Gaëtan Cane

Laboratoire De Mathématiques

J A Dieudonné

Vue De L'

Obtention Du

Grade De

Soutenue Le

M François

Huveneers Maître

UNIVERSITÉ D E Conférences

Université Paris-Dauphine

M Tomasz

Komorowski Professeur

Université De Lublin

M Cédric

Bernardin Professeur

Université Côte D'azur

Directeur De

Thèse M François

Delarue Professeur

Président Du

Jury M François

Paris-Dauphine Rapporteur

Université Ellen De

Saada

Examinatrice Mme Marielle Simon

Je Voudrais Remercier Estelle

Cuong, Bochra, Sami, Lyly, Thomas Alexia Audrey

TRANSPORT D'ÉNERGIE DANS LES CHAÎNES D'OSCILLATEURS SOUMISES À UN CHAMP MAGNÉTIQUE

Keywords: Chaînes d'oscillateurs, diffusion, limites hydrodynamiques, limites cinétiques, processus d'interpolation, processus de Lévy, localisation, champ magnétique, superdiffusion Chains of oscillators, diffusion, hydrodynamic limits, kinetic limits, interpolation process, Lévy processes, localization, magnetic field, superdiffusion. I. FROM MICROSCOPIC

EXAMINATRICE

1 Chapitre dont l'existence fut ô combien compliquée, n'est-ce pas Cédric ? :p 2 Merci de m'avoir fait confiance pour représenter les doctorants au conseil du laboratoire. 3 Malheureusement nous n'avons pas pu discuter autant que je l'aurais souhaité... 4 Ah ces heures passées sur AMC !!! 5 Assisté de Cédric bien sûr ! Que serait une thèse sans des collègues doctorants incroyables ? Billel, Dahmane 6 , Lamine 7 , Zakaria 8 , Thibault : 9 on en aura passé des moments à faire autre chose que de la recherche. Manger des pizzas au bureau tout en faisant semblant de rédiger sa thèse, c'est une expérience que tout bon doctorant doit avoir vécu. Les discussions mathématiques ou bien sur la vie en général furent légions avec vous tous, un grand merci. Je ne pourrai pas suffisamment vous remercier en écrivant ces quelques lignes mais sachez que je sais ce que je vous dois. Cela sera très étrange l'année prochaine, je ne pourrai plus aller prendre un thé avec vous et faire une pause, ladite pause durant une demi-journée...

REMERCIEMENTS

Avant d'entamer la partie scientifique de ce manuscrit j'aimerais remercier les personnes qui ont rendu possible sa rédaction.

Mes premiers remerciements vont à Cédric, je n'aurais jamais imaginé il y a 6 ans quand nous nous sommes rencontrés que cela nous mènerait où nous sommes aujourd'hui. Avant d'être un grand scientifique tu es un homme incroyable et je mesure la chance d'être tombé sur un directeur comme toi. Tu m'as fait confiance dès la première année de master et conseillé du mieux que tu pouvais. Bon tu m'as un peu arnaqué au début, je t'avais dit « je n'aime pas la physique » et tu es arrivé avec un sujet de physique statistique. . . Après presque 5 années à travailler sur le sujet, je suis d'accord avec toi, c'est très intéressant mais quand même je me suis fait arnaquer ! Malgré le fait que je puisse être assez difficile à vivre tu n'as pas perdu patience, dans les moments difficiles qu'ils soient personnels ou professionnels tu as toujours été là pour me tendre la main. . . Merci infiniment. Dans la vie, il y a des rencontres qui changent à jamais notre existence, merci d'avoir été l'une d'entre elles pour moi.

Je voudrais remercier David pour ses précieux conseils, les heures qu'il a passées à m'aider, me soutenir. C'est une personne incroyable, un mathématicien de grand talent, un enseignant exceptionnel et le monde gagnerait beaucoup à avoir plus de David Chiron. . . Merci encore David d'avoir relu le premier chapitre 1 et fait des remarques très pertinentes le concernant.

Il m'est impossible de ne pas remercier Billel, il y a quatre ans quand je suis arrivé au bureau je ne pensais pas trouver une personne aussi incroyable. Tu es gentil, humain compétent et prêt à faire passer le bonheur des autres avant le tien. Merci infiniment pour ce que tu as fait, notamment durant l'année scolaire 2019/2020. Cela fera bizarre de ne plus être dans le même bureau l'année prochaine... Les nombreux encouragements de mon comité de thèse ont été très importants, je remercie donc chaleureusement François Delarue, Roland Diel ainsi que François Huveneers. Vous avez été parfaits, chacun dans des rôles très différents mais c'était parfait ! Je tiens à remercier tout particulièrement François Delarue car sa porte était toujours ouverte pour moi, je n'oublierai pas non plus le grand rôle que vous avez eu durant ma seconde année de master.

Il y a de très nombreuses personnes qui m'ont soutenu au LJAD durant ma thèse, je voudrais toutes les remercier, Roland, Stephane, Maxime, Yves 2 , Clemens, Yannick 3 , Haroune. Je remercie Narymen pour son travail remarquable durant ma thèse, c'est une personne incroyablement professionnelle et gentille.

Je voudrais remercier les membres de mon jury d'avoir accepté de lire ce manuscrit, je remercie tout particulièrement Tomasz Komorowski et François Huveneers d'avoir accepté de rapporter ma thèse, leurs remarques ayant permis d'améliorer très nettement la qualité de cet ouvrage.

Parfois durant plusieurs années vous travaillez dans le même endroit qu'une personne incroyablement cool, drôle et compétente mais vous ne la découvrez que trop tard. Ce fut mon cas, je n'ai découvert que trop tard que Raphaël Chétrite est de ceux-là. J'ai été ravi d'être ton chargé de TD 4 . Tu m'as donné une autre vision du monde de la recherche, je te remercie de m'avoir torturé 5 avant mes entretiens de post-doc car sans ça je n'aurais peut-être pas réussi à être recruté ! J'apprécie beaucoup les discussions que nous avons eues et espère collaborer un jour avec toi, il est certain que j'apprendrai beaucoup.

TABLE OF CONTENTS

II.5 DISORDERED SYSTEMS

configuration of all atoms. Newton's laws 3 state that ∀t ≥ 0, d dt x i (t) = v i (t) , and m i d dt

v i (t) = F i (x) = -∑ j̸ =i ∇ x [U](x j -x i ) , (I.1)
where F i (x) is a force acting on the i th particle. We assumed that all the particles interact pairwise in the same way and that the interaction force derives from a symmetric potential U : R 3 → R.

Observe two important facts:

• The system of particles conserves three physical quantities: the number N of particles, the global momentum ∑ N i=1 m i v i (t) and the total energy ∑ N i=1 m i

|v i (t)| 2
2 + ∑ j̸ =i U(x j (t)x i (t)) . Hence, the evolution takes place on the 6N -4-dimensional submanifold of R 3 × R 3 N defined by the previous constraints (which are fixed by the initial conditions).

• Newton's laws are reversible in time: if we change t into -t and v i (t) into -v i (-t) in Eq. (I. [START_REF] Ajanki | Rigorous scaling law for the heat current in disordered harmonic chain[END_REF] then the equations remain the same. In practice, this means that we can rewind the film and do not see any difference.

However everyday's life experiments show the opposite: we cannot rewind the film without being totally confused. It means that at a larger time-space scale, an arrow of time appears. For example if we drop a ball and watch it until it comes to rest as on the left-hand side of practice: the evolution of the ball is irreversible. This means that the ball will remain in this state as long as it has not been supplied with energy.

I.1.2 ENTROPY AND IRREVERSIBILITY

ENTROPY

To understand how the macroscopic irreversibility can result from the microscopic reversibility, Boltzmann introduced fundamental concepts, in particular the notion of statistical entropy, at the origin of statistical physics. By the way, Boltzmann was able to prove the second principle (in any thermodynamical process, entropy can only increase). In the next section, we give a very short introduction to it. We refer the interested reader to [START_REF] Terrell | An introduction to statistical thermodynamics[END_REF][START_REF] Huang | Introduction to statistical physics[END_REF] for a more complete study.

Consider a gas composed of N ≫ 1 atoms contained on the left-hand side of a box which is thermally and mechanically isolated. We assume that there is a piston between the right-hand side and the left-hand side of the box which prevents the gas from passing through 4 . When the piston is removed we observe that the gas spreads to occupy all the space, as described in Fig. I.2. After a sufficiently long time the density, temperature and pressure of the gas will be homogeneous in the box. The final state is called a thermodynamic equilibrium of the system. In this thermodynamic equilibrium, particles are not at rest. Only the empirical distribution of the global position-velocity configuration is time-invariant. Since Newton's laws are reversible, it should be possible that the gas comes back to its initial position on the left-hand side of the box. However, we never observe this situation in practice: the transformation in which the gas spreads from the left-hand side of the box to the entire box is called irreversible. How can we explain this since the fundamental laws of physics make it possible? It is in fact due to the separation of space and time scales between microscopic and macroscopic worlds.

One way to show in a more quantitative way this macroscopic irreversibility is to introduce the statistical entropy of the system. But before that we have to define the notion of macro-state and micro-state.

Let us divide the full box in K = O(ε -3 ) small cubes of length ε where ε is very small with respect to the length of the box. At the microscopic level however there is typically a number of particles in each cube which is huge. In the experiment above a macro-state is given by the data composed of few thermodynamic parameters in each small box such as the energy (empirical average of the energies of particles), the temperature (empirical average of the kinetic energies of particles), pressure (force acting on the sides of the cube) and density (number of particles in the box divided by total number of particles) etc. Since ε is very small, these data can be seen as some (space-time dependent) energy, temperature, pressure and density functions E, T, P, ρ respectively, defined on the macroscopic space occupied by the box. In the experiment given above, for example, the initial macro-state corresponds to a density function which is 0 on the right of the box while the final macro-state corresponds to a density which is constant in the full box. After a long time the system will reach an equilibrium state where the functions E, T, P, ρ will be constant in time and space 5 . A micro-state is simply the position-velocities configuration of all the particles in the box. Observe that there is a huge number of micro-states compatible with a given macro-state. As an example we can think about the gas described on Fig. I.2. In this one, the left-hand side describes a macro-state which can be described by a lot of different micro-states. Indeed, if we assume that the gas is composed of N particles and that we can label each particle then two possible micro-states are presented in have changed positions without affecting the macro-state, these two micro-states correspond to the same macro-state. Hence, to a macro state corresponds a lot of micro-states.

The concept of entropy was first developed by Clausius in the mid-nineteenth century as a thermodynamic property that predicts that certain spontaneous processes are irreversible or impossible. Statistical (or Boltzmann's) entropy was introduced in [START_REF] Boltzmann | Sitzungberichte der kaiserlichen akademie der wissenschaften[END_REF] by Boltzmann in 1877 6 . Let Σ be a macro-state and assume that it is compatible with Ω ∈ N micro-states. The Boltzmann entropy is defined as

S B (Σ) = k B log(Ω) ,
where k B is the Boltzmann's constant. Some other entropy, the Shannon's entropy S S , is also very useful in statistical mechanics and has some interpretation in the information's theory. Imagine we have a macro-state Σ but we ignore from which micro-state it comes from. Then we associate a probability P defined on the micro-state and the Shanonn entropy S S (Σ) is defined as

S S (Σ) = -k B ∑ i∈Ω P(i) log[P(i)] . (I.2)
Observe that the uniform distribution on Ω maximizes the Shannon entropy and that in this case S S (Σ) = S B (Σ). The Shanonn entropy measures the microscopic information 7 we have on the macrosystem Σ. Higher it is, less information we have. Hence, if P(i) = card(Ω) -1 this means that we have no information about the real micro-state of the system. For example, if a macro-state of Σ has only one compatible micro-state j which is possible then S S (Σ) = 0 because P(i) = 0 for every i ̸ = j and P( j) = 1.

The fundamental claim of Boltzmann is that with this definition of the entropy, the latter always increases with time (second law of thermodynamics).

As mentioned previously, when the piston of the box is removed, the gas spreads to occupy all the space of the box, this is the equilibrium state of the gas. Thus, why do we never observe the return of gas in the left part of the box? Initially, each particle is on the left-hand side of the box, as it is described on the left-hand side of Fig. I.4. This is one initial micro-state of the system compatible with the initial macro-state. A micro-state of the equilibrium macro-state can be described by the right-hand side of Fig. I. [START_REF] Aoki | Energy transport in weakly anharmonic chains[END_REF]. In this one, there is almost an equal number of particles on the left and right-hand sides of the box. In fact it is possible to evaluate the Boltzmann entropy of the initial macro-state and of the final macro-state showing that the initial one is considerably smaller than the second one. By the second law, it is thus impossible to observe an evolution from the final macro-state to the initial macro-state. This difference with the microscopic world results from the huge number of particles and the coarse-graining procedure to define a macro-state.

MOLECULAR CHAOS

In the previous section, we introduced the notion of statistical entropy in order to show how the microscopic description of the system can explain the irreversibility of the system at the macroscopic scale. There is another phenomenon which explains this: the molecular chaos.

Every system is subject to disturbances, these disturbances are weak but their effects are important because of the huge number of particles present in the system and the colossal number of collision 10 9 per second in a gas . Hence, at the microscopic level, the movement of particles is chaotic and looks like random. This implies that the system forgets where it comes from, it loses its memory very quickly thus involving the irreversibility of the system observed at the macroscopic level.

One way to model this is to use the probabilistic representation of micro-states. Since initially we assume that the system is distributed according to some probability measure, this implies that at any time t the evolution is probabilistic. Hence, it becomes clear that we can not rewind the time, this simulates the system memory loss.

LITTLE SUMMARY

From the two previous sections it is clear that in order to understand the macroscopic world we have to start from the microscopic one. However because of the important number of particles, solving Newton's equations is almost impossible. Hence, the idea is to study some functional of the microscopic dynamics in order to obtain the macroscopic properties of the studied system . One way to do that is to use scaling limits. We will use this in Chapter III to study the superdiffusion of energy in a noisy harmonic chain. This is why we decided to give a short introduction to this concept in the next section.

I.2 SPACE-TIME SCALES AND SCALING LIMITS

We can often divide the description of the physical world according to three scales, the microscopic, mesoscopic and macroscopic one. Each of them gives a different view of the same system and a fascinating fact is that these views can be very different. An illustration is presented in Fig. I.5.

In the microscopic scale, the description is provided by the global position-velocity configuration of all particles and its evolution is described by the Newton's laws (Eq. (I.1)). In the mesoscopic one, we do no longer study each particle individually but describe the configuration of the system by a function f where f (t, x, v) represents the probability density function for a particle to be at position x with velocity v at time t. f is called the "space-velocity" density. Newton's laws are then replaced by an equation on f . Finally, in the macroscopic scale the system is described by some macroscopic quantities like the density ρ, the temperature T or the velocity V which evolve in time and space according to some coupled PDE's. To go from one scale to another we use a tool called scaling limit. In the next section, we present how we can formally use this tool in a simple case. A summary of this section, is presented in TRANSPORT EQUATION As a trivial example, we first study a gas of N identical particles leaving in the three-dimensional unit torus8 T 3 := [0, 1) 3 . Assume first that it is a very diluted gas so that the N particles evolve without crossing any other one, i.e. do not interact. We denote by x(t) := (x i (t)) i≤N and v(t) := (v i (t)) i≤N the position and velocity vectors respectively. If we assume that each particle i has a constant velocity v i then the particles are not submitted to any force and by Newton's laws (cf. Eq. (I.1)) we have Here we assumed that each particle has a mass equal to one. We define the empirical measure on T 3 × R 3 as follows

∀(x, v) ∈ T 3 × R 3 , f N (t, x, v) = 1 N N ∑ i=1 δ (x i (t),v i (t)) (x, v) . (I.4)
The function f N (t, x, v) is, up to a constant, the probability for the N particles to be at position x with velocity v at time t. We assume that the gas is isolated so that the total number of particles is constant in time hence the integral of f N with respect to the space and the velocity is constant in time and by differentiating Eq. (I.4) with respect to t, we have that f N is a weak solution of

∂ t f N + div x v f N = 0 . (I.5)
Observe that we have replaced 6N ODE (Newton's laws) by a single PDE (Eq. (I.5)). Moreover, observe that Eq. (I.5) is still reversible in time! This is due to the assumption we made on the system that the particles evolve without crossing any other one. The solution of this equation is given by

f N (t, x, v) = f N 0 (x -tv, v) , (I.6)
where f N 0 is the initial condition of Eq. (I.5). Hence, at mesoscopic scale the space-velocity density follows a transport equation and we say that the particles have a ballistic trajectory.

I.2.1 FROM NEWTON'S EQUATIONS TO BOLTZMANN'S EQUATION

INTRODUCTION

Let us now consider a more realistic representation of the gas where the particles interact with each other. Let us denote by ε ≪ 1 the radius of interactions of the particles. The derivation is performed in the so-called Boltzmann-Grad regime, i.e. by assuming that Nε 2 is of order one 9 . This assumption means that a particle travelling during a time t at velocity of order 1 will typically interact only with a finite number of particles.

A key insight applied by Boltzmann was to determine the interaction term resulting solely from two-body interactions between particles that are assumed to be uncorrelated prior to the interaction. This assumption was referred to by Boltzmann as the "Stosszahlansatz" and is also known as the "molecular chaos assumption". Boltzmann argued that the empirical measure defined in Eq. (I.4) converges to a function f solution of the following mesoscopic equation (which replaces the previous transport equation (cf. Eq. (I.5)):

∂ t f + div x (v f ) = Q [ f , f ] . (I.7)
Here Q is the so-called "collisional" operator representing the interaction between two particles. For simplicity we consider the model of hard spheres, i.e. particles travel ballistically and collide elastically when they are at distance ε of some other particles. If we denote by (v, v * ) the velocities of the particles before they collide and by (v ′ , v ′ * ) the one after we have only on the velocities and defined for suitable functions g :

v ′ = v -⟨v -v * ,
v ∈ R 3 → g(v) ∈ R by Q[g, g](v) = R 3 S 2 B (v -v * , ω) g v ′ g v ′ * -g (v) g (v * ) dωdv * , (I.9)
with v ′ and v ′ * defined in Eq. (I.8) and B a function which characterizes the collision between two particles.

The derivation of the Boltzmann's equation (Eq. (I.7)) from Newton's laws is a challenging open mathematical problem. The first step in the mathematical derivation of the Boltzmann's equation was given by Lanford in [START_REF] Lanford | Time evolution of large classical systems[END_REF] for the model of hard spheres, to our knowledge the last advanced for this model was made in [START_REF] Gallagher | From Newton to Boltzmann: hard spheres and short-range potentials[END_REF]. We refer the interested reader on this topic to the reviews [START_REF] Esposito | From particles to fluids[END_REF][START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF].

Observe that Eq. (I.9) is not reversible! Hence, at this mesoscopic scale, we lose the reversibility induced by the particles at the microscopic scale. To see it, introduce the entropy of the system at the mesoscopic scale given by which is a form of the second law of thermodynamics. It means that an arrow of time appears at the mesoscopic scale while it did not appear at the microscopic one. This apparent contradiction has been the subject of many controversies in physics. It is the consequence of the huge number of particles involved and the coarse-graining induced in the definition of the empirical measure. The huge number of particles induces a behavior which seems random and hence the loose of memory of the microscopic particles as we mentioned at the end of Sec. I.1.2.

H[ f ](t) = - T 3 R 3 f (t, x, v)log [ f (t,

THERMODYNAMIC EQUILIBRIUM

As we saw in Sec. I.1.2, a thermodynamic equilibrium of an isolated system corresponds to a maximum of the entropy of the system. This claim can be recovered here by using the H-Theorem, see Eq. (I.11). Assume that f (t) converges to a time independent distribution F in the long time-limit. Since Q satisfies the following properties

R 3 Q[ f , f ](t, x, v)φ (v)dv = 0 (I.12)
for φ (v) equal to either 1, v or v 2 2 , by using Eq. (I.7) and Eq. (I.12) we obtain the following conservation laws10

∂ t R 3 f (t, x, v)φ (v)dv + div x R 3 v f (t, x, v)φ (v)dv = 0 . (I.13)
Hence, the global thermodynamic equilibrium F(x, v) is to search among the probability distribution functions maximizing H[F] (by H-Theorem) and respecting the constraints

T 3 ×R 3 φ (v)F(x, v)dxdv = T 3 ×R 3 φ (v) f 0 (x, v)dxdv .
One can prove that F is then in the form

F(x, v) = 1 √ 2πT exp - |v -V | 2 2T , (I.14)
where the temperature T and the global velocity V are two positive constants fixed by the initial condition. Such a probability distribution is called a (global) thermodynamic equilibrium.

I.2.2 FROM BOLTZMANN'S EQUATION TO EULER'S EQUATIONS

After a longer but not infinite time, the system will reach a local thermodynamic equilibrium before to reach the global thermodynamic equilibrium discussed above. To describe this intermediate regime, i.e. to pass from the mesoscopic scale (Eq. (I.7)) to the macroscopic one we need to consider a longer time-space scale. Hence, we perform the following change of space and time variables

t ′ = τ(ε)t and u = εx , (I.15)
where ε is a small parameter and τ a positive function which goes to zero as ε goes to zero. Hence, we define f ε such that

f ε (t, u, v) = f t ′ [τ(ε)] -1 , uε -1 , v ,
where f is solution of Eq. (I.7). In this new variable we obtain

τ(ε)∂ t f ε + εv∇ u • f ε = Q [ f ε , f ε ] . (I.16)
If we consider here the Euler time scale corresponding to τ(ε) = ε and if we assume that f ε converges to some distribution g, then, by sending ε to zero in Eq. (I. [START_REF] Bernardin | Interpolation process between standard diffusion and fractional diffusion[END_REF]) we obtain

Q[g, g] = 0 .
It can be shown that such a g takes the form of a local thermodynamic equilibrium given by

g(t, x, v) = ρ(t, x) 2πT (t, x) exp - |v -V (t, x)| 2 2T (t, x) , (I.17)
with ρ, T and V defined by

ρ(t, x) = R 3 g(t, x, v)dv , V (t, x) = ρ(t, x) -1 R 3 vg(t, x, v)dv , (I.18) T (t, x) = ρ(t, x) -1 R 3 |v -V (x)| 2 g(t, x, v)dv .
Thus, formally we have that

f ε = g + o(ε) , (I.19)
where g is defined in Eq. (I.17). Then, by using the fact that f ε satisfies the conservation laws (Eq. (I.13)) we obtain by sending ε to zero in these equations that

∂ t ρ + div u (ρV ) = 0 , ∂ t (ρV ) + div u (ρV ⊗V ) + ∇ u P = 0 , ∂ t (ρE) + div u ((E + T )ρV ) = 0 . (I.20)
These are the (compressible) Euler's equations of a monoatomic ideal gas where the pressure P is equal to P = ρT , the energy

E equal to E = |V | 2 2 + 3T 2 -sum of the kinetic energy |V | 2 2
and of the internal energy 3T 2 .

At first sight these equations seem to be reversible (by changing t into -t and V into -V the equations are still satisfied). Moreover, by denoting by S := ρ log ρ 2/3 T the thermodynamical (macroscopic) entropy of the gas, we have that ∂ t S = 0. But they should be in fact irreversible because the previous computation implicitly assumes the smoothness of the solution of the compressible Euler's equations and it is known that after some time the solution will develop shocks. In fact H-theorem and the assumption that f ε converges nicely to g only implies that ∂ t S ≤ 0. We refer the interested reader to [START_REF] Bardos | Fluid dynamic limits of kinetic equations. i. formal derivations[END_REF] for a more detailed argument on the derivation of these equations.

I.2.3 FROM NEWTON'S EQUATION TO EULER'S EQUATIONS

We consider now the situation in which we want to derive some compressible Euler's equations in a single scaling limit (we call it one-step hydrodynamic limit) without first performing a kinetic limit and then a two-step hydrodynamic limit. This derivation is even more challenging than the previous scaling limits. However, some rigorous work in this direction has been done in [START_REF] Olla | Hydrodynamical limit for a hamiltonian system with weak noise[END_REF]. For technical reasons the authors replaced the unbounded classical velocity by a bounded relativistic velocity. Since Euler equations develop shocks the authors of [START_REF] Olla | Hydrodynamical limit for a hamiltonian system with weak noise[END_REF] are only able to derive their results in a short time interval in which Euler's equations do not have time to develop shocks.

In 1993, Olla, Varadhan and Yau studied in [START_REF] Olla | Hydrodynamical limit for a hamiltonian system with weak noise[END_REF] a system of N = O(ε -3 ) ≫ 1 particles living in T 3 whose Hamiltonian is given by

H OVY = N ∑ x=1     3 ∑ i=1 |p i (x)| 2 + 1 1/2 -1   + 1 2 ∑ x̸ =y U q(x) -q(y) ε   . (I.21)
Here U is a potential with compact support. Observe that in this system, there are five (globally) conserved quantities, the total number of particles, the energy of the system and the three components of the total momentum, but it is not clear that they are the only ones. We expect however that for generic potentials V they are the only ones and the aim is then to study the macroscopic equations governing the local evolution of these globally conserved quantities. Let us define the empirical measures associated to these conserved quantities in the Euler time scale tε -1

ξ ε 0 (t, du) = 1 N N ∑ x=1 δ u -q(tε -1 , x) du (I.22) ξ ε i (t, du) = 1 N N ∑ x=1 δ u -q(tε -1 , x) p i (tε -1 , x)du, ∀i ∈ {1, 2, 3} , (I.23) ξ ε 4 (t, du) = 1 N N ∑ x=1 δ u -q(tε -1 , x) e(tε -1 , x)du , (I.24) with e(t, x) =   3 ∑ i=1 |p i (t, x)| 2 + 1 1/2 -1   + 1 2 N ∑ y=1,y̸ =x U q(t, x) -q(t, y) ε (I.25)
Assume that at initial time, these quantities converge to some absolutely continuous probability measures w.r.t. Lebesgue measure whose densities are denoted by ρ 0 , V 0 i and E 0 respectively. Then, under some ergodic assumption 11 , it is proved in [START_REF] Olla | Hydrodynamical limit for a hamiltonian system with weak noise[END_REF] that at time t > 0, these quantities converge to absolutely continuous probability measures w.r.t. Lebesgue measure whose densities denoted by ρ, V i and E respectively are solution of Eq. (I.20), but with a pressure function P, function of ρ,V, E, which is not necessarily equal to ρT .

LITTLE SUMMARY

As we saw, to describe the system in different scales (micro, meso, macro) and make the connection between them, we have to perform some scaling limits. To go from microscopic to mesoscopic scale, we call this scaling limit a kinetic limit and to go from mesoscopic scale to macroscopic one we call it a two-step hydrodynamic limit. Finally, to derive a macroscopic equation for the quantities of interest (density, energy etc.) from Newton's law, we can perform some other scaling limit, the hydrodynamic limit in one-step. There is no reason a priori that the resulting Euler's equations from the hydrodynamic limit in one-step coincides with the Euler's equations (ideal gas) resulting from the two successive limits, kinetic limit and then hydrodynamic limit in two steps. The information presented in this section about scaling limits is summarised in Fig. I.7. Let us mention that we presented here only few possible scaling limits looking always the system in some Euler's time scale of order ε -1 . Of course, it makes sense to describe the system at longer time scales, e.g. diffusive time scales of order ε -2 , or to precise the next order correction term o(ε) in Eq. (I.19). For example, starting from suitable initial conditions, it is possible to derive heuristically by some scaling limits procedure the so-called incompressible Navier-Stokes-Fourier's system

∂ t V + div u (V ⊗V ) + ∇ u P = ν∆V, div u V = 0 , ∂ t T + div u (V T ) = κ∆ u T,
where the viscosity ν and the thermal conductivity κ are some constants. The last equation describes the time-space evolution of the temperature, whose evolution is the consequence of advection (corresponding to the term div u (V T )) and diffusion (corresponding to the term κ∆T ), and is therefore called an advection-diffusion equation.

In the next section, we discuss in more detail the time-space evolution of the temperature and in particular the heat equation (which is nothing but a diffusion equation) and Fourier's law. They will play a crucial role in Chapter III and Chapter IV.

I.3 FOURIER'S LAW AND HEAT DIFFUSION EQUATION I.3.1 FOURIER'S LAW

We consider a one-dimensional 12 macroscopic system which is outside of equilibrium. For example we can heat the left end to temperature T L while the right end is cooled to temperature T R < T L ; or we can start with an isolated system by preparing it with a non-homogeneous temperature. In these situations, the temperature T (t, u) of the system at time t and position u is not homogeneous in time and space and a heat current J(t, u) appears. This situation is described in Fig. I.8. The heat current, J, satisfies the conservation law

∂ t T + div u J = 0 . (I.26)
In his seminal paper [START_REF] Fourier | Théorie analytique de la chaleur[END_REF], Fourier stated that there is a strong link between the current J and the temperature T of the system given by

J = -∇ u [κ (T ) T ] , (I.27)
where κ := κ(T ) is the thermal conductivity of the system. This phenomenological law given by Eq. (I.27) is called Fourier's law. In a three-dimensional space, we experimentally know that Fourier's law is true since engineers use it to produce efficient systems of heating, refrigerators etc. However, we do not know precisely which conditions on microscopical models of heat conduction are necessary to ensure the validity of Eq. (I.27). Hence, deriving this equation from physical microscopic models of heat conduction, in particular from interacting particle systems whose evolution is ruled out by Newton's equations, is still a famous open challenging problem. In Sec. II.5 of Chapter II we will introduce a two-dimensional model where the Fourier's law is violated. This system will be deeply studied in Chapter IV.

I.3.2 HEAT DIFFUSION EQUATION

Even if we do not know precisely the conditions for the validity of Fourier's law, we can try to understand the consequences on the system. Combining Eq. (I.27) and Eq. (I.26) we get that for any u inside the box,

∂ t T = div u (κ (T ) ∇ u T ) . (I.28)
Eq. (I.28) is the heat diffusion equation 13 which is nothing but a nonlinear parabolic diffusion equation.

Observe that once again, the equation is not reversible in time. On Fig. I.9 we plot the spatial evolution of the heat diffusion equation (Eq. (I.28)) at six different times for a system with initial temperature equal to zero and where T L = 30 and T R = 0. On the left-hand side of Fig. I.9, the thermal conductivity κ is constant equal to 0.5 whereas on the right-hand side κ = 0.2. In both cases the equilibrium is reached when T is linear. This is logical, because at equilibrium, T does not depend on t and hence ∆ u [T ] = 0. Observe the differences between the evolution at time t = 14.4. The difference is due to the fact that when κ = 0.2 (which corresponds to the right-hand side of Fig. I.9) the equilibrium is reached later than for κ = 0.5. This is why κ is called the thermal conductivity, the larger it is, the more conductive the system is.

∀t > 0, ∀u ∈ R, ∂ t T (t, u) = κ∆ u T (t, u) , T (0, u) = T 0 (u) . (I.29)
Then, one can show that for any positive time t and u in R

T (t, u) = √ κ √ πt R T 0 (z) exp - κ|z -u| 2 t dz := T 0 * p κ t (u) . (I.30)
The function

p κ t (•) = √ πtκ -1 -1 exp -κ|•| 2 t
is called the heat kernel of Eq. (I.29). Deriving the linear heat equation (Eq. (I.29)) from Newton's laws is a mathematical open problem. However, instead of dealing with deterministic dynamics we want to introduce some randomness. From this point of view, it is easier to obtain the diffusion equation. In Sec. I.3.3 we will obtain it using random walks. In Sec. I.4 we will prove how we can derive Eq. (I.29), in two different ways, from the linear Boltzmann's equation (Eq. (I.39)) which is a mesoscopic equation.

I.3.3 RANDOM WALKS, BROWNIAN MOTION AND LINEAR DIFFUSION EQUATION

FROM RANDOM WALKS TO BROWNIAN MOTION

We consider a particle living in Z, starting from x in Z, and whose position Z x (t) at time t ≥ 0 is given by

Z x (t) = x + ⌊t⌋ ∑ n=1 X n , (I.31)
where (X n ) n≥1 is a sequence of i.i.d. random variables such that P (X 1 = -1) = P (X 1 = 1) = 1 2 . Such process is called a symmetric simple random walk and we refer the reader to the two books [START_REF] Andrej N Borodin | Limit theorems for functionals of random walks[END_REF][START_REF] Klafter | First steps in random walks: from tools to applications[END_REF] for a more complete introduction to random walks. Some trajectories of the particle are represented in Fig. I.10 . We interpret Z as the miscroscopic space and we want to observe the macroscopic behavior of our particle. To do it we have to perform a space-time scaling limit. In order to understand the macroscopic behavior of the particle we have to perform some space-time scaling as presented in Sec. I.2.2. Hence, for any integer N ≥ 1, we define with α a positive real number to be determined. To make the link with Sec I.2.2, N -1 plays the role of ε and N -α the one of τ(ε). To study the macroscopic behavior of the particle, one can study p N x (t, •), where p N x (t, •) is the probability distribution of Z N x (t). By Fourier's inverse formula we have

Z N x (•) by ∀t ≥ 0, Z N x (t) := N -1 Z Nx (N α t) = x + N -1 ⌊N α t⌋ ∑ n=1 X n , (
∀u ∈ R, p N x (t, u) = 1 2π R exp (-iuθ ) Φ N x (t, θ )dθ , (I.33)
where

Φ N x (t, •) is the characteristic function of the random variable Z N x (t) defined by 14 ∀θ ∈ R, Φ N x (t, θ ) = E exp iθ Z N x (t) = exp (iθ x) E exp iθ N -1 X 0 ⌊N α t⌋ , (I.34)
where in the last inequality we used that the random variables (X n ) n∈N * are identically distributed and independent. By performing a Taylor expansion in N in Eq. (I.34) we get

Φ N x (t, θ ) = exp iθ x - ⌊N α t⌋θ 2 2N 2 + O N α-3 . (I.35)
Hence, when N goes to infinity there are three cases to consider i) If α < 2, then Φ N x (t, θ ) converges to exp (iθ x), which is the Fourier's transform of the Dirac measure δ x , and hence p N x (t, u) converges to δ x (u). Thus, in this time scale, the particle does not move at the macroscopic scale. The time scale is too short to observe some macroscopic evolution.

ii

) If α = 2, then Φ N x (t, θ ) converges to exp iθ x -tθ 2 2
which is the Fourier's transform of a Gaussian random variable of density p t (•) where for any u in R

p t (u) = 1 √ 2πt exp - |u -x| 2 2t . (I.36)
Hence, p N x (t, u) converges to p t (u) which is the diffusion kernel of Eq. (I.29) with κ = 1/2. iii) If α > 2, then Φ N (t, θ ) converges to 0. In this time scale, the particle is oscillating too much at bigger and bigger distances to observe a macroscopic behavior.

In fact Donsker's Theorem [START_REF] David | An invariance principle for certain probability limit theorems[END_REF] implies that N -1 Z N xN (N 2 •) converges in law to a stochastic process B x (•) which is called the Brownian motion.

On Fig. I.11, we plot a numerical simulation of p N 0 (1, u) with α = 1 in red and α = 2 in blue. As we formally proved, when α is less than one, the probability distribution function of Z N 0 (t) is concentrated around the initial point x (here x = 0). When α = 2, the probability distribution limit is more diffusive in space. In the simulations, the random variables X n are chosen such that σ 2 = 1.

BROWNIAN MOTION AND LINEAR DIFFUSION

The name Brownian motion has its origin in the observation done by Robert Brown in 1822 of the surprising 15 cahotic movement of very fine particles on the surface of the water at rest [START_REF] Brown | a brief account of microscopical observations made in the months of june, july and august 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies[END_REF]. The aim of this section is not to give the complete story of the Brownian motion since its discovery by Brown (we refer the interested reader to [START_REF] Duplantier | Le mouvement brownien,"divers et ondoyant[END_REF]). However we would like to explain briefly the link between the linear diffusion equation (Eq. (I.29)) and the random walks theory (Eq. (I.31)) studied previously.

As seen previously Brownian motion B x (•) can be defined as the limit (in some sense) of the scaled random walks N -1 Z Nx (N 2 •) defined in Eq. (I.31). From this point of view, the Brownian motion starting from x is a Markov process (i.e. conditionally to its present history, the past history and the future history are independent) with Gaussian stationary independent increments and continuous trajectories. At any time t, B x (t) is distributed according to the probability distribution p B x (t, •) where

∀u ∈ R, p B x (t, u) = 1 √ 2πt exp - |u -x| 2 2t . (I.37)
Observe that p B x (t, •) is the heat kernel of Eq. (I.29) with κ = 1/2. Therefore we have that the solution T (t, u) of Eq. (I.29) is equal to

∀t ≥ 0, ∀u ∈ R, T (t, u) = E [T 0 (B u (t))] = R T 0 (z) 1 √ 2πtσ 2 exp - |z -u| 2 2t dz . (I.38)
Hence, Brownian motion induces diffusion.

In the next section, we will go back to mesoscopic equations with the linear Boltzmann's equation (cf. Eq. (I.39)). We will prove that from this equation we can obtain, at least formally, with a space-time scaling the diffusion equation (cf. Eq. (I.29)).

I.4 LINEAR BOLTZMANN'S EQUATION AND DIFFUSION EQUATION I.4.1 LINEAR BOLTZMANN'S EQUATION

In Sec. I.2.1, we presented the general Boltzmann's equation (Eq. I.9). In this P.h.D thesis we will be more interested into the linear one. Thus, we decided to present it and to show the strong link with the diffusion equation (Eq. (I.29)).

Consider a gas with two different types of particles, the blue ones and the red ones (see Fig. I.12). The density of the blue particles is denoted by f and we assume that the red particles are at thermodynamic equilibrium and so, distributed according to F. Formally, if we assume that the collisions between red particles can be neglected then when the blue and the red particles collide, the velocity of the blue particles goes from v to v ′ and red one from F (v ′ ) to F(v). In this context one can prove that f is no longer solution of the Boltzmann's equation (I.7) but is solution of the linear one defined as

∂ t f (t, x, v) + v∇ x f (t, x, v) = R 3 b(v, v ′ ) f t, x, v ′ F (v) -f (t, x, v) F v ′ dv ′ := Q F [ f ](t, x, v) , (I.39)
where b is a positive function. For a detailed study of Eq. (I.39) we refer the reader to [34, Chapter 3, Sec. 3] and [START_REF] Bardos | The diffusion approximation for the linear boltzmann equation with vanishing scattering coefficient[END_REF]. Deriving Eq. (I.39) from Newton's laws is a challenging problem, however we refer the interested reader on this subject to [START_REF] Ayi | From newton's law to the linear boltzmann equation without cut-off[END_REF] who proved this, starting from a system of particles interacting via a potential with infinite range.

Observe that Q F [F] = 0 and since F does not depend on x, the local equilibrium should be reached by a function of the form ρF with ρ defined in Eq. (I.18). In the next two sections we derive the macroscopic behavior of the function f , solution of Eq. (I.39) using two different methods.

I.4.2 MACROSCOPIC BEHAVIOR VIA ANALYTICAL TOOLS

In this section 16 , in order to derive the macroscopic diffusion equation from Eq. (I.39), we follow the strategy presented in Sec. I.2.2. Let f ε be the solution of Eq. (I.39) in the coordinates (t ′ , u) presented in Eq. (I.15) with τ(ε) = ε 2 , then

ε 2 ∂ t f ε (t, u, v) + εv div u f ε (t, u, v) = Q [ f ε ] (t, u, v) . (I.40)
We assume that we can write

f ε = f 0 + ε f 1 + ε 2 f 2 + R(ε)
, where R(ε) are the error terms that we will not discuss in this formal presentation. Then, by plugging this into Eq. (I.40) and grouping terms with the same power of ε we get

Q f 0 = 0 , Q f 1 = v div u f 0 , and Q f 2 = ∂ t f 0 + v div u f 1 . (I.41) Since Q f 0 = 0, we deduce that f 0 is a local equilibrium and thus f 0 (t, u, v) = ρ(t, u)F(v).
Observe that since b is a symmetric function, the integral of Q with respect to v is zero. Hence, by integrating the right-hand side of Eq. (I.41) with respect to v we obtain

∂ t R 3 f 0 (t, u, v)dv + div u R 3 v f 1 (t, u, v)dv = 0 . (I.42)
Since Q is an operator which acts only on the variable v using the fact that f 0 = ρF and that

Q f 1 = div u (v f 0 ) we get 17 f 1 = Q -1 [div u (vρF)] = Q -1 [vF] • ∇ u ρ .
(I.43) 16 I would like to thank my friend Dahmane Dechicha who showed me this method years ago. 17 In this formal presentation we assumed that Q is invertible.

By Eq. (I.42) and Eq. (I. [START_REF] Dhar | Heat transport in low-dimensional systems[END_REF]) we obtain that

∂ t ρ(t, u) = div u [D∇ u ρ(t, u)] where D = - R 3 v • Q -1 [vF] dv . (I.44)
This is a diffusion equation. Observe that if we assume that b is the constant function equal to 1 then since F is an even function we get that Q -1 [vF] = -vF and hence

∂ t ρ(t, u) = D∆ u [ρ](t, u) where D = R 3 |v| 2 F(v)dv . (I.45)
Hence, if F has a moment of order 2, D is finite and ρ is the solution of the diffusion equation (Eq. (I.29)) with diffusion coefficient D. In the case where F does not have a moment of order 2, the previous derivation fails and it turns out that the diffusion equation is replaced by a fractional diffusion equation. We will give more information about this phenomena in Sec. II.4 of Chapter II 18 . We refer the interested reader to [START_REF] Goudon | Homogenization and diffusion asymptotics of the linear boltzmann equation[END_REF] for the complete proof of Eq. (I.45) from Eq. (I.39) 19 .

I.4.3 MACROSCOPIC BEHAVIOR VIA PROBABILISTIC TOOLS

In the previous section, we studied the linear Boltzmann's equation (Eq. (I.39)) with analytical tools but in order to make the link with Chapter III we decided to show how we can also get Eq. (I.45) using a probabilistic approach. Since the functions b and F in Eq. (I.39) are symmetric we can write Eq. (I.39) in the following way

∂ t f + div x (v f ) = L[ f ] , (I.46)
where the operator L acts only on the v variable according to 20 . The operator L is the infinitesimal generator of a continuous time jump Markov process V (•). Let τ n be a sequence of i.i.d random variables such that τ 0 ∼ E (1) 21 .

L[g](v) = R 3 p(v, v ′ ) g v ′ -g (v) dv ′ , with p (v, v ′ ) = b (v, v ′ ) F (v ′ )
Then we can define the process V (•) as follows. The process starts at V 0 and is constant to V 0 during a random time τ 0 , then the process takes the value V 1 with probability p (V 0 ,V 1 ). During a random time

τ 1 , V (•) is constant equal to V 1 .
The process continues like this. Hence, if we define for any n in N the random variable T n by

T n = n ∑ k=0 τ k , (I.47)
we have

∀n ∈ N, ∀t ∈ [T n , T n+1 [ , V (t) = V n+1 and ∀t ∈ [0, T 0 [ , V (t) = V 0 . (I.48)
Such process is described in Fig. (I.13) and is called a jump Markov process. We say that the discrete process (V n , T n ) n∈N is the embedded chain of V (•) in the sense that it is sufficient to know it to recover V (•). Eq. (I.46) describes the density of a particle which starts from a position x with a velocity v. At some random time τ 0 , the particle changes its velocity to v ′ with probability p (v, v ′ ). Hence, on the time interval [0, τ 0 ] if we denote by Z x (t) the position of the particle starting initially from x at time t we have Then during the random time τ 1 , the particle has its velocity equals to v ′ hence for any time t in [T 0 , T 1 ]

Z x (t) = x -vt .
Z x (t) = Z x (τ 0 ) -v ′ (t -τ 0 ) . (I.49)
Observe that

Z x (τ 1 ) = Z x (τ 0 ) -v ′ τ 1 = x -vτ 0 -v ′ τ 1 = x -V 0 τ 0 -V 1 τ 1 . (I.50)
From this formula we deduce that if we denote by N (t) the N (t)-th renewal time we have

Z x (t) = x - N (t) ∑ n=0 V n τ n , (I.51)
where V n is the n-th velocity and τ n the n-th waiting times. Since V (•) is a jump process constant equal to V n on the interval [T n-1 , T n [ we have

Z x (t) = x - t 0 V (s)ds . (I.52)
Using Eq. (I.52) we have a "semi-explicit" formula for the solution of Eq. (I.46) given by

f (t, x, v) = E v [ f 0 (Z x (t),V (t))] , (I.53)
where E v means that we take expectation with the process V starting from v. Here f 0 is the initial condition of Eq. (I.46). Eq. (I.53) is similar to Eq. (I.38), the Brownian motion B x (•) of Eq. (I.38) is replaced by the stochastic process (Z x (•),V (•)). In order to get the macroscopic behavior of f we need to understand the one of Z x (•) as we did in Sec. I.3.3 to study the random walk. Observe that Eq. (I.51) is close to Eq. (I.32). Indeed if we denote by X n the random variable τ n V n then we can write Eq. (I.51) in the following way

Z x (t) = x - N (t) ∑ n=0 X n , (I.54)
which is an expression really close to the one of Eq. (I.32). Thus, as we formally saw in Section. I.3.3, if the random variables X n are in L 2 up to a space-time scaling Z x (•) converges to a Brownian motion and induce some diffusive behavior. Indeed, one can prove that if we scale defined ρ N in the following way

ρ N (t, x) := R f Nt, N 2 x, v dv , (I.55)
then ρ N converges (in some sense) to ρ B x where

ρ B x (t, u) = 1 √ 2πtσ 2 exp - |u -x| 2 2tσ 2 where σ 2 = V (X 0 ) = V (τ 0 V 0 ) . (I.56)
If we assume, as we did at the end of Sec. I.4.2 that b is the constant function equals to 1 we have that p (v, v ′ ) = F(v) and then

V (X 0 ) = E τ 2 0 E V 2 0 -[E (τ 0 ) E (V 0 )] 2 = +∞ 0 y 2 exp(-y)dy R 3 |v| 2 F(v)dv - +∞ 0 y 2 exp(-y)dy R 3 vF(v)dv = 2 R 3 |v| 2 F(v)dv .
Hence, ρ B x is solution of Eq. (I.45) with D = σ 2 /2. We will give an explicit proof of this kind of results in Chapter III. If the chain (X n ) n∈N is not in L 2 , then instead of having a diffusion equation at macroscopic scale, we will have a fractional diffusion equation. Some probabilistic references on this topic are [START_REF] Durrett | Functional limit theorems for dependent variables[END_REF][START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF].

II. INTRODUCTION TO THE THESIS

The aim of the thesis is to study the transport of energy in harmonic systems of coupled oscillators submitted to a magnetic field. In Sec. II.1, we give a general presentation of anharmonic chain of coupled oscillators without the presence of a magnetic field. We also present some results which seem to us very important to understand the scientific interest and the phenomena studied in Chapter III. In Sec. II.2, we describe four setups to study the transport of energy in anharmonic systems. Two of them, presented in Sec. II.2.1 and in Sec. II.2.4, will be mainly focused in Chapter IV and III respectively. Sec. II.3 introduce some historical results on anharmonic systems. In Sec. II.4, we present the results of our paper [START_REF] Cane | Superdiffusion transition for a noisy harmonic chain subject to a magnetic field[END_REF] about the superdiffusion of energy in a noisy harmonic chain submitted to a constant magnetic field. Finally, in Sec. II.5 we introduce the results of Chapter IV extracted from our papers [START_REF] Majeed Bhat | Heat transport in an ordered harmonic chain in presence of a uniform magnetic field[END_REF][START_REF] Cane | Localization effects due to a random magnetic field on heat transport in a harmonic chain[END_REF] where we studied a harmonic chain of size N submitted to a random magnetic field.

II.1 FPUT CHAINS

In isolating crystals, heat is transported by lattice vibrations and systems of coupled anharmonic oscillators have been used as microscopic models for heat conduction in metals. We consider a onedimensional chain of coupled anharmonic oscillators. At rest, the atoms are aligned according to a lattice1 Λ ⊂ Z d with d ≥ 1 and each x in Λ represents the rest position of one atom2 . We denote the displacement from its rest position of the atom with rest position x by q(x) ∈ R, the momentum by p(x) = m x v(x) ∈ R where m x denotes its mass and v(x) its velocity. We denote a typical configuration of the system by (q, p) := (q(x), p(x)) x∈Λ ∈ (R × R) Λ and the configuration over time by {(q(t), p(t)) | t ≥ 0} := (q(t, x), p(t, x)) x∈Λ | t ≥ 0 . The Hamiltonian of a generic system is given by

H := H (q, p) = ∑ x∈Λ     |p(x)| 2 2m x +U(q(x)) + ∑ |x-y|=1 x,y∈Λ V (q(x) -q(y))     . (II.1)
Here, V is the interaction potential between nearest neighbor atoms and U is an external (also called on site or pining) potential representing for example the interaction with a substrate. Newton's laws give II.1 FPUT CHAINS that for any time t ≥ 0 and x in Λ d dt q(t, x) = ∂ p(x) H (q(t), p(t)) and d dt p(t, x) = -∂ q(x) H (q(t), p(t)) .

(II.2)

Above, we considered free boundary conditions but other physical boundary conditions could be implemented in this equations (e.g. fixing the positions of the atoms at the boundaries of Λ). Liouville's operator associated to the Hamiltonian defined in Eq. (II.1) is given by

A = ∑ x∈Λ ∂ p(x) [H ] ∂ q(x) -∂ q(x) [H ] ∂ p(x) (II.3)
and is in fact the infinitesimal generator of the interacting particle system.

CONSERVATION LAWS

The total energy of the system at time t, denoted by E (t), is defined by

E (t) := H (q(t), p(t)) = ∑ x∈Λ e(t, x) , (II.4)
where e(t, x) denotes the energy 3 of the particle x at time t. It is defined by

e(t, x) = 1 2 p(t, x) 2 m x
+V (q(t, x + 1)q(t, x)) +U(q(t, x)) .

(II.5)

The total energy E is a constant function, i.e. for any time t we have

E (t) = E (0) . (II.6)
If the chain is unpinned, i.e. U = 0, there are two other conserved quantities associated to the dynamics. In this case, it is easier to study the configuration (r, p) instead of (q, p) where for any x and any time t we have r(t, x) = q(t, x + 1)q(t, x) .

(II.7)

Then, the two following quantities are also invariant (in the bulk, i.e. by ignoring the boundaries effects of Λ or by taking Λ infinite or periodic) over time

P(t) := ∑ x∈Λ p(t, x) total momentum , (II.8) R(t) := ∑ x∈Λ r(t, x) total elongation . (II.9)
Observe that a priori there could exist other conserved quantities and it is a very non trivial task to know if the ones mentioned above are the only ones. It is widely accepted that apart from very special cases of potentials, they are the only conserved ones, at least when Λ is very large (mathematically when Λ = Z). The question of the existence of other conserved quantities is intrinsically related to the ergodic properties of the chain. 3 It could also be defined by e(t, x) = 1 4 2p(t,x) 2 m x +V (q(t, x + 1)q(t, x)) +V (q(t, x)q(t, x -1)) +U(q(t, x)) but for mathematical reasons it is simpler to define it by putting all the interaction energy of the bond (x, x + 1) to the particle x. Moreover, the given definition has to be modified at the right boundary of Λ by if x, x + 1 ∈ Λ, and by e(t, x) = p(t,x) 2 2m x +U(q(t, x)).

GIBBS ENSEMBLES AND ERGODIC HYPOTHESIS

Ergodic hypothesis is a crucial assumption for the foundation of statistical mechanics. We will not give a complete introduction to the ergodic hypothesis but we refer the interested reader to [START_REF] Joel | Modern ergodic theory[END_REF][START_REF] Patrascioiu | The ergodic hypothesis[END_REF] and to [START_REF] Walters | An introduction to ergodic theory[END_REF] for a mathematical introduction to ergodic theory.

Assume the chain is unpinned 4 , i.e. U = 0, and that Λ = T 1 N := Z/NZ is the one-dimensional discrete torus5 of length N. Let Σ E,P,R := {(r, p) ∈ (R × R) Λ |E = E, P = P, R = R} be the surface of total energy E > 0, total momentum P ∈ R and total elongation R ∈ R, which is left invariant by the dynamics since the energy, the momentum and the elongation are conserved by the dynamics. Liouville's theorem states that the uniform probability measure ν := ν E,P,R on Σ E,P,R , called micro-canonical measure, is a stationary measure of the system. Consequently the canonical Gibbs (probability) measure µ := µ β ,λ ,γ defined by

µ(dr, d p) = 1 Z β ,λ ,γ exp (-β E -λ P -γR) drd p , (II.10)
is also invariant for any β > 0 (inverse of the temperature), λ ∈ R and γ ∈ R. The partition function Z β ,λ ,γ is a normalization constant. Since other conserved quantities could exist, it is not clear that ν is the unique stationary measure of the dynamics restricted to Σ E,P,R but the ergodic hypothesis stated below implies it is the only one (if we restrict our study to measures absolutely continuous with respect to the Lebesgue measure).

We say that the system satisfies the ergodic hypothesis if for any values of E, P, R and for any smooth function

F : (R × R) Λ → R, lim T →∞ 1 T T 0 F (q(t), p(t)) dt = Σ E,P,R F (q, p) dν (q, p) (II.11)
holds for almost every initial configuration (q(0), p(0)) ∈ Σ E,P,R . Hence, ergodic hypothesis states that average in space is equal to average in time. The left-hand side of Eq. (II.11) is a quantity we can obtained from simulation or doing numerical/real experiments whereas the right-hand side is the one we can compute explicitly using statistical physics formalism. Hence, in some sense, ergodic hypothesis is the bridge between the theory and the experiment.

II.1.1 HARMONIC CHAIN AND THE ERGODIC HYPOTHESIS

The first natural model of chain of coupled oscillators to study is when it is unpinned (U = 0) and the interaction potential V in Eq. (II.1) is quadratic, namely V (q) = q 2 2 with lattice6 Λ = T 1 N . This is the main system studied in this thesis.

In order to lighten and simplify the presentation, we assume that for every x, m x = 1. The equations of motion are then given by d dt q(t, x) = p(t, x) and d dt p(t, x) = q(t, x + 1) + q(t, x -1) -2q(t, x) .

(II.12)

Using Eq. (II.7) and Eq. (II.5) the energy of a particle x at time t is given by e(t, x) = p(t, x) The harmonic chain does not satisfy the ergodic hypothesis in the sense above. Indeed, observe that Eq. (II.12) is linear, then it is natural to use Fourier's transform to study these equations. In Fourier's space these equations become

d dt q(t, k) = p(t, k) and d dt p(t, k) = -4π 2 sin 2 πk N q(t, k) , (II.16)
where for any function f :

T 1 N → R, the function f : T 1 N → C is the discrete Fourier's transform of f defined as ∀k ∈ T 1 N , f (k) = N ∑ x=1 exp 2iπkx N f (x) .
(II.17)

We define ω 0 :

T → R by ∀k ∈ T, ω 0 (k) = 4π 2 sin 2 (πk) = 2π |sin (πk)| , (II.18)
where T is the one-dimensional (continuous) torus. For any configuration (q, p) we define the wave function ψ :

T 1 N → C by ∀k ∈ T 1 N , ψ(k) := ψ [q, p] (k) = 1 √ 2 p(k) -iω 0 kN -1 q(k) . (II.19)
The evolution of ψ(t, k) := ψ [q(t), p(t)] (k) encapsulates the dynamics of the harmonic chain. Using Eq. (II.16) we have

d dt ψ(t, k) = -iω 0 kN -1 ψ(t, k) which leads to ψ(t, k) = exp -iω 0 kN -1 t ψ(0, k) . (II.20)
For each k in T 1 N , ψ(k) is a fictive particle called phonon. The energy of the phonon is defined by | ψ(k)| 2 and since for any positive time t and k in T

1 N we have | ψ(t, k)| 2 = | ψ(0, k)| 2 , (II.21)
the energy of each phonon with wave number k is conserved during the time evolution. Using Plancherel's formula we have that

∥ ψ(t)∥ 2 L 2 := ∑ k∈T 1 N | ψ(t, k)| 2 = E (t) = E (0) . (II.22)
Due to the linearity of the system, each phonon evolves without any interactions with the other ones, like if we had an ideal gas of phonons. Assume the harmonic chain is ergodic. Then for a given k ∈ T 1 N , in Eq. (II.11), we could take F(q, p) = | ψ[q, p](k)| 2 and by using Eq. (II.21) we would obtain

| ψ(0, k)| 2 = Σ E,P,R | ψ[q, p](k)| 2 dν (q, p) .
(II.23)

The right-hand side is some positive number depending on the initial total energy, the total momentum and the total elongation. But, if we choose the initial condition such that ψ(0, k) = 0 then, the left hand side is equal to zero but not the right hand side. Hence, this equality is false. This proves that the harmonic chain does not satisfy the ergodic hypothesis.

II.1.2 ANHARMONIC CHAINS AND ERGODIC HYPOTHESIS

Eq. (II.11) is the historical statement of the ergodic hypothesis but we know that this statement is not correct for the harmonic chain. This is due to the existence of other conserved quantities apart from the total energy, momentum and elongation (the energies of the phonons). In fact, the harmonic chain belongs to the class of integrable systems which are never ergodic in the sense of the ergodic hypothesis.

This observation motivates the idea of introduce some nonlinearity in the model to prove that the ergodic hypothesis is restored, as we usually assume in the foundations of statistical mechanics. In 1955, Fermi, Pasta, Ulam and Tsingou [START_REF] Fermi | Studies of the nonlinear problems[END_REF] studied a system of 64 particles with periodic boundary conditions where in Eq. (II.1) we have U = 0 and

V (r) = r 2 2 + α r 3 3 + β r 4 4 , (II.24)
with α and β two positive number. This chain is named as the FPUT chain, more precisely we speak about FPUT-α (resp. FPUT-β ) chains if β = 0 (resp. α = 0). In such system, the equations of motion of the particles are no longer linear and then phonons ψ(k) interact with each other. From this observation, Fermi expected, as it is presented in [60, Chapter 1], that the nonlinearity will lead to the ergodic hypothesis. However the numerical simulations they performed did not allow to observe this. An explanation of this result could be that the system size N = 64 was too small. However, it is not clear that this strong ergodic hypothesis is valid for generic anharmonic chains. For example the Toda's chain [START_REF] Toda | Recent advances in the theory of nonlinear lattices[END_REF] given by Eq. (II.1) with U = 0 and

V (r) = exp(-r) + r -1
is a completely nonlinear integrable system and therefore does not satisfy the ergodic hypothesis.

Since this seminal numerical experiments, 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. Before introducing further the models studied in this thesis, we give a brief summary of the different ways to study energy transport in FPUT models.

II.2 SETUPS TO STUDY THE TRANSPORT OF ENERGY

In this section, we describe four different setups to study the diffusion of the energy in a FPUT system. Two of them will be used in Chapter III and IV.

II.2.1 NON EQUILIBRIUM STATIONARY STATE

The first way to study the transport of energy in a chain of oscillators is to follow the study made by Rieder, Lieb and Lebowitz in their seminal paper [START_REF] Rieder | Properties of a harmonic crystal in a stationary nonequilibrium state[END_REF]. In this setup we connect the lattice to two heat baths with different temperatures at the boundaries, then a heat current inside the system appears. Waiting for a long time, the system will reach a stationary state called by physicists a Non Equilibrium Stationary State (NESS). The term Non Equilibrium being due to the presence of a non zero energy current. In order to study the diffusive character of the system it "is sufficient" to obtain some properties on the NESS of the system. Mathematically, this is a difficult task since, most of the time, we do not have an explicit expression for the NESS (if the temperatures are equal the NESS is given by the explicit canonical ensemble7 defined by Eq. (II.10) ). Let ⟨•⟩ denote the average w.r.t. the NESS. Since we will use this approach in Chapter IV, we decided to recall the historical study of Rieder, Lieb and Lebowtiz in [START_REF] Rieder | Properties of a harmonic crystal in a stationary nonequilibrium state[END_REF].

In 1967, Rieder, Lieb and Lebowitz studied in [START_REF] Rieder | Properties of a harmonic crystal in a stationary nonequilibrium state[END_REF] a one-dimensional homogeneous (i.e. m x = 1 for all x) harmonic chain of size N connected to two heat baths at the left-hand side and right-hand side with respective temperature T L = T + 1 2 ∆T and T R = T -1 2 ∆T where ∆T > 0, i.e. T L > T R . The situation is described in

H = N ∑ x=0 |p(x)| 2 2 + (q(x + 1) -q(x)) 2 2 , (II.25)
supplemented by fixed boundary conditions q (t, 0) = q (t, N + 1). Let δ x,y be the Kronecker symbol, using Eq. (II.2), we have that

d p(t, x) = (q(t, x + 1) + q(t, x -1) -2q(t, x)) dt + η L (t)δ x,1 + η R (t)δ x,N dt -(δ x,1 + δ x,N ) p(t, x)dt . (II.26)
Here, η L/R are white noise and follow the fluctuation-dissipation relation

η L/R (t) η L/R t ′ = 2T L/R δ t -t ′ ,
The blue term in Eq. (II.26) models the heat baths as Langevin baths.

As we saw at the beginning of Sec. II.1 the energy of the system is preserved in the bulk, but not at the boundaries. If we denote the energy of particle x at time t by e(t, x), there exists a microscopic energy current from particle x to particle x + 1, denoted by j x,x+1 (t), such that

d dt e(t, x) = j x-1,x (t) -j x,x+1 (t) . (II.27)
The expression of the current is given by Eq. (II.14) in the bulk and has to be modified suitably at the boundaries of the chain to take into account the energy transfer between the reservoirs and the system which leads to

j 0,1 = T L -p 2 1 2 and j N,N+1 = - T R -p 2 N 2 .
(II.28)

Then we have that for each x, ⟨ j x,x+1 ⟩ = ⟨ j 0,1 ⟩ . As we saw in Sec. I.3.1 of Chapter I, if Fourier's law (cf. Eq. (I.27)) is valid, we should have as N → ∞ and then ∆T → 0 that

⟨ j 0,1 ⟩ ∼ -κ(T ) ∆T N . (II.29)
This microscopic current defines a macroscopic current J := J N in the NESS by Hence, the heat conductivity of the finite system κ N = J N /∆T ∼ N is independent of T but is proportional to the size of the system and not of order one. It follows that κ(T ) = ∞, i.e. the harmonic chain is a superconductor and does not satisfy Fourier's law. Since κ N ∼ N, we say that the transport of energy in the chain is ballistic.

J N = N ⟨ j 0,

II.2.2 GREEN-KUBO FORMULA

An other way to study the conductivity in the FPUT-chains is to use the Green-Kubo formula which is the corner stone of the linear response theory. Even if we will not use this tool in Chapter III and IV, we decided to recall this formula since our model was first studied in this framework.

If the difference of the temperatures ∆T is equal to zero the NESS coincides with the Gibbs canonical ensemble. If ∆T ̸ = 0, but is small, we expect that the NESS can be approximated by performing an order one perturbation in ∆T in the stationary Fokker-Planck equation defining the NESS. We can then derive an alternative expression for the conductivity κ(T ) which has the big advantage to invoke only the dynamics at Gibbs equilibrium with temperature T , but the disadvantage to be in infinite volume.

Therefore, we consider here the dynamics in infinite volume (Λ = Z) with initial condition distributed according to a Gibbs measure, see Eq (II.10), µ with inverse temperature β = T -1 and λ = γ = 0. Recall the definition of the energy current j x,x+1 in Eq. (II.27). Linear response theory predicts that if the system is diffusive then the value κ(T ) of the system is given by the Green-Kubo formula

κ(T ) = 1 T 2 ∑ x∈Z ∞ 0 E µ [ j x,x+1 (t) j 0,1 (0)] dt := 1 T 2 ∑ x∈Z ∞ 0 C j (t)dt , (II.33)
where for any positive time t

C j (t) = E µ [ j x,x+1 (t) j 0,1 (0)] . (II.34)
Therefore, a way to prove that a system is diffusive (κ(T ) < ∞) or superdiffusive (κ(T ) = ∞) is to study the decay of the correlation function C j of the infinite system at equilibrium. Moreover it is believed that if we truncate the integral in Eq. (II.33) at time of order N, then we get a good approximation of the conductivity κ N of the finite system of size N, defined in the previous setup.

II.2.3 HYDRODYNAMIC LIMIT IN ONE STEP

Consider the unpinned anharmonic chain and assume that the lattice is Z and that each particle x has mass equal to one. In this section, we perform the study using the variable (r, p) defined in Eq. (II.7). We recall that the energy, the momentum and the elongation satisfy the conservation laws presented in Eq. (II.15) and in Eq. (II.14) 8 .

Let ε be a positive number. We say that the sequence of probability measures (µ ε ) ε>0 on (R × R) Z is a local equilibrium associated to the macroscopic energy profile e 0 : R → (0, ∞), the macroscopic deformation profile r 0 : R → R and the macroscopic momentum profile p 0 : R → R if for every local 9function F := F(r, p) and any u ∈ R, we have lim ε→0 τ [ε -1 u] F (r, p)dµ ε (r, p) -F(r, p)dµ β 0 (u),λ 0 (u),γ 0 (u) (r, p) = 0 , where µ β ,λ ,γ is the canonical Gibbs ensemble defined by Eq. (II.10) and τ z is the usual shift operator on the lattice 10 . The relations between the profiles e 0 , r 0 , p 0 and the profiles β 0 , λ 0 , γ 0 are obtained through the thermodynamic relations e(x) dµ β 0 (u),λ 0 (u),γ 0 (u) = e 0 (u), r(x) dµ β 0 (u),λ 0 (u),γ 0 (u) = r 0 (u), p(x) dµ β 0 (u),λ 0 (u),γ 0 (u) = p 0 (u) .

(II.35) It implies in particular that for every smooth functions J : R → R and any δ > 0

lim ε→0 µ ε ε ∑ x∈Z J (εx) e(0, x) - R J(u)e 0 (u)du > δ = 0 , (II.36) lim ε→0 µ ε ε ∑ x∈Z J (εx) r(0, x) - R J(u)r 0 (u)du > δ = 0 , (II.37) lim ε→0 µ ε ε ∑ x∈Z J (εx) p(0, x) - R J(u)p 0 (u)du > δ = 0 . (II.38)
For example, the probability measures (µ ε ) ε>0 defined by

dµ ε (dr, d p) = Z -1 ε exp -∑ x∈Z {β 0 (εx)e(x)) + λ 0 (εx)p(x) + γ 0 (εx)r x } drd p ,
where β 0 , λ 0 , γ 0 are three (macroscopic) functions, form a sequence of local equilibria (called local Gibbs states) associated to the macroscopic energy profile e 0 : R → (0, ∞), the macroscopic deformation profile r 0 : R → R and the macroscopic momentum profile p 0 : R → R related to β 0 , λ 0 , γ 0 through the one to one mapping (II. [START_REF] Cesbron | Fractional diffusion limit for a kinetic equation in the upper-half space with diffusive boundary conditions[END_REF].

One way to study the transport of energy in FPUT chain is therefore to start from a local equilibrium like above at initial time and to prove that this local equilibrium property propagates until time tε -a with new macroscopic profiles e(t, •), r(t, •), p(t, •) instead of e 0 , r 0 , p 0 , solutions of suitable coupled PDE's. The parameter a > 0 defines the time scale.

If we assume that in the time scale tε -1 the probability distribution of the system is close to a local equilibrium associated to macroscopic profiles e(t, •), r(t, •), p(t, •), by using Eq. (II.15) and Eq. (II.14), we have immediately that for any smooth function J

lim ε→0 ε ∑ x∈Z J (εx) E µ ε e ε -1 t, x = R J(u)e(t, u)du , lim ε→0 ε ∑ x∈Z J (εx) E µ ε p ε -1 t, x = R J(u)p(t, u)du , lim ε→0 ε ∑ x∈Z J (εx) E µ ε r ε -1 t, x = R J(u)r(t, u)du ,
where r, p and e are solutions of the following compressible Euler's equations

∂ t r(t, u) = ∂ u p(t, u), ∂ t p(t, u) = ∂ u τ [r] (t, u), ∂ t e(t, u) = ∂ u [τ(r)p] (t, u) , (II.39)
with initial conditions r 0 , p 0 , e 0 . Here τ is a suitable thermodynamic function (equals to τ(r) = r in the harmonic case).

Obtaining local equilibrium propagation in nonlinear systems is very difficult due to the lack of control of the ergodic properties of the system. For example, for the harmonic chain it is known that the Euler's equations do not hold [START_REF] Rl Dobrushin | One-dimensional harmonic lattice caricature of hydrodynamics[END_REF]. Moreover, a well defined theory of existence and uniqueness of solutions to compressible Euler's equations is missing and it is known that the solutions will not be smooth (they will develop shocks after a finite time). We will see in Sec. II.3 how to perturb the system by a stochastic noise in order to have a better control of the ergodic properties of the system without destroying the conservation laws we are interested in. Euler's equations have been derived in noisy systems in [START_REF] Olla | Hydrodynamical limit for a hamiltonian system with weak noise[END_REF] and in disordered ones [START_REF] Bernardin | Hydrodynamic limit for a disordered harmonic chain[END_REF].

Assume now that we start with a local equilibrium με such that the momentum does not evolve in Euler's time scale. For example, assume that the initial distribution of the system με satisfies that for any smooth functions J

lim ε→0 ∑ x∈Z J(εx)E με [e(0, x)] = R J(u)e 0 (u)du , lim ε→0 ∑ x∈Z J(εx)E με [p(0, x)] = 0 , and lim ε→0 ∑ x∈Z J(εx)E με [r(0, x)] = 0 .
Then, in such settings we should not see any evolution of the energy profile in the Euler time scale

lim ε→0 ∑ x∈Z J(εx)E με e tε -1 , x = R J(u)e 0 (u)du , (II.40)
but it is expected that in the diffusive time scale we have

lim ε→0 ∑ x∈Z J(εx)E με e tε -2 , x = R J(u)e(t, u)du , (II.41)
where e is solution of some diffusion equation with initial condition e 0 . The diffusion coefficient appearing in this equation should coincide with the Green-Kubo formula defined in Eq. (II.33). A result as in Eq. (II.41) gives the macroscopic behavior of the energy starting from a microscopic description of the system in the diffusive time scale. This is again a challenging problem and outside of the range of mathematical technics for deterministic systems.

We saw in Chapter I different ways to obtain macroscopic equations. The most ambitious way is to derive these equations starting directly from the Newton's laws as we explained just above. An other way is to use the two-step hydrodynamics limit as we explained in Chapter I. This is the idea that motivates the use of the Wigner's functional. This tool will be central in Chapter III.

II.2.4 WIGNER'S FUNCTION AND PHONONIC BOLTZMANN'S EQUATION

In 1929 , Peierls [START_REF] Peierls | Zur kinetischen theorie der wärmeleitung in kristallen[END_REF] proposed to study the energy transport by perturbative methods, i.e. for weakly anharmonic chains, seen as a small perturbation of harmonic systems. He stated that this will lead to a kinetic description of a gas of phonons in interaction and then to a phononic Boltzmann's equation. His approach was used to study the conductivity of anharmonic systems and we refer the interested reader to [82, Section 5] for more information. In 2005, Spohn [START_REF] Spohn | The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics[END_REF] revisited and developed the idea of Peierls [START_REF] Peierls | Zur kinetischen theorie der wärmeleitung in kristallen[END_REF] to study a weakly anharmonic lattice. Namely the Hamiltonian of the system is defined with

U = 0 , V (q) = q 2 + γ √ εV 1 (q) and Λ = Z in Eq. (II.1) , (II.42)
where ε is a scaling parameter, γ ≥ 0 and V 1 an anharmonic potential 11 . Let ψ be the wave function associated to this system defined by

∀k ∈ T, ψ(k) = 1 √ 2 ( p(k) -iω 0 (k) q(k)) with ω 0 (k) = 4π 2 sin 2 (πk) . (II.43)
Here for any function f : Z → R, the function f is the Fourier's transform of f defined by

∀k ∈ T, f (k) = ∑ x∈Z f (x) exp (2iπkx) , (II.44)
where we recall that T = [0, 1[ is the one-dimensional (continuous) torus. Observe that

1 2 ∥ ψ∥ 2 L 2 (T) = 1 2 T | ψ(k)| 2 dk = E , (II.45)
where we recall that E is the total energy of the system that we assumed to be finite. Let (ν ε ) ε>0 be a family of probability measure on the phase space such that for any

(k, k ′ ) in T 2 E ν ε [ ψ(k)] = 0 , E ν ε ψ(k) ψ k ′ = 0 and E ν ε ∥ ψ(•)∥ 2 L 2 (T) = O ε -1 . (II.46)
The one point Wigner's function, W ε associated to this dynamics is defined as

∀(x, k) ∈ (Z/2) × T, W ε (t, x, k) = 1 2 2T exp (2iπxp) E ν ε ψ t, k - p 2 * ψ t, k + p 2 d p .
Hence, formally W ε measures the correlation between ψ(k) and ψ(k ′ ) for any k, k ′ . The idea of Spohn by introducing this function was to follow the derivation of the Boltzmann's equation (cf. Eq. (I.7) of Chapter I) in the small density regime. To derive the phononic Boltzmann's equation, the idea is to replace the role played by the density function f in the kinetic theory of gas by the Wigner's function in the system of phonons (recall that when the system is without nonlinearities the phonons are not interacting similarly to what happens when the density is small for the atoms in a usual gas). Hence, in [START_REF] Spohn | The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics[END_REF] Spohn studied the behavior of W ε when ε goes to zero. Since we want to obtain a kinetic equation, we study W ε rescaled in the following way

W ε tε -1 , y, k = ε 2 2Tε -1 exp (2iπyp) E ν ε ψ tε -1 , k - ε p 2 * ψ tε -1 , k + ε p 2 d p . (II.47)
Here y is in (εZ/2) and k is in T. Instead of studying W ε , it is easier to study its (continuous) Fourier's inverse W ε where

∀(p, k) ∈ ε -1 T × T, W ε tε -1 , p, k = E ν ε ψ tε -1 , k - ε p 2 * ψ tε -1 , k + ε p 2 . (II.48)
Assume, in a first time, that γ = 0, then the system is harmonic and using the equations of motion we have

d dt ψ(t, k) = -iω 0 (k) ψ(t, k) which leads to ψ(t, k) = exp (-iω 0 (k)t) ψ(0, k) . (II.49)
Using Eq. (II.49) we have

W ε (p, k) = exp -itε -1 ω 0 k + ε p 2 -ω 0 k - ε p 2 W ε (0, p, k) .
(II.50)

II.3 DIFFUSION AND SUPERDIFFUSION IN ANHARMONIC SYSTEMS

We assume that for any p in R

lim ε→0 W ε 0, ε p ε , k = W (p, k) . (II.51)
Then, using Eq. (II.50) and Eq. (II.51) we formally obtain that W ε converges to W , where for any positive time t and

(p, k) in R × T ∂ t W (t, p, k) + ipω ′ 0 (k) W (t, p, k) = 0 . (II.52)
By inverting the (continuous) Fourier's transform we obtain for any u in R and k in T that

∂ t W (t, u, k) + 1 2π ω ′ 0 (k)∇ u W (t, u, k) = 0 . (II.53)
The energy at the kinetic scale is defined by

E(t, u) = T W (t, u, k)dk . (II.54)
Hence, using Eq. (II.53) we have the following conservation law

∂ t E(t, u) + ∂ u J e (t, u) = 0 , where J e (t, u) = (2π) -1 T W (t, u, k)ω ′ 0 (k)dk . (II.55)
Assuming now that γ ̸ = 0, Spohn argued that when ε goes to zero, the transport equation (II.53) is replaced by a phononic Boltzmann's equation in the form

∀(u, k) ∈ R × T, ∂ t W (t, u, k) + 1 2π ω ′ 0 (k)∇ u W (t, u, k) = Q [W , W ] (t, u, k) . (II.56)
Here, Q is a collisional operator whose expression is not important in this presentation. This equation is similar to Eq. (I.7) of Chapter I but here instead of studying the evolution of a gas of physical particles, we study the evolution of a gas of phonons (indexed by their wave number k in T).

This paper of Spohn was fundamental in the sense that it gives a tool to study anharmonic systems. We refer the interested reader to [START_REF] Lukkarinen | Kinetic theory of phonons in weakly anharmonic particle chains[END_REF] for more information on this topic.

II.3 DIFFUSION AND SUPERDIFFUSION IN ANHARMONIC SYSTEMS

In this section, we shortly present some results obtained by physicists about the transport of energy in anharmonic chains. Most of the results mentioned in Sec. II.3.1 are formal and based on the study of the Green-Kubo formula. Sec. II.3.2 is devoted to a very short review about the (non rigorous) nonlinear fluctuating hydrodynamics theory of Spohn to predict the form of the anomalous diffusion of energy in anharmonic lattices.

II.3.1 TWO FUNDAMENTAL PROPER TIES

At the end of the twentieth century, a lot of study were performed to understand the behavior of the heat conductivity of anharmonic chains. Due to the non linearity of the system, theoretical studies are very complicated. However, using Green-Kubo formula (cf. Eq. (II.33)), some progress has been made using numerical simulations. All of these studies allow the scientific community to highlight two crucial properties at the origin of the normal or anomalous diffusion of energy in chains of oscillators, the conservation of momentum (see however [START_REF] Komorowski | Diffusive propagation of energy in a non-acoustic chain[END_REF] ) and the dimension12 of the lattice.

As we mentioned in Sec. II.2, one way to study the conductivity of the system is to use the Green-Kubo formula. More precisely, the conductivity of the system is completely determined by the current-current correlation functions C j (t) defined in Eq. (II.34). Hence, the idea at the end of the twentieth century, to study transport properties of anharmonic chains, was to simulate them at equilibrium (without heat baths) and in the NESS (with heat baths). The aim was to understand, for large system size N, how the power law for the conductivity was depending on the form of the potentials i.e. κ N ∼ N δ .

FPUT-α (resp.FPUT-β ) chains were studied numerically and different authors suggested some values of δ in (0, 1), without a global agreement. In [START_REF] Lepri | On the anomalous thermal conductivity of one-dimensional lattices[END_REF], using the results of the numerical simulation and a formal argument based on the Green-Kubo's formula, Lepri, Livi and Politi stated that the conservation of the momentum plays a crucial role in this anomalous diffusion of energy as well as the dimension of the lattice (d in Eq. (II.1)). Namely, they stated that 1. for d = 1, the conductivity κ N should be of order N δ with δ in (0, 1) if the momentum is preserved, and of order one if momentum is not preserved (pinned chains), 2. for d = 2, the conductivity κ N should be of order log N if the momentum is preserved and of order one if momentum is not preserved, 3. for d ≥ 3, the conductivity should be finite whatever the momentum is preserved or not.

For more information about the study of the conductivity in the chains of coupled oscillators, we refer the interested reader to the following reviews [START_REF] Dhar | Heat transport in low-dimensional systems[END_REF][START_REF] Lepri | Thermal conduction in classical low-dimensional lattices[END_REF] and to references therein.

II.3.2 UNIVERSALITY CLASSES

As we mentioned above, the numerical simulations were not completely conclusive about the values of δ and how it was depending on the microscopic details of the system. Moreover, numerical simulations and theoretical arguments were not sufficient to really understand the form of the superdiffusion. The most elaborated theory to describe the form of this anomalous transport is probably the recent nonlinear fluctuating hydrodynamics theory initiated by Spohn [START_REF] Spohn | The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics[END_REF][START_REF] Spohn | Nonlinear fluctuating hydrodynamics for anharmonic chains[END_REF]. This theory 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 [START_REF] Quastel | The one-dimensional kpz equation and its universality class[END_REF] 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 Euler's equations (cf. Sec. II.2.3 and Eq. (II.39)) associated to the interacting particle system under investigation. For example, this theory states that if an unpinned anharmonic chain has an even interaction potential (hence preserving the momentum) its diffusion should satisfies a fractional diffusion equation with exponent 3/4.

II.4 DIFFUSION AND SUPERDIFFUSION OF ENERGY IN NOISY SYSTEMS

Since the study of anharmonic systems is too difficult, in particular to control their ergodic properties, an idea to study it is to add a stochastic noise. Then, even in the harmonic case, the system has interesting features. The aim of this section is to introduce Chapter III in which we study a noisy harmonic chain which preserves some quantities (pseudo-momentum, energy, volume). We first review the origin of these models. In Sec. II.4.1, we recall the first derivation of Euler's equations for a gas of particles. Sec. II.4.2 is a short reminder on the study of noisy harmonic chains which preserve the energy and not the momentum. Sec. II.4.3 focuses on noisy harmonic systems which preserve both of these quantities. Sec. II.4.4 introduces the question of transition between two universality classes. Sec. II.4.5 presents the model studied in Chapter III. We end this section by Sec. II.4.6 which presents the main contribution of Chapter III extracted from [START_REF] Cane | Superdiffusion transition for a noisy harmonic chain subject to a magnetic field[END_REF].

II.4.1 INTRODUCTION OF A STOCHASTIC NOISE

As we mentioned in Chapter I, in 1993 Olla, Varadhan and Yau [START_REF] Olla | Hydrodynamical limit for a hamiltonian system with weak noise[END_REF] studied a gas of particles living in R 3 , interacting via a short-range potential, by adding an energy-momentum conserving noise to the dynamics. Hence, in this system the energy, the momentum and the number of particles are conserved. Using the ergodicity added by the noise Olla, Varadhan and Yau were able to prove that the empirical densities associated to these conserved quantities converge to the following Euler's equations (before the appearance of the first shock)

d dt ρ(x) + 3 ∑ j=1 ∂ ∂ x j [ρ(x)π j ] = 0 , d dt (ρ(x)π i (x)) + 3 ∑ j=1 ∂ ∂ x j [ρ(x)π i (x)π j (x) + δ i j P] = 0 , (II.57) d dt (ρ(x)E(x)) + 3 ∑ j=1 ∂ ∂ x j [ρ(x)E(x)π j (x) -π j (x)P] = 0 .
Here, ρ is the density, π is the velocity per particle, E is the energy and P is the pressure. Their system was not completely satisfactory from a physical point of view because they needed to replace the usual velocity by a relativist velocity in order to have uniformly bounded velocities.

Historically, this is the first study where a conservative noise is added to a deterministic dynamics. At this time, the authors were only interested in the derivation of Euler equations without looking at the energy diffusion problem.

II.4.2 DIFFUSION IN NOISY ENERGY CONSERVING CHAINS

In 2005, Bernardin and Olla [START_REF] Bernardin | Fourier's law for a microscopic model of heat conduction[END_REF] studied a one-dimensional harmonic chain of N oscillators connected to two heat baths at the left-hand side and right-hand side with the temperatures T L and T R respectively, with T L > T R . They perturbed the bulk of the dynamics by a stochastic noise conserving the total energy of the system but not the total momentum. The noise consisted in a continuous exchange of the momenta of nearest neighbors particles. Let S e be the surface of constant kinetic energy between two particles, i.e.

S e = [p(x), p(y)] ∈ R 2d 1 2 |p(x)| 2 + |p(y)| 2 = e . (II.58)
Observe that the following vector field is tangent to S e X(x, y

) = p(y)∂ p(x) -p(x)∂ p(y) . (II.59)
As a consequence X(x, y) 2 generates a diffusion on S e . The infinitesimal generator of the stochastic noise introduced in [START_REF] Bernardin | Fourier's law for a microscopic model of heat conduction[END_REF] is given by

S BO = N-1 ∑ x=1 X(x, x + 1) 2 . (II.60)
The equations of motion are given by

∀x ∈ {1, . . . , N -2} d p(t, x) = {q(t, x + 1) + q(t, x -1) -2q(t, x)}dt -γ p(t, x)dt + √ γ [p(t, x -1)dB x-1,x (t) -p(t, x + 1)dB x,x+1 (t)] , d p(t, 1) = {q(t, 2) -q(t, 1)}dt + δ x,1 η L (t) -p(t, 1) dt (II.61) -γ p(t, 1)dt + √ γ p(t, 2)dB 1,2 (t) , d p(t, N -1) = {q(t, N -2) -q(t, N -1)}dt + δ x,N-1 η R (t) -p (t, N -1) dt -γ p(t, N -1)dt + √ γ p(t, N -2)dB N-2,N-1 (t) .
Here γ > 0, η L/R are Gaussian white noise terms such that

η L/R (t)η L/R (t ′ ) = 2T L/R δ (t -t ′ ) and B x,x+1 (
t) are independent standard Brownian motions. In Eq. (II.61), the black terms represent the deterministic dynamics, the blue terms represent the modelling of the baths by Langevin baths as in Eq. (II.26) and the red terms represent the introduction of the noise in the equation of motion. The positive parameter γ represents the strength of the random exchange of momenta.

Since the dynamics does not preserve the total momentum, the Euler's equation are trivial, i.e. p = 0 in Eq. (II.39). The energy of the system is preserved and then as we explained in Sec. II.2, the macroscopic current J N (see Eq. (II.30)), should be proportional to (T L -T R ) if the system satisfies the Fourier's law. Bernardin and Olla proved in [22, Theorem 1] that the system satisfies the Fourier's law lim

N→∞ J N = γ 2 + 1 2γ (T L -T R ) . (II.62)
Moreover, in [START_REF] Bernardin | Fourier's law for a microscopic model of heat conduction[END_REF]Theorem 3] they proved that for any bounded functions G on [0, 1] we have

lim N→∞ 1 N N-1 ∑ x=1 G xN -1 e(x) = 1 0 G(u)T (u)du = 1 0 G(u) (T L + [T R -T L ]u) du . (II.63)
Observe the Eq. (II.63) states that the energy profile converges to the linear profile of temperature T (•) which is the stationary solution to the following diffusion equation

∂ t T (t, u) = γ 2 + 1 2γ ∆ u T (t, u) with T (t, 0) = T L and T (t, 1) = T R . (II.64)
Hence, in this model where the introduced noise does not preserve the momentum, the Fourier's law is satisfied.

The study of harmonic system submitted to an energy conserving noise attracted a lot of interest during the last years (see for example [START_REF] Bernardin | Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise[END_REF][START_REF] Simon | Hydrodynamic limit for the velocity-flip model[END_REF] for the derivation of hydrodynamic limits without boundaries, [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF][START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF][START_REF] Komorowski | Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF][START_REF] Komorowski | Hydrodynamic limit for a chain with thermal and mechanical boundary forces[END_REF] for the derivation of kinetic or hydrodynamic limits with thermostats at the boundaries using Wigner's transform and [START_REF] Bernardin | Stationary nonequilibrium properties for a heat conduction model[END_REF] for large deviations for the system in contact with thermostats 13 ). Few results exist also in the anharmonic case [START_REF] Aoki | Energy transport in weakly anharmonic chains[END_REF][START_REF] Lukkarinen | Anomalous energy transport in the fpu-β chain[END_REF][START_REF] Marchesani | Hydrodynamic limit for an anharmonic chain under boundary tension[END_REF][START_REF] Olla | Macroscopic energy diffusion for a chain of anharmonic oscillators[END_REF][START_REF] Olla | Equilibrium fluctuation for an anharmonic chain with boundary conditions in the euler scaling limit[END_REF].

Until now in this section, we discussed only the noisy harmonic chain in the case where U = 0 in Eq. (II.1) (i.e. unpinned chain). When the chain is pinned (i.e. U ̸ = 0), the total momentum is not preserved and it has been proved in [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF][START_REF] Komorowski | Long time, large scale limit of the wigner transform for a system of linear oscillators in one dimension[END_REF] that in this case, the energy diffuses normally. At the end of Sec. II.4.5, we will give a formal argument to explain this diffusive behavior of pinned chain.

II.4.3 SUPERDIFFUSION IN NOISY MOMENTUM CONSERVING HARMONIC CHAINS

As we mentioned previously, the anomalous diffusivity of the anharmonic chain seems to come from the conservation of the momentum in such systems, but due to the non linearity of the system their mathematical study is very difficult. As we saw in Sec. II.4.2, the introduction of noise to replace the non linearity of the dynamics is helpful to obtain some mathematical results on the behavior of the energy. Hence, in 2006 Basile, Bernardin and Olla introduced in [9] a noise which preserves the energy of the system and the momentum.

We recall that T d N is the d-dimensional discrete torus of length N. We assume moreover that the particles are living in R d , the Hamiltonian of the system is defined as (all the masses are equal to one)

H = ∑ x∈T d N d ∑ i=1 |p i (x)| 2 2 +U (q i (x)) + 1 2 ∑ |x-y|=1
V (q i (x)q i (y)) . Observe that S e,p is a d -1 dimensional space and that the following vector fields are tangent to it

X i, j (x, y) = (p j (y) -p j (x)) ∂ p i (y) -∂ p i (x) -(p i (y) -p i (x)) ∂ p j (y) -∂ p j (x) . (II.67)
As a consequence, ∑ i, j X i, j (x, y) 2 generates a diffusion on S e,p . For d ≥ 2, the noise added to the system is generated by

S = 1 4(d -1) ∑ x,y∈T d N |x-y|=1 ∑ i, j (X i, j (x, y)) 2 . (II.68)
Observe that the total momentum and the energy are conserved by the noise since we have

S (P) = 0 and S (E ) = 0 . (II.69)
For d = 1, the manifold S e,p is reduced to a point. In order to introduce a noise which preserves the energy and the momentum, Basile, Bernardin and Olla introduced a random exchange of momentum between three consecutive particles. In this case, the infinitesimal generator of the noise S is defined as

S = 1 6 ∑ x∈T 1 N (Y (x, x + 1)) 2 , (II.70)
where for any x in T 1 N we have

Y (x, x + 1) = (p(x) -p(x + 1)) ∂ p(x-1) + (p(x + 1) -p(x -1)) ∂ p(x) + (p(x -1) -p(x)) ∂ p(x+1) .
Observe that Y (x, x + 1) is the vector field tangent to the surface of constant energy and momentum of the three particles x, x + 1 and x -1 involved. Let κ N be the conductivity of the system obtained by the truncation at time N of the Green-Kubo formula (II.33). In [START_REF] Basile | Thermal conductivity for a momentum conservative model[END_REF]Theorem 2] it is proved that if the chain is unpinned, (i.e. U = 0) and V is a harmonic potential i.e. V (q) = q 2 we have

1. κ N ∼ N 1/2 if d = 1. 2. κ N ∼ log N if d = 2. 3. κ N is bounded and converges to some finite number κ if d ≥ 3.
The results obtained by Basile, Bernardin and Olla are consistent with the predictions of Levi, Lipri and Politi in [START_REF] Lepri | On the anomalous thermal conductivity of one-dimensional lattices[END_REF]. Moreover, in [START_REF] Basile | Thermal conductivity for a momentum conservative model[END_REF]Theorem 3] it is proved that if V is harmonic and U is not equal to zero then κ N is bounded. To prove these results, Basile, Bernardin and Olla proved that the correlations functions C j defined in Eq. (II.34) decay like t -d/2 . Hence, the Green-Kubo formula (c.f. Eq. (II.33)) is finite if and only if d ≥ 3.

From these results we can make some observations: if we assume that the introduced noise correctly mimics the anharmonicity of the system then we should have a behavior of the conductivity similar to the ones predicted by physicists (see Sec. II.3.1). This is exactly what is obtained in [START_REF] Basile | Thermal conductivity for a momentum conservative model[END_REF].

To go further, once the anomalous energy transport is established, it is of interest to describe the form of this diffusion.We have seen in Sec. II.2 several setups to do it. The macroscopic behavior of the energy of this system has been derived using a hydrodynamic limit in two steps. First, a (linear) phononic Boltzmann equation has been obtained by Basile, Olla and Spohn in [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF] using the Wigner's distribution, by sending the intensity of the noise to 0 (instead of the anharmonicity), and then a fractional diffusion equation with exponent 3/4 has been obtained by Jara, Komorowski and Olla in [66] by a second scaling limit performed in the phononic Boltzmann equation (seel also [START_REF] Komorowski | Ballistic and superdiffusive scales in the macroscopic evolution of a chain of oscillators[END_REF]). A hydrodynamic limit in one step has been obtained by Jara, Komorowksi and Olla in [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF]. Observe that the exponent 3/4 is consistent with the nonlinear fluctuating hydrodynamics theory of Spohn. Let us also mention that a family of fractional diffusion equations have been derived from stochastic harmonic chains with long-range interaction by Suda in [START_REF] Suda | A family of fractional diffusion equations derived from stochastic harmonic chains with long-range interactions[END_REF] and a skew-fractional diffusion equation with exponent 3/4 in [START_REF] Bernardin | 3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise[END_REF] for a model we describe below.

Before introducing the model studied in Chapter III we would like to mention a study where a transition appears between two universality classes.

II.4.4 AN INTERPOLATION BETWEEN A DIFFUSIVE AND A SUPERDIFFUSIVE SYSTEM

As we mentioned in Sec II.3.2, there exists some universality classes that describe the anomalous transport of interacting particle systems with several conservation laws. A very interesting question is to understand the potential mechanisms to pass from one universality class to an other universality class. Often, in the microscopic interacting particle system, there is a parameter whose intensity (depending on the scaling parameter) is responsible to the belonging to one of the different universality classes. For a critical value of this intensity, the system does not belong to the universality classes but in some transition class interpolating between the universality classes and it is of interest to understand what are the possible transitions classes. For example, for systems with only one conservation law, there are only two universality classes, the Edward-Wilkinson class (normal diffusion) and the KPZ-fixed point universality class (anomalous diffusion) and the transition class corresponds to the KPZ-equation (which seems to be the only mechanism to transit between the two classes). For systems with several conservation laws, there are more classes and the problem becomes harder.

A result due to Bernardin, Gonçalves, Jara, Sasada and Simon answers this question for a particular model with two conservation laws. Since, in Chapter III we will describe such phenomenon, we decided to recall the study made in [START_REF] Bernardin | From normal diffusion to superdiffusion of energy in the evanescent flip noise limit[END_REF][START_REF] Bernardin | Nonlinear perturbation of a noisy Hamiltonian lattice field model: universality persistence[END_REF].

In 2011 Bernardin and Stoltz [START_REF] Bernardin | Anomalous diffusion for a class of systems with two conserved quantities[END_REF], studied the infinite noisy harmonic chain but they changed the variable of interest. Instead of studying a configuration (q, p) they decided to study η where ∀x ∈ Z, η(t, 2x + 1) = q(t, x + 1)q(t, x) and η(t, 2x) = p(t, x) .

(II.71)

In this system of coordinates, the equation of evolution of η is given by

dη(t, x) = (η(t, x + 1) -η(t, x -1)) dt . (II.72)
In the coordinates (q, p) we saw that the interesting quantities are the energy (E ) of the system and the momentum (P). In these new coordinates, the energy is still a quantity of interest but the momentum is replaced by the volume V where

V = ∑ x∈Z η(x) . (II.73)
This quantity is preserved by the deterministic dynamics. In [START_REF] Bernardin | Anomalous diffusion for a class of systems with two conserved quantities[END_REF], it is proved that if the dynamics is perturbed by an energy-volume conserving noise then the system is superdiffusive. The macroscopic equation has been derived in [START_REF] Bernardin | 3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise[END_REF] by Bernardin, Gonçalves and Jara. They proved that after a suitable space-time scaling the energy superdiffuses as a skew version of a fractional Lapalcian with exponent 3/4. This result is also consistent with the theory of Spohn we presented in Sec. II.3.2. In [START_REF] Bernardin | From normal diffusion to superdiffusion of energy in the evanescent flip noise limit[END_REF], the authors perturbed the dynamics of [START_REF] Bernardin | 3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise[END_REF] by adding a second noise with intensity γ which does not preserve the volume. If γ = 0, then the energy superdiffuses and if γ ̸ = 0 the energy diffuses normally since the volume is not preserved. In [START_REF] Bernardin | Nonlinear perturbation of a noisy Hamiltonian lattice field model: universality persistence[END_REF] it is proved that playing with the intensity of γ it is possible to pass from the diffusive regime to the superdiffusive regime by passing through an interpolation regime between the two classes, the transition regime being described by a (non-stable) Lévy process. In [START_REF] Bernardin | Interpolation process between standard diffusion and fractional diffusion[END_REF], an other interpolating process is studied.

In next section, we present the main model studied in this thesis and explain the results obtained on it in [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF][START_REF] Saito | Thermal conductivity for coupled charged harmonic oscillators with noise in a magnetic field[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF]. In 2018, Saito and Sasada in [START_REF] Saito | Thermal conductivity for coupled charged harmonic oscillators with noise in a magnetic field[END_REF] studied the model presented in [START_REF] Basile | Momentum conserving model with anomalous thermal conductivity in low dimensional systems[END_REF] when the deterministic system is submitted to a constant magnetic field of intensity B in R * . 14 We will only consider the case where the dimension of the lattice is d = 1 but particles live in R 2 . Hence, the deterministic equations of motion are given for i in {1, 2} by

II.4.5 INTRODUCTION OF A MAGNETIC FIELD

d dt q i (t, x) = p i (t, x) , d dt p i (t, x) = q i (t, x + 1) + q i (t, x -1) -2q i (t, x) + B (δ i,1 p 2 (t, x) -δ i,2 p 1 (t, x)) . (II.74)
This model does not preserve the momentum contrary to the system studied in [START_REF] Basile | Momentum conserving model with anomalous thermal conductivity in low dimensional systems[END_REF][START_REF] Basile | Thermal conductivity for a momentum conservative model[END_REF][START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF][START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF] but it preserves the pseudo-momentum denoted by P m which is defined by

P m (t) = ∑ x∈Z p 1 (t, x) -Bq 2 (t, x), ∑ x∈Z p 2 (t, x) + Bq 1 (t, x) . (II.75)
If B = 0, the pseudo-momentum coincides with the momentum. In [START_REF] Saito | Thermal conductivity for coupled charged harmonic oscillators with noise in a magnetic field[END_REF], it is proved that the currentcurrent correlation function appearing in the Green Kubo integral decays with time like t -1/4 with a constant magnetic field whereas without a magnetic field, Basile, Bernardin and Olla proved in [START_REF] Basile | Thermal conductivity for a momentum conservative model[END_REF] that it decays like t -1/2 . From this observation, it is natural to try to understand if this difference in the divergence of the Green-Kubo formula can be seen macroscopically. In others words, is the fractional diffusion equation with exponent 3/4 obtained in [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF] is changed into a new one? Saito, Sasada and Suda answered to this question in [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] and proved that the presence of a magnetic field changes the exponent 3/4 into 5/6.

The aim of Chapter III is to understand the interpolation phenomenon between these two universality classes. Hence, we first recall how the authors of [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF][START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] obtained these macroscopic equations.

BATTLE PLAN AND INITIAL CONDITIONS

To obtain the macroscopic equation satisfied by the energy of the system, the idea developed in [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF] was to follow the approach of Spohn with the Wigner's function. In Sec. II.2.4, we recalled that Spohn in [START_REF] Spohn | The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics[END_REF] studied a weakly anharmonic system where the intensity of the anharmonicity was of order ε. By sending ε to zero, Spohn was able to obtain some phononic Boltzmann's equation (cf. Eq. (II.56)). Hence, the idea is to add a small noise with intensity ε instead of a small anharmonicity. In this context the formal infinitesimal generator of this system is given by

A + BG + γεS , (II.76)
where

A = 1 2 ∑ x∈Z 2 ∑ i=1 p i (x)∂ q i (x) + ∑ x∈Z 2 ∑ i=1 [q i (x + 1) + q i (x -1) -2q i (x)] ∂ p i (x) , G = ∑ x∈Z (p 2 (x)∂ p 1 (x) -p 1 (x)∂ p 2 (x) ) , S = 1 4 ∑ x∈Z 2 ∑ i, j=1 i̸ = j (Y i, j (x, x + 1)) 2 ,
and

Y i, j (x, x + 1) = (p j (x + 1) -p j (x))(∂ p i (x+1) -∂ p i (x) ) -(p i (x + 1) -p i (x))(∂ p j (x+1) -∂ p j (x) ) .
Here ε is the scaling parameter and γ is a positive number.

The introduction of a magnetic field in the deterministic dynamics induces a different definition of the waves functions of the system. For every configuration (q, p), the waves functions ψ i,B : T → C 2 , i ∈ {1, 2} are defined for any k in T by

ψ 1,B (k) := ψ 1,B (q, p)[k] := θ 1,B (k) p 1 (k) -iω 2,B (k) q 1 (k) + i p 2 (k) + ω 2,B (k) q 2 (k) ,
(II.77)

ψ 2,B (k) := ψ 2,B (q, p)[k] := θ 2,B (k) p 1 (k) -iω 1,B (k) q 1 (k) -i p 2 (k) -ω 1,B (k) q 2 (k) , (II.78)
where for any k in T and i in {1, 2} we have α(k) = 4π 2 sin 2 (πk) , (II.79)

ω 1,B (k) = α(k) + B 2 4 + B 2 and ω 2,B (k) = α(k) + B 2 4 - B 2 , (II.80) θ i,B (k) = ω i,B (k) ω 1,B (k) + ω 2,B (k) and v B (k) = d dk ω 1,B (k) = d dk ω 2,B (k) = α′ (k) 2 α(k) + B 2 4
. (II.81)

In this system, ψi,B (k) represents the phonon with wave number k in T of type i. Since the energy of the system E is preserved we have that for any time t

E = 1 2 T | ψ1,B (t, k)| 2 + | ψ2,B (t, k)| 2 dk = 1 2 T | ψ1,B (0, k)| 2 + | ψ2,B (0, k)| 2 dk . (II.82)
We assume initially that the system is distributed according to a measure µ ε which satisfies the following property

sup ε>0 ε 2 ∑ i=1 E µ ε ∥ ψi (0, •)∥ 2 L 2 (T) = sup ε>0 ε E µ ε T | ψ1,B (0, k)| 2 + | ψ2,B (0, k)| 2 dk < K 0 . (II.83)
Observe that Eq. (II.83) means that at initial time, the energy of the system is of order ε -1 . Since the energy of the system is preserved, Eq. (II.83) is true at any time t, i.e. replacing ψi,B (0, k) by ψi,B (t, k) in Eq. (II.83).

WIGNER'S DISTRIBUTION

In Sec. II.2.4, we recalled (see Eq. (II.47)) the definition of the Wigner's function. In order to study its behavior, we use it as a distribution on a space of smooth functions. Hence, in this context, the Wigner's distribution W ε is a two-dimensional distribution whose action on a couple of smooth functions

J := (J 1 , J 2 ) : (R × T) 2 → C × C is defined for any time t by ⟨W ε (t), J⟩ = 2 ∑ i=1 ⟨W ε i (t), J i ⟩ , (II.84)
where

⟨W ε i (t), J i ⟩ = R T W ε i tε -1 , p, k J i (u, k)dkdu (II.85) = ε 2 R T E µ ε ψi,B tε -1 , k - ε p 2 * ψi,B tε -1 , k + ε p 2 F [J i ](p, k)dkd p , (II.86)
where F [J i ] is the continuous Fourier's transform of J i defined as

∀p ∈ R, F [J i ] (p) = R J i (u) exp(2iπ pu)du . (II.87)
Here, ⟨•⟩ denotes the distribution bracket. In [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF]Theorem 5]

(B = 0) and in [104, Theorem 1] (B ̸ = 0) it is proved that W ε converges to the unique solution f B (•) := ( f B (•, 1), f B (•, 2
)), of the following phononic linear Boltzmann's equation

∂ t f B (t, u, k, i) + v B (k) 2π ∂ u f B (t, u, k, i) = C B [ f B ](t, u, k, i) , (II.88) with C B [ f B ](t, u, k, i) = γ 2 ∑ j=1 T θ 2 i,B (k)R k, k ′ θ 2 j,B k ′ f B t, u, k ′ , j -f B (t, u, k, i) dk ′ , (II.89) where R k, k ′ = 16 sin 2 (πk) sin 2 πk ′ . (II.90)
Here, u ∈ R represents the position along the chain after the kinetic limit, t ∈ [0, T ] is the time and k ∈ T is the wave number of a phonon, whereas i is the type of phonon. C B is a collisional operator due to the introduced noise on the system and v B is the dispersion relation. Hence, if the microscopic system is not perturbed by the noise, which corresponds to γ = 0, W ε converges to the transport equation as we recalled in Sec. II.2.4 (see Eq. (II.53)). Observe that the introduction of the noise (instead of anharmonicity) changes the general phononic Boltzmann's equation obtained by Spohn in [START_REF] Spohn | The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics[END_REF] (cf.Eq. (II.56)) into a linear one.

INTERPRETATION OF THE BOLTZMANN'S EQUATION

Let L B be the operator acting on a function f :

T × {1, 2} → R by L B [ f ](k, i) = γλ -1 B (k, i) 2 ∑ j=1 T P B (k, i, dk ′ , j) f k ′ , j -f (k, i) dk ′ , (II.91)
where for any (k, k ′ ) in T 2 and (i, j) in {1, 2} we have

P B (k, i, dk ′ , j) = γλ B (k, i)θ 2 i,B (k)θ 2 j,B k ′ R k, k ′ dk ′ , (II.92)
and with

λ B (k, i) = γθ 2 i,B (k)R(k) -1 and R(k) = T R k, k ′ dk ′ . (II.93)
Then, Eq. (II.91) can be written in the following way

∂ t f B (t, u, k, i) + v B (k) 2π ∂ u f B (t, u, k, i) = L B [ f B ](t, u, k, i) . (II.94)
Let X B n n∈N := K B n , I B n n∈N be the Markov chain on T × {1, 2} with transition probability P B defined in Eq. (II.92). Let (τ n ) n∈N be an i.i.d sequence of random variables, independent of X B n n∈N such that τ 0 ∼ E (1). We define the random variable T N by

T 0 = 0 and ∀N ∈ N * , T N = N ∑ n=1 λ B X B n-1 τ n-1 . (II.95)
Then we can define a pure jump Markovian process K B (•), I B (•) with values in T × {1, 2} where for any positive time t in [0, T ] [START_REF] O'connor | Heat conduction and sound transmission in isotopically disordered harmonic crystals[END_REF]. Observe that the process K B (•), I B (•) is a jump process as the process V (•) defined in Eq. (I.48) of Chapter I. We can formally describe the process K B (•), I B (•) as follows 1. The process starts at point (k, i).

K B (t) = K B n , and I B (t) = I B n , ∀t ∈ [T n , T n+1 [ . The infinitesimal generator of K B (•), I B (•) is L B defined in Eq. (II.
2. During a mean time λ B (k, i), the process is constant to its initial position. 3. After a mean time λ B (k, i), the process jumps from (k, i) to (k ′ , j) with probability P B .

Once we defined the stochastic process K B (•), I B (•) , we can interpret Eq. (II.94) as follows. Eq. (II.94) describes the evolution of a wave which starts from u in R and which is of type i. This wave moves with velocity v B (k) during a mean time λ B (k, i), after this time the wave change its type from i to j and its velocity from v B (k) to v B (k ′ ) with probability P B (k, i, dk ′ , j). Hence, the motion of the particle defines a PDMP (Piecewise Deterministic Markov Process) 15 . As we saw in Sec. I.4.3 of Chapter I, the position of the particle at time t in [0, T ] starting from u in R is given by Z B u (t) where

Z B u (t) = u - t 0 v B K B (s) 2π ds . (II.96) Z B u (•)
is a stochastic process defined as an additive functional of Markov process. Then by Dynkin's formula, we get that for any i in {1, 2} and

(u, k) in R × T f B (t, u, k, i) = E (k,i) f 0 Z B u (t), K B (t), I B (t) , (II.97)
where f 0 is the initial condition of Eq. (II.94). Observe that this expression has the same form as the one of T in Eq. (I.38) of Chapter I. In Eq. (II.97) we replaced the Brownian motion B u (•) of Eq. (I.38) by the stochastic process

Z B u (•), K B (•), I B (•) .

SHORT REMINDER ABOUT LÉVY PROCESSES

Before studying the asymptotic behavior of f B we make a short reminder about Lévy processes [START_REF] Bertoin | Lévy processes[END_REF]. Given a measure ν on R * , we say that ν is a Lévy measure if and only if

R * min 1, r 2 dν(r) < ∞ . (II.98)
Let Y u (•) be a real valued stochastic process starting from u in R. We say that Y u (•) is a Lévy process with (Lévy) measure ν if and only if it has stationary independent increments, is continuous in probability and for any positive t and θ in R,

E [exp (iθY u (t))] = exp (tΦ Y (θ ) + iθ u) .
Here, Φ Y denotes the Lévy exponent associated to the Lévy process Y u (•) and is given for any θ in R by

Φ Y (θ ) = aθ 2 + R * exp (iθ r) -1 + iθ r1 {|r|<1} dν(r) , (II.99)
where a is in R + . The action of the infinitesimal generator L of Y u (•) on a smooth function φ : R → R which decays sufficiently fast is given by

∀p ∈ R, L [φ ] (p) = R F [φ ] (ξ )Φ Y (ξ ) exp (2iπ pξ ) dξ . (II.100)
In this thesis we will only study Lévy processes where a = 0. Let α in (1, 2), if dν(r) := |r| -α-1 then we have that for any

θ in R θ R * r1 {|r|<1} dν(r) := U < ∞ . (II.101)
Hence, if we change the initial point of the process from u to u -U we have that (up to a constant)

Φ Y (θ ) = -|θ | α .
Hence, for this choice of ν, the infinitesimal generator of Y u is (up to a constant) the pseudo-differential

operator -(-∆) α 2 .
In this case, we say that Y u is a α-stable Lévy process.

TWO-STEP HYDRODYNAMIC LIMIT

The hydrodynamic behavior of f B (defined in Eq. (II.94)) is completely determined by the one of the process Z B u (•). Hence, in the following, we recall how the authors of [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] studied this process. As we presented in Sec. I.4.3 of Chapter I, we can interpret Z B u (•) as a random walk. Let (τ n ) n∈N be a sequence of i.i.d random variables such that τ 0 ∼ E (1). Let t be a positive time, we can decompose Z B u (t) into two parts, the first one is given by all the jumps made by the particle until time t and the second part is given by the displacement made by the particle since the last jump. Hence, if we denote by N (t) the N (t)-th renewal time we can write

Z B u (t) = u - N (t)-1 ∑ n=0 λ B X B n v B K B n 2π τ n -(t -N (t)) v B K B N (t) 2π . (II.102)
We define the stochastic process ZB u (•) as follows

∀t ∈ [0, T ] , ZB u (t) = u - ⌊t⌋ ∑ n=0 λ B X B n v B K B n 2π τ n . (II.103)
Let π B the probability measure on T × {1, 2} defined by

π B (dk, di) = 2 ∑ j=1 λ B (k, j) -1 γ R dkδ j (di) with R = T R(k)dk . (II.104)
We can prove that π B is a reversible probability measure of the Markov chain X B n n∈N . Observe that

P B (k, i, dk ′ , j) = π B (dk ′ , j) , (II.105)
where P B is defined in Eq. (II.92). From this observation we deduce that X B n n≥1 is a i.i.d sequence of random variables. We define a function

Ψ B by ∀k ∈ T, ∀i ∈ {1, 2}, Ψ B (k, i) = v B (k)λ B (k, i) .
(II.106)

Observe that ZB u (•) gives us the trajectory of the particle until the time ⌊t⌋ and that its expression has the same form as Eq. (I.51) in Chapter I. Hence, we can write that for any t in [0, T ]

Z B u (t) = u - N (t)-1 ∑ n=0 Ψ B K B n 2π τ n -(t -N (t)) v B K B N (t) 2π and ZB u (t) = u - ⌊t⌋ ∑ n=0 Ψ B K B n 2π τ n .
To obtain the asymptotic behavior of Z B u (•) the idea developed by Jara, Komorowski and Olla in [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF] was to understand the one of ZB u (•) and to prove that the process ZB u (•) and Z B u (•) are close. For a complete proof of these results we refer the reader to [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF]. Since we will need to adapt their proof to our study in Chapter III we decided to briefly recall the main idea of the convergence of ZB u (•). As we formally proved in Sec. I.4.3 of Chapter I, the asymptotic behavior of the random walk Z x (•) defined in Eq. (I.51) is determined by the tails of the random variables V n . Here, the random variables V n of Eq. (I.51) are replaced by Ψ B X B n . Hence, in this study we need to compute the tails of the random variables Ψ B X B n . In [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF], it is proved that

∀r > 0, lim N→∞ N α B π 0 k ∈ T Ψ B (k) > Nr = κB |r| -α B := ν B (r, ∞) , ∀r < 0, lim N→∞ N α B π 0 k ∈ T Ψ B (k) < Nr = κB |r| -α B := ν B (-∞, r) , (II.107)
where

α B = 3 2 if B = 0 and α B = 5 3 if B ̸ = 0 . (II.108) Here κB =        κ1 ( α ′′ (0)) 3 4 γ -3 2 if B = 0 , κ2 |B| -1/3 α ′′ (0)γ -5 3 if B ̸ = 0 , (II.109)
with κ1 and κ2 two positive constants.

Let Y B u (•) be the Lévy process with Lévy measure ν B . Then as we saw previously, the infinitesimal generator of

Y B u is given (up to a constant) by -(-∆ u ) α B 2 . Using Eq. (II.107), it is proved in [66, 104] that if k ̸ = 0 under P (k,i) , the finite-dimensional distributions of N -1 ZNu (N α B •) (II.
∀u ∈ R, ∀t ∈]0, T ], ∂ t ρ B (t, u) = -D B (-∆) α B 2 ρ B (t, u) . (II.111)
When B = 0, we recover a fractional diffusion equation with exponent 3/4 and this is consistent with Spohn's theory. In the case B ̸ = 0 since the hydrodynamic limits of this chain seem to be trivial in the Euler time scale, we do not see how the nonlinear fluctuating hydrodynamics theory of Spohn could give some prediction for this model.

This two-steps argument consisting to first prove the convergence of the Wigner's distribution to the unique solution f B of a phonon Boltzmann's equation and then the convergence of f B to the solution ρ B of a fractional Laplacian equation gives the nature of the superdiffusion of energy proved in [START_REF] Basile | Thermal conductivity for a momentum conservative model[END_REF][START_REF] Saito | Thermal conductivity for coupled charged harmonic oscillators with noise in a magnetic field[END_REF]. Fractional diffusion equations can also be derived from some Boltzmann's equations using purely analysis arguments, see for example [START_REF] Cesbron | Fractional diffusion limit for a kinetic equation in the upper-half space with diffusive boundary conditions[END_REF][START_REF] Cesbron | Fractional diffusion limit of a kinetic equation with diffusive boundary conditions in a bounded interval[END_REF][START_REF] Mellet | Fractional diffusion limit for collisional kinetic equations[END_REF] 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 [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF] proved that we can obtain Eq. (II.111) in one step in the case B = 0, from the microscopic model in a suitable time scale.

PHYSICAL INTERPRETATION

We can formally explain the reason of the presence of a different exponent when

B ̸ = 0, say B ≥ 0. Let (k, i) in T × {1, 2}, λ B (k, i)
is the mean time that the particle waits with velocity v B (k) before changing its speed. Using Eq. (II.93) and Eq. (II.81) we have that for small k

v B (k) ∼ k λ B (k, 1) ∼ k -2 and λ B (k, 2) ∼ k -4 if B > 0 , v 0 (k) ∼ 1 λ 0 (k, 1) ∼ k -2 and λ 0 (k, 2) ∼ k -2 if B = 0 .
We denote by d B (k, i) the distance traveled by the wave starting from the state (k, i) before changing the speed. Then we have that

d B (k, 1) ∼ k -1 and d B (k, 2) ∼ k -3 if B > 0 , d 0 (k, 1) ∼ k -2 and d 0 (k, 2) ∼ k -2 if B = 0 .
Thus, the introduction of a magnetic field allows the wave to travel longer distances before scattering. Before introducing Chapter III, we would like to say a few words about pinning lattice. Assume that B = 0, then using Eq. (II.81) we have ω i,0 (k) = 2π| sin(πk)|. Assume now that the chain is pinned and that U(q) = C q 2 2 , then ω 0 is changed into ω0 where for any k, ω0 (k) = C + 4π 2 sin 2 (πk). In this situation, we have for small k

ṽ0 (k) ∼ k and λ 0 (k, i) ∼ k -2 .
Hence, in this situation, the random walk Z 0 u converges, up to a rescaling, to a Brownian motion and the fractional Laplacian is replaced by a Laplacian.

II.4.6 CONTRIBUTION TO SUPERDIFFUSION

In this section, we introduce Chapter III where we will present the results we obtained in [START_REF] Cane | Superdiffusion transition for a noisy harmonic chain subject to a magnetic field[END_REF] where we studied the transition between the study of [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF] and the one of [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF]. As we have seen in Sec. II.4.5, the presence of the magnetic field moves the model from the 3/4-fractional universality class to the 5/6-fractional universality class. Then 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. The mechanism proved in Chapter III is quite similar to the one proved in [START_REF] Bernardin | Nonlinear perturbation of a noisy Hamiltonian lattice field model: universality persistence[END_REF] and recalled in Sec. II.4.4. In Chapter III, instead of proving an interpolation between a diffusion process and a superdiffusive process as in [START_REF] Bernardin | Nonlinear perturbation of a noisy Hamiltonian lattice field model: universality persistence[END_REF], we prove that there exists an interpolation between two superdiffusive processes. We recall that the system studied is formally described by the operator defined in Eq. (II.76).

ABSENCE OF TRANSITION AT KINETIC SCALE ε -1

In a first time, we assume that the intensity of the magnetic field in the equation of motion (Eq. (II.74)) is Bε δ with δ > 0 and B ̸ = 0. In Theorem III.4 of Chapter III, we prove that at the kinetic time scale of order ε -1 , the transition is trivial in the sense that for δ = 0, the Wigner's distribution (defined in Eq. (II.84)) converges to the solution of the Boltzmann's equation (cf. Eq. (II.94)). For B ̸ = 0 and δ > 0, the Wigner's distribution converges to the solution with B = 0. We believe however that the effect of a small magnetic field could be seen in a longer time scale. We refer the interested reader to Chapter V where we develop some open problems in this direction.

TRANSITION IN THE HYDRODYNAMIC LIMIT IN TWO STEPS

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 (II.94) when B is changed into B N := BN -δ with δ in R + and B ̸ = 0. Namely, f B N is the solution of the following Boltzmann's equation

∂ t f B N (t, u, k, i) + v B N (k) 2π ∂ u f B N (t, u, k, i) = L B N [ f B N ] (t, u, k, i) , (II.112)
where

L B N [ f ] (k, i) = γλ -1 B N (k, i) 2 ∑ j=1 T P B N k, i, dk ′ , j f B N k ′ , j -f B N (k, i) dk ′ . (II.113)
Here, we recall that P B N and λ B N are defined in Eq. (II.91) and Eq. (II.93) respectively. As we explained in Sec. II.4.5, we can define a stochastic process

K B N (•), I B N (•) with infinitesimal generator L B N .
Then, the solution of Eq. (II.112) is given for any i in {1, 2} and

(u, k) in R × T by f B N (t, u, k, i) = E (k,i) f 0 Z B N u (t), K B N (t), I B N (t) , (II.114)
where

Z B N u (t) = u - t 0 v B N K B N (s) 2π ds .
The asymptotic behavior of Z B N u (•) depends on the one of Ψ B N defined in Eq. (II.106). Let π B N , defined in Eq. (II.104), be the invariant probability measure of the Markov chain

X B N n n∈N = K B N n , I B N n n∈N . Let α δ defined as follows α δ = 3 2 if δ ≥ 1 2 and α δ = 5 -δ 3 if δ < 1 2 . (II.115)
In Chapter III, we prove (Proposition III.4.5) that ∀r > 0, lim

N→∞ N α δ π B N (k, i) Ψ B N (k, i) > Nr = ν δ (r) , ∀r < 0, lim N→∞ N α δ π B N (k, i) Ψ B N (k, i) < Nr = ν δ (r) ,
where

dν δ (r) =                κ 0 E τ α δ 0 |r| -3 2 -1 dr if δ > 1 2 , 2γE τ -1 0 g B,+ 2πr τ 0 + g B,- 2πr τ 0 dr if δ = 1 2 , κ ∞ |B| -1/3 E τ α δ 0 |r| -5 3 -1 dr if δ < 1 2 ,
(II.116)

with

κ 0 = 1 2 10 π 2 γ 1 2 and κ ∞ = 1 2 13 27π 3 γ 2 1 3
.

(II.117)

Here g B,± is a positive function on R * whose precise definition is given in Eq. (III.41) of Chapter III and τ 0 ∼ E (1).

Let Y δ u (•) be the Lévy process starting from u in R with Lévy measure ν δ defined in Eq. (II.116). In Theorem III.5 of Chapter III we prove that under P (k,i) the finite-dimensional distributions of the process

N -1 Z B N u N (N α δ •) (II.118)
converge weakly to the finite-dimensional distributions of Y δ u (•). Once we have obtained the asymptotic behavior of Z B N u we can obtained the one of f B N . Let Φ δ be the Lévy exponent associated to the Lévy process

Y δ u (•), then ∀θ ∈ R, Φ δ (θ ) = R * exp (iθ r) -1 + iθ r1 {|r|<1} dν δ (r) . (II.119) Observe that for δ > 1 2 resp.δ < 1 2 , Y δ u (•)
is the Lévy process obtained by Jara, Komorowski and Olla in [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF] (resp. by Saito, Sasada and Suda in [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF]). For δ = 1 2 we have an interpolation process denoted by Y B u (•).

We recall that α δ is defined in Eq. (II.115), in Theorem III.6 of Chapter III we prove that f B N (N α δ t, Nu, k) converges in L 1 (T) to ρ δ which is the solution of

∂ t ρ δ (t, u) = L δ [ρ δ ] (t, u) = R F [ρ δ ] (t, ξ )Φ δ (ξ ) exp (2iπuξ ) dξ .
(II.120)

Observe that for any smooth functions

∀u ∈ R, L δ [φ ] (u) =        -Γ 1 + 3 2 D 0 (-∆) 3 4 [φ ] (u) if δ > 1 2 , -Γ 1 + 5 3 D ∞ (-∆) 5 6 [φ ] (u) if δ < 1 2 ,
(II.121)

where

D 0 = 2κ 0 +∞ 0 1 -cos(r) r 5 2
dr and

D ∞ = 2κ ∞ |B| -1/3 +∞ 0 1 -cos(r) r 8 3
, (II.122) with κ 0 and κ ∞ defined in Eq. (II.117). Hence, we recover the case of [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF]. Moreover here we have the explicit constant in front of the fractional Laplacian. For δ = 1/2, we have an interpolation operator L B whose behavior when B goes to zero and B goes to infinity is studied in the next section.

STUDY OF THE INTERPOLATION PROCESS

For δ = 1/2, we observe that the Lévy measure ν δ defined in Eq. (II.116) depends on B so that we denote it now by L B instead of L 1/2 . Let ρB be the solution of

∂ t ρB (t, u) = L B [ ρB ] (t, u) .
In Theorem III.7 of Chapter III, we prove that i) lim

B→0 ρB = ρ 0 in L 1 [0, T ], L 2 (R)
where ρ 0 is the solution of

∂ t ρ 0 (t, u) = -Γ 1 + 3 2 D 0 (-∆) 3 4 ρ 0 (t, u) . ii) lim B→+∞ ρB B 1 3 •, • = ρ ∞ in L 1 [0, T ], L 2 (R) where ρ ∞ is the solution of ∂ t ρ ∞ (t, u) = -Γ 1 + 5 3 D ∞ (-∆) 5 6 ρ ∞ (t, u) . (II.123)
Hence, we recover the two fractional diffusion equations obtained previously, thus L B is the infinitesimal generator of a Lévy process which interpolates between the two fractional universality classes. These results are summarized in Fig. II.4. On the horizontal axis, δ represents the intensity of the magnetic field. On the vertical axis α δ represents the scaling in space to obtain the hydrodynamic limit of f B N .

II.5 DISORDERED SYSTEMS

As mentioned in Sec. II.2.1, a way to study the transport of energy is to connect the system to two heat baths at the left-hand side and right-hand side with temperatures T L and T R respectively where T L > T R . In Chapter IV, we will consider this setup for a (deterministic) harmonic chain with random charges in the presence of a magnetic field (or equivalently particles with constant charges in the presence of a random magnetic field). These results will be introduced in Sec. II.5.4. In order to clarify the presentation of Sec. II.5.4, we decided to briefly recall the study performed by Casher and Lebowitz in [START_REF] Casher | Heat flow in regular and disordered harmonic chains[END_REF] in the context of a (deterministic) harmonic chain with random masses in contact with two heat baths at different temperatures. This is the content of Sec. II.5.3. But first in Sec. II.5.1 we review the famous concept of Anderson's localisation and then in Sec. II.5.2 its relevance in the context of harmonic chains. In condensed matter physics, Anderson's localization refers to the absence of diffusion of waves in a disordered medium. This phenomenon has been discovered by the Nobel prize Anderson in the context of electric diffusion. The electric current inside a metal is due to the movement of the electrons it contains. During their journeys, the electrons undergo shocks with the impurities of the metal which leads to a diffusive behavior of the current. Hence, if τ denotes the average distance traveled by an electron between two shocks, then if τ increases, the electric conductivity of the system increases as well. In 1958, Anderson [START_REF] Anderson | Absence of diffusion in certain random lattices[END_REF] suggested that if the number of impurities is sufficiently high, then the system can become insulating (i.e. does not transport electric charges). The original work of Anderson was for a specific quantum system, but it is in fact a general wave phenomenon that applies to the transport of electromagnetic waves, acoustic waves, quantum waves, spin waves, etc. and in particular to energy transport in disordered chains of oscillators where the energy is transported by phonons which are nothing but waves. The waves propagate ballistically and change direction when they encounter an impurity. If τ is less that the wave length then the wave is localised and the system becomes insulating. Nowadays, Anderson localization is seen as a generic phenomenon present in disordered media. Whereby the addition of random defects in the medium has the tendency to localize in space the normal modes of the system.

II.5.2 FROM ORDERED SYSTEMS TO DISORDERED ONES

Since the seminal work of Rieder, Lebowitz and Lieb (RLL) in [START_REF] Rieder | Properties of a harmonic crystal in a stationary nonequilibrium state[END_REF] heat transport in harmonic chains connected to heat reservoirs has been extensively studied. The main results of RLL were the demonstration that, in the nonequilibrium steady state (NESS), the heat current across the chain saturates with increase in system size while the temperature profile is flat in the bulk of the chain. They provided exact expressions for the asymptotic values of the current and the temperature profile. In later work Nakazawa extended these results to systems with on-site potentials, not considered by RLL, and to higher dimensions in [START_REF] Nakazawa | Energy flow in harmonic linear chain[END_REF][START_REF] Nakazawa | On the lattice thermal conduction[END_REF]. RLL and Nakazawa obtained the steady state current by an exact solution of the correlation matrix which described the Gaussian NESS measure.

In the 70's Lebowitz and others [START_REF] Casher | Heat flow in regular and disordered harmonic chains[END_REF][START_REF] Rubin | Abnormal lattice thermal conductivity of a oneâdimensional, harmonic, isotopically disordered crystal[END_REF][START_REF] O'connor | Heat conduction and sound transmission in isotopically disordered harmonic crystals[END_REF][START_REF] Verheggen | Transmission coefficient and heat conduction of a harmonic chain with random masses: asymptotic estimates on products of random matrices[END_REF] started to investigate the effect of impurities (random masses) on the transport properties of a one-dimensional harmonic chain, arguing in particular that the conductivity κ N of the chain (which is proportional to the system size N for a purely harmonic chain), loses some order of magnitude because of disorder (κ N ∼ N a with a < 1). An alternative approach to compute NESS properties of harmonic chains was used by [START_REF] Casher | Heat flow in regular and disordered harmonic chains[END_REF][START_REF] Roy | Heat transport in ordered harmonic lattices[END_REF] and this is essentially based on the so called Landauer and nonequilibrium Green's function formalism (NEGF) where the current is expressed as an integral over frequency of a phonon transmission coefficient. The NEGF approach has been successfully applied to various classical and quantum systems [START_REF] Dhar | Heat conduction in the disordered harmonic chain revisited[END_REF][START_REF] Dhar | Heat transport in harmonic lattices[END_REF][START_REF] Roy | Heat transport in ordered harmonic lattices[END_REF][START_REF] Venkateshan Kannan | Nonequilibrium stationary state of a harmonic crystal with alternating masses[END_REF][START_REF] Dhar | Nonequilibrium green's function formalism and the problem of bound states[END_REF][START_REF] Majeed | Transport in spinless superconducting wires[END_REF] and will be presented in Sec. II.5.3 since we will use it to study the harmonic chain with random charges.

The aim of Casher and Lebowitz in [START_REF] Casher | Heat flow in regular and disordered harmonic chains[END_REF] was to study the effects of the disorder on the heat current of the chain. Hence, they studied the harmonic chain where the particles have i.i.d. random masses m x . As we mentioned, the conductivity of the chain κ N loses some order due do the impurities of the system, i.e. κ N ∼ N a with a < 1. But at the difference with respect to original Anderson localization, the conductivity does not become exponentially small in the system size and, depending on the physical boundary conditions and the form of the thermostats, it can vanish (a < 0), diverge (0 < a < 1) or even converge (a = 0) as it presented in [START_REF] Dhar | Heat conduction in the disordered harmonic chain revisited[END_REF][START_REF] Roy | Role of pinning potentials in heat transport through disordered harmonic chains[END_REF][START_REF] De Roeck | Step density profiles in localized chains[END_REF]. The reason for this is roughly due to the fact that for disordered harmonic chains, normal modes with frequency ω becomes localized but with a length of localization ℓ(ω) ∼ ω -2 . The role of thermostats and boundary conditions is more difficult to explain without going into computational details.

II.5.3 STUDY OF THE HEAT CURRENT IN THE HARMONIC CHAIN WITH RANDOM MASSES

Consider the one-dimensional harmonic chain of size N connected to two heat baths at the left-hand side and right-hand side with the temperatures T L and T R respectively, with T L > T R . The Hamiltonian of the system is given by

H = N ∑ x=0 |p(x)| 2 2m x + (q(x + 1) -q(x)) 2 2 , (II.124)
completed with the boundary conditions q(t, 0) = q (t, N + 1). Here (m x ) x∈N are a sequence of i.i.d random variables in L 2 . We recall that for any x in {1, • • • , N} we have

p(t, x) = m x v(t, x) .
The equations of motion are given by d dt p(t, 1) = q(t, 2) -2q(t, 1) + η L (t)v(t, 1) , ∀x ∈ {2, . . . , N -1}, d dt p(t, x) = q(t, x + 1) + q(t, x -1) -2q(t, x) , (II.125)

d dt p(t, N) = q (t, N -1) -2q (t, N) + η R (t) -v(t, N) .
(II.126)

Here η L and η R are Gaussian white noise terms acting on the 1 st and N th oscillators respectively and follow the fluctuation-dissipation relation

η L/R (t) η L/R t ′ = 2T L/R δ t -t ′ . (II.127)
In order to simplify the notations we prefer to change the definition of the continuous Fourier's transform. Hence, for f : R → R we denote by f the continuous Fourier's transform of f where for any

ω in R f (ω) = 1 2π R f (t) exp (-iωt) dt . (II.128)
The noise correlations in Fourier space satisfies

ηL/R (ω) ηL/R ω ′ = T L/R π -1 δ ω + ω ′ . (II.129)
The heat current, denoted by J N , entering the wire from the left bath is given by ⟨F L • v(t, 1)⟩ where F L is the force on the the 1 st particle having velocity v(t, 1) and ⟨•⟩ denotes the steady state average. Here F L is defined as

F L = η L (t) -v(t, 1)
.

Hence, we have that

J N = η L (t)v(t, 1) -v 2 (t, 1) . (II.130)
Since the dynamic is linear we can obtain, using the continuous Fourier's transform, an explicit formula for J N . Namely, we want to prove that

J N = (T L -T R ) R T N (ω) , (II.131)
where T N is called the transmission amplitude across the harmonic chain. This is the aim of the next section.

EXPLICIT FORMULA FOR THE HEAT CURRENT

Let us define the column vectors of size N

Q(ω) =        q(ω, 1) q(ω, 2) . . . q(ω, N -1) q(ω, N)        and η(ω) =        ηL (ω) 0 . . . 0 ηR (ω)        .
Then using Fourier's transform in Eq. (II.125) we obtain that

G -1 (ω) Q(ω) = η(ω) with G -1 (ω) = Φ(ω) -ω 2 M -A(ω) , (II.132)
where Φ(ω), M and A(ω) are N × N square matrices with entries given by

[Φ(ω)] x,y = 2δ x,y -δ x,y+1 -δ x,y-1 , [A(ω)] x,y = δ x,y (iγωδ x,1 + iγωδ x,N ) , (II.133) [M] x,y = m x δ x,y .
From Eq. (II.132), we have for every x in {1, . . . , N}

q(ω, x) = N ∑ y=1 G (ω) x,y [ η(ω)] y , (II.134) where G (ω) = Φ(ω) -ω 2 M -A(ω) -1 .
(II.135) By Eq. (II.134) and the definition of the vector η(ω) we have

q(ω, 1) = G (ω) 1,1 ηL (ω) + G (ω) 1,N ηR (ω) .
(II.136) By Fourier's inverse formula we deduce that

q(t, 1) = R G (ω) 1,1 ηL (ω) + G (ω) 1,N ηR (ω) exp (iωt) dω . (II.137)
Using the fact that v(t, 1) is the derivative of q(t, 1) and Eq. (II.129), Eq. (II.130) and Eq. (II.137) we can prove that

J N = (T L -T R ) R T N (ω)dω , (II.138)
where T N is defined by

T N (ω) = 2 π ω 2 G (ω) 1,N 2
.

(II.139)

We recall that

G (ω) 1,N = C 1,N ∆ N (ω) , (II.140)
where

∆ N (ω) = det Φ(ω) -ω 2 M -A(ω) and C 1,N is the cofactor of Φ(ω) -ω 2 M -A(ω) 1,N .
Observe that C 1,N = 1 hence, using Eq. (II.139) and Eq. (II.140) we obtain

T N (ω) = 2 π ω 2 |∆ N (ω)| -2 .
(II.141) By Eq. (II.138) we conclude that

E [J N ] = (T L -T R ) R 2 π ω 2 E |∆ N (ω)| -2 .
(II.142)

EFFECT OF LOCALIZATION DUE TO RANDOM MASSES ON THE NET TRANSMISSION

Now we have an explicit expression for the heat current (cf. Eq. (II.142)) we would like to understand the effect of random masses on the transmission amplitude T N defined in Eq. (II.141).

Let D l,m be the determinant of the sub-matrix of Φω 2 M beginning with the l th row and column and ending with the m th row and column. Then we have

∆ N (ω) = D 1,N -iω (D 2,N + D 1,N-1 ) -ω 2 D 2,N-1 = (1, -iω) D 1,N -D 1,N-1 D 2,N -D 2,N-1 1 iω = (1, -iω) N ∏ x=1 2 -m x ω 2 -1 1 0 1 iω . (II.143)
Let (T x ) x∈N be the two-dimensional Markov chain defined for any x by

T x+1 = 2 -m x ω 2 -1 1 0 T x , where T x = t x t x-1 and T 0 ∈ R 2 . (II.144)
Observe that the discrete time process (t x ) x∈N satisfies for any ω

t x+1 = (2 -m x ω 2 )t x -t x-1 . (II.145)
Formally, the behavior of ∆ N (ω) is related to the growth of |T N (ω)| which will be in the form e 2λ RM (ω)N where is the so-called Lyapunov exponent of the process (T N (ω)) N∈N . Here E [•] denotes the average with respect to the disorder. The limit exists by Furstenberg's theorem [START_REF] Furstenberg | Noncommuting random products[END_REF], is non-negative and independent of the initial condition T 0 . In [START_REF] Matsuda | Localization of normal modes and energy transport in the disordered harmonic chain[END_REF] it is proved that for ω close to zero we have

λ RM (ω) = lim N→∞ 1 2N E [log |T N (ω)|] = lim N→∞ 1 N E [log |t N (ω)|] , (
λ RM (ω) ∼ σ 2 m 8E[m] ω 2 , (II.147)
where E[m] and σ 2 m denote the expectation and the variance of the random variable m 1 respectively. As it can be seen from Fig. II.5, the randomness causes suppression of the net transmission T N (ω) due to localization of the normal modes of the system. The normal modes of frequency ω get exponentially localized due to randomness with a localization length ℓ RM (ω) given by

ℓ RM (ω) = [λ RM (ω)] -1 .
(II.148)

In Eq. (II.147) we see that λ RM (ω) goes to zero when ω goes to zero. Hence, the normal modes with energies closer to ω = 0 are not localized and contribute to the transmission. Hence, formally we could reduce the integration limit of Eq. (II.142) to values of ω for which the localization length is greater than the system size. For the remaining ω values, for which the localization length is less than the system size, the transmission will be negligible. Hence, we cut off the integral limit to ω N max where ℓ RM ω N max = N. Asymptotically the current is formally given by

E [J N ] ∼ 2 (T L -T R ) ω N max 0 lim N→∞ E [T N (ω)] dω ∼ 2 (T L -T R ) ω N max 0 T ∞ (ω)dω . (II.149)
Observe that the frequency ω N max will be very small for large N and for such small frequencies we expect T ∞ (ω) to have a weak dependence on disorder since in the Eq. (II.145), the randomness is multiplied by ω 2 . Hence, in the above equation T ∞ (ω) is written without a disorder average and can be determined by considering the chain with constant masses equal to E[m]. By Eq. (II.147) and Eq. (II.149) we deduce that formally

E [J N ] ∼ 2 (T L -T R ) N -1/2 0 T ∞ (ω)dω .
(II.150)

Hence, to obtain the size of the current it remains to compute T ∞ , this has been done in [START_REF] Dhar | Heat conduction in the disordered harmonic chain revisited[END_REF] where the authors proved that

T ∞ (ω) ∼ ω 2 .
(II.151)

Hence, we finally have

E [J N ] ∼ N -3/2 . (II.152)
We recall that in [START_REF] Rieder | Properties of a harmonic crystal in a stationary nonequilibrium state[END_REF], Rieder, Lieb and Lebowitz proved that when all the masses are equal the heat current is of order N. Hence, the impurities due to disorder make the system less conductive. The results presented in this section were only formal until Ajanki and Huveneers proved rigorously, in [START_REF] Ajanki | Rigorous scaling law for the heat current in disordered harmonic chain[END_REF] that the heat current is of order

N -3/2 .
The case where the disorder is in the interparticle springs instead of the masses has recently been addressed in [START_REF] Amir | Thermal conductivity in 1d: Disorder-induced transition from anomalous to normal scaling[END_REF][START_REF] Ash | Thermal conductance of one-dimensional disordered harmonic chains[END_REF]. In higher dimensions the situation is less understood, we refer the reader to [START_REF] Chaudhuri | Heat transport and phonon localization in mass-disordered harmonic crystals[END_REF][START_REF] Wee | Heat conduction in a two-dimensional harmonic crystal with disorder[END_REF] for more information. More recently there has been a renewed interest for these questions with respect to the effect of nonlinearities [START_REF] Dhar | Effect of phonon-phonon interactions on localization[END_REF][START_REF] Dhar | Heat conduction in the disordered fermi-pasta-ulam chain[END_REF] or of an energy conserving noise [START_REF] Bernardin | Thermal conductivity for a noisy disordered harmonic chain[END_REF][START_REF] Bernardin | Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential[END_REF][START_REF] Bernardin | Green-Kubo formula for weakly coupled systems with noise[END_REF][START_REF] Dhar | Heat conduction in disordered harmonic lattices with energy-conserving noise[END_REF][START_REF] Erignoux | Equilibrium fluctuations for the disordered harmonic chain perturbed by an energy conserving noise[END_REF]. In the next section, we introduce the results of Chapter IV where the disorder in the chain is due to a random magnetic field.

II.5.4 CONTRIBUTION TO THE STUDY OF HEAT CURRENT

In this section, we introduce Chapter IV where we will present the results obtained in [START_REF] Majeed Bhat | Heat transport in an ordered harmonic chain in presence of a uniform magnetic field[END_REF][START_REF] Cane | Localization effects due to a random magnetic field on heat transport in a harmonic chain[END_REF].

SHORT PRESENTATION OF THE MODEL

We consider a homogeneous (m x = 1 for all x) one-dimensional chain of N harmonic oscillators each having two transverse degree's of freedom so that every oscillator is free to move in a plane. The oscillators are assumed to have masses equal to one. We consider a site-dependent magnetic field ⃗ B x = B x ⃗e 3 , perpendicular to the plane of motion 16 , which can be obtained from a vector potential ⃗ A x = (-B x q 2 (x), B x q 1 (x), 0) at each lattice site. The Hamiltonian of the chain is given by

H = N ∑ x=1 (p 1 (x) + B x q 2 (x)) 2 + (p 2 (x) -B x q 1 (x)) 2 2 + N ∑ x=0 |q(x + 1) -q(x)| 2 2 .
We will consider the two different boundary conditions i) fixed boundaries with q(t, 0) = q (t, N + 1) = 0. ii) free boundaries with q(t, 0) = q(t, 1) and q(t, N) = q(t, N + 1). In order to study heat current through this system, we consider the 1 st and the N th oscillators to be connected to heat reservoirs at temperatures T L and T R respectively with T L > T R . The heat reservoirs are modelled using dissipative and noise terms leading to the following Langevin equations of motion

d dt p i (t, x) = q i (t, x + 1) + q i (t, x -1) -c x q i (t, x) + B x (δ i,1 p 2 (t, x) -δ i,2 p 1 (t, x)) + η L i (t)δ x,1 + η R i (t)δ x,N -(δ x,1 + δ x,N )p i (t, x) , (II.153)
for i in {1, 2} and x in {1, 2, . . . , N}. Here η L (t

) := (η L 1 (t), η L 2 (t)) and η R (t) := (η R 1 (t), η R 2 (t)
) are Gaussian white noise terms acting on the 1 st and N th oscillators respectively. These follow the regular white noise correlations. The coefficients c x fix the boundary conditions of the problem. For fixed boundaries c x = 2 for all x, while for free boundary conditions c x = 2δ x,1δ x,N . Fig. II.6 gives a picture of the model. In this study, since the particles live in R 2 , the heat current is defined as 17 Figure II.6: Two dimensional harmonic chain submitted to a magnetic field.

J N = η L (t)v 1 (t, 1) -v 2 1 (t, 1) + η L (t)v 2 (t, 1) -v 2 2 (t, 1) . (II.154)
The aim is to obtain a equation for J N as the one obtained in Eq. (II.138). We will follow the strategy presented in the previous section.

EXPLICIT FORMULA OF THE HEAT CURRENT

For i in {1, 2} and ω in R we define

Qi (ω) =        qi (ω, 1) qi (ω, 2) . . . qi (ω, N -1) qi (ω, N)        and ηi (ω) =        ηL i (ω) 0 . . . 0 ηR i (ω)       
. Using Fourier's transform in Eq. (II.153) we obtain then

G -1 (ω) Q1 (ω) Q2 (ω) = η1 (ω) η2 (ω) with G -1 (ω) = Π(ω) K(ω) -K(ω) Π(ω) , (II.155)
where Π(ω) and K(ω) are square matrices with matrix elements given by

[Π(ω)] x,y = (c x -ω 2 )δ x,y -δ x,y+1 + δ x,y-1 + iωδ x,1 δ y,1 + iωδ x,N δ y,N , [K(ω)] x,y = -iB x ωδ x,y .
Observe that Π = Φω 2 M -A(ω) defined in Eq. (II.133) with M = Id. Using Eq. (II.155) we obtain

q1 (ω, x) = N ∑ y=1 G + 1 (ω) x,y [ η1 (ω)] y + N ∑ y=1 G + 2 (ω) x,y [ η2 (ω)] y , (II.156) q2 (ω, x) = - N ∑ y=1 G + 2 (ω) x,y [ η1 (ω)] y + N ∑ y=1 G + 1 (ω) x,y [ η2 (ω)] y , (II.157)
where

G + 1 = Π + KΠ -1 K -1 and G + 2 = -G + 1 KΠ -1 .
(II.158)

These two last matrices form the 2 × 2 block structure of the matrix G (ω) as

G (ω) = G + 1 (ω) G + 2 (ω) -G + 2 (ω) G + 1 (ω)
.

Observe that if for all x B x = 0,we recover the expression given in Eq. (II.134). The magnetic field coupled the evolution of q 1 and q 2 by the presence of G + 2 . Using the same method we presented to obtain Eq. (II.138) we obtained that

J N = (T L -T R ) R dωT N (ω) , (II.159)
where T N is defined by

T N (ω) = 2 π ω 2 G + 1 (ω) 1,N 2 + G + 2 (ω) 1,N 2 .
(II.160)

Contrary to the study performed by Casher and Lebowitz presented in Sec. II.5.3 using the cofactor is not sufficient to obtain a simpler expression for the transmission. In order to rewrite the components of the two Green's functions [G + 1/2 (ω)] 1,N we had to use a transfer matrix approach which is presented in Sec. IV.2.2 of Chapter IV. We finally obtained that

J n = (T L -T R ) π R ω 2 1 F + N (ω) 2 + 1 F - N (ω) 2 , (II.161)
where for any ω we have

F ± N (ω) = f ± N + iω g ± N + f ± N-1 -ω 2 g ± N-1 . (II.162)
Here the process ( f ± x ) x∈N and (g ± x ) x∈N are defined by Observe that the stochastic processes ( f - x ) x∈N and (g - x ) x∈N are defined in terms of the two-dimensional discrete time Markov chain (U x ) x∈N given by

f ± x+1 = (c x+1 -ω 2 ± ωB x+1 ) f ± x -f ± x-1 , f ± 0 = 1, f ± 1 = c 1 -ω 2 ± ωB 1 , g ± x+1 = (c x+1 -ω 2 ± ωB x+1 )g ± x -g ± x-1 , g ± 0 = 0, g ± 1 = 1 .
U x+1 = 2 -ω 2 -ωB x+1 -1 1 0 U x , where U x := u x u x-1 , (II.164)
by choosing suitable initial conditions. By replacing the B x 's by -B x 's in the last display, we see that ( f + x ) x∈N and (g + x ) x∈N can also be expressed in terms of (U x ) x∈N . In Sec. II.5.3, in order to obtain an estimation of the size of the heat current we studied the asymptotic behavior of the discrete process (t x ) x∈N defined in Eq. (II.145). Here, we have to study the behavior of the discrete process (u x ) x∈N . We denote by λ the Lyapunov exponent associated to the Markov chain (U x ) x∈N , which is defined by

λ (ω) = lim N→∞ 1 2N E [log ∥U N (ω)∥] = lim N→∞ 1 N E [log |u N (ω)|] > 0 , (II.165)
with E [•] denoting the average with respect to the random variable B 1 . As it is showed in Fig. II.7 the effects of the random magnetic field are similar to the one of random masses in the sense that the normal modes are localized with a localization length ℓ defined in Eq. (II.148) where λ RM is replaced by λ . Hence, formally, following the argument presented in Sec. II.5.3 we have

E [J N ] = 2 (T L -T R ) ω N max 0 T ∞ (ω)dω , (II.166)
where ℓ ω N max = N. In Eq. (II.166), T ∞ is the net transmission for the harmonic chain submitted to a constant magnetic field of intensity E [B].

To obtain the size of the heat current, it remains to compute the value of T ∞ and the Lyapunov exponent λ . In Sec. IV.3 of Chapter IV we prove 18 that if E [B] ̸ = 0 T ∞ (ω) ∼ ω 3/2 and T ∞ (ω) ∼ ω 1/2 for fixed and free boundaries respectively.

(II.167)

If E [B] = 0 then the harmonic chain is not submitted to a magnetic field and in [START_REF] Dhar | Heat conduction in the disordered harmonic chain revisited[END_REF][START_REF] Rubin | Abnormal lattice thermal conductivity of a oneâdimensional, harmonic, isotopically disordered crystal[END_REF] it is proved that T ∞ (ω) ∼ ω 2 and T ∞ (ω) ∼ ω 0 for fixed and free boundaries respectively. (II.168)

LYAPUNOV EXPONENT AND SIZE OF THE CURRENT

In order to obtain the size of the current it remains to compute the Lyapunov exponent λ . We formally derive it in Sec. IV.4.4 of Chapter IV and obtained 19

λ (ω) ∼ ω 2/3 if E [B] = 0 , λ (ω) ∼ ω if E [B] > 0 , (II.169) λ (ω) ∼ ω 1/2 if E [B] < 0 .
In Sec. IV.4.5 of Chapter IV, we present some numerical simulations of the Lyapunov exponent which seem to confirm our formal results. Using Eq. (II.166), Eq. (II.167), Eq. (II.168) and Eq. (II.169) we can obtain the size of the current. In order to lighten the presentation we group all of these results in In Sec. IV.4.5 of Chapter IV we present some numerical simulations of the heat current. We observe reasonable agreement with the theoretically expected power laws except for the case E [B] = 0 for free boundary conditions. For this case, from numerical simulations it seems that E [J N ] ∼ N -2 instead of the expected N -3/2 . This case seems to be quite subtle because of the following two reasons 18 These results are extracted from [START_REF] Majeed Bhat | Heat transport in an ordered harmonic chain in presence of a uniform magnetic field[END_REF]. 19 These results are extracted from [START_REF] Cane | Localization effects due to a random magnetic field on heat transport in a harmonic chain[END_REF].

Boundary Conditions Average magnetic field T ∞ (ω) λ (ω) Power law for the current . Thus, for any finite but large N, we have ω N max > ω N s and there are a sufficient number of conducting modes. On the other hand, for E [B] = 0, both ω N max and ω N s scale as N -3/2 and this could be the reason why our heuristic approach for current scaling fails for this case. In this chapter, 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 [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF][START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF] 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 [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] that if B ̸ = 0 it is described by a 5/6-fractional diffusion. In this chapter 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. These results are extracted from [START_REF] Cane | Superdiffusion transition for a noisy harmonic chain subject to a magnetic field[END_REF].

E [J N ] Fixed E [B] ̸ = 0 ∼ ω 3/2 ∼ ω ∼ N -5/2 Fixed E [B] = 0 ∼ ω 2 ∼ ω 2/3 ∼ N -9/2 Free E [B] ̸ = 0 ∼ ω 1/2 ∼ ω ∼ N -3/2 Free E [B] = 0 ∼ ω 0 ∼ ω 2/3 ∼ N -3/2
In Sec. III.1, we precise the notations of the chapter. To facilitate the reading of this chapter, we have decided to recall the studied Hamiltonian dynamics that we presented in Sec. II.4.5 of Chapter II. This is the subject of Sec. III.2 where we also give a precise definition of the Wigner's distribution. In Sec. III.3, we recall the historical results obtained in [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF][START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF]. In Sec. III.4, we state the main results of this paper which are proved in Sec. III.5. In order to make the reading easier, intermediate results are shown in Sec. III.6, Sec. III.7 and Sec. III.8.

III.1 NOTATIONS

Let a and b be two positive real numbers, we will write a ≲ b when there exists a positive constant C such that a ≤ Cb. We recall that the conjugate of a complex number z is denoted by z * and i denotes the complex number of modulus 1. We denote the one-dimensional torus by T := R/Z := -1 2 , 1 2 and the euclidean norm on R n by | • |.

If X is a topological space we denote the Borelian σ -field of X by B(X). We denote by F ([0, T ], X) the set of X-valued functions on [0, T ], by C ([0, T ], X) the subspace of X-valued continuous functions on [0, T ] and by C b ([0, T ], X) the subspace of X-valued bounded continuous functions on [0, T ]. Let n in N ∪ {∞}, the space of R-valued functions on X with compact support and n times differentiable is denoted by C n c (X). The space of R-valued càdlàg functions on [0, T ] is denoted by D ([0, T ], R) .

For f in ℓ 1 (Z), we define its (discrete) Fourier's transform f :

T → C by ∀k ∈ T, f (k) = ∑ x∈Z f (x) exp(2ikπx) .
As usual we extend this notation for all functions in ℓ 2 (Z). In order to study the Wigner's distribution defined in Sec. III.2.4, we introduce the set of test functions S given by

S := H ∈ C ∞ c (R × T) ∀(n, m, l) ∈ N 3 , sup k∈T sup u∈R (1 + u) l ∂ n k ∂ m u H(u, k) < ∞ .
For any H in S we recall that its (continuous) Fourier's transform in the first variable is denoted by

F [H] : R × T → C where ∀(p, k) ∈ R × T, F [H] (p, k) = R H(u, k) exp(2iπ pu)du .
The set S is stable under the action of F . In the whole chapter, for H in S , we denote the Laplacian of H in the first variable by ∆ [H] and for α in (1, 2) the fractional Laplacian of H in the first variable by -(-∆)

α 2 [H] where we recall that for any p in R and k in T

F [∆ [H]](p, k) = -(2π p) 2 F [H] (p, k) , F -(-∆) α 2 [H] (p, k) = -|2π p| α F [H] (p, k) .
The space S × S is equipped with the norm ∥ • ∥ defined by

∀J := (J 1 , J 2 ) ∈ S × S , ∥J∥ = 2 ∑ i=1 R sup k |F [J i ] (p, k)|d p .
The space (S × S , ∥ • ∥) is then a separable space. We denote by (S × S ) ′ the dual of S × S for the weak-* topology (we refer the reader to Sec. III.2.4 for a precise definition). For W in (S × S ) ′ and J in S × S we denote by ⟨W , J⟩ the duality bracket between W and J.

Throughout the chapter, the random variables are defined on an underlying probability space (Ω, F , P) and T denotes a fixed positive time.

III.2 MICROSCOPIC DYNAMICS

In this section, we briefly recall the microscopic dynamics studied in this chapter which was presented in Sec. II.4.5 of Chapter II. This dynamic was first introduced in [START_REF] Basile | Momentum conserving model with anomalous thermal conductivity in low dimensional systems[END_REF][START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF] without a magnetic field and later in [START_REF] Saito | Thermal conductivity for coupled charged harmonic oscillators with noise in a magnetic field[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] with a magnetic field.

III.2.1 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 Z and each x in Z represents the balance position of one atom. We denote the velocity of the atom with rest position x in Z by p(x) = (p 1 (x), p 2 (x)) in R 2 and the displacement from its rest position by q(x) = (q 1 (x), q 2 (x)) in R 2 . We denote the strength of the magnetic field by B ∈ R. The deterministic dynamics is defined at any positive time t and i in {1, 2}

d dt q i (t, x) = p i (t, x) , d dt p i (t, x) = q i (t, x + 1) + q i (t, x -1) -2q i (t, x) + δ i,1 Bp 2 (t, x) -δ i,2 Bp 1 (t, x) . (III.1)
We denote a typical configuration of the system by (q, p) := (q(x), p(x)) x∈Z and the configuration over time by {(q(t), p(t);t ≥ 0)} := {(q(x,t), p(x,t)) x∈Z | t ≥ 0}. The initial configuration (q(x, 0), p(x, 0)) x∈Z is denoted by (q 0 , p 0 ). Let α be the function defined on Z by

∀z ∈ Z, α(z) =    2 if z = 0, -1 if |z| = 1, 0 otherwise.
Observe that for any k in T, α(k) = 4 sin 2 (πk) , (III.2) with α(0) = α′ (0) = 0 and α′′ (0) = 8π 2 . The formal infinitesimal generator of this Markovian dynamics is given by A + BG where for every smooth and local functions 1 φ we have

A[φ ] = 1 2 ∑ x∈Z 2 ∑ i=1 p i (x)∂ q i (x) [φ ] -∑ x,x ′ ∈Z 2 ∑ i=1 α(x -x ′ )q i (x ′ )∂ p i (x) [φ ] , G[φ ] = ∑ x∈Z (p 2 (x)∂ p 1 (x) [φ ] -p 1 (x)∂ p 2 (x) [φ ]) .
The total energy of the configuration (q, p) is denoted by E (q, p) where

E (q, p) = 1 2 ∑ x∈Z |p(x)| 2 + ∑ x∈Z |q(x) -q(x + 1)| 2 .
In the whole chapter, we will study only configurations (q, p) with finite total energy. Observe that the energy is conserved during the time evolution, i.e.

∀t ∈ [0, T ], d dt E (q(t), p(t)) = 0 . (III.3)

III.2.2 EIGENVALUES AND EIGENVECTORS OF THE DETERMINISTIC DYNAMICS

We define on T the following real valued functions

ω 1,B (k) = α(k) + B 2 4 + B 2 and ω 2,B (k) = α(k) + B 2 4 - |B| 2 , (III.4) ∀i ∈ {1, 2}, θ i,B (k) = ω i,B (k) ω 1,B (k) + ω 2,B (k) , (III.5) v B (k) := dω 1,B dk (k) = dω 2,B dk (k) = α′ (k) 2 α(k) + B 2 4 . (III.6)
Observe that for any i in {1, 2} we have

∀k ∈ T, |θ i,B (k)| ≤ 1 .
For every configuration (q, p) we recall that the waves functions ( ψ 1,B , ψ 2,B ) : T → C 2 are defined by

ψ 1,B (q, p)[k] := θ 1,B (k) p 1 (k) -iω 2,B (k) q 1 (k) + i p 2 (k) + ω 2,B (k) q 2 (k) , ψ 2,B (q, p)[k] := θ 2,B (k) p 1 (k) -iω 1,B (k) q 1 (k) -i p 2 (k) -ω 1,B (k) q 2 (k) .
1 Here smooth and local means that for any i in {1, 2}, φ depends only on a finite number of the sequence (q i (x), p i (x)) x∈Z and is smooth with respect to these coordinates.

Lemma III.2.1

For each k ∈ T and each i in {1, 2}, ψ i,B (k) is an eigenvector of (A + BG) corresponding to the eigenvalue -i ω i,B (k), i.e.

∀i ∈ {1, 2}, (

A + BG) { ψ i,B [q, p]} (k) = -i ω i,B (k) ψ i,B [q, p](k) .

III.2.3 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 functions φ is given by

S [φ ] = 1 4 ∑ x∈Z 2 ∑ i, j=1 i̸ = j (Y i, j (x, x + 1)) 2 [φ ] ,
where

Y i, j (x, x + 1) = (p j (x + 1) -p j (x))(∂ p i (x+1) -∂ p i (x) ) -(p i (x + 1) -p i (x))(∂ p j (x+1) -∂ p j (x) ) .
The operator S conserves the total energy and the total pseudomomentum P m of the system defined by

P m = ∑ x∈Z p 1 (x) -Bq 2 (x), ∑ x∈Z p 2 (x) + Bq 1 (x) . (III.7)
We introduce a scaling parameter ε > 0 and we take γ > 0 which represents the intensity of the stochastic noise. We denote the formal infinitesimal generator of the dynamics by L ε where

L ε = A + BG + εγS . (III.8)
Let (q ε (t), p ε (t)) t≥0 be the Markovian dynamics generated by the formal infinitesimal generator L ε . For a rigorous definition of the dynamics we refer the reader to [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF]Sec 3.4] or to [START_REF] Da | Stochastic equations in infinite dimensions[END_REF]Chapter 6].

We can check that L ε conserves the total energy and the total pseudo-momentum. Since the energy of the dynamics (q ε , p ε ) is constant we denote it by E ε . Remark however that since (q ε (0), p ε (0)) is a random variable, E ε is also a random variable constant in time. In order to simplify the notations for all i in {1, 2}, k in T and positive time t, we will write ψε i (t, k) instead of ψi,B (q ε (t), p ε (t)) [k]. Using the conservation of the energy, we can express the total energy E ε in terms of ( ψ ε i ) i∈{1,2} in the following way

E ε = 1 2 T | ψ ε 1 (0, k)| 2 + | ψ ε 2 (0, k)| 2 dk .
We assume that the initial configuration of the system (q 0 , p 0 ) is distributed according to a measure µ ε which satisfies the following condition

K 0 = sup 0<ε<1 ε T E µ ε | ψ ε 1 (0, k)| 2 + | ψ ε 2 (0, k)| 2 dk < ∞ . (III.9)
Since the energy of the system is preserved, this condition is true at any time t, i.e.

sup 0<ε<1 sup t≥0 ε T E µ ε | ψ ε 1 (t, k)| 2 + | ψ ε 2 (t, k)| 2 dk = K 0 . (III.

III.2.4 WIGNER'S DISTRIBUTION

The space (S × S ) ′ is equipped with the weak-* topology, i.e. a sequence

W N N∈N = W N 1 , W N 2 N∈N
in (S × S ) ′ converges to W in (S × S ) ′ if and only if for any

J := (J 1 , J 2 ) in S × S and i in {1, 2} lim N→∞ W N i , J i -⟨W i , J i ⟩ = 0 .
We say then that a sequence

W N N∈N in F ([0, T ], (S × S ) ′ ) converges pointwise to W if and only if for any t in [0, T ], W N (t) N∈N converges to W (t) in (S × S ) ′ .
For each t in [0, T ], we define the Wigner's distribution, denoted by W ε (t), as the element of (S × S ) ′ defined for any J = (J 1 , J 2 ) in S × S by

⟨W ε (t), J⟩ = 2 ∑ i=1 ⟨W ε i (t), J i ⟩ ,
where, for i in {1, 2},

⟨W ε i (t), J i ⟩ = ε 2 ∑ x,x ′ ∈Z E µ ε ψ ε i tε -1 , x ′ * ψ ε i tε -1 , x T e 2iπ(x ′ -x)k J i ε 2 (x + x ′ ), k * dk = ε 2 R T E µ ε ψ ε i tε -1 , k -ε p 2 * ψ ε i tε -1 , k + ε p 2 F [J i ] (p, k) * dkd p . (III.11)
The integrability conditions justifying the existence of this expression is proved in Sec. III.6.1.

The following lemma proves the well posedness of the Wigner's distribution as an element of C ([0, T ], (S × S ) ′ ) and gives some of its properties.

Lemma III.2.2

The Wigner's distribution satisfies the following properties i) For all t in [0, T ], W ε (t) belongs to (S × S ) ′ . ii) For any J in S × S , the family

(⟨W ε , J⟩) ε>0 is bounded in (C ([0, T ], C), ∥ • ∥ ∞ ) and ∥ ⟨W ε , J⟩ ∥ ∞ ≲ K 0 ∥J∥ .
Furthermore, the application t → W ε (t) belongs to C ([0, T ], (S × S ) ′ ).

Proof : We refer the reader to Sec. III.6.1.

To study the asymptotic behavior of (W ε ) ε>0 we need to introduce two distributions on S × S denoted by Γ ε and Γε defined for any

J := (J 1 , J 2 ) in S × S by ⟨Γ ε , J⟩ = 2 ∑ i=1 ⟨Γ ε i , J i ⟩ and ⟨ Γε , J⟩ = 2 ∑ i=1 ⟨ Γε i , J i ⟩ , (III.12)
where, for any i in {1, 2}, by letting i * = 3i, we set

⟨Γ ε i (t), J i ⟩ = ε 2 R T E µ ε ψ ε i tε -1 , ε p 2 -k ψ ε i * tε -1 , k + ε p 2 F [J i ] (p, k) * dkd p . (III.13) Γε i (t), J i = ε 2 R T E µ ε ψ ε i tε -1 , k - ε p 2 * ψ ε i * tε -1 , -k - ε p 2 * F [J i ] (p, k) * dkd p . (III.14)

III.3 REVIEW OF PREVIOUS RESULTS

In this section, we recall the results of [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF][START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] that we formally presented in Sec. II.4.5 of Chapter II. We first recall Theorem III.1 from [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] which states the convergence of the Wigner's distribution to the solution f B of some linear Boltzmann's equation (cf. Eq.(III.18)). Then we recall Theorem III.2 and Theorem III.3 which states the convergence of f B in some hydrodynamic scaling to the solution of some fractional diffusion equation (Eq. (III.34)). These results are proved in [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF][START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF].

III.3.1 KINETIC LIMIT OF THE WIGNER'S DISTRIBUTION

Let J := (J 1 , J 2 ) in S × S and recall Eq. (III.4), (III.5) and (III.6). We define a collisional operator C B : S × S → S × S in the following way. For any u in R, k in T, i in {1, 2} and

J := (J 1 , J 2 ) in S × S , [C B J] i (u, k) = 2 ∑ j=1 T θ 2 i,B (k)R k, k ′ θ 2 j,B k ′ J j (u, k ′ ) -J i (u, k) dk ′ , (III.15) with ∀ k, k ′ ∈ T × T, R k, k ′ = 16 sin 2 (πk) sin 2 (πk ′ ) . (III.16)
Let W in C ([0, T ], (S × S ) ′ ) and W 0 in (S × S ) ′ . i) We say that W is a weak solution on [0, T ] of the linear Boltzmann's equation

∂ t W + 1 2π v B ∂ u W = γC B W , (III.17)
with W 0 as initial condition if and only if for any J in S × S and any t in [0, T ]

⟨W (t), J⟩ -⟨W (0), J⟩ = t 0 1 2π ⟨W (s), v B ∂ u J⟩ ds + γ t 0 ⟨W (s),C B J⟩ ds . (III.18)
ii) We say that W is a measure valued weak solution on [0, T ] of the linear Boltzmann's equation (III.17) if it is a weak solution of Eq. (III.17) such that for all t in [0, T ] and i in {1, 2}, W i (t) is a bounded positive measure on R × T.

Lemma III.3.1

Let µ 0 := (µ 0 1 , µ 0 2 ) in (S × S ) ′ be a couple of bounded positive measure on R × T. Then, there exists a unique measure valued weak solution µ on [0, T ] to the linear Boltzmann's equation (III.17) with µ 0 as initial condition.

Proof : We refer the reader to Sec. III.6.5.

The following theorem summarizes the results obtained by the authors of [104, Theorem 1] and by those of [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF]Theorem 5] respectively.

Theorem III.1 ([11, 104])

Let T > 0. Assume that the condition (III.9) holds and that (W ε (0)) ε>0 converges in (S × S ) ′ to a bounded positive distribution W 0 . i) If B ̸ = 0, then there exists W in C ([0, T ], (S × S ) ′ ) such that (W ε ) ε>0 converges pointwise to W in F ([0, T ], (S × S ) ′ ). Moreover for each time t in [0, T ], the limit W (t) and W 0 can be extended to a couple of bounded positive measures on R × T respectively denoted by µ(t) := (µ 1 (t), µ 2 (t)) and µ 0 := (µ 0 1 , µ 0 2 ). Furthermore, µ belongs to C ([0, T ], (S × S ) ′ ) and is the unique measure valued weak solution of the Boltzmann's equation (III.17) with initial condition µ 0 . ii) If B = 0 and furthermore

lim κ→0 lim sup ε→0 ε 2 |k|<κ E µ ε | ψ ε i (0, k)| 2 dk = 0 , (III.19)
then the same conclusion as in i) holds with B = 0 in Eq. (III.17).

Remark III.3.2

When B = 0, the assumption (III. [START_REF] Bernardin | Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential[END_REF]) is required to compensate the lack of differentiability at k = 0 of the function v 0 defined in Eq. (III.6).

III.3.2 HYDRODYNAMIC LIMIT OF THE BOLTZMANN'S EQUATION

In this section, we recall the study made in [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] to obtain the hydrodynamic limit of the solution of the Boltzmann's equation (cf. Eq. (III.18)). We can summarize their study as follows; first, the Boltzmann's equation is interpreted as the infinitesimal generator of a Piecewise Deterministic Markov Process, then they studied this Markov process and proved the convergence to some α-stable Lévy process. Finally, they proved that the solution of the Boltzmann's equation converges to some ρ B which is solution of a fractional diffusion equation, its exponent depends on the presence or absence of the magnetic field. For a short reminder about α stable Lévy process, we refer the reader to Sec. II.4.5 of Chapter II.

For i in {1, 2} and (k, k ′ ) in T 2 , we recall that θ i,B (k) and R (k, k ′ ) are defined in Eq. (III.5) and Eq. (III.16) respectively. Let f : T × {1, 2} → R be a real function such that for any i in {1, 2} f (•, i) ∈ C (T, R). We define an operator L B acting on f by

L B [ f ](k, i) = λ -1 B (k, i) 2 ∑ j=1 T P B k, i, dk ′ , j f k ′ , j -f (k, i) dk ′ . (III.20)
Here for any (k, k ′ ) in T 2 and (i, j) in {1, 2} 2

P B k, i, dk ′ , j = γλ B (k, i)θ 2 i,B (k)θ 2 j,B k ′ R k, k ′ dk ′ , (III.21)
where

λ B (k, i) = γθ 2 i,B (k)R(k) -1 and R(k) = T R k, k ′ dk ′ . (III.22)
Let µ be the measure valued weak solution of the Boltzmann's equation (III.17) with initial condition µ 0 . We assume that µ = (µ 1 , µ 2 ) and µ 0 = (µ 0 1 , µ 0 2 ) have a density denoted by f B (•) := ( f B (•, 1), f B (•, 2)) and f 0 (•) := f 0 (•, 1), f 0 (•, 2) respectively with respect to the measure dkdδ . Then we have for any time t in [0, T ], k in T, u in R and i in {1, 2}

∂ t f B (t, u, k, i) + v B (k) 2π ∂ u f B (t, u, k, i) = L B [ f B ] (t, u, k, i) , (III.23) with f 0 (•) := ( f 0 (•, 1), f 0 (•, 2
)) as initial condition. To study the hydrodynamic behavior of f B we interpret the operator L B as the infinitesimal generator of a pure jump continuous time Markov process on T × {1, 2} as follows.

Let X B n n∈N := K B n , I B n n∈N be the Markov chain on T × {1, 2} with transition probability P B defined in Eq. (III.21). Let (τ n ) n∈N be an i.i.d sequence of random variables, independent of X B n n∈N such that τ 0 ∼ E (1). We define the random variable T N by2

T 0 = 0 and ∀N ∈ N * , T N = N ∑ n=1 λ B X B n-1 τ n-1 . (III.24)
Then we can define a pure jump Markovian process K B (•), I B (•) with values in T × {1, 2} where for any positive time t in [0, T ] [START_REF] Bernardin | Green-Kubo formula for weakly coupled systems with noise[END_REF]. From this process, we can define an additive functional of Markov process Z B u (•) such that for any time t in [0, T ] and u in R

K B (t) = K B n , and I B (t) = I B n , ∀t ∈ [T n , T n+1 [ . The infinitesimal generator of K B (•), I B (•) is L B defined in Eq. (III.
Z B u (t) = u - t 0 v B K B (s) 2π ds .
Then by Dynkin's formula we get that for any i in {1, 2} and

(u, k) in R × T f B (t, u, k, i) = E (k,i) f 0 Z B u (t), K B (t), I B (t) . (III.25)
The hydrodynamic behavior of f B is completely determined by the one of the process K B (•), I B (•) . Hence, in the following we recall how the authors of [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] studied this process. Let π B the probability measure on T × {1, 2} defined as follows

π B (dk, di) = 2 ∑ j=1 λ B (k, j) -1 γ R dkδ j (di) with R = T R(k)dk . (III.26)
We can prove that π B is a reversible probability measure of the Markov chain X B n n∈N . Observe that

P B (k, i, dk ′ , j) = π B (dk ′ , j) , (III.27)
where P B is defined in Eq. (III.21). Hence, X B n n≥1 is an i.i.d sequence of random variables on T × {1, 2} but observe that X B n n≥0 is not because of X B 0 . We define a function Ψ B in the following

way ∀k ∈ T, ∀i ∈ {1, 2}, Ψ B (k, i) = v B (k)λ B (k, i) . (III.28)
The asymptotic behavior of the process Z B u (•) is fully determined by the tails of the function Ψ B . In [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] it is proved that

∀r > 0, lim N→∞ N α B π B (k, i) Ψ B (k, i) > Nr = κB |r| -α B , (III.29) ∀r < 0, lim N→∞ N α B π B (k, i) Ψ B (k, i) < Nr = κB |r| -α B , (III.30)
where

α B = 3 2 if B = 0 and α B = 5 3 , if B ̸ = 0 , (III.31)
Figure III.2: Summary of the studies made in [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF][START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] end this section with Theorem III.7 which shows that the interpolation process Y B (•) converges to the fractional process studied in [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF] when B goes to zero and to the one studied in [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] when B is sent to infinity.

III.4.1 TRIVIALITY OF THE TRANSITION IN THE KINETIC TIME SCALE

In this section, we assume that the microscopic dynamics (cf. Eq. (III.1)) is submitted to a magnetic field of intensity B ε := Bε δ where δ > 0 and B > 03 . Theorem III.4

Let δ > 0, η in ]0, 1[ such that δ > η and κ > 0. We define κ ε := κε η and we assume that the condition (III.9) holds and that (W ε (0)) ε>0 converges in (S × S ) ′ to a bounded positive distribution W 0 . We assume furthermore that

lim sup ε→0 ε 2 |k|<κ ε E µ ε | ψ ε i (0, k)| 2 = 0 . (III.35)
Then, there exists

W in C ([0, T ], (S × S ) ′ ) such that (W ε ) ε>0 converges pointwise to W in F ([0, T ], (S × S ) ′ ).
Moreover for each time t in [0, T ], the limit W (t) and W 0 can be extended to couples of bounded positive measures on R × T denoted respectively by µ(t) := (µ 1 (t), µ 2 (t)) and µ 0 := (µ 0 1 , µ 0 2 ). Furthermore, µ belongs to C ([0, T ], (S × S ) ′ ) and is the unique measure valued weak solution on [0, T ] of the Boltzmann's equation (III.17) with B = 0 and initial condition µ 0 := (µ 0 1 , µ 0 2 ).

Proof : We refer the reader to Sec. III.5.1 for the proof of Theorem III.4.

From Theorem III.4 we deduce that under assumption (III.35) the transition in the kinetic time scale ε -1 between the case of [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF] (zero magnetic field) and [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] (magnetic field of order one) is trivial in the sense that it holds for δ = 0. We show in Theorem III.6 that it is not the case in a longer time scale. To prove Theorem III.4 we need the following lemma which shows that the assumption (III.35) can be extended to times tε -1 . Lemma III.4.1

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

lim sup ε→0 ε 2 |k|<κ ε E µ ε ψ ε i tε -1 , k 2 dk = 0 ,
where we recall that κ ε = κε η with η in ]0, 1[ such that δ > η and κ > 0.

Proof : We refer the reader to Sec. III.6.4 for the proof of Lemma III.4.1.

Remark III.4.2

Observe that assumption (III. [START_REF] Cesbron | Fractional diffusion limit for a kinetic equation in the upper-half space with diffusive boundary conditions[END_REF]) is weaker than the assumption (III. [START_REF] Bernardin | Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential[END_REF]) made in [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF] to compensate the lack of differentiabilty of the function v 0 .

III.4.2 TRANSITION IN THE HYDRODYNAMIC TIME SCALE

In this section, we study the Markov chain X B n n∈N introduced in Sec. III.3.2 but with a magnetic field of intensity B N := BN -δ where δ is in R + and B > 04 . Hence, for each N we will denote this chain by X B N n n∈N := K B N n , I B N n n∈N . In Eq. (III.4), Eq. (III.5) and Eq. (III.6) we replace B by B N and in all the functions defined in Sec. III.3.2. The aim of this section, is to study the behavior of

f B N (•) := ( f B N (•, 1), f B N (•, 2)) where for any positive time t, k in T, u in R and i in {1, 2} ∂ t f B N (t, u, k, i) + v B N (k) 2π ∂ u f B N (t, u, k, i) = L B N [ f B N ] (t, u, k, i) , (III.36)
where we recall that v B N and L B N are defined in Eq. (III.6) and Eq. (III.20) respectively. By Feynman Kac's formula we get that for any time t in [0, T ], u in R, k in T and i in {1, 2}

f B N (t, u, k, i) = E (k,i) f 0 Z B N u (t), K B N (t), I B N (t) . (III.37)
Here for t in [0, T ] u in R and N in N we recall that

Z B N u (t) = u - t 0 v B N K B N (s) 2π ds ,
where

K B N (t) = K B N n , and
I B N (t) = I B N n , ∀t ∈ T B N n , T B N n+1 , with T B N 0 = 0 and ∀n ∈ N * , T B N n = n ∑ m=1 λ B N X B N m-1 τ N m-1 .
Here for any integer N, τ N n n≥0 is a sequence of i.i.d random variables independent of X B N n n∈N with τ N 0 ∼ E (1).

For every N, we recall that the invariant probability measure of the Markov chain X B N n n∈N is denoted by π B N and defined in Eq. (III.26). We define the function Ψ B N on T × {1, 2} by

∀(k, i) ∈ T × {1, 2}, Ψ B N (k, i) = v B N (k) λ B N (k, i) .
(III.38)

As we recalled in Sec. III.3.2, the asymptotic behavior of f B N is fully determined by the one of the process Z B N u (•). The next propositions allow us to compute the tails of the function Ψ B N and to determine the asymptotic behavior of the process Z B N u (•).

Proposition III.4.3

We define implicitly two functions x B,± on R * by

2 x 2 B,± (r) + B 2 4 ± B x B,± (r) = π γr , r ̸ = 0 . (III.39)
Then i) x B,± are odd functions.

ii) The functions x B,± and x ′ B,± converge pointwise when B goes to zero and for any r

̸ = 0 lim B→0 x B,± (r) = sign(r) π 2γ |r| -1 2 and lim B→0 x ′ B,± (r) = - √ π 2 √ 2γ |r| -3 2 .
iii) The functions x B,+ and x ′ B,+ converge pointwise to zero when B goes to infinity and for any r ̸ = 0, we have

lim B→∞ Bx B,+ (r) = sign(r) π 2γ |r| -1 and lim B→∞ Bx ′ B,+ (r) = - π 2γ |r| -2 .
The functions x B,-and x ′ B,-diverge pointwise to infinity when B goes to infinity and for any r ̸ = 0, we have

lim B→∞ B -1 3 x B,-(r) = sign(r)C|r| -1 3 and lim B→∞ B -1 3 x ′ B,-(r) = - C 3 |r| -4 3 , with C = π 2γ 1 3
.

Proof : We refer the reader to Sec. III.7.1.

Proposition III.4.4

Let h B,± and g B± be the four reals functions defined for any r ̸ = 0 as follows h B,± (r) = 1 4π

x B,± (r)

0   1 2 ± B 4 x 2 + B 2 4   x 2 dx1 r>0 (III.40) + 1 4π 0 x B,± (r)   1 2 ± B 4 x 2 + B 2 4   x 2 dx1 r<0 . g B,± (r) = -h ′ B,± (r)1 r>0 + h ′ B,± (r)1 r<0 . (III.41)
Then i) g B,± are positive even functions.

ii) The measure ν B with density g B,+ + g B,-with respect to the Lebesgue measure is a Lévy measure on R * . iii) The functions g B,± converge pointwise to g 0 when B goes to zero where for any r ̸ = 0

g 0 (r) = π 2 11 γ 3 1 2 |r| -3 2 -1 . (III.42)
Moreover, g 0 is the density of a Lévy measure on R * .

iv) The function B 1 3 g B,+ converges almost everywhere to zero and B 1 3 g B,-converges pointwise to g ∞ when B goes to infinity where for any r ̸ = 0

g ∞ (r) = π 2 2 11 27γ 5 1 3 |r| -5 3 -1 . (III.43)
Moreover, g ∞ is the density of a Lévy measure on R * .

Proof : We refer the reader to Sec. III.7.2.

Proposition III.4.5

Let π B N be the invariant probability measure defined in Eq. (III.26) and Ψ B N the function defined in Eq. (III.38) then we have

∀r > 0, lim N→∞ N α δ π B N (k, i) Ψ B N (k, i) > Nr = κ δ (r) , ∀r < 0, lim N→∞ N α δ π B N (k, i) Ψ B N (k, i) < Nr = κ δ (r) ,
where for any r ̸ = 0

       α δ = 3 2 and κ δ (r) = κ 1 γ -3 2 |r| -3 2 if δ > 1 2 , α δ = 3 2 and κ δ (r) = (h B,-(r) + h B,+ (r)) if δ = 1 2 , α δ = 5-δ 3 and κ δ (r) = κ 2 |B| -1 3 γ -5 3 |r| -5 3 if δ < 1 2 , (III.44)
with κ 1 and κ 2 two positive constants defined respectively in Eq. (III.107) and in Eq. (III.108).

Proof : We refer the reader to Sec. III.7.3.

Theorem III.5

Let α δ and κ δ defined in Eq. (III.44) and τ 0 ∼ E (1). Let u in R, we define u N := Nu and we assume that X B N 0 = (k, i) with k ̸ = 0 and i in {1, 2}. We denote by Z δ u (•) the Lévy process starting from point u with Lévy measure ν δ where

dν δ (r) :=                γ -1 2 κ 0 E τ α δ 0 |r| -3 2 -1 dr if δ > 1 2 , 2γE τ -1 0 g B,+ 2πr τ 0 + g B,- 2πr τ 0 dr if δ = 1 2 , γ -2 3 |B| -1/3 κ ∞ E τ α δ 0 |r| -5 3 -1 dr if δ < 1 2 ,
(III.45)

with κ 0 = 1 2 10 π 2 1 2
and κ ∞ = 1 2 13 27π 3 .

(III.46)

Then, under P (k,i) the finite-dimensional distributions of the process N -1 Z B N u N (N α δ •) converge weakly to the finite-dimensional distributions of Z δ u (•).

Observe that E τ

α δ 0 = Γ(1 + α δ )
where Γ is the Gamma function. Theorem III.5 allows us to obtain the hydrodynamic limit of the solution f B N of the Boltzmann's equation (III.36). Let Z δ u (•) be the Lévy process starting from u in R with Lévy exponent Φ δ given by

∀θ ∈ R, Φ δ (θ ) = R * exp (iθ r) -1 + iθ r1 {|r|<1} dν δ (r) , (III.47)
where ν δ is the Lévy measure defined in Eq. (III.45). Observe that for δ > 1 2 resp.δ < 1 2 , Z δ u (•) is the Lévy process obtained in [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF] (resp. [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF]). For δ = 1 2 we have an interpolation process denoted by

Y B u (•). Theorem III.6 Let f B N (•) := ( f B N (•, 1), f B N (•, 2
)) be the solution of Eq. (III.36) with initial condition

f 0 N (•) := ( f 0 N (•, 1), f 0 N (•, 2
)) defined as follows

∀(u, k) ∈ R × T, f 0 N (u, k) = f 0 u N , k, 1 , f 0 u N , k, 2 .
(III.48)

Here ( f 0 (•, 1), f 0 (•, 2)) belongs to C ∞ c (R × T) 2 .
We define on R the function f 0 as follows

∀u ∈ R, f 0 (u) = 2 ∑ i=1 T f 0 (u, k, i)dk . (III.49)
We define the operator L δ on the space C ∞ c (R) in the following way

∀φ ∈ C ∞ c (R), ∀p ∈ R, L δ [φ ] (p) = R F [φ ] (ξ )Φ δ (ξ ) exp (2iπ pξ ) dξ , (III.50)
where Φ δ is defined in Eq. (III.47). Let α δ defined by Eq. (III.44), then

∀t ∈]0, T ], ∀u ∈ R, lim N→∞ 2 ∑ i=1 T f B N (N α δ t, Nu, k, i) - 1 2 ρ δ (t, u) dk = 0 ,
where ρ δ is the solution of

∀t ∈]0, T ], ∀u ∈ R, ∂ t ρ δ (t, u) = L δ [ρ δ ] (t, u) , ρ δ (0, u) = f 0 (u) . (III.

51)

Proof : We refer the reader to Sec. III.5.3.

Remark III.4.6

Observe that when δ

> 1 2 , L δ = -Γ 1 + 3 2 γ -1 2 D 0 (-∆) 3
4 and when δ < 1 2 we have that

L δ = -|B| -1/3 Γ 1 + 5 3 γ -2 3 D ∞ (-∆) 5 6
where dr .

D 0 = 2κ 0 +∞ 0 1 -cos(r)
(III.52)

Hence, we recover the different cases studied in [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF] and [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] respectively. . The next proposition shows that by sending B to zero (resp. infinity), ρB converges to the fractional diffusion equations of exponent 3/4 (resp. 5/6).

When δ = 1 2 , L 1

Theorem III.7

Let f 0 be the function defined in Eq. (III.49), D 0 and D ∞ the constants defined in Eq. (III.52), then i) lim This section, is devoted to the proof of Theorem III.4. The strategy of our proof is similar to the one developed by the authors of [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF], so we will only give the ideas of the different steps and we refer the reader to the corresponding sections of [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] for detailed proofs.

B→0 ρB = ρ 0 in L 1 [0, T ], L 2 (R) where ρ 0 is the solution of ∀t ∈]0, T ], ∀u ∈ R, ∂ t ρ 0 (t, u) = -Γ 1 + 3 2 γ -1 2 D 0 (-∆) 3 4 ρ 0 (t, u) , ρ 0 (0, u) = f 0 (u) . (III.53) ii) lim B→+∞ ρB B 1 3 •, • = ρ ∞ in L 1 [0, T ], L 2 (R) where ρ ∞ is the solution of ∀t ∈]0, T ], ∀u ∈ R, ∂ t ρ ∞ (t, u) = -Γ 1 + 5 3 γ -2 3 D ∞ (-∆) 5 6 ρ ∞ (t, u) , ρ ∞ (0, u) = f 0 (u) . (
To prove Theorem III.4 we first prove that there exists W in C [0, T ], (S × S ) ′ and a sequence (ε n ) n∈N such that for any J in S × S and t in [0, T ]

lim n→∞ |⟨W ε n (t), J⟩ -⟨W (t), J⟩| = 0 . (III.55)
Let J := (J 1 , J 2 ) in S × S , to prove Eq. (III.55) we start to show that for any i in {1, 2} and t in [0, T ] we have

∂ t ⟨W ε i (t), J i ⟩ = 1 2π ⟨W ε i (t), v 0 ∂ u J i ⟩ + γ ⟨W ε i (t), [C 0 J] i ⟩ + γ ⟨Γ ε i (t), [C 0 J] i ⟩ + ⟨ Γε i (t), [C 0 J] i ⟩ + O t,J (ε) , (III.56)
where Γ ε and Γε are the anti-Wigner's distributions defined in Eq. (III.13) and Eq. (III.14) and |O t,J (t, ε)| ≤ K(J) × ε with K(J) a constant independent of ε and t. The derivation of this equation is presented in Sec. III.6.2 and Sec. III.6.3. To end the proof we need the following lemma. Lemma III.5.1

Let t in [0, T ] and

J := (J 1 , J 2 ) in S × S then i) ∥⟨Γ ε , J⟩∥ ∞ ≲ K 0 ∥J∥ and ∥⟨ Γε , J⟩∥ ∞ ≲ K 0 ∥J∥. ii) For any i in {1, 2} lim ε→0 T 0 ⟨Γ ε i (t), [C 0 J] i ⟩ = 0 and lim ε→0 T 0 ⟨ Γε i (t), [C 0 J] i ⟩ = 0 .
Proof : We refer the reader to [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF]Sec. 3.4].

By Eq. (III.56), item ii) of Lemma III.2.2 and item i) of Lemma III.5.1 we deduce that for all J in S × S the family of functions (⟨W ε , J⟩) ε>0 is equicontinuous and bounded in

(C ([0, T ], C) , ∥ • ∥ ∞ ) .
Hence, for any J in S × S , there exists

W J (•) ∈ (C ([0, T ], C) , ∥ • ∥ ∞ ) such that ⟨W ε (•), J⟩ converges (up to a subsequence) to W J (•) in (C ([0, T ], C) , ∥ • ∥ ∞ ).
At this point, the subsequence depends on J but since (S × S , ∥ • ∥) is a separable space, we can use a diagonal argument to erase this dependency. Hence, there exists a countable subset D = (J r ) r∈N ⊂ S × S dense in (S × S , ∥ • ∥), a subsequence (ε n ) n∈N going to zero as n goes to infinity and a sequence (W J (•)

) J∈D ∈ (C ([0, T ], C) , ∥ • ∥ ∞ ) such that ∀J ∈ D, lim n→∞ ⟨W ε n (•), J⟩ = W J (•) . Using item ii) of Lemma III.2.2, we deduce that ∀t ∈ [0, T ], ∀J ∈ D, |W J (t)| ≤ K 0 ∥J∥ .

Hence the application

J ∈ D ⊂ S × S → W J (•) ∈ (C ([0, T ], C) , ∥ • ∥ ∞ )
is Lipschitz and linear. It can be uniquely extended into a linear Lipschitz function on S × S . Hence, for any t in [0, T ],

J ∈ S × S → W J (t)
defines an element of (S × S ) ′ denoted by ⟨W (t), J⟩. It remains to prove that W (•) belongs to C [0, T ], (S × S ) ′ .

Let s in [0, T ] and (s p ) p≥0 in [0, T ] N which converges to s when p goes to infinity. Let J in S × S , η > 0 and J r in D such that ∥J -J r ∥ < η .

We have that

|⟨W (s p ) , J⟩ -⟨W (s) , J⟩| ≤ |⟨W (s p ) -W (s), J r ⟩| + |⟨W (s p ) , J -J r ⟩| + |⟨W (s) , J -J r ⟩| ≤ |⟨W (s p ) -W (s), J r ⟩| + 2K 0 ∥J -J r ∥ ≤ |⟨W (s p ) -W (s), J r ⟩| + 2K 0 η . Recall that t ∈ [0, T ] → ⟨W , J r ⟩ = W J r (t) ∈ C (III.57)
is continuous because J r is in D. Hence, it exists p 0 in N * such that

∀p > p 0 , |⟨W (s p ) -W (s), J r ⟩| ≤ η .
Consequently, we have

|⟨W (s p ) , J⟩ -⟨W (s) , J⟩| ≤ (2K 0 + 1) η .
This proves that W (•) is in C [0, T ], (S × S ) ′ and ends the proof.

Hence, we deduce that there exists W in C [0, T ], (S × S ) ′ such that (W ε ) ε>0 converges pointwise (up to a subsequence) to W . Using item ii) of Lemma III.5.1 and sending ε to zero in Eq. (III.56) we get that for any t in [0, T ] and J in S × S

⟨W (t), J⟩ -⟨W (0), J⟩ = t 0 1 2π ⟨W (s), v 0 ∂ u J⟩ ds + γ t 0 ⟨W (s),C 0 J⟩ ds . (III.58)
This proves that W is a weak solution of the Boltzmann's equation (III.17).

Let t in [0, T ]. It remains to extend W (t) into a couple µ(t) := (µ 1 (t), µ 2 (t)) of bounded positive measure on R × T. This is proved in [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF]Lemma D.1]. The idea is to prove that the Wigner's distribution is a positive linear form on C c ((R × T) 2 ) and then by using Riesz's representation theorem we can extend W (t) into a couple of positive measure on R × T denoted by µ(t) := (µ 1 (t), µ 2 (t)).

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

∀u ∈ R, J λ ,r (u) := exp - λ r 2 -u 2 1 [-r,r] (u), λ > 0, r > 0 .
Let λ be fixed, by using Eq. (III.10) and item ii) of Lemma III.2.2 we get that for any i in {1, 2} lim ε→0

W ε i (t), J λ ,r = R×T J λ ,r (u)µ i (t, du, dk) ≤ K 0 .
By sending first λ to zero and then r to infinity we get

µ i (t, R, T) ≤ K 0 .
Hence, µ is a measure valued weak solution of the Boltzmann's equation (III.17) with initial condition µ 0 := (µ 0 1 , µ 0 2 ). By Lemma III.3.1, there is a unique measure valued weak solution of Eq. (III.17) hence we deduce that (W ε ) ε>0 converges pointwise to µ := (µ 1 , µ 2 ) in F ([0, T ], (S × S ) ′ ).

III.5.2 SKETCH OF THE PROOF OF THEOREM III.5

Let u in R, to lighten the notations we define for each time t, Z N u (t) := N -1 Z B N u N (N α δ t) where we recall that α δ is defined in Eq. (III.44) and u N = Nu.

III.5.3 SKETCH OF THE PROOF OF THEOREM III.6

In this section, we prove Theorem III.6, to prove it we need the following lemma. Lemma III.5.5

Let ρ be the counting measure on {1, 2}. Observe that the measure σ := 1 2 dkdρ is an invariant measure for the process

K B N (•), I B N (•) . Let a ∈ (0, 1) such that sup N T×{1,2} λ a B N (k, i)dσ (k, i) < ∞ , (III.70)
where λ B N is defined in Eq. (III.22) and define dσ a N := λ a B N dσ . Then for any centered f in L 1 (σ ) and positive time t

P t N f L 1 (σ ) ≤ C (1 + t) a ∥ f ∥ L 1 (σ 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

K B N (•), I B N (•) .
Proof : This result is proved in a general setting in [START_REF] Komorowski | Long time asymptotics of a degenerate linear kinetic transport equation[END_REF]Theorem 1.1]. In our context, following the proof of [START_REF] Komorowski | Long time asymptotics of a degenerate linear kinetic transport equation[END_REF] we can check that C does not depend on N.

We follow the strategy developed in [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF] and [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF]. The fundamental tool is Theorem III.5. Let K B N (0), I B N (0) = (k, i) with k ̸ = 0 and i in {1, 2}. By definition for any positive time t, u in R, i in {1, 2} and k in T

f B N (N α t, Nu, k, i) = E (k,i) f 0 Z N u (t), K B N (N α t) , I B N (N α t) ,
where we recall that For any i in {1, 2},

Z N u (t) := N -1 Z B N u N (N α δ t) = u - 1 2πN N α t 0 dsv B N K B N (s) .
f 0 (i) ∈ C ∞ c (R × T) hence f 0 is in C ∞ c (R)
where f 0 is defined in Eq. (III.49).

We have then

1 2 lim N→∞ E (k,i) f 0 Z N u ( 1 -N -α δ m N t) = 1 2 ρ δ (u,t) P.a.s , (III.72)
where ρ δ is the unique solution of Eq. (III.51). By using Fourier's inverse formula we have

f B N (N α δ t, Nu, k, i) = E (k,i) f 0 Z N u (t) , K B N (N α δ t) , I B N (N α δ t) = ∑ r∈Z R d p 2 ∑ j=1 F [ f 0 ](p, r, j)E (k,i) e i(pZ N u (t)+rK B N (N α δ t)) 1 I B N (N α δ t)= j ,
where f 0 (i) is the function defined for any z in R, i in {1, 2}, by

f 0 (z, k, i) := ∑ r∈Z f 0 (z, r, i) exp (2iπkr) . (III.73)
By using Fourier's inverse formula we get

1 2 E (k,i) f 0 Z N u t -N -α δ m N t = 2 ∑ j=1 E (k,i) 1 2 T×{1,2} dk ′ dρ( j) f 0 Z N u t -N -α δ m N t , k ′ , j = 2 ∑ j=1 ∑ r∈Z R d pF [ f 0 ](p, r, j)E (k,i) e ipZ N u (t-N -α δ m N t) 1 2 T dk ′ e ik ′ .
By using Eq. (III.72) and the dominated convergence theorem we get lim sup

N→∞ T×{1,2} dkdρ(i) f B N (N α δ t, Nu, k, i) - 1 2 ρ δ (u,t) ≤ lim sup N→∞ T×{1,2} dkdρ(i) |A i,r,u,k,N | , where A i,r,u,k,N := f B N (N α δ t, Nu, k, i) - 1 2 E (k,i) f 0 Z N u 1 -N -α δ m N t .
By using Fourier's inverse formula we get

|A i,r,u,k,N | ≤ ∑ r∈Z R d p 2 ∑ j=1 F [ f 0 j ](p, r) I N 1 (t, k, i, j, p, r) + I N 2 (t, k, i, j, p, r) .
where

I N 1 (t, k, i, j, p, r) = E (k,i) e i(pZ N u (t)+rK B N (N α δ t)) 1 I B N (N α δ t)= j -E e i(pZ N u (t-N -α δ m N t)+rK B N (N α δ t)) 1 I B N (N α δ t)= j , and
I N 2 (t, k, i, j, p, r) = E (k,i) e i(pZ N u (t-N -α δ m N t)+rK B N (N α δ t)) 1 I B N (N α δ t)= j -E e ipZ N u (t-N -α δ m N t) 1 2 T e ik ′ dk ′ .
To conclude the proof it is sufficient to show that for any positive time t, k ̸ = 0, (i, j) in {1, 2} 2 , p in R and r in Z we have ∀q ∈ {1, 2}, lim

N→∞ I N q (t, k, i, j, p, r) = 0 . (III.74)
We start to deal with I N 1 . We recall that for any a in R we have 1e ia ≤ |a| and that the function v B N is a bounded function independently of N (see Eq. (III.6)). Hence, for any positive t, p in R, r in Z and j in {1, 2} we have

I N 1 (t, k, i, j, p, r) ≤ E k,i 1 -e ip(Z N u (t)-Z N u (t-N -α δ m N t)) ≤ E k,i p Z N u (t) -Z N u t -N -α δ m N t ≲ ∥v B N ∥ ∞ t|p|m N N -1 ,
which goes to zero when N goes to infinity by definition of the sequence (m N ) N∈N (cf. Eq. (III.71)). This proves Eq. (III.74) for i = 1. It remains to prove the convergence of I N 2 . Using Markov Property we get

E (k,i) e i(pZ N u (t-N -α δ m N t)+rK B N (N α δ t)) 1 I B N (N α δ t)= j = E (k,i) e i(pZ N u (t-N -α δ m N t)) E (K B N (N α δ t-m N t),I B N (N α δ t-m N t)) e i(rK B N (m N t)) 1 I B N (m N t)= j .
Hence, by Markov Property we get

I N 2 (t, k, i, j, p, r) ≤ E (k,i) E (K B N (N α δ t-m N t),I B N (N α δ t-m N t)) g r K B N , I B N ,
where

g r K B N , I B N = e irK B N (m N t) 1 I B N (m N t)= j - 1 2 T e irk ′ dk ′ .
Let r in N and g r the centered function defined for any k in T and i in {1, 2} by

g r (k, i) = e irk 1 j (i) - 1 2 T e irk ′ dk ′ .
Using the fact that 1 2 dkdρ(i) is the reversible probability measure of the stochastic process

K B N (•), I B N (•) we get E (k,i) E (K B N (N α δ t-m N t),I B N (N α δ t-m N t)) e irK B N (m N t) 1 I B N (m N t)= j - 1 2 T e irk ′ dk ′ = P m N t N g L 1 (σ ) ,
which goes to zero when N goes to infinity according to Lemma III.5.5. This ends the proof.

III.5.4 SKETCH OF THE PROOF OF THEOREM III.7

We recall that τ 0 ∼ E (1). By applying Fourier's formula to Eq. (III.51) we get for any p in R and positive time t in [0, T ]

F [ ρB ] (t, p) = exp (tΦ B (p)) F f 0 (p) , (III.75)
where we recall that for any p in R

Φ B (p) = 2 ∞ 0 (cos(pr) -1) E τ -1 0 g B,+ 2πr
τ 0 + g B,- 2πr
τ 0 dr , since g B,± are even functions by item i) of Proposition III.4.4. Observe that Φ B is negative, to conclude the proof we need the following lemma, the proof of which is left to the readers.

Lemma III.5.6

There exists functions h ± and f ± such that for almost every r

∀B > 1, B 1 3 g B,± (r) < f ± (r) , (III.76) ∀B < 1, g B,± (r) < h ± (r) , (III.77)
where

1 0 f ± (r)r 2 dr < ∞ and ∞ 1 f ± (r)dr < ∞ , (III.78) 1 0 h ± (r)r 2 dr < ∞ and ∞ 1 h ± (r)dr < ∞ . (III.79)
By using item iii), item iv) of Proposition III.4.4, Lemma III.5.6 and the dominated convergence theorem we get that for almost every

p in R lim B→0 Φ B (p) = -Γ 1 + 3 2 γ -1 2 D 0 |p| 3 2
and lim

B→∞ B 1 3 Φ B (p) = -Γ 1 + 5 3 γ -2 3 D ∞ |p| 5 3 ,
where D 0 and D ∞ are defined in Eq. (III.52). Since the proofs of item i) and ii) are similar we only prove item i). By applying Fourier's formula to Eq. (III.53) we obtain

F [ρ 0 ] (t, p) = exp -tΓ 1 + 3 2 γ -1 2 D 0 |p| 3 2 F f 0 (p) . (III.80) We recall that f 0 is in C ∞ c (R) hence is in L 2 (R)
. By the dominated convergence theorem we deduce that for any positive time t in [0, T ]

lim B→0 ∥ ρB (t, •) -ρ 0 (t, •)∥ 2 L 2 (R) = lim B→0 ∥F [ ρB ] (t, •) -F [ρ 0 ] (t, •)∥ 2 L 2 (R) = lim B→0 R F f 0 (p) exp (tΦ B (p)) -exp -tΓ 1 + 3 2 γ -1 2 D 0 |p| 3 2 2 d p = R lim B→0 F f 0 (p) exp (tΦ B (p)) -exp -tΓ 1 + 3 2 γ -1 2 D 0 |p| 3 2 2 d p = 0 .
This proves that for each positive time t, ρ B (t, •) converges in L 2 (R) to ρ 0 (t, •). We conclude the proof by the dominated convergence theorem.

III.6 APPENDIX A. PROPERTIES OF THE WIGNER'S DISTRIBUTION

In this section, we start to prove Lemma III.2.2 in Sec. III.6.1. In Sec. III.6.2 and Sec. III.6.3 we prove the evolution equation of the Wigner's distribution (cf. Eq. (III.56)). In Sec. III.6.4 and Sec. III.6.5 we prove Lemma III.4.1 and Lemma III.3.1 respectively.

III.6.1 PROOF OF LEMMA III.2.2

The aim of this section, is to prove Lemma III.2.2. We start to prove that the integral in Eq. (III.11) is well defined. Let J := (J 1 , J 2 ) in S × S and i in {1, 2}, by using Cauchy-Schwarz's inequality, Fubini's theorem and the periodicity of

ψ ε i we have ε 2 R d p |F [J i ] (p, k)| T dk E µ ε ψ ε i tε -1 , k + ε p 2 ψ ε i tε -1 , k - ε p 2 * ≤ ε 2 R d p sup k∈T |F [J i ] (p, k)| E µ ε T dk ψ ε i tε -1 , k 2 ≤ K 0 ∥J∥ ,
where we used Eq. (III.10) to obtain the last inequality. This proves the existence of the integral in Eq. (III.11). The others expressions of the Wigner's distribution presented in Sec. III.2.4 are consequences of Fubini's theorem and Fourier's inverse formula. Observe that we proved

∀t ∈ [0, T ], |⟨W ε i (t), J⟩| ≤ K 0 ∥J∥ . (III.81)
Now we prove Lemma III.2.2. By Eq. (III.81) and the linearity of the Fourier's transform the proof of item i) is immediate. To prove item ii) we use the evolution equation (Eq. (III.56)) satisfied by the Wigner's distribution. Let J in S × S , in Sec. III.6.2 and Sec. III.6.3 we prove that

⟨W ε (t), J⟩ -⟨W ε (0), J⟩ = t 0 ds ⟨W ε (s), v 0 ∂ u J⟩ + γ t 0 ds ⟨W ε (t),C 0 J⟩ + γ t 0 ds⟨Γ ε n (s),C 0 J⟩ + t 0 ds⟨ Γε n (s),C 0 J⟩ + t 0 dsO s,J (ε) .
with |O s,J | ≤ K J a constant independent of ε and time s. Hence, for any t in [0, T ], ⟨W ε (t), J⟩ is the integral of a bounded function, and so it is a continuous function. Using Eq. (III.81) we conclude the proof of Lemma III.2.2.

III.6.2 BEHAVIOR OF THE TRANSPOR T TERM

We recall that B ε = Bε δ with δ > 0. To prove Eq. (III.56) we use Dynkin's formula, Lemma III.2.2 and Lemma III.5.1. Let J := (J 1 , J 2 ) in S × S , t in [0, T ] and i in {1, 2} we want to prove that

∂ t ⟨W ε i (t), J i ⟩ = 1 2π ⟨W ε i (t), v 0 ∂ u J i ⟩ + ⟨W ε i (t), [C 0 J] i ⟩ + ⟨Γ ε i (t), [C 0 J] i ⟩ + ⟨ Γε i (t), [C 0 J] i ⟩ + O t,J (ε) .
For any i in {1, 2} we define φ i by

φ i (q ε (t), p ε (t)) = R d p T dk ψ ε i tε -1 , k - ε p 2 * ψ ε i tε -1 , k + ε p 2 F [J i ] (p, k) * . (III.82)
In order to lighten notation we denote

ψ ε i tε -1 , k ± ε p 2 by ψ ε i k ± ε p 2 .
By Eq. (III.82) we can write that

⟨W ε i (t), J i ⟩ = ε 2 E µ ε [φ i (q ε (t) , p ε (t))] . (III.83)
By Dynkin's formula we have

∂ t ⟨W ε i (t), J i ⟩ = ε -1 E µ ε {(A + B ε G) [φ i (q ε (t) , p ε (t))]} + γE µ ε {S [φ i (q ε (t) , p ε (t))]} .
Let i in {1, 2}, by Lemma III.2.1 and the fact that A + Bε δ G is a first order differential operator we have

ε -1 E µ ε {(A + B ε G) [φ i (q ε (t) , p ε (t))]} = ε 2 R d p T dkE µ ε ψ ε i k - ε p 2 * ψ ε i k + ε p 2 F [J i ] (p, k) * × i ε ω i,B ε k + ε p 2 -ω i,B ε k - ε p 2 . (III.84)
Following the proof of [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF] we can prove

ε -1 E µ ε {(A + B ε G) [φ i (q ε (t) , p ε (t))]} = 1 2π ⟨W ε i (t), v B ε ∂ u J⟩ + O J (ε) , (III.85)
where |O J (ε)| ≤ κ J (B)×ε and κ J (B) is a constant which depends only on J and B. To prove Eq. (III.85), we need the following lemma which is proved at the end of this section. Lemma III.6.1

We recall that the functions ω 1/2,B ε and v B ε are defined in Eq. (III.4) and Eq. (III.6) respectively. Then we have the following properties. i) v B ε is a bounded function on T.

ii) There exists a positive constant C such that

∀p ∈ R, ∀k ∈ T, ε -1 ω i,B ε k + ε p 2 -ω i,B ε k - ε p 2 ≤ C|p| .
iii) For all p in R and k in T such that |k| > ε|p| we have

ε -1 ω i,B ε k + ε p 2 -ω i,B ε k - ε p 2 -pv B ε (k) ≲ ε |p| 2 |k| .
Let κ > 0 and η in ]0, 1[ with δ > η. We denote by κ ε , the real κε η . We cut the right-hand side of Eq. (III.84) into two terms denoted by I ε < (κ ε ) and I ε > (κ ε ) where

I ε < (κ ε ) = ε 2 R d p |k|<κ ε dkE µ ε ψ ε i k - ε p 2 * ψ ε i k + ε p 2 F [J i ] (p, k) * × i ε ω i,B ε k + ε p 2 -ω i,B ε k - ε p 2 -ε pv B ε (k) .
By using point i) and point ii) of Lemma III.6.1 and the fact that J i belongs to S we have

|I ε < (κ ε )| ≲ |k|<κ ε dkE | ψ ε i (k)| 2 .
By Lemma III.4.1 we obtain then lim sup ε→0

I ε < (κ ε ) = 0 .
It remains to deal with I ε > (κ ε ). We have

I ε > (κ ε ) = ε 2 ε|p|>κ ε d p |k|>κ ε dkE µ ε ψ ε i k - ε p 2 * ψ ε i k + ε p 2 F [J i ] (p, k) * × i ε ω i,B ε k + ε p 2 -ω i,B ε k - ε p 2 -ε pv B ε (k) + ε 2 ε|p|<κ ε d p |k|>κ ε dkE µ ε ψ ε i k - ε p 2 * ψ ε i k + ε p 2 F [J i ] (p, k) * × i ε ω i,B ε k + ε p 2 -ω i,B ε k - ε p 2 -ε pv B ε (k) = I ε >,1 (κ ε ) + I ε >,2 (κ ε ) .
By the points i) and ii) of Lemma III.6.1 and Eq. (III.10) we have

I ε >,1 (κ ε ) ≲ K 0 |p|> κε ε d p(p + 1) sup k∈T |F [J i ] (p, k)| = K 0 |p|>κε η-1 d p(p + 1) sup k∈T |F [J i ] (p, k)| ,
which goes to zero when ε goes to zero since η < 1. For the remaining term we have that |k| > κ ε > ε|p| hence by point iii) of Lemma III.6.1 we get

I ε >,2 (κ ε ) ≲ ε|p|<κ ε d p |k|>κ ε dkE µ ε ψ ε i k - ε p 2 * ψ ε i k + ε p 2 ε|p| 2 |k| F [J i ] (p, k) * ≲ K 0 R d p ε|p| 2 κ ε sup k∈T |F [J i ] (p, k)| ,
which goes to zero since η < 1. In order to end the computations for the transport term it remains to prove that

⟨W ε i (t), v B ε ∂ u J i ⟩ = ⟨W ε i (t), v 0 ∂ u J i ⟩ + O J (ε) . (III.86)
We define R ε in the following way

R ε = ⟨W ε i , [v B ε -v 0 ] ∂ u J i ⟩ .
Hence, we want to prove that (R ε ) ε>0 converges to zero when ε goes to zero. As before, we cute

R ε into two terms R ε < (κ ε ) and R ε > (κ ε ) where R ε < (κ ε ) = W ε i , [v B ε -v 0 ] ∂ u J i 1 {k≤κ ε } .
We start to show that (R ε > (κ ε )) ε>0 converges to zero. Using the fact that v 0 is bounded and the fact that k > κ ε we can bound

|R ε > (κ ε ) | by ε 2 R d p |k|>κ ε dkE µ ε ψ ε i k - ε p 2 * ψ ε i k + ε p 2 sup k∈T |F [J i ] (p, k)| |p| κ ε α(k) -α(k) + B 2 ε 2δ 4 .
Since k > κ ε , α(k) > 0 and by using a Taylor expansion we get

|R ε > (ρ)| ≲ ε 2 R d p |k|>κ ε dkE µ ε ψ ε i k - ε p 2 * ψ ε i k + ε p 2 sup k∈T |F [J i ] (p, k)| × |p|B 2 ε 2δ κ ε α(k) ≲ ε 2 R d p |k|>κ ε dkE µ ε ψ ε i k - ε p 2 * ψ ε i k + ε p 2 sup k∈T |F [J i ] (p, k)| × |p|B 2 ε 2δ |κ ε | 2 ≲ K 0 ε 2(δ -η) ,
which goes to zero when ε goes to zero since δ > η by assumption. It remains to show that (R ε < (κ ε )) ε>0 converges to zero. By using the fact that v 0 and v B ε are bounded and the fact that J i belongs to S we obtain for any time t in [0, T ]

|R ε < (κ ε )| ≲ ε 2 |k|<κ ε dkE µ ε | ψ ε i (k)| 2 = ε 2 |k|<κ ε dkE µ ε ψ ε i tε -1 , k 2 ,
which goes to zero by Lemma III.4.1. Hence, we proved that for any i in {1, 2}

E µ ε {(A + B ε G) [φ i (q ε (t) , p ε (t))]} = ⟨W ε i (t), v 0 ∂ u J i ⟩ + O J (ε) .
So far we obtained the first term of Eq. (III.56) which corresponds to the transport term of the Boltzmann's equation (cf. Eq. (III.17)). In the next section, we obtain the last terms of Eq. (III.56). Before doing this, we give the sketch of the proof of Lemma III.6.1.

SKETCH OF THE PROOF OF LEMMA III.6.1

We recall that α(k) = 4π 2 sin 2 (πk). Hence, by definition we have for every

k in T v B ε (k) = α′ (k) 2 α(k) + B 2 ε 4 = 4π 3 cos (πk) sin (πk) sin 2 (πk) + B 2 ε 4 ≲ 4π 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| > ε|p| hence it is sufficient to discuss the case |k| > 0. Let f i and g i be two real functions defined on ]0, ε[ by

f i (s) = ω i,B ε k + sp 2 and g i (s) = ω i,B ε k - sp 2 .
Then we have

ε -1 ω i,B ε k + ε p 2 -ω i,B ε k - ε p 2 -pv B ε (k) = ε -1 f i (ε) -f i (0) -ε f ′ i (0) -ε -1 g i (ε) -g i (0) -εg ′ i (0) .
We conclude the proof by a Taylor expansion and using the fact that α(k) > C|k| with C a positive constant.

III.6.3 BEHAVIOR OF THE COLLISIONAL TERM

Let i in {1, 2} and i * = 3i, following the proof did by the authors of [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] we can prove that

γE µ ε S φ i q ε tε -1 , p ε tε -1 = γ (⟨W ε i (t), [C B ε J] i ⟩ + ⟨Γ ε i (t), [K B ε J] i ⟩ + ⟨(Γ ε i ) * (t), [K B ε J] i ⟩) + O J (ε) ,
where for any

(u, k) in R × T [C B ε J] i (u, k) = 2 ∑ j=1 T dk ′ θ 2 i,B ε (k)R k, k ′ θ 2 j,B ε k ′ J j (u, k ′ ) -J i (u, k) , [K B ε J] i (u, k) = T dk ′ θ 1,B ε (k) θ 2,B ε (k)R k, k ′ θ 2 i * ,B ε k ′ J i * (u, k ′ ) - J i (u, k) 2 .
When we add the term that we had for the contribution due to the operator A + B ε G and the term due to S we get

∂ t ⟨W ε i (t), J⟩ = 1 2π ⟨W ε i (t), v 0 ∂ u J i ⟩ + γ (⟨W ε i (t), [C B ε J] i ⟩ + ⟨Γ ε i (t), [K B ε J] i ⟩⟨(Γ ε i ) * (t), [K B ε J] i ⟩) + O J (ε) .
To conclude the proof of Eq. (III.56) we have to replace K B ε J and C B ε J in the previous equation by C 0 J where for any

(u, k) in R × T [C 0 J] i (u, k) = 2 ∑ j=1 T dk ′ 1 4 R k, k ′ J j u, k ′ -J i (u, k) .
Since the arguments are similar we only prove that

⟨W ε i (t), [C B ε J] i ⟩ = ⟨W ε i (t), [C 0 J] i ⟩ + O J (ε) . (III.87)
We define an operator CB ε on S × S where for any i in {1, 2} and

(u, k) in R × T [ CB ε J] i (u, k) = 2 ∑ j=1 T dk ′ θ 2 i,B ε (k) 2 R k, k ′ J j u, k ′ -J i (u, k) .
Let i in {1, 2} we first prove that

W ε i (t), [C B ε J] i -[ CB ε J] i = O J (ε) . (III.88) Let (p, k) in R × T and j in {1, 2}, we recall that R (k, k ′ ) = 16 sin 2 (πk) sin 2 (πk ′ ) then T dk ′ R k, k ′ θ 2 i,B ε (k) θ 2 j,B ε k ′ - 1 2 F [J j ] p, k ′ * -F [J i ] (p, k) * ≲ |Bε δ | × κ J (p),
where

κ J (p) = sup k∈T |F [J i ] (k, p)| + sup k∈T F [J j ] (k, p) .
Hence, using the fact that J i belongs to S and the usual argument we conclude that

W ε i (t), [C B ε J] i -[ CB ε J] i ≲ K 0 ∥J∥Bε δ .
This proves Eq. (III.88). Similarly , one can prove that

W ε i (t), [ CB ε J] i -[C B ε J] i = O J (ε) .
By the triangle inequality we obtain Eq. (III.87) and conclude the proof.

III.6.4 PROOF OF LEMMA III.4.1

In this section, we prove Lemma III.4.1 which allows us to extend the assumption (III.35), on the initial distribution µ ε to times tε -1 . The proof of this Lemma follows the one of [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF]Lemma 7]. Let J = (J 1 , J 2 ) be a bounded measurable function on T. Then for any i in {1, 2} and t in [0, T ]

⟨W ε i (t), J⟩ = ε 2 T E µ ε ψ ε i tε -1 , k 2 J i (k) * dk .
(III.89)

⟨Γ ε i (t), J⟩ = ε 2 T E µ ε ψ ε i tε -1 , k ψ ε i tε -1 , -k J i (k) * dk . (III.90)
Since J is a bounded measurable function the previous objects are well defined. Indeed, we have

|⟨W ε i (t), J⟩| ≤ K 0 × sup k∈T |J(k)| and |⟨Γ ε i (t), J⟩| ≤ K 0 sup k∈T |J(k)| . (III.91)
Using Lemma III.2.1 we get

(A + B ε G) | ψ ε i | 2 (k) = 0 . (III.92)
Following the proof presented in Sec. III.6.3 and using Eq. (III.92) we obtain that

∂ t ⟨W ε i (t), J i ⟩ = γ ⟨W ε i (t), [C 0 J] i ⟩ + ⟨Γ ε i (t), [C 0 J] i ⟩ + Γε i (t), [C 0 J] i + O t,J (ε) ,
where we recall that for any i in {1, 2},

[C 0 J] i is defined in Eq. (III.15), κ is a positive constant and η is in ]0, 1[. Let κ ε := κε η and J κ ε := (1 [-κ ε ,κ ε ] , 1 [-κ ε ,κ ε ] ). Then we have |[C 0 J κ ε ] i (k)| ≲ κ ε + J κ ε i (k) and ⟨Γ ε i (t) + (Γ ε )(t) * , [C 0 J κ ε ] i ⟩ ≲ |⟨W ε i (t), [C 0 J κ ε ] i ⟩| .
Hence, using Eq. (III.56) we obtain that for any i in {1, 2} and t in [0, T ]

W ε i (t), J κ ε i ≲ W ε i (0), J κ ε i + (C 1 ρ +C 2 ε)t + t 0 W ε i (s), J κ ε i ds ,
where C 1 and C 2 are two positive constants. By Gronwall's lemma we conclude that

W ε i (t), J κ ε i ≲ exp(C 3 ) C 1 κ ε +C 2 ε + W ε i (0), J κ ε i . (III.93)
Using assumption (III.35) we get that lim sup ε→0

W ε i (t), J κ ε i = lim sup ε→0 ε 2 |k|<κ ε E µ ε ψ ε i tε -1 , k 2 = 0 .
This ends the proof of Lemma III.4.1.

III.6.5 PROOF OF THE UNIQUENESS IN THE BOLTZMANN'S EQUATION

In this section, we want to prove Lemma III.3.1. Let µ be a measure valued solution of Eq. (III.17).

Then, for any

J := (J 1 , J 2 ) ∈ S × S we have that 2 ∑ i=1 R×T E [J(u, k)dµ i (u, k)] = 2 ∑ i=1 R×T E J ĨB (t) ZB u (t), KB (t) dµ i (0, u, k) , (III.94)
where ZB

u (•), ĨB (•), KB (•) is the dual process to Z B u (•), I B (•), K B (•) starting from (u, k, i).
Hence, the infinitesimal generator of the dual process is defined by L B + v B ∂ u where L B is defined in Eq. (III.20). Since the PDMP ZB u (•), ĨB (•), KB (•) is unique, we conclude the proof.

III.7 APPENDIX B. STUDY OF THE TAILS OF Ψ B N

In this section, we prove Proposition III.4.3, Proposition III.4.4 and Proposition III.4.5.

III.7.1 PROOF OF PROPOSITION III.4.3

We recall that B > 0. To simplify the reading of this section, we repeat the definition of x B,± which is given by

2 x 2 B,± (r) + B 2 4 ± B x B,± (r) = π γr , r ̸ = 0 . (III.95)
Observe that sign(x B,± (r)) = sign(r), moreover Eq. (III.95) has exactly two solutions with opposite sign. This proves that x B,± (-r) = -x B,± (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. (III.95) we get that for any r ̸ = 0

|x B,+ (r)| < π 2γ |r| -1 2 . (III.96)
Hence, for each r ̸ = 0, (x B,+ (r)) B>0 is a bounded sequence and therefore admits an accumulation point. Thus, for any r ̸ = 0, there exists a subsequence (B(n

)) n∈N such that lim n→∞ x B(n),+ (r) = sign(r) π 2γ |r| -1 2 .
As each convergent subsequence satisfies this limit we deduce that the whole sequence converges which means that for any r ̸ = 0 we have lim

B→0 x B,+ (r) = sign(r) π 2γ |r| -1 2 = x 0 (r) .
To prove the convergence of the sequence (x ′ B,+ ) B>0 we derive Eq. (III.95) with respect to r and get

x ′ B,± (r)   2 x 2 B,± (r) + B 2 4 + B + 2x 2 B,± (r) x 2 B,± (r) + B 2 4   = - π γr 2 . (III.97)
By sending B to zero, we end the proof of item ii). To prove item iii) we write Eq. (III.95) in the following way

  4x 2 B,± (r) B 2 + 1 ± 1   x B,± (r) = π Bγr . (III.98)
Using Eq. (III.98) we observe that the sequence

x B,± (r) B B>0
converges to zero when B goes to infinity. Hence, by performing a Taylor expansion in Eq. (III.98) we obtain the first part of item iii). To conclude the proof we derive Eq. (III.98) with respect to r and send B to infinity.

III.7.2 PROOF OF PROPOSITION III.4.4

Let r ̸ = 0 we have

g B,± (r) = - x ′ B,± (r) 4π   1 2 ± B 4 x 2 B,± (r) + B 2 4   x 2 B,± (r) . (III.99)
By Proposition III.4.3, x B,± 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

∞ 1 h ′ B,± (r)dr < ∞ and 1 0 r 2 h ′ B,± (r)dr < ∞ . (III.100) Let A > 1, then A 1 h ′ B,± (r)dr ≲ 3 -1 (x 3 B,± (1) -x 3 B,± (A)) .
By Eq. (III.95) we deduce that (x B,± (A)) A>1 goes to zero when A goes to infinity. The monotone convergence theorem ends the proof. Let ε < 1, we have

1 ε r 2 h ′ B,± (r)dr ≲ 3 -1 (x 3 B,± (1) -ε 2 x 3 B,± (ε)) + 2 3 1 ε rx 3 B,± (r)dr .
By sending r to zero in Eq. (III.95) we obtain that for any B and r in V (0)

x B,± (r) ∼ |r| -1 2 C(B) ,
where C(B) is a constant which depends on B. Hence, we deduce that

lim ε→0 x 3 B,± (ε)ε 2 = 0 and lim ε→0 1 ε rx 3 B,± (r)dr < ∞ .
This ends the proof of item ii). Using item ii) (resp. item iii)), of Proposition III.4.3 and sending B to zero (resp. to infinity) in Eq. (III.99) we get item iii) (resp. item iv)) of Proposition III.4.4. This ends the proof.

III.7.3 PROOF OF PROPOSITION III.4.5

We recall that Ψ B N is defined in Eq. (III.38). Since for any i, Ψ B N (•, i) is an odd function and that the density of π B N with respect to the Lebesgue measure on T is even we have for any r ̸ = 0

π B N ({(k, i), Ψ B N (k, i) > Nr}) = π B N ({(k, i), Ψ B N (k, i) < -Nr}) .
Hence, we only prove the result for r > 0. We perform the change of variable x = sin(πk)N δ for k in T and we get for r > 0

π B N ({(k, i), Ψ B N (k, i) > Nr}) = N -3δ 4π N δ 0 1 A N,B,+ (r) (x)   1 2 + B 4 x 2 + B 2 4   x 2 1 -x 2 N -2δ dx + N -3δ 4π N δ 0 1 A N,B,-(r) (x)   1 2 - B 4 x 2 + B 2 4   x 2 1 -x 2 N -2δ dx = I N B,+ (r) + I N B,-(r) ,
where

A N,B,± (r) =        x 2 x 2 + B 2 4 ± B x 1 -x 2 N -2δ < πN 2δ -1 γr       
.

Let r ̸ = 0, we define x N,B,± (r) the solutions on [-N δ , N δ ] of the following equations

2 x 2 N,B,± (r) + B 2 4 + B x N,B,± (r) 1 -x 2 N,B,± (r)N -2δ = πN 2δ -1 γr . (III.101)
Observe that sign(x N,B,± (r)) = sign(r). To complete the proof of Proposition III.4.5 we need the following lemma which is proved at the end of this section. Lemma III.7.1

Let r ̸ = 0, we have the following results

i) If δ > 1 2 then lim N→∞ N 1 2 -δ x N,B,± (r) = √ π(2γ) -1 2 |r| -1 2 sign(r) . (III.102) ii) If δ = 1 2 then lim N→∞ x N,B,± (r) = x B,± (r) , (III.103)
where x B,± is defined in Eq. (III.95

). iii) If δ < 1 2 then lim N→∞ N 1-2δ x N,B,+ (r) = π(rγ) -1 B . (III.104) lim N→∞ N 1-2δ 3 x N,B,-(r) = (π) 1 3 (2γ) -1 3 |r| -1 3 B 1 3 sign(r) . (III .105)
From Lemma III.7.1 we deduce that for any r ̸ = 0

I N B,± (r) = N -3δ 4π x N,B,± (r) 0   1 2 ± B 4 x 2 + B 2 4   x 2 1 -x 2 N -2δ
dx1 r>0 (r) .

For δ > 1 2 we perform the change of variable y = xN .

From this we deduce that for r > 0

lim N→∞ N 3 2 π B N ({(k, i)| Ψ B N (k, i) > Nr}) = κ 1 r -3 2 γ -3 2 , (III.106) with κ 1 = √ π 2 7 2 3
.

(III.107)

• Let δ = 1 2 , by using the dominated convergence theorem we get lim

N→∞ N 3δ I N B,± (r) = 1 4π x B,1 (r) 0   1 2 ± B 4 x 2 + B 2 4   x 2 dx = h B,± (r) , u lim N→∞ E ⌊N α t⌋ ∑ n=1 E g Z N n N G N n-1 -t R g(r)dν(r) = 0 . (III.113) lim N→∞ NE E g Z N 1 N G N 0 2 = 0 . (III.114)
Then the finite-dimensional distributions of

N -1 M N (•) N∈N converge weakly to Z(•) in D ([0, T ], R)
where Z(•) is a Lévy process with Lévy measure ν.

Proof : We refer the reader to [65, Appendix.A] for the proof of Proposition III.8.1.

We recall that π B N is the stationary measure of the chain

K B N n , I B N n
. Let α δ defined in Eq. (III.44). We recall that for any u in R and positive time t in [0, T ]

N -1 Z N u N (N α δ t) = u + N -1 ⌊N α δ t⌋ ∑ n=0 Ψ B N X B N n τ N n .
We introduce the array

Z N n (n,N)∈N 2 := Ψ B N X B N n τ N n (n,N)∈N
∑ n=1 E π B N g Z N n N G N n-1 -t R g(r)dν δ (r) = 0 , (III.115)
where ν δ is the Lévy measure defined in Eq. (III.44). We have

E π B N ⌊N α δ t⌋ ∑ n=1 E π B N g Z N n N G N n-1 = N α δ +∞ 0 e -u du T×{1,2} g Ψ B N (k, i)u 2πN dπ B N (k, i) = N α δ +∞ 0 e -u du ∞ 0 (2πN) -1 g ′ r N π B N Ψ B N > 2πr u dr = N α δ +∞ 0 e -u du ∞ 0 g ′ (r)π B N Ψ B N ≥ 2πNr u dr .
Using the tails condition (item i) of Proposition III.4.5) we get that lim

N→∞ N α δ +∞ 0 e -u du ∞ 0 g ′ (r)π B N Ψ B N ≥ Nr u dr = R g(r)dν δ (r
P π B N 1 ⌊N α u⌋ ⌊N α u⌋ ∑ n=0 λ B N X B N n τ N n - u 2 > σ = 0 . (III.116)
PROOF OF LEMMA III.8.2

In order to prove the convergence in probability we prove the convergence in L 1 . Let N be fixed, we define the sequence of function λ ε B N ε>0 where for any k in T and i in {1, 2} we have

λ ε B N (k, i) = λ B N (k, i)1 - 1 2 +ε ∪ ε, 1 2 -ε and λ ε B N = T×{1,2} λ ε B N (k, i)dπ B N (k, i) .
Observe that for any ε, λ ε B N is in L 2 (π B N ). Using the stationnarity of the Markov chain X B N n n∈N and the definition of π B N given by Eq. (III.26) we have

1 ⌊N α u⌋ ⌊N α u⌋ ∑ n=0 λ B N X B N n τ N n -λ ε B N X B N n τ N n L 1 ≤ C(ε) and 1 ⌊N α u⌋ ⌊N α u⌋ ∑ n=0 λ ε B N - u 2 L 1 ≤ C(ε) ,
where |C(ε)| ≲ ε. By the triangle inequality, to end the proof it is sufficient to prove that for any ε we have

lim N→∞ 1 ⌊N α u⌋ ⌊N α u⌋ ∑ n=0 λ ε B N X B N n τ N n -λ ε B N L 1 = 0 .
Using Cauchy-Schwarz's inequality and that for any N the random variables X B N n , τ N n are i.i.d and centered we get

1 ⌊N α u⌋ ⌊N α u⌋ ∑ n=0 λ ε B N X B N n τ N n -λ ε B N L 1 ≤ K(ε) √ N α δ ,
where K(ε) ≲ ε. Finally we get that lim sup

N→∞ 1 ⌊N α u⌋ ⌊N α u⌋ ∑ n=0 λ B N X B N n τ N n -λ ε B N L 1 ≲ lim sup N→∞ 1 ⌊N α u⌋ ⌊N α u⌋ ∑ n=0 λ B N X B N n τ N n -λ ε B N X B N n τ N n L 1 + lim sup N→∞ 1 ⌊N α u⌋ ⌊N α u⌋ ∑ n=0 λ N ε -λ N L 1 + lim sup N→∞ 1 ⌊N α u⌋ ⌊N α u⌋ ∑ n=0 λ ε B N X B N n τ N n -λ ε B N L 2 ≤ C(ε) .
By sending ε to zero we end the proof of Lemma III.8.2. From this result we can prove Lemma III.5.3.

u PROOF OF LEMMA III.5.3

Let t 0 > 0, T ≥ t 0 and ε > 0.

P π B N sup t∈[t 0 ,T ] |S N (t) -S(t)| > ε ≤ P π B N ∃t ∈ [t 0 , T ], S N (t) ≤ S(t) - ε 2
+ P π B N ∃t ∈ [t 0 , T ], S N (t) ≥ S(t) + ε 2 .
Let m in N, since S is a continuous function there exists a subdivision

(t i ) i∈{0,••• ,m} such that t 0 < t 1 < • • • < t m = T with ∀i ∈ {0, • • • , m}, S(t i+1 ) -S(t i ) ≤ ε 10 .
Using the fact that S and S N are increasing functions we get

P π B N ∃t ∈ [t 0 , T ], S N (t) ≤ S(t) - ε 2 ≤ m-1 ∑ i=0 P π B N ∃t ∈ [t i ,t i+1 ], S N (t) ≤ S(t) - ε 2 ≤ m-1 ∑ i=0 P π B N S N (t i ) ≤ S(t i+1 ) - ε 2 ≤ m-1 ∑ i=0 P π B N S N (t i ) ≤ S(t i ) - 2ε 5 .
Using the same techniques we get

P π B N ∃t ∈ [t 0 , T ], S N (t) ≥ S(t) + ε 2 ≤ m-1 ∑ i=0 P π B N S N (t i+1 ) -S(t i+1 ) ≥ 2ε 5 .
Hence, we proved that

P π B N sup t∈[t 0 ,T ] |S N (t) -S(t)| > ε ≤ 2m sup i=0,••• ,m P π B N |S N (t i ) -S(t i )| ≥ 2ε 5 .
To conclude the proof it is sufficient to prove that ∀t ∈ [t 0 , T ], ∀δ > 0, lim

N→∞ P π B N [|S N (t) -S(t)| > δ ] = 0 .
This result follows from Lemma III.8.2, indeed let δ > 0 then

P π B N [|S N (t) -S(t)| > δ ] ≤ P π B N [S N (t) < S(t) + δ ] + P π B N [S N (t) > S(t) + δ ] ≤ P π B N 1 ⌊N α u⌋ ⌊N α [S(t)+δ ]⌋ ∑ n=0 λ B N X B N n τ N n > t + P π B N 1 ⌊N α u⌋ ⌊N α [S(t)-δ ]⌋ ∑ n=0 λ B N X B N n τ N n ≤ t .
Using the dominated convergence theorem and Lemma III.8.2, we conclude the proof.

In this chapter, we consider a harmonic chain of N oscillators in presence of a disordered magnetic field. The ends of the chain are connected to heat baths at the left-hand side and right-hand side with temperature T L and T R respectively with T L > T R . Following the study made by Rieder, Lieb and Lebowitz in [START_REF] Rieder | Properties of a harmonic crystal in a stationary nonequilibrium state[END_REF] we obtained analytic expressions for the current for free and fixed boundary conditions when N, the number of particles, goes to infinity in the case where the magnetic field is uniform. Then we study the system in the presence of a random magnetic field following the study made by Casher and Lieb in [START_REF] Casher | Heat flow in regular and disordered harmonic chains[END_REF] and perform analytic predictions of different system-size dependence of the current, depending on the expectation value of the magnetic field and the boundary conditions. These results are extracted from [START_REF] Majeed Bhat | Heat transport in an ordered harmonic chain in presence of a uniform magnetic field[END_REF][START_REF] Cane | Localization effects due to a random magnetic field on heat transport in a harmonic chain[END_REF].

In Sec. IV.1, we present the model studied in this chapter. In Sec. IV.2, we derive the expression of the heat current of the system using NEGF. In Sec. IV.3, we study the heat current when the magnetic field is uniform and obtain its asymptotic limits when N goes to infinity for fixed and free boundaries conditions. When the system is submitted to a random magnetic field, the steady of the heat current is harder, we need to compute a Lyapunov exponent. This is the aim of Sec. IV.4 where we formally derive it and present some numerical simulations. We end Sec. IV.4 by giving the different power laws of the current that we get. Finally in Sec. IV.5, we recall the proof of the derivation of the Lyapunov exponent associated to a harmonic oscillator with parametric noise that we used in Sec. IV.4.

IV.1 PRESENTATION OF THE MODEL

We consider a chain of N harmonic oscillators each having two transverse degree's of freedom so that every oscillator is free to move in a plane perpendicular to the length of the chain. We denote the positions and momenta of the x th oscillator by (q 1 (x), q 2 (x)) and (p 1 (x), p 2 (x)) respectively, with x in {1, 2, . . . , N}. The oscillators are assumed to have masses, m, and each carry a positive charge e. We consider a site-dependent magnetic field ⃗ B x = B x ⃗e 3 , perpendicular to the plane of motion 1 , which can be obtained from a vector potential ⃗ A x = (-B x q 2 (x), B x q 1 (x), 0) at each lattice site. The Hamiltonian of the chain is given by

H = N ∑ x=1 (p 1 (x) + eB x q 2 (x)) 2 + (p 2 (x) -eB x q 1 (x)) 2 2m + k N ∑ x=0 |q(x + 1) -q(x)| 2 2 ,
where k denotes the inter particle spring constant. We will consider the two different boundary conditions i) fixed boundaries with q(t, 0) = q (t, N + 1) = 0. ii) free boundaries with q(t, 0) = q(t, 1) and q(t, N) = q(t, N + 1). In order to study heat current through this system, we consider the 1 st and the N th oscillators to be connected to heat reservoirs at temperatures T L and T R respectively with T L > T R . The heat reservoirs are modelled using dissipative and noise terms leading to the following Langevin equations of motion

m d 2 dt 2 q i (t, x) = k(q i (t, x + 1) + q i (t, x -1) -c x q i (t, x)) + eB x δ i,1 d dt q 2 (t, x) -δ i,2 d dt q 1 (t, x) + η L i (t)δ x,1 + η R i (t)δ x,N -(γδ x,1 + γδ x,N ) d dt q i (t, x) , (IV.1)
for i in {1, 2} and x in {1, 2, . . . , N}. Here η L (t) := (η L 1 (t), η L 2 (t)) and η R (t) := (η R 1 (t), η R 2 (t)) are Gaussian white noise terms acting on the 1 st and N th oscillators respectively. These follow the regular white noise correlations, η L/R (t)η L/R (t ′ ) = 2γT L/R δ (tt ′ ) (Boltzmann's constant is fixed to one to simplify), where γ is the dissipation strength at the reservoirs. The coefficients c x fix the boundary conditions of the problem. For fixed boundaries c x = 2 for all x, while for free boundary conditions

IV.2 DERIVATION OF THE HEAT CURRENT IV.2.1 HEAT CURRENT USING NEGF

For heat current in the setup considered here, we need to obtain the steady state solution of the equations of motion given by Eq. (IV.1). In order to lighten the notations we decided to change the definition of the continuous Fourier's transform introduced in Chapter III. Hence, in this chapter the Fourier's transform of a function u : R → R will be denoted by ũ where

ũ(ω) = 1 2π R exp (-iωt) u(t)dt . (IV.2)
We rewrite the equations of motion (cf. Eq. (IV.1)) in Fourier space as

ηL 1 (ω)δ x,1 + ηR 1 (ω)δ x,N = (-mω 2 + c x k + iγωδ x,1 + iγωδ x,N ) q1 (ω, x) -iωeB x q2 (ω, x) -(k q1 (ω, x + 1) + q2 (ω, x -1)) , (IV.3) ηL 2 (ω)δ x,1 + ηR 2 (ω)δ x,N = (-mω 2 + c x k + iγωδ x,1 + iγωδ x,N ) q2 (ω, x) + iωeB x q1 (ω, x) -k ( q2 (ω, x + 1) + q2 (ω, x -1)) . (IV.4)
The noise correlations in Fourier space now satisfy

ηL/R (ω) ηL/R ω ′ = γT L/R π -1 δ ω + ω ′ .
For i in {1, 2}, let us define the column vectors

Qi (ω) =        qi (ω, 1) qi (ω, 2) . . . qi (ω, N -1) qi (ω, N)        and ηi (ω) =        ηL i (ω) 0 . . . 0 ηR i (ω)       
.

Then, we can write Eq. (IV.3) and Eq. (IV.4) jointly in the block-matrix form

G -1 (ω) Q1 (ω) Q2 (ω) = η1 (ω) η2 (ω) with G -1 (ω) = Π(ω) K(ω) -K(ω) Π(ω) , (IV.5)
where Π(ω) and K(ω) are square matrices with matrix elements given by [Π(ω)] x,y = (c x kmω2 +)δ x,yk(δ x,y+1 + δ x,y-1 ) + iγωδ x,1 δ y,1 + iγωδ x,N δ y,N , [K(ω)] x,y = -ieB x ωδ x,y .

From Eq. (IV.5) we have

q1 (ω, x) = N ∑ y=1 G + 1 (ω) x,y [ η1 (ω)] y + N ∑ y=1 G + 2 (ω) x,y [ η2 (ω)] y , (IV.6) q2 (ω, x) = - N ∑ y=1 G + 2 (ω) x,y [ η1 (ω)] y + N ∑ y=1 G + 1 (ω) x,y [ η2 (ω)] y , (IV.7)
where

G + 1 = Π + KΠ -1 K -1 and G + 2 = -G + 1 KΠ -1 . (IV.8)
These two last matrices form the 2 × 2 block structure of the matrix G (ω) as

G (ω) = G + 1 (ω) G + 2 (ω) -G + 2 (ω) G + 1 (ω)
.

Defining the square matrices 2 G - 1/2 = [G + 1/2 ] † and Γ x,y (ω) = [Π † (ω) -Π(ω)]
x,y = -2iω(γδ x,1 δ y,1 + γδ x,N δ y,N ), one gets from Eq. (IV.8) that [G

- 1 ] -1 -[G + 1 ] -1 = Π † -Π+KΠ † -1 K -KΠ -1 K.
Multiplying this on the left by G + 1 and on the right by G - 1 we get, after some manipulations, the following relation

G + 1 (ω) -G - 1 (ω) = G + 1 ΓG - 1 + G + 2 ΓG - 2 .
(IV.9)

cases. We start by rewriting the equations of motion in a way such that the 2N × 2N matrix G -1 (ω) appearing in Eq. (IV.5) gets restructured into a 2 × 2-block tri-diagonal matrix, G (ω). To that end, we define for each x = 1, . . . , N the column vectors

Rx (ω) = q1 (ω, x) q2 (ω, x) , ηx (ω) = [ η1 (ω)] x [ η2 (ω)] x ,
and notice that the Eq. (IV.3) and Eq. (IV.4) can then be written as

N ∑ y=1 [G -1 (ω)] x,y Ry (ω) = ηx (ω) , (IV.14)
where [G -1 (ω)] x,y are 2 × 2 matrices defined via G -1 (ω) given by

G -1 (ω) =              A 1 (ω) + iγωI 2 -
(ω) + iγωI 2             
.

Here I x refers to a x × x identity matrix, while A y (ω) is the 2 × 2 matrix defined by

A y := A y (ω) = -mω 2 + c y k -iωeB y iωeB y -mω 2 + c y k y = 1, . . . , N .
Observe that Eq. (IV. [START_REF] Bernardin | Thermal conductivity for a noisy disordered harmonic chain[END_REF] gives

Ry (ω) = N ∑ z=1 [G (ω)] y,z ηz (ω) .
Hence, on comparison with the solution in Eq. (IV.6) and Eq. (IV.7) we can write the components of the 2 × 2 matrix [G (ω)] x,y to be

[G (ω)] x,y = G + 1 (ω) x,y G + 2 (ω) x,y -G + 2 (ω) x,y G + 1 (ω) x,y .
(IV.15)

Thus we now require the 2×2 block [G (ω)] 1,N to calculate the components G + 1 (ω) 1,N and [G + 2 (ω)] 1,N . Since the matrix G -1 (ω) is tri-diagonal, G + 1 (ω) 1,N can be expressed as products of matrices using a transfer matrix approach. This may be achieved by writing down the first column of equations from the identity

G (ω)G -1 (ω) = I 2N , G 1,1 (A 1 + iγωI 2 ) -kG 1,2 = I 2 , (IV.16) G 1,y-1 + G 1,y+1 -k -1 G 1,y A y = 0 , 1 < y < N , (IV.17) G 1,N (A N + iγωI 2 ) -kG 1,N-1 = 0 . (IV.18)
We now write these equations in the form

I 2 G 11 = G 11 G 12 A 1 + iγωI 2 I 2 -kI 2 0 , (IV.19) G 1,y-1 G 1,y = G 1,y G 1,y+1 k -1 A y I 2 -I 2 0 1 < y < N , (IV.20) G 1,N-1 G 1,N = G 1,N 0 k -1 (A N + iγωI 2 ) I 2 0 0 . (IV.21)
We then use Eq. (IV.20) in Eq. (IV. [START_REF] Bernardin | Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential[END_REF]) repeatedly and finally use Eq. (IV.21) to get

I 2 G 11 = G 1N 0 Ω L Π N Ω R , (IV.22)
where Ω L , Π N , Ω R are 4 × 4 matrices defined by Thus we have expressed the required components of the Green's function as a product of 4 × 4 matrices Ω y . This product can further be simplified by making a unitary transformation such that σ p = U † σ z U with σ z = 1 0 0 -1 the third Pauli's matrix, in order to diagonalise the matrix a y I 2 + b y σ p . This makes the product to be

Ω L = I 2 -i γω k I 2 0 0 , Ω R = kI 2 0 iγωI 2 I 2 , Π N = N ∏ y=1 Ω y = N ∏ y=1 k -1 A y I 2 -I 2 0 = N ∏
Π N = N ∏ y=1 a y I 2 + b y σ p I 2 -I 2 0 = U † 0 0 U † N ∏ y=1 a y I 2 + b y σ z I 2 -I 2 0 U 0 0 U .
The product in this equation is now composed of 2 × 2 diagonal blocks and therefore we have that for any 1

≤ x ≤ N, x ∏ y=1 a y I 2 + b y σ z I 2 -I 2 0 =     f + x 0 g + x 0 0 f - x 0 g - x -f + x-1 0 -g + x-1 0 0 -f - x-1 0 -g - x-1     ,
where the numbers f ± x , g ± x , defined for x in = {0, 1, .., N}, follow the same second order recursive equation but with different initial conditions. More exactly we have that for

f ± x+1 = (a x+1 ± b x+1 ) f ± x -f ± x-1 , f ± 0 = 1, f ± 1 = a 1 ± b 1 , g ± x+1 = (a x+1 ± b x+1 )g ± x -g ± x-1 , g ± 0 = 0, g ± 1 = 1 .
(IV.25)

Hence, we obtain that

f ± N f ± N-1 = N ∏ y=2 a y ± b y -1 1 0 a 1 ± b 1 1 and g ± N g ± N-1 = N ∏ y=2 a y ± b y -1 1 0 1 0 .
Therefore, the 4 × 4 matrices in the product are effectively reduced to 2 × 2 matrices. The expressions for f ± x , g ± x could be exactly found for the two boundary conditions. We do this in the next section, for now we conclude this section by expressing required components of the Green's function using the variables f ± x , g ± x . We use

U = 1 √ 2 i 1 -i 1 to rewrite Π N as, Π N = 1 2i     i( f + N + f - N ) ( f + N -f - N ) i(g + N + g - N ) (g + N -g - N ) -( f + N -f - N ) i( f + N + f - N ) -(g + N -g - N ) i(g + N + g - N ) -i( f + N-1 + f - N-1 ) -( f + N-1 -f - N-1 ) -i(g + N-1 + g - N-1 ) -(g + N-1 -g - N-1 ) ( f + N-1 -f - N-1 ) -i( f + N-1 + f - N-1 ) (g + N-1 -g - N-1 ) -i(g + N-1 + g - N-1 )     . (IV.26)
We define the matrices P N and Q N as follows,

P N = 1 2i i( f + N + f - N ) f + N -f - N -( f + N -f - N ) i( f + N + f - N ) and Q N = 1 2i i(g + N + g - N ) g + N -g - N -(g + N -g - N ) i(g + N + g - N )
.

(IV.27)

Then

Π N = P N Q N -P N-1 -Q N-1 .
(IV.28)

Let F ± N defined as follows

F ± N := F ± N (ω) = k f ± N + i γ k ω(g ± N + f ± N-1 ) -γ 2 k 2 ω 2 g ± N-1 . (IV.29)
Then, substituting Π N from Eq. (IV.26) in Eq. (IV.22), we can show that

I 2 = G 1,N kP N + iγω(Q N + P N-1 ) - γ 2 ω 2 k Q N-1 = G 1,N 1 2 (F + N + F - N ) 1 2i (F + N -F - N ) -1 2i (F + N -F - N ) 1 2 (F + N + F - N ) , (IV.30)
and

G 1,1 = G 1,N Q N + i γω k Q N-1 . (IV.31)
Using Eq. (IV.15) and inverting Eq. (IV.30) gives us the required components of the Green's functions

[G + 1 ] 1,N = 1 2 1 F + N + 1 F - N and [G + 2 ] 1,N = - 1 2i 1 F + N - 1 F - N .
These then give, using Eq. (IV.12) and Eq. (IV.13), the heat current to be

J N = (T L -T R ) γ 2 π ∞ -∞ dω ω 2 1 F + N (ω) 2 + 1 F - N (ω) 2 .
(IV.32)

Thus we have now obtained a new expression for the net transmission amplitude T N (ω). In the next section, we use this form and for the case of a uniform magnetic field, derive analytical expressions for the current in the thermodynamic limit. Before that, we take a quick digression to discuss the temperature profile.

TEMPERATURE PROFILE

We can also obtain the temperature profile of the chain, which is defined as

T y = ⟨v 1 (t, y) + v 2 (t, y)⟩ .
Using the steady state expression for q 1 (t, y) and q 2 (t, y), we can show that this is given by

T y = T L Λ y + T R (1 -Λ y ) where Λ y = ∞ -∞ dω π mγω 2 G + 1 (ω) 1,y 2 + G + 2 (ω) 1,y 2 .
Using Eq. (IV.19), Eq. (IV.20) and Eq. (IV.21), we could obtain the matrix block G 1,y , containing the required components for the temperature profile, to be, where we recall that Π y is defined by Eq. (IV.26) with N replaced by y and Ω R is defined in Eq. (IV.23). Using Eq. (IV.33) we can write linear equations for G 1,y and G 1,y+1 in the block form as,

I 2 G 1,1 = G 1,y G 1,y+1 Π y Ω R = G 1,y G 1,y+1 kP y + iγωQ y Q y -kP y-1 -iγωQ y+1 -Q y-1 , (
I 2 -iγωG 1,1 = G 1,y kP y -G 1,y+1 kP y-1 , (IV.33) G 1,1 = G 1,y Q y -G 1,y+1 Q y-1 , (IV.34)
where we used Eq. (IV.31) to obtain the expression of G 1,1 . This set of equations seems complicated to simplify further but when evaluated numerically, for an ordered chain, we obtain the results given in Fig ( IV.2). The temperature in the bulk is the same as for zero magnetic field case, (T L + T R )/2 and the magnetic field mostly effects the profile near the ends of the chain.

IV.3 STUDY OF THE HEAT CURRENT FOR A CONSTANT MAGNETIC FIELD

We consider a uniform chain (B x = B for all x in {1, 2 . . . , N}) and derive the expressions for the current in the thermodynamic limit,

J ∞ = lim N→∞ J N ,
for the cases of fixed boundary conditions and free boundary conditions. For the infinite system, the phonon spectrum consists of two bands {ω ± (q) ; q ∈ (0, π)} where 2mω ± (q) = ±eB + (eB) 2 + 8mk(1cos q). The bands are gapped for eB > √ 2mk. In In the former, the bands overlap while in the latter they are gapped. We expect the transmission to be zero outside the bandwidth of the two bands which becomes explicit in the thermodynamic limit. We will show that in the thermodynamic limit, the current expression in Eq. (IV.32) is the sum of two terms due to the two frequency bands of the system. Also, in the small ω limit we will find that T ∞ (ω) = lim N→∞ T N (ω) is equivalent to ω 3/2 and ω 1/2 for free and open boundary conditions respectively. We consider the two boundary conditions separately and set k = e = 1 without loosing generality.

IV.3.1 FIXED BOUNDARY CONDITIONS

We have then c x = 2 for all x in {1, . . . , N}. The expressions for f ±

x for free boundary conditions can be found exactly from Eq. (IV.25). Recalling Eq. (IV.24) let us denote q ± := q ± (ω) ∈ C such that

2 cos q ± (ω) = a x (ω) ± b x (ω) = 2 -mω 2 ± Bω .
(IV.35) We obtain

f ± x =
sin[q ± (x + 1)] sin q ± and g ± x = f ± x-1 (for x > 1) .

(IV.36)

Recall Eq. (IV.32). If ω is not in the frequencies band defined by ω + (resp. ω -) then q + (ω) (resp.q -(ω)) has a non-vanishing imaginary part and F + N (ω) resp. F - N (ω) becomes exponentially large in N, so that these ω's will not contribute in the thermodynamic limit to the heat current. We therefore obtain in the thermodynamic limit,

J ∞ = (T L -T R ) γ 2 π lim N→∞ R dω ω 2 1 F + N (ω) 2 + 1 F - N (ω) 2 = (T L -T R ) 2γ 2 π lim N→∞ π 0 dω + (q) [ω + (q)] 2 F + N (ω + (q)) 2 + π 0 dω -(q) [ω -(q)] 2 F - N (ω -(q)) 2 .
where in the second equality, the 2 comes from the fact that by symmetry we restricted us to positive frequencies.

To obtain the limits, we follow the steps given in [START_REF] Roy | Heat transport in ordered harmonic lattices[END_REF]. By using Eq. (IV.29) and Eq. (IV.36) we can express F ± N (ω ± (q)) as

F ± N (ω ± (q)) = 1 sin(q)
[α ± (q) sin(qN) + β ± (q) cos(qN)] ,

α ± (q) = (1γ 2 [ω ± (q)] 2 ) cos(q) + 2iγω ± (q), β ± (q) = (1 + γ 2 [ω ± (q)] 2 ) sin(q) .

(IV.37)

We have then to study the limit as

N → ∞ of π 0 dq ∂ ω ± ∂ q (q) sin 2 (q)[ω ± (q)] 2 |α ± (q) sin(qN) + β ± (q) cos(qN)| 2 = π 0 dq H ± (q) 1 + R ± (q) sin(2qN + ϕ ± (q))
,

where

H ± (q) = 2 |α ± (q)| 2 + |β ± (q)| 2 ∂ ω ± ∂ q (q) sin 2 (q)[ω ± (q)]
= 0.2, m = k = e = 1, T L = 1, T R = 0. and R ± (q) cos ϕ ± (q) = 2 Re(α ± (q)β ± (q)) |α ± (q)| 2 + |β ± (q)| 2 , R ± (q) cos ϕ ± (q) = |α ± (q)| 2 -|β ± (q)| 2 |α ± (q)| 2 + |β ± (q)| 2 .
β ± (q) is the complex conjugate of β ± (q). It can be shown [START_REF] Roy | Heat transport in ordered harmonic lattices[END_REF][START_REF] Casher | Heat flow in regular and disordered harmonic chains[END_REF][START_REF] Venkateshan Kannan | Nonequilibrium stationary state of a harmonic crystal with alternating masses[END_REF] that

lim N→∞ π 0 dq H ± (q) 1 + R ± (q) sin(2qN + ϕ ± (q)) = π 0 dq H ± (q) 1 -[R ± (q)] 2 .
Since we have that 1

1 -[R ± ] 2 = |α ± | 2 + |β ± | 2 2| Im(α ± β ± )|
and Im α ± (q)β ± (q) = 2γω ± (q) 1 + γ 2 [ω ± (q)] 2 sin(q) ≥ 0 for q ∈ (0, π) ,

we get that lim N→∞ π 0 dq ∂ ω ± ∂ q (q) sin 2 (q)[ω ± (q)] 2 |α ± (q) sin(qN) + β ± (q) cos(qN)| 2 = 1 2γ π 0 dq ∂ ω ± ∂ q (q) ω ± (q) sin(q) 1 + γ 2 [ω ± (q)] 2 = 1 2γ ω ± (π) ω ± (0) dω ω sin(q ± (ω)) 1 + γ 2 ω 2 . (IV.38)
We conclude that

J ∞ = (T L -T R ) γ π ω + (π) ω + (0) dω ω sin(q + (ω)) 1 + γ 2 ω 2 + ω -(π) ω -(0) dω ω sin(q -(ω)) 1 + γ 2 ω 2 . (IV.39)
The two integrals run over the two bands of the spectrum:

[ω -(0), ω -(π)] = [0, (-B+ √ B 2 + 16m)/2m] and [ω + (0), ω + (π)] = [B/2m, (B+ √ B 2 + 16m)/2m
]. We see that in the thermodynamic limit the transmission is exactly zero at energy values outside the two bands of the spectrum and also the current is explicitly expressed as sum of two terms coming from the two bands. For small ω behaviour of the transmission, T ∞ (ω), we take the contribution due to the lower band (depending on the sign of eB we have ω + (0) = 0 or ω -(0) = 0). It is straightforward to see from Eq. (IV.39) that T ∞ (ω) ∼ ω 3/2 for B ̸ = 0 while for B = 0, T ∞ (ω) ∼ ω 2 .

IV.3.2 FREE BOUNDARY CONDITIONS

For free boundary conditions we have c x = 2δ x,1δ x,N . Recalling Eq. (IV.24) and the definition of q ± := q ± (ω) ∈ C the numbers f ± x , g ± x can once again be obtained with from Eq. (IV.25). We have

f ± N = 2
(cos(q ± ) -1) sin(q ± ) sin(q ± N), g ± N-1 = sin(q ± (N -1)) sin(q ± ) , and

g ± N = f ± N-1 = 1 sin(q ± ) (sin(q ± N) -sin(q ± (N -1))) ,
where q ± is defined in Eq. (IV. [START_REF] Cesbron | Fractional diffusion limit for a kinetic equation in the upper-half space with diffusive boundary conditions[END_REF]. Using these we can express F ± N defined by Eq. (IV.29) as

F ± N (ω ± (q)) = 1 sin(q) [α ± (q) sin(qN) + β ± (q) cos(qN)] ,
where α ± (q) = 2(cos(q) -1)γ 2 [ω ± (q)] 2 cos(q) + 2iγω ± (q)(1cos(q)) , β ± (q) = γ 2 [ω ± (q)] 2 sin(q) + 2iγω ± (q) sin(q) .

It has the same form as F ± N appearing in Eq. (IV.37) but with different expressions for α ± and β ± . Hence, using the same method, and noticing that Im α ± (q)β ± (q) = 2γ[ω ± (q)] 2 sin(q) ∓B + γ 2 + m ω ± (q) , we deduce that

J ∞ = (T L -T R ) γ π ω + (π) ω + (0) dω sin(q + (ω)) -B + (γ 2 + m) ω + ω -(π) ω -(0) dω sin(q -(ω)) B + (γ 2 + m) ω . (IV.40)
As in the case of fixed boundary condition, we have expressed the current as the sum of two integrals running over the two bands of the spectrum. However, from this expression, for small ω behaviour of T ∞ (ω), the lower band gives T ∞ (ω) ∼ ω 1/2 and ∼ ω 0 for B ̸ = 0 and B = 0 respectively. In ), we show a comparison between T ∞ (ω) derived for the two boundary conditions with the respect transmission obtained numerically for N = 20. It can be seen that the transmission in the thermodynamic limit looks exactly like the envelope covered by the transmission for finite N. Table 1 shows the comparison of the numerically obtained current for N = 10, 20 and B = 1, 2 with the value of the current calculated from the Eq. (IV.39) and Eq. (IV.40) for the two boundary conditions respectively. These show a good agreement.

We also show in Fig. (IV.5) the variation of the current in thermodynamic limit J ∞ with respect to the magnetic field and we find that it decreases monotonically to 0 with the magnetic field B, as 1/B 2 for large B, independently of the boundary conditions. We can also check easily that the limit B → 0 and N → ∞ commute, i.e. the limit of J ∞ as B → 0 is equal to the normalised current of the ordered harmonic chain without magnetic field considered in [START_REF] Rieder | Properties of a harmonic crystal in a stationary nonequilibrium state[END_REF][START_REF] Nakazawa | Energy flow in harmonic linear chain[END_REF][START_REF] Nakazawa | On the lattice thermal conduction[END_REF][START_REF] Roy | Role of pinning potentials in heat transport through disordered harmonic chains[END_REF], for free and fixed boundary conditions.

IV.4 STUDY OF THE HEAT CURRENT FOR A RANDOM MAGNETIC FIELD IV.4.1 SHORT INTRODUCTION TO THE MODEL

In this section, we study the model introduced in Sec. IV.1 but here, we assume that the particles are submitted to a random magnetic field. The intensity of the magnetic field (B x ) 0<x≤N form a sequence of independent identically distributed random variables with average denoted by E [B] and a finite variance σ 2 > 0. In order to simplify the presentation we decided to set k = e = m = 1 in the following.

IV.4.2 LYAPUNOV EXPONENT

In Sec. IV.2, by using the non-equilibrium Green function formalism, we obtained an exact expression for the heat current J N in the steady state of the chain. For the convenience of the reader we recall the expression in this section. We recall that the processes ( f ±

x ) x∈N and (g ± x ) x∈N are defined by

f ± x+1 = (c x+1 -ω 2 ± ωB x+1 ) f ± x -f ± x-1 , f ± 0 = 1, f ± 1 = c 1 -ω 2 ± ωB 1 , g ± x+1 = (c x+1 -ω 2 ± ωB x+1 )g ± x -g ± x-1 , g ± 0 = 0, g ± 1 = 1 , (IV.41)
where c x = 2δ x,1δ x,N for free boundaries and c x = 2 for fixed boundaries. We recall that

F ± N (ω) = f ± N + iγω(g ± N + f ± N-1 ) -γ 2 ω 2 g ± N-1 , (IV.42)
and the heat current is equal to

J N = 2(T L -T R ) ∞ 0 dω T N (ω) , (IV.43)
with the net transmission function T N defined for any frequency ω by

T N (ω) := γ 2 π ω 2 1 F + N (ω) 2 + 1 F - N (ω) 2 .
(IV.44)

We denote by E [J N ] the expectation of the heat current with respect to the magnetic field distribution and our goal is to understand its scaling behavior in N.

Observe that the stochastic processes ( f - x ) x∈N and (g - x ) x∈N are defined in terms of the twodimensional discrete time Markov chain (U x ) x∈N given by

U x+1 = 2 -ω 2 -ωB x+1 -1 1 0 U n , where U x := u x u x-1 , (IV.45)
by choosing suitable initial conditions. By replacing the B x 's by -B x 's in the last display, we see that ( f + x ) x∈N and (g + x ) x∈N can also be expressed in terms of (U x ) x∈N . The state of the Markov chain is nothing but the result of a product of 2 × 2 product of independent and identically distributed random matrices. Roughly, the behaviour of F ± N is related to the growth of ∥U N (ω)∥ which will be in the form e 2λ (ω)N , where

λ (ω) = lim N→∞ 1 2N E [log ∥U N (ω)∥] = lim N→∞ 1 N E [log |u N (ω)|] > 0 , (IV.46)
with E [•] denoting a disorder average, is the Lyapunov exponent associated to the Markov chain (U x ) x≥0 , or equivalently of the corresponding product of random matrices. The limit exists by Furstenberg's theorem [START_REF] Furstenberg | Noncommuting random products[END_REF], is non-negative, independent of the initial condition U 0 and the limit holds in fact also for any realisation and not only by averaging over the magnetic field distribution.

For now we quickly discuss the effect of localization due to the random magnetic field on the heat transport and the need for calculating the Lyapunov exponent λ (ω) for small frequencies ω.

IV.4.3 EFFECT OF LOCALIZATION DUE TO RANDOM MAGNETIC FIELD ON THE NET TRANSMISSION

Using Eq. (IV.44), we can calculate the net transmission T N (ω) for any spatial configuration of the magnetic field using a computer program. In ) we plot the net transmission function with ω for a uniform magnetic field and for a random magnetic field for different system sizes respectively. On comparison of the two plots, we can see that the randomness causes suppression of the net transmission and also the net transmission for the random magnetic field case goes down with system size while the system size has nearly no effect on the transmission for the uniform magnetic field. The suppression in case of random magnetic field is due to localization of the normal modes of the system. The normal modes of frequency ω get exponentially localized due to randomness with a localization length given by 1/λ (ω) where λ (ω) is the Lyapunov exponent defined in Eq. (IV.46). As a result of this they a priori do not contribute to the transmission. However, note that the transmission for random magnetic field is higher near ω = 0 and goes down as we move away which means that the normal modes with energies closer to ω = 0 have a larger localization length, i.e. λ (ω) → 0 as ω → 0. Since we are eventually interested in the size dependence of the current, for large N, which is the integral of the transmission over all ω, we can reduce the integration limit to values of ω for which the localization length is greater than the system size. For the remaining ω values for which the localization length is less than the system size, the transmission would be negligible. Hence, we cut off the integral limit to ω = ω N max where 1/λ ω N max = N and the current is then given by

E [J N ] ≈ 2(T L -T R ) ω N max 0 dω lim N→∞ E [T N (ω)] = 2(T L -T R ) ω N max 0 dωT ∞ (ω) . (IV.47)
Note that the frequency ω N max would be very small for large N, and for such small frequencies we expect T ∞ (ω) to have a weak dependence on disorder [since in the recursion Eq. (IV.45), the randomness is multiplied by ω] -hence in the above equation T ∞ (ω) is written without a disorder average and can in fact be determined by considering the chain in a constant magnetic field of strength E [B]. In Sec. IV.3, we proved that for constant magnetic field E [B] ̸ = 0, T ∞ (ω) ∼ ω 3/2 and ∼ ω 1/2 for fixed and free boundaries respectively, while for E [B] = 0 it goes as ω 2 and ω 0 for the two boundary conditions respectively. To determine the size dependence of the current in addition to the small ω behaviour of T ∞ (ω) we also need the small ω behaviour of λ (ω). We now proceed to the next section where we discuss the Lyapunov exponents of this equation.

IV.4.4 THEORETICAL RESULTS FOR LYAPUNOV EXPONENTS

In this section, we present theoretical results on the asymptotic of Lyapunov exponents for small ω for the Markov processes defined by Eq. (IV.45). The Lyapunov exponents are independent of the boundary conditions -so for this section we only work with fixed boundary conditions by setting c x = 2 for all x -and of the initial condition of the process, i.e. it is the same for f ± x and g ± x . We show that Eq. (IV.45) has three different behaviors for the Lyapunov exponent depending on the expected value E [B] of the random magnetic field. For E [B] > 0 the Lyapunov exponent satisfies λ (ω) ∼ ω and for E [B] < 0, λ (ω) ∼ ω 1/2 . However, for E [B] = 0, λ (ω) ∼ ω 2/3 . Similar Lyapunov exponent behaviours are found for a harmonic oscillator with parametric noise, [START_REF] Crauel | Bifurcations of One-Dimensional Stochastic Differential Equations[END_REF] and we will see that Eq. (IV.45) could be written exactly in this form in the continuum limit.

Let z(•) in R 2 be the solution of the following stochastic differential equation (with arbitrary initial condition)

∀t ≥ 0, d dt z(t) = A 0 z(t) + εσ A 1 z(t)ξ (t) , (IV.48)
where ε is a small positive parameter, σ > 0 a constant, ξ (•) a one-dimensional standard white noise and A 0 and A 1 are 2 × 2 matrices such that

A 0 = 0 1 -c 0 , A 1 = 0 0 -1 0 with c ∈ R .
The Lyapunov exponent λ z (ε) of the process z(•) is defined by The following result, proved in [START_REF] Volker | Perturbation Methods for Lyapunov Exponents[END_REF], gives the behaviour of the Lyapunov exponent λ z (ε) for small

λ z (ε) =
noise i) If c = 0 then λ z (ε) = λ (σ )ε 2/3 where λ (σ ) is defined in Eq. (IV.67) . ii) If c > 0 then λ z (ε) ∼ σ 2 8c ε 2 . iii) If c < 0 then λ z (ε) ∼ |c| .
A sketch of the proof of this result is given in Sec. IV.5.

Consider now Eq. (IV.45) defining the discrete time Markov chain U x = (u x , u x-1 ) ⊤ and rewrite it in the following form, for small ω,

u x+1 + u x-1 -2u x = -ωE [B] u x -ω(B x+1 -E [B])u x + O(ω 2 ) .
In the continuum limit, the discrete time process (u x ) x∈N becomes then the continuous time process

u(•) solution of d 2 dt 2 u(t) = -ωE [B] u(t) -ωσ u(t)ξ (t) , (
(t) = 0 1 -E [B] 0 z(t) + σ ω 1/4 0 0 -1 0 z(t)ξ (t) . (IV.55)
With the previous notation we have

λ w (ω) = √ ω λ z(ω 1/4 ) . (IV.56)
Eq. (IV.55) fits perfectly Eq. (IV.48) with c = E [B] and ε = ω 1/4 . Then using point (i), (ii) and (iii) of Eq. (IV.48) and Eq. (IV.56) we get

i) If E [B] = 0, λ w (ω) = λ (σ )ω 2/3 where λ (σ ) is defined in Eq. (IV.67) . ii) If E [B] > 0, λ w (ω) ∼ σ 2 8⟨B⟩ ω . iii) If E [B] < 0, λ w (ω) ∼ |E [B] |ω 1/2 .
It makes sense to believe that λ (ω) defined by Eq. (IV.46) and λ w (ω) defined by Eq. (IV.53) have roughly the same behaviour as ω → 0 but a strong theoretical argument supporting this belief is missing. However, in the case E [B] > 0, we can obtain directly the behaviour of λ (ω) by following the approach of [START_REF] Matsuda | Localization of normal modes and energy transport in the disordered harmonic chain[END_REF] and we observe then a good agreement at first order between λ (ω) and λ w (ω), not only at the level of the exponent in ω but also at the level of the prefactor, see Table IV.2. We refer the reader to Chapter V for a derivation of λ using the method of [START_REF] Matsuda | Localization of normal modes and energy transport in the disordered harmonic chain[END_REF]. Unfortunately we were not able to carry this approach for E [B] < 0 or E [B] = 0 and we decided to not pursue it. However numerical results presented in the next section support strongly the claim that λ (ω) ∼ λ w (ω) for ω → 0.

IV.4.5 NUMERICAL RESULTS FOR LYAPUNOV EXPONENTS

We numerically calculate the Lyapunov exponents by using Eq. (IV.25) to generate u N for 100 realizations of the random magnetic field. The Lyapunov exponent would then be given by λ

= 1 N E [log |u N |],
where N is the number of oscillators. We plot in Fig. (IV.7), the numerical data thus obtained for different ω and the power law fit, Cω s , for the data with C and s as fitting parameters. We see that the values of s obtained for the three casses, E [B] > 0, E [B] < 0 and E [B] = 0, agree reasonably well with the theoretically expected values. The prefactor, C, obtained for the three cases also seems to agree with the expected values from theory, see Table IV

.2.
We now have the behaviour of the Lyapunov exponents at small ω for Eq. (IV.25) and we found this to be different depending on the expectation value of the random magnetic field. The transmission is determined by f + N as well as f - N and these two have different Lyapunov exponents for E [B] ̸ = 0, therefore the larger of the two exponents will dominate in the transmission. This is the Lyapunov exponent for f - N for E [B] > 0, while for E [B] = 0, f + N and f - N have the same Lyapunov exponent. In the next section, we determine the size dependence of the current using these results for the Lyapunov exponents.

Case

Range of B x s :

λ (ω) ∼ Cω s C C theoretical E [B] > 0 λ (ω) ∼ σ 2 8E[B] ω (
[J N ] Fixed E [B] ̸ = 0 ∼ ω 3/2 ∼ ω ∼ N -5/2 Fixed E [B] = 0 ∼ ω 2 ∼ ω 2/3 ∼ N -9/2 Free E [B] ̸ = 0 ∼ ω 1/2 ∼ ω ∼ N -3/2 Free E [B] = 0 ∼ ω 0 ∼ ω 2/3 ∼ N -3/2
Table IV.3: Power law for the current for different boundary conditions and average magnetic fields.

IV.4.6 SIZE DEPENDENCE OF THE CURRENT

We now have the small ω behaviour of λ (ω) for the transmission. We found this to be different for E [B] ̸ = 0 and E [B] = 0, so we expect different power laws for the current for the two cases. The boundary conditions will also play a role in the power law via the small ω behaviour of T ∞ (ω). We therefore take the cases E [B] ̸ = 0 and E [B] = 0 separately for the two boundary conditions.

FIXED BOUNDARY CONDITIONS

(a) For E [B] ̸ = 0, T ∞ (ω) ∼ ω 3/2 and λ (ω) ∼ ω. Therefore using these in Eq. (IV.47) we have

E [J N ] ∼ N -5/2 . (b) For E [B] = 0, T ∞ (ω) ∼ ω 2 and λ (ω) ∼ ω 2/3 which gives E [J N ] ∼ N -9/2 . FREE BOUNDARY CONDITIONS (a) For E [B] ̸ = 0, T ∞ (ω) ∼ ω 1/2 and λ (ω) ∼ ω which gives E [J N ] ∼ N -3/2 . (b) For E [B] = 0, T ∞ (ω) ∼ ω 0 and λ (ω) ∼ ω 2/3 which gives E [J N ] ∼ N -3/2 .
The results are summarized in Table 1. Fig (IV.8) shows the numerically obtained power laws for E [B] ̸ = 0 and E [B] = 0. Numerically, the power laws are obtained by calculating T N (ω) for different ω and then performing the integration numerically. We expect to see the power law behaviour at some large enough N. We see a reasonable agreement with the theoretically expected power laws except for the case with E [B] = 0 and free BC, where we get E

[J N ] ∼ N -2 instead of the expected E [J N ] ∼ N -3/2 .
The case with E [B] = 0 seems to be quite subtle because of the following reasons:

• The assumption that T ∞ (ω) may be replaced by the transmission for the uniform case for small ω does not hold good for E [B] = 0 case. This can be clearly seen from

B] ̸ = 0, ω N max ∼ N -1 while ω N s ∼ N -2
. Thus, for any finite but large N, we have ω N max > ω N s and there are a sufficient number of conducting modes. On the other hand, for E [B] = 0, both ω N max and ω N s scale as N -3/2 and this could be the reason why our heuristic approach for current scaling fails for this case.

IV.5 APPENDIX A. COMPUTATION OF THE LYAPUNOV EXPONENT λ z

Let z(•) be the solution of the following stochastic differential equation (with arbitrary initial condition)

∀t ≥ 0, d dt z(t) = A 0 z(t) + εσ A 1 z(t)ξ (t) , (IV.57)
where ε is a small positive parameter, σ > 0 a constant, ξ (•) a one-dimensional standard white noise and A 0 and A 1 are 2 × 2 matrices such that

A 0 = 0 1 -c 0 , A 1 = 0 0 -1 0 with c ∈ R .
In Sec. IV.4.4 we stated that we can compute explicitly the Lyapunov exponent of z(•) denoted by λ z (ε). This was done by Pardoux et al. in [START_REF] Pardoux | Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion[END_REF] and by Wihstutz in [START_REF] Volker | Ergodic Theory of Linear Parameter-Excited Systems[END_REF]. For the convenience of the reader we decided to give a sketch of the proof. The first step of the proof is to use the ergodic theorem to obtain an explicit formula (cf. Eq. (IV.63)) for λ z (ε) instead of Eq. (IV.49). In the second step we perform a perturbation analysis in ε with this new expression. Observe that ∥z(t)∥ = R(t). Moreover, since in Eq. (IV.58) the noise is vanishing exactly at the points θ * k = (2k + 1)π/2, k ∈ Z, and that the drift in Eq. (IV.58) at θ * k is equal to -1, we see that starting from θ 0 ∈ [θ * k-1 , θ * k ) the process θ (•) will pass successively in the intervals θ 0 ∈ [θ * ℓ , θ * ℓ+1 ) for ℓ ≤ k -1 without coming back to an interval previously visited. This defines a sequence of random times t ℓ = inf{t ≥ 0 ; θ (t) ∈ [θ * k-ℓ-1 , θ * k-ℓ )} for ℓ ≥ 0 with t 0 = 0. The process is thus clearly not ergodic. A simple way to restore this ergodicity (that will be needed later) is to consider the process θ (•), living in [-π/2, π/2), and defined by θ (t) = θ (t) + (k -ℓ)π for t ∈ [t ℓ ,t ℓ+1 ). The process θ (•) satisfies the same stochastic differential equation as θ (•) but when it reaches -π/2 it is immediately reseted to π/2. Equivalently θ (•) is solution of Eq. (IV.58) but seen as a SDE on the torus [-π/2, π/2) where the two end points of the interval have been identified. The process θ (•) has now the nice property to be ergodic. We denote by ρ ε (θ )dθ its invariant measure which is computed below. Observe moreover that Eq. (IV.60) still holds by replacing θ by θ because the functions h 3 , h 4 , r are π-periodic. In order to keep notation simple we denote in the sequel the process θ by θ . where we recall that ρ ε (θ )dθ is the invariant measure of the process θ (•).

IV.5.1 EXPLICIT FORMULA OF THE LYAPUNOV EXPONENT

The expansion in ε for λ z (ε) can then be obtained from the expansion of ρ ε . Before doing this we prove Eq. (IV.50), i.e. that the process z(•) = (u(•), v(•)) ⊤ and the process u( This proves the claim.

IV.5.2 PERTURBATION ANALYSIS IN ε

Let us now compute ρ ε which is the solution of the stationary Fokker-Planck equation

∂ θ ε 2 2 ∂ θ (h 2 1 ρ ε ) -(h 0 + ε 2 2 h 1 ∂ θ h 1 )ρ ε = 0 . (IV.65)
If we look for a solution such that ε 2 2 ∂ θ (h 2 1 ρ ε ) -(h 0 + ε 2 2 h 1 ∂ θ h 1 )ρ ε = 0 we get ρ ε (θ ) ∝ cos -2 (θ )e -2ε -2 3σ 2 tan 3 (θ )-2cε -2 σ 2 tan(θ ) , which is not normalisable. Hence, we have to look for a normalisable solution such that

ε 2 2 ∂ θ (h 2 1 ρ ε ) -(h 0 + ε 2 2 h 1 ∂ θ h 1 )ρ ε = A ,
for some constant A. We get then that and Z ε the partition function making ρ ε a probability. Injecting this in Eq. (IV.63) we may derive the results claimed by a careful saddle point analysis. We prefer instead to rely on a more heuristic analysis to bypass boring computations.

ρ ε (θ ) = Z -1 ε v ε (θ ) cos -2 (θ ) tan(θ ) -∞ exp 2ε -2 3σ 2 u 3 + 2cε -2
It is natural to expect that as ε → 0 the stationary measure ρ ε (θ )dθ will converge to the one of d dt θ (t) = h 0 [θ (t)] i.e. Eq. (IV.58) with ε = 0 .

However as we will see this deterministic dynamical system has different behaviours depending on the value of c and that in some cases we have also to compute the next order corrections.

STUDY OF THE CASE c > 0

If c > 0, the deterministic dynamical system has a unique invariant state ρ 0 (θ )dθ with ρ 0 (θ ) = -√ c π h -1 0 (θ ) because h 0 never vanishes on [-π/2, π/2). Hence, ρ ε → ρ 0 as ε → 0. However, since π 0 q 0 (θ )ρ 0 (θ )dθ = 0, we have to expand ρ ε at order ε 2 to obtain the behavior of λ ε in Eq. (IV.63).

Let us assume that ρ ε = ρ 0 + ε 2 δ ρ 0 + o(ε 2 ) . (IV.66)

Injecting this in Eq. (IV.65) and identify the powers in ε, we obtain that Hence, we finally get

∂ θ [h 0 (δ ρ 0 )] = 1 2 ∂ θ ∂ θ (h 2 1 ρ 0 ) -(h 1 ∂ θ h 1 )ρ 0 .
λ z (ε) = ε 2 σ 2 8c + o(ε 2 ) .
This ends the proof of the case c > 0. There are two solutions θ * > 0 and -θ * < 0. The deterministic dynamical has two extremal invariant probability measures given by δ ±θ ⋆ . Since h ′ 0 (θ * ) < 0 < h ′ 0 (-θ * ), δ -θ * is unstable while δ θ * is stable. By introducing noise in this dynamical system the stable stationary state is selected when the intensity of the noise is sent to zero afterwards, i.e. ρ ε (θ )dθ → δ θ * . We conclude that lim ε→0 λ z (ε) = q 0 (θ * ) = |c| . This ends the proof of the case c < 0.

to an energy, momentum conserving noise. The formal generator of the dynamics is given by L ε where L ε = ∑ x∈Z p(x)∂ q(x) + [q(x + 1) + q(x -1) -2q(x)] ∂ p(x) + γ T ∂ 2 Here ε is a scaling parameter, T is the temperature of the thermostat and γ is the dissipation strength at the reservoir. We recall that the last term of Eq. (V.1) is the formal infinitesimal generator of a stochastic noise which preserves the energy and the momentum of the system.

Let ψ : T → C be the wave function associated to the dynamics defined for any k in T by

ψ(k) = 1 √ 2 { p(t, k) -iω 0 (k) q(t, k)} , (V.2)
where we recall that ω 0 (k) = 4π 2 sin 2 (πk) = 2π| sin(πk)| . (V.3)

We recall that the energy of the system is preserved and equals to half of the L 2 norm of ψ. Let W ε be the Wigner's distribution associated to ψ. Then, in [START_REF] Komorowski | High frequency limit for a chain of harmonic oscillators with a point langevin thermostat[END_REF] it is proved that W ε converges to W where W : Let us define W N (t, u, k) = W N 3/2 t, Nu, k , then W N is solution for u ̸ = 0 of

∂ t W N N 3/2 (t, u, k) + v 0 (k) 2πN ∂ u W N (t, u, k) = T R k, k ′ W N t, u, k ′ -W N (t, u, k) dk ′ . (V.8)
When γ = 0 (no thermostat) then g = p -= 0 and p + = 1 and then Eq. (V.8) is satisfied for any u in R. Hence, we recover the linear Boltzmann's equation obtained (cf. Eq. (III.18) of Chapter III with B = 0 ) by Basile, Olla and Spohn in [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF]. In this case, as explained in Chapter III, by interpreting the linear Boltzmann's equation as the infinitesimal generator of a Piecewise Deterministic Markov Process we are able to obtain the macroscopic behavior of W N as N goes to infinity.

In the presence of an interface, γ ̸ = 0, Olla, Komorowski and Ryshik were able in [START_REF] Komorowski | Fractional diffusion limit for a kinetic equation with an interface[END_REF] to obtain the macroscopic behavior of W N when N goes to infinity. They proved (we refer the reader to [START_REF] Komorowski | Fractional diffusion limit for a kinetic equation with an interface[END_REF]Theorem 1.1] for the precise statement) that for any smooth functions J lim N→∞ R×T W N (t, u, k)J(u, k)dudk = R×T ρ(t, u)J(u, k)dudk .

(V.9)

where ρ is (up to a constant) a weak solution of

∂ t ρ(t, u) = p.v uu ′ >0 h u -u ′ ρ t, u ′ -ρ(t, u) du ′ + g × [T -ρ(t, u)] uu ′ >0 h u -u ′ du ′ + p + uu ′ <0 h u -u ′ ρ t, u ′ -ρ(t, u) du ′ + p - uu ′ <0
h uu ′ ρ t, -u ′ρ(t, u) du ′ . (V.10)

Here the integral appearing in Eq. (V.10) are understood in the principal value sense and (V.12)

Observe that if γ = 0, we recover the fractional diffusion equation of exponent 3/4 (cf. Eq. (III.34) of Chapter III with B = 0) and if we send γ to infinity then we have only reflexion since p + = g = 0.

MAGNETIC INTERFACE

Based on the results presented in the previous section, we study a deterministic infinite harmonic chain of coupled oscillators with two degrees of freedom subject to the action of a magnetic field of intensity γB 0 at site 0. The equations of motions are given for any i in {1, 2} by d dt q i (t, x) = p i (t, x) , d dt p i (t, x) = q i (t, x + 1) + q i (t, x -1) -2q i (t, x) + γB 0 [p 1 (t, 0)δ i,2p 2 (t, 0)δ i,1 ] δ 0,x . (V.13)

For any configurations (q, p) we define the wave functions ψ1/2,0 : T → C where for any i in {1, 2} ψ1,0 (V.17)

(k) = 1 √ 2 { p1 (k) -iω 0 (k) q1 (k) + i ( p2 (
Then we can prove that ψ1,0 (t, k) = e -iω 0 (k)t ψ1,0 (0, k) - We can formally prove that the Wigner's distribution, W ε , associated to this system converges to W 0 (•) := (W 0 (•, 1), W 0 (•, 2)) where for any i in {1, 2} k in T, any time t and u in R * ∂ t W 0 (t, u, k, i)

+ v 0 (k) 2π ∂ u W 0 (t, u, k, i) = 0 . (V.19)
For any i in {1, 2}, at the interface u = 0, W 0 (•, i) satisfies ∀k ∈ 0, 1 2 , W 0 t, 0 + , k, i = p-,0 (k)W 0 t, 0 + , -k, i + p+,0 (k)W 0 t, 0 -, k, i , (V.20) ∀k ∈ -1 2 , 0 , W 0 t, 0 -, k, i = p-,0 (k)W 0 t, 0 -, -k, i + p+,0 (k)W 0 t, 0 + , k, i , (V. From this result we can suspect that if we add to the dynamics (V.13) a stochastic noise, which preserves the energy and the momentum, then the transport equation (cf. Eq. (V.19)) is changed, for any u ̸ = 0, into the following linear Boltzmann's equation

∂ t W 0 (t, u, k, i) + v 0 (k) 2π ∂ u W 0 (t, u, k, i) = 2 ∑ j=1 T
R k, k ′ W 0 t, u, k ′ , j -W 0 (t, u, k, i) dk ′ , (V. [START_REF] Bernardin | Anomalous diffusion for a class of systems with two conserved quantities[END_REF] supplemented with the boundary conditions given in Eq. (V.22). Here for any (k, k ′ ) we have R k, k ′ = 16 sin 2 (πk) sin 2 πk ′ . (V.24) Indeed, as mentioned in Chapter II and Chapter III, the introduction of noise changes the transport equation into a linear phononic Boltzmann's equation.

Hence, the interface due to the Langevin bath in the study of Komorowski, Olla Ryzhik and Spohn in [START_REF] Komorowski | High frequency limit for a chain of harmonic oscillators with a point langevin thermostat[END_REF] is replaced by some magnetic interface. From this observation, we believe that following the study made by Komorowski, Olla and Ryzhik in [START_REF] Komorowski | Fractional diffusion limit for a kinetic equation with an interface[END_REF] we can obtain in a second scaling limit a 3/4 fractional diffusion equation with boundary conditions p+ and plike in Eq. (V.10) where p+,0 = lim k→0 p+,0 (k) = In other words, we believe that (up to a constant) we can obtain Eq. (V.10) with p + and p -replaced by p+ and prespectively and g = 0.

From this observation and the results of Chapter III, we guess that we can obtain an interpolation between two fractional Laplacians with a magnetic interface. We develop this approach in the next section.

TRANSITION IN TWO STEPS BETWEEN TWO FRACTIONAL LAPLACIANS WITH BOUNDARY CONDITIONS

Consider an infinite harmonic chain of coupled oscillators with two degrees of freedom subject to the action of a magnetic field of intensity B on each particle and submitted to an energy-momentum preserving noise. Moreover we assume that the particle labelled zero is submitted to a second magnetic field of intensity γB 0 . The infinitesimal generator is formally given by

L ε = 2 ∑ i=1
∑ x∈Z p i (x)∂ q i (x) + [q i (x + 1) + q i (x -1) -2q i (x) + Bδ i,1 p 2 (x) -Bδ i,2 p 1 (x)] ∂ If B = 0 then we recover the system studied in the previous section 1 , if B 0 = 0 we recover the system studied by Saito, Sasada and Suda in [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF].

If B 0 = 0 since the harmonic chain is submitted to a magnetic field of intensity B, we know, as we recalled in Chapter III that the Wigner's distribution converges to some linear Boltzmann's equation given by Eq. (III.17) of Chapter III. As seen previously, if B = 0 and B 0 ̸ = 0 the magnetic field at site zero implies the existence of some magnetic interface in the Boltzmann's linear equation. Hence, we can guess that in the case B 0 ̸ = 0, the Wigner's distribution converges to W B 0 (•) := (W B 0 (•, 1), W B 0 (•, 2)) where for any u ̸ = 0

∂ t W B (t, u, k, i) + v B (k) 2π ∂ u W B (t, u, k, i) = 2 ∑ j=1 T R B k, k ′ , i, j W B t, u, k ′ , j -W B (t, u, k, i) dk ′ ,
where

R B k, k ′ , i, j = θ 2 i,B (k)R k, k ′ θ 2 j,B k ′ . (V.27)
We recall that the expressions of θ 1/2 and v B are given in Eq. (III.5) and Eq. (III.6) of Chapter III. For any i in {1, 2}, at the interface u = 0, W B (•, i) satisfies ∀k ∈ 0, 1 2 , W B t, 0 + , k, i = p-,B (k)W B t, 0 + , -k, i + p+,B (k)W B t, 0 -, k, i , (V.28) ∀k ∈ -1 2 , 0 , W B t, 0 -, k, i = p-,B (k)W B t, 0 -, -k, i + p+,B (k)W B t, 0 + , k, i . (V.29)

1 But here we are in dimension two.

where η(•) is a standard white noise and σ 2 is the variance of the (B x ) x∈N . Instead of studying the Lyapunov exponent λ , associated to (u x ) x∈N , we studied the one of u(•), denoted λ . As we mentioned in Sec. IV.4.5 of Chapter IV numerical simulations allow us to believe that λ = λ . When E[B] > 0, following the approach presented in [START_REF] Matsuda | Localization of normal modes and energy transport in the disordered harmonic chain[END_REF] we can explicitly compute λ , and its derivation is presented in this section.

Let (ξ x ) x∈N be the Markov chain defined for each x in N by From the last equality we deduce that

∂ φ Ψ (ω, B, φ ) = 1 + ∂ φ Ψ 1 (B, φ )ω 1/2 + ∂ φ Ψ 2 (B, φ )ω + O ω 3/2 . (V.46)
We can expect that F has the following power expansion We expand at the second order in x the previous equation and equalize the two sides with respect to the powers in x. To do this we perform first a second order Taylor expansion

F x, Ψ(x, B, φ ) = F 0 (φ ) + F 1 (φ ) + (∂ φ F 0 )(φ )Ψ 1 (B, φ ) x + F 2 (φ ) + 2(∂ φ F 1 )(φ )Ψ 1 (B, φ ) + (∂ 2 φ F 0 )(φ )Ψ 2 1 (b, φ ) + (∂ φ F 0 )(φ )Ψ 2 (B, φ ) x 2 + O x 3 . (V.49)
Then, we multiply it by

∂ φ Ψ (x, B, φ ) = 1 + ∂ φ Ψ 1 (B, φ )x + ∂ φ Ψ 2 (B, φ )x 2 + O x 3 ,
and average in B. The equalization of the zeroth order term does not provide any information. The equalization of the first order term provides that F 0 (φ ) is constant. Since F(ω, φ ) is a probability density for any ω we have that F 0 (φ ) = π -1 . Using this and equalizing the second order terms, we obtain

(∂ φ F 1 )(φ ) = - 1 2πγ 1 ∂ φ (E [Ψ 2 (B, φ )]) = 1 2π E (B -E [B]) 2 E [B]
∂ φ (cos 3 (φ ) sin(φ )) .

Observe that π/2 -π/2 F 1 (φ )dφ = 0 , then we get that Observe that the prefactor is also the same. Unfortunately we were not able to perform this argument when E [B] ≤ 0 since, as we mentioned in Sec. IV.5 of Chapter IV, the invariant distribution is converging to a Dirac measure as ω goes to zero. Hence, we think that an interesting problem could be to perform the rigorous derivation of λ when E [B] ≤ 0.

F 1 (φ ) = - 1 2πγ 1 σ 2 E [B]

V.2.2 TWO OTHER OPEN PROBLEMS

In this section, we would like to describe two open problems which seem to us very interesting. Unfortunately we did not have time to get any results on it.

As we saw, if we assume that the particles of the harmonic chain have random masses the heat current has some behavior which is different from the one observed when the masses are constants and the chain submitted to a random magnetic field. An interesting system to study is the harmonic chain submitted to a random magnetic field where all the particles have random masses, in order to observe the impact of these two phenomena on the heat current In [START_REF] Bernardin | Hydrodynamic limit for a disordered harmonic chain[END_REF], Bernardin, Huveneers and Olla studied the hydrodynamic behavior of the disordered harmonic chain. They proved that the empirical measures associated to the momentum, the energy and the elongation converge, using the same notations used in Sec. II. Assume now that all the particles have constant masses but the chain is submitted to a random magnetic field. In this system, the momentum is not preserved but the pseudo-momentum is. Hence, an interesting problem could be to understand if some Euler's equations like in [START_REF] Bernardin | Hydrodynamic limit for a disordered harmonic chain[END_REF] could be derived in this case.

I

  FROM MICROSCOPIC TO MACROSCOPIC . . . . . . . . . . . . . . . . . . . . . . . I.1 FEW ELEMENTS OF STATISTICAL PHYSICS I.1.1 NEWTON'S LAWS AND REVERSIBILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.1.2 ENTROPY AND IRREVERSIBILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Figure I.1: Mechanical reversibility.

Figure I. 2 :

 2 Figure I.2: Joule expansion macroscopic point of view.

Fig. I. 3 .

 3 On the left-hand side of Fig. I.3 the particles have given positions in the box and on the right-hand side some particles (in red)

Figure I. 3 :

 3 Figure I.3: Two accessible micro-states.

Figure I. 4 :

 4 Figure I.4: Joule expansion microscopic point of view.

  Fig. I.7.

Figure I. 5 :

 5 Figure I.5: Summary of Scales.

  Figure I.6: Collision of two particles.

  x, v)] dx dv ≥ 0 . (I.10) Under some regularity assumptions of the solution of the Boltzmann equation, Boltzmann showed the famous H-Theorem d dt H[ f ](t) ≥ 0 , (I.11)

Figure I. 7 :

 7 Figure I.7: Summary of Scaling Limits.

Figure I. 8 :

 8 Figure I.8: Fourier's Law.

Figure I. 9 :

 9 Figure I.9: Numerical simulations of the one-dimensional heat diffusion equation.

I. 32 )Figure I. 10 :

 3210 Figure I.10: Two trajectories of the random walk Z 0 (•).

Figure I. 11 :

 11 Figure I.11: Plots of the distribution of a random walk.

Figure I. 12 :

 12 Figure I.12: A gas with two types of particles.

Figure I. 13 :

 13 Figure I.13: Plots of a jump process.

  Figure II.1: Rieder, Lieb and Lebowitz's system

(II. 65 )N 2 ,

 652 For a given (x, y) in T d we denote the surface of constant kinetic energy and momentum by S e,p where S e,p = [p(x), p(y)] ∈ R 2d 1 2 |p(x)| 2 + |p(y)| 2 = e and p(x) + p(y) = p . (II.66)

Figure II. 2 :

 2 Figure II.2: Two dimensional harmonic chain noised submitted to a magnetic field.

  [START_REF] Suda | A family of fractional diffusion equations derived from stochastic harmonic chains with long-range interactions[END_REF] converge weakly to the finite-dimensional distributions of the Lévy process Y B u (•). Observe that, in Sec. I.4.3 of Chapter I, the random walk converged to the Brownian motion B u (•), but here, since the random variables Ψ B X B n do not have a moment of order 2, the Brownian motion is replaced by the Lévy process Y B u . After the convergence of the random walk, in Chapter I, to the Brownian motion, we obtained the convergence of the probability distribution p Nx defined in Eq. (I.33) to the solution of the diffusion equation (cf. Eq. (I.29)). Here, the Laplacian which is the infinitesimal generator of the Brownian motion, is replaced by a fractional Laplacian since the infinitesimal generator of Y B u (•) is a fractional Laplacian. In [66, Theorem 3.1] and in [104, Theorem 2], it is proved that f B (N α B t, Nu, •) converges in L 2 (T) (the precise statement is recalled in Theorem III.3 of Chapter III) to the unique solution ρ B of the following fractional diffusion equation

  Fig. II.3 summarizes the results presented in this section.

Figure II. 3 :

 3 Figure II.3: Summary of the studies made in [11, 66, 104].

Figure II. 4 :

 4 Figure II.4: Transition graph of the superdiffusion of energy

  Figure II.5: Variation of the net transmission with ω for constant masses, panel (a), and random masses, panel (b). The masses in (a) are set to be 1 on all oscillators and in (b) it was chosen uniformly from the interval (0, 2).

(

  Figure II.7: Variation of the net transmission with ω for uniform magnetic field, panel (a), and random magnetic field, panel (b). The magnetic field in (a) is set to be 1 on all oscillators and in (b) it was chosen uniformly from the interval (0, 2). As can be seen clearly from the plots, the localization effects cause suppression of the transmission.

Fig. II. 8 .

 8

Figure II. 8 :

 8 Figure II.8: Power law for the current for different boundary conditions and average magnetic fields.

Figure II. 9 :

 9 Figure II.9: Comparison for the transmission for disordered and uniform cases for the two boundary conditions. For E [B] ̸ = 0, B x is chosen from (1, 3) while for E [B] = 0, B x is chosen from (-1, 1). These are compared with the transmission for the uniform cases with B x = E [B] respectively.

Figure II. 10 :

 10 Figure II.10: Scaling of lowest allowed normal mode, ω N s with the system size, N. For E [B] ̸ = 0, B x is chosen from (1, 3) while for E [B] = 0, B x is chosen from (-1, 1). The B = 0 plot corresponds to the ordered chain (the ordered case B ̸ = 0 is not shown and has the scaling N -2 ).

10 )A

 10 representation of the stochastic dynamics is given in Fig. III.1.

Figure III. 1 :

 1 Figure III.1: Two dimensional noisy harmonic chain submitted to a magnetic field.

2 depends 2 ,

 22 on B hence we denote by L B the operator L 1 by Φ B the Lévy exponent Φ 1 2 and by ρB the function ρ 1 2

III. 54 )

 54 Proof : We refer the reader to Sec. III.5.4.These results are summarized in Fig. III.3. On the horizontal axis δ represents the intensity of the magnetic field and on the vertical axis α δ represents the scaling in space to obtain the hydrodynamic limit of f B N .

Figure III. 3 :

 3 Figure III.3: Transition graph of the superdiffusion of energy

Let

  (m N ) n∈N be an increasing sequence of positive numbers such that lim N→∞ m N = +∞ and lim N→∞ m N N = 0 . (III.71) Since X B N 0 = K B N (0), I B N (0) = (k, i) with k ̸ = 0 by Theorem III.5 under P (k,i) the finite-dimensional distributions of Z N u ([1 -N -α δ m N ] •) converge weakly to the finite-dimensional distributions of a Lévy process Z δ u (•) generated by L δ where L δ is defined in Eq. (III.50).

1 2

 1 -δ in I N B,± and by the dominated convergence theorem we get that lim

  c x = 2δ x,1δ x,N . A picture of the model is presented in Fig. IV.1.

Figure IV. 1 :

 1 Figure IV.1: Two dimensional harmonic chain submitted to a magnetic field.

  := a y (ω) = (-mω 2 + c y k) k and b y := b y (ω) = ωeB y k . (IV.24)

5 Figure IV. 2 :

 52 Figure IV.2: Temperature distribution for the ordered chain. Parameter values -e = m = k = 1, N = 32, T L = 3.5, T R = 1.5.

  Fig. (IV.3a) and Fig. (IV.3b) we show the spectrum for eB < √ 2mk and eB > √ 2mk respectively.

2 Figure IV. 3 :

 23 Figure IV.3: Spectrum of the chain in the bulk. Parameter values -e = m = k = 1.

  Figure IV.4: Comparison of the transmission T N (ω) for fixed and free boundary for N = 20 with T ∞ (ω). Parameter values -m = k = e = 1, γ = 0.2 and B = 2.

Figure IV. 5 :

 5 Figure IV.5: Variation of the current with the magnetic field. Parameter values -e = m = k = 1, γ = 0.2, T L = 1, T R = 0.

  Fig. (IV.4a) and Fig. (IV.4b

  Fig. (IV.6a) and Fig. (IV.6b

  Figure IV.6: Variation of the net transmission, in units of k B = 1, with ω for uniform magnetic field, panel (a), and random magnetic field, panel (b). The axes are in log scale and γ = 0.2. The magnetic field in (a) is set to be 1 on all oscillators and in (b) it was chosen uniformly from the interval (0, 2). As can be seen clearly from the plots, the localization effects cause suppression of the transmission.

Figure IV. 7 :

 7 Figure IV.7: Variation of numerically calculated Lyapunov exponent, λ = 1 N E [log |u N |], with ω. E [log |u N |] denotes average of log |u N | over 100 realizations of the random magnetic field. For (a), (b) and (c) the magnetic fields were chosen randomly from the intervals (0, 1), (-1, 1) and (-1, 0) respectively. The solid line is the data from the simulation while the dashed line is a power law fit, Cω s , to the data with C and s as fitting parameters. The obtained values of the fitting parameters agree appreciably with the theoretical values.

  Fig. (IV.9), where we show a comparison of the transmission for small ω for E [B] ̸ = 0 and E [B] = 0 with their respective uniform cases. While E [B] ̸ = 0 shows a clear agreement with the corresponding uniform case, E [B] = 0 case shows a clear disagreement. It is not clear how to estimate T ∞ (ω) for this case.• Interestingly we note that the transmission coefficient has peaks at much lower frequencies than

  Figure IV.8: Numerically obtained power laws for the average current, averaged over 100 realizations of the disorder, with fixed and free boundary conditions. For E [B] > 0, B x is chosen from (1, 3) while for E [B] = 0, B x is chosen from (-2, 2).

  Figure IV.9: Comparison for the transmission for disordered and uniform cases for the two boundary conditions. For E [B] ̸ = 0, B x is chosen from (1, 3) while for E [B] = 0, B x is chosen from (-1, 1). These are compared with the transmission for the uniform cases with B x = E [B] respectively.

Figure IV. 10 :

 10 Figure IV.10: Scaling of lowest allowed normal mode, ω N s with the system size, N. For E [B] ̸ = 0, B x is chosen from (1, 3) while for E [B] = 0, B x is chosen from (-1, 1). The B = 0 plot corresponds to the ordered chain (the ordered case B ̸ = 0 is not shown and has the scaling N -2 ).

First 0 dτ h 3 0 dτh 4

 0304 we express the solution of the 2-dimensional stochastic differential equation z(•) in terms of a 1-dimensional stochastic differential equation. Let θ (•) be the solution ofd dt θ (t) = h 0 [θ (t)] + 1 2 ε 2 ∂ θ h 1 [θ (t)]h 1 [θ (t)] + εh 1 [θ (t)]ξ (t) , (IV.58) with h 0 (θ ) = sin 2 (θ )(c -1)c and h 1 (θ ) = -σ cos 2 (θ ) . (IV.59)One can check thatz(t) = R(t) (cos(θ (t)), sin(θ (t))) ⊤ ,whereR(t) = ∥z 0 ∥ exp t [θ (τ)] + ε 2 r [θ (τ)]ε t [θ (τ)] ξ (τ) , (IV.60) with h 3 (θ ) = (1c) cos(θ ) sin(θ ) , h 4 (θ ) = σ 2 cos(θ ) sin(θ ) ,(IV.61) r(θ ) = σ 2 cos 2 (θ ) 2 2 cos 2 (θ ) -1 . (IV.62)

3 0 dτ h 4 2 -π/ 2 dθ h 3

 304223 By definition (IV.49) of Lyapunov exponent and Eq. (IV.60) we get thatλ z (ε) = lim t→∞ 1 t t 0 dτ h 3 [θ (τ)] + ε 2 r [θ (τ)] + ε t 0 dτ h 4 [θ (τ)] ξ (τ) [θ (τ)] + ε 2 r [θ (τ)] , since t [θ (τ)] ξ (τ) = 0 .Then by using the ergodic theorem we obtain λ z (ε) = π/(θ ) + ε 2 r(θ ) ρ ε (θ ) , (IV.63)

σ 2 2 σ

 22 u du , with v ε (θ ) = exp -2ε -2 3σ 4 tan 3 (θ ) -2cε -4 tan(θ ) ,

Since π/ 2 -π/ 2 ( 2 -π/ 2 2 0+ 2 π/ 2 -π/ 2 ( 2 - 2 -π/ 2 q 0 4 √ c + 1 )(c - 1

 222222222220411 δ ρ 0 )(θ )dθ = 0 , we getδ ρ 0 = A h 0 + 1 2h 0 ∂ θ (h 2 1 ρ 0 ) -(h 1 ∂ θ h 1 )ρ 0 . (δ ρ 0 )(θ )dθ = 0 we obtain A = 0 and δ ρ 0 = σ 2 √ c π sin(θ ) cos 3 (θ ) h (c -1) cos 5 (θ ) sin(θ ) h 3 0 (θ ).Hence, we get thatλ z (ε) = ε r(θ )ρ 0 (θ ) + h 3 (θ )δ ρ 0 (θ )) dθ + o(ε 2 ) .By the change of variable x = tan(θ ) we get π/(θ )δ ρ 0 (θ )dθ = σ 2 (

STUDY OF THE CASE c < 0 If c < 0 then c c- 1 ∈

 01 (0, 1) and the function h 0 vanishes on[-π/2, π/2) if and only if θ ∈ [-π/2, π/2) is solution of sin 2 (θ ) = c c -1 .

Y

  (x, y) = (p(y)p(x)) ∂ p(y) -∂ p(x) -(p(y)p(x)) ∂ p(y) -∂ p(x) .

2 , 2 , 2 ,

 222 [0, T ] × R × T is the solution of the following linear Boltzmann's equation with interface at u = 0∀u ̸ = 0, ∂ t W (t, u, k) + v 0 (k) 2π ∂ u W (t, u, k) = T R k, k ′ W t, u, k ′ -W (t, u, k) dk ′ , (V.4)where v 0 is the derivative of ω 0 . At the interface u = 0, W satisfies∀k ∈ 0, 1 W t, 0 + , k = p -(k)W t, 0 + , -k + p + (k)W t, 0 -, k + T g(k) , (V.5) ∀k ∈ -1 2 , 0 , W t, 0 -, k = p -(k)W t, 0 -, -k + p + (k)W t, 0 + , k + T g(k) , (V.6)where for any k ̸ = 0p + (k) = cos(πk) cos(πk) + γ p -(k) = γ cos(πk) + γ and g(k) = 2γ cos(πk) cos(πk) + γ . (V.7)Here p + (k), p -(k) and g(k) are the probabilities of the phonon with wave number k to be reflected, transmitted or created respectively. Observe that p + (k) + p -(k) + g(k) = 1 .

p

  ± = lim k→0 p ± (k) , g = lim k→0 g(k) and h(u) = 1 |u| 5/2 . (V.11)Using Eq. (V.7) we obtain thatp + = 1 1 + γ and p -= γ 1 + γ and g = 2γ (1 + γ) 2 .

2 {

 2 k) -iω 0 (k) q2 (k))} , p1 (k) -iω 0 (k) q1 (k) -i ( p2 (k) -iω 0 (k) q2 (k))} . (V.15)Let J : R + → R and F γ : R + → C be two functions defined byJ(s) = T cos [ω 0 (k)s] dk and F γ (s) = iγB 0 2 T e -iω 0 (k)s ψ1,0 (0, k) + e iω 0 (k)s ψ2,0 (0, k) * dk .Let H + = {λ ∈ C ; Re(λ ) > 0}, for h a function defined on R + its Laplace's transform L [h] : H + :→ C is defined for any λ in H + byL [h](λ ) = ∞ 0 e -λ s h(s)ds . (V.16)Let g γ (ds) be the signed measure on (0, ∞) whose Laplace transform is given for any λ in H + by L [g γ ](λ ) = {1 + γB 0 L [J](λ )} -1 .

t 0 F 0 F 0 e

 000 γ (s) φ γ,+ (ts, k)ds , ψ2,0 (t, k) = e -iω 0 (k)t ψ2,0 (0, k) -t * γ (s) φ γ,-(ts, k)ds , (V.18)whereφ γ,+ (s, k) = s -iω 0 (k)(s-τ) g γ (dτ) .

0 e

 0 φ γ,-(s, k) = s -iω 0 (k)(s-τ) g * γ (dτ) .

.

  [START_REF] Bernardin | Hydrodynamic limit for a disordered harmonic chain[END_REF] where for any k in T,p+,0 (k) = γ 2 B 2 0 [1cos 2 (πk)] + cos 2 (πk) cos 2 (πk) + γ 2 B 2 0 and p-,0 (k) = γ 2 B 2 0 cos 2 (πk) cos 2 (πk) + γ 2 B 2 0 (k) + p-,0 (k) = 1 .

2 ∑

 2 , j (x, x + 1)) 2 + γB 0 i=1 {δ i,1 p 2 (x)δ i,2 p 1 (x)}∂ p i (x) , (V.26)whereY i, j (x, x + 1) = (p j (x + 1)p j (x))(∂ p i (x+1) -∂ p i (x) ) -(p i (x + 1)p i (x))(∂ p j (x+1) -∂ p j (x)) .

ξ x+1 = u x+1 u x = 2 -ω 2 - 2 -π 2 F 2 -π 2 {F 1 2 -γ 3 ω 3 / 2 + O(ω 5 2 )= φ -γ 1 + 1 2ω

 222222132211 ωB x -1 ξ x . (V.34)To use Furstenberg's theorem we have to compute the invariant distribution of the Markov chain (ξ x ) x∈N .To simplify the study, we make the following change of variableξ = cos[γ(ω)]tan[φ ] sin[γ(ω)] = cos [φ + γ(ω)] cos[φ ] , for φ ∈ -For small ω this change of variable is valid since E [B] ω ≥ 0.Let ω be fixed and B ∈ R then we define the function Φ(ω, B, •) andΨ(ω, B, •) for any φ in [-π 2 , π 2 ] by Φ(ω, B, φ ) = arctan ω sin(γ(ω)) (B -|E [B]|) + tan (φ + γ(ω)) , (V.36) Ψ(ω, B, φ ) = arctan tan[φ ] -ω sin[γ(ω)] (B -E [B])γ(ω) . (V.37)Observe that for any φΦ [Ψ (ω, B, φ )] = Ψ [Φ (ω, B, φ )] = φ . (V.38)Using this change of variable, Eq. (V.34) and defining for any x in N,φ x = Φ (ξ x ) we obtain tan[φ x+1 ] = ω sin[γ(ω)] (B x -E [B]) + tan [φ x + γ(ω)] . (V.39)From Eq. (V.39) we getφ x+1 = arctan ω sin [γ(ω)] (B x -E [B]) + tan [φ x + γ(ω)] . (V.40)Let F(ω, •) be the density of the invariant measure of the Markov chain (φ x ) x∈N thenF(ω, φ ) = E F (ω, Ψ(ω, B, φ )) ∂ φ Ψ(ω, B, φ ) . (V.41)One can prove, using Furstenberg's theorem[START_REF] Furstenberg | Noncommuting random products[END_REF] Theorem 8.5] that the Lyapunov exponent of (u x ) x∈N is given byλ (ω) = π (ω, φ ) log cos [φ + γ(ω)] cos[φ ] dφ . (V.42)Using a change of variable and the periodicity of F we get thatλ (ω) = π [ω, φγ(ω)] -F[ω, φ ]} log [|cos(φ )|] dφ . (V.43)This formula shows that the behavior of λ (ω) as ω goes to zero, is completely determined by the behavior of γ(ω) anf F (ω, •, •) as ω goes to zero. For small positive ω we haveγ(ω) = |E [B]|ω 1/2 + |E [B]| 3/2 ω 3/2 + O(ω 5/2 ) = γ 1 ω 1/2 + γ 2 ω 3/2 + O ω 5/2 . (V.44)Let φ in -π 2 , π 2 and B in R. We perform a Taylor expansion on both sides of the fixed point Eq. (V.41). Let us start to perform a Taylor expansion when ω goes to zero for Ψ. Using Eq. V.44 we haveΨ(ω, B, φ ) = arctan tan(φ ) -ω γ 1 ω 1/2 + γ 3 ω 3/2 + O(ω 5/2 ) (B -E [B])γ 1 ω cos 2 (φ ) γ 1 (B -E [B]) ω 1/2cos 3 (φ ) sin(φ ) B -E [B]γ + O(ω 3/2 ) .Hence, we obtain that Ψ(ω, B, φ ) = φ + Ψ 1 (B, φ )ω 1/2 + Ψ 2 (B, φ )ω + O ω 3/2 . (V.45)

F

  (ω, φ ) = F 0 (φ ) + F 1 (φ )ω 1/2 + F 2 (φ )ω + O ω 3/2 . (V.47)Denote by F and Ψ the functions defined for any positive x byF(x, •) = F x 2 , • and Ψ(x, •, •) = Ψ x 2 , •, • .Then Eq. (V.41) can be rewrittenF(x, φ ) = E F x, Ψ(x, B, φ ) ∂ φ Ψ(x, B, φ ) . (V.[START_REF] Dhar | Heat conduction in disordered harmonic lattices with energy-conserving noise[END_REF]

cos 3 2 ∂ φ cos 3

 323 (φ ) sin(φ ) , where σ 2 is the variance of (B x ) x∈N . Plugging this in (V.[START_REF] Dhar | Heat transport in low-dimensional systems[END_REF] we get that as ω → 0,λ (ω) ∼ -(φ ) sin(φ ) log | cos(φ )|dφ σ 2 E [B] ω . (V.50)Hence, we recover the Lyapunov exponent behavior obtained in Sec. IV.4.4 of Chapter IV.

  II.3.1 TWO FUNDAMENTAL PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.3.2 UNIVERSALITY CLASSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.4.1 INTRODUCTION OF A STOCHASTIC NOISE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.4.2 DIFFUSION IN NOISY ENERGY CONSERVING CHAINS . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.4.3 SUPERDIFFUSION IN NOISY MOMENTUM CONSERVING HARMONIC CHAINS . . . . . . . . . . . . . II.4.4 AN INTERPOLATION BETWEEN A DIFFUSIVE AND A SUPERDIFFUSIVE SYSTEM . . . . . . . . . . . . . II.4.5 INTRODUCTION OF A MAGNETIC FIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.4.6 CONTRIBUTION TO SUPERDIFFUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	II.3	DIFFUSION AND SUPERDIFFUSION IN ANHARMONIC SYSTEMS
	II.4	DIFFUSION AND SUPERDIFFUSION OF ENERGY IN NOISY SYSTEMS
	I.2	SPACE-TIME SCALES AND SCALING LIMITS

  2 and its natural array of filtration(G N n ) (n,N)∈N 2 .To prove Proposition III.5.2 it is sufficient to show that the array(Z N n ) (n,N)∈N2 satisfies the assumption of Proposition III.8.1. Since Ψ B N is a centered function, Eq. (III.110) is satisfied. By Proposition III.4.5, Eq. (III.111) and Eq. (III.112) are satisfied. It remains to prove Eq. (III.113) and Eq. (III.114).

	Let g in C ∞ c (R * ), we want to prove that
		⌊N α δ t⌋
	lim N→∞	E π B N

  ) .We recall that π B N is the stationary measure of the chain K B N n , I B N To prove Lemma III.5.3 we need the following result.

	u
	Lemma III.8.2
	Let u > 0 then
	∀σ > 0, lim N→∞
	This proves Eq. (III.113). From this result we deduce that Eq. (III.114) is satisfied and the proof of
	Proposition III.5.2 is complete.
	III.8.2 PROOF OF LEMMA III.5.3

n

.

Table IV .

 IV 1: Comparison of numerical values of the current for finite N and the value of the current in the thermodynamic limit for γ

		J N		J ∞
		Fixed BC Free BC Fixed BC Free BC
	N = 10, B = 1	0.179	0.160	0.179	0.158
	N = 20, B = 1	0.179	0.158	0.179	0.158
	N = 10, B = 2	0.163	0.121	0.163	0.131
	N = 20, B = 2	0.163	0.131	0.163	0.131
					2 ,

  lim denotes the expectation with respect to the white noise. It is proved in Sec. IV.5 that if we denote z(•) = (u(•), v(•)) ⊥ then the Lyapounov exponents u(•) and z(•) are equal. Namely,

	t→∞	1 t	⟨log ∥z(t)∥⟩ ,	(IV.49)
	where ⟨•⟩ λ z (ε) = lim t→∞	1 t	⟨log |u(t)|⟩ .	(IV.50)

  IV.51) where ξ (•) is a standard white noise and σ 2 the variance of the (B x ) x∈N * . Let w(•) be the stochastic process defined as follows Eq. (IV.52) looks similar to Eq. (IV.48) but to fit perfectly with Eq. (IV.48) we perform the time scaling

	w(•) = Using Eq. (IV.51) we have that w(•) is solution of the following stochastic differential equation u(•) d dt u(•) . d dt w(t) = 0 1 -ωE [B] 0 w(t) + σ ω 0 0 -1 0 t √ ω , in Eq. (IV.51) which gives by scaling invariance of white noise d 2 dt 2 ũ(t) = -E [B] ũ(t) -ω 1/4 σ ξ (t) ũ(t) , (IV.54) or equivalently for z(t) = ũt d dt ũ(t) , the equation w ũ(t) = u d dt z

t ξ (t) . (IV.

[START_REF] Durrett | Functional limit theorems for dependent variables[END_REF]

We are interested in the Lyapunov exponent of the process u(•) (or equivalently of the process w(•) as mentioned before)

λ w (ω) =

lim t→∞ 1 t ⟨log ∥w(t)∥⟩ = lim t→∞ 1 t ⟨log |u(t)|⟩ . (IV.53)

Table IV .

 IV 2: Comparison of analytical prefactor for the three cases with the numerical prefactor. For this table, N = 10 7 .

		0, 0.25)	0.986	0.0045	0.0052
		(0, 0.5)	0.999	0.0102	0.0104
		(0, 0.75)	1.0005	0.0156	0.0156
	E [B] < 0 λ (ω) ∼ |E [B] |ω 1/2	(-0.25, 0) (-0.5, 0) (-0.75, 0)	0.492 0.492 0.491	0.315 0.444 0.532	0.353 0.5 0.612
	E [B] = 0 λ (ω) = λ (σ )ω 2/3	(-0.25, 0.25) (-0.5, 0.5) (-0.75, 0.75)	0.658 0.658 0.649	0.073 0.115 0.136	0.079 0.127 0.167
	Boundary Conditions Average magnetic field T ∞ (ω)	λ (ω) Power law for the current E

  •) have the same Lyapunov exponent. By definition we have lim

	Since θ (•) is an ergodic process we obtain that
	lim t→∞	1 t	⟨log | cos [θ (t)] |⟩ = lim t→∞	1 t	π/2 -π/2	ρ

t→∞ 1 t ⟨log |u(t)|⟩ = lim t→∞ 1 t ⟨log ∥z(t)∥⟩ + lim t→∞ 1 t ⟨log | cos [θ (t)] |⟩ . (IV.64) ε (θ ) log (| cos(θ )|) dθ = 0 .

  2.3 of Chapter II, in an hydrodynamic scaling to the following Euler's equations2

	∂ t r(t, u) =	1 E [m]	∂

u p(t, u) ∂ t p(t, u) = ∂ u r(t, u) ∂ t e(t, u) = 1 E [m] ∂ u [rp] (t, u) .

These mechanical laws have been published in Philosophiae naturalis principia mathematica in 1687. This book marks the birth of classical mechanics.

This is called Joule expansion.

This is only because the system is isolated. Otherwise, some inhomogeneity in space of these functions could be observed.

An English translation of his paper can be found in[START_REF] Sharp | Translation of ludwig boltzmann's paper "on the relationship between the second fundamental theorem of the mechanical theory of heat and probability calculations regarding the conditions for thermal equilibrium" sitzungberichte der kaiserlichen akademie der wissenschaften[END_REF].

We refer the interested reader on this topic to[START_REF] Elwood | A mathematical theory of communication[END_REF] where Shannon makes the link between the theory of communication and the entropy.

The advantage to consider particles living in T 3 instead of R 3 or a bounded domain of R 3 is that particles evolve in a compact manifold without boundaries.

Mathematically, we send ε to 0 by assuming that the number of particles N := N(ε) = ε -2 .

In the formula, if φ (v) = v, the term vφ (v) has to be interpreted as v ⊗ v.

The ergodic assumption required in their work is that the regular translation invariant in time and space probability measures of the infinite dynamics (N = ∞) are mixture of Gibbs measures.

To simplify the notation.

Heat diffusion equation is often considered with a constant thermal conductivity. Here we call heat diffusion equation the (potentially nonlinear) equation governing the time evolution of the temperature.

Observe that for any time t, Φ(t, •) is the Fourier's transform of p N x (t, •).

At this time the existence of atoms was not fully accepted and this motion was therefore unexpected.

We refer the impatient reader to[START_REF] Mellet | Fractional diffusion limit for collisional kinetic equations[END_REF] for more information on the fractional diffusion equation.

To be honest, the proof of[START_REF] Goudon | Homogenization and diffusion asymptotics of the linear boltzmann equation[END_REF] is more general.

For each v, p(v, •) could not be a probability density function, if it is the case we divide it by its integral to get one.

Here E (1) is an exponential law which means that for anyA ∈ B (R), P [E (1) ∈ A] = A exp (-x) 1 x≥0 (x)dx.

We assume here Λ finite but later we will consider also Λ infinite.

We may also consider atoms evolving in R n , n ≥ 1.

If the chain is pinned we have only to consider energy instead of the three conserved quantities.

Here for any d ≥ 1, T d N is the d-dimensional discrete torus of length N where for any i in {1, . . . , d} and y in Z, we identify x and x + Nye i where (e 1 , . . . , e d ) is the canonical basis of R d .

Here to lighten and simplify the presentation, we decide to work on the torus to avoid boundaries terms but we will see later that we can consider the harmonic chain on others lattices.

Slightly modified to take into account the boundary conditions.

In the nonlinear case we have to replace the energy current by j x,x+1 (t) = V ′ (r(t, x))p(t, x + 1).

It means the function F depends only on the sites {y ∈ Z ; |y| ≤ k} for some k ≥ 1.

It means that τ z F(r, p) = F(τ z r, τ z p) with (τ z r)(x) = r(x + z), (τ z p)(x) = p(x + z).

In his article, Spohn studied this system for V 1 (q) = q 3 but his approach is general for anharmonic potentials.

In the previous section, we only presented the case of the one-dimensional lattice since we will consider only this case but the reader will generalize easily to a d-dimensional lattice.

The study of[START_REF] Bernardin | Stationary nonequilibrium properties for a heat conduction model[END_REF] is formal.

Here R * = R\{0}.

We refer the interested reader on this subject to[START_REF] De Saporta | Numerical methods for simulation and optimization of piecewise deterministic Markov processes: application to reliability[END_REF].

Here, e 3 is the third vector of the canonical basis of R 3 .

We recall that since the masses of the particles are equal to one, p(t, x) = v(t, x) for every x.

We decided to not write T B N in order to lighten the notations.

In 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.

As in the previous section, the case B < 0 can be studied in a similar way.

Proof : We refer the reader to Sec. III.5.2.

III.8 APPENDIX C. ASYMPTOTIC STUDY OF Y N u

Here e 3 is the third vector of the canonical basis of R 3 .

Here a matrix U, U † = U ⊥ .

In[START_REF] Hannani | Hydrodynamic limit for a disordered quantum harmonic chain[END_REF], Hannani studied the quantum version of this model.

 [START_REF] Cane | Localization effects due to a random magnetic field on heat transport in a harmonic chain[END_REF]with κ1 and κ2 two positive constants. Then the following results are proved in [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF]Theorem 3.1] and [START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF]Theorem 3] respectively. Theorem III.2 ( [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF]) Let u in R and define u N := Nu. We assume that X B (0) = K B (0), I B (0) = (k, i) with k ̸ = 0 and i in {1, 2}. Then, under P (k,i) the finite-dimensional distributions of the scaled process N -1 Z B u N (N α B •) converge weakly to the finite-dimensional distributions of a Lévy process Y B u (•) generated by -2γ κB (-∆)

where κB and α B are defined in Eq. (III.31) and Eq. (III.32) respectively.

From these results, the authors of [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF] have been able to prove the following theorem.

Theorem III. 3 ([66, 104])

)) be the solution of Eq. (III.23) with initial condition

)) defined as follows

(III. [START_REF] Casher | Heat flow in regular and disordered harmonic chains[END_REF]) 2 . Let f 0 : R → R be the real function defined by

f 0 (u, k, i)dk .

Then we have

∀t ∈]0, T ], ∀u ∈ R, lim

where ρ B is the solution of

with D B a positive constant.

Remark III. 3.3 In Eq. (III.52) we give the explicit values of the constant D B which were not given in [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF].

We summarized the results of [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF][START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF][START_REF] Saito | 5/6-superdiffusion of energy for coupled charged harmonic oscillators in a magnetic field[END_REF]

III.4 STATEMENT OF THE RESULTS

In this section, we state our main results. Theorem III.4 shows that at kinetic time scale ε -1 under assumption (III.35) there is no transition in the convergence of the Wigner's distribution. Theorem III.5 and Theorem III.6 show that a transition can be observed at some hydrodynamic time scale with the appearance of an interpolation process Y B (•) for some critical intensity of the magnetic field. We

BATTLE PLAN

The proof is divided into two steps. We first show that the finite-dimensional distributions of Z N u (•) converge weakly under P π B N to the finite-dimensional distributions of a Lévy process Z δ u (•) where for any positive time t and θ in R E e iθ Z δ u (t) = exp (tΦ δ (θ ) + iθ u) .

(III.59)

where ν δ is defined in Eq. (III.44). Then we prove that we can extend this result under P (k,i) when k is in T * and i is in {1, 2}.

FIRST STEP OF THE PROOF

We start to show the first step, we follow the strategy of [START_REF] Jara | Limit theorems for additive functionals of a markov chain[END_REF]Sec. 6] and only prove that one-dimensional distribution of Z N u (•) N∈N converges weakly to one-dimensional distribution of Z δ u (•). Let t > 0, we define the random integer j N (t) as follows

where we recall that for any integer N, τ N n n∈N is a sequence of i.i.d random variables independent of the Markov chain X N n n∈N and such that τ N 0 ∼ E (1) and λ B N is defined in Eq. (III.22). Let t ≥ 0, we define

Let t be a positive time, we can decompose Z N u (t) into two parts, the first one is given by all the jumps made by the particle until time N α δ t and the second part is given by the displacement made by the particle since the last jump. Hence, using the definition of Ψ B N given in Eq. (III.38) we can write

We first prove that Y N u (S N (t)) converges weakly to Z δ u (t), then we prove that Z N u (t) and Y N u (S N (t)) are close. Proposition III.5.2

Let ν δ defined in Eq. (III.45). Then, under P π B N the finite-dimensional distributions of Y N u ((•) N∈N converge weakly to the finite-dimensional distributions of a Lévy process Y δ u where for any positive time t and θ in R E e iθY u (t) = exp (2γ) -1 tΦ δ (θ ) + iθ u , (III. [START_REF] Huang | Introduction to statistical physics[END_REF] where the expression of Φ δ is recalled in Eq. (III.60).

Proof : We refer the reader to Sec. III.8.1.

To complete the proof of Theorem III.5 we need the following two lemmas.

Lemma III.5.3

Let t 0 > 0 and T ≥ t 0 then ∀ε > 0, lim

65)

Proof : We refer the reader to Sec. III.8.2.

Lemma III.5.4

For any time t and ε > 0 be fixed we have

Proof of Lemma III.5.4 : We recall only the main ideas and we refer the reader to [66, Lemma 6.2] for a complete proof. Let σ > 0 be fixed, then by Lemma III.5.3 we have

Using the stationarity of the chain

we have

This ends the proof.

We recall that if a sequence

then the convergence follows in the sense of Skorohod topology. Hence, from Lemma III.5.3, we conclude that

Using Proposition III.5.2 and Lemma III.5.3 we can conclude that

From this we deduce that there exists a sequence of increasing homeomorphisms

Then we have

SECOND STEP OF THE PROOF

To conclude the proof, it remains to extend this convergence under P (k,i) when k ̸ = 0 and i is in {1, 2}.

Since k ̸ = 0 using Eq. (III.38) we have

Hence, we deduce that

From Markov inequality we deduce that ∀ε > 0, lim

We define the stochastic process ZN u (•) for any positive time t by

Whatever the distribution of X B N 0 is, by using Eq. (III.27), we observe that X B N n n≥1 is a sequence of i.i.d random variables distributed according to π B N . Hence, since ZN u (•) depends only on X B N n n≥1 we deduce by Proposition III.5.2 that under P (k,i) the finite-dimensional distributions of ZN u (•) converge weakly to the finite-dimensional distributions of the Lévy process Z δ u (•) defined in Eq. (III.60). Using Eq. (III.68) we deduce that for any positive time t ∀ε > 0, lim

This concludes the proof.

where h B,± is defined in Eq. (III.40).

• Let δ < 1 2 , then we perform the change of variable y = xN From this we deduce that lim

This ends the proof of Proposition III.4.5.

PROOF OF LEMMA III.7.1

Since the proofs are similar we will only prove the results for (x N,B,-(r)) N∈N and r > 0. When 2δ -1 > 0 , N 2δ -1 goes to infinity when N goes to infinity hence by Eq. (III.101) the sequence (x N,B,-(r)) N∈N is not bounded. Since r > 0, (x N,B,-(r)) N∈N is positive we deduce that (x N,B,-(r)) N∈N goes to infinity when N goes to infinity. By a Taylor expansion in N in Eq. (III.95) we get Eq. (III.103).

When 2δ -1 ≤ 0, N 2δ -1 is bounded, then we deduce that the sequence (x N,B,-(r)) N∈N is bounded and has an accumulation point denoted by l B (r). By sending N to infinity in Eq. (III.101) we get that for 2δ -1 < 0 (resp. 2δ = 1), l B (r) = 0 (resp. l B (r) = x B,± (r) defined in Eq. (III.39)). By a Taylor expansion in Eq. (III.101) we obtain Eq. (III.103) and Eq. (III.105). This ends the proof of Lemma III.7.1.

III.8 APPENDIX C. ASYMPTOTIC STUDY OF Y N u

In this section, first we prove Proposition III.5.2 and then Lemma III.5.3 which allow us to prove Theorem III.5.

III.8.1 PROOF OF PROPOSITION III.5.2

To prove Proposition III.5.2 we need the following result which is adapted from [

where α is in (1, 2). Let g in C ∞ c (R * ), r ̸ = 0 and ν a Lévy measure on R * we assume that

This will be useful later on to put the heat current in the Landauer or NEGF form. Having obtained the steady state solution, we now proceed to calculate the average heat current J N in the steady state. We can compute the current at any point on the chain since the steady state value will be the same everywhere.

Let us consider the current from the left reservoir into the system. This is given by taking the steady state average ⟨•⟩ of the dot product of the velocity v(t, 1) = (v 1 (t, 1), v 2 (t, 1)) of the first particle with the force on it from the left reservoir, thus

The first term on simplification gives

and the sum of the other two terms is given by

where in the last step we used Eq. (IV.9). Adding Eq. (IV.10) and Eq. (IV.11) we finally get

where T N is the net transmission amplitude across the harmonic chain defined by

Note that in the absence of the magnetic field, B x = 0, the Green's function, G + 2 , vanishes and we recover the current due to two uncoupled oscillator chains [START_REF] Roy | Role of pinning potentials in heat transport through disordered harmonic chains[END_REF]. The magnetic field couples the two transverse modes. In fact, we see from Eq. (IV.6) that G + 2 (ω) connects the displacements of the first coordinate with the displacements of the second one and it can be interpreted as the scattering matrix for a polarized incident plane wave on the first coordinate to be scattered into a polarized wave on the second coordinate. The term involving G + 1 (ω) is the normal transmission amplitude which is attributed to scattering of the incoming plane wave without change of polarization. The combination of the two terms leads to the rotation of the polarization of the incoming plane wave.

It is interesting to note that the mathematical structure of the Green's functions obtained here is of the same form as that found for electron transport in superconducting wires [START_REF] Majeed | Equivalence of NEGF and scattering approaches to electron transport in the Kitaev chain[END_REF][START_REF] Majeed | Transport in spinless superconducting wires[END_REF]. Analogous to the scattering between transverse modes that we see here, in superconducting wires, the superconducting order causes scattering between particle and hole electronic states.

IV.2.2 GREEN'S FUNCTION AS PRODUCT OF MATRICES

In this section, we rewrite the components of the two Green's functions, G + 1 (ω) 1,N and G + 2 (ω) 1,N defined by Eq. (IV.8), as a product of matrices using a transfer matrix approach. This will give us explicit expressions for the two components enabling us to obtain analytic closed-form results in special

The case c = 0 is more delicate. Since h 0 (•) =sin 2 (•), the unique invariant measure for the deterministic dynamical system is δ 0 (stable) and we expect that ρ ε (θ )dθ → δ 0 as ε → 0. Observe however that h 3 (0) = 0 so that we have to find the first correction to the approximation of ρ ε to δ 0 . Due to the singularity of the Dirac mass we cannot perform an expansion analysis in ε. Hence, we will use another argument to get item (i). Consider the following linear transformation T ε where

which is such that ε 2/3 ∥z∥ ≤ ∥T ε z∥ ≤ ∥z∥ for any z in R 2 and ε ≤ 1. This implies that z(•) and T ε z(•) have the same Lyapunov exponent. Expressing as we did before

which implies by scaling invariance of the white noise that θt = α tε 2/3 where

If ρ(α)dα is the unique invariant measure for (α t ) t≥0 we have by a scaling argument that the Lyapunov exponent satisfies

where h 3 and r are defined respectively in Eq. (IV.61) and Eq. (IV.62) with c = 0. To obtain the value of λ (σ ) it is sufficient to find ρ which is the unique normalisable function of the Fokker-Planck equation associated to the process α(•), i.e.

where Ẑ is the normalisation constant making ρ a probability measure.

V. OPEN PROBLEMS CHAPTER STRUCTURE

In this chapter, we present some open problems on superdiffusion of energy and localization phenomena in harmonic chain of coupled oscillators. This chapter contains few rigorous results and should be seen as a trailer for future research projects.

V.1 SUPERDIFFUSION OF ENERGY

V.1.1 TRANSITION IN ONE STEP BETWEEN TWO FRACTIONAL LAPLACIANS

Let us consider the system studied in Chapter III, i.e. the noisy harmonic chain of oscillators submitted to a magnetic field of intensity B. In Chapter III, we saw that depending on the intensity B of the magnetic field, the energy satisfies either a fractional diffusion equation with exponent 3/4 (when B = 0) or 5/6 (when B ̸ = 0) or an interpolation equation. Our argue was made by the usual two steps limits, first a kinetic limit and then a hydrodynamic one. The hydrodynamic limit in one-step has been done by Jara, Komorowski and Olla in [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF] when the intensity of the magnetic field is zero. We believe that following their study we can perform the hydrodynamic limit in one-step when the microscopic system is submitted to a magnetic field of intensity Bε δ where ε is the scaling parameter and δ a positive number. Depending on the value of δ , we should obtain either a fractional diffusion equation with exponent 3/4 or 5/6 or an interpolation equation.

V.1.2 TRANSITION IN TWO STEPS BETWEEN TWO FRACTIONAL LAPLACIANS WITH BOUNDARY CONDITIONS

In this section, we describe an ongoing research project where the aim is to obtain, using a hydrodynamic limit in two steps, an interpolation between two fractional diffusion equations with boundary conditions. This study will follow the work of Komoromski, Olla, Ryzhik ans Spohn in [START_REF] Komorowski | Fractional diffusion limit for a kinetic equation with an interface[END_REF][START_REF] Komorowski | High frequency limit for a chain of harmonic oscillators with a point langevin thermostat[END_REF].

SHORT REMINDER ABOUT THE STUDY MADE IN [START_REF] Komorowski | High frequency limit for a chain of harmonic oscillators with a point langevin thermostat[END_REF][START_REF] Komorowski | Fractional diffusion limit for a kinetic equation with an interface[END_REF] In [START_REF] Komorowski | High frequency limit for a chain of harmonic oscillators with a point langevin thermostat[END_REF], Komorowski, Olla, Ryzhik, and Spohn studied the one-dimensional infinite chain of coupled harmonic oscillators with a Langevin thermostat attached at the origin at temperature T and submitted If B = 0, then we should recover Eq. (V.23). Let us defined W N B (t, u, k) = W B N 5/6 t, Nu, k , then W N B is solution for any i in {1, 2} and u ̸ = 0 of

Following the study of Komorowski, Olla and Ryzhik in [START_REF] Komorowski | Fractional diffusion limit for a kinetic equation with an interface[END_REF], it should be possible to prove that W N B converges to the solution of some fractional diffusion equation with exponent 5/6 with some boundary conditions p±,B .

Hence, we formally have two fractional diffusion equations with boundary conditions with exponent either 3/4 or 5/6. An interesting problem is to study the transition between them. Indeed, when B 0 = 0, from our results in Chapter III we know that if we put in the Boltzmann's equation a magnetic field of intensity B N := N -δ then according to the value of δ , we have either a fractional diffusion equation with exponent 3/4 or 5/6 or an interpolation process. Hence, if B 0 ̸ = 0 we should obtain the same behavior away from zero and get some boundary conditions at zero. Namely, for a magnetic field of intensity B N = BN -δ , if we define W δ B N as

where α δ is defined as in Chapter III i.e.

(V.31) Then 1. For δ < 1 2 , W δ B N should converge to the solution of the fractional diffusion equation with exponent 5/6 and probability of reflexion and transmission given by p-,B and p+,B respectively. 2. For δ > 1 2 , W δ B N should converge to the solution of the fractional diffusion equation with exponent 3/4 and probability of reflexion and transmission given by p-,0 and p+,0 respectively. 3. For δ = 1 2 , W δ B N should converge to the solution of some interpolation equation given in Chapter III with some intermediate probability of reflexion and transmission. Moreover in this interpolation equation, by sending B to infinity (resp. to zero) the interpolation function should converge to the 5/6 (resp. 3/4) fractional diffusion equation of item 1 (resp. 2 ) as we proved in Chapter III.

V.2 LOCALIZATION V.2.1 RIGOROUS STUDY OF THE HARMONIC CHAIN SUBMITTED TO A RANDOM MAGNETIC FIELD

In Chapter IV, we studied a harmonic chain submitted to a random magnetic field and we formally obtained the scaling of the heat current with the system size. We can expect that following the strategy of Ajanki and Huveneers in [START_REF] Ajanki | Rigorous scaling law for the heat current in disordered harmonic chain[END_REF] we could perform the rigorous derivation of the heat current.