,i)=\mathbb{E}_{(k,i)}\left[f^{0}\left(Z^{B_{N}}_{u}(t),K^{B_{N}}(t),I^{B_{N}}(t)\right)\right]\
.$

Here for $t$ in $[0,T]$ $u$ in ${\mathbb{R}}$ and $N$ in $\mathbb{N}$ we
recall that

$\displaystyle Z^{B_{N}}_{u}(t)$
$\displaystyle=u-\int_{0}^{t}\frac{\textbf{v}_{B_{N}}\left(K^{B_{N}}(s)\right)}{2\pi}ds\
,$

where

$\displaystyle K^{B_{N}}(t)$ $\displaystyle=K_{n}^{B_{N}},\quad\text{and}\quad
I^{B_{N}}(t)=I_{n}^{B_{N}},\quad\forall
t\in\left[{\mathcal{T}}_{n}^{B_{N}},{\mathcal{T}}_{n+1}^{B_{N}}\right[\ ,$

with

$\displaystyle{\mathcal{T}}^{B_{N}}_{n}$
$\displaystyle=\sum_{m=0}^{n}\lambda_{B_{N}}\left(X_{m}^{B_{N}}\right)\tau_{m}^{N}\
.$

Here for any integer $N$, $\left(\tau_{n}^{N}\right)_{n\geq 0}$ is a sequence
of i.i.d random variables independent of
$\left(X_{n}^{B_{N}}\right)_{n\in{\mathbb{N}}}$ with
$\tau_{0}^{N}\sim\mathcal{E}(1)$. For every $N$, we recall that the invariant
probability measure of the Markov chain
$\left(X_{n}^{B_{N}}\right)_{n\in{\mathbb{N}}}$ is denoted by $\pi_{B_{N}}$
and defined in Eq. (40). We define the function $\Psi_{B_{N}}$ on
$\mathbb{T}\times\\{1,2\\}$ by

(55)
$\forall(k,i)\in\mathbb{T}\times\\{1,2\\},\quad\Psi_{B_{N}}(k,i)=\textbf{v}_{B_{N}}\left(k\right)\lambda_{B_{N}}\left(k,i\right)\
.$

As we recalled in Sec. 4.2, the asymptotic behavior of $f_{B_{N}}$ is fully
determined by the one of the process $Z_{u}^{B_{N}}(\cdot)$. The next
propositions allow us to compute the tails of the function $\Psi_{B_{N}}$ and
to determine the asymptotic behavior of the process $Z_{u}^{B_{N}}(\cdot)$.

######  Proposition 5.4.

We define implicitly two functions $x_{B,\pm}$ on $\mathbb{R}^{*}$ by

(56) $\displaystyle\left(2\sqrt{x_{B,\pm}^{2}(r)+\frac{B^{2}}{4}}\pm
B\right)x_{B,\pm}(r)$ $\displaystyle=\frac{\pi}{\gamma r},\quad r\neq 0\ .$

Then

  * i)

$x_{B,\pm}$ are odd functions.

  * ii)

The functions $x_{B,\pm}$ and $x_{B,\pm}^{\prime}$ converge pointwise when $B$
goes to zero and for any $r\neq 0$

$\displaystyle\lim\limits_{B\rightarrow
0}x_{B,\pm}(r)=\text{sign}(r)\sqrt{\frac{\pi}{2\gamma}}|r|^{-\frac{1}{2}}\quad\text{and}\quad\lim\limits_{B\rightarrow
0}x_{B,\pm}^{\prime}(r)=-\frac{\sqrt{\pi}}{2\sqrt{2\gamma}}|r|^{-\frac{3}{2}}\
.$

  * iii)

The functions $x_{B,+}$ and $x_{B,+}^{\prime}$ converge pointwise to zero when
$B$ goes to infinity and for any $r\neq 0$, we have

$\displaystyle\lim\limits_{B\rightarrow\infty}Bx_{B,+}(r)=\text{sign}(r)\frac{\pi}{2\gamma}|r|^{-1}\quad\text{and}\quad\lim\limits_{B\rightarrow\infty}Bx_{B,+}^{\prime}(r)=-\frac{\pi}{2\gamma}|r|^{-2}\
.$

The functions $x_{B,-}$ and $x_{B,-}^{\prime}$ diverge pointwise to infinity
when $B$ goes to infinity and for any $r\neq 0$, we have

$\displaystyle\lim\limits_{B\rightarrow\infty}B^{-\frac{1}{3}}x_{B,-}(r)=\text{sign}(r)C|r|^{-\frac{1}{3}}\quad\text{and}\quad\lim\limits_{B\rightarrow\infty}B^{-\frac{1}{3}}x_{B,-}^{\prime}(r)=-\frac{C}{3}|r|^{-\frac{4}{3}}\
,$

with $C=\left(\frac{\pi}{2\gamma}\right)^{\frac{1}{3}}$.

###### Proof.

We refer the reader to Appendix B.1. ∎

######  Proposition 5.5.

Let $h_{B,\pm}$ and $g_{B\pm}$ be the four real-valued functions defined for
any $r\neq 0$ as follows

(57) $\displaystyle h_{B,\pm}(r)$
$\displaystyle=\frac{1}{4\pi}\int_{0}^{x_{B,\pm}(r)}\left(\frac{1}{2}\pm\frac{B}{4\sqrt{x^{2}+\frac{B^{2}}{4}}}\right)x^{2}dx\mathds{1}_{r>0}$
$\displaystyle+\frac{1}{4\pi}\int_{x_{B,\pm}(r)}^{0}\left(\frac{1}{2}\pm\frac{B}{4\sqrt{x^{2}+\frac{B^{2}}{4}}}\right)x^{2}dx\mathds{1}_{r<0}\
.$ (58) $\displaystyle g_{B,\pm}(r)$
$\displaystyle=-h_{B,\pm}^{\prime}(r)\mathds{1}_{r>0}+h_{B,\pm}^{\prime}(r)\mathds{1}_{r<0}\
.$

Then

  * i)

$g_{B,\pm}$ are positive even functions.

  * ii)

The measure $\nu_{B}$ with density $g_{B,+}+g_{B,-}$ with respect to the
Lebesgue measure is a Lévy measure on $\mathbb{R}^{*}$.

  * iii)

The functions $g_{B,\pm}$ converge pointwise to $g_{0}$ when $B$ goes to zero
where for any $r\neq 0$

(59)
$g_{0}(r)=\left(\frac{\pi}{2^{11}\gamma^{3}}\right)^{\frac{1}{2}}|r|^{-\frac{3}{2}-1}.$

Moreover, $g_{0}$ is the density of a Lévy measure on $\mathbb{R}^{*}$.

  * iv)

The function $B^{\frac{1}{3}}g_{B,+}$ converges pointwise to zero and
$B^{\frac{1}{3}}g_{B,-}$ converges pointwise to $g_{\infty}$ when $B$ goes to
infinity where for any $r\neq 0$

(60)
$g_{\infty}(r)=\left(\frac{\pi^{2}}{2^{11}27\gamma^{5}}\right)^{\frac{1}{3}}|r|^{-\frac{5}{3}-1}\
.$

Moreover, $g_{\infty}$ is the density of a Lévy measure on $\mathbb{R}^{*}$.

###### Proof.

We refer the reader to Appendix B.2. ∎

######  Proposition 5.6.

Let $\pi_{B_{N}}$ be the invariant probability measure defined in Eq. (40) and
$\Psi_{B_{N}}$ the function defined in Eq. (55) then we have

$\displaystyle\forall
r>0,\quad\lim\limits_{N\rightarrow\infty}N^{\alpha_{\delta}}\;\pi_{B_{N}}\left(\left\\{(k,i)\;\big{|}\quad\Psi_{B_{N}}(k,i)>Nr\right\\}\right)$
$\displaystyle=\kappa_{\delta}(r)\ ,$ $\displaystyle\forall
r<0,\quad\lim\limits_{N\rightarrow\infty}N^{\alpha_{\delta}}\;\pi_{B_{N}}\left(\left\\{(k,i)\;\big{|}\quad\Psi_{B_{N}}(k,i)<Nr\right\\}\right)$
$\displaystyle=\kappa_{\delta}(r)\ ,$

where for any $r\neq 0$

(61)
$\left\\{\begin{array}[]{ll}\alpha_{\delta}=\frac{3}{2}\quad\text{and}\quad\kappa_{\delta}(r)=\gamma^{-\tfrac{3}{2}}\kappa_{1}|r|^{-\frac{3}{2}}\quad\text{if}\quad\delta>\frac{1}{2}\
,\vspace{0.1cm}\\\
\alpha_{\delta}=\frac{3}{2}\quad\text{and}\quad\kappa_{\delta}(r)=\left(h_{B,-}(r)+h_{B,+}(r)\right)\quad\text{if}\quad\delta=\frac{1}{2}\
,\vspace{0.1cm}\\\
\alpha_{\delta}=\frac{5-\delta}{3}\quad\text{and}\quad\kappa_{\delta}(r)=\gamma^{-\tfrac{5}{3}}\kappa_{2}|B|^{-\frac{1}{3}}|r|^{-\frac{5}{3}}\quad\text{if}\quad\delta<\tfrac{1}{2}\
,\end{array}\right.$

with $\kappa_{1}$ and $\kappa_{2}$ two positive constants defined respectively
in Eq. (118) and in Eq. (119).

###### Proof.

We refer the reader to Appendix C.1. ∎

######  Theorem 5.7.

Let $\alpha_{\delta}$ and $\kappa_{\delta}$ defined in Eq. (61) and
$\tau_{0}\sim\mathcal{E}\left(1\right)$. Let $u$ in ${\mathbb{R}}$, we define
$u_{N}:=Nu$ and we assume that $\left(X^{B_{N}}_{0}\right)=(k,i)$ with $k\neq
0$ and $i$ in $\\{1,2\\}$. We denote by $Z_{u}^{\delta}(\cdot)$ the Lévy
process starting from point $u$ with Lévy measure $\nu_{\delta}$ where

(62)
$d\nu_{\delta}(r):=\left\\{\begin{array}[]{ll}\gamma^{-\tfrac{1}{2}}\kappa_{0}\mathbb{E}\left[\tau_{0}^{\alpha_{\delta}}\right]|r|^{-\frac{3}{2}-1}dr\quad\text{if}\quad\delta>\frac{1}{2}\
,\\\ \\\ 2\gamma\mathbb{E}\left[\tau_{0}^{-1}\left(g_{B,+}\left(\frac{2\pi
r}{\tau_{0}}\right)+g_{B,-}\left(\frac{2\pi
r}{\tau_{0}}\right)\right)\right]dr\quad\text{if}\quad\delta=\frac{1}{2}\ ,\\\
\\\
\gamma^{-\tfrac{2}{3}}|B|^{-1/3}\kappa_{\infty}\mathbb{E}\left[\tau_{0}^{\alpha_{\delta}}\right]|r|^{-\frac{5}{3}-1}dr\quad\text{if}\quad\delta<\frac{1}{2}\
,\end{array}\right.\\\ $

with

(63)
$\kappa_{0}=\left(\frac{1}{2^{10}\pi^{2}}\right)^{\frac{1}{2}}\quad\text{and}\quad\kappa_{\infty}=\left(\frac{1}{2^{13}27\pi^{3}}\right)^{\frac{1}{3}}\
.$

Then, under ${\mathbb{P}}_{(k,i)}$ the finite-dimensional distributions of the
process $N^{-1}Z_{u_{N}}^{B_{N}}\left(N^{\alpha_{\delta}}\cdot\right)$
converge weakly to the finite-dimensional distributions of
$Z_{u}^{\delta}(\cdot)$.

###### Proof.

We refer the reader to Sec. 6.2. ∎

######  Remark 5.8.

Observe that the process $Z_{u}^{1/2}$ defined in Eq. (62), which is not a
Lévy stable process, belongs nevertheless to the domain of attraction of a
Lévy stable law of exponent $5/3$. Indeed, using the definition of $g_{B,\pm}$
given in Eq. (58), one can prove that there exists a positive constant $C_{B}$
such that

$\displaystyle\nu_{1/2}(r)\underset{\infty}{\sim}C_{B}|r|^{-\tfrac{5}{3}}\ ,$

where $C_{B}$ converges to zero when $B$ goes to infinity.

Observe that
$\mathbb{E}\left[\tau_{0}^{\alpha_{\delta}}\right]=\Gamma(1+\alpha_{\delta})$
where $\Gamma$ is the Gamma function. Theorem 5.7 allows us to obtain the
hydrodynamic limit of the solution $f_{B_{N}}$ of the Boltzmann’s equation
(53). Let $Z_{u}^{\delta}(\cdot)$ be the Lévy process starting from $u$ in
${\mathbb{R}}$ with Lévy exponent $\Phi_{\delta}$ defined by

(64)
$\forall\theta\in{\mathbb{R}},\quad\Phi_{\delta}(\theta)=\int_{\mathbb{R}^{*}}\left(\exp\left(\mathbf{i}\theta
r\right)-1+\mathbf{i}\theta r\mathds{1}_{\\{|r|<1\\}}\right)d\nu_{\delta}(r)\
,$

where $\nu_{\delta}$ is the Lévy measure defined in Eq. (62). Observe that for
$\delta>\frac{1}{2}$ $\left(\text{resp.}\delta<\frac{1}{2}\right)$,
$Z_{u}^{\delta}(\cdot)$ is the Lévy process obtained in [21] (resp. [30]). For
$\delta=\frac{1}{2}$ we have an interpolation process denoted by
$Y_{u}^{B}(\cdot)$.

######  Theorem 5.9.

Let $f_{B_{N}}(\cdot):=(f_{B_{N}}(\cdot,1),f_{B_{N}}(\cdot,2))$ be the
solution of Eq. (53) with initial condition $f_{N}^{0}(\cdot)$
$:=(f_{N}^{0}(\cdot,1),f_{N}^{0}(\cdot,2))$ defined as follows

(65) $\displaystyle\forall(u,k)\in\mathbb{R}\times\mathbb{T},\quad
f_{N}^{0}(u,k)=\left(f^{0}\left(\frac{u}{N},k,1\right),f^{0}\left(\frac{u}{N},k,2\right)\right)\
.$

Here $(f^{0}(\cdot,1),f^{0}(\cdot,2))$ in
$\mathcal{C}^{\infty}_{c}\left(\mathbb{R}\times\mathbb{T}\right)^{2}$. We
define on ${\mathbb{R}}$ the function $\overline{f}^{0}$ as follows

(66) $\forall
u\in{\mathbb{R}},\quad\overline{f}^{0}(u)=\sum_{i=1}^{2}\int_{\mathbb{T}}f^{0}(u,k,i)dk\
.$

We define the operator $\mathfrak{L}_{\delta}$ on the space
${\mathcal{C}}_{\textbf{c}}^{\infty}\left({\mathbb{R}}\right)$ in the
following way

(67) $\forall\phi\in{\mathcal{C}}_{c}^{\infty}({\mathbb{R}}),\quad\forall
p\in{\mathbb{R}},\quad\mathfrak{L}_{\delta}\left[\phi\right](p)=\int_{\mathbb{R}}{\mathcal{F}}\left[\phi\right](\xi)\Phi_{\delta}(\xi)\exp\left(2\mathbf{i}\pi
p\xi\right)d\xi\ ,$

where $\Phi_{\delta}$ is defined in Eq. (64). Let $\alpha_{\delta}$ defined by
Eq. (61), then

$\forall t\in[0,T],\quad\forall
u\in{\mathbb{R}},\quad\lim\limits_{N\rightarrow\infty}\
\sum_{i=1}^{2}\int_{\mathbb{T}}\left|f_{B_{N}}\left(N^{\alpha_{\delta}}t,Nu,k,i\right)-\frac{1}{2}\rho_{\delta}(t,u)\right|dk=0\
,$

where $\rho_{\delta}$ is the solution of

(68) $\forall t\in[0,T],\quad\forall
u\in{\mathbb{R}},\quad\left\\{\begin{array}[]{ll}\partial_{t}\rho_{\delta}(t,u)=\mathfrak{L}_{\delta}\left[\rho_{\delta}\right](t,u)\
,\\\ \rho_{\delta}(0,u)=\overline{f}^{0}(u)\ .\end{array}\right.$

###### Proof.

We refer the reader to Sec. 6.3. ∎

######  Remark 5.10.

Observe that when $\delta>\frac{1}{2}$,
$\mathfrak{L}_{\delta}=-\Gamma\left(1+\frac{3}{2}\right)\gamma^{-\tfrac{1}{2}}D_{0}(-\Delta)^{\frac{3}{4}}$
and when $\delta<\frac{1}{2}$ we have that
$\mathfrak{L}_{\delta}=-|B|^{-1/3}\Gamma\left(1+\frac{5}{3}\right)\gamma^{-\tfrac{2}{3}}D_{\infty}(-\Delta)^{\frac{5}{6}}$
where

(69)
$D_{0}=2\kappa_{0}\int_{0}^{+\infty}\frac{1-\cos(r)}{r^{\frac{5}{2}}}dr\quad\text{and}\quad
D_{\infty}=2\kappa_{\infty}\int_{0}^{+\infty}\frac{1-\cos(r)}{r^{\frac{8}{3}}}dr\
.$

Hence, we recover the different cases studied in [21] and [30] respectively.

When $\delta=\frac{1}{2}$, $\mathfrak{L}_{\frac{1}{2}}$ depends on $B$ hence
we denote by $\mathfrak{L}_{B}$ the operator $\mathfrak{L}_{\frac{1}{2}}$, by
$\Phi_{B}$ the Lévy exponent $\Phi_{\frac{1}{2}}$ and by $\tilde{\rho}_{B}$
the function $\rho_{\frac{1}{2}}$. The next proposition shows that by sending
$B$ to zero (resp. infinity), $\tilde{\rho}_{B}$ converges to the fractional
diffusion equations of exponent $3/4$ $\left(\text{resp.}\;5/6\right)$.

######  Theorem 5.11.

Let $\overline{f}^{0}$ be the function defined in Eq. (66) and $D_{0}$ and
$D_{\infty}$ the constants defined in Eq. (69). Then

  * i)

$\lim\limits_{B\rightarrow 0}\tilde{\rho}_{B}=\rho_{0}$ in
$\mathbb{L}^{1}\left([0,T],\mathbb{L}^{2}(\mathbb{R})\right)$ where $\rho_{0}$
is the solution of

(70) $\forall t\in[0,T],\quad\forall
u\in{\mathbb{R}},\quad\left\\{\begin{array}[]{ll}\partial_{t}\rho_{0}(t,u)=-\Gamma\left(1+\frac{3}{2}\right)\gamma^{-\tfrac{1}{2}}D_{0}(-\Delta)^{\tfrac{3}{4}}\rho_{0}(t,u)\
,\\\ \rho_{0}(0,u)=\overline{f}^{0}(u)\ .\end{array}\right.$

  * ii)

$\lim\limits_{B\rightarrow+\infty}\tilde{\rho}_{B}\left(B^{\frac{1}{3}}\cdot,\cdot\right)=\rho_{\infty}$
in $\mathbb{L}^{1}\left([0,T],\mathbb{L}^{2}(\mathbb{R})\right)$ where
$\rho_{\infty}$ is the solution of

(71) $\forall t\in[0,T],\quad\forall
u\in{\mathbb{R}},\quad\left\\{\begin{array}[]{ll}\partial_{t}\rho_{\infty}(t,u)=-\Gamma\left(1+\frac{5}{3}\right)\gamma^{-\tfrac{2}{3}}D_{\infty}(-\Delta)^{\tfrac{5}{6}}\rho_{\infty}(t,u)\
,\\\ \rho_{\infty}(0,u)=\overline{f}^{0}(u)\ .\end{array}\right.$

###### Proof.

We refer the reader to Sec. 6.4. ∎

##  6\. Proof of Theorem 5.1, Theorem 5.7, Theorem 5.9 and Theorem 5.11

###  6.1. Sketch of the proof of Theorem 5.1

This section, is devoted to the proof of Theorem 5.1. The strategy of our
proof is similar to the one developed by the authors of [30], so we will only
give the ideas of the different steps and we refer the reader to the
corresponding sections of [30] for detailed proofs.

To prove Theorem 5.1, we first prove that there exists $\mathcal{W}$ in
${\mathcal{C}}\left([0,T],\left({\mathcal{S}}\times{\mathcal{S}}\right)^{\prime}\right)$
and a sequence $(\varepsilon_{n})_{n\in{\mathbb{N}}}$ such that for any $J$ in
$\mathcal{S}\times\mathcal{S}$ and $t$ in $[0,T]$

(72)
$\lim\limits_{n\rightarrow\infty}\left|\left\langle\mathcal{W}^{\varepsilon_{n}}(t),J\right\rangle-\left\langle\mathcal{W}(t),J\right\rangle\right|=0\
.$

Let $J:=(J_{1},J_{2})$ in ${\mathcal{S}}\times{\mathcal{S}}$, to prove Eq.
(72) we start to show that for any $i$ in $\\{1,2\\}$ and $t$ in $[0,T]$ we
have

$\displaystyle\partial_{t}\left\langle\mathcal{W}_{i}^{\varepsilon}(t),J_{i}\right\rangle$
$\displaystyle=\frac{1}{2\pi}\left\langle\mathcal{W}_{i}^{\varepsilon}(t),\textbf{v}_{0}\partial_{u}J_{i}\right\rangle+\gamma\left\langle\mathcal{W}_{i}^{\varepsilon}(t),[C_{0}J]_{i}\right\rangle$
(73)
$\displaystyle+\gamma\left(\langle\Gamma_{i}^{\varepsilon}(t),[C_{0}J]_{i}\rangle+\langle\tilde{\Gamma}_{i}^{\varepsilon}(t),[C_{0}J]_{i}\rangle\right)+\mathcal{O}_{t,J}(\varepsilon)\
,$

where $\Gamma^{\varepsilon}$ and $\left(\Gamma^{\varepsilon}\right)*$ are the
anti-Wigner’s distributions defined in Eq. (23) and Eq. (24) and
$|\mathcal{O}_{t,J}(t,\varepsilon)|$ $\leq K(J)\times\varepsilon$ with $K(J)$
a constant independent of $\varepsilon$ and $t$. The derivation of this
equation is presented in Appendix A.2 and Appendix A.3. To end the proof we
need the following lemma.

######  Lemma 6.1.

Let $t$ in $[0,T]$ and $J:=(J_{1},J_{2})$ in
${\mathcal{S}}\times{\mathcal{S}}$ then

  * i)

$\|\langle\Gamma^{\varepsilon},J\rangle\|_{\infty}\lesssim K_{0}\|J\|$ and
$\|\langle\tilde{\Gamma}^{\varepsilon},J\rangle\|_{\infty}\lesssim
K_{0}\|J\|.$

  * ii)

For any $i$ in $\\{1,2\\}$

$\lim\limits_{\varepsilon\rightarrow
0}\left|\int_{0}^{T}\langle\Gamma_{i}^{\varepsilon}(t),[C_{0}J]_{i}\rangle\right|=0\quad\text{and}\quad\lim\limits_{\varepsilon\rightarrow
0}\left|\int_{0}^{T}\langle\tilde{\Gamma}_{i}^{\varepsilon}(t),[C_{0}J]_{i}\rangle\right|=0\
.$

###### Proof.

We refer the reader to [4, Sec. 3.4]. ∎

By Eq. (73), item ii) of Lemma 3.2 and item i) of Lemma 6.1 we deduce that for
all $J$ in ${\mathcal{S}}\times{\mathcal{S}}$ the family of functions
$\left(\left\langle{\mathcal{W}}^{\varepsilon},J\right\rangle\right)_{\varepsilon>0}$
is equicontinuous and bounded in
$\left({\mathcal{C}}\left([0,T],{\mathbb{C}}\right),\|\cdot\|_{\infty}\right).$
Hence, for any $J$ in $\mathcal{S}\times\mathcal{S}$, there exists
$\mathcal{W}_{J}(\cdot)\in\left(\mathcal{C}\left([0,T],\mathbb{C}\right),\|\cdot\|_{\infty}\right)$
such that $\left\langle\mathcal{W}^{\varepsilon}(\cdot),J\right\rangle$
converges (up to a subsequence) to $\mathcal{W}_{J}(\cdot)$ in
$\left(\mathcal{C}\left([0,T],\mathbb{C}\right),\|\cdot\|_{\infty}\right)$.

At this point, the subsequence depends on $J$ but since
$\left(\mathcal{S}\times\mathcal{S},\|\cdot\|\right)$ is a separable space, we
can use a diagonal argument to erase this dependency. Hence, there exists a
countable subset $\mathcal{D}=(J^{r})_{r\in\mathbb{N}}$ $\subset$
$\mathcal{S}\times\mathcal{S}$ dense in
$\left(\mathcal{S}\times\mathcal{S},\|\cdot\|\right)$, a subsequence
$\left(\varepsilon_{n}\right)_{n\in{\mathbb{N}}}$ going to zero as $n$ goes to
infinity and a sequence
$\left(\mathcal{W}_{J}(\cdot)\right)_{J\in\mathcal{D}}$ $\in$
$\left(\mathcal{C}\left([0,T],\mathbb{C}\right),\|\cdot\|_{\infty}\right)$
such that

$\displaystyle\forall
J\in\mathcal{D},\quad\lim\limits_{n\rightarrow\infty}\langle\mathcal{W}^{\varepsilon_{n}}(\cdot),J\rangle=\mathcal{W}_{J}(\cdot)\
.$

Using item ii) of Lemma 3.2, we deduce that

$\forall t\in[0,T],\quad\forall
J\in\mathcal{D},\quad\left|{\mathcal{W}}_{J}(t)\right|\leq K_{0}\|J\|\ .$

Hence the application

$\displaystyle
J\in\mathcal{D}\subset\mathcal{S}\times\mathcal{S}\mapsto\mathcal{W}_{J}(\cdot)\in\left(\mathcal{C}\left([0,T],\mathbb{C}\right),\|\cdot\|_{\infty}\right)$

is Lipschitz and linear. It can be uniquely extended into a linear Lipschitz
function on $\mathcal{S}\times\mathcal{S}$. Hence, for any $t$ in $[0,T]$,

$\displaystyle J\in\mathcal{S}\times\mathcal{S}\mapsto\mathcal{W}_{J}(t)$

defines an element of $\left(\mathcal{S}\times\mathcal{S}\right)^{\prime}$
denoted by $\left\langle\mathcal{W}(t),J\right\rangle$. It remains to prove
that $\mathcal{W}(\cdot)$ belongs to
$\mathcal{C}\left([0,T],\left(\mathcal{S}\times\mathcal{S}\right)^{\prime}\right)$.

Let $s$ in $[0,T]$ and $\left(s_{p}\right)_{p\geq 0}$ in
$\left[0,T\right]^{\mathbb{N}}$ which converges to $s$ when $p$ goes to
infinity. Let $J$ in $\mathcal{S}\times\mathcal{S}$, $\eta>0$ and $J^{r}$ in
$\mathcal{D}$ such that

$\displaystyle\|J-J^{r}\|<\eta\ .$

We have that

$\displaystyle\left|\left\langle\mathcal{W}\left(s_{p}\right),J\right\rangle-\left\langle\mathcal{W}\left(s\right),J\right\rangle\right|$
$\displaystyle\leq\left|\left\langle\mathcal{W}\left(s_{p}\right)-\mathcal{W}(s),J^{r}\right\rangle\right|+\left|\left\langle\mathcal{W}\left(s_{p}\right),J-J^{r}\right\rangle\right|$
$\displaystyle+\left|\left\langle\mathcal{W}\left(s\right),J-J^{r}\right\rangle\right|$
$\displaystyle\leq\left|\left\langle\mathcal{W}\left(s_{p}\right)-\mathcal{W}(s),J^{r}\right\rangle\right|+2K_{0}\|J-J^{r}\|$
$\displaystyle\leq\left|\left\langle\mathcal{W}\left(s_{p}\right)-\mathcal{W}(s),J^{r}\right\rangle\right|+2K_{0}\eta\
.$

Recall that

(74) $\displaystyle
t\in[0,T]\mapsto\left\langle\mathcal{W},J^{r}\right\rangle=\mathcal{W}_{J^{r}}(t)\in\mathbb{C}$

is continuous because $J^{r}$ is in $\mathcal{D}$. Hence, it exists $p_{0}$ in
$\mathbb{N}^{*}$ such that

$\displaystyle\forall
p>p_{0},\quad\left|\left\langle\mathcal{W}\left(s_{p}\right)-\mathcal{W}(s),J^{r}\right\rangle\right|\leq\eta\
.$

Consequently, we have

$\displaystyle\left|\left\langle\mathcal{W}\left(s_{p}\right),J\right\rangle-\left\langle\mathcal{W}\left(s\right),J\right\rangle\right|\leq\left(2K_{0}+1\right)\eta\
.$

This proves that $\mathcal{W}(\cdot)$ is in
$\mathcal{C}\left([0,T],\left(\mathcal{S}\times\mathcal{S}\right)^{\prime}\right)$
and ends the proof.

Hence, we deduce that there exists ${\mathcal{W}}$ in
${\mathcal{C}}\left([0,T],\left({\mathcal{S}}\times{\mathcal{S}}\right)^{\prime}\right)$
such that $(\mathcal{W}^{\varepsilon})_{\varepsilon>0}$ converges pointwise
(up to a subsequence) to ${\mathcal{W}}$. Using item ii) of Lemma 6.1 and
sending $\varepsilon$ to zero in Eq. (73) we get that for any $t$ in $[0,T]$
and $J$ in ${\mathcal{S}}\times{\mathcal{S}}$

(75)
$\displaystyle\left\langle{\mathcal{W}}(t),J\right\rangle-\left\langle{\mathcal{W}}(0),J\right\rangle$
$\displaystyle=\int_{0}^{t}\frac{1}{2\pi}\left\langle{\mathcal{W}}(s),\textbf{v}_{0}\partial_{u}J\right\rangle
ds+\gamma\int_{0}^{t}\left\langle{\mathcal{W}}(s),C_{0}J\right\rangle ds\ .$

This proves that ${\mathcal{W}}$ is a weak solution of the Boltzmann’s
equation (27).

Let $t$ in $[0,T]$. It remains to extend ${\mathcal{W}}(t)$ into a couple
$\mu(t):=(\mu_{1}(t),\mu_{2}(t))$ of bounded positive measure on
${\mathbb{R}}\times\mathbb{T}$. This is proved in [30, Lemma D.1]. The idea is
to prove that the Wigner’s distribution is a positive linear form on
${\mathcal{C}}_{\textbf{c}}((\mathbb{R}\times\mathbb{T})^{2})$ and then by
using Riesz’s representation theorem we can extend ${\mathcal{W}}(t)$ into a
couple of positive measure on ${\mathbb{R}}\times\mathbb{T}$ denoted by
$\mu(t):=(\mu_{1}(t),\mu_{2}(t))$.

To end the proof, it remains to prove that for any $t$ in $[0,T]$ and $i$ in
$\\{1,2\\}$, $\mu_{i}(t)$ is a bounded measure on
${\mathbb{R}}\times\mathbb{T}$. Let $J$ in $\mathcal{S}$ defined by
$J(u,k):=J^{\lambda,r}(u)$ where

$\forall u\in{\mathbb{R}},\quad
J^{\lambda,r}(u):=\exp\left(-\frac{\lambda}{r^{2}-u^{2}}\right)\mathds{1}_{[-r,r]}(u),\quad\lambda>0,\quad
r>0\ .$

Let $\lambda$ be fixed, by using Eq. (20) and item ii) of Lemma 3.2 we get
that for any $i$ in $\\{1,2\\}$

$\lim\limits_{\varepsilon\rightarrow
0}\left\langle\mathcal{W}_{i}^{\varepsilon}(t),J^{\lambda,r}\right\rangle=\int_{\mathbb{R}\times\mathbb{T}}J^{\lambda,r}(u)\mu_{i}(t,du,dk)\leq
K_{0}\ .$

By sending first $\lambda$ to zero and then $r$ to infinity we get

$\displaystyle\mu_{i}\left(t,\mathbb{R},\mathbb{T}\right)\leq K_{0}\ .$

Hence, $\mu$ is a measure valued weak solution of the Boltzmann’s equation
(27) with initial condition $\mu^{0}:=(\mu_{1}^{0},\mu_{2}^{0})$. By Lemma
4.1, there is a unique measure valued weak solution of Eq. (27) hence we
deduce that $(\mathcal{W}^{\varepsilon})_{\varepsilon>0}$ converges pointwise
to $\mu:=(\mu_{1},\mu_{2})$ in
${\mathcal{F}}\left([0,T],({\mathcal{S}}\times{\mathcal{S}})^{\prime}\right)$.

###  6.2. Sketch of the proof of Theorem 5.7

Let $u$ in ${\mathbb{R}}$, to lighten the notations we define for each time
$t$, $Z_{u}^{N}(t):=N^{-1}Z_{u_{N}}^{B_{N}}\left(N^{\alpha_{\delta}}t\right)$
where we recall that $\alpha_{\delta}$ is defined in Eq. (61) and $u_{N}=Nu$.
The proof is divided into two steps.

We first show that the finite-dimensional distributions of $Z_{u}^{N}(\cdot)$
converge weakly under $\mathbb{P}_{\pi_{B_{N}}}$ to the finite-dimensional
distributions of a Lévy process $Z_{u}^{\delta}(\cdot)$ where for any positive
time $t$ and $\theta$ in ${\mathbb{R}}$

(76) $\mathbb{E}\left[e^{\mathbf{i}\theta
Z_{u}^{\delta}(t)}\right]=\exp\left(t\Phi_{\delta}(\theta)+\mathbf{i}\theta
u\right)\ .$

Here

(77)
$\Phi_{\delta}(\theta)=\int_{{\mathbb{R}}}\left(e^{\mathbf{i}r\theta}-1+\mathbf{i}\theta
r\mathds{1}_{\\{|r|<1\\}}\right)d\nu_{\delta}(r)\ ,$

where $\nu_{\delta}$ is defined in Eq. (61). Then we prove that we can extend
this result under $\mathbb{P}_{(k,i)}$ when $k$ is in $\mathbb{T}^{*}$ and $i$
is in $\\{1,2\\}$.

We start to show the first step, we follow the strategy of [21, Sec. 6] and
only prove that one-dimensional distribution of
$\left(Z^{N}_{u}(\cdot)\right)_{N\in{\mathbb{N}}}$ converges weakly to one-
dimensional distribution of $Z_{u}^{\delta}(\cdot)$. Let $t>0$, we define the
random integer $j_{N}(t)$ as follows

(78)
$\sum_{n=0}^{j_{N}(t)}\lambda_{B_{N}}\left(X_{n}^{B_{N}}\right)\tau_{n}^{N}\leq
t<\sum_{n=0}^{j_{N}(t)+1}\lambda_{B_{N}}\left(X_{n}^{B_{N}}\right)\tau_{n}^{N}\
,$

where we recall that for any integer $N$,
$\left(\tau_{n}^{N}\right)_{n\in{\mathbb{N}}}$ is a sequence of i.i.d random
variables independent of the Markov chain
$\left(X_{n}^{N}\right)_{n\in{\mathbb{N}}}$ and such that
$\tau_{0}^{N}\sim\mathcal{E}(1)$ and $\lambda_{B_{N}}$ is defined in Eq. (36).
Let $t\geq 0$, we define

$\displaystyle Y_{u}^{N}(t)$ $\displaystyle=u-N^{-1}\sum_{n=0}^{\lfloor
N^{\alpha_{\delta}}t\rfloor}\frac{\textbf{v}_{B_{N}}\left(K_{n}^{B_{N}}\right)}{2\pi}\lambda_{B_{N}}\left(X_{n}^{B_{N}}\right)\tau_{n}^{N}\
.$ $\displaystyle S_{N}(t)$
$\displaystyle=\frac{j_{N}(N^{\alpha_{\delta}}t)}{N^{\alpha_{\delta}}}=\inf\left\\{u>0\;\Bigg{|}\;\;\sum_{n=0}^{\lfloor
N^{\alpha_{\delta}}u\rfloor}\lambda_{B_{N}}\left(X_{n}^{B_{N}}\right)\tau_{n}^{N}>t\right\\}\
.$ $\displaystyle S(t)$ $\displaystyle=t\overline{\lambda}^{-1}=2\gamma t\ .$

Let $t$ be a positive time, we can decompose $Z_{u}^{N}(t)$ into two parts,
the first one is given by all the jumps made by the particle until time
$N^{\alpha_{\delta}}t$ and the second part is given by the displacement made
by the particle since the last jump. Hence, using the definition of
$\Psi_{B_{N}}$ given in Eq. (55) we can write

$\displaystyle
Z_{u}^{N}(t)=u-\sum_{n=0}^{j_{N}\left(N^{\alpha_{\delta}}t\right)-1}\tfrac{\Psi_{B_{N}}\left(X_{n}^{B_{N}}\right)\tau_{n}^{N}}{2\pi
N}-\left(\tfrac{N^{\alpha_{\delta}}t-t_{j_{N}(N^{\alpha_{\delta}}t)}}{2\pi
N}\right)\textbf{v}_{B_{N}}\left(K_{j_{N}\left(N^{\alpha_{\delta}}t\right)}^{B_{N}}\right)\
.$ $\displaystyle
Y_{u}^{N}\left(S_{N}(t)\right)=u-\sum_{n=0}^{j_{N}\left(N^{\alpha_{\delta}}t\right)}\tfrac{\Psi_{B_{N}}\left(X_{n}^{B_{N}}\right)\tau_{n}^{N}}{2\pi
N}\ .$

We first prove that $Y_{u}^{N}(S_{N}(t))$ converges weakly to
$Z_{u}^{\delta}(t)$, then we prove that $Z_{u}^{N}(t)$ and
$Y_{u}^{N}(S_{N}(t))$ are close.

######  Proposition 6.2.

Let $\nu_{\delta}$ defined in Eq. (62). Then, under $\mathbb{P}_{\pi_{B_{N}}}$
the finite-dimensional distributions of
$\left(Y_{u}^{N}((\cdot)\right)_{N\in{\mathbb{N}}}$ converge weakly to the
finite dimensional distributions of a Lévy process $Y_{u}^{\delta}$ where for
any positive time $t$ and $\theta$ in ${\mathbb{R}}$

(79) $\mathbb{E}\left[e^{\mathbf{i}\theta
Y_{u}(t)}\right]=\exp\left((2\gamma)^{-1}t\Phi_{\delta}(\theta)+\mathbf{i}\theta
u\right)\ ,$

where the expression of $\Phi_{\delta}$ is recalled in Eq. (77).

###### Proof.

We refer the reader to Appendix C.2. ∎

To complete the proof of Theorem 5.7 we need the following two lemmas.

######  Lemma 6.3.

Let $t_{0}>0$ and $T\geq t_{0}$ then

(80)
$\forall\varepsilon>0,\quad\lim\limits_{N\rightarrow\infty}\mathbb{P}_{\pi_{B_{N}}}\left[\sup_{t\in[t_{0},T]}|S_{N}(t)-S(t)|>\varepsilon\right]=0\
.$

###### Proof.

We refer the reader to Appendix D. ∎

######  Lemma 6.4.

For any time $t$ and $\varepsilon>0$ be fixed we have

(81)
$\lim\limits_{N\rightarrow\infty}\mathbb{P}_{\pi_{B_{N}}}\left[\left|Z_{u}^{N}(t)-Y_{u}^{N}\left(S_{N}(t)\right)\right|>\varepsilon\right]=0\
.$

###### Proof of Lemma 6.4.

We recall only the main ideas and we refer the reader to [21, Lemma 6.2] for a
complete proof. Let $\sigma>0$ be fixed, then by Lemma 6.3 we have

$\displaystyle\mathbb{P}_{\pi_{B_{N}}}\left[\left|Z_{u}^{N}(t)-Y_{u}^{N}\left(S_{N}(t)\right)\right|>\varepsilon\right]$
$\displaystyle\leq{\mathbb{P}}_{\pi_{B_{N}}}\left[\left|S_{N}(t)-t\right|>\sigma\right]$
$\displaystyle+{\mathbb{P}}_{\pi_{B_{N}}}\left[\left|S_{N}(t)-t\right|\leq\sigma\cap\left|Y_{N}(t)-Y_{u}^{N}\left(S_{N}(t)\right)\right|>\varepsilon\right]$
$\displaystyle\leq{\mathbb{P}}_{\pi_{B_{N}}}\left[\left|S_{N}(t)-t\right|\leq\sigma\
,\
\left|\tfrac{\Psi_{B_{N}}\left(X_{j_{N}\left(N^{\alpha_{\delta}}t\right)}^{B_{N}}\right)}{N}\right|>\varepsilon\right]+\sigma\
.$

Using the stationarity of the chain
$\left(\Psi_{B_{N}}\left(X_{n}^{B_{N}}\right)\tau_{n}^{N}\right)_{n\in{\mathbb{N}}}$
we have

$\displaystyle{\mathbb{P}}_{\pi_{B_{N}}}\left[\left|S_{N}(t)-t\right|\leq\sigma\
,\
\left|N^{-1}\Psi_{B_{N}}\left(X_{j_{N}(N^{\alpha_{\delta}}t)}^{B_{N}}\right)\right|>\varepsilon\right]$
$\displaystyle\leq{\mathbb{P}}_{\pi_{B_{N}}}\left[\sup\left\\{\Psi_{B_{N}}\left(X_{n}^{B_{N}}\right)\tau_{n}^{N},\;n\in\left[(S(t)-\sigma)N^{\alpha_{\delta}},(S(t)+\sigma)N^{\alpha_{\delta}}\right]\right\\}>N\varepsilon\right]$
$\displaystyle={\mathbb{P}}_{\pi_{B_{N}}}\left[\sup\left\\{\Psi_{B_{N}}\left(X_{n}^{B_{N}}\right)\tau_{n}^{N},\;n\in\left[0,2\sigma
N^{\alpha_{\delta}}\right]\right\\}>N\varepsilon\right]$
$\displaystyle=N^{\alpha_{\delta}}\int_{0}^{+\infty}e^{-\tau}\pi_{B_{N}}\left(\left\\{u\;\big{|}\;\Psi_{B_{N}}(u)>N\tau^{-1}\varepsilon\right\\}\right)d\tau$
$\displaystyle\lesssim\frac{\sigma}{\varepsilon^{\alpha_{\delta}}}\ .$

This ends the proof. ∎

We recall that if a sequence $(x_{n})_{n\in{\mathbb{N}}}$ in
$\mathcal{D}\left([t_{0},T],\mathbb{R}\right)$ converges uniformly to $x$ in
$\mathcal{D}\left([t_{0},T],\mathbb{R}\right)$ then the convergence follows in
the sense of Skorohod topology. Hence, from Lemma 6.3, we conclude that
$S_{N}(\cdot)$ converges in probability to $S(\cdot)$ in
$\mathcal{D}\left([t_{0},T],\mathbb{R}\right)$.

Using Proposition 6.2 and Lemma 6.3 we can conclude that
$\left(Y_{u}^{N}(\cdot),S_{N}(\cdot)\right)$ converges in law to
$(Y_{u}^{\delta}(\cdot),S(\cdot))$ in $\mathcal{D}([t_{0},T],\mathbb{R})$.
Hence, by Skorokhod’s representation theorem there exists
$\left(\tilde{Y}_{u}^{N}(\cdot),S_{N}(\cdot)\right)$ with values in
$\mathcal{D}([0,\infty)\times[0,\infty))$ such that the pair
$\left(Y_{u}^{N}(\cdot),S_{N}(\cdot)\right)$ and
$\left(\tilde{Y}_{u}^{\delta}(\cdot),\tilde{S}_{N}(\cdot)\right)$ have the
same law and $\left(\tilde{Y}_{u}^{N}(\cdot),\tilde{S}_{N}(\cdot)\right)$
converges almost surely to $\left(Y_{u}^{\delta}(\cdot),S(\cdot)\right)$ in
the Skorokhod topology.

From this we deduce that there exists a sequence of increasing homeomorphisms
$(h_{N})_{N\in{\mathbb{N}}}$ from $[t_{0},T]$ to $[t_{0},T]$ such that

(82)
$\displaystyle\lim\limits_{N\rightarrow\infty}\sup_{t\in[t_{0},T]}\left|h_{N}(t)-t\right|$
$\displaystyle=0\quad\text{and}\quad\lim\limits_{N\rightarrow\infty}\sup_{t\in[t_{0},T]}\left|\tilde{Y}_{u}^{N}\left(h_{N}(t)\right)-Y^{\delta}_{u}\left(S(t)\right)\right|=0\
.$

Then we have

$\displaystyle\left|\tilde{Y}_{u}^{N}\left(\tilde{S}_{N}(t)\right)-Y_{u}^{\delta}\left(S(t)\right)\right|$
$\displaystyle\leq\left|\tilde{Y}_{u}^{N}\left(\tilde{S}_{N}(t)\right)-Y_{u}^{\delta}\left(h_{N}^{-1}\left(\tilde{S}_{N}(t)\right)\right)\right|$
$\displaystyle+\left|Y_{u}^{\delta}\left(h_{N}^{-1}\left(\tilde{S}_{N}(t)\right)\right)-Y_{u}^{\delta}\left(S(t)\right)\right|\
,$

which goes to zero when $N$ goes to infinity. Hence,
$\left(\tilde{Y}_{u}^{N}\left(\tilde{S}_{N}(t)\right)\right)_{N\in{\mathbb{N}}}$
converges to $Y_{u}^{\delta}(S(t))$. Using that
$\tilde{Y}_{u}^{N}\left(\tilde{S}_{N}(t)\right)$ and
$Y_{u}^{N}\left(\tilde{S}_{N}(t)\right)$ have the same law we conclude that
$\left(Y_{u}^{N}\left(\tilde{S}_{N}(t)\right)\right)_{N\in{\mathbb{N}}}$
converges in law to $Y_{u}^{\delta}(S(t))$. Using Eq. (81) we conclude that
$\left(Z_{u}^{N}(t)\right)_{N\geq 0}$ converges weakly to
$Y_{u}^{\delta}(S(t))$. Observe that $Y_{u}^{\delta}(S(t))$ and
$Z_{u}^{\delta}(t)$ have the same law hence we proved that under
${\mathbb{P}}_{\pi_{B_{N}}}$ the finite-dimensional distribution of
$\left(Z_{u}^{N}(\cdot)\right)_{N\geq 0}$ converge to the finite-dimensional
distribution of $Z_{u}^{\delta}(\cdot)$.

To conclude the proof, it remains to extend this convergence under
$\mathbb{P}_{(k,i)}$ when $k\neq 0$ and $i$ is in $\\{1,2\\}$. Since $k\neq 0$
using Eq. (55) we have

$\lim\limits_{N\rightarrow\infty}N^{-1}\Psi_{B_{N}}\left(X_{0}^{B_{N}}\right)=\lim\limits_{N\rightarrow\infty}N^{-1}\Psi_{B_{N}}\left(k,i\right)=0\;\;\;\mathbb{P}_{(k,i)}\;\;\;\text{a.s.}$

Hence, we deduce that
$\left(N^{-1}\Psi_{B_{N}}\left(X_{0}^{B_{N}}\right)\tau_{0}^{N}\right)_{N\in{\mathbb{N}}}$
goes to zero in $\mathbb{L}_{{\mathbb{P}}_{(k,i)}}^{1}$. From Markov
inequality we deduce that

(83)
$\forall\varepsilon>0,\quad\lim\limits_{N\rightarrow\infty}{\mathbb{P}}_{(k,i)}\left(N^{-1}\left|\Psi_{B_{N}}\left(X_{0}^{B_{N}}\right)\right|\tau_{0}^{N}>\varepsilon\right)=0\
.$

We define the stochastic process $\tilde{Z}_{u}^{N}(\cdot)$ for any positive
time $t$ by

$\displaystyle{\tilde{Z}}_{u}^{N}(t)=u-\sum_{n=1}^{j_{N}(N^{\alpha_{\delta}}t)-1}\tfrac{\Psi_{B_{N}}\left(X_{n}^{B_{N}}\right)\tau_{n}^{N}}{2\pi
N}-\left(\tfrac{N^{\alpha_{\delta}}t-t_{j_{N}\left(N^{\alpha_{\delta}}t\right)}}{2\pi
N}\right)\textbf{v}_{B_{N}}\left(K_{j_{N}\left(N^{\alpha_{\delta}}t\right)}^{B_{N}}\right)\
.$

Whatever the distribution of $X_{0}^{B_{N}}$ is, by using Eq. (41), we observe
that $\left(X_{n}^{B_{N}}\right)_{n\geq 1}$ is a sequence of i.i.d random
variables distributed according to $\pi_{B_{N}}$. Hence, since
$\tilde{Z}_{u}^{N}(\cdot)$ depends only on $\left(X_{n}^{B_{N}}\right)_{n\geq
1}$ we deduce by Proposition 6.2 that under ${\mathbb{P}}_{(k,i)}$ the finite-
dimensional distributions of $\tilde{Z}_{u}^{N}(\cdot)$ converge weakly to the
finite-dimensional distributions of the Lévy process $Z_{u}^{\delta}(\cdot)$
defined in Eq. (77).
Using Eq. (83) we deduce that for any positive time $t$

(84)
$\forall\varepsilon>0,\quad\lim\limits_{N\rightarrow\infty}{\mathbb{P}}_{(k,i)}\left(\left|Z_{u}^{N}(t)-\tilde{Z}_{u}^{N}(t)\right|>\varepsilon\right)=0\
.$

This concludes the proof.

###  6.3. Sketch of the proof of Theorem 5.9

In this section, we prove Theorem 5.9, to prove it we need the following
lemma.

######  Lemma 6.5.

Let $\rho$ be the counting measure on $\\{1,2\\}$. Observe that the measure
$\sigma:=\frac{1}{2}dkd\rho$ is an invariant measure for the process
$\left(K^{B_{N}}(\cdot),I^{B_{N}}(\cdot)\right)$. Let $a\in(0,1)$ such that

(85)
$\sup_{N}\int_{\mathbb{T}\times\\{1,2\\}}\lambda^{a}_{B_{N}}(k,i)d\sigma(k,i)<\infty\
,$

where $\lambda_{B_{N}}$ is defined in Eq. (36) and define
$d\sigma_{N}^{a}:=\lambda_{B_{N}}^{a}d\sigma$. Then for any centered $f$ in
$\mathbb{L}^{1}\left(\sigma\right)$ and positive time $t$

$\left\|P_{N}^{t}f\right\|_{\mathbb{L}^{1}\left(\sigma\right)}\leq\frac{C}{(1+t)^{a}}\|f\|_{\mathbb{L}^{1}(\sigma^{a}_{N})}\
,$

where $C$ is a positive constant which dos not depends on $N$ and $P_{N}$ is
the semi-group associated to the process
$\left(K^{B_{N}}(\cdot),I^{B_{N}}(\cdot)\right)$.

###### Proof.

This result is proved in a general setting in [24, Theorem 1.1]. In our
context, following the proof of [24] we can check that $C$ does not depend on
$N$. ∎

We follow the strategy developed in [21] and [30]. The fundamental tool is
Theorem 5.7. Let $\left(K^{B_{N}}(0),I^{B_{N}}(0)\right)=(k,i)$ with $k\neq 0$
and $i$ in $\\{1,2\\}$. By definition for any positive time $t$, $u$ in
${\mathbb{R}}$, $i$ in $\\{1,2\\}$ and $k$ in $\mathbb{T}$

$\displaystyle
f_{B_{N}}\left(N^{\alpha}t,Nu,k,i\right)=\mathbb{E}_{(k,i)}\left[f^{0}\left(Z_{u}^{N}(t),K^{B_{N}}\left(N^{\alpha}t\right),I^{B_{N}}\left(N^{\alpha}t\right)\right)\right]\
,$

where we recall that

$\displaystyle
Z_{u}^{N}(t):=N^{-1}Z_{u_{N}}^{B_{N}}\left(N^{\alpha_{\delta}}t\right)=u-\frac{1}{2\pi
N}\int_{0}^{N^{\alpha}t}ds\textbf{v}_{B_{N}}\left(K^{B_{N}}(s)\right)\ .$

Let $(m_{N})_{n\in{\mathbb{N}}}$ be an increasing sequence of positive numbers
such that

(86)
$\displaystyle\lim\limits_{N\rightarrow\infty}m_{N}=+\infty\quad\text{and}\quad\lim\limits_{N\rightarrow\infty}\frac{m_{N}}{N}=0\
.$

Since $X_{0}^{B_{N}}=\left(K^{B_{N}}(0),I^{B_{N}}(0)\right)=(k,i)$ with $k\neq
0$ by Theorem 5.7 under $\mathbb{P}_{(k,i)}$ the finite-dimensional
distributions of
$Z_{u}^{N}\left(\left[1-N^{-\alpha_{\delta}}m_{N}\right]\cdot\right)$ converge
weakly to the finite-dimensional distributions of a Lévy process
$Z_{u}^{\delta}(\cdot)$ generated by $\mathfrak{L}_{\delta}$ where
$\mathfrak{L}_{\delta}$ is defined in Eq. (67).

For any $i$ in $\\{1,2\\}$,
$f^{0}(i)\in{\mathcal{C}}_{c}^{\infty}\left({\mathbb{R}}\times\mathbb{T}\right)$
hence $\overline{f}^{0}$ is in ${\mathcal{C}}_{c}^{\infty}({\mathbb{R}})$
where $\overline{f}^{0}$ is defined in Eq. (66). We have then

(87)
$\frac{1}{2}\lim\limits_{N\rightarrow\infty}\mathbb{E}_{(k,i)}\left[\overline{f}^{0}\left(Z_{u}^{N}(\left[1-N^{-\alpha_{\delta}}m_{N}\right]t)\right)\right]=\frac{1}{2}\rho_{\delta}(u,t)\quad\mathbb{P}.a.s\
,$

where $\rho_{\delta}$ is the unique solution of Eq. (68). By using Fourier’s
inverse formula we have

$\displaystyle f_{B_{N}}\left(N^{\alpha_{\delta}}t,Nu,k,i\right)$
$\displaystyle=\mathbb{E}_{(k,i)}\left[f^{0}\left(Z_{u}^{N}\left(t\right),K^{B_{N}}\left(N^{\alpha_{\delta}}t\right),I^{B_{N}}\left(N^{\alpha_{\delta}}t\right)\right)\right]$
$\displaystyle=\sum_{r\in{\mathbb{Z}}}\int_{\mathbb{R}}dp\sum_{j=1}^{2}\mathcal{F}[\tilde{f}^{0}](p,r,j)\mathbb{E}_{(k,i)}\left[e^{\mathbf{i}\left(pZ_{u}^{N}\left(t\right)+rK^{B_{N}}\left(N^{\alpha_{\delta}}t\right)\right)}\mathds{1}_{I^{B_{N}}\left(N^{\alpha_{\delta}}t\right)=j}\right]\
,$

where $\tilde{f}^{0}(i)$ is the function defined for any $z$ in $\mathbb{R}$,
$i$ in $\\{1,2\\}$, by

(88)
$f^{0}(z,k,i):=\sum_{r\in{\mathbb{Z}}}\tilde{f}^{0}(z,r,i)\exp\left(2\mathbf{i}\pi
kr\right).$

By using Fourier’s inverse formula we get

$\displaystyle\frac{1}{2}\mathbb{E}_{(k,i)}\left[\overline{f}^{0}\left(Z_{u}^{N}\left(t-N^{-\alpha_{\delta}}m_{N}t\right)\right)\right]$
$\displaystyle=\sum_{j=1}^{2}\mathbb{E}_{(k,i)}\left[\frac{1}{2}\int_{\mathbb{T}\times\\{1,2\\}}dk^{\prime}d\rho(j)f^{0}\left(Z_{u}^{N}\left(t-N^{\alpha_{\delta}}m_{N}t\right),k^{\prime},j\right)\right]$
$\displaystyle=\sum_{j=1}^{2}\sum_{r\in{\mathbb{Z}}}\int_{{\mathbb{R}}}dp\mathcal{F}[\tilde{f}^{0}](p,r,j)\mathbb{E}_{(k,i)}\left[e^{\mathbf{i}pZ_{u}^{N}\left(t-N^{-\alpha_{\delta}}m_{N}t\right)}\frac{1}{2}\int_{\mathbb{T}}dk^{\prime}e^{\mathbf{i}k^{\prime}}\right]\
.$

By using Eq. (87) and the dominated convergence theorem we get

$\displaystyle\limsup_{N\rightarrow\infty}\int_{\mathbb{T}\times\\{1,2\\}}dkd\rho(i)\left|f_{B_{N}}\left(N^{\alpha_{\delta}}t,Nu,k,i\right)-\frac{1}{2}\rho_{\delta}(u,t)\right|$
$\displaystyle\leq\limsup_{N\rightarrow\infty}\int_{\mathbb{T}\times\\{1,2\\}}dkd\rho(i)\left|A_{i,r,u,k,N}\right|\
,$

where

$A_{i,r,u,k,N}:=f_{B_{N}}\left(N^{\alpha_{\delta}}t,Nu,k,i\right)-\frac{1}{2}\mathbb{E}_{(k,i)}\left[\overline{f}^{0}\left(Z_{u}^{N}\left(\left[1-N^{-\alpha_{\delta}}m_{N}\right]t\right)\right)\right]\
.$

By using Fourier’s inverse formula we get

$\displaystyle\left|A_{i,r,u,k,N}\right|$
$\displaystyle\leq\sum_{r\in{\mathbb{Z}}}\int_{{\mathbb{R}}}dp\sum_{j=1}^{2}\left|\mathcal{F}[\tilde{f}^{0}_{j}](p,r)\right|\left(\left|I_{1}^{N}(t,k,i,j,p,r)\right|+\left|I_{2}^{N}(t,k,i,j,p,r)\right|\right)\
.$

where

$\displaystyle I_{1}^{N}(t,k,i,j,p,r)$
$\displaystyle=\mathbb{E}_{(k,i)}\left[e^{\mathbf{i}\left(pZ_{u}^{N}\left(t\right)+rK^{B_{N}}\left(N^{\alpha_{\delta}}t\right)\right)}\mathds{1}_{I^{B_{N}}\left(N^{\alpha_{\delta}}t\right)=j}\right]$
$\displaystyle-\mathbb{E}\left[e^{\mathbf{i}\left(pZ_{u}^{N}\left(t-N^{-\alpha_{\delta}}m_{N}t\right)+rK^{B_{N}}\left(N^{\alpha_{\delta}}t\right)\right)}\mathds{1}_{I^{B_{N}}\left(N^{\alpha_{\delta}}t\right)=j}\right]\
,$

and

$\displaystyle I_{2}^{N}(t,k,i,j,p,r)$
$\displaystyle=\mathbb{E}_{(k,i)}\left[e^{\mathbf{i}\left(pZ_{u}^{N}\left(t-N^{-\alpha_{\delta}}m_{N}t\right)+rK^{B_{N}}\left(N^{\alpha_{\delta}}t\right)\right)}\mathds{1}_{I^{B_{N}}\left(N^{\alpha_{\delta}}t\right)=j}\right]$
$\displaystyle-\mathbb{E}\left[e^{\mathbf{i}pZ_{u}^{N}\left(t-N^{-\alpha_{\delta}}m_{N}t\right)}\frac{1}{2}\int_{\mathbb{T}}e^{\mathbf{i}k^{\prime}}dk^{\prime}\right]\
.$

To conclude the proof it is sufficient to show that for any positive time $t$,
$k\neq 0$, $(i,j)$ in $\\{1,2\\}^{2}$, $p$ in ${\mathbb{R}}$ and $r$ in
${\mathbb{Z}}$ we have

(89) $\forall
q\in\\{1,2\\},\quad\lim\limits_{N\rightarrow\infty}\left|I_{q}^{N}(t,k,i,j,p,r)\right|=0\
.$

We start to deal with $I_{1}^{N}$.
We recall that for any $a$ in $\mathbb{R}$ we have
$\left|1-e^{\mathbf{i}a}\right|\leq|a|$ and that the function
$\textbf{v}_{B_{N}}$ is a bounded function independently of $N$ (see Eq.
(15)). Hence, for any positive $t$, $p$ in ${\mathbb{R}}$, $r$ in
${\mathbb{Z}}$ and $j$ in $\\{1,2\\}$ we have

$\displaystyle\left|I_{1}^{N}(t,k,i,j,p,r)\right|$
$\displaystyle\leq\mathbb{E}_{k,i}\left[\left|1-e^{\mathbf{i}p\left(Z_{u}^{N}\left(t\right)-Z_{u}^{N}\left(t-N^{-\alpha_{\delta}}m_{N}t\right)\right)}\right|\right]$
$\displaystyle\leq\mathbb{E}_{k,i}\left[\left|p\left(Z_{u}^{N}(t)-Z_{u}^{N}\left(t-N^{-\alpha_{\delta}}m_{N}t\right)\right)\right|\right]$
$\displaystyle\leq\left\|\textbf{v}_{B_{N}}\right\|_{\infty}t|p|m_{N}N^{-1}\
,$

which goes to zero when $N$ goes to infinity by definition of the sequence
$(m_{N})_{N\in{\mathbb{N}}}$ $\left(\text{cf.
Eq.}\;\eqref{def_sequence_m_N}\right)$. This proves Eq. (89) for $i=1$. It
remains to prove the convergence of $I_{2}^{N}$. Using Markov Property we get

$\displaystyle\mathbb{E}_{(k,i)}\left[e^{\mathbf{i}\left(pZ_{u}^{N}\left(t-N^{-\alpha_{\delta}}m_{N}t\right)+rK^{B_{N}}\left(N^{\alpha_{\delta}}t\right)\right)}\mathds{1}_{I^{B_{N}}\left(N^{\alpha_{\delta}}t\right)=j}\right]$
$\displaystyle=\mathbb{E}_{(k,i)}\left[e^{\mathbf{i}\left(pZ_{u}^{N}\left(t-N^{-\alpha_{\delta}}m_{N}t\right)\right)}\mathbb{E}_{\left(K^{B_{N}}\left(N^{\alpha_{\delta}}t-m_{N}t\right),I^{B_{N}}\left(N^{\alpha_{\delta}}t-m_{N}t\right)\right)}\left[e^{\mathbf{i}\left(rK^{B_{N}}(m_{N}t)\right)}\mathds{1}_{I^{B_{N}}(m_{N}t)=j}\right]\right]\
.$

Hence, by Markov Property we get

$\displaystyle\left|I_{2}^{N}(t,k,i,j,p,r)\right|\leq\mathbb{E}_{(k,i)}\left[\left|\mathbb{E}_{\left(K^{B_{N}}\left(N^{\alpha_{\delta}}t-m_{N}t\right),I^{B_{N}}\left(N^{\alpha_{\delta}}t-m_{N}t\right)\right)}g_{r}\left(K^{B_{N}},I^{B_{N}}\right)\right|\right],$

where

$\displaystyle
g_{r}\left(K^{B_{N}},I^{B_{N}}\right)=e^{\mathbf{i}rK^{B_{N}}(m_{N}t)}\mathds{1}_{I^{B_{N}}(m_{N}t)=j}-\frac{1}{2}\int_{\mathbb{T}}e^{\mathbf{i}rk^{\prime}}dk^{\prime}\
.$

Let $r$ in ${\mathbb{N}}$ and $g_{r}$ the centered function defined for any
$k$ in $\mathbb{T}$ and $i$ in $\\{1,2\\}$ by

$g_{r}(k,i)=e^{\mathbf{i}rk}\mathds{1}_{j}(i)-\frac{1}{2}\int_{T}e^{\mathbf{i}rk^{\prime}}dk^{\prime}.$

Using the fact that $\frac{1}{2}dkd\rho(i)$ is the reversible probability
measure of the stochastic process
$\left(K^{B_{N}}(\cdot),I^{B_{N}}(\cdot)\right)$ we get

$\displaystyle\mathbb{E}_{(k,i)}\left[\left|\mathbb{E}_{\left(K^{B_{N}}\left(N^{\alpha_{\delta}}t-m_{N}t\right),I^{B_{N}}\left(N^{\alpha_{\delta}}t-m_{N}t\right)\right)}\left[e^{\mathbf{i}rK^{B_{N}}(m_{N}t)}\mathds{1}_{I^{B_{N}}(m_{N}t)=j}-\frac{1}{2}\int_{\mathbb{T}}e^{\mathbf{i}rk^{\prime}}dk^{\prime}\right]\right|\right]$
$\displaystyle=\left\|P_{N}^{m_{N}t}g\right\|_{L^{1}(\sigma)}\ ,$

which goes to zero when $N$ goes to infinity according to Lemma 6.5. This ends
the proof.

###  6.4. Sketch of the proof of Theorem 5.11

We recall that $\tau_{0}\sim{\mathcal{E}}\left(1\right)$. By applying
Fourier’s formula to Eq. (68) we get for any $p$ in $\mathbb{R}$ and positive
time $t$ in $[0,T]$

(90)
$\mathcal{F}\left[\tilde{\rho}_{B}\right](t,p)=\exp\left(t\Phi_{B}(p)\right)\mathcal{F}\,\left[\overline{f}^{0}\right](p)\
,$

where we recall that for any $p$ in $\mathbb{R}$

$\displaystyle\Phi_{B}(p)$
$\displaystyle=2\int_{0}^{\infty}\left(\cos(pr)-1\right)\left(\mathbb{E}\left[\tau_{0}^{-1}\left(g_{B,+}\left(\frac{2\pi
r}{\tau_{0}}\right)+g_{B,-}\left(\frac{2\pi
r}{\tau_{0}}\right)\right)\right]\right)dr\ ,$

since $g_{B,\pm}$ are even functions by item i) of Proposition 5.5. Observe
that $\Phi_{B}$ is negative, to conclude the proof we need the following Lemma
the proof of which is left to the readers.

######  Lemma 6.6.

There exists functions $h_{\pm}$ and $f_{\pm}$ such that for almost every $r$

(91) $\displaystyle\forall B>1,\quad B^{\frac{1}{3}}g_{B,\pm}(r)<f_{\pm}(r)\
,$ (92) $\displaystyle\forall B<1,\quad g_{B,\pm}(r)<h_{\pm}(r)\ ,$

where

(93)
$\displaystyle\int_{0}^{1}f_{\pm}(r)r^{2}dr<\infty\quad\text{and}\quad\int_{1}^{\infty}f_{\pm}(r)dr<\infty\
,$ (94)
$\displaystyle\int_{0}^{1}h_{\pm}(r)r^{2}dr<\infty\quad\text{and}\quad\int_{1}^{\infty}h_{\pm}(r)dr<\infty\
.$

By using item iii), item iv) of Proposition 5.5, Lemma 6.6 and the dominated
convergence theorem we get that for almost every $p$ in ${\mathbb{R}}$

$\displaystyle\lim\limits_{B\rightarrow 0}\Phi_{B}(p)$
$\displaystyle=-\Gamma\left(1+\frac{3}{2}\right)\gamma^{-\tfrac{1}{2}}D_{0}|p|^{\frac{3}{2}}\
,$ $\displaystyle\lim\limits_{B\rightarrow\infty}B^{\frac{1}{3}}\Phi_{B}(p)$
$\displaystyle=-\Gamma\left(1+\frac{5}{3}\right)\gamma^{-\tfrac{2}{3}}D_{\infty}|p|^{\frac{5}{3}}\
,$

where $D_{0}$ and $D_{\infty}$ are defined in Eq. (69). Since the proofs of
item i) and ii) are similar we only prove item i). By applying Fourier’s
formula to Eq. (70) we obtain

(95)
$\mathcal{F}\left[\rho_{0}\right](t,p)=\exp\left(-t\Gamma\left(1+\frac{3}{2}\right)\gamma^{-\tfrac{1}{2}}D_{0}|p|^{\tfrac{3}{2}}\right)\mathcal{F}\left[\overline{f}^{0}\right](p)\
.$

We recall that $\overline{f}^{0}$ is in
$\mathcal{C}_{c}^{\infty}\left(\mathbb{R}\right)$ hence is in
$\mathbb{L}^{2}(\mathbb{R})$. By the dominated convergence theorem we deduce
that for any positive time $t$ in $[0,T]$

$\displaystyle\lim\limits_{B\rightarrow
0}\|\tilde{\rho}_{B}(t,\cdot)-\rho_{0}(t,\cdot)\|_{\mathbb{L}^{2}(\mathbb{R})}^{2}$
$\displaystyle=\lim\limits_{B\rightarrow
0}\|\mathcal{F}\left[\tilde{\rho}_{B}\right](t,\cdot)-\mathcal{F}\left[\rho_{0}\right](t,\cdot)\|_{\mathbb{L}^{2}(\mathbb{R})}^{2}$
$\displaystyle=\lim\limits_{B\rightarrow
0}\int_{{\mathbb{R}}}\left|\mathcal{F}\,\left[\overline{f}^{0}\right](p)\left(\exp\left(t\Phi_{B}(p)\right)-\exp\left(-t\Gamma\left(1+\tfrac{3}{2}\right)\gamma^{-\tfrac{1}{2}}D_{0}|p|^{\frac{3}{2}}\right)\right)\right|^{2}dp$
$\displaystyle=0\ .$

This proves that for each positive time $t$, $\rho_{B}(t,\cdot)$ converges in
$\mathbb{L}^{2}({\mathbb{R}})$ to $\rho_{0}(t,\cdot)$. By the dominated
convergence theorem we conclude the proof.

##  Appendix A Proof of the evolution equation of the Wigner’s distribution

Let $i$ in $\\{1,2\\}$. We start to prove that the integral in Eq. (21) is
well defined. In order to lighten the notations we denote
$\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},k\pm\frac{\varepsilon
p}{2}\right)$ by $\widehat{\psi}^{\varepsilon}_{i}\left(k\pm\frac{\varepsilon
p}{2}\right)$. Let $J:=(J_{1},J_{2})$ in ${\mathcal{S}}\times{\mathcal{S}}$
and $i$ in $\\{1,2\\}$, by using Cauchy-Schwarz’s inequality, Fubini’s theorem
and the periodicity of $\widehat{\psi}^{\varepsilon}_{i}$ we have

$\displaystyle\frac{\varepsilon}{2}\int_{\mathbb{R}}dp\left|{\mathcal{F}}\left[J_{i}\right](p,k)\right|\int_{\mathbb{T}}dk\;\;\mathbb{E}_{\mu^{\varepsilon}}\left[\left|\widehat{\psi}^{\varepsilon}_{i}\left(k+\frac{\varepsilon
p}{2}\right)\widehat{\psi}^{\varepsilon}_{i}\left(k-\frac{\varepsilon
p}{2}\right)^{*}\right|\right]$
$\displaystyle\leq\frac{\varepsilon}{2}\left(\int_{\mathbb{R}}dp\sup_{k\in\mathbb{T}}\left|{\mathcal{F}}\left[J_{i}\right](p,k)\right|\right)\left(\mathbb{E}_{\mu^{\varepsilon}}\left[\int_{\mathbb{T}}dk\left|\widehat{\psi}^{\varepsilon}_{i}\left(k\right)\right|^{2}\right]\right)$
$\displaystyle\leq K_{0}\|J\|\ ,$

where we used Eq. (20) to obtain the last inequality. This proves the
existence of the integral in Eq. (21). The others expressions of the Wigner’s
distribution presented in Sec. 3.4 are consequences of Fubini’s theorem and
Fourier’s inverse formula. Observe that we proved in fact

(96) $\displaystyle\forall
t\in[0,T],\quad\left|\left\langle\mathcal{W}^{\varepsilon}_{i}(t),J\right\rangle\right|\leq
K_{0}\|J\|\ .$

###  A.1. Proof of Lemma 3.2

By Eq. (96) and the linearity of the Fourier’s transform the proof of item i)
is immediate. To prove item ii) we use the evolution equation (73) satisfied
by the Wigner’s distribution. Let $J$ in ${\mathcal{S}}\times{\mathcal{S}}$,
in Appendix A.2 and Appendix A.3 we prove that

$\displaystyle\left\langle\mathcal{W}^{\varepsilon}(t),J\right\rangle-\left\langle\mathcal{W}^{\varepsilon}(0),J\right\rangle$
$\displaystyle=$
$\displaystyle\int_{0}^{t}ds\left\langle\mathcal{W}^{\varepsilon}(s),\textbf{v}_{0}\partial_{u}J\right\rangle+\gamma\int_{0}^{t}ds\left\langle\mathcal{W}^{\varepsilon}(t),C_{0}J\right\rangle$
$\displaystyle+$
$\displaystyle\gamma\left(\int_{0}^{t}ds\langle\Gamma_{n}^{\varepsilon}(s),C_{0}J\rangle+\int_{0}^{t}ds\langle\tilde{\Gamma}_{n}^{\varepsilon}(s),C_{0}J\rangle\right)$
$\displaystyle+$ $\displaystyle\int_{0}^{t}ds\mathcal{O}_{s,J}(\varepsilon)\
,$

with $|{\mathcal{O}}_{s,J}|\leq K_{J}$ a constant independent of $\varepsilon$
and time $s$. Hence for any $t$ in $[0,T]$,
$\left\langle\mathcal{W}^{\varepsilon}(t),J\right\rangle$ is the integral of a
bounded function, hence this is a continuous function. Using Eq. (96) we
conclude the proof of Lemma 3.2.

###  A.2. Behavior of the transport term

We recall that $B_{\varepsilon}=B\varepsilon^{\delta}$ with $\delta>0$. To
prove Eq. (73) we use Dynkin’s formula, Lemma 3.2 and Lemma 6.1. Let
$J:=(J_{1},J_{2})$ in ${\mathcal{S}}\times{\mathcal{S}}$, $t$ in $[0,T]$ and
$i$ in $\\{1,2\\}$ we want to prove that

$\displaystyle\partial_{t}\left\langle\mathcal{W}_{i}^{\varepsilon}(t),J_{i}\right\rangle$
$\displaystyle=\frac{1}{2\pi}\left\langle\mathcal{W}_{i}^{\varepsilon}(t),\textbf{v}_{0}\partial_{u}J_{i}\right\rangle+\left\langle\mathcal{W}_{i}^{\varepsilon}(t),[C_{0}J]_{i}\right\rangle$
$\displaystyle+\langle\Gamma_{i}^{\varepsilon}(t),[C_{0}J]_{i}\rangle+\langle\tilde{\Gamma}_{i}^{\varepsilon}(t),[C_{0}J]_{i}\rangle+\mathcal{O}_{t,J}(\varepsilon)\
.$

For any $i$ in $\\{1,2\\}$ we define $\phi_{i}$ by

$\displaystyle\phi_{i}\left(q^{\varepsilon}(t),p^{\varepsilon}(t)\right)$
$\displaystyle=\int_{\mathbb{R}}dp\int_{\mathbb{T}}dk\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},k-\tfrac{\varepsilon
p}{2}\right)^{*}\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},k+\tfrac{\varepsilon
p}{2}\right){\mathcal{F}}\left[J_{i}\right](p,k)^{*}\ .$

In order to lighten notation we denote
$\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},k\pm\frac{\varepsilon
p}{2}\right)$ by $\widehat{\psi}^{\varepsilon}_{i}\left(k\pm\frac{\varepsilon
p}{2}\right)$. By the equation above we can write that

(97)
$\displaystyle\left\langle\mathcal{W}_{i}^{\varepsilon}(t),J_{i}\right\rangle=\frac{\varepsilon}{2}{\mathbb{E}}_{\mu^{\varepsilon}}\left[\phi_{i}\left(q^{\varepsilon}\left(t\right),p^{\varepsilon}\left(t\right)\right)\right]\
.$

By Dynkin’s formula we have

$\displaystyle\partial_{t}\left\langle\mathcal{W}_{i}^{\varepsilon}(t),J_{i}\right\rangle$
$\displaystyle=\varepsilon^{-1}\mathbb{E}_{\mu^{\varepsilon}}\left\\{(A+B_{\varepsilon}G)\left[\phi_{i}\left(q^{\varepsilon}\left(t\right),p^{\varepsilon}\left(t\right)\right)\right]\right\\}$
$\displaystyle+\gamma\mathbb{E}_{\mu^{\varepsilon}}\left\\{S\left[\phi_{i}\left(q^{\varepsilon}\left(t\right),p^{\varepsilon}\left(t\right)\right)\right]\right\\}\
.$

Let $i$ in $\\{1,2\\}$, by Lemma 3.1 and the fact that
$A+B\varepsilon^{\delta}G$ is a first order differential operator we have

$\displaystyle\varepsilon^{-1}\mathbb{E}_{\mu^{\varepsilon}}\left\\{(A+B_{\varepsilon}G)\left[\phi_{i}\left(q^{\varepsilon}\left(t\right),p^{\varepsilon}\left(t\right)\right)\right]\right\\}$
$\displaystyle=\frac{\varepsilon}{2}\int_{\mathbb{R}}dp\int_{\mathbb{T}}dk\mathbb{E}_{\mu^{\varepsilon}}\left[\widehat{\psi}^{\varepsilon}_{i}\left(k-\frac{\varepsilon
p}{2}\right)^{*}\widehat{\psi}^{\varepsilon}_{i}\left(k+\frac{\varepsilon
p}{2}\right)\right]{\mathcal{F}}\left[J_{i}\right](p,k)^{*}$ (98)
$\displaystyle\times\frac{\mathbf{i}}{\varepsilon}\left[\omega_{i,B_{\varepsilon}}\left(k+\frac{\varepsilon
p}{2}\right)-\omega_{i,B_{\varepsilon}}\left(k-\frac{\varepsilon
p}{2}\right)\right]\ .$

Following the proof of [4] we can prove

(99)
$\varepsilon^{-1}\mathbb{E}_{\mu^{\varepsilon}}\left\\{(A+B_{\varepsilon}G)\left[\phi_{i}\left(q^{\varepsilon}\left(t\right),p^{\varepsilon}\left(t\right)\right)\right]\right\\}=\frac{1}{2\pi}\langle\mathcal{W}_{i}^{\varepsilon}(t),\textbf{v}_{B_{\varepsilon}}\partial_{u}J\rangle+\mathcal{O}_{J}(\varepsilon)\
,$

where $|\mathcal{O}_{J}(\varepsilon)|\leq\kappa_{J}(B)\times\varepsilon$ and
$\kappa_{J}(B)$ is a constant which depends only on J and $B$. To prove Eq.
(99), we need the following lemma which is proved at the end of this section.

######  Lemma A.1.

We recall that the functions $\omega_{1/2,B_{\varepsilon}}$ and
$\mathbf{v}_{B_{\varepsilon}}$ are defined in Eq. (13) and Eq. (15)
respectively. Then we have the following properties.

  * i)

$\mathbf{v}_{B_{\varepsilon}}$ is a bounded function on $\mathbb{T}$.

  * ii)

There exists a positive constant $C$ such that

$\displaystyle\forall p\in\mathbb{R},\quad\forall
k\in\mathbb{T},\quad\varepsilon^{-1}\left|\omega_{i,B_{\varepsilon}}\left(k+\frac{\varepsilon
p}{2}\right)-\omega_{i,B_{\varepsilon}}\left(k-\frac{\varepsilon
p}{2}\right)\right|\leq C|p|\ .$

  * iii)

For all $p$ in $\mathbb{R}$ and $k$ in $\mathbb{T}$ such that
$|k|>\varepsilon|p|$ we have

$\displaystyle\varepsilon^{-1}\left|\omega_{i,B_{\varepsilon}}\left(k+\frac{\varepsilon
p}{2}\right)-\omega_{i,B_{\varepsilon}}\left(k-\frac{\varepsilon
p}{2}\right)-p\mathbf{v}_{B_{\varepsilon}}(k)\right|\lesssim\varepsilon\frac{|p|^{2}}{|k|}\
.$

Let $\kappa>0$ and $\eta$ in $]0,1[$ with $\delta>\eta$. We denote by
$\kappa_{\varepsilon}$, the real $\kappa\varepsilon^{\eta}$. We cut the right-
hand side of Eq. (98) into two terms denoted by
$I^{\varepsilon}_{<}\left(\kappa_{\varepsilon}\right)$ and
$I^{\varepsilon}_{>}\left(\kappa_{\varepsilon}\right)$ where

$\displaystyle I_{<}^{\varepsilon}\left(\kappa_{\varepsilon}\right)$
$\displaystyle=\frac{\varepsilon}{2}\int_{\mathbb{R}}dp\int_{|k|<\kappa_{\varepsilon}}dk\mathbb{E}_{\mu^{\varepsilon}}\left[\widehat{\psi}^{\varepsilon}_{i}\left(k-\frac{\varepsilon
p}{2}\right)^{*}\widehat{\psi}^{\varepsilon}_{i}\left(k+\frac{\varepsilon
p}{2}\right)\right]{\mathcal{F}}\left[J_{i}\right](p,k)^{*}$
$\displaystyle\times\frac{\mathbf{i}}{\varepsilon}\left[\omega_{i,B_{\varepsilon}}\left(k+\frac{\varepsilon
p}{2}\right)-\omega_{i,B_{\varepsilon}}\left(k-\frac{\varepsilon
p}{2}\right)-\varepsilon p\textbf{v}_{B_{\varepsilon}}(k)\right]\ .$

By using point i) and point ii) of Lemma A.1 and the fact that $J_{i}$ belongs
to $\mathcal{S}$ we have

$\displaystyle\left|I^{\varepsilon}_{<}\left(\kappa_{\varepsilon}\right)\right|\lesssim\int_{|k|<\kappa_{\varepsilon}}dk{\mathbb{E}}\left[\left|\widehat{\psi}^{\varepsilon}_{i}(k)\right|^{2}\right]\
.$

By Lemma 5.2 we obtain then

$\displaystyle\limsup_{\varepsilon\rightarrow
0}I^{\varepsilon}_{<}\left(\kappa_{\varepsilon}\right)=0\ .$

It remains to deal with
$I^{\varepsilon}_{>}\left(\kappa_{\varepsilon}\right)$. We have

$\displaystyle I_{>}^{\varepsilon}\left(\kappa_{\varepsilon}\right)$
$\displaystyle=\frac{\varepsilon}{2}\int_{\varepsilon|p|>\kappa_{\varepsilon}}dp\int_{|k|>\kappa_{\varepsilon}}dk\mathbb{E}_{\mu^{\varepsilon}}\left[\widehat{\psi}^{\varepsilon}_{i}\left(k-\frac{\varepsilon
p}{2}\right)^{*}\widehat{\psi}^{\varepsilon}_{i}\left(k+\frac{\varepsilon
p}{2}\right)\right]{\mathcal{F}}\left[J_{i}\right](p,k)^{*}$
$\displaystyle\times\frac{\mathbf{i}}{\varepsilon}\left[\omega_{i,B_{\varepsilon}}\left(k+\frac{\varepsilon
p}{2}\right)-\omega_{i,B_{\varepsilon}}\left(k-\frac{\varepsilon
p}{2}\right)-\varepsilon p\textbf{v}_{B_{\varepsilon}}(k)\right]$
$\displaystyle+\frac{\varepsilon}{2}\int_{\varepsilon|p|<\kappa_{\varepsilon}}dp\int_{|k|>\kappa_{\varepsilon}}dk\mathbb{E}_{\mu^{\varepsilon}}\left[\widehat{\psi}^{\varepsilon}_{i}\left(k-\frac{\varepsilon
p}{2}\right)^{*}\widehat{\psi}^{\varepsilon}_{i}\left(k+\frac{\varepsilon
p}{2}\right)\right]{\mathcal{F}}\left[J_{i}\right](p,k)^{*}$
$\displaystyle\times\frac{\mathbf{i}}{\varepsilon}\left[\omega_{i,B_{\varepsilon}}\left(k+\frac{\varepsilon
p}{2}\right)-\omega_{i,B_{\varepsilon}}\left(k-\frac{\varepsilon
p}{2}\right)-\varepsilon p\textbf{v}_{B_{\varepsilon}}(k)\right]$
$\displaystyle=I^{\varepsilon}_{>,1}\left(\kappa_{\varepsilon}\right)+I^{\varepsilon}_{>,2}\left(\kappa_{\varepsilon}\right)\
.$

By the points i) and ii) of Lemma A.1 and Eq. (20) we have

$\displaystyle\left|I^{\varepsilon}_{>,1}\left(\kappa_{\varepsilon}\right)\right|\lesssim$
$\displaystyle
K_{0}\int_{|p|>\kappa\varepsilon^{\eta-1}}dp(p+1)\sup_{k\in\mathbb{T}}|{\mathcal{F}}\left[J_{i}\right](p,k)|\
,$

which goes to zero when $\varepsilon$ goes to zero since $\eta<1$. For the
remaining term we have that $|k|>\kappa_{\varepsilon}>\varepsilon|p|$ hence by
point iii) of Lemma A.1 we get

$\displaystyle\left|I^{\varepsilon}_{>,2}\left(\kappa_{\varepsilon}\right)\right|$
$\displaystyle\lesssim
K_{0}\int_{\mathbb{R}}dp\frac{\varepsilon|p|^{2}}{\kappa_{\varepsilon}}\sup_{k\in\mathbb{T}}|{\mathcal{F}}\left[J_{i}\right](p,k)|\
,$

which goes to zero since $\eta<1$.

In order to end the computations for the transport term it remains to prove
that

(100)
$\displaystyle\left\langle{\mathcal{W}}_{i}^{\varepsilon}(t),\textbf{v}_{B_{\varepsilon}}\partial_{u}J_{i}\right\rangle=\left\langle{\mathcal{W}}_{i}^{\varepsilon}(t),\textbf{v}_{0}\partial_{u}J_{i}\right\rangle+\mathcal{O}_{J}(\varepsilon)\
.$

We define $R^{\varepsilon}$ in the following way

$\displaystyle
R^{\varepsilon}=\left\langle{\mathcal{W}}_{i}^{\varepsilon},\left[\textbf{v}_{B_{\varepsilon}}-\textbf{v}_{0}\right]\partial_{u}J_{i}\right\rangle\
.$

Hence, we want to prove that $(R^{\varepsilon})_{\varepsilon>0}$ converges to
zero when $\varepsilon$ goes to zero. As before, we cute $R^{\varepsilon}$
into two terms $R^{\varepsilon}_{<}(\kappa_{\varepsilon})$ and
$R^{\varepsilon}_{>}(\kappa_{\varepsilon})$ where

$\displaystyle R_{<}^{\varepsilon}\left(\kappa_{\varepsilon}\right)$
$\displaystyle=\left\langle{\mathcal{W}}_{i}^{\varepsilon},\left[\textbf{v}_{B_{\varepsilon}}-\textbf{v}_{0}\right]\partial_{u}J_{i}\mathds{1}_{\\{k\leq\kappa_{\varepsilon}\\}}\right\rangle\
.$

We start to show that
$(R^{\varepsilon}_{>}\left(\kappa_{\varepsilon}\right))_{\varepsilon>0}$
converges to zero. Using the fact that $\textbf{v}_{0}$ is bounded and the
fact that $k>\kappa_{\varepsilon}$ we can bound
$|R_{>}^{\varepsilon}\left(\kappa_{\varepsilon}\right)|$ by

$\displaystyle\frac{\varepsilon}{2}\int_{\mathbb{R}}dp\int_{|k|>\kappa_{\varepsilon}}dk\mathbb{E}_{\mu^{\varepsilon}}\left[\left|\widehat{\psi}^{\varepsilon}_{i}\left(k-\frac{\varepsilon
p}{2}\right)^{*}\widehat{\psi}^{\varepsilon}_{i}\left(k+\frac{\varepsilon
p}{2}\right)\right|\right]\sup_{k\in\mathbb{T}}\left|\mathcal{F}\left[J_{i}\right](p,k)\right|\frac{|p|}{\kappa_{\varepsilon}}$
$\displaystyle\times\left|\sqrt{\widehat{\alpha}(k)}-\sqrt{\widehat{\alpha}(k)+\frac{B^{2}\varepsilon^{2\delta}}{4}}\right|\
.$

Since $k>\kappa_{\varepsilon}$, $\hat{\alpha}(k)>0$ and by using a Taylor
expansion we get

$\displaystyle\left|R^{\varepsilon}_{>}(\rho)\right|$
$\displaystyle\lesssim\frac{\varepsilon}{2}\int_{\mathbb{R}}dp\int_{|k|>\kappa_{\varepsilon}}dk\mathbb{E}_{\mu^{\varepsilon}}\left[\left|\widehat{\psi}^{\varepsilon}_{i}\left(k-\frac{\varepsilon
p}{2}\right)^{*}\widehat{\psi}^{\varepsilon}_{i}\left(k+\frac{\varepsilon
p}{2}\right)\right|\right]$
$\displaystyle\times\sup_{k\in\mathbb{T}}\left|{\mathcal{F}}\left[J_{i}\right](p,k)\right|\times\left(\frac{|p|B^{2}\varepsilon^{2\delta}}{\kappa_{\varepsilon}\sqrt{\widehat{\alpha}(k)}}\right)$
$\displaystyle\lesssim\frac{\varepsilon}{2}\int_{\mathbb{R}}dp\int_{|k|>\kappa_{\varepsilon}}dk\mathbb{E}_{\mu^{\varepsilon}}\left[\left|\widehat{\psi}^{\varepsilon}_{i}\left(k-\frac{\varepsilon
p}{2}\right)^{*}\widehat{\psi}^{\varepsilon}_{i}\left(k+\frac{\varepsilon
p}{2}\right)\right|\right]$
$\displaystyle\times\sup_{k\in\mathbb{T}}\left|{\mathcal{F}}\left[J_{i}\right](p,k)\right|\times\left(\frac{|p|B^{2}\varepsilon^{2\delta}}{|\kappa_{\varepsilon}|^{2}}\right)$
$\displaystyle\lesssim K_{0}\varepsilon^{2\left(\delta-\eta\right)}\ ,$

which goes to zero when $\varepsilon$ goes to zero since $\delta>\eta$ by
assumption. It remains to show that
$\left(R^{\varepsilon}_{<}\left(\kappa_{\varepsilon}\right)\right)_{\varepsilon>0}$
converges to zero. By using the fact that $\textbf{v}_{0}$ and
$\textbf{v}_{B_{\varepsilon}}$ are bounded and the fact that $J_{i}$ belongs
to ${\mathcal{S}}$ we obtain for any time $t$ in $[0,T]$

$\displaystyle\left|R^{\varepsilon}_{<}\left(\kappa_{\varepsilon}\right)\right|\lesssim\frac{\varepsilon}{2}\int_{|k|<\kappa_{\varepsilon}}dk\mathbb{E}_{\mu^{\varepsilon}}\left[\left|\widehat{\psi}^{\varepsilon}_{i}\left(k\right)\right|^{2}\right]=\frac{\varepsilon}{2}\int_{|k|<\kappa_{\varepsilon}}dk\mathbb{E}_{\mu^{\varepsilon}}\left[\left|\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},k\right)\right|^{2}\right]\
,$

which goes to zero by Lemma 5.2. Hence, we proved that for any $i$ in
$\\{1,2\\}$

$\displaystyle\mathbb{E}_{\mu^{\varepsilon}}\left\\{(A+B_{\varepsilon}G)\left[\phi_{i}\left(q^{\varepsilon}\left(t\right),p^{\varepsilon}\left(t\right)\right)\right]\right\\}=\left\langle{\mathcal{W}}_{i}^{\varepsilon}(t),\textbf{v}_{0}\partial_{u}J_{i}\right\rangle+\mathcal{O}_{J}(\varepsilon)\
.$

So far we obtained the first term of Eq. (73) which corresponds to the
transport term of the Boltzmann’s equation $\left(\text{cf.
Eq.}\;\eqref{BE}\right)$. In the next section, we obtain the last terms of Eq.
(73). Before doing this, we give the sketch of the proof of Lemma A.1.

###### Sketch of the proof of Lemma A.1.

We recall that $\hat{\alpha}(k)=4\pi^{2}\sin^{2}\left(\pi k\right)$.
Hence, by definition we have for every $k$ in $\mathbb{T}$

$\displaystyle\textbf{v}_{B_{\varepsilon}}(k)=\frac{\hat{\alpha}^{\prime}(k)}{2\sqrt{\hat{\alpha}(k)+\frac{B_{\varepsilon}^{2}}{4}}}=\frac{4\pi^{3}\cos\left(\pi
k\right)\sin\left(\pi k\right)}{\sqrt{\sin^{2}\left(\pi
k\right)+\frac{B_{\varepsilon}^{2}}{4}}}\lesssim 4\pi^{3}\ .$

This ends the proof of point i). Using a Taylor expansion, the proof of point
ii) follows using the item i). It remains to deal with point iii). For $p=0$,
the results is obvious since $|k|>\varepsilon|p|$ hence it is sufficient to
discuss the case $|k|>0$. Let $f_{i}$ and $g_{i}$ be two real-valued functions
defined on $]0,\varepsilon[$ by

$\displaystyle
f_{i}(s)=\omega_{i,B_{\varepsilon}}\left(k+\frac{sp}{2}\right)\quad\text{and}\quad
g_{i}(s)=\omega_{i,B_{\varepsilon}}\left(k-\frac{sp}{2}\right)\ .$

Then we have

$\displaystyle\varepsilon^{-1}\left|\omega_{i,B_{\varepsilon}}\left(k+\frac{\varepsilon
p}{2}\right)-\omega_{i,B_{\varepsilon}}\left(k-\frac{\varepsilon
p}{2}\right)-p\textbf{v}_{B_{\varepsilon}}(k)\right|$
$\displaystyle=\varepsilon^{-1}\left[f_{i}(\varepsilon)-f_{i}(0)-\varepsilon
f_{i}^{\prime}(0)\right]-\varepsilon^{-1}\left[g_{i}(\varepsilon)-g_{i}(0)-\varepsilon
g_{i}^{\prime}(0)\right]\ .$

We conclude the proof by a Taylor expansion and using the fact that
$\hat{\alpha}(k)>C|k|$ where $C$ is a positive constant. ∎

###  A.3. Behavior of the collisional term

Let $i$ in $\\{1,2\\}$ and $i^{*}=3-i$, following the proof did by the authors
of [30] we can prove that

$\displaystyle\gamma\mathbb{E}_{\mu^{\varepsilon}}\left[S\left[\phi_{i}\left(q^{\varepsilon}\left(t\varepsilon^{-1}\right),p^{\varepsilon}\left(t\varepsilon^{-1}\right)\right)\right]\right]$
$\displaystyle=\gamma\left(\left\langle{\mathcal{W}}^{\varepsilon}_{i}(t),[C_{B_{\varepsilon}}J]_{i}\right\rangle+\left\langle\Gamma^{\varepsilon}_{i}(t),[K_{B_{\varepsilon}}J]_{i}\right\rangle\right)$
$\displaystyle+\gamma\left(\left\langle(\Gamma^{\varepsilon}_{i})^{*}(t),[K_{B_{\varepsilon}}J]_{i}\right\rangle\right)+\mathcal{O}_{J}(\varepsilon)\
,$

where for any $(u,k)$ in $\mathbb{R}\times\mathbb{T}$

$\displaystyle[C_{B_{\varepsilon}}J]_{i}(u,k)$
$\displaystyle=\sum_{j=1}^{2}\int_{\mathbb{T}}dk^{\prime}\theta_{i,B_{\varepsilon}}^{2}(k)R\left(k,k^{\prime}\right)\theta_{j,B_{\varepsilon}}^{2}\left(k^{\prime}\right)\left[J_{j}(u,k^{\prime})-J_{i}(u,k)\right]\
,$ $\displaystyle[K_{B_{\varepsilon}}J]_{i}(u,k)$
$\displaystyle=\int_{\mathbb{T}}dk^{\prime}\theta_{1,B_{\varepsilon}}\left(k\right)\theta_{2,B_{\varepsilon}}(k)R\left(k,k^{\prime}\right)\left[\theta_{i^{*},B_{\varepsilon}}^{2}\left(k^{\prime}\right)J_{i^{*}}(u,k^{\prime})-\tfrac{J_{i}(u,k)}{2}\right]\
.$

When we add the term that we had for the contribution due to the operator
$A+B_{\varepsilon}G$ and the term due to $S$ we get

$\displaystyle\partial_{t}\langle{\mathcal{W}}^{\varepsilon}_{i}(t),J\rangle$
$\displaystyle=\frac{1}{2\pi}\left\langle{\mathcal{W}}_{i}^{\varepsilon}(t),\textbf{v}_{0}\partial_{u}J_{i}\right\rangle+\gamma\left(\langle{\mathcal{W}}_{i}^{\varepsilon}(t),[C_{B_{\varepsilon}}J]_{i}\rangle+\langle\Gamma_{i}^{\varepsilon}(t),[K_{B_{\varepsilon}}J]_{i}\rangle\right)$
$\displaystyle+\gamma\left(\langle(\Gamma_{i}^{\varepsilon})^{*}(t),[K_{B_{\varepsilon}}J]_{i}\rangle\right)+\mathcal{O}_{J}(\varepsilon)\
.$

To conclude the proof of Eq. (73) we have to replace $K_{B_{\varepsilon}}J$
and $C_{B_{\varepsilon}}J$ in the previous equation by $C_{0}J$ where for any
$(u,k)$ in $\mathbb{R}\times\mathbb{T}$

$[C_{0}J]_{i}(u,k)=\sum_{j=1}^{2}\int_{\mathbb{T}}dk^{\prime}\frac{1}{4}R\left(k,k^{\prime}\right)\left[J_{j}\left(u,k^{\prime}\right)-J_{i}(u,k)\right]\
.$

Since the arguments are similar we only prove that

(101)
$\left\langle{\mathcal{W}}_{i}^{\varepsilon}(t),[C_{B_{\varepsilon}}J]_{i}\right\rangle=\left\langle{\mathcal{W}}_{i}^{\varepsilon}(t),[C_{0}J]_{i}\right\rangle+\mathcal{O}_{J}(\varepsilon)\
.$

We define an operator $\tilde{C}_{B_{\varepsilon}}$ on
${\mathcal{S}}\times{\mathcal{S}}$ where for any $i$ in $\\{1,2\\}$ and
$(u,k)$ in ${\mathbb{R}}\times\mathbb{T}$

$[\tilde{C}_{B_{\varepsilon}}J]_{i}(u,k)=\sum_{j=1}^{2}\int_{\mathbb{T}}dk^{\prime}\frac{\theta_{i,B_{\varepsilon}}^{2}(k)}{2}R\left(k,k^{\prime}\right)\left[J_{j}\left(u,k^{\prime}\right)-J_{i}(u,k)\right]\
.$

Let $i$ in $\\{1,2\\}$ we first prove that

(102)
$\displaystyle\left\langle{\mathcal{W}}^{\varepsilon}_{i}(t),[C_{B_{\varepsilon}}J]_{i}-[\tilde{C}_{B_{\varepsilon}}J]_{i}\right\rangle=\mathcal{O}_{J}(\varepsilon)\
.$

Let $(p,k)$ in ${\mathbb{R}}\times\mathbb{T}$ and $j$ in $\\{1,2\\}$, we
recall that

$\displaystyle R\left(k,k^{\prime}\right)=16\sin^{2}(\pi k)\sin^{2}\left(\pi
k^{\prime}\right)$

then

$\displaystyle\left|\int_{\mathbb{T}}dk^{\prime}R\left(k,k^{\prime}\right)\theta_{i,B_{\varepsilon}}^{2}(k)\left[\theta_{j,B_{\varepsilon}}^{2}\left(k^{\prime}\right)-\frac{1}{2}\right]\left[\mathcal{F}\left[J_{j}\right]\left(p,k^{\prime}\right)^{*}-{\mathcal{F}}\left[J_{i}\right](p,k)^{*}\right]\right|$
$\displaystyle\lesssim|B\varepsilon^{\delta}|\times\kappa_{J}(p)\ ,$

where

$\displaystyle\kappa_{J}(p)=\left(\sup_{k\in\mathbb{T}}\left|{\mathcal{F}}\left[J_{i}\right](k,p)\right|+\sup_{k\in\mathbb{T}}\left|\mathcal{F}\left[J_{j}\right](k,p)\right|\right)\
.$

Hence, using the fact that $J_{i}$ belongs to ${\mathcal{S}}$ and the usual
argument we conclude that

$\displaystyle\left|\left\langle{\mathcal{W}}^{\varepsilon}_{i}(t),[C_{B_{\varepsilon}}J]_{i}-[\tilde{C}_{B_{\varepsilon}}J]_{i}\right\rangle\right|\lesssim
K_{0}\|J\|B\varepsilon^{\delta}\ .$

This proves Eq. (102). Similarly , one can prove that

$\displaystyle\left\langle{\mathcal{W}}^{\varepsilon}_{i}(t),[\tilde{C}_{B_{\varepsilon}}J]_{i}-[C_{B_{\varepsilon}}J]_{i}\right\rangle=\mathcal{O}_{J}(\varepsilon)\
.$

By the triangle inequality we obtain Eq. (101) and conclude the proof.

###  A.4. Proof of Lemma 5.2

In this section, we prove Lemma 5.2 which allows us to extend the assumption
(52), on the initial distribution $\mu^{\varepsilon}$ to times
$t\varepsilon^{-1}$. The proof of this Lemma follows the one of [4, Lemma 7].
Let $J=(J_{1},J_{2})$ be a bounded measurable function on $\mathbb{T}$. Then
for any $i$ in $\\{1,2\\}$ and $t$ in $[0,T]$

(103)
$\displaystyle\left\langle{\mathcal{W}}^{\varepsilon}_{i}(t),J\right\rangle$
$\displaystyle=\frac{\varepsilon}{2}\int_{\mathbb{T}}\mathbb{E}_{\mu^{\varepsilon}}\left[\left|\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},k\right)\right|^{2}\right]J_{i}(k)^{*}dk\
.$ (104) $\displaystyle\left\langle\Gamma^{\varepsilon}_{i}(t),J\right\rangle$
$\displaystyle=\frac{\varepsilon}{2}\int_{\mathbb{T}}\mathbb{E}_{\mu^{\varepsilon}}\left[\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},k\right)\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},-k\right)\right]J_{i}(k)^{*}dk\
.$

Since $J$ is a bounded measurable function the previous objects are well
defined. Indeed, we have

(105)
$\displaystyle\left|\left\langle{\mathcal{W}}_{i}^{\varepsilon}(t),J\right\rangle\right|\leq
K_{0}\times\sup\limits_{k\in\mathbb{T}}|J(k)|\quad\text{and}\quad\left|\left\langle\Gamma_{i}^{\varepsilon}(t),J\right\rangle\right|\leq
K_{0}\sup_{k\in\mathbb{T}}\left|J(k)\right|\ .$

Using Lemma 3.1 we get

(106)
$\displaystyle(A+B_{\varepsilon}G)\left[\left|\widehat{\psi}^{\varepsilon}_{i}\right|^{2}\right]\left(k\right)=0\
.$

Following the proof presented in Appendix A.3 and using Eq. (106) we obtain
that

$\displaystyle\partial_{t}\left\langle\mathcal{W}_{i}^{\varepsilon}(t),J_{i}\right\rangle$
$\displaystyle=\gamma\left(\left\langle\mathcal{W}_{i}^{\varepsilon}(t),[C_{0}J]_{i}\right\rangle+\left\langle\Gamma_{i}^{\varepsilon}(t),[C_{0}J]_{i}\right\rangle+\left\langle\tilde{\Gamma}_{i}^{\varepsilon}(t),[C_{0}J]_{i}\right\rangle\right)$
$\displaystyle+\mathcal{O}_{t,J}(\varepsilon)\ ,$

where we recall that for any $i$ in $\\{1,2\\}$, $[C_{0}J]_{i}$ is defined in
Eq. (25), $\kappa$ is a positive constant and $\eta$ is in $]0,1[$. Let
$\kappa_{\varepsilon}:=\kappa\varepsilon^{\eta}$ and
$J^{\kappa_{\varepsilon}}:=(\mathds{1}_{[-\kappa_{\varepsilon},\kappa_{\varepsilon}]},\mathds{1}_{[-\kappa_{\varepsilon},\kappa_{\varepsilon}]})$.
Then we have

$\displaystyle\left|\left[C_{0}J^{\kappa_{\varepsilon}}\right]_{i}(k)\right|\lesssim\left(\kappa_{\varepsilon}+J^{\kappa_{\varepsilon}}_{i}(k)\right)\
,$

and

$\displaystyle\left\langle\Gamma_{i}^{\varepsilon}(t)+(\Gamma^{\varepsilon})(t)^{*},\left[C_{0}J^{\kappa_{\varepsilon}}\right]_{i}\right\rangle\lesssim\left|\langle{\mathcal{W}}_{i}^{\varepsilon}(t),[C_{0}J^{\kappa_{\varepsilon}}]_{i}\rangle\right|\
.$

Hence, using Eq. (73) we obtain that for any $i$ in $\\{1,2\\}$ and $t$ in
$[0,T]$

$\displaystyle\left\langle{\mathcal{W}}_{i}^{\varepsilon}(t),J^{\kappa_{\varepsilon}}_{i}\right\rangle$
$\displaystyle\lesssim\left\langle{\mathcal{W}}_{i}^{\varepsilon}(0),J^{\kappa_{\varepsilon}}_{i}\right\rangle+(C_{1}\rho+C_{2}\varepsilon)t+\int_{0}^{t}\left\langle{\mathcal{W}}_{i}^{\varepsilon}(s),J^{\kappa_{\varepsilon}}_{i}\right\rangle
ds\ ,$

where $C_{1}$ and $C_{2}$ are two positive constants. By Gronwall’s lemma we
conclude that

(107)
$\displaystyle\left\langle{\mathcal{W}}_{i}^{\varepsilon}(t),J^{\kappa_{\varepsilon}}_{i}\right\rangle$
$\displaystyle\lesssim\exp(C_{3})\left(C_{1}\kappa_{\varepsilon}+C_{2}\varepsilon+\left\langle{\mathcal{W}}_{i}^{\varepsilon}(0),J^{\kappa_{\varepsilon}}_{i}\right\rangle\right)\
.$

Using assumption (52) we get that

$\limsup_{\varepsilon\rightarrow
0}\left\langle{\mathcal{W}}_{i}^{\varepsilon}(t),J^{\kappa_{\varepsilon}}_{i}\right\rangle=\limsup_{\varepsilon\rightarrow
0}\frac{\varepsilon}{2}\int_{|k|<\kappa_{\varepsilon}}\mathbb{E}_{\mu^{\varepsilon}}\left[\left|\widehat{\psi}^{\varepsilon}_{i}\left(t\varepsilon^{-1},k\right)\right|^{2}\right]=0\
.$

This ends the proof of Lemma 5.2.

###  A.5. Proof of the uniqueness in the Boltzmann’s equation

In this section, we want to prove Lemma 4.1. Let $\mu$ be a measure valued
solution of Eq. (27). Then, for any
$J:=(J_{1},J_{2})\in\mathcal{S}\times\mathcal{S}$ we have that

$\displaystyle\sum_{i=1}^{2}\int_{\mathbb{R}\times\mathbb{T}}\mathbb{E}\left[J(u,k)d\mu_{i}(u,k)\right]=\sum_{i=1}^{2}\int_{\mathbb{R}\times\mathbb{T}}\mathbb{E}\left[J_{\tilde{I}^{B}(t)}\left(\tilde{Z}^{B}_{u}(t),\tilde{K}^{B}(t)\right)d\mu_{i}(0,u,k)\right]$

where
$\left(\tilde{Z}_{u}^{B}(\cdot),\tilde{I}^{B}(\cdot),\tilde{K}^{B}(\cdot)\right)$
is the dual process to
$\left(Z_{u}^{B}(\cdot),I^{B}(\cdot),K^{B}(\cdot)\right)$ starting from
$(u,k,i)$. Hence, the infinitesimal generator of the dual process is defined
by $\mathcal{L}+\textbf{v}_{B}\partial_{u}$ where $\mathcal{L}_{B}$ is defined
in Eq. (34). Since the PDMP
$\left(\tilde{Z}^{B}_{u}(\cdot),\tilde{I}^{B}(\cdot),\tilde{K}^{B}(\cdot)\right)$
is unique, we conclude the proof.

##  Appendix B Proof of Proposition 5.4 and Proposition 5.5

In this section, we prove Proposition 5.4 and Proposition 5.5.

###  B.1. Proof of Proposition 5.4

We recall that $B>0$ and that $x_{B,\pm}$ is defined in Eq. (56). Observe that
$\text{sign}(x_{B,\pm}(r))=\text{sign}(r)$, moreover Eq. (56) has exactly two
solutions with opposite sign. This proves that $x_{B,\pm}(-r)=-x_{B,\pm}(r)$
and ends the proof of item i).

We prove item ii), since the proof are similar we only give the details for
$x_{B,+}$. Using Eq. (56) we get that for any $r\neq 0$

$|x_{B,+}(r)|<\sqrt{\frac{\pi}{2\gamma}}|r|^{-\frac{1}{2}}\ .$

Hence for each $r\neq 0$, $(x_{B,+}(r))_{B>0}$ is a bounded sequence and
therefore admits an accumulation point. Observing that there is only one
accumulation point we conclude that the whole sequence converges and

$\lim\limits_{B\rightarrow
0}x_{B,+}(r)=\text{sign}(r)\sqrt{\frac{\pi}{2\gamma}}|r|^{-\frac{1}{2}}=x_{0}(r)\
.$

To prove the convergence of the sequence $(x_{B,+}^{\prime})_{B>0}$ we derive
Eq. (56) with respect to $r$ and we send $B$ to zero. This concludes the proof
of item ii).

We prove item iii). We can write Eq. (56) in the following way

(108) $\left(\sqrt{\frac{4x_{B,\pm}^{2}(r)}{B^{2}}+1}\pm
1\right)x_{B,\pm}(r)=\frac{\pi}{B\gamma r}\ .$

Using Eq. (108) we observe that the sequence
$\left(B^{-1}x_{B,\pm}(r)\right)_{B>0}$ converges to zero when $B$ goes to
infinity. Hence by performing a Taylor expansion in Eq. (108) we obtain the
first part of item iii). To conclude the proof we derive Eq. (108) with
respect to $r$ and send $B$ to infinity.

###  B.2. Proof of Proposition 5.5

Let $r\neq 0$ we have

$\displaystyle g_{B,\pm}(r)$
$\displaystyle=-\frac{x_{B,\pm}^{\prime}(r)}{4\pi}\left(\frac{1}{2}\pm\frac{B}{4\sqrt{x_{B,\pm}^{2}(r)+\frac{B^{2}}{4}}}\right)x_{B,\pm}^{2}(r)\mathds{1}_{r>0}$
(109)
$\displaystyle-\frac{x_{B,\pm}^{\prime}(r)}{4\pi}\left(\frac{1}{2}\pm\frac{B}{4\sqrt{x_{B,\pm}^{2}(r)+\frac{B^{2}}{4}}}\right)x_{B,\pm}^{2}(r)\mathds{1}_{r<0}\
.$

By Proposition 5.4, $x_{B,\pm}$ is an odd function, hence its derivative is an
even function and this proves item i).
Since the arguments are similar, to prove item ii) it is sufficient to show
that

(110)
$\int_{1}^{\infty}g_{B,\pm}^{\prime}(r)dr<\infty\quad\text{and}\int_{0}^{1}r^{2}g_{B,\pm}^{\prime}(r)dr<\infty\
.$

Let $A>1$, then

$\displaystyle\int_{1}^{A}g_{B,\pm}^{\prime}(r)dr\lesssim
3^{-1}(x_{B,\pm}^{3}(1)-x_{B,\pm}^{3}(A))\ .$

By Eq. (56) we deduce that $(x_{B,\pm}(A))_{A>1}$ goes to zero when $A$ goes
to infinity. The monotone convergence theorem ends the proof. Let
$\varepsilon<1$, we have

$\displaystyle\int_{\varepsilon}^{1}r^{2}h_{B,\pm}^{\prime}(r)dr$
$\displaystyle\lesssim