Dans cette thèse, on étudie des processus stochastiques issus de la physique statistique. Le mouvement Brownien fractionnaire, objet central des premiers chapitres, généralise le mouvement Brownien aux cas où la mémoire est importante pour la dynamique. Ces effets de mémoire apparaissent par exemple dans les systèmes complexes et la diffusion anormale. L’absence de la propriété de Markov rend difficile l’étude probabiliste du processus. On développe une approche perturbative autour du mouvement Brownien pour obtenir de nouveaux résultats, sur des observables liées aux statistiques des extrêmes. En plus de leurs applications physiques, on explore les liens de ces résultats avec des objets mathématiques, comme les lois de Lévy et la constante de Pickands. / In this thesis, we study stochastic processes appearing in different areas of statistical physics: Firstly, fractional Brownian motion is a generalization of the well-known Brownian motion to include memory. Memory effects appear for example in complex systems and anomalous diffusion, and are difficult to treat analytically, due to the absence of the Markov property. We develop a perturbative expansion around standard Brownian motion to obtain new results for this case. We focus on observables related to extreme-value statistics, with links to mathematical objects: Levy’s arcsine laws and Pickands’ constant. Secondly, the model of elastic interfaces in disordered media is investigated. We consider the case of a Brownian random disorder force. We study avalanches, i.e. the response of the system to a kick, for which several distributions of observables are calculated analytically. To do so, the initial stochastic equation is solved using a deterministic non-linear instanton equation. Avalanche observables are characterized by power-law distributions at small-scale with universal exponents, for which we give new results.

Les propriétés magnétiques détaillées des solides peuvent être vu comme le résultat de l'interaction de plusieurs sous-systèmes: celui des spins effectifs, portant l'aimantation, celui des électrons et celui du réseau crystallin. Différents processus permettent à ces sous-systèmes d'échanger de l'énergie. Parmis ceux-ci, les phénomènes de relaxation jouent un rôle prépondérants. Cependant, la complexité de ces processus en rend leur modélisation ardue. Afin de prendre en compte ces interactions de façon abordable aux calculs, l'approche de Langevin est depuis longtemps appliquée à la dynamique d'aimantation, qui peut être vue comme la réponse collective des spins. Elle consiste à modéliser les interactions entre les trois sous-systèmes par des interactions effectives entre le sous-système d'intérêt, les spins, et un bain thermique, dont seulement la densité de probabilité constituerait une quantité pertinente. Après avoir présenté cette approche, nous verrons en quoi elle permet de bâtir une dynamique atomique de spin. Une fois son implémentation détaillée, cette méthodologie sera appliquée à un exemple tiré de la littérature et basé sur le superparamagnétisme de nanoaimants de fer. / Detailed magnetic properties of solids can be regarded as the result of the interaction between three subsystems: the effective spins, that will be our focus in this thesis, the electrons and the crystalline lattice. These three subsystems exchange energy, in many ways, in particular, through relaxation processes. The nature of these processes remains extremely hard to understand, and even harder to simulate. A practical approach, for performing such simulations, involves adapting the description of random processes by Langevin to the collective dynamics of the spins, usually called the magnetization dynamics. It consists in describing the, complicated, interactions between the subsystems, by the effective interactions of the subsystem of interest, the spins, and a thermal bath, whose probability density is only of relevance. This approach allows us to interpret the results of atomistic spin dynamics simulations in appropriate macroscopic terms. After presenting the numerical implementation of this methodology, a typical study of a magnetic device based on superparamagnetic iron monolayers is presented, as an example. The results are compared to experimental data and allow us to validate the atomistic spin dynamics simulations.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Este trabalho apresenta uma família de novos métodos numéricos euleriano-lagrangeanos localmente conservativos para leis de conservação hiperbólicas escalares. Estes métodos não utilizam soluções analíticas de problemas de Riemann e são bastante precisos na captura de saltos nas soluções. Estes métodos foram introduzidos, implementados computacionalmente e testados para leis de conservação em uma e duas dimensões espaciais. Foram consideradas as equações de Burgers e Buckley-Leverett. Nossos experimentos numéricos indicaram que os métodos são pouco difusivos e que as soluções não apresentam oscilações espúrias.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Este trabalho apresenta uma família de novos métodos numéricos euleriano-lagrangeanos localmente conservativos para leis de conservação hiperbólicas escalares. Estes métodos não utilizam soluções analíticas de problemas de Riemann e são bastante precisos na captura de saltos nas soluções. Estes métodos foram introduzidos, implementados computacionalmente e testados para leis de conservação em uma e duas dimensões espaciais. Foram consideradas as equações de Burgers e Buckley-Leverett. Nossos experimentos numéricos indicaram que os métodos são pouco difusivos e que as soluções não apresentam oscilações espúrias.