The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of convergence towards equilibrium in the L²-distance. This result was first obtained by Bovier et. al. in 2004 relying on potential theory. The presented approach in the thesis generalizes to obtain also asymptotic sharp estimates of the constant in the logarithmic Sobolev inequality. The optimal constant in the logarithmic Sobolev inequality characterizes the convergence rate to equilibrium with respect to the relative entropy, which is a stronger distance as the L²-distance and slightly weaker than the L¹-distance. The optimal constant has here no direct spectral representation. The proof makes use of the scale separation present in the dynamics. The Eyring-Kramers formula follows as a simple corollary from the two main results of the work: The first one shows that the associated Gibbs measure restricted to a basin of attraction has a good Poincaré and logarithmic Sobolev constants providing the fast convergence of the diffusion to metastable states. The second main ingredient is a mean-difference estimate. Here a weighted transportation distance is used. It contains the main contribution to the Poincaré and logarithmic Sobolev constant, resulting from exponential long waiting times of jumps between metastable states of the diffusion.

This doctoral study consists of two parts. In the first part, the quantum statistical effects on the formation process of the heavy ion fusion reactions have been investigated by using the c-number quantum Langevin equation approach. It has been shown that the quantum effects enhance the over-passing probability at low temperatures. In the second part, we have developed a simulation technique for the quantum noises which can be approximated by two-term exponential colored noise.

Quantification of stochastic or quantum systems by a joint probability density or wave function is a notoriously difficult computational problem, since the solution depends on all possible states (or realizations) of the system. Due to this combinatorial flavor, even a system containing as few as ten particles may yield as many as $10^{10}$ discretized states. None of even modern supercomputers are capable to cope with this curse of dimensionality straightforwardly, when the amount of quantum particles, for example, grows up to more or less interesting order of hundreds. A traditional approach for a long time was to avoid models formulated in terms of probabilistic functions, and simulate particular system realizations in a randomized process. Since different times in different communities, data-sparse methods came into play. Generally, they aim to define all data points indirectly, by a map from a low amount of representers, and recast all operations (e.g. linear system solution) from the initial data to the effective parameters. The most advanced techniques can be applied (at least, tried) to any given array, and do not rely explicitly on its origin. The current work contributes further progress to this area in the particular direction: tensor product methods for separation of variables. The separation of variables has a long history, and is based on the following elementary concept: a function of many variables may be expanded as a product of univariate functions. On the discrete level, a function is encoded by an array of its values, or a tensor. Therefore, instead of a huge initial array, the separation of variables allows to work with univariate factors with much less efforts. The dissertation contains a short overview of existing tensor representations: canonical PARAFAC, Hierarchical Tucker, Tensor Train (TT) formats, as well as the artificial tensorisation, resulting in the Quantized Tensor Train (QTT) approximation method. The contribution of the dissertation consists in both theoretical constructions and practical numerical algorithms for high-dimensional models, illustrated on the examples of the Fokker-Planck and the chemical master equations. Both arise from stochastic dynamical processes in multiconfigurational systems, and govern the evolution of the probability function in time. A special focus is put on time propagation schemes and their properties related to tensor product methods. We show that these applications yield large-scale systems of linear equations, and prove analytical separable representations of the involved functions and operators. We propose a new combined tensor format (QTT-Tucker), which descends from the TT format (hence TT algorithms may be generalized smoothly), but provides complexity reduction by an order of magnitude. We develop a robust iterative solution algorithm, constituting most advantageous properties of the classical iterative methods from numerical analysis and alternating density matrix renormalization group (DMRG) techniques from quantum physics. Numerical experiments confirm that the new method is preferable to DMRG algorithms. It is as fast as the simplest alternating schemes, but as reliable and accurate as the Krylov methods in linear algebra.

Lagrangian stochastic models are a largely used tool in the study of passive substances dispersion inside the Atmospheric Boundary Layer. Its application is related to the trajectory computation of thousands of particles, that numerically simulate the dispersion of suspense substances in the atmosphere. In this study, the basic concepts related to the Lagrangian stochastic modelling are presented and discussed together with its main characteristics and its computational implementation, to the study of particles dispersion in the atmosphere. In a computational experiment, the obtained results are compared with observational data from the TRACT experiment, that took place in Europe in 1992. The input data needed for the dispersion model are extracted from simulations with the numerical weather forecast model RAMS. Dispersion over Rio de Janeiro region is also tested in a second experiment. / Modelos Lagrangianos estocásticos constituem ferramenta muito utilizada no estudo da dispersão de substâncias passivas na Camada Limite Atmosférica. Sua aplicação consiste em calcular a trajetória de milhares de partículas, que simulam numericamente a dispersão de uma substância em suspensão na atmosfera. Nesta tese, são apresentados e discutidos os conceitos básicos relacionados à Modelagem Lagrangiana Estocástica de Partículas, bem como suas principais características e sua implementação computacional, para o estudo da dispersão de partículas na atmosfera. Numa experimentação computacional, comparam-se os resultados obtidos com dados observacionais provenientes do experimento TRACT, realizado na Europa em 1992. Os dados de entrada necessários ao modelo de dispersão são extraídos de simulações do modelo de previsão numérica do tempo RAMS. A dispersão sobre o Estado do Rio de Janeiro é também testada em um segundo experimento.

Dans cette thèse, nous étudions la généralisation à une infinité de particules d'un modèle hamiltonien décrivant les interactions entre une particule et son environnement. Le milieu est considéré comme une superposition continue de membranes vibrantes. Au bout d'un certain temps, tout se passe comme si la particule était soumise à une force de frottement linéaire. Les équations obtenus pour un grand nombre de particules sont proches des équations de Vlasov. Dans un premier chapitre, on montre d'abord l'existence et l'unicité des solutions puis on s'intéresse à certains régimes asymptotiques; en faisant tendre la vitesse des ondes dans le milieu vers l'infini et en redimensionnant les échelles, on obtient à la limite une équation de Vlasov, on montre que si l'on modifie en plus une fonction paramètrisant le système, on obtient l'équation de Vlasov-Poisson attractive. Dans un deuxième chapitre, on ajoute un terme de diffusion à l'équation. Cela correspond à prendre en compte une agitation brownienne et un frottement linéaire sur les particules. Le principal résultat de ce chapitre est la convergence de la distribution de particules vers une unique distribution stationnaire. On montre la limite de diffusion pour ce nouveau système en faisant tendre simultanément la vitesse de propagation vers l'infini. On obtient une équation plus simple pour la densité spatiale. Dans le chapitre 3, nous montrons la validité des équations déjà étudiées par une limite de champ moyen. Dans le dernier chapitre, on étudie l'asymptotique en temps long de l'équation décrivant l'évolution de la densité spatiale obtenue dans le chapitre 2, des résultats faibles de convergence sont obtenus / The goal of this PhD is to study a generalisation of a model describing the interaction between a single particle and its environment. We consider an infinite number of particles represented by their distribution function. The environment is modelled by a vibrating scalar field which exchanges energy with the particles. In the single particle case, after a large time, the particle behaves as if it were subjected to a linear friction force driven by the environment. The equations that we obtain for a large number of particles are close to the Vlasov equation. In the first chapter, we prove that our new system has a unique solution. We then care about some asymptotic issues; if the wave velocity in the medium goes to infinity, adapting the scaling of the interaction, we connect our system with the Vlasov equation. Changing also continuously a function that parametrizes the model, we also connect our model with the attractive Vlasov-Poisson equation. In the second chapter, we add a diffusive term in our equation. It means that we consider that the particles are subjected to a friction force and a Brownian motion. Our main result states that the distribution function converges to the unique equilibrium distribution of the system. We also establish the diffusive limit making the wave velocity go to infinity at the same time. We find a simpler equation satisfied by the spatial density. In chapter 3, we prove the validity of both equations studied in the two first chapters by a mean field limit. The last chapter is devoted to studying the large time asymptotic properties of the equation that we obtained on the spatial density in chapter 2. We prove some weak convergence results

Dans cette thèse, j'étudie des équations aux dérivées partielles qui modélisent des phénomènes biologiques.Dans la première partie, je m'intéresse à une variante des équations de Keller-Segel qui modélise la chimiotaxie des micro-organismes et vise à expliquer la façon dont des colonies bactériennes s'auto-organisent et forment, en fonction de la quantité de nutriments disponibles, différents motifs géométriques. Pour le modèle en question, je construis des solutions et j'étudie leur comportement en temps long. Je montre que certaines solutions explosent en temps fini.Dans la deuxième partie, je m'intéresse au modèle Intègre et tire avec bruit et fuite non-linéaire, une équation de type Fokker-Planck qui décrit l'activité d'un réseau de neurones. J'améliore certaines estimées sur l'existence globale et l'explosion en temps fini et je démontre des résultats pour une variante du modèle avec un délai synaptique : existence globale, comportement en temps long, recherche de solutions périodiques.Dans la troisième partie, je propose une modélisation d'abord stochastique, ensuite déterministe, pour le phénomène d'adaptation des dommages à l'ADN chez les eucaryote. Des simulations numériques sont proposées et commentées. / In this thesis, I study some partial differential equations modelling biological phenomena.In the first part, I am concerned with a variant of the Keller-Segel equations which models chemotaxis in microorganisms and aims at understanding the way they self-organise and form, depending upon the density of nutrients, different geometrical patterns. For this model, I construct solutions and I study their long time behaviour. I show that some solutions blow-up in finite time.In the second part, I study the model Nonlinear Noisy leaky integrate and fire, a Fokker-Planck type equation which describes the activity of a neural network. I upgrade some estimates on global existence and finite time blow-up and I prove results for a variant of the model in which a synaptic delay is added : global existence, long time behaviour, search of periodic solutions.In the third part, I propose a stochastic model, and then a deterministic model, for the phenomenon of adaptation to DNA damage in eukaryotes. Numerical simulations are proposed and discussed.

Les satellites sont vulnérables aux particules de forte énergie piégées dans les ceintures de Van Allen. Afin d’en assurer la protection, il est nécessaire de prédire avec précision la dynamique des électrons au sein de la magnétosphère. Dans un premier temps nous proposons une méthode originale de résolution analytique de l’équation de Fokker-Planck réduite qui modélise le transport et les pertes des électrons de la magnétosphère interne. La résolution repose sur une technique de décomposition spectrale. Si la solution analytique s’avère utile pour mettre en exergue certaines propriétés physiques des ceintures de radiation, elle est également pertinente pour valider le code numérique de résolution de l’équation de Fokker-Planck réduite, développé durant la thèse. Ce dernier nous amène à généraliser l’étude précédente en illustrant l’évolution des flux d’électrons pour diverses énergies et positions. Nous prouvons notamment que la structure des ceintures de radiation ainsi que leur temps d’évolution ne dépendent que de quelques facteurs bien choisis. Dans la perspective de reproduire un événement particulier de retour au calme après un orage magnétique, mesuré par les satellites de la NASA dédiés aux ceintures de radiation, nous sommes en mesure de simuler la précipitation des électrons dans l’atmosphère terrestre causée par les interactions avec les ondes électromagnétiques de la magnétosphère. L’utilisation de conditions bâties sur des données empiriques et spécifiques à la période en question nous permet de corroborer les flux observés. Enfin, l’influence du champ magnétique terrestre sur la dynamique des ceintures de radiation est étudiée sous divers aspects. Nous nous concentrons sur la ceinture externe pour comprendre comment les asymétries du champ magnétique, considérablement façonnées par l’activité solaire, affectent notre manière de concilier théorie et observations. Nous explorons également l’importance de certains processus diffusifs nouveaux et cachés, qui émergent à cause de l’irrégularité naturelle du champ magnétique au plus proche voisinage de la Terre. / Satellites are vulnerable to high-energy particles trapped in the Van Allen belts. To ensure their protection, it is necessary to predict properly the electron dynamics in the magnetosphere. We first propose an original method to find the analytical solution of the reduced Fokker-Planck equation that models the transport and loss of electrons in the inner magnetosphere. The resolution relies on an eigenfunction expansion approach. If the analytical solution is proven to be useful at uncovering some of the physical properties of the radiation belts, it is also relevant to validate the numerical code that solves the reduced Fokker-Planck equation, which has been developed during the PhD. The latter code is used to generalize the previous study in illustrating the evolution of the electron fluxes for various energies and locations. We demonstrate that the structure of the radiation belts as well as their dynamical timescales only depend on a few well-chosen parameters. In the perspective of reproducing a specific storm-recovery event reported by the NASA Van Allen Probes, we are able to simulate the electron scattering in the Earth’s atmosphere due to the interaction with magnetospheric electromagnetic waves. The consideration of data-driven and event-specific conditions enables us to corroborate the observed fluxes. Finally, various influences of the Earth’s magnetic field on the dynamics of the radiation belts are investigated. We focus on the outer belt to see how the magnetic field asymmetries, which are strongly shaped by solar activity, affect the way of conciliating theory and observations. We also explore the importance of new hidden diffusive processes that emerge due to the natural irregularity of the magnetic field in the closest vicinity of the Earth.

No description available.

In dieser Arbeit werden mithilfe der Fokker-Planck-Gleichung die Statistiken, vor allem die Leistungsspektren, von Punktprozessen berechnet, die von mehrdimensionalen Integratorneuronen [Engl. integrate-and-fire (IF) neuron], Netzwerken von IF Neuronen und Entscheidungsfindungsmodellen erzeugt werden. Im Gehirn werden Informationen durch Pulszüge von Aktionspotentialen kodiert. IF Neurone mit radikal vereinfachter Erzeugung von Aktionspotentialen haben sich in Studien die auf Pulszeiten fokussiert sind als Standardmodelle etabliert. Eindimensionale IF Modelle können jedoch beobachtetes Pulsverhalten oft nicht beschreiben und müssen dazu erweitert werden. Im erste Teil dieser Arbeit wird eine Theorie zur Berechnung der Pulszugleistungsspektren von stochastischen, multidimensionalen IF Neuronen entwickelt. Ausgehend von der zugehörigen Fokker-Planck-Gleichung werden partiellen Differentialgleichung abgeleitet, deren Lösung sowohl die stationäre Wahrscheinlichkeitsverteilung und Feuerrate, als auch das Pulszugleistungsspektrum beschreibt. Im zweiten Teil wird eine Theorie für große, spärlich verbundene und homogene Netzwerke aus IF Neuronen entwickelt, in der berücksichtigt wird, dass die zeitlichen Korrelationen von Pulszügen selbstkonsistent sind. Neuronale Eingangströme werden durch farbiges Gaußsches Rauschen modelliert, das von einem mehrdimensionalen Ornstein-Uhlenbeck Prozess (OUP) erzeugt wird. Die Koeffizienten des OUP sind vorerst unbekannt und sind als Lösung der Theorie definiert. Um heterogene Netzwerke zu untersuchen, wird eine iterative Methode erweitert. Im dritten Teil wird die Fokker-Planck-Gleichung auf Binärentscheidungen von Diffusionsentscheidungsmodellen [Engl. diffusion-decision models (DDM)] angewendet. Explizite Gleichungen für die Entscheidungszugstatistiken werden für den einfachsten und analytisch lösbaren Fall von der Fokker-Planck-Gleichung hergeleitet. Für nichtliniear Modelle wird die Schwellwertintegrationsmethode erweitert. / This thesis is concerned with the calculation of statistics, in particular the power spectra, of point processes generated by stochastic multidimensional integrate-and-fire (IF) neurons, networks of IF neurons and decision-making models from the corresponding Fokker-Planck equations. In the brain, information is encoded by sequences of action potentials. In studies that focus on spike timing, IF neurons that drastically simplify the spike generation have become the standard model. One-dimensional IF neurons do not suffice to accurately model neural dynamics, however, the extension towards multiple dimensions yields realistic behavior at the price of growing complexity. The first part of this work develops a theory of spike-train power spectra for stochastic, multidimensional IF neurons. From the Fokker-Planck equation, a set of partial differential equations is derived that describes the stationary probability density, the firing rate and the spike-train power spectrum. In the second part of this work, a mean-field theory of large and sparsely connected homogeneous networks of spiking neurons is developed that takes into account the self-consistent temporal correlations of spike trains. Neural input is approximated by colored Gaussian noise generated by a multidimensional Ornstein-Uhlenbeck process of which the coefficients are initially unknown but determined by the self-consistency condition and define the solution of the theory. To explore heterogeneous networks, an iterative scheme is extended to determine the distribution of spectra. In the third part, the Fokker-Planck equation is applied to calculate the statistics of sequences of binary decisions from diffusion-decision models (DDM). For the analytically tractable DDM, the statistics are calculated from the corresponding Fokker-Planck equation. To determine the statistics for nonlinear models, the threshold-integration method is generalized.

В работе реализован численный алгоритм для решения уравнения Фоккера-Планка, позволяющий получать значения первой и третьей гармоники ансамбля взаимодействующих частиц для различных амплитуд переменного поля. В формулы первой и третьей гармоники вводились функции, зависящие от параметра и восприимчивости Ланжевена, с неопределенными коэффициентами. Неопределенные коэффициенты находились методом наименьших квадратов. Выражения для функций приближались данными из численного решения уравнения Фоккера-Планка и затем минимизировались относительно неопределённых коэффициентов. Получившиеся формулы сравнивались с численным решением и с известными теориями. / In this work, a numerical algorithm for solving the Fokker-Planck equation, which allows to obtain the values of the first and third harmonics of the ensemble of interacting particles for different amplitudes of the alternating field, was implemented. The functions depending on the parameter and Langevin susceptibility with uncertain coefficients were introduced into the formulas for the first and third harmonics. The uncertain coefficients were found by the least-squares method. Expressions for the functions were approximated with data from the numerical solution of the Fokker-Planck equation and then minimized with respect to the uncertain coefficients. The resulting formulas were compared with the numerical solution and with known theories.