1,720,992 research outputs found
Mean field limit and propagation of chaos for particle system with discontinuous interaction
No full textDans cette thèse, on étudie des problèmes de propagation du chaos et de limite de champ moyen pour des modèles relatant le comportement collectif d'individus ou de particules. Particulièrement, on se place dans des cas où l'interaction entre ces individus/particules est discontinue. Le premier travail établit la propagation du chaos pour l'équation de Vlasov-Poisson-Fokker-Planck 1d. Plus précisément, on montre que la distribution des particules évoluant sur la droite des réels interagissant via la fonction signe, converge vers la solution de l'équation de VPFP 1d, en probabilité par des techniques de type grandes déviations, et en espérance par des techniques de loi des grands nombre. Dans le second travail, on étudie une variante du modèle de Cucker-Smale, où le noyau de communication est l'indicatrice d'un cône dont l'orientation dépend de la vitesse de l'individu. Une estimation de stabilité fort-faible en distance de M.K.W. est obtenue, qui implique la limite de champ moyen. Le troisième travail a consisté à introduire de la diffusion en vitesse dans le modèle précédemment cité. Cependant, il faut ajouter une diffusion tronquée afin de préserver un système dans lequel les vitesses restent uniformément bornées. Finalement, on étudie une variante de l'équation d'agrégation où l'interaction entre individus est donnée par un cône dont l'orientation dépend de la position de l'individu. Dans ce cas on peut seulement donner une estimation de stabilité fort-faible en distance W∞, et le modèle doit être posé dans un domaine borné dans le cas avec diffusion.In this thesis, we study some propagation of chaos and mean field limit problems arising in modelisation of collective behavior of individuals or particles. Particularly, we set ourselves in the case where the interaction between the individuals/particles is discontinuous. The first work establihes the propagation of chaos for the 1d Vlasov-Poisson-Fokker-Planck equation. More precisely, we show that the distribution of particles evolving on the real line and interacting through the sign function converges to the solution of the 1d VPFP equation, in probability by large deviations-like techniques, and in expectation by law of large numbers-like techniques. In the second work, we study a variant of the Cucker-Smale, where the communication weight is the indicatrix function of a cone which orientation depends on the velocity of the individual. Some weak-strong stability estimate in M.K.W. distance is obtained for the limit equation, which implies the mean field limit. The third work consists in adding some diffusion in velocity to the model previously quoted. However one must add some truncated diffusion in order to preserve a system in which velocities remain unifomrly bounded. Finally we study a variant of the aggregation equation where the interaction between individuals is also given by a cone which orientation depends on the position of the individual. In this case we are only able to provide some weak-strong stability estimate in W∞ distance, and the problem must be set in a bounded domain for the case with diffusion
On Liouville Transport Equation with Force Field in BVloc
No full textWe prove the existence and uniqueness of renormalized solutions of the Liouville equation for n particles with an interaction potential in BVloc except at the origin. This implies the existence and uniqueness of a a.e. flow solution of the associated ODE.ou
Mathematical study of quantum decoherence on a simplified cloud chamber model
No full textDans cette thèse, on reprend l'analyse effectuée par le physicien N.F. Mott dans un article ancien (1929), qui donnait une première réponse à un paradoxe soulevé à l'époque par de nombreux physiciens: l'apparition des trajectoires rectilignes pour des particules alpha dans une chambre à brouillard, alors que les noyaux radioactifs émettent des ondes alpha sphériques. Cette analyse a été reprise récemment par plusieurs mathématiciens. A leur suite, nous introduisons un modèle quantique simplifié, unidimensionnel de chambre à brouillard. Ce modèle simplifié nous permet d'étudier les effets de décohérence liés à l'émission d'une onde alpha. Nous obtenons un résultat rigoureux qui permet de justifier mathématiquement l'approximation faite par Mott, et de l'étendre au cas où l'onde inter-agit avec de nombreuses particules. Ce résultat est confirmé par nos simulations numériques. Ces dernières mettent en évidence que le phénomène de décohérence s'observe dans des conditions physiques précises : dans certains régimes on observe en effet un comportement intrinsèquement quantiqueIn this thesis, we take up the analysis carried out by the physicist N.F.Mott in an old article (1929), which gave a first response to a paradox raised at the time by many physicists: the appearance of rectilinear trajectories for alpha particles in a cloud chamber, while radioactive nuclei emit spherical alpha waves. This analysis was recently taken up by several mathematicians. Following them, we introduce a simplified, one-dimensional cloud chamber quantum model. This simplified model allows us to study the effects of decoherence linked to the emission of an alpha wave. We obtain a rigorous result which makes it possible to mathematically justify the approximation made by Mott, and to extend it to the case where the wave interacts with many particles. This result is confirmed by our numerical simulations. The latter show that the phenomenon of decoherence is observed under specific physical conditions: in certain regimes, we indeed observe an intrinsically quantum behavio
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
A new proof of the uniqueness of the flow for ordinary differential equations with BV vector fields
Full text linkWe provide in this article a new proof of the uniqueness of the flow solution to ordinary differential equations with BV vector-fields that have uniformly bounded divergence (or with divergence in L1 and nearly incompressible flow, see the text for the definition of this term). The novelty of the proof lies in the fact it does not use the associated transport equation
Mathematical analysis of kinetic models with strong magnetic field
No full textCette thèse propose une analyse mathématique des modèles cinétiques en présence d'un champ magnétique intense.L'objectif de ce projet est le développement d'outils mathématiques nécessaires à la modélisation des plasmas de fusion. Les phénomènes physiques rencontrés dans les plasmas de fusion mettent en jeu des échelles caractéristiques disparates. L'interaction entre ces ordres de grandeurs est un enjeu important et requiert une analyse multi-échelle. Il s'agit d'un problème d'homogénéisation par rapport au mouvement rapide de rotation des particules autour des lignes de champ magnétique. Nous étudions le régime du rayon de Larmor fini pour le système de Vlasov-Poisson, dans le cadre de champs magnétiques uniformes, en appliquant les méthodes de gyro-moyenne. Nous donnons l'expression explicite du champ d'advection effectif de l'équation de Vlasov, dans laquelle nous avons substitué le champ électrique auto-cohérent, via la résolution de l'équation de Poisson moyennée à l'échelle cyclotronique. Nous mettons en évidence la structure hamiltonienne du modèle limite et présentons ses propriétés : conservations de la masse, de l'énergie cinétique, de l'énergie électrique, etc.Nous généralisons ensuite cette étude dans le cadre de champs magnétiques non uniformes. Comme précédemment, les principales propriétés des modèles limites sont mises en évidence : conservations de la masse, de l'énergie, structure hamiltonienne.Nous prenons en compte également les effets collisionnels, en présence d'un champ magnétique intense. Après identification des équilibres et invariants du noyau de collision moyenné, on s'intéresse à la dérivation de modèles fluides.This thesis proposes a mathematical analysis of kinetic models in the presence of strong magnetic fields.The objective of this project is the development of mathematical tools required for modelisation of fusion plasmas. The physical phenomena encountered in fusion plasmas involve disparate characteristic scales. The interaction between these orders of magnitude is an important issue and requires a multi-scale analysis. We appeal to homogenization techniques with respect to the fast rotation motion around the magnetic field lines.We study the finite Larmor radius regime for the Vlasov-Poisson system, in the framework of uniform magnetic fields, by appealing to gyro-average methods. We indicate the explicit expression of the effective advection field entering the Vlasov equation, after substituting the self-consistent electric field, obtained by the resolution of the averaged (with respect to the cyclotronic time scale) Poisson equation. We emphasize the hamiltonian structure of the limit model and present its properties : conservation of mass, of kinetic energy, of electric energy, etc.Then we generalize this study to general magnetic shapes. As before, the main properties of the limit model are emphasized : mass and energy balances, hamiltonian structure.We also take into account the collisional effects, under strong magnetic fields. After identifying the equilibria and the invariants of the average collision operator, we inquire about fluid models
Approximation of Euler-type equations by systems of vortices
Full text linkWe prove the weak convergence for any time of a system of quasi-vortex with positive and negative signs and without any trucature of the kernel to the solution of the Euler equation. Quasi-vortex means here that the kernel has a singularity in 1/|x|α with α ≤ 1 instead of diverging in 1/|x| near the origin. We also give some bounds on the force field for the true vortex case and explain why our technic fails in this case. <br
Uniform Contractivity in Wasserstein Metric for the Original 1D Kac’s Model
No full text5 pagesInternational audienceWe study here a very popular 1D jump model introduced by Kac: it consists of velocities encountering random binary collisions at which they randomly exchange energy. We show the uniform (in ) exponential contractivity of the dynamics in a non-standard Monge-Kantorovich-Wasserstein: precisely the MKW metric of order 2 on the energy. The result is optimal in the sense that for each , the contractivity constant is equal to the spectral gap of the generator associated to Kac's dynamic. As a corollary, we get an uniform but non optimal contractivity in the MKW metric of order . We use a simple coupling that works better that the parallel one. The estimates are simple and new (to the best of our knowledge)
Mean field limit for the one dimensional Vlasov-Poisson equation
No full text16 pages, to appear in Séminaire Laurent Schwartz (2012-2013)We consider systems of particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in \cite{Tro86}. Here we shall give a simpler proof of this result, and explain why it implies the so-called "Propagation of molecular chaos". More precisely, both results will be a direct consequence of a weak-strong stability result on the one dimensional Vlasov-Poisson equation that is interesting by it own. We also prove the existence of global solutions to the particles dynamic starting from any initial positions and velocities, and the existence of global solutions to the Vlasov-Poisson equation starting from any measures with bounded first moment in velocity
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