Cette thèse s'intéresse au vieillissement des métaux au niveau microstructural. On étudie en particulier les défauts (amas de lacunes, interstitiels ou solutés) via un modèle de dynamique d'amas (DA), qui permet de prédire l'évolution des concentrations de défauts sur des temps longs (plusieurs dizaines d'années). Ce modèle est décrit par un système d'équations différentielles ordinaires (EDOs) de très grande taille, pouvant excéder la centaine de milliards d'équations. Les méthodes numériques classiques de simulation d'EDOs ne sont alors pas efficaces pour de tels systèmes. On montre dans un premier temps que la DA est bien posée et qu'elle vérifie certaines bonnes propriétés physiques comme la conservation de la quantité de matière et la positivité de la solution. On s'intéresse également à une approximation de la DA, qui prend la forme d'une équation aux dérivées partielles, de type Fokker--Planck. On caractérise en particulier l'erreur d'approximation entre la DA et cette approximation. Dans un second temps, on introduit un algorithme de simulation de la DA. Cet algorithme est basé sur un splitting de la dynamique ainsi que sur une interprétation probabiliste des équations de la DA (sous la forme d'un processus de saut) ou de son approximation de Fokker--Planck (sous la forme d'un processus de Langevin). Le but est de réduire le nombre d'équations à résoudre et d'accélérer par conséquent les simulations. On utilise enfin cet algorithme de simulation à différents modèles physiques. On confirme l'intérêt de ce nouvel algorithme pour des modèles complexes. On montre également que cet algorithme permet d'enrichir le modèle de dynamique d'amas à moindre coût / We study ageing of materials at a microstructural level. In particular, defects such as vacancies, interstitials and solute atoms are described by a model called Cluster Dynamics (CD), which characterize the evolution of the concentrations of such defects, on period of times as long as decades. CD is a set of ordinary differential equations (ODEs), which might contain up to hundred of billions of equations. Therefore, classical methods used for solving system of ODEs are not suited in term of efficiency. We first show that CD is well-posed and that physical properties such as the conservation of matter and the preservation of the sign of the solution are verified. We also study an approximation of CD, namely the Fokker--Planck approximation, which is a partial differential equation. We quantify the error between CD and its approximation. We then introduce an algorithm for simulating CD. The algorithm is based on a splitting of the dynamics and couples a deterministic and a stochastic approach of CD. The stochastic approach interprets directly CD as a jump process or its approximation as a Langevin process. The aim is to reduce the number of equations to solve, hence reducing the computation time. We finally apply this algorithm to physical models. The interest of this approach is validated on complex models. Moreover, we show that CD can be efficiently improved thanks to the versatility of the algorithm

The present work is about measure-valued diffusion processes, which are aligned with two distinct geometries on the set of probability measures. In the first part we focus on a stochastic partial differential equation, the Dean-Kawasaki equation, which can be considered as a natural candidate for a Langevin equation on probability measures, when equipped with the Wasserstein distance. Apart from that, the dynamic in question appears frequently as a model for fluctuating density fields in non-equilibrium statistical mechanics. Yet, we prove that the Dean-Kawasaki equation admits a solution only in integer parameter regimes, in which case the solution is given by a particle system of finite size with mean field interaction. For the second part we restrict ourselves to positive probability measures on a finite set, which we identify with the open standard unit simplex. We show that Brownian motion on the simplex equipped with the Aitchison geometry, can be interpreted as a replicator dynamic in a white noise fitness landscape. We infer three approximation results for this Aitchison diffusion. Finally, invoking Fokker-Planck equations and Wasserstein contraction estimates, we study the long time behavior of the stochastic replicator equation, as an example of a non-gradient drift diffusion on the Aitchison simplex.

Voltage-driven translocation, the core concept of nanopore sensing for biomolecules, has been extensively studied in silico and in vitro over the past two decades. However, the theories of analyte capture are still not complete due to the complex dynamics resulting from the coupling of multiple physical processes such as di usion, electrophoresis, and electroosmotic flow. In this thesis, I build and design translocation simulations for analytes ranging from point-like particles to rod-like molecules and long flexible polymers. The primary goal is to test, clarify and complete the existing capture theories. For example, we revisit and revise the existing definitions of the capture radius, clarify the concept of depletion zones, and investigate the impacts of the flat field near the pore. Earlier theories of translocation underestimate the importance of the electric field out- side the nanopore. In our work, we analyze the non-equilibrium dynamics during the cap- ture process originating from the converging field lines, i.e., rod orientation and polymer deformation. We characterize the rod orientation and quantify its impact on capture time both with and without Electrohydrodynamic interactions. We investigate the polymer chain deformation and calculate the translocation time by taking the electric field outside the nanopore into account as opposed to the conventional simulation approaches. Besides nanopore sensing, there are many undiscovered possibilities for nanopore trans- location technologies. We test two proof-of-concept ideas in which we suggest to use capture and translocation to separate molecules of di erent physical properties. For example, we show how one could selectively capture particles sharing the same mobility but di erent di usion coe cients using a pulsed field. Moreover, we demonstrate that it is possible to build a ratchet using pulsed fields and a nanopore to change the concentration ratios of a polymer mixture of different sized polyelectrolytes.