In order to optimize limited infrastructure, many power systems are frequently operated close to critical, or bifurcation, points. While operating close to such critical points can be economically advantageous, doing so increases the probability of a blackout. With the continued deployment of Phasor Measurement Units (PMUs), high sample rate data are dramatically increasing the real time observability of the power grids. Prior research has shown that the statistics of these data can provide useful information regarding network stability and associated bifurcation proximity. Currently, it is not common practice for transmission and distribution control centers to leverage the higher order statistical properties of PMU data. If grid operators have the tools to determine when these statistics warrant control action, though, then the otherwise unused statistical data present in PMU streams can be transformed into actionable information. In order to address this problem, we present two methods that aim to gauge and improve system stability using the statistics of PMU data. The first method shows how sensitivity factors associated with the spectral analysis of the reduced power flow Jacobian can be used to weight and filter incoming PMU data. We do so by demonstrating how the derived participation factors directly predict the relative strength of bus voltage variances throughout a system. The second method leverages an analytical solver to determine a range of "critical" bus voltage variances. The monitoring and testing of raw statistical data in a highly observable load pocket of a large system are then used to reveal when control actions are needed to mitigate the risk of voltage collapse. A simple reactive power controller is then implemented that pushes the stability of the system back to a stable operating paradigm. Full order dynamic time domain simulations are used in order to test this method on both the IEEE 39 bus system and the 2383 bus Polish system. We also compare this method to two other, more conventional, controllers. The first relies on voltage magnitude signals, and the second depends only on local control of a reactive power resource. This comparison illustrates how the use of statistical information from PMU measurements can substantially improve the performance of voltage collapse mitigation methods.

The motivation for this research is to leverage the increasing deployment of the phasor measurement unit (PMU) technology by electric utilities in order to improve situational awareness in power systems. PMUs provide unprecedentedly fast and synchronized voltage and current measurements across the system. Analyzing the big data provided by PMUs may prove helpful in reducing the risk of blackouts, such as the Northeast blackout in August 2003, which have resulted in huge costs in past decades. In order to provide deeper insight into early warning signs (EWS) of catastrophic events in power systems, this dissertation studies changes in statistical properties of high-resolution measurements as a power system approaches a critical transition. The EWS under study are increases in variance and autocorrelation of state variables, which are generic signs of a phenomenon known as critical slowing down (CSD). Critical slowing down is the result of slower recovery of a dynamical system from perturbations when the system approaches a critical transition. CSD has been observed in many stochastic nonlinear dynamical systems such as ecosystem, human body and power system. Although CSD signs can be useful as indicators of proximity to critical transitions, their characteristics vary for different systems and different variables within a system. The dissertation provides evidence for the occurrence of CSD in power systems using a comprehensive analytical and numerical study of this phenomenon in several power system test cases. Together, the results show that it is possible extract information regarding not only the proximity of a power system to critical transitions but also the location of the stress in the system from autocorrelation and variance of measurements. Also, a semi-analytical method for fast computation of expected variance and autocorrelation of state variables in large power systems is presented, which allows one to quickly identify locations and variables that are reliable indicators of proximity to instability.

Snap-through is a striking instability in which an elastic object rapidly jumps from one state to another. It is seen in the leaves of the Venus flytrap plant and umbrellas flipping on a windy day among many other examples. Similar structures that snap-through are used to generate fast motions in soft robotics, switches in micro-scale electronics and artificial heart valves. Despite the ubiquity of snap-through in nature and engineering, its dynamics is usually only understood qualitatively. In this thesis we develop analytical understanding of this dynamics, focussing on how the mathematical structure underlying the snap-through transition controls the timescale of instability. We begin by considering the dynamics of 'pull-in' instabilities in microelectromechanical systems (MEMS) - a type of snap-through caused by electrostatic forces in which the motions are dominated by fluid damping. Using a lumped-parameter model, we show that the observed time delay near the pull-in transition is a type of critical slowing down - a so-called 'bottleneck' due to the 'ghost' of a saddle-node bifurcation. We obtain a scaling law describing this slowing down, and, in the process, unify a large range of experiments and simulations that exhibit delay phenomena during pull-in. We also investigate the pull-in dynamics of MEMS microbeams, extending the lumped-parameter approach to incorporate the details of the beam geometry. This provides a model system in which to understand snap-through of a continuous elastic structure due to external loading. We develop a perturbation method that systematically exploits the proximity to pull-in to reduce the governing equations to a simpler evolution equation, with a structure that highlights the saddle-node bifurcation. This allows us to analyse the bottleneck dynamics in detail, which we compare with previous experimental and numerical data. The remainder of the thesis is concerned with the dynamics of snap-through in macroscopic systems. In particular, we explore the extent to which dissipation is required to explain anomalously slow snap-through. Considering an elastic arch as an archetype of a snapping system, we use the perturbation method developed earlier to show that two bottleneck regimes are possible, depending delicately on the relative importance of external damping. In particular, we show that critical slowing down occurs even in the absence of damping, leading to a new scaling law for the snap-through time that is confirmed by elastica simulations and experiments. In many real systems material viscoelasticity is present to some degree. Finally, we examine how this influences the snap-through dynamics of a simple truss-like structure. We present a regime diagram that characterises when the timescale of snap-through is controlled by viscous, elastic or viscoelastic effects.

Les glaces de spin, comme Dy2Ti2O7 et Ho2Ti2O7, sont des matériaux présentant un magnétisme particulièrement exotique. Ils constituent les premiers composés cristallins ferromagnétiques frustrés à avoir été découverts. Cette frustration permet la fractionnalisation des degrés de liberté de spin et l’émergence de monopôles magné-tiques, dont la physique est formalisée par le modèle des haltères.Dans cette thèse, nous étudions dans un premier temps le diagramme de phase de ce modèle grâce à un parallèle avec le modèle de Blume-Capel S = 2. On identifie dans ce diagramme la phase fragmentée observée expérimentalement dans Ho2Ir2O7,et on localise le point critique de la transition entre la phase glace de spin et la phase fragmentée.Dans un second temps, on montre numériquement que la dynamique du système autour de ce point critique appartient à la classe d’universalité du modèle d’Ising 3D. On utilise pour cela deux outils : les lois d’échelle de Kibble-Zurek et le rapport de fluctuation-dissipation. L’obtention de ce dernier a nécessité l’introduction d’une méthode novatrice pour le calcul des fonctions de réponse. Nous soulignons également que ces outils sont spécifiquement intéressants dans le cas des glaces de spin où les temps microscopiques sont de l’ordre de 1 μs, rendant le ralentissement critique observable expérimentalement.Dans un troisième temps, nous employons à nouveau la violation du théorème de fluctuation-dissipation pour caractériser un régime fortement hors équilibre de la phase glace de spin, où les degrés de liberté sont cinétiquement bloqués du fait de l’attraction coulombienne entre les monopôles. / Spin ices, such as Dy2Ti2O7 and Ho2Ti2O7, are materials exhibiting exotic magnetic properties. They were the first frustrated ferromagnetic crystalline compounds to be discovered. The frustration leads to the fractionnalisation of the spin degrees of freedom and the emergence of magnetic monopoles, whose physics is formalised in the dumbbell model. In this thesis, we study the full phase diagram of this model in analogy with theS=2 Blume-Capel model. We identify in this diagram the fragmented phase observed experimentally in Ho2Ir2O7, and we localise the critical point of the transition between the spin ice phase and the fragmented phase.In a second part, we show numerically that the dynamics of this system at thecritical point belongs to the 3D Ising university class. We use for this two tools :the Kibble-Zurek scaling law and the fluctuation-dissipation ratio. For the latter, ithas been necessary to introduce a novel method to measure response functions. Wealso emphasize that these tools are specifically interesting for spin ice materials, as the unusually long microscopic time scale (1 μs) should make it possible to experimentallyobserve out-of-equilibrium phenomena related to critical slowing down.In a third part, we use the violation of the fluctuation-dissipation theorem to characterise a strongly out-of-equilibrium regime of spin ice - a thermal quench from high to low temperature, where degrees of freedom are kinetically blocked because ofthe Coulombic attraction between the monopoles.

No description available.

The physics of ultracold quantum gases has been the subject of a long-lasting and intense research activity, which started almost a century ago with purely theoretical studies and had a fluorishing experimental development after the implementation of laser and evaporative cooling techniques that led to the first realization of a Bose Einstein condensate (BEC) over 25 years ago. In recent years, a great interest in ultracold atoms has developed for their use as platforms for quantum technologies, given the high degree of control and tunability offered by ultracold atom systems. These features make ultracold atoms an ideal test bench for simulating and studying experimentally, in a controlled environment, physical phenomena analogous to those occurring in other, more complicated, or even inaccessible systems, which is the idea at the heart of quantum simulation. In the rapidly developing field of quantum technologies, it is highly important to acquire an in-depth understanding of the state of the quantum many-body system that is used, and of the processes needed to reach the desired state. The preparation of the system in a given target state often involves the crossing of second order phase transitions, bringing the system strongly out-of-equilibrium. A better understanding of the out-of-equilibrium processes occurring in the vicinity of the transition, and of the relaxation dynamics towards the final equilibrium condition, is crucial in order to produce well-controlled quantum states in an efficient way. In this thesis I present the results of the research activity that I performed during my PhD at the BEC1 laboratory of the BEC center, working on ultracold gases of 23Na atoms in an elongated harmonic trap. This work had two main goals: the accurate determination of the equilibrium properties of a Bose gas at finite temperature, by the measurement of its equation of state, and the investigation of the out-of-equilibrium dynamics occurring when a Bose Einstein condensate is prepared by cooling a thermal cloud at a finite rate across the BEC phase transition.To study the equilibrium physics of a trapped atomic cloud, it is crucial to be able to observe its density distribution in situ. This requires a high optical resolution to accurately obtain the density profile of the atomic distribution, from which thermodynamic quantities can then be extracted. In particular, in a partially condensed atomic cloud at finite temperature, it is challenging to resolve well also the boundaries of the BEC, where the condensate fraction rapidly drops in a narrow spatial region. This required an upgrade of the experimental apparatus in order to obtain a high enough resolution. I designed, tested and implemented in the experimental setup new imaging systems for all main directions of view. Particular attention was paid for the vertical imaging system, which was designed to image the condensates in trap with a resolution below 2 μm, with about a factor 4 improvement compared to the previous setup. The implementation of the new imaging systems involved a partial rebuilding of the experimental apparatus used for cooling the atoms. This created the occasion for an optimization of the whole system to obtain more stable working conditions. Concurrently I also realized and included in the experiment an optical setup for the use of a Digital Micromirror Device (DMD) to project time-dependent arbitrary light patterns on the atoms, creating optical potentials that can be controlled at will. The use of this device opens up exciting future scenarios where it will be possible to locally modify the trapping potential and to create well-controlled barriers moving through the atomic cloud. Another challenge in imaging the density distribution in situ is determined by the fact that the maximum optical density (OD) of the BEC, in the trap center, exceeds the low OD of the thermal tails by several orders of magnitude. In order to obtain an accurate image of the whole density profile, we developed a minimally destructive, multi-shot imaging technique, based on the partial transfer of a fraction of atoms to an auxiliary state, which is then probed. Taking multiple images at different extraction fractions, we are able to reconstruct the whole density profile of the atomic cloud avoiding saturation and maintaining a good signal to noise ratio. This technique, together with the improvements in the imaging resolution, has allowed us to accurately obtain the optical density profile of the Bose gas in trap, from which the 3D density profile was then calculated applying an inverse Abel transform, taking advantage of the symmetry of the trap. From images of the same cloud after a time-of-flight expansion, we measured the temperature of the gas. From these quantities we could find the pressure as a function of the density and temperature, determining the canonical equation of state of the weakly interacting Bose gas in equilibrium at finite temperature. These measurements also allowed us to clearly observe the non-monotonic temperature behavior of the chemical potential near the critical point for the phase transition, a feature that characterizes also other superfluid systems, but that had never been observed before in weakly interacting Bose gases. The second part of this thesis work is devoted to the study of the dynamical processes that occur during the formation of the BEC order parameter within a thermal cloud. The cooling at finite rate across the Bose-Einstein condensation transition brings the system in a strongly out-of-equilibrium state, which is worth investigating, together with the subsequent relaxation towards an equilibrium state. This is of interest also in view of achieving a better understanding of second order phase transitions in general, since such phenomena are ubiquitous in nature and relevant also in other platforms for quantum technologies. A milestone result in the study of second order phase transitions is given by the Kibble-Zurek mechanism, which provides a simple model capturing important aspects of the evolution of a system that crosses a second-order phase transition at finite rate. It is based on the principle that in an extended system the symmetry breaking associated with a continuous phase transition can take place only locally. This causes the formation of causally disconnected domains of the order parameter, at the boundaries of which topological defects can form, whose number and size scale with the rate at which the transition is crossed, following a universal power law. It was originally developed in the context of cosmology, but was later successfully tested in a variety of systems, including superfluid helium, superconductors, trapped ions and ultracold atoms. The BEC phase transition represents in this context a paradigmatic test-bench, given the high degree of control at which this second-order phase transition can be crossed by means of cooling ramps at different rates. Already early experiments investigated the formation of the BEC order parameter within a thermal cloud, after quasi-instantaneous temperature quenches or very slow evaporative cooling. In the framework of directly testing the Kibble-Zurek mechanism, further experiments were performed, both in 2D and 3D systems, focusing on the emergence of coherence and on the statistics of the spontaneously generated topological defects as a function of the cooling rate. The Kibble-Zurek mechanism, however, does not fully describe the out-of-equilibrium dynamics of the system at the transition, nor the post-quench interaction mechanisms between domains that lead to coarse-graining. Most theoretical models are based on a direct linear variation of a single control parameter, e.g. the temperature, across the transition. In real experiments, the cooling process is controlled by the tuning of other experimental parameters and a global temperature might not even be well defined, in a thermodynamic sense, during the whole process. Moreover, the temperature variation is usually accompanied by the variation of other quantities, such as the number of atoms and the collisional rate, making it difficult to accurately describe the system and predict the post-quench properties. Recent works included effects going beyond the Kibble-Zurek mechanism, such as the inhomogeneity introduced by the trapping potential, the role of atom number losses, and the saturation of the number of defects for high cooling rates. These works motivate further studies, in particular of the dynamics taking place at early times, close to the crossing of the critical point. The aim of the work presented in this thesis is to further investigate the timescales associated to the formation and evolution of the BEC order parameter and its spatial fluctuations, as a function of the rate at which the transition point is crossed. We performed experiments producing BECs by means of cooling protocols that are commonly used in cold-atom laboratories, involving evaporative cooling in a magnetic trap. We explored a wide range of cooling rates across the transition and found a universal scaling for the growth of the BEC order parameter with the cooling rate and a finite delay in its formation. The latter was already observed in earlier works, but for a much more limited range of cooling rates. The evolution of the fluctuations of the order parameter was also investigated, with an analysis of the timescale of their decay during the relaxation of the system, from an initial strongly out-of-equilibrium condition to a final equilibrium state. This thesis is structured as follows: The first chapter presents the theoretical background, starting with a brief introduction to the concept of Bose Einstein condensation and a presentation of different models describing the thermodynamics of an equilibrium Bose gas. The second part of this chapter then deals with the out-of-equilibrium dynamics that is inevitably involved in the crossing of a second-order phase transition such as the one for Bose-Einstein condensation. The Kibble-Zurek mechanism is briefly reviewed and beyond KZ effects are pointed out, motivating a more detailed investigation of the timescales involved in the BEC formation. In the second chapter, I describe the experimental apparatus that we use to cool and confine the atoms. Particular detail is dedicated to the parts that have been upgraded during my PhD, such as the imaging system. In the third chapter I show our experimental results on the measurement of the equation of state of the weakly interacting uniform Bose gas at finite temperature. In the fourth chapter I present our results on the out-of-equilibrium dynamics in the formation of the condensate order parameter and its spatial fluctuations, as a function of different cooling rates.