The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of convergence towards equilibrium in the L²-distance. This result was first obtained by Bovier et. al. in 2004 relying on potential theory. The presented approach in the thesis generalizes to obtain also asymptotic sharp estimates of the constant in the logarithmic Sobolev inequality. The optimal constant in the logarithmic Sobolev inequality characterizes the convergence rate to equilibrium with respect to the relative entropy, which is a stronger distance as the L²-distance and slightly weaker than the L¹-distance. The optimal constant has here no direct spectral representation. The proof makes use of the scale separation present in the dynamics. The Eyring-Kramers formula follows as a simple corollary from the two main results of the work: The first one shows that the associated Gibbs measure restricted to a basin of attraction has a good Poincaré and logarithmic Sobolev constants providing the fast convergence of the diffusion to metastable states. The second main ingredient is a mean-difference estimate. Here a weighted transportation distance is used. It contains the main contribution to the Poincaré and logarithmic Sobolev constant, resulting from exponential long waiting times of jumps between metastable states of the diffusion.

Dans le cadre de la discrétion acoustique des navires militaires, cette thèse contribue à la modélisation de structures amorties par des matériaux viscoélastiques, sous forme de couches contraintes. Afin de pouvoir optimiser et dimensionner la structure et le matériau amortissant, il est souhaitable de disposer d'un outil numérique prédictif et efficace. Dans un premier temps, une stratégie de caractérisation et de modélisation du comportement de matériaux viscoélastiques est proposée. Elle inclue une méthode de recalage de résultats de mesures par DMA basée sur le respect des relations de Kramers-Kronig, permettant ainsi de construire des courbes maîtresses du matériau en accord avec le principe de causalité. Dans un deuxième temps, un code de calcul par éléments finis est développé, puis des essais de vibration sur structures sandwich sont réalisés afin de valider la modélisation par éléments finis de structures munies de matériaux viscoélastiques. Dans le cas d'une fine couche viscoélastique insérée dans une structure maillée par des éléments volumiques, deux éléments d'interface sont développés : ils permettent de tester simplement plusieurs configurations. Enfin, dans un troisième temps, deux catégories de méthodes de réduction adaptées au calcul de la réponse fréquentielle de structures fortement amorties par des matériaux viscoélastiques sont étudiées : les méthodes de projection sur bases modales et la méthode de reconstruction par approximants de Padé. Dans le cadre d'études paramétriques pour l'optimisation des performances acoustiques des traitements viscoélastiques contraints, les avantages des méthodes développées sont mis avant à travers deux cas d'application. / In the context of acoustic discretion of naval ships, this thesis contributes to the modelling of viscoelastically damped structures by means of thin constrained layers. In order to optimize and design the structure and the damping material, a predictive and efficient numerical tool is desirable. Firstly, a characterization and modelling strategy of the behaviour of viscoelastic materials is proposed. A shifting procedure of DMA measurements based on the fulfillment of the Kramers-Kronig relations is developed in order to build master curves of the material which are consistent with the causality principle. Secondly, a finite element code is developed, and vibration experiments are realized in order to validate the finite element modelling of structures with viscoelastic materials. In the case of thin constrained viscoelastic layers applied to a structure meshed using brick elements, two interface finite elements are developed, which facilitate parametric studies. Finally, two families of reduction methods adapted to the calculation of the frequency response of structures highly damped by viscoelastic materials are studied: modal projection methods and Padé approximants reconstruction method. The advantages of the proposed methods, in the frame of parametric studies for the optimization of the acoustic performances of constrained viscoelastic layers, are highlighted through two applications.

Cette thèse comporte un résumé avec des formules mathématiques. Vous pouvez le consulter via le texte intégral du document à la dernière page. / This thesis contains an abstract with mathematical formulae. You can consult it via the complete text of the document in the back page.

G-quadruplex forming human telomere sequence (HTS) DNA, has been widely studied due to the telomere's implied role in biological processes, including cellular ageing and cancer physiology. The goal of these previous efforts has been to characterize the physiologically relevant structures and their stability and dynamics in order to develop therapeutic applications. Unfortunately, understanding the biologically relevant form of the human telomere DNA is complicated by the fact that HTS-derived sequences are highly polymorphic. To further complicate the issue, recent investigations have demonstrated the ability of "cell-like" co-solvents to alter the preferred G-quadruplex fold of HTS DNA. However, the origins of G-quadruplex structure selection, the relative contributions of crowding versus dehydration, and the possible effects of co-solvents on kinetically determined folding pathways remain unresolved. Towards answering these questions, I investigated HTS DNA G-quadruplex in extreme anhydrous and high viscosity conditions utilizing a deep eutectic solvent (DES) consisting of choline chloride and urea. Herein I report that the water-free DES supports an extremely stable parallel stranded structure, consistent with observations that diminished water activity is the main cause of structural transitions to the "parallel-propeller" form. Furthermore, my research shows that the highly viscous nature of the solvent enables significant diffusion based control over HTS g-quadruplex folding rates and topology, fully consistent with Kramers rate theory. To the best of my knowledge, this is the first example of the kinetic exploration of G-quadruplex folding utilizing high friction solvent; the results of which display a decreased intramolecular folding rate of HTS DNA to a never before encountered time scale on the order of days at physiological temperature. Moreover, I have demonstrated that the folding pathway of a G-quadruplex can be altered with increased solvent friction. These discoveries are important because they highlight the need to consider the viscosity when exploring the dynamics of human telomeres specifically drug binding and folding of G-quadruplexes in vivo where cellular viscosity has been reported to be as high as 140cP. Lastly, it appears that tuning solvent viscosity could prove useful to the continued study of G-quadruplex dynamics.

Made available in DSpace on 2014-06-11T19:25:32Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-03-28Bitstream added on 2014-06-13T19:53:24Z : No. of bitstreams: 1 santos_afs_me_ilha.pdf: 529659 bytes, checksum: add7c6495e005e9cdef13e0ca3baf349 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Na caracterização das propriedades ópticas em materiais, a determinação dos parâmetros ópticos como o índice de refração (n) e o coeficiente de extinção (k) são fundamentais. A análise da dispersão dos parâmetros ópticos com a freqüência fornece informações do comportamento da polarização do material com a radiação e as relações de Kramers-Kronig representam uma ferramenta básica de estudo. Neste trabalho, o principal objetivo é determinar parâmetros ópticos (n e k) e dielétricos do sistema vítreo telureto ((20-x)Li2O-xWO3-80TeO2) através da análise de espectro infravermelho pelas relações de Kramers-Kronig. Para tanto, amostras de vidro na composição x = 0, 5, 10 e 15 %mol foram sintetizadas no laboratório do grupo Vidros e Cerâmicas. As amostras foram polidas opticamente e seus espectros de refletância foram obtidos nos espectrômetros Bomem DA8 e Nexus 670 da Nicolet, na região entre 40 e 4000 cm-1. Os espectros obtidos apresentam picos de reflexão característicos observados em outros sistemas vítreos telureto, o que permitiu identificar as bandas vibracionais. As constantes ópticas foram obtidas utilizando as relações de Kramers-Kronig. Para todas as composições estudadas foi observado um aumento no índice de refração quando se aumentou a quantidade de WO3. Do espectro obtido para a constante dielétrica foi possível determinar as freqüências correspondentes aos modos ópticos transversais e longitudinais. As freqüências obtidas se ajustam com aquelas obtidas pelo modelo analítico dos quatro parâmetros semi-quânticos da constante dielétrica, principalmente para a região acima de 500 cm-1. / In the study of optical and dielectric properties of materials, the determination of complex indices of refraction and complex dielectric function is fundamental. The analysis of the dispersion of the optical parameters with the frequency provides information about the behavior of the polarization of the material with the radiation, and the Kramers-Kronig relations represent a basic tool of the study. In this work, the main objective was to determine the optical (n and k) and dielectric parameters of a tellurite glasses system ((20-x)Li2O-xWO3-80TeO2) through the infrared spectrum analysis by Kramers-Kronig dispersion relations. For this, glass samples with x=0, 5, 10, and 15 %mol were prepared at the laboratory of Glasses and Ceramics group. The glass samples were polished and their reflectance spectra were obtained in the Bomem DA8 and Nicolet Nexus 670 spectrometers, in the spectral range of 40 to 4000 cm-1. The obtained spectra present allowed us to assign the vibrational bands. The optical and dielectric constants were obtained by the Kramers-Kronig method. For all studied glass compositions an increase was observed in the real part of refractive index when the WO3 content is increased. From de obtained dielectric function it was possible to extract transversal and longitudinal optical phonons frequencies. The frequencies agree with those obtained by the four-parameter semiquantum model, mainly for the region above 500 cm-1.

En microscopie à force atomique (AFM), l'étude des échantillons est réalisée à l'aide d'une pointe montée sur un microlevier. Le coeur de la technique est la mesure de la force d'interaction pointe-surface, directement proportionnelle à la déflexion du levier. Plus généralement, la compréhension profonde des propriétés mécaniques des microstructures joue un rôle significatif dans le développement des microsystèmes électromécaniques (MEMS), ou encore de capteurs chimiques ou biologiques miniatures.<br /><br />Au delà du dispositif traditionnel de mesure de déflexion angulaire, nous avons conçu et réalisé un AFM avec une détection interférométrique différentielle (entre la base encastrée et l'extrémité libre du levier). La résolution ultime est de 10^-14 m/Hz^1/2, la mesure est de plus intrinsèquement calibrée, indifférente aux dérives thermiques lentes et sans limitation de la plage d'amplitude de la déflexion. <br /><br />Grâce à notre dispositif, nous mesurons le bruit thermique le long du levier. Une reconstruction de la forme spatiale des quatre premiers modes propres en flexion révèle un excellent accord avec le modèle de poutre de Euler-Bernoulli. Un ajustement simultané sur les quatre résonances thermiquement excitées est réalisé à l'aide d'un seul paramètre libre : la raideur du levier, qui est ainsi mesurée avec une grande précision et robustesse. <br /><br />Les fluctuations thermiques de déflexion à basse fréquence démontrent qu'un modèle d'oscillateur harmonique avec dissipation visqueuse n'est plus pertinent hors résonance. De plus, on observe des différences substantielles entre les leviers avec et sans revêtement métallique. Pour ces derniers, l'approche hydrodynamique de Sader rend compte fidèlement du comportement des fluctuations en dessous de la résonance dans l'air. La présence du revêtement introduit une deuxième source de dissipation : la viscoélasticité. Elle se manifeste comme un bruit en 1/f à basse fréquence. L'utilisation du Théorème Fluctuation-Dissipation (TFD) et des relations de Kramers-Kronig permettent une caractérisation complète de la réponse du levier à l'aide des spectres de fluctuations. Une estimation quantitative de la viscoélasticité et de sa dépendance en fréquence est notamment obtenue.

In the thesis work we investigate four problems connected with dynamical processes in condensed medium, using different techniques of equilibrium and non-equilibrium statistical mechanics. Biology is rich in dynamical events ranging from processes involving single molecule [1] to collective phenomena [2]. In cell biology, translocation and transport processes of biological molecules constitute an important class of dynamical phenomena occurring in condensed phase. Examples include protein transport through membrane channels, gene transfer between bacteria, injection of DNA from virus head to the host cell, protein transport thorough the nuclear pores etc. We present a theoretical description of the problem of protein transport across the nuclear pore complex [3]. These nuclear pore complexes (NPCs) [4] are very selective filters that monitor the transport between the cytoplasm and the nucleoplasm. Two models have been suggested for the plug of the NPC. The first suggests that the plug is a reversible hydrogel while the other suggests that it is a polymer brush. In the thesis, we propose a model for the transport of a protein through the plug, which is treated as elastic continuum, which is general enough to cover both the models. The protein stretches the plug and creates a local deformation, which together with the protein is referred to as the bubble. The relevant coordinate describing the transport is the center of the bubble. We write down an expression for the energy of the system, which is used to analyze the motion. It shows that the bubble executes a random walk, within the gel. We find that for faster relaxation of the gel, the diffusion of the bubble is greater. Further, on adopting the same kind of free energy for the brush too, one finds that though the energy cost for the entry of the particle is small but the diffusion coefficient is much lower and hence, explanation of the rapid diffusion of the particle across the nuclear pore complex is easier within the gel model. In chemical physics, processes occurring in condensed phases like liquid or solid often involve barrier crossing. Simplest possible description of rate for such barrier crossing phenomena is given by the transition state theory [5]. One can go one step further by introducing the effect of the environment by incorporating phenomenological friction as is done in Kramer’s theory [6]. The “method of reactive flux” [7, 8] in chemical physics allows one to calculate the time dependent rate constant for a process involving large barrier by expressing the rate as an ensemble average of an infinite number of trajectories starting at the barrier top and ending on the product side at a specified later time. We compute the time dependent transmission coefficient using this method for a structureless particle surmounting a one dimensional inverted parabolic barrier. The work shows an elegant way of combining the traditional system plus reservoir model [9] and the method of reactive flux [7] and the normal mode analysis approach by Pollak [10] to calculate the time dependent transmission coefficient [11]. As expected our formula for the time dependent rate constant becomes equal to the transition state rate constant when one takes the zero time limit. Similarly Kramers rate constant is obtained by taking infinite time limit. Finally we conclude by noting that the method of analyzing the coupled Hamiltonian, introduced by Pollak is very powerful and it enables us to obtain analytical expressions for the time dependent reaction rate in case of Ohmic dissipation, even in underdamped case. The theory of first passage time [12] is one of the most important topics of research in chemical physics. As a model problem we consider a particle executing Brownian motion in full phase space with an absorbing boundary condition at a point in the position space we derive a very general expression of the survival probability and the first passage time distribution, irrespective of the statistical nature of the dynamics. Also using the prescription adopted elsewhere [13] we define a bound to the actual survival probability and an approximate first passage time distribution which are expressed in terms of the position-position, velocity-velocity and position-velocity variances. Knowledge of these variances enables one to compute the survival probability and consequently the first passage distribution function. We compute both the quantities for gaussian Markovian process and also for non-Markovian dynamics. Our analysis shows that the survival probability decays exponentially at the long time, irrespective of the nature of the dynamics with an exponent equal to the transition state rate constant [14]. Although the field of equilibrium thermodynamics and equilibrium statistical mechanics are well explored, there existed almost no theory for systems arbitrarily far from equilibrium until the advent of fluctuation theorems (FTs)[15] in mid 90�s. In general, these fluctuation theorems have provided a general prescription on energy exchanges that take place between a system and its surroundings under general nonequilibrium conditions and explain how macroscopic irreversibility appears naturally in systems that obey time reversible microscopic dynamics. Based on a Hamiltonian description we present a rigorous derivation [16] of the transient state work fluctuation theorem and the Jarzynski equality [17] for a classical harmonic oscillator linearly coupled to a harmonic heat bath, which is dragged by an external agent. Coupling with the bath makes the dynamics dissipative. Since we do not assume anything about the spectral nature of the harmonic bath the derivation is valid for a general non-Ohmic bath.

A discrete analogue of the Witten Laplacian on the n-dimensional integer lattice is considered. After rescaling of the operator and the lattice size we analyze the tunnel effect between different wells, providing sharp asymptotics of the low-lying spectrum. Our proof, inspired by work of B. Helffer, M. Klein and F. Nier in continuous setting, is based on the construction of a discrete Witten complex and a semiclassical analysis of the corresponding discrete Witten Laplacian on 1-forms. The result can be reformulated in terms of metastable Markov processes on the lattice. / In dieser Arbeit wird auf dem n-dimensionalen Gitter der ganzen Zahlen ein Analogon des Witten-Laplace-Operatoren eingeführt. Nach geeigneter Skalierung des Gitters und des Operatoren analysieren wir den Tunneleffekt zwischen verschiedenen Potentialtöpfen und erhalten vollständige Aymptotiken für das tiefliegende Spektrum. Der Beweis (nach Methoden, die von B. Helffer, M. Klein und F. Nier im Falle des kontinuierlichen Witten-Laplace-Operatoren entwickelt wurden) basiert auf der Konstruktion eines diskreten Witten-Komplexes und der Analyse des zugehörigen Witten-Laplace-Operatoren auf 1-Formen. Das Resultat kann im Kontext von metastabilen Markov Prozessen auf dem Gitter reformuliert werden und ermöglicht scharfe Aussagen über metastabile Austrittszeiten.

Biological processes at the cellular level take place in heterogeneous environments, and usually involve only a small number of molecules. They tend to exhibit strong time dependent fluctuations, as a result, and are, therefore, intrinsically stochastic. The present thesis describes some of the efforts I have made during the course of my research work to develop simple, analytically tractable models of a selection of biologically-inspired problems in which this kind of stochasticity is a central ingredient. These problems are: (i) single molecule enzyme activity (ii) intermittency in single enzymes, (iii) liquids crystal dynamics (iv) modulation of electron transfer kinetics during photosynthesis, and (v) anomalous polymer translocation dynamics. All of these problems can be defined in terms of quantity that changes randomly in time because of environmental fluctuations with broad distributions of relaxation times. In this thesis I show that a generalization of a model that describes simple Brownian Motion can be used to understand many of the dynamical aspects of these problems.

The reflectivity spectrum of SbSeI crystals in the spectral range of 10 – 300 cm–1 over a wide range of temperatures (10 – 297 K) with light polarization E||c and Ec was experimentally studied. Measurements were carried out in Germany. Also experimental research was carried out with the reflectivity spectrum of SbSI crystals in the spectral range of 10-450 cm–1 with light polarization E||c over temperature range (273 – 350 K) . The spectra of optical constants and optical functions were calculated using the Kramers–Kronig (KK) technique and the method of optical parameter fitting (OP). Tables and diagrams are presented. Vibrational spectra of SbSeI crystals in harmonic approximation were theoretically investigated. The oscillation frequencies of normal vibrational modes and the amplitudes of normal coordinates along z(c) axis and in x-y plane by diagonalization of the dynamic matrix were calculated. Experimental studies of reflectivity spectrum of SbSeI crystals revealed, that there is no ferroelectric phase transition in SbSeI crystals in the temperature range of 10 - 297 K. And experimental studies of reflectivity spectrum of SbSeI crystals proved, that ferroelectric phase transition in SbSI crystals grown by us occurs at temperature TC = 293 K. SbSI crystals in range of temperatures T > TC are in paraelectric phase, and in range of temperatures T < TC are in ferroelectric phase. It was experimentally proved, that low-frequency IR mode of SbSeI, when E ׀׀ c is... [to full text]