Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

xii, 112 p. : ill. (some col.) / Biological molecular motors exist in an interesting regime of physics where momentum is unimportant and diffusive motion is large. While only exerting small forces, these motors still manage to achieve directed motion and do work. Brownian motors induce directed motion of diffusive particles and are used as models for biological and artificial molecular motors. A flashing ratchet is a Brownian motor that rectifies thermal fluctuations of diffusive particles through the use of a time-dependent, periodic, and asymmetric potential. It has been predicted that a feedback-controlled flashing ratchet has a center of mass speed as much as one order of magnitude larger than the optimal periodically flashing ratchet. We have successfully implemented the first experimental feedback ratchet and observed the predicted order of magnitude increase in velocity. We experimentally compare two feedback algorithms for small particle numbers and find good agreement with Langevin dynamics simulations. We also find that existing algorithms can be improved to be more tolerant to feedback delay times. This experiment was implemented by a scanning line optical trap system. In a bottom-up approach to understanding molecular motors, a synthetic protein-based molecular motor, the "tumbleweed", is being designed and constructed. This design uses three ligand dependent DNA repressor proteins to rectify diffusive motion of the construct along a DNA track. To predict the behavior of this artificial motor one needs to understand the binding and unbinding kinetics of the repressor proteins at a single-molecule level. An assay, similar to tethered particle motions assays, has been developed to measure the unbinding rates of these three DNA repressor proteins. In this assay the repressor is immobilized to a surface in a microchamber. Long DNA with the correct recognition sequence for one of the repressors is attached to a microsphere. As the DNA-microsphere construct diffuses through the microchamber it will sometimes bind to the repressor protein. Using brightfield microscopy and a CCD camera the diffusive motion of the microsphere can be characterized and bound and unbound states can be differentiated. This method is tested for feasibility and shown to have sufficient resolution to measure the unbinding rates of the repressor proteins. / Committee in charge: Dr. Raghu Parthasarathy, Chair; Dr. Heiner Linke, Research Advisor; Dr. Dan Steck; Dr. John Toner; Dr. Brad Nolan

xii, 122 p. : ill. (some col.) A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / Biological molecular motors achieve directed motion and perform work in an environment dominated by thermal noise and in most cases incorporate thermally driven motion into the motor process. Inspired by bio-molecular motors, many other motor systems that incorporate thermal motion have been developed and studied. These motors are broadly referred to as Brownian motors. This dissertation presents simulation studies of two particular Brownian motors, the feedback-controlled flashing ratchet and an artificial molecular motor concept, the results of which not only drive experimental considerations but also illuminate physical behaviors that may be applicable to other Brownian motors. A flashing ratchet rectifies the motion of diffusive particles using a time dependent, asymmetric potential energy landscape, and the transport speed of the ratchet can be increased if information about the particle distribution is incorporated as feedback in the time dependency of the landscape. Using a Langevin Dynamics simulation, we compare two implementations of feedback control, a discrete algorithm and a continuous algorithm, and find that the discrete algorithm is less sensitive to fluctuations in the particle distribution. We also model an experimental system with time delay and find that the continuous algorithm can be improved by adjusting the feedback criteria to react to the expected state of the system after the delay time rather than the real-time state of the system. Motivated by the desire to understand bio-molecular linear stepping motors, we present a bottom-up approach of designing an artificial molecular motor. We develop a coarse-grained Molecular Dynamics model that is used to understand physical contributions to the diffusive stepping time of the motor and discover that partially reducing the diffusional space from 3D to 1D can dramatically increase motor speed. We also develop a stochastic model based on the classical Master equation for the system and explore the sensitivity of the motor to currently undetermined experimental parameters. We find that a reduced diffusional stepping time is critical to maintain motor attachment for many successive steps and explore an experimental design effect that leads to motor misstepping. / Committee in charge: Stephen Kevan, Chairperson, Physics; Heiner Linke, Member, Physics; John Toner, Member, Physics; Raghuveer Parthasarathy, Member, Physics; Marina Guenza, Outside Member, Chemistry

Este texto apresenta alguns dos elementos básicos envolvidos em um estudo introdutório das equações diferencias estocásticas. Tais equações modelam problemas a tempo contínuo em que as grandezas de interesse estão sujeitas a certos tipos de perturbações aleatórias. Em nosso estudo, a aleatoriedade nessas equações será representada por um termo que envolve o processo estocástico conhecido como Movimento Browniano. Para um tratamento matematicamente rigoroso dessas equações, faremos uso da Integral Estocástica de Itô. A construção dessa integral é um dos principais objetivos do texto. Depois de desenvolver os conceitos necessários, apresentaremos alguns exemplos e provaremos existência e unicidade de solução para equações diferenciais estocásticas satisfazendo certas hipóteses. / This text presents some of the basic elements involved in an introductory study of stochastic differential equations. Such equations describe certain kinds of random perturbations on continuous time models. In our study, the randomness in these equations will be represented by a term involving the stochastic process known as Brownian Motion. For a mathematically rigorous treatment of these equations, we use the Itô Stochastic Integral. The construction of this integral is one of the main goals of the text. After developing the necessary concepts, we present some examples and prove existence and uniqueness of solution of stochastic differential equations satisfying some hypothesis.

Made available in DSpace on 2014-06-11T19:25:30Z (GMT). No. of bitstreams: 0 Previous issue date: 2005-04Bitstream added on 2014-06-13T20:48:05Z : No. of bitstreams: 1 bessada_dfa_me_ift.pdf: 3052096 bytes, checksum: bfe2b25d2283cf5ec06ca7dc7407c70c (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Realizamos neste trabalho uma exposição geral da Teoria do Movimento Browniano, desde suas primeiras observações, feitas no âmbito da Biologia, até sua completa descrição seundo as leis da Mecânica estatística, formulação esta efetuada por Einstein em 1905. Com base nestes princípios físicos analisamos a Teoria do Movimento Browniano de Einstein como sendo um processo estocástico, o que permite sua generalização para um processo de Lévy. Fazemos uma exposição da Teoria de Lévy, e aplicamo-la em seguida na análise de dados provenientes do índice IBOVESPA. Camparamos os resultados com as distribuições empíricas e a modelada via distribuição gaussiana, demonstrando efetivamente que a série financeira analisada apresenta um comportamento não-gaussiano. / Abstracts: We review in this work the foundations of the Theory of Brownian Motion, from the first observations made in Biology to its complete description according to the laws of Statistical Mechanics performed by einstein in 1905. Afterwards we discuss the Einstein's Theory of Brownian Motion as a stochastic process, since this connection allows its generalization to a Lévy process. After a brief review of Lévy Theory we analyse IBOVESPA data within this framework. We compare the outcomes with the empirical and gaussian distributions, showing effectively that the analyzed financial series behaves exactly as a non-gaussian stochastic process.

Made available in DSpace on 2014-06-11T19:25:31Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-02-17Bitstream added on 2014-06-13T20:14:03Z : No. of bitstreams: 1 bauke_fc_me_rcla.pdf: 1413465 bytes, checksum: 5695187aaf8a438767e3a8684e26c073 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Neste trabalho estudamos o modelo do movimento Browniano de uma partícula carregada sob a ação de campos elétrico e magnético, externos e homogêneos, no formalismo de Langevin. Calculamos a energia cinética média através do teorema da flutuação-dissipação e obtivemos uma expressão para a temperatura efetiva das partículas Brownianas em função da temperatura do reservatório e dos campos externos. Esta temperatura efetiva mostrou-se sempre maior que a temperatura do reservatório, o que explica a expressão “portadores quentes”. Estudamos essa temperatura efetiva no regime assintótico, ou seja, no estado estacionário atingido em tempos muito longos (quando comparado com o tempo de colisão) e a utilizamos para escrever as equações de transporte em semicondutores, denominadas equações de Shockley generalizadas sendo que incluem nesse caso também a ação do campo magnético. Uma aplicação direta e relevante foi a modelagem para o já conhecido efeito Gunn para portadores assumidos como Brownianos. A temperatura efetiva calculada por nós no regime transiente permitiu estudar também os efeitos do reservatório na relaxação da temperatura efetiva à temperatura terminal (de não equilíbrio e estacionária). Nossos resultados no que diz respeito ao efeito Gunn, embora seja o modelo mais simples de um portador Browniano, mostrou uma surpreendente concordância com resultados experimentais, sugerindo que modelos mais sofisticados devam incluir os elementos apresentados neste estudo / We present a Brownian model for a charged particle in a field of forces, in particular, electric and magnetic external homogeneous fields, within the Langevin formalism. We compute the average kinetic energy via the fluctuation dissipation and obtain an expression for the Brownian particle´s effective temperature. The latter is a function of the heat bath temperature and both external fields. This effective temperature is always greater than the heat bath temperature, therefore the expression “hot carriers”. This effective temperature, in the asymptotic regime, the stationary state at long times (greater than the collision time), is used to write down the transport equations for semiconductors, namely the generalized Shockley equations, now incorporating the magnetic field effect. A direct and relevant application follows: a model for the well known Gunn effect, assuming a Brownian scheme. In the transient regime the computed effective temperature also allow us to probe some features of the heat bath, as the effective temperature relaxes to its terminal stationary value. As for our results in the Gunn effect model, the simplest of all in a Brownian scheme, we obtain a surprisingly good agreement with experimental data, suggesting that more involved models should include our minimal assumptions

Este texto apresenta alguns dos elementos básicos envolvidos em um estudo introdutório das equações diferencias estocásticas. Tais equações modelam problemas a tempo contínuo em que as grandezas de interesse estão sujeitas a certos tipos de perturbações aleatórias. Em nosso estudo, a aleatoriedade nessas equações será representada por um termo que envolve o processo estocástico conhecido como Movimento Browniano. Para um tratamento matematicamente rigoroso dessas equações, faremos uso da Integral Estocástica de Itô. A construção dessa integral é um dos principais objetivos do texto. Depois de desenvolver os conceitos necessários, apresentaremos alguns exemplos e provaremos existência e unicidade de solução para equações diferenciais estocásticas satisfazendo certas hipóteses. / This text presents some of the basic elements involved in an introductory study of stochastic differential equations. Such equations describe certain kinds of random perturbations on continuous time models. In our study, the randomness in these equations will be represented by a term involving the stochastic process known as Brownian Motion. For a mathematically rigorous treatment of these equations, we use the Itô Stochastic Integral. The construction of this integral is one of the main goals of the text. After developing the necessary concepts, we present some examples and prove existence and uniqueness of solution of stochastic differential equations satisfying some hypothesis.

Submitted by Claudinei Pracidelli (cpracide@ipen.br) on 2015-06-11T17:44:49Z No. of bitstreams: 0 / Made available in DSpace on 2015-06-11T17:44:49Z (GMT). No. of bitstreams: 0 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Dissertação (Mestrado em Tecnologia Nuclear) / IPEN/D / Instituto de Pesquisas Energeticas e Nucleares - IPEN-CNEN/SP / FAPESP:13/05492-9

Le premier chapitre de cette thèse introduit les différentes notions que nous utiliserons et présente les travaux qui constituent ce mémoire.Dans le deuxième chapitre de cette thèse nous donnons une construction ainsi que les principales propriétés de l'intégrale stochastique par rapport au mBm harmonisable. Y sont également établies des formules d'Itô et une formule de Tanaka pour l'intégrale stochastique par rapport à ce mBm..Dans le troisième chapitre nous donnons une nouvelle définition, à la fois plus simple et plus générale, du mouvement brownien multifractionnaire. Nous montrons ensuite que le mBm apparaît naturellement comme limite de suite de somme de mouvement brownien fractionnaire (fBm) d’indices de Hurst différents.Nous appliquons alors cette idée pour tenter de construire une intégrale stochastique par rapport au mouvement brownien multifractionnaire à partir d’intégrales par rapport au fBm. Cela fait nous appliquons cette définition d’intégrale par rapport au mBm pour une méthode d’intégration donnée aux deux méthodes que sont le calcul de Malliavin et la théorie du bruit blanc.Dans ce dernier cas nous comparons alors l’intégrale ainsi construite à celle obtenue au chapitre 2. Le quatrième et dernier chapitre est une application du calcul stochastique développé dans les chapitres précédents. Nous y proposons un modèle à volatilité multifractionnaire où le processus de volatilité est dirigée par un mBm. L’intérêt résidant dans le fait que l’on peut ainsi prendre en compte à la fois la dépendance à long terme des accroissements de la volatilité mais aussi le fait que la trajectoire de ces accroissements varie au cours du temps.Utilisant alors la théorie de la quantification fonctionnelle pour, entre autres, approximer la solution de certaines des équations différentielles stochastiques, nous parvenons à calculer le prix d’option à départ forward et implicitons ainsi une nappe de volatilité que l’on représente graphiquement pour différentes maturités. / The aim of this PhD Thesis was to build and develop a stochastic calculus (in particular a stochastic integral) with respect to multifractional Brownian motion (mBm). Since the choice of the theory and the tools to use was not fixed a priori, we chose the White Noise theory which generalizes, in the case of fractional Brownian motion (fBm) , the Malliavin calculus. The first chapter of this thesis presents several notions we will use in the sequel.In the second chapter we present a construction as well as the main properties of stochastic integral with respect to harmonizable mBm.We also give Ito formulas and a Tanaka formula with respect to this mBm. In the third chapter we give a new definition, simplier and generalier of multifractional Brownian motion. We then show that mBm appears naturally as a limit of a sequence of fractional Brownian motions of different Hurst index.We then use this idea to build an integral with respect to mBm as a limit of sum of integrals with respect ot fBm. This being done we particularize this definition to the case of Malliavin calculus and White Noise theory. In this last case we compare the integral hence defined to the one we got in chapter 2. The fourth and last chapter propose a multifractional stochastic volatility model where the process of volatility is driven by a mBm. The interest lies in the fact that we can hence take into account, in the same time, the long range dependence of increments of volatility process and the fact that regularity vary along the time.Using the functional quantization theory in order to, among other things, approximate the solution of stochastic differential equations, we can compute the price of forward start options and then get and plot the implied volatility nappe that we graphically represent.

Dans cette thèse, nous examinons les propriétés de régularité locale de certains processus stochastiques multiparamètres définis sur RN + , sur une collection d’ensembles, ou encore sur des fonctions de L2. L’objectif est d’étendre certains outils standards de la théorie des processus stochastiques, en particulier concernant la régularité hölderienne locale, à des ensembles d’indexation qui ne sont pas totalement ordonnés. Le critère de continuité de Kolmogorov donne classiquement une borne inférieure pour la régularité hölderienne d’un processus stochastique indicé par un sous-ensemble de R ou RN . Tirant partie de la structure de treillis des ensembles d’indexations dans la théorie des processus indicés par des ensembles de Ivanoff et Merzbach, nous étendons le critère de Kolmogorov dans ce cadre. Différents accroissements pour les processus indicés par des ensembles sont considérés, et leur sont attachés en conséquence des exposants de Hölder. Pour les processus gaussiens, ces exposants sont, presque surement et uniformément le long des trajectoires, déterministes et calculés en fonction de la loi des accroissements du processus. Ces résultats sont appliqués au mouvement brownien fractionnaire set-indexed, pour lequel la régularité est constante. Afin d’exhiber un processus pour lequel la régularité n’est pas constante, nous utilisons la structure d’espace de Wiener abstrait pour introduire un champ brownien fractionnaire indicé par (0, 1=2]_L2(T,m), relié à une famille de covariances kh, h 2 (0, 1=2]. Ce formalisme permet de décrire un grand nombre de processus gaussiens fractionnaires, suivant le choix de l’espacemétrique (T,m). Il est montré que la loi des accroissements d’un tel champ est majorée par une fonction des accroissements en chacun des deux paramètres. Les techniques développées pour mesurer la régularité locale s’appliquent alors pour prouver qu’il existe dans ce cadre des processus gaussiens indicés par des ensembles ou par L2 ayant une régularité prescrite. La dernière partie est consacrée à l’étude des singularités produites par le processus multiparamètre défini par kh sur L2([0, 1]_,dx). Ce processus est une extension naturelle du mouvement brownien fractionnaire et du drap brownien. Au point origine de RN+, ce mouvement brownien fractionnaire multiparamètre possède une régularité hölderienne différente de celle observée en tout autre point qui ne soit pas sur les axes. Une loi du logarithme itéré de Chung permet d’observer finement cette différence. / In this thesis, local regularity properties of some multiparameter, set-indexed and eventually L2-indexed random fields are investigated. The goal is to extend standard tools of the theory of stochastic processes, in particular local Hölder regularity, to indexing collection which are not totally ordered.The classic Kolmogorov continuity criterion gives a lower estimate of the Hölder regularityof a stochastic process indexed by a subset of R or RN . Using the lattice structure of the indexing collections in the theory of set-indexed processes of Ivanoff and Merzbach, Kolmogorov’scriterion is extended to this framework. Different increments for set-indexed processes are considered,and several Hölder exponents are defined accordingly. For Gaussian processes, these exponents are, almost surely and uniformly along the sample paths, deterministic and related to the law of the increments of the process. This is applied to the set-indexed fractional Brownian motion, for which the regularity is constant. In order to exhibit a process having a variable regularity,we resorted to structures of Abstract Wiener Spaces, and defined a fractional Brownian field indexed by a product space (0, 1=2]_L2(T,m), based on a family of positive definite kernels kh, h 2 (0, 1=2]. This field encompasses a large class of existing multiparameter fractional Brownian processes, which are exhibited by choosing appropriate metric spaces (T,m). It is proven that the law of the increments of such a field is bounded above by a function of the increments in both parameters of the field. Applying the techniques developed to measure the local Hölder regularity, it is proven that this field can lead to a set-indexed, or L2-indexed, Gaussian process with prescribed local regularity.The last part is devoted to the study of the singularities induced by the multiparameter process defined by the covariance kh on L2([0, 1]_,dx). This process is a natural extension of the fractional Brownian motion and of the Brownian sheet. At the origin 0 of RN+, this multiparameter fractional Brownian motion has a different regularity behaviour. A Chung (or lim inf ) law of the iterated logarithm permits to observe this.