Fokker-Planck equations, along with stochastic differential equations, play vital roles in physics, population modeling, game theory and optimization (finite or infinite dimensional). In this thesis, we study three topics, both theoretically and computationally, centered around them. In part one, we consider the optimal transport for finite discrete states, which are on a finite but arbitrary graph. By defining a discrete 2-Wasserstein metric, we derive Fokker-Planck equations on finite graphs as gradient flows of free energies. By using dynamical viewpoint, we obtain an exponential convergence result to equilibrium. This derivation provides tools for many applications, including numerics for nonlinear partial differential equations and evolutionary game theory. In part two, we introduce a new stochastic differential equation based framework for optimal control with constraints. The framework can efficiently solve several real world problems in differential games and Robotics, including the path-planning problem. In part three, we introduce a new noise model for stochastic oscillators. With this model, we prove global boundedness of trajectories. In addition, we derive a pair of associated Fokker-Planck equations.

Real neurons, and their networks, are far too complex to be described exactly by simple deterministic equations. Any description of their dynamics must therefore incorporate noise to some degree. It is my thesis that the nervous system is organized in such a way that its performance is optimal, subject to this constraint. I further contend that neuronal dynamics may even be enhanced by noise, when compared with their deterministic counter-parts. To support my thesis I will present and analyze three case studies. I will show how noise might (i) extend the dynamic range of mammalian cold-receptors and other cells that exhibit a temperature-dependent discharge; (ii) feature in the perception of ambiguous figures such as the Necker cube; (iii) alter the discharge pattern of single cells.

Ce manuscrit est dédié au transport relativiste cinétique sous influence de champs magnétiques, identifié comme obstacle pour la modélisation et la simulation intégrée, dans le cadre de la Fusion par Confinement Inertiel (FCI). Une réalisation importante concerne le développement d'un code déterministe de référence, 2Dx-3Dv, de type Maxwell-Fokker-Planck-Landau, permettant la prise en compte de fonctions de distribution à large degré d'anisotropie. Ce travail se situe à l'interface de l'analyse numérique, des mathématiques appliquées, et de la physique des plasmas. Un deuxième résultat marquant concerne la dérivation d'un modèle collisionel multi-échelle, pour le transport d'électrons relativistes dans la matière dense. Des processus importants sont mis en évidence pour la FCI, et une analogie est menée vis-à-vis des processus de transport collisionels connus en radiothérapie. Enfin, un modèle mésoscopique aux moments angulaires, avec fermeture entropique, a été dérivé et utilisé pour le dépôt de dose pour la radiothérapie. Des schémas numériques précis, d'ordre élevé, et robustes, ont été développé dans ce cadre. / This manuscript is dedicated to the relativistic kinetic transport, under the influence of the magnetic field, identified as a barrier for the modeling and integrated simulations, in the frame of the Inertial Confinement Fusion (ICF). An important achievement concerns the development of a deterministic, reference code, 2Dx-3Dv, of Maxwell-Fokker-Planck-Landau type, that permits the treatment of distribution functions with large anisotropy degree. This work is at the interface between the numerical analysis, applied mathematics, and plasma physics. Another milestone result concerns the derivation of a multi-scale, collisional model, for the transport of relativistic electrons in dense matter. A set of processes is demonstrated to be of importance for ICF, and an analogy is conducted with respect to well-known collisional transport processes in radiotherapy. Finally, a mesoscopic angular moment model, with entropic closure, is derived and employed for radiotherapy dose computation. High order precise and robust numerical schemes are then developed in this framework.

Made available in DSpace on 2014-06-11T19:30:54Z (GMT). No. of bitstreams: 0 Previous issue date: 2003-10-31Bitstream added on 2014-06-13T18:41:05Z : No. of bitstreams: 1 castro_grpb_dr_sjrp.pdf: 1005126 bytes, checksum: 6a95db71dd7b9909e6062d9a618753c2 (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Neste trabalho a equação de Schrödinger associada a equação de Fokker-Planck é estudada através do formalismo de supersimetria. O procedimento usado é buscar a função de onda através da determinação do superpotencial. Quando esta função não pode ser determinada exatamente, procuram-se soluções aproximadas que podem ser usadas no método variacional. Através do formalismo supersimétrico é possível construir uma hierarquia de Hamiltonianos efetivos, e deste modo determinar as autofunções aproximadas e os autovalores variacionais. A equação de Fokker-Planck é analisada envolvendo dois potenciais biestáveis unidimensionais, um simétrico e outro assimétrico. Os resultados obtidos são comparados com aqueles encontrados por outros métodos. Finalmente, um potencial com características adequadas para o estudo do enovelamento de proteína é analisado. As funções de onda e os autovalores de energia obtidos variacionalmente são utilizados para o cálculo da probabilidade de transição. Algumas quantidades dinâmicas do processo são descritas. / In this work the Schrödinger and Fokker-Planck equation are studied through the supersymmetric formalism. The method introduced here is based on an ansatz for the superpotential and it gives the analytical wave function. When this function cannot be determined exactly the formalism supplies a trial function to be used in the variational method. Using the supersymmetric formalism it is possible to build an effective hierarchy of Hamiltonians, then for a given potential, the approximated eigenfunctions and variational eigenvalues are determined. We analyze the Fokker-Planck equation with two one-dimensional bistable potential, symmetric and asymmetric. The results are compared with the values found by others methods. Finally, a potential with characteristics adapted for the study of the protein folding is analyzed. The wave functions and the eigenvalues of energy are used for the calculation of the transition probability. Some dynamic variables used in the description of the process are discussed.

Plasma, the fourth state of matter, is a gas in which a significant portion of the atoms are ionized. It is estimated that more than 99% of the material in the visible universe is in the plasma state. The process that stars, including our sun, combine atomic nuclei and produce large amounts of energy is called thermonuclear fusion. It is anticipated future energy demands will be met by large terrestrial devices harnessing the energy of nuclear fusion. A gas hot enough to produce the number of atomic collisions needed for fusion is necessarily in the plasma state. Therefore, plasmas are of great interest to researchers studying nuclear fusion. Stars are massive enough that the gravitational attraction heats and confines the plasma. Gravitational confinement cannot be used to confine fusion plasmas on Earth. Material containers cause cooling, which prevent a plasma from maintaining the high temperature needed for fusion. Fortunately plasmas have electrical properties, which allow them to be controlled by strong magnetic fields. Although serious research into controlled thermonuclear fusion began over 60 years ago, only a couple of man-made devices are even close to obtaining more energy from fusion than is put into them. One difficulty lies in understanding the physics of particle collisions. A relative few particle collisions result in the fusion of atomic nuclei, while the vast majority of collisions are understood in terms of the electrostatic force between particles. My work has been to create an a computer code, which can be executed in parallel on supercomputers, to quickly and accurately calculate the evolution of a plasma due to particle collisions. This work explains the physics and mathematics underlying our code, as well as several tests which demonstrate the code is working as expected.

L'objectif de notre thèse de doctorat est d’étudier et de décrire les propriétés chimiques et mé- caniques du moteur moléculaire F1 -ATPase. Le moteur F1 -ATPase est un moteur rotatif, d’aspect sphérique et d’environ 10 nanomètre de rayon, qui utilise l’énergie de l’hydrolyse de l’ATP comme car- burant moléculaire. Des questions fondamentales se posent sur le fonctionnement de ce moteurs et sur la quantité de travail qu’il peut fournir. Il s’agit de questions qui concernent principalement la thermodynamique des processus irréversibles. De plus, comme ce moteur est de taille nanométrique, il est fortement influencé par les fluctuations moléculaires, ce qui nécessite une approche stochastique. C’est en créant deux modéles stochastiques complémentaires de ce moteur que nous avons contribué à répondre à ces questions fondamentales. Le premier modèle discuté au chapitre 5 de la thèse, est un mod- èle continu dans le temps et l’espace, décrit par des équations de Fokker-Planck, est construit sur des résultats expérimentaux. Ce modèle tient compte d’une description explicite des fluctua- tions affectant le degré de liberté mécanique et décrit les tran- sitions entre les différents états chimiques discrets du moteur, par un processus de sauts aléatoires entre premiers voisins. Nous avons obtenus des résultats précis concernant la chimie d’hydrolyse et de synthèse de l’ATP, et pour les dépendences du moteur en les différentes variables mécaniques, à savoir, la friction et le couple de force extérieur, ainsi que la dépendence en la température. Les résultats que nous avons obtenus avec ce modèle sont en ex- cellent accord avec les observations expérimentales. Le second modèle est discret dans l’espace et continu dans le temps et est décrit dans le chapitre 6. L’analyse des résultats obtenus par simulations numériques montre que le modèle est en accord avec les observations expérimentales et il permet en outre de dériver des grandeurs thermodynamiques analytique- ment, décrites au chapitre 4, ce que le modèle continu ne permet pas. La comparaison des deux modèles révele la nature du fonction- nement du moteur, ainsi que son régime de fonctionnement loin de l’équilibre. Le second modèle a éte soumis récemment pour publication.

Für mikroskopische Teilchen, die sich durch eine überdämpfte Fockker-Planck-Gleichung beschreiben lassen, werden thermodynamische Größen definiert. Es wird ein Ausdruck für die irreversible Arbeit berechnet. Weiterhin wird ein Kreisprozess konstruiert und für diesen der Wirkungsrad am Punkt maximaler Leistung berechnet. Als Spezialfall wird dann ein Teilchen in einem harmonischen Potential betrachtet. Alle Ergebnisse stammen bereits aus einer Veröffentlichung, es werden jedoch hier alle Berechnungen angegeben.

Thesis (MSc.(Physics)--University of Pretoria, 2007. / Includes bibliographical references.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2004. / Includes bibliographical references (p. 64-65).

Este trabalho constitui-se numa monografia para ser apresentada como requisito para o título de Doutor em Ciências dentro do Programa de Pós-Graduação em Física da Universidade Federal do Rio Grande do Sul. Faz parte da pesquisa teórica desenvolvida pelo Grupo de Física de Plasmas do IF-UFRGS, e tem por objetivo participar do esforço de tornar comercialmente viável a tecnologia de fusão termonuclear como fonte geradora de energia elátrica. Conhecimento científico e tecnológico que é imprescindível para um país em desenvolvimento como o Brasil. Dentro deste contexto, o trabalho desenvolvido segue a linha de pesquisa do estudo de processos de geraçãao de correntes em tokamaks por meio de interações onda-partícula. Assim, estudamos os efeitos devido a ocorrência de transporte radial de partículas em um tokamak na eficiência de geração de corrente devido a ondas de radio freqüência (RF), na presença de barreiras internas de transporte (ITB - Internal Transport Barriers). Para isso, considerou-se dois casos: no primeiro, a geração de corrente ocorre devido somente a ondas do tipo híbrida inferior (LH - Lower Hybrid), e no segundo, a geração com ação combinada das ondas LH e do tipo eletrônica de cíclotron (EC - Electron Cyclotron). Os resultados foram obtidos por solução numérica da equação de Fokker-Planck que rege a evolução da função distribuição de elétrons. Nós consideramos que o transporte radial de partículas pode ser devido a utuações magnéticas ou eletrostáticas. Considerando ambos os tipos de utuações, a eficiência da geração de corrente mostrou um crescimento com o crescimento das utuações que originam o transporte. A dependência da eficiência da geração de corrente na profundidade e posição da barreira também foi investigada. / This work constitutes a monograph to be presented as requirement to the title of PhD at the Physics Graduate Program at Federal University of Rio Grande do Sul. It is part of a theoretical research developed by the Plasma Physics Group of the IF-UFRGS, and it have the purpose to participe on the e ort to make feasible the thermonuclear fusion technology as a source of electric energy. Scienti c and technological knowledge that are indispensable to a developing country as the Brazil. In that context, the work under development follows the research line of the study of the process of current drive in tokamaks by means of wave-particle interaction. Therefore, we studied the e ects due to the occurrence of radial transport of particles in a tokamak on the e ciency of current drive due the radio frequency waves (RF), in the presence of an Internal Transport Barrier (ITB). For this, we consider two cases: at rst, the current drive occur due to Lower Hybrid waves (LH) only, and at second case, the current drive with combined action of LH and Electron Cyclotron (EC) waves. The present results are obtained by numerical solution of the Fokker-Planck equation which rules the evolution of the electron distribution function. We assume that the radial transport of particles can be due to magnetic or to eletrostatic utuations. It considering both the utuations types, the e ciency of current drive is shown to increase with the increase of the utuations which originate the transport. The dependence of the current drive e ciency on the depth and position of the barrier is also investigated.