Micro and nanoscale systems are rapidly evolving to improve the resolution of experimental measurements. Experiments involving such small scale devices are difficult and expensive, and the available analytical theory to describe their dynamics is idealized. The dynamics of microscopic cantilevers in fluid are complicated and include significant contributions from many sources in an actual experiment. Some examples are: complex cantilever geometries, near-wall effects, thermal and external actuation techniques, and a variety of measurement techniques. Numerical simulations are a powerful approach to describe the dynamics of micro and nanoscale systems for the precise conditions of experiment. This thesis provides a numerical approach capable of addressing these inherent challenges and quantifies the dynamics of microscopic cantilevers in fluid for experimentally relevant conditions. A thermodynamic approach based upon the fluctuation-dissipation theorem allows for the calculation of stochastic dynamics from deterministic dynamics. Using numerical simulations, the thermal motion can be described for the precise conditions of experiment. It is found that the measured dynamics of cantilevers differs depending on the quantity being measured. In particular, the dynamics of displacement and angle of the cantilever tip distribute energy differently to the higher flexural modes. The externally driven dynamics of microscale cantilevers in fluid are also considered. The driven dynamics are calculated using numerical simulations of the cantilever response to a force impulse. It is found that the driven dynamics depend upon the type of actuation in addition to the quantity measured. A comparison of the driven dynamics to the corresponding stochastic dynamics yields insight into the nature of the Brownian force acting on the cantilever. Another experimentally relevant condition is the use of cantilevers with V-shaped planforms in fluid. The resulting flow field is three-dimensional and complex in contrast to what is found for a long and slender rectangular cantilever. Despite the flow complexity, the stochastic and driven dynamics of the fundamental mode can be predicted using a two-dimensional model with an appropriately chosen length scale. An experimentally motivated magnetomotive actuation technique is investigated. Results show that this approach generates power spectra nearly equivalent to the noise spectra. Furthermore, the case of a V-shaped cantilever in fluid and oscillating in proximity of a solid boundary is investigated. In the presence of a solid surface the fluid damping and added mass of fluid on the cantilever are larger than for a cantilever far from boundaries. This results in a lower frequency and quality factor for the fundamental resonance. This can impede experimental efforts because broad peaks lack distinct features that can be used to identify experimental signals. An option to overcome the large viscous damping is to take advantage of higher modes of cantilever oscillation. The higher frequency oscillations of the higher modes generate a smaller viscous boundary layer and have a reduced added mass. As a result, the quality factor increases with increasing mode number. The frequency dependence of the fluid dynamics around a fluctuating microscale cantilever is also studied. The mass of fluid entrained by the cantilever and the viscous damping quantify the interaction of a cantilever with the surrounding fluid and are computed. Analytical expressions for these parameters are derived for moderate mode number. The techniques and findings of this thesis have broad applicability to a wide range of micro and nanotechnologies that rely upon understanding the dynamics of small scale structures in fluid. / Ph. D.

A number of microcantilever systems of fundamental importance are explored using theoretical and numerical methods to quantify and provide physical insights into the dynamics of experimentally accessible systems that include a variety of configurations and viscous fluids. It is first shown that the correlated dynamics of both a laterally and vertically offset cantilever pair can be accurately predicted by numerical simulations. This is verified by comparing the correlated dynamics yielded by numerical simulations with experimental measurement. It is also demonstrated that in order to obtain these accurate predictions, geometric details of the cantilever must be included in the numerical simulation to directly reflect the experimental cantilever. A microrheology technique that utilizes the fluctuation-dissipation theorem is proposed. It is shown that by including the frequency dependence of the fluid damping, improvements in accuracy of the predictions of the rheological properties of the surrounding fluid are observed over current techniques. The amplitude spectrum of a 2-D cantilever in a power-law fluid is studied. The resulting amplitude spectrum yielded a curve similar to an overdamped system. It is observed that the amplitude and noise spectrum yield the same qualitative response for a 2-D cantilever in a shear thinning, power-law fluid. The correlated dynamics of a tethered vertically offset cantilever pair is investigated. It is shown that for a range of stiffness ratios, which is the ratio of the spring constant of the tethering relative to the cantilever spring constant, the change in the correlated dynamics of a Hookean spring tethered cantilever pair can be seen in the presence of fluid coupling. The dynamics of a spring-mass tethered, vertically offset cantilever pair is qualitatively studied by simplifying the model to an array of springs and masses. The resulting study found that the correlated dynamics of the displacement of mass of the tethered object yielded newly observed features and characteristics. It is shown that the curve shape of the cross-correlation of the displacement of the mass of the tethered object is similar to that of the auto-correlation of the displacement of the mass representing a step forced cantilever. The cross-correlation of the displacement of the mass of the tethered object, however, is found to be significantly more dependent on the stiffness ratio than the auto-correlation of the displacement of the mass representing a cantilever for t > 0. At t = 0, it is observed that the mass of the tethered object yields the same finite value for the cross-correlation for all studied values of the stiffness ratio. This characteristic is a result of the symmetry of the studied spring-mass system. / Ph. D.

Soumise à une sollicitation sismique importante, une structure de génie civil peut être endommagée. Cela se traduit par l'apparition de fissures et donc une réduction de ses propriétés élastiques de la structure. Cependant, dans le cas d'un événement sismique d'amplitude inférieure, la variation observée est seulement transitoire. Elle consiste en une perturbation brutale de la fréquence et de l'amortissement suivie d'un recouvrement lent vers leurs valeurs initiales. Ce phénomène non linéaire de relaxation est appelé Dynamique Lente. Il s'explique par la fermeture progressive des fissures initialement présentes dans la structure et qui ont été ouvertes par la sollicitation. Nous avons observé en laboratoire que l'analyse de la Dynamique Lente dans une poutre avant et après son endommagement permet de détecter cette augmentation de la densité de fissures. La sensibilité différentielle des modes à une dégradation locale indique, de plus, une piste pour le développement d'une méthode de localisation de l'endommagement. L'étude de la Dynamique Lente que nous avons menée dans deux structures de génie civil a montré que nous pouvions également y détecter l'apparition de l'endommagement. Le suivi de l'amortissement de manière continue a mis en évidence une relation linéaire entre l'amortissement et l'intensité des vibrations ambiantes à la fois dans les poutres et les structures de génie civil. Nous expliquons celle-ci via l'application du théorème Fluctuation-Dissipation à ces systèmes. Les résultats en laboratoire et la proximité de l'expression de la relation avec les équations utilisées dans le cas de la Dynamique Lente suggère une dépendance de ce phénomène vis à vis de la densité des hétérogénéités dans le matériau composant la structure. De plus amples recherches sont cependant nécessaires afin d'expliquer complètement nos observations et ainsi pouvoir les utiliser dans le cadre de la surveillance des structures uniquement à partir de données de très faible amplitude. / During strong seismic loadings, a structure may be damaged. This results in the appearing of cracks and then a reduction of the elastic properties of the structure. The degradation remains only transitory in the case of smaller seismic events. It consists in a sharp disruption of both the frequency and damping followed by their slow recovery to their initial values. This non linear phenomenon is called Slow Dynamics. It is explained by the gradual closing of the cracks which were initially present in the material and which were opened during the loading. We observed in the laboratory that the analysis of the Slow Dynamics in a beam before and after it is damaged allows to detect the increase of the crack density. The different sensitivties of the modes regarding a local damaging indicates a track to develop a method to locate the damages. The study of the Slow Dynamics in civil engineering structures demonstrated the possibility to detect the damaging also in this kind of system. The continuous monitoring of the damping highlighted a linear relationship between damping and the intensity of the ambiante vibration in the case of both the beams and real case structures. We explain this relation by applying the Fluctuation-Dissipation to these systems. Laboratory results and the proximity of the expression of the linear relationship wit hthe equations used in the theory of the Slow Dynamics suggest a dependency of this phenonmenon on the density of heterogeneities in the structure. Further research is however required in order to fully explain our observations and thus, to use them to monitor the state of health of structures.

The problem under consideration in the thesis is a two level atom in a photonic crystal and a pumping laser. The photonic crystal provides an environment for the atom, that modifies the decay of the exited state, especially if the atom frequency is close to the band gap. The population inversion is investigated als well as the emission spectrum. The dynamics is analysed in the context of open quantum systems. Due to the multiple reflections in the photonic crystal, the system has a finite memory that inhibits the Markovian approximation. In the Heisenberg picture the equations of motion for the system variables form a infinite hierarchy of integro-differential equations. To get a closed system, approximations like a weak coupling approximation are needed. The thesis starts with a simple photonic crystal that is amenable to analytic calculations: a one-dimensional photonic crystal, that consists of alternating layers. The Bloch modes inside and the vacuum modes outside a finite crystal are linked with a transformation matrix that is interpreted as a transfer matrix. Formulas for the band structure, the reflection from a semi-infinite crystal, and the local density of states in absorbing crystals are found; defect modes and negative refraction are discussed. The quantum optics section of the work starts with the discussion of three problems, that are related to the full resonance fluorescence problem: a pure dephasing model, the driven atom and resonance fluorescence in free space. In the lowest order of the system-environment coupling, the one-time expectation values for the full problem are calculated analytically and the stationary states are discussed for certain cases. For the calculation of the two time correlation functions and spectra, the additional problem of correlations between the two times appears. In the Markovian case, the quantum regression theorem is valid. In the general case, the fluctuation dissipation theorem can be used instead. The two-time correlation functions are calculated by the two different methods. Within the chosen approximations, both methods deliver the same result. Several plots show the dependence of the spectrum on the parameters. Some examples for squeezing spectra are shown with different approximations. A projection operator method is used to establish two kinds of Markovian expansion with and without time convolution. The lowest order is identical with the lowest order of system environment coupling, but higher orders give different results. / Die Arbeit befasst sich mit der Emission eines 2-Niveau-Atoms in einem photonischen Kristall mit einem treibenden Laser. Der photonische Kristall stellt für das Atom eine Umgebung dar, die seinen Zerfall verändert, insbesondere wenn die Übergangsfrequenz des Atoms nahe an der Bandkante ist. Es werden sowohl die Besetzungen als auch das Emissionsspektrum untersucht. Die Dynamik wird im Kontext offener Quantensysteme analysiert. Durch die vielfachen Reflexionen im photonischen Kristall hat das System ein endliches Gedächtnis, das die Markov-Näherung verhindert. Im Heisenberg-Bild stellen die Bewegungsgleichungen für die Systemvariablen eineunendliche Hierachie von Integro-Differentialgleichungen dar. Um ein geschlossenes System zu erhalten, sind Näherungen wie eine schwache Kopplung nötig. Zunächst wird ein einfacher photonischer Kristall betrachtet.: Der eindimensionale photonische Kristall, der aus wechselnden Lagen besteht. Die Blochmoden innerhalb und die Vakuummoden außerhalb des endlichen photonischen Kristalls sind durch eine Transformationsmatrix, die als Transfermatrix interpretiert werden kann, miteinander verbunden. Einfache Formeln für die Bandstruktur, Reflexion eines halb-unendlichen Kristalls, die lokale Zustandsdichte im absorbierenden Kristall werden gefunden; außerdem werden Defektmoden und negative Brechung diskutiert. Im quantenoptischen Teil der Arbeit werden zu Anfang drei Probleme diskutiert, die im Zusammenhang zum Problem der Resonanzfluoreszenz stehen und die analytisch berechnet werden können: Ein Dephasierungsmodell, das getriebenen Atom und Resonanzfluoreszenz im freien Raum. In der niedrigsten Ordnung der System-Bad-Kopplung werden die Erwartungswerte analytisch berechnet und die stationären Zustände werden für bestimmte Fälle diskutiert. Bei der Berechnung der Zweizeitkorrelationsfunktion und der Spektren taucht das zusätzliche Problem der Korrelationen zwischen den beiden Zeiten auf. Im Markov-Fall gilt das Quantenregressionstheorem. Im allgemeinen Fall kann stattdessen das Fluktuations-Dissipations-Theorem benutzt werden. Die Korrelationsfunktionen werden mit zwei verschiedenen Methoden berechnet. Innerhalb der gewählten Näherungen liefern beide Methoden dasselbe Resultat. Einige Plots zeigen die Abhängigkeit des Spektrums von den verschiedenen Parametern. Mehrere Beispiele für Squeezing-Spektren werden mit den verschiedenen Näherungen gezeigt. Eine Projektions-Operator-Methode wird benutzt, um zwei Arten einer Markov-Entwicklung zu implementieren, mit und ohne Faltungsintegral. Die niedrigste Ordnung ist identisch mit der niedrigsten Ordnung der System-Bad-Kopplung, wohingegen höhere Ordnungen andere Resultate ergeben.

Le vieillissement physique est un sujet très actif en physique statistique car il ouvre la perspective d’une généralisation de la physique statistique à l’équilibre à des cas faiblement hors équilibre. Dans ce contexte où d’importants travaux ont été réalisés sur les polymères et les verres de spin, des travaux théoriques ont montré l’intérêt que pouvait apporter l’étude de système subissant une transition de phase du second ordre, les ingrédients théoriques étant plus simples. Nous avons donc étudié dans le détail la transition de Fréedericksz dans un cristal liquide qui est décrite par une transition du second ordre afin d’utiliser ensuite ce système pour faire une étude expérimentale du vieillissement au point critique. Nous avons alors montré que les équations usuellement utilisées (développement de l’énergie libre à la Landau) pour décrire la dynamique de cette transition ont un domaine de validité extrêmement mince qui n’est pas accessible expérimentalement. Il faut donc tenir compte des termes non linéaires de l’énergie libre pour décrire la dynamique du système, même dans le cadre de la réponse linéaire. Nous avons montré dans ce cadre le très bon accord entre la simulation numérique sans paramètre ajustable et les résultats expérimentaux. Nous avons ensuite étudié le comportement des fluctuations au voisinage du seuil de la transition et montré que lorsque la normalisation tenait bien compte du fait que la mesure est celle d’une variable quadratique, alors les fluctuations étaient d’amplitude maximale au seuil de la transition de Fréedericksz, comme on l’attend d’une transition du second ordre. L’étude de ces fluctuations permet alors de mesurer précisément la valeur du champ critique, ce qui est une mesure totalement nouvelle. Nous avons ensuite réalisé une étude de la dynamique des fluctuations du système lors de la réponse à une marche du paramètre de contrôle de la transition de Fréedericksz, et en particulier lors de trempes au point critique. Nous avons alors retrouvé les résultats prédits sur les systèmes de verres de spin, et nous montrons le lien entre la violation du théorème de fluctuation dissipation et l’évolution de la variance. Enfin nous avons monté un dispositif permettant de faire l’étude spatio-temporelle du système, nous avons montré que les 2 dispositifs mis en œuvre étaient limités par leur sensibilité dépendant de la valeur moyenne du paramètre d’ordre et nous proposons donc un autre système de mesure que nous n’avons pas eu le temps d’implémenter. / Physical aging is an active subject in statistical physics as it could lead to the generalization of equilibrium statistical physics to weakly out of equilibrium systems. In this context, where polymers and spin glasses have already been extensively studied despite not still well understood, theoretical works have shown new interests in systems undergoing a second order phase transition, where model ingredients are based on simpler physical arguments. Therefore, we studied in details a second order phase transition in liquid crystals : the Fréedericksz transition, in order to monitor experimentally aging at its critical point. We showed that the equations usually proposed to describe the dynamics of the transition (Landau like development of the free energy) have a very limited domain of validity, not accessible experimentally. In fact, one has to take into account the non-linearities, even in the vicinity of the Fréedericksz threshold and in the linear response regime. In this framework, we found a very good agreement between numerical simulations (no adjustable parameters) and experimental results. Then, we studied the behavior of fluctuations in the vicinity of the transition threshold and showed that, when ones takes into account the quadratic dependence of the measured variable, the fluctuations have a maximal amplitude at the threshold, as expected in a second order phase transition. This experimental study of fluctuations provides a new method to precisely measure of the value of the Fréedericksz threshold. A detailed analysis of the system dynamic fluctuations during quenches, and in particular, critical quenches was also performed, and we found the same behavior as predicted on spin glasses. The relation between fluctuation-dissipation theorem violation and variance evolution during the quench could be established. In parallel, we designed an experimental set-up to study the spatio-temporal fluctuations and also used a classical one, which both have sensitivity limits due to the mean value of the order parameter. We therefore propose a third set-up which could not be implemented due to a lack of time, that should overcome these difficulties.

Depuis quelques décennies, la technologie de contrôle de la fréquence a été au coeur de l'électronique des tempsmodernes grâce à son vaste domaine d'applications dans les systèmes de communication, les ordinateurs, les systèmesde navigation ou de défense militaire. Les dispositifs temps-fréquence fournissent des stabilités de fréquence et despuretés spectrales élevées dans le domaine de la stabilité court-terme. L'amélioration de la performance de cesdispositifs reste un grand défi pour les chercheurs. La réduction du bruit afin d'augmenter cette stabilité court-terme etd'éviter les commutations non souhaitées entre les canaux est donc très souhaitable. Il est communément admis que lalimitation fondamentale à cette stabilité court-terme est due au bruit flicker de fréquence des résonateurs. Dans cemanuscrit, un premier chapitre rappelle quelques faits de base sur l’acoustique, la cristallographie et les définitions dudomaine temps-fréquence nécessaires à l’étude des résonateurs et oscillateurs ultra-stables. Le deuxième chapitre estconsacré à un résumé de la littérature sur le bruit de fréquence en 1/f. Ensuite, le troisième chapitre concerne nos étudessur le modèle quantique de bruit en 1/f du Pr. Handel, qui, bien que critiqué par beaucoup, est encore le seul qui fournitune estimation de l'amplitude de plancher de bruit en 1/f et qui n'est pas infirmé par les données expérimentales. Dans lequatrième chapitre, une autre approche, basée sur le théorème de fluctuation-dissipation, est utilisée afin de mettre descontraintes numériques sur un modèle de bruit en 1/f causé par une dissipation interne (ou de structure) proportionnelleà l'amplitude, et non à la vitesse. Le dernier chapitre est consacré aux résultats expérimentaux. Le design et lesparamètres du résonateur ultra-stable utilisé lors de cette étude sont décrits. Les mesures de bruit de phase sur plusieurslots de résonateurs sont données. Les mesures des paramètres de résonateur ont été effectuées à basse température afinde les corréler avec les résultats de bruit. Afin d'évaluer rapidement la qualité des différents résonateurs, une autreapproche dans le domaine temporel a été testée. Elle utilise des oscillations pseudo-périodiques transitoires mettant lesoscilloscopes numériques actuellement disponibles à leurs limites de capacité. Enfin, les conclusions et perspectivessont présentées. / Since a few decades, frequency control technology has been at the heart of modern day electronics due to its huge areaof applications in communication systems, computers, navigation systems or military defense. Frequency controldevices provide high frequency stabilities and spectral purities in the short term domain. However, improvement of theperformance of these devices, in terms of frequency stability, remains a big challenge for researchers. Reducing noise inorder to increase the short term stability and avoid unwanted switching between channels is thus very desirable. It iscommonly admitted that the fundamental limitation to this short-term stability is due to flicker frequency noise in theresonators. In this manuscript, a first chapter recalls some basic facts about acoustic, crystallography and definitions oftime and frequency domain needed to explore ultra-stable resonators and oscillators. The second chapter is devoted to asummary of the literature on flicker frequency noise. Then, the third chapter concerns our studies on Handel’s quantum1/f noise model, which although criticized by many, is still the only one that provides an estimation of the flooramplitude of 1/f noise that is not invalidated by experimental data. In the fourth chapter, another approach, based on thefluctuation-dissipation theorem, is used in order to put numerical constraints on a model of 1/f noise caused by aninternal (or structural) dissipation proportional to the amplitude and not to the speed. The last chapter is devoted toexperimental results. An ultra-stable resonator used during this study is described. Phase noise measurements on severalbatches of resonators are given. Measurements of resonator parameters have been done at low temperature in order tocorrelate them with noise results. Another approach with a procedure that use transient pseudo periodic oscillations andput to their limits the capacities of presently available digital oscilloscopes, is presented, in order to assess rapidly thequality of various resonators. Finally, conclusions and perspectives are given.

Community Climate Model 3 (CCM3) is run on an idealized all land zonally symmetric planet (Terra Blanda) with no seasonality, no snow and fixed soil moisture to obtain a stationary time series effectively much longer than conventional runs with geography and seasons. The surface temperature field generated is studied to analyze the spatial and temporal spectra, estimate the length scale and time scale of the model, and test the linearity of response to periodic and steady heat source forcings. The length scale of the model is found to be in the range of 1000-2000 km and the time scale is estimated to be 24 days for the global average surface temperature field. The response of the global average temperature is found to be fairly linear to periodic and the steady heat source forcings. The results obtained are compared with the results of a similar study that used CCM0. Fluctuation Dissipation theorem is also tested for applicability on CCM3. The response of the surface temperature field to a step function forcing is demonstrated to be very similar to the decay of naturally occurring anomalies, and the auto-correlation function. Return period of surface temperature anomalies is also studied. It is observed that the length of the data obtained from CCM3, though sufficient for analysis of first and second moments, is significantly deficient for return period analysis. An AR1 process is simulated to model the global averaged surface temperature of Terra Blanda 3 to assess the sampling error associated with short runs.

This thesis describes an experimental study on fluctuations of a Brownian particle immersed in a fluid, confined by optical tweezers and subject to two different kinds of non-equilibrium conditions. We aim to gain a rather general understanding of the relation between spontaneous fluctuations, linear response and total entropy production for processes away from thermal equilibrium. The first part addresses the motion of a colloidal particle driven into a periodic non-equilibrium steady state by a nonconservative force and its response to an external perturbation. The dynamics of the system is analyzed in the context of several generalized fluctuation-dissipation relations derived from different theoretical approaches. We show that, when taking into account the role of currents due to the broken detailed balance, the theoretical relations are verified by the experimental data. The second part deals with fluctuations and response of a Brownian particle in two different aging baths relaxing towards thermal equilibrium: a Laponite colloidal glass and an aqueous gelatin solution. The experimental results show that heat fluxes from the particle to the bath during the relaxation process play the same role of steady state currents as a non-equilibrium correction of the fluctuation-dissipation theorem. Then, the present thesis provides evidence that the total entropy production constitutes a unifying concept which links the statistical properties of fluctuations and the linear response function for non-equilibrium systems either in stationary or non stationary states.

Ces travaux de thèse présentent une étude expérimentale des fluctuations d'une particule Brownienne soumise à deux différentes conditions hors-équilibre dans un fluide . Le but est de comprendre d'une manière générale la relation entre les fluctuations spontanées, la fonction de réponse linéaire et la production totale d'entropie des processus loin de l'équilibre thermique. La première partie est consacrée à l'étude du mouvement d'une particule colloïdale dans un état stationnaire périodique hors-équilibre induit par une force non-conservative et à sa réponse à une perturbation externe. Nous analysons la dynamique du système dans le contexte des différentes approches généralisées de fluctuation-dissipation. Nous montrons que ces relations théoriques sont satisfaites par les données expérimentales quand on prend en compte le rôle du courant du à la rupture du bilan détaillé. Dans une deuxième partie nous étudions les fluctuations et la réponse d'une particule Brownienne dans deux types de bains vieillissants qui relaxent vers l'équilibre thermique: un verre colloïdal de Laponite et une solution aqueuse de gélatine. Dans ce cas-là nous montrons que le flux de chaleur de la particule vers le bain pendant sa relaxation représente une correction hors-équilibre du théorème de fluctuation-dissipation. Donc, le flux de chaleur joue le même rôle que le courant dans un état stationnaire. En conséquence, les résultats de la thèse mettent en évidence l'importance générale de la production totale d'entropie pour quantifier les relations de fluctuation-dissipation généralisées dans les systèmes hors-équilibre. / This thesis describes an experimental study on fluctuations of a Brownian particle immersed in a fluid, confined by optical tweezers and subject to two different kinds of non-equilibrium conditions. We aim to gain a rather general understanding of the relation between spontaneous fluctuations, linear response and total entropy production for processes away from thermal equilibrium. The first part addresses the motion of a colloidal particle driven into a periodic non-equilibrium steady state by a nonconservative force and its response to an external perturbation. The dynamics of the system is analyzed in the context of several generalized fluctuation-dissipation relations derived from different theoretical approaches. We show that, when taking into account the role of currents due to the broken detailed balance, the theoretical relations are verified by the experimental data. The second part deals with fluctuations and response of a Brownian particle in two different aging baths relaxing towards thermal equilibrium: a Laponite colloidal glass and an aqueous gelatin solution. The experimental results show that heat fluxes from the particle to the bath during the relaxation process play the same role of steady state currents as a non-equilibrium correction of the fluctuation-dissipation theorem. Then, the present thesis provides evidence that the total entropy production constitutes a unifying concept which links the statistical properties of fluctuations and the linear response function for non-equilibrium systems either in stationary or non stationary states.