The pseudospectral method is a family of numerical methods for the solution of differential equations based on the expansion of basis functions defined on a set of grid points. In this thesis, the relationship between the distribution of grid points and the accuracy and convergence of the solution is emphasized. The polynomial and sinc pseudospectral methods are extensively studied along with many applications to quantum and statistical mechanics involving the Fokker-Planck and Schroedinger equations. The grid points used in the polynomial methods coincide with the points of quadrature, which are defined by a set of polynomials orthogonal with respect to a weight function. The choice of the weight function plays an important role in the convergence of the solution. It is observed that rapid convergence is usually achieved when the weight function is chosen to be the square of the ground-state eigenfunction of the problem. The sinc method usually provides a slow convergence as the grid points are uniformly distributed regardless of the behaviour of the solution. For both polynomial and sinc methods, the convergence rate can be improved by redistributing the grid points to more appropriate positions through a transformation of coordinates. The transformation method discussed in this thesis preserves the orthogonality of the basis functions and provides simple expressions for the construction of discretized matrix operators. The convergence rate can be improved by several times in the evaluation of loosely bound eigenstates with an exponential or hyperbolic sine transformation. The transformation can be defined explicitly or implicitly. An explicit transformation is based on a predefined mapping function, while an implicit transformation is constructed by an appropriate set of grid points determined by the behaviour of the solution. The methodologies of these transformations are discussed with some applications to 1D and 2D problems. The implicit transformation is also used as a moving mesh method for the time-dependent Smoluchowski equation when a function with localized behaviour is used as the initial condition.

Orientador: Elso Drigo Filho / Banca: Nelson Augusto Alves / Banca: José Roberto Ruggiero / Resumo: Este trabalho explora a relação entre a equação de Fokker-Planck e a equação de Schrödinger para estudar soluções da primeira equação. O ponto de partida é o estudo do potencial de Morse, seguido pela geração de potenciais isoespectrais ao potencial de Morse, usando o formalismo de Supersimetria em Mecânica Quântica. Os potenciais quânticos isoespectrais possuem os mesmos autovalores de energia do potencial original, mas as funções de onda são distintas. Dessa forma, a probabilidade de transição resultante da equação de Fokker-Planck, que pode ser escrita como uma expansão destas funções de onda conduz a resultados diferentes daqueles obtidos para o potencial original gerando toda uma classe de resultados novos. / Abstract: This work explores the relation between the Fokker-Planck equation and the Schrödinger equation in order to study solutions for the first one. The starting point is the study of the Schrödinger equation for Morse potential. The next step is to determine the isospectral potential by using the formalism of Supersymmetric Quantum Mechanics. Quantum isospectral potentials have the same energy spectrum of the original Morse potential, but the wave functions are different. Therefore, the transition probability that results from the Fokker-Planck equation, leads to different results from those obtained for the original potential. / Mestre

The pseudospectral method is a family of numerical methods for the solution of differential equations based on the expansion of basis functions defined on a set of grid points. In this thesis, the relationship between the distribution of grid points and the accuracy and convergence of the solution is emphasized. The polynomial and sinc pseudospectral methods are extensively studied along with many applications to quantum and statistical mechanics involving the Fokker-Planck and Schroedinger equations. The grid points used in the polynomial methods coincide with the points of quadrature, which are defined by a set of polynomials orthogonal with respect to a weight function. The choice of the weight function plays an important role in the convergence of the solution. It is observed that rapid convergence is usually achieved when the weight function is chosen to be the square of the ground-state eigenfunction of the problem. The sinc method usually provides a slow convergence as the grid points are uniformly distributed regardless of the behaviour of the solution. For both polynomial and sinc methods, the convergence rate can be improved by redistributing the grid points to more appropriate positions through a transformation of coordinates. The transformation method discussed in this thesis preserves the orthogonality of the basis functions and provides simple expressions for the construction of discretized matrix operators. The convergence rate can be improved by several times in the evaluation of loosely bound eigenstates with an exponential or hyperbolic sine transformation. The transformation can be defined explicitly or implicitly. An explicit transformation is based on a predefined mapping function, while an implicit transformation is constructed by an appropriate set of grid points determined by the behaviour of the solution. The methodologies of these transformations are discussed with some applications to 1D and 2D problems. The implicit transformation is also used as a moving mesh method for the time-dependent Smoluchowski equation when a function with localized behaviour is used as the initial condition. / Science, Faculty of / Mathematics, Department of / Graduate

The primary objective of this thesis is to make a quantitative study of complex biological networks. Our fundamental motivation is to obtain the statistical dependency between modules by injecting external noise. To accomplish this, a deep study of stochastic dynamical systems would be essential. The first chapter is about the stochastic dynamical system theory. The classical estimation of invariant measures of Fokker-Planck equations is improved by the level set method. Further, we develop a discrete Fokker-Planck-type equation to study the discrete stochastic dynamical systems. In the second part, we quantify systematic measures including degeneracy, complexity and robustness. We also provide a series of results on their properties and the connection between them. Then we apply our theory to the JAK-STAT signaling pathway network.

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.

Neste trabalho introduzimos a representação de estados coerentes de fônons e desenvolvemos o formalismo para o tratamento do campo sujeito a excitações lineares. Obtemos a equação de Fokker-Planck para a função distribuição associada ao operador densidade assim como as amplitudes dos estados devidas à excitação. Como aplicação, usamos a interação fóton-núcleo para o estudo de algumas propriedades fundamentais do efeito Mössbauer. Obtivemos, de uma maneira muito simples, a energia e o momento linear transferidos ao cristal devido à absorção ou emissão de um raio gama e a relação entre o operador criação do estado coerente /β> e a função de espalhamento Fs(k, t). Mostramos claramente que o efeito Mössbauer é devido a transições de zero-fonons O efeito da temperatura sobre probabilidades de trasições e sobre medições também aparece de um modo claro / We introduce the representation of coherent states of phonons and develop the procedures to treat the field with linear excitation. We obtain the Fokker-Planck equation of the distribution function associated with the density operator as well as the amplitudes of the states due to the excitation. As an application, we use the interaction photon-nucleus in the study of some fundamental properties of Mössbauer effect. We obtain, in a very simple way, the energy and linear momentum transferred to the crystal due to absorption or emission of a gamma ray and the connection between the creation operator of the /β> coherent state and the scattering function Fs(k, t). We show clearly that Mössbauer effect is due to zero-phonon transitions. The effect of temperature on transition probabilities and measurements arises in a clear way

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.

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

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

Orientador: Elso Drigo Filho / Resumo: Tem-se como objetivo estudar a relação da equação de Fokker-Planck mapeada em uma equação tipo Schrödinger e assim usar supersimetria para resolução de alguns potenciais polinomiais encontrando sua distribuição de probabilidade P(x,t) e o tempo de passagem entre barreiras de potenciais e a partir destes dados compreender melhor o sistema físico proposto. / Abstract: The objective of this work is to study the relationship Fokker-Planck equation a Schrödinger-type equation . Thus, it is used supersymmetry for is to solve some polynomial potential in order to find the probability distribution, P (x, t), and the passage time between barriers of potential. These data prit us a better understand of the proposed physical system. / Mestre