Дисертації з теми "Disordered and aperiodic systems"

The aim of this article is to give a pedagogical introduction to the exact equilibrium and nonequilibrium properties of free fermionic quantum spin chains. In a first part we present in full details the canonical diagonalisation procedure and review quickly the equilibrium dynamical properties. The phase diagram is analysed and possible phase transitions are discussed. The two next chapters are concerned with the effect of aperiodicity and quenched disorder on the critical properties of the quantum chain. The remaining part is devoted to the nonequilibrium dynamical behaviour of such quantum chains relaxing from a nonequilibrium pure initial state. In particular, a special attention is made on the relaxation of transverse magnetization. Two-time linear response functions and correlation functions are also considered, giving insights on the nature of the final nonequilibrium stationnary state. The possibility of aging is also discussed.

Consideramos os efeitos de desordem ou aperiodicidade sobre três sistemas magnéticos distintos. Inicialmente, apresentamos um modelo fenomenológico para descrever a dependência térmica da magnetização remanente induzida por diluição numa classe de antiferromagnetos quase-unidimensionais. O modelo trata exatamente as correlações ao longo da direção dominante, levando em conta as demais interações por meio de um campo efetivo. Em seguida, utilizamos uma aproximação autoconsistente de Bethe-Peierls para avaliar os efeitos de um campo cristalino aleatório sobre os diagramas de fases de um modelo de Ising de spins mistos. Mostramos que a desordem é capaz de modificar a natureza dos pontos multicríticos existentes no limite uniforme do modelo. Finalmente, estudamos os efeitos de interações aleatórias ou aperiódicas sobre o comportamento da cadeia XX quântica em baixas temperaturas, através de câlculos numéricos baseados no mapeamento do sistema em um modelo de férmions livres. Apontamos evidências de que, em temperatura zero, existe um único ponto fixo universal, característico de uma fase de singleto aleatório, que governa o comportamento do modelo na presença de interações desordenadas. No caso de interações aperiódicas,obtemos resultados consistentes com previsões de grupo de renormalização, indicando, para uma certa classe de seqüências de substituição, um comportamento semelhante àquele associado à desordem.
We consider effects of disorder or aperiodicity on three different magnetic systems. First, we present a phenomenological model to describe the thermal dependence of the dilution-induced remanent magnetization in a class of quasi-one-dimensional antiferromagnets. The model treats correlations along the dominant direction in an exact way, while including the remaining inte-. i ractions via an effective field. Then, we use a self-consistent Bethe-Peierls ~ j .. approximation to gauge the effects of a random crystal field on the phase diagram of a mixed-spin Ising mode!. We show that disorder may have profound effects on the multicritical behavior associated with the uniform limit of the mo de!. Finally, we study effects of random or aperiodic interactions on the behavior of the quantum XX chain at low temperatures, by performing numerical calculations based on a mapping of the system onto a free-fermion mo de!. . We present evidence that, at zero temperature, there exists a single, universal fixed-point, associated with a random-singlet phase, which governs the behavior of the model in the presence of disordered interactions. In the case of aperiodic interactions, our results are consistent with renormalizationgroup predictions, indicating, for a certain class of substitution sequences, a behavior similar to the one induced by disorder.

The use of dynamical system models is commonplace in many areas of science and engineering. One is often interested in whether the attracting solutions in these models are robust to perturbations of the equations of motion. This question is extremely important in situations where it is undesirable to have a large response to perturbations for reasons of safety. An especially interesting case occurs when the perturbations are aperiodic and their exact form is unknown. Unfortunately, there is a lack of theory in the literature that deals with this situation. It would be extremely useful to have a practical technique that provides an upper bound on the size of the response for an arbitrary perturbation of given size. Estimates of this form would allow the simple determination of safety criteria that guarantee the response falls within some pre-specified safety limits. An excellent area of application for this technique would be engineering systems. Here one is frequently faced with the problem of obtaining safety criteria for systems that in operational use are subject to unknown, aperiodic perturbations. In this thesis I show that such safety criteria are easy to obtain by using the concept of persistence of hyperbolicity. This persistence result is well known in the theory of dynamical systems. The formulation I give is functional analytic in nature and this has the advantage that it is easy to generalise and is especially suited to the problem of unknown, aperiodic perturbations. The proof I give of the persistence theorem provides a technique for obtaining the safety estimates we want and the main part of this thesis is an investigation into how this can be practically done. The usefulness of the technique is illustrated through two example systems, both of which are forced oscillators. Firstly, I consider the case where the unforced oscillator has an asymptotically stable equilibrium. A good application of this is the problem of ship stability. The model is called the escape equation and has been argued to capture the relevant dynamics of a ship at sea. The problem is to find practical criteria that guarantee the ship does not capsize or go through large motions when there are external influences like wind and waves. I show how to provide good criteria which ensure a safe response when the external forcing is an arbitrary, bounded function of time. I also consider in some detail the phased-locked loop. This is a periodically forced oscillator which has an attracting periodic solution that is synchronised (or phase-locked) with the external forcing. It is interesting to consider the effect of small aperiodic variations in the external forcing. For hyperbolic solutions I show that the phase-locking persists and I give a method by which one can find an upperbound on the maximum size of the response.

Uncertainties and constraints are present in most control systems. For example, robot motion planning and building climate regulation can be modeled as uncertain constrained systems. In this thesis, we develop mathematical and computational tools to analyze and synthesize controllers for such systems. As our first contribution, we characterize when a set is a probabilistic controlled invariant set and we develop tools to compute such sets. A probabilistic controlled invariantset is a set within which the controller is able to keep the system state with a certainprobability. It is a natural complement to the existing notion of robust controlled invariantsets. We provide iterative algorithms to compute a probabilistic controlled invariantset within a given set based on stochastic backward reachability. We prove that thesealgorithms are computationally tractable and converge in a finite number of iterations. The computational tools are demonstrated on examples of motion planning, climate regulation, and model predictive control. As our second contribution, we address the control design problem for uncertain constrained systems with aperiodic sensing and actuation. Firstly, we propose a stochastic self-triggered model predictive control algorithm for linear systems subject to exogenous disturbances and probabilistic constraints. We prove that probabilistic constraint satisfaction, recursive feasibility, and closed-loop stability can be guaranteed. The control algorithm is computationally tractable as we are able to reformulate the problem into a quadratic program. Secondly, we develop a robust self-triggered control algorithm for time-varying and uncertain systems with constraints based on reachability analysis. In the particular case when there is no uncertainty, the design leads to a control system requiring minimum number of samples over finite time horizon. Furthermore, when the plant is linear and the constraints are polyhedral, we prove that the previous algorithms can be reformulated as mixed integer linear programs. The method is applied to a motion planning problem with temporal constraints.

QC 20181016

Real-time systems require a-priori temporal guarantees. While most of the normal operation in such a system is modeled using time-driven, hard-deadline sporadic tasks, event-driven behavior is modeled using aperiodic jobs with soft or no deadlines. To provide good Quality-of- Service for aperiodic jobs in the presence of sporadic tasks, aperiodic servers were introduced. Aperiodic servers act as a sporadic task and reserve a quota periodically to serve aperiodic jobs. The use of aperiodic servers in systems with caches is unsafe because aperiodic servers do not take into account, the indirect cache-related preemption delays that the execution of aperiodic jobs might impose on the lower-priority sporadic tasks, thus jeopardizing their safety. To solve this problem, we propose an enhancement to the aperiodic server that we call a Cache Delay Server. Here, each lower-priority sporadic task is assigned a delay quota to accommodate the cache-related preemption delay imposed by the execution of aperiodic jobs. Aperiodic jobs are allowed to execute at their assigned server priority only when all the active lower-priority sporadic tasks have a sufficient delay quota to accommodate it. Simulation results demonstrate that a Cache Delay Server ensures the safety of sporadic tasks while providing acceptable Quality-of-Service for aperiodic jobs. We propose a Integer Linear Program based approach to calculate delay quotas for sporadic tasks within a task set where Cache Delay Servers have been pre-assigned. We then propose algorithms to determine Cache Delay Server characteristics for a given sporadic task set. Finally, we extend the Cache Delay Server concept to multi-core architectures and propose approaches to schedule aperiodic jobs on appropriate Cache Delay Servers. Simulation results demonstrate the effectiveness of all our proposed algorithms in improving aperiodic job response times while maintaining the safety of sporadic task execution.

Disordered systems are ubiquitous in nature and their study is complicated and often leads to controversial results. In any case, the important role of such systems in science and technological applications should not be ignored. The characteristic properties of such systems seem to be driven by a fundamental feature, the degrees of freedom. Although many problems still remain matter of debate, the challenge posed in recent decades in the understanding of the impact of disorder in the physical behavior of materials is of considerable scientic interest. An exact description of a disordered phase is not possible since it is a many-body problem hard to model. However, for some materials it is possible, upon cooling, to preserve the disordered liquid-like structure having a state of high regularity. Therefore, for the so-called glass formers, it is possible to freeze some degrees of freedom obtaining a glass that presents the irregularity of a liquid with the high viscosity of a solid below the melting temperature Tm. The aim of the present PhD thesis is to present our understanding of disorder in related experimental approaches using three different pure compounds: two plastic crystals (1-Chloroadamantane and Freon113) and a liquid (Glycerol). To understand the behavior of these kinds of materials neutron scattering and dielectric spectroscopy have been used. These two powerful techniques allow us to investigate the dynamics of disordered phases on a picosecond time scale. Furthermore, given the complexity of these disordered phases, data analysis and model selection have been performed with a Bayesian approach that provides a solid statistical ground bases on probability distribution functions. Such methods have been applied to study of the above mentioned compounds dynamics in order to give an explanation of some open questions: the microscopic origin of the plastic-plastic transition in 1-chloroadamantane (C10H15Cl), the high fragility and the correlation between kinetic and thermodynamic fragility in freon113 (Cl2FC-CClF2) and the dynamics, accompanied by a robust model selection, of one of the most studied glass former compound, glycerol (C3H8O3). In addition, a brief overview of the theoretical background for neutron scattering and dielectric spectroscopy, as well as a description of the experimental setup and the consequent data treatment and analysis, are given to deliver a comprehensive and consistent view of the topic under consideration. The results, presented in this work of thesis, represent a small step in a deeper understanding of disordered phases dynamics, giving a base for further investigations.
Los sistemas desordenados son ubicuos en la naturaleza y su estudio es complicado y con frecuencia conduce a resultados controvertidos. En cualquier caso, el papel importante de este tipo de sistemas en aplicaciones científicas y tecnológicas no debe ser ignorada. Las propiedades características de tales sistemas parecen estar impulsadas por una característica fundamental, los grados de libertad. Aunque muchos problemas siguen siendo materia de debate, el desafío planteado en las últimas décadas en el entendimiento del impacto del desorden en el comportamiento físico de los materiales es de considerable interés científico. Una descripción exacta de una fase desordenada no es posible, ya que es un problema de muchos cuerpos difícil de modelar. Sin embargo, para algunos materiales, es posible, tras el enfriamiento, conservar la estructura desordenada del líquido con un estado de alta regularidad. Por lo tanto, para los denominados glass-formers, es posible congelar algunos grados de libertad obteniendo un vidrio que presenta la irregularidad de un líquido con la alta viscosidad de un sólido por debajo de la temperatura de fusión Tm. El objetivo de la presente tesis doctoral es presentar nuestra comprensión del desorden en los enfoques experimentales relacionados utilizando tres diferentes compuestos puros: dos cristales de plástico (1-Chloroadamantane y Freon113) y un líquido (Glycerol). Para entender el comportamiento de este tipo de materiales se han utilizado scattering de neutrones y espectroscopía dieléctrica. Estas dos técnicas nos permiten investigar la dinámica de las fases desordenadas en una escala de tiempo de picosegundos. Por otra parte, dada la complejidad de estas fases desordenadas, análisis de datos y la selección del modelo se han realizado con un enfoque bayesiano que proporciona una sólida base estadística basada sobre las funciones de distribución de probabilidad. Tales métodos se han aplicado al estudio de la dinámica de los compuestos antes mencionados con el fin de dar una explicación de algunas preguntas abiertas: el origen microscópico de la transición plástico-plástico en 1-chloroadamantane (C10H15Cl), la alta fragilidad y la correlación entre la fragilidad cinética y termodinámica en freon113 (Cl2FC-CClF2) y la dinámica, acompañada por una robusta selección de modelo, de uno de los compuestos más estudiados, glycerol (C3H8O3). Los resultados, presentados en este trabajo de tesis, representan un pequeño paso para una comprensión más profunda de la dinámica de las fases desordenadas, dando una base para futuras investigaciones.

In this master thesis we propose an aperiodic formulation of Model Predictive Control for distributed agents with additive bounded disturbances. In this control method, each agent solves an optimal control problem only when certain control performances are not guaranteed according to several triggering rules. This may lead not only to the dramatic reduction of energy expenditures but also to the alleviation of communication loads among them. The problem will be considered to be general and practical; it handles the non-linearity of the each agent which is perturbed by additive bounded disturbances, where the triggering rule is derived from several robust stability criterion. The triggering rule will be addressed for event-based control and self-triggered control, which are the two main different aperiodic control approaches. Finally some simulation results verify our proposal.

The nature of the glass transition and of the glassy state is a fundamental and still unsolved problem of condensed matter physics. Many liquids can be supercooled below their melting point without crystallizing, that is, without acquiring translational and orientational order. As the temperature of a supercooled liquid is lowered, the characteristic timescale of moleuclar motions, called relaxation time, increases until it becomes comparable to the timescale of human experimentation. This takes place at the glass transition temperature and leads to a non-equilibrium state of matter, called a ¿structural glass¿, in which a liquid-like lack of order is combined with solid-like elastic properties. Glass transitions are also observed in systems where there is only orientational disorder, such as orientationally disordered (OD) crystals or plastic crystals, which are translationally ordered solids in which the constituent molecules display reorientational motions about their centres of mass. Upon supercooling an OD crystal, the orientational disorder can ¿freeze¿, yielding a so-called ¿orientational glass¿. In molecular materials forming structural or orientational glasses, the most important molecular dynamics process is the cooperative motion of the molecules, referred to as primary relaxation, whose freezing marks the transition to the glass state characterized by static disorder. The main difference between orientational and structural glasses is that in the former the freezing involves exclusively the rotational degrees of freedom of the molecules, while in the latter all six molecular degrees of freedom (i.e., both orientational and translational ones) are frozen. Orientational glasses are therefore systems with fewer degrees of freedom than structural glasses. This simplification, together with the fact that many OD phases are characterized by a crystal lattice with high symmetry, makes OD phases a model playground to investigate the nature of the glass transition. Other than the primary relaxation, there can be also so-called ¿secondary relaxations¿, usually characterized by shorter relaxation time than the primary process. Secondary relaxations may have different origins; for example, they can be due to conformational fluctuations or intramolecular vibrations; in many cases a special kind of secondary relaxation is observed, which is the single-molecule precursor process of the primary relaxation. This thesis focuses on the effect of pressure and temperature on the dynamics of several pure compounds and binary mixtures forming structural or orientational glasses. We present a comparative study between two structural glass formers (ternidazole and the mixture of m-fluoroaniline with m-xylene), a plastic binary mixed crystal (neopenthyl alchol and neopentyl glycol), and two materials displaying statistical orientational disorder (2-adamantanone and pentachloronitrobenzene). In all cases a primary relaxation is present, associated with the collective motion of the molecules, and in most cases also secondary relaxations are observed. For each material, we analyse the temperature- and pressure-dependence of the various molecular relaxation and discuss the origin of secondary processes. One of the most important results of the thesis is the presence of secondary relaxations also in systems with low-dimensional disorder that behave similarly to the secondary relaxations observed in structural glasses.
La naturaleza de la transición vítrea es un problema fundamental y aún no resuelto de la física de la materia condensada. Muchos líquidos pueden ser superenfriados por debajo de su temperatura de fusión sin que cristalicen, es decir, sin que adquieran orden traslacional y orientacional. Cuando la temperatura de un líquido superenfriado baja, el tiempo característico de los movimientos moleculares, llamado tiempo de relajación, aumenta hasta llegar a tiempos comparables con el tiempo característico de los experimentos y de la observación humana. Esto ocurre a una temperatura llamada temperatura de transición vítrea y lleva a un estado de non-equilibrio del material llamado ¿vidrio estructural¿, en el que la ausencia de orden de largo alcance típica del estado líquido se combina con las propiedades elásticas propias de un sólido ordenado. Las transiciones vítreas se pueden observar también en sistemas caracterizados por desorden exclusivamente orientacional, como en los cristales orientacionalmente desordenados (OD) o cristales plásticos. Estos son sólidos traslacionalmente ordenados en los que las moléculas tienen movimientos de reorientación alrededor de sus centros de masa, que están fijos. Superenfriando un cristal OD se obtiene un ¿vidrio orientacional¿ en el cual este desorden orientacional está congelado. El proceso dinámico más importante que caracteriza los materiales moleculares que forman vidrios estructurales u orientacionales es el movimiento cooperativo de las moléculas conocido como relajación primaria. Su congelamiento marca la transición al estado vítreo caracterizado por un desorden estático. La diferencia principal entre los vidrios orientacionales y estructurales es que en los primeros el congelamiento involucra sólo los grados de libertad de rotación, mientras que en los segundos todos los seis grados de libertad moleculares (orientacionales y traslacionales) están congelados. Por tanto, los vidrios orientacionales son sistemas con menos grados de libertad respecto los vidrios estructurales y pueden considerarse como sistemas modelo para investigar la transición vítrea, ya que además muchas fases OD están caracterizadas por redes cristalinas de alta simetría. Además de la relajación primaria, existen también relajaciones secundarias caracterizadas por tiempos de relajación más cortos con respecto al proceso primario. Estas relajaciones secundarias pueden tener diferentes orígenes: por ejemplo, pueden ser debidas a fluctuaciones de la conformación molecular o a vibraciones de enlaces intramoleculares; en muchos casos se observa una relajación secundaria que es considerada como la precursora del proceso primario (relajación Johari-Goldstein). Esta tesis está enfocada en el estudio de los efectos de la presión y de la temperatura sobre la dinámica de algunos compuestos puros y mezclas binarias, los cuales forman vidrios estructurales u orientacionales. Se presenta un estudio comparativo entre dos vidrios estructurales (ternidazole y la mezcla de m-fluoroanilina con m-xileno), un cristal plástico binario (formado por neopenthyl alcohol y neopentyl glycol), y dos materiales que presentan desorden estadístico (2-adamantanona y pentacloronitrobenceno). En todos los casos se observa una relajación primaria asociada a los movimientos colectivos de las moléculas y en la mayoría de los casos se observa también relajaciones secundarias. Para cada material se analiza la dependencia de diferentes relajaciones con la temperatura y con la presión y se discute el origen de los procesos secundarios. Uno de los resultados importantes de la tesis es que en sistemas con desorden de baja dimensionalidad, pueden aparecer relajaciones secundarias que obecen a patrones similares a las encontradas en vidrios estructurales

Disorder is present in many systems in nature and in many different versions. For instance, the dislocations of a crystal lattice, or the randomness of the interaction between magnetic moments. One of the most studied examples is that of spin glasses because they are simple to model but keep most of the very complex features that many disordered systems have. The frustration of the ground state configuration is responsible for the existence of a gap less spectrum of excitations and a rugged and complex free-energy landscape which bring about a very slow relaxation towards the equilibrium state. The main concern of the thesis has been to study what the properties of the typical excitation, i.e. those excitations that are large and contribute dominantly to the physics in the frozen phase.
The existence of these large excitations brings about large fluctuations of the order parameter, and we have shown in these theses that this feature can be exploited to study the transition of any spin glass model. Moreover, we have shown that the information about these excitations can be extracted from the statistics of the lowest lying excitations. This is because due to the random nature of spin glasses, the physics obtained from averaging over the whole spectrum of excitations of an infinite sample is equivalent to averaging over many finite systems where only the ground state and the first excitation are considered. The novelty of this approach is that we do not need to make any assumption on what are typical excitations like because we can compute them exactly using numerical methods. Finally, we have investigated the dynamics and more specifically the link between the problem of chaos and the rejuvenation phenomena observed experimentally. Rejuvenation means that when lowering the temperature the aging process restarts again from scratch. This is potentially linked with the chaos assumption which states that equilibrium configurations at two different properties are not correlated. Chaos is a large scale phenomenon possible if entropy fluctuations are large. However, in this thesis we have shown that the response to temperature changes can be large in the absence of chaos close to a localization transition where the Boltzmann weight condenses in a few states. This has been observed in simulation of the Sinai model in which this localization is realized dynamically. In this model, since at low temperatures the system gets trapped in the very deep states, the dynamics is only local, so that only small excitations contribute to the rejuvenation signal that we have been able to observe. Thus, in agreement with the hierarchical picture, rejuvenation is possible even in the absence of chaos and reflects the start of the aging process of small length scales.

In this thesis we investigate the employ of the renormalization group for glassy systems. More precisely, we focus on models of spin glasses and structural glasses. Spin-glass models represent disordered uniaxial magnetic materials, such as a dilute solution of Mn in Cu, modeled by an array of spins on the Mn arranged at random in the matrix of Cu, and interacting with a potential which oscillates as a function of the separation of the spins. Structural glasses are liquids that have been cooled fast enough to avoid crystallization, like o-Terphenyl or Glycerol. Spin and structural glasses are physically interesting because their critical properties are known only in the limit where the space dimensionality tends to infinity, i. e. in the mean-field approximation. A fundamental question is whether the physical properties characterizing these systems in the mean-field case still hold for real spin or structural glasses, which live in a space with a finite number of dimensions. The spin and structural glasses that we study in this thesis are models built up on hierarchical lattices, which are the simplest non-mean field systems where the renormalisation group approach can be implemented in a natural way. The features emerging from this implementation clarify the critical behavior of these systems. As far as the finite-dimensional spin glass studied in this thesis is concerned, we developed a new technique to implement the renormalization group transformation for finite-dimensional spin glasses. This technique shows that the system has a finite-temperature phase transition characterized by a critical point where the system's correlation length is infinite. As far as the structural glass studied in this thesis is concerned, this is the first structural glass model where we showed the existence of a phase transition beyond mean field. The ideas introduced in this work can be further developed in order to understand the structure of the low-temperature phase of these systems, and in order to establish whether the properties of the low-temperature phase holding in the mean-field case still hold for finite-dimensional glassy systems.

Thesis (Ph. D.)--University of Oregon, 2006.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 126-128). Also available for download via the World Wide Web; free to University of Oregon users.

This thesis presents results from large scale Monte Carlo simulations of systems subject to a superfluid phase transition in the presence of disorder. The simulations are performed by state-of-the-art, collective Monte Carlo algorithms treating phase degrees of freedom in effective models with amplitude fluctuations integrated out. In Paper I a model system for the possible solid to supersolid transition in 4He is presented.The Wolff cluster algorithm is used to study how the presence of linearly correlated random defects is able to alter the universality class of the 3-dimensional XY-model. In the pure case the superfluid density and heat capacity have singular onsets, which are not seen in the supersolid experiments where instead a smooth onset is obtained. Using finite size scaling of Monte Carlo data, we find a similar smooth onset in our simulations, governed by exponents  ν=1 for the superfluid density and α=-1 for the heat capacity. These results are in qualitative agreement with experiments for the observed transition in solid 4He. In Paper II a systematic investigation of the scaling result z=d for the dynamic critical exponentat the Bose glass to superfluid quantum phase transition is performed. The result z=d has been believed to be exact for about 20 years, but although it has been questioned lately no accurate estimate of z has been available. An effective link current model of quantum bosons at T=0 with disorder in 2D is simulated using highly effective worm Monte Carlo simulations.The data analysis is based on a finite size scaling approach todetermine the quantum correlation time from simulationdata for boson world lines without any a priori assumption on the critical parameters. The resulting critical exponents are z=1.8 \pm 0.05, ν=1.15 \pm 0.03, and η=-0.3 \pm 0.1. This suggests that z=d is not satisfied.

QC 20111206

Transport phenomena in nature occur in spatially complex and heterogeneous systems, such as the lung and the placenta. Motivated by such examples, we seek to characterise the effects of and interaction between multiscale spatial structure and disorder in models of transport and flow. We first investigate an individual-based transport model which incorporates two sources of noise and transport processes occurring over disparate lengthscales. Particles can hop stochastically between sites on a lattice in one spatial dimension and are taken up via first-order kinetics at discrete sinks with random strengths, sparsely but periodically located along the domain. Simulations indicate that disordered sink strengths induce long-range correlations in the particle concentration which are absent when sinks are of fixed strength. Furthermore, mean concentrations are elevated by disorder. Exploiting the separation of lengthscales and properties of the stochastic hopping, we use a continuum limit to derive approximate, homogenized expressions describing the correlations and elevated mean. These involve non-local combinations of the leading-order concentration profile and the Green's function associated with the corrections. We show that the correction to the mean concentration is always non-negative and so the leading-order classical homogenization approximation underestimates the mean. We finally classify the regions of parameter space according to the dominant physical processes in each region and put bounds on the validity of the homogenization approach by analysing the sizes of fluctuations due to disorder. We next study a related model of solute transport in one dimension where sinks of equal strength are distributed randomly along a line. Extending the Green's function approach, we investigate the interactions between the discrete nature of the sinks and their disordered locations. We find that classical two-scale expansion-based homogenization fails to accurately predict higher-order corrections and develop refined predictions. These methods extend to handling spatially disordered sinks, and we demonstrate that our predictions of the spatial correlations and corrections to leading-order means agree well with simulations in large areas of parameter space and for both weak and strong disorder. The strength of the spatial disorder can affect whether the discrete-to-continuous or disorder effects are the most important. Finally, we study the flow through a two-dimensional disordered porous medium. We perform direct numerical simulations of a low Reynolds number flow past a series of obstacles. Applying spatial perturbations to individual obstacles in an otherwise periodic arrangement, and comparing with a completely periodic array, we investigate the long-range effects in the velocity and pressure fields. This is effective a numerical approximation of a series of Green's functions, inspired by the theoretical framework used in our previous models. We then generate large ensembles of weakly and strongly disordered porous media and compare the statistics of the flow rate with predictions made by Darcy's law, typically applicable to large, periodic arrays. Depending on the type of spatial disorder, mean flow rates can be either elevated or diminished compared with a periodic array and Darcy's law. Furthermore, we analyse the decay in the fluctuations around the mean flow rate as the size of the array is increased.

Landau theory of Fermi liquids foresees that the charge and heat are tranported by the same objects: the fermionic Landau quasi-particles. In a very general way, it is true if the screening among particles is quite strong to can consider the system as composed yet by independent particles. This is the case for the electron sea in an ordinary bulk metal. The existance of just one responsible for charge and heat transport is expressed by the Wiedemann-Franz (WF) law. It states that the ratio between the thermal and electrical conductivity depends on temperature by a constant which is more or less the same for several metals. The constant of proportionality is called the Lorenz number. What happens if the above-mentioned condition on screening is no longer satisfied, as for example in low-dimensional systems or electronic low-density systems ? The thesis is diveded into two parts. In the first one, we studied the thermal and electrical transport in a disordered quantum wire; in the second one, the influence of superconducting fluctuations on thermal conductivity in granular metals
La théorie de Landau des liquides de Fermi prévoit que la charge et la chaleur sont transportées par les mêmes objets: les quasi-particules fermionics de Landau. De façon très général, ceci est vrai, si l'écrantage parmi les particules dans le système est assez fort pour pouvoir continuer à considérer le système comme composé de particules indépendantes. C'est le cas, par exemple, pour la mer d'électrons dans un métal ordinaire. L'existence d'un même responsable pour le transport de la charge et de la chaleur est exprimé par la lois de Wiedemann-Franz (WF) qui affirme que le rapport entre la conductivité thermique et électrique dépend de la température par une constante qui est plus au moins la même pour plusieurs métaux. La constante de proportionnalité est appelé nombre de Lorenz. Que se passe-t-il si les conditions concernant l'écrantage que nous avons mentionnées ne sont plus satisfaites, comme par exemple dans les systèmes à dimensionalité réduite, ou des système à basse densité électronique ? Le travail de thèse est divisé en deux parties. Dans la première partie, nous avons étudié le transport thermique et électrique dans un fil quantique désordonné; dans la deuxième, l'influence des fluctuations supraconductives sur la conductivité thermique dans un métal granulaire

Two mathematical problems in disordered systems are studied: geodesics in first-passage percolation and conductivity of random resistor networks. In first-passage percolation, we consider a translation-invariant ergodic family {t(b): b bond of Z²} of nonnegative random variables, where t(b) represent bond passage times. Geodesics are paths in Z², infinite in both directions, each of whose finite segments is time-minimizing. We prove part of the conjecture that geodesics do not exist in any fixed half-plane and that they have to intersect all straight lines with rational slopes. In random resistor networks, we consider an independent and identically distributed family {C(b): b bond of a hierarchical lattice H} of nonnegative random variables, where C(b) represent bond conductivities. A hierarchical lattice H is a sequence {H(n): n = 0, 1, 2} of lattices generated in an iterative manner. We prove a central limit theorem for a sequence x(n) of effective conductivities, each of which is defined on lattices H(n), when a system is in a percolating regime. At a critical point, it is expected to have non-Gaussian behavior.

Dans cette thèse nous avons étudié de nombreux aspects de la théorie des systèmes désordonnés. En particulier, nous avons étudié les systèmes vitreux. La description détaillée des systèmes désordonnés et vitreux est un problème ouvert en physique de la matière condensée. Dans le cadre de la théorie de champ moyen pour les verres structuraux nous avons étudié la théorie des fluctuations proche de la transition vitreuse dynamique. L’étude des fluctuations peut etre fait avec le formalisme statique de la théorie de répliques. Nous avons fait cela en introduisant une théorie des champs pour la transition vitreuse à partir du potentiel microscopique entre les particules. Nous avons étudié dans ce cadre les fluctuations au niveau gaussien et nous avons évalués les exposants critiques dans ces approximations. Nous avons aussi étudié la région de validité de la prédiction gaussienne avec l’introduction d’un critère de Ginzburg pour la transition vitreuse. Les résultats que nous avons obtenues ne sont valides que dans la région β. Pour obtenir des resultats dans la région α nous avons étudié la pseudodynamique de Boltzmann que a été introduit par Franz and Parisi. Nous sommes parti des équations de Ornstein-Zernike et nous avons obtenu un ensemble d’équations dynamiques. En utilisant l’approximation Hypernetted Chain nous avons obtenu un ensemble complet d’équations qui sont très similaires aux équations de la théorie de mode-coupling. La troisième partie de la thèse porte sur l’étude des états amorphes des sphères dures en hautes dimensions. Pour obtenir les exposants dynamique dans ce cas, nous avons étudié la stabilité du diagramme de phase 1RSB (one-step-replica-symmetry-breaking). Nous avons découvert que ce diagramme de phase possède une région où la solution 1RSB est instable. La région où la solution 1RSB est instable est connectée avec la description théorique de la physique de jamming des sphères dures et nous avons montré que l’instabilité 1RSB est responsable d’une transition de phase en haute densité. Cette transition s’appelle la transition de Gardner. Nous avons cherché une solution 2RSB et nous avons vu qu’il existait un point en densité après lequel on peut avoir une solution 2RSB (et aussi fullRSB). Nous avons étudié le diagramme de phase 2RSB dans la limite de jamming où la pression devient infini. Après la solution 2RSB nous avons cherché à décrire la solution fullRSB. Nous avons écrit les équations fullRSB et nous avons découvert qu’elles sont identiques aux equations que l’on a dans le cas de un modèle de verres de spins qui s’appelle modèle de Sherrington et Kirkpatrick. Nous avons aussi étudié la solution numerique des équations fullRSB dans la limite de jamming. Cette solution montre beaucoup des choses intéressantes. La plus importante est le comportement du mean square displacement dans la limite de jamming. Si l’on regard les résultats numériques et éxperimentaux, il semble que le plateau de le mean square displacement s’approche a zero comme la pression à un exposant proche de −3/2. Nous avons vu que la solution numérique des équations fullRSB est en mesure de reproduire ce comportement. La quatrième partie de la thése a porté sur la dynamique de mode-coupling dans le régime où la transition vitreuse devient continue
In this thesis we have studied many aspects of the physics of disordered and glassy systems. The first part of the work is about the theory of dynamical fluctuations in the beta regime. When a system undergoes a dynamical arrest, it can be studied by introducing an appropriate dynamical correlation function that plays the role of the order parameter of the transition. To understand the collective effects underlying the glass transition we have studied the fluctuations of the order parameter on a time scale where the system is relaxed in a typical metastable glassy state. To do this we have seen that coming from the glass phase the system develops critical fluctuations with a diverging correlation length at the mean field level. We have thus derived an effective field theory by focusing only on them. This field theory can be used firstly to derive the mode-coupling exponent parameter that controls the relaxation of the dynamical correlation function when the system relaxes in a metastable glassy state. Moreover we can give a Ginzburg Criterion that can be used to determine the region of validity of the Gaussian approximation. These considerations are valid in the beta regime. To clarify what happens in the alpha regime we have studied a quasi-equilibrium construction, called Boltzmann-Pseudodynamics, recently introduced in order to describe with static techniques the long time regime of glassy dynamics. We have extended this formalism to structural glasses by producing a new set of dynamical equations. We have done this in the simplest approximation scheme that is called Hypernetted Chain. Two results have been obtained : firstly, we have computed the mode-coupling exponent parameter and we have shown that it coincides with the one obtained with the formalism of the first part of the thesis ; secondly we have studied the aging regime and we have derived that the condition that determines the fluctuation-dissipation ratio is a marginal stability one. In the third part of the thesis we have studied the theory of amorphous states of hard spheres in high dimensions. Hard spheres provide simple models of glasses and they are extensively studied for the jamming transition. In our framework jammed states can be thought as infinite pressure limit of metastable glassy states. During the last years it has been derived a mean field theory of hard spheres based on the 1RSB assumption on the structure of the free energy landscape. However it has been realized that this construction is inconsistent for what concerns the property of the packings at jamming. In the present work we have firstly investigated the possibility of an instability of the 1RSB solution and we have actually found that the 1RSB solution is unstable in the jamming part of the phase diagram. At the same time we have been able to compute the mode-coupling exponent parameter for this system. In order to go beyond the 1RSB solution we have first tried a 2RSB ansatz and then a fullRSB solution. We have derived a set of variational equations that are very close to the ones that have been derived in the Sherrington-Kirkpatrick model. We have solved numerically the equations and we have shown that the fullRSB solution seems to predict that the plateau value of the mean square displacement scale as the pressure to a power close to 3/2 as it seems to be predicted by scaling arguments and in contrast with the 1RSB predictions that show a scaling with the inverse of the pressure. The last chapter of the thesis is on the mode-coupling theory when the glass transition is becoming continuous. We have been able to show that in such a situation a detailed characterization of the solution of the equations can be obtained in the long time regime

Embedded/real-time systems are becoming ubiquitous in today's world and their pervasive nature is increasing with the advent of cyber-physical systems. Providing temporal guarantees is paramount in such systems. Most of the normal operation in real-time systems is modelled using periodic tasks. Event-driven behaviour is modelled using aperiodic jobs. To ensure an acceptable Quality of Service for aperiodic jobs without jeopardizing safety of periodic tasks, aperiodic servers were introduced [2], [3]. Aperiodic servers are used to reserve a quota for the execution of aperiodic jobs. However, they do not take into account, cache-related delays that the execution of aperiodic jobs could impose on periodic tasks, thereby making their use in systems with caches unsafe. In this thesis, we introduce Cache Related Delay Servers to solve this problem. Statically, every periodic task's worst-case execution time includes a pre-determined delay quota for delay caused by aperiodic jobs. During system operation, the aperiodic server is allowed to execute only if periodic jobs that may be affected by it have sufficient delay quota to accommodate its execution. Otherwise, the priority of the aperiodic server is temporarily decreased to the level of the lowest-priority periodic job with insufficient quota, thereby ensuring safe execution of periodic tasks.

The transition from quantum to classical behavior of complex systems, known as dephasing, has fascinated physicists during the last decades. Disordered systems provide an insightful environment to study the dephasing time \tau_\varphi, since electron interference leads to quantum corrections to classical quantities, such as the weak- localization correction \Delta g to the conductance, whose magnitude is governed by \tau_\varphi. In this thesis, we study one of the fundamental questions in this field: How does Pauli blocking influence the interaction-induced dephasing time at low temperatures? In general, Pauli blocking limits the energy transfer \omega of electron interactions to \omega \ll T, which leads to an increase of \tau_\varphi. However, the so-called 0D regime of dephasing, reached at T \ll E_{Th}, is practically the only relevant regime, in which Pauli blocking significantly influences the temperature dependence of \tau_\varphi. Despite of its fundamental physical importance, 0D dephasing has not been observed experimentally in the past. We investigate several possible scenarios for verifying its existence: (1) We analyze the temperature dependence of \Delta g in open and confined systems and give detailed instructions on how the crossover to 0D dephasing can be reliably detected. Two concrete examples are studied: an almost isolated ring and a new quantum dot model. However, we conclude that in transport experiments, 0D dephasing unavoidably occurs in the universal regime, in which all quantum corrections to the conductance depend only weakly on \tau_\varphi, and hence carry only weak signatures of 0D dephasing. (2) We study the quantum corrections to the polarizability \Delta \alpha of isolated systems, and derive their dependence on \tau_\varphi and temperature. We show that \tZeroD dephasing occurs in a temperature range, in which \Delta \alpha depends strongly (as a power-law) on \tau_\varphi, making the quantum corrections to the polarizability an ideal candidate to study dephasing at low temperatures and the influence of Pauli blocking.

In many systems of interest, both physical and biological, disorder inhibits the organization and cooperative properties of the system. Disorder can originate from a variety of system defects and the degree of disorder also varies. Geometric frustration introduces disorder into a system in which all the preferred interactions between the elements of the system cannot be satisfied due to the topology of an underlying lattice that describes the position of these elements. Recently, geometric frustration has been recognized as an important organizing principle in a diverse range of systems from superconducting networks to neural computation. The correlation behavior of such systems is often complicated and poorly understood. The myosin lattice of higher vertebrate muscle is a geometrically frustrated system, and the presence of this kind of disorder has prevented a rigorous interpretation of X-ray diffraction patterns from muscle fibres for the purposes of studying muscle molecular structure. This thesis investigates the correlation behavior of two geometrically frustrated systems, the triangular Ising antiferromagnet (TIA) and the fully frustrated square Ising model (FFS), and its use to interpret X-ray fibre diffraction patterns. A combination of numerical evaluation of exact expressions and Monte Carlo simulation is used to study a number of aspects of the two-point correlation function of the TIA and FFS. In the case of the TIA, a simple functional expression is developed that allows accurate calculation of the correlation function. Theory is developed for calculating diffraction by polycrystalline fibres of helical molecules, in which the constituent crystallites contain correlated substitution disorder. The theory was used to study the characteristics of diffraction by fibres with TIA-type substitution disorder statistics. A quantitative model of the disorder in the myosin filament array is developed and the above theory is used to calculate X-ray fibre diffraction from low resolution models of the myosin filament array in higher vertebrate muscle. The calculated diffraction is compared to measured diffraction data, showing good agreement.

CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
Nesta tese estudamos as propriedades físicas de materiais fortemente correlacionados e desordenados, usando Hamiltonianos modelos para descrevê-los. A tese está dividida em duas partes. Na primeira, estudamos o modelo de Anderson periódico para descrever as propriedades de um isolante Kondo. Em particular tomamos o composto de Ce3Bi4Pt3 como paradigma deste tipo de materiais caracterizados por apresentar um pequeno gap(da ordem dos meV ). Na presença de pequenas concentrações de impurezas metálicas como íons de La substituindo os de Ce, como é o caso da liga (Ce1-xLax)Bi4Pt3, sofre uma transição metal-isolante. O Hamiltoniano de Anderson periódico é resolvido a partir da solução de um único sítio atômico que logo é embebido numa rede de Bethe. Este modelo consegue explicar qualitativamente os resultados experimentais como a resistividade em função da temperatura para diferentes concentrações de íons de La, assim como as propriedades óticas do sistema puro. A influência da localização de Anderson nesta transição é analisada a partir do estudo da condutividade elétrica do sistema. A segunda parte está dedicada ao estudo das propriedades de sistemas descritos pelo Hamiltoniano de Falicov- Kimball, largamente utilizado para estudar fenômenos como a transição de valência e metal- isolante, também em compostos de Metais de Transição e Terras Raras. Neste modelo, o caráter destas transições ainda não está bem estabelecido já que o resultado é muito dependente da aproximação utilizada. Utilizamos o Hamiltoniano de Falicov-Kimball sem spin onde a banda de condução é tratada de forma exata já que mostramos a sua equivalência com o problema de uma liga. Os estados f são resolvidos em forma aproximada a partir da equação de movimento, aproximação que chamamos de Aproximação do Estreitamento Dinâmico(AED). Estudamos as propriedades eletrônicas como a ocupação dos estados localizados em função da energia local. Também neste caso, analisamos um sistema desordenado estudando o contraponto entre a correlação eletrônica e a desordem. As diferentes fases que aparecem no sistema como, metálica, isolante de Anderson e de Mott são investigadas em função dos parâmetros que definem o sistema.
In this thesis we study the properties of strongly correlated and disordered materials, using model Hamiltonians to describe them. The thesis is divided in two parts. The first one studies the periodic Anderson model used to describe the properties of a Kondo insulator. In particular we take Ce3Bi4Pt3 as a paradigmatic compound, characterized by a small gap(of the order of meV ). For small concentration of metallic impurities, ions of La substituting Ce, the alloy (Ce1-xLax)Bi4Pt3 suffers a metal- insulator transition. The periodic Anderson Hamiltonian is solved using the atomic solution that is embedded into a Bethe lattice. This model explains the experimental results as the resistivity as a function of temperature for different concentrations of ions of La, as well as, the optical properties of the pure system. The Anderson localization is analyzed studying the electric conductivity of the system. The second part of the thesis is dedicated to study the property of a system described by the Falicov- Kimball Hamiltonian. This Hamiltonian has been used to study the valence and metal-insulator transitions in Transitions Metal and Rare Earth compounds. In this model, the character of these transitions is still not well understood, since it is very dependent of the approximation used. We study the Falicov-Kimball Hamiltonian without spin. The conduction band is exactly described since we show its equivalence with the problem of an alloy. The f states are studied using the equation of motion for the Green functions, decoupling them in a way defined as the Dynamic Narrowing Approximation(DNA). We study the occupation of the local states as a function of energy and other electronic properties. For an alloy the interplay between the electronic correlation and disorder is analized. The different phases that appear in the system, as metallic and Anderson and Mott insulating, are investigated as a function of the parameters that define the system.

In this thesis we study the problem of computing probability distribution of rare events on disordered systems, i.e. systems made up of heterogeneous interacting agents, using tools from statistical physics. We explore several areas of statistical physics, from traditional subjects such as the random field Ising model to random walkers on graphs to more recent developments such as systemic risk in financial networks, using simple models to explore how interactions and network structure shape the occurrence of rare events, and how these rare events in turn shape our macroscopic perceptions. We present exact analytical results using generating functional methods when possible, and propose approximations where needed, and we compare our results with simulations. Finally, we discuss how our models can be modified and extended to answer new questions.

We investigate the properties of two distinct disordered systems: the two-species predator-prey Lotka-Volterra model with rate variability, and an elastic line model to simulate vortex lines in type-II superconductors. We study the effects of intrinsic demographic variability with inheritance in the reaction rates of the Lotka-Volterra model via zero-dimensional Monte Carlo simulations as well as two-dimensional lattice simulations. Individuals of each species are assigned inheritable predation efficiencies during their creation, leading to evolutionary dynamics and thus population-level optimization. We derive an effective subspecies mean-field theory and compare its results to our numerical data. Furthermore, we introduce environmental variability via quenched spatial reaction-rate randomness. We investigate the competing effects and relative importance of the two types of variability, and find that both lead to a remarkable enhancement of the species densities, while the aforementioned optimization effects are essentially neutral in the densities. Additionally, we collected extinction time histograms for small systems and find a marked increase in the stability of the populations against extinction due to the presence of variability. We employ an elastic line model to investigate the steady-state properties and non-equilibrium relaxation kinetics of magnetic vortex lines in disordered type-II superconductors. To this end, we developed a versatile and efficient Langevin molecular dynamics simulation code, allowing us to do a careful study of samples with or without vortex-vortex interactions or disorder allows us to disentangle the various complex relaxational features present in this system and investigate their origin. In particular, we compare disordered samples with randomly distributed point defects versus correlated columnar defects. We extract two-time quantities such as the mean-square displacement, the height and density correlations, to investigate the relaxation kinetics of the system of flux lines. Additionally, we compare the steady-state mean velocity and gyration radius as a function of an external driving current in the presence of point-like and columnar disorder. We validate our simulation algorithm by matching our results against a previously-used Monte Carlo algorithm, verifying that these microscopically quite distinct methods yield similar results even in out-of-equilibrium settings.
Ph. D.

Thesis (Ph. D.)--University of Oregon, 2000.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 199-204). Also available for download via the World Wide Web; free to UO users.

The transport properties of low dimensional systems (especially wires) are investigated when the dominant scattering is due to the impurities and is elastic. Such a situation is expected to be relevant to experiments carried out at very low (liquid Helium) temperatures. Initially a Boltzmann formalism is used to illustrate the effects of multiple sub-band occupancy. Structure is found in the electrical conductivity, thermal conductivity and thermopower when plotted as a function of chemical potential, due to the lateral quantisation of the electron states. These quantum size effects (QSE) are most pronounced in the thermopower, which is expected to show sign changes when the chemical potential sweeps through a sub-band minimum. A more sophisticated treatment based on Green's function methods reveals the importance of lifetime broadening in quasi-one-dimensional systems, which smears out the single-particle density of states and the QSE. The type of behaviour expected of realistic devices is explored, and it is shown that the thermopower offers the best chance of observing confinement effects. The formal theory may also be applied to the weak localisation corrections in multi-sub-band systems. An expression for the correction term is obtained which is valid for arbitrary channel width, and enables the crossover from a linear to logarithmic scaling in L. to be demonstrated. A transverse inelastic length is derived, and shown to be the length scale which controls the system dimensionality rather than Lφ. The implication for experiment in narrow channels is discussed. Weak localisation corrections are also calculated for the thermopower and the thermal conductivity. This corrects a result due to Ting et al (1982) that there are no weak localisation corrections to the thermopower in 2D. These results are shown to be a consequence of a rather general scaling theory of thermal transport which has wider implications, such as for the behaviour expected near a Metal-Insulator transition for example. Comparison with the single parameter scaling theory of the zero temperature conductance is made. Fluctuation effects for thermal transport in mesoscopic samples are also explored (both numerically and analytically), and the analogue of universal conductance fluctuations explicitly demonstrated.

2013/2014
The present work of thesis is situated within the framework of the study of disordered systems as liquids and glasses. A liquid is a system characterized by long range translational invariance and by a short range ordered structure. In the liquid state, contrarily to the crystalline one, there is not structural periodicity and all we learnt from solid state physics (Block’s theorem, phonons, Brillouin’s zones, eigen-states of plane waves, etc.) must be fully revised. The macroscopic collective properties of condensed matter are the result of inter and intra-molecular interactions that are typified by characteristic time and space scales. A longstanding and powerful tool to investigate the collective nature of the microscopic processes inside the system is the acoustic spectroscopy. While in the case of crystalline phase we could take advantage of the periodical structure of the system, limiting our investigation to the so called first Brillouin zone, in order to characterize a disordered system we need to explore a widest as possible spectral range to access the all time and space scales in which the dynamical phenomena occur. This approach can be called Broad Band Acoustic Spectroscopy . My PhD activity was devoted to the development of new experimental methods and techniques allowing the exploration with continuity of dynamics evolving with timescales from tenth’s of ns’s to ps’s. I could test such Broad Band Acoustic Spectroscopy on a prototypical sample: acetonitrile, the liquid with highest dipole moment, known for its many different inter- and intra-molecular dynamics. Using several experimental and computational approaches I could characterize the main dynamical processes for such compound in its whole liquid phase. Thanks to the crossing of the all acquired information, it was possible to identify a mutual influence between different relaxations whose behaviour otherwise was not possible to correctly understand. After a brief introduction to contextualize the Broad Band Acoustic Spectroscopy in disordered systems, in Chapter 1 is presented an overview of the experimental techniques used to perform the measurements within this work. In this chapter, the first my original contribution to the extension of acoustic spectral range finds the place. Thanks to the design and realization of particular spatial filters it is now possible to perform Brillouin Light Scattering experiments with angle tunabil- ity without incurring in annoying spurious contributions issues. In the chapter are also recalled the main physical principles at the ground of every presented tech- nique, in particular stressing the complementarity of the energy and time resolved spectroscopies. Because all the information from the inner dynamics of the sample are mediated by the acoustic modes interaction, Chapter 2 is dedicated to the formalism of the density fluctuations, highlighting the differences between the hydrodynamic model, valid at macroscopic length-scales, and the memory functions approach, necessary to describe the mesoscopic region where the characteristic length-scales of the acoustic modes are comparable to the inter-particles distances. In Chapter 3 are shown the experimental results obtained thanks to the Broad Band Spectroscopy in the whole temperature range of the liquid phase of acetoni- trile. We could first measure some thermodynamical quantities by a non linear spectroscopy named Transient Grating, shedding light on the literature debate on them. Among the obtained results, we appreciated a temperature dependency of the heat capacity ratio which is usually considered a constant. Starting from these results and using the filtering approach introduced in Chapter 1, we could profitably study the vibrational relaxation of acetonitrile discovering some relevant discrepan- cies with previous studies. In this way we demonstrated the importance to avoid eventual spurious contributions and to have complementary information from dif- ferent spectral domains. Then there will be shown the results for the so called structural relaxation, obtained in the high energy domain by Inelastic X-ray Scat- tering. A really interesting correlation has been found with the result obtained by the ultrafast response of the Transient Grating method. Once again, to obtain such comparison was necessary to combine the information arising from many different spectral ranges, fact that highlights the utility and endorses the multi-techniques broad band acoustic spectroscopy method. The temperature dependence of the aforementioned relaxation processes sug- gested to evaluate if any coupling there exist between the two. This is exactly what we were able to appreciate in the deep-UV domain and we proposed a phenomeno- logical model to give a picture of the occurring interaction. To further investigate such dynamics coupling effect, could be extremely useful to explore the soft-UV spectral range but so far any technique was able to access such energy domain. In Chapter 4 we thus present our innovative solution to the problem: a novel table top Fabry-Perot interferometer conceived to operate with UV laser source and with only reflecting optics. The last chapters of the thesis describe my contribution to the forthcoming Free Electron Laser (FEL) based experiment to further increase the accessible spectral range to the acoustic spectroscopy: the TIMER project. In particular I will show the result obtained with the pilot experiment named “mini-TIMER” in which we could demonstrate the feasibility of a Transient Grating experiment in the Extreme Ultra Violet (EUV) domain. When TIMER will be operative it will be the first experimental setup able to probe the EUV mesoscopic region of crucial importance for the study of glasses and nano-structures. This result paves the way to a new class of intriguing experiments only matter of theoretical considerations so far: the so called four wave mixing experiments with elemental selectivity.
Il presente lavoro di tesi si colloca nell’ambito dello studio dei sistemi disordi- nati quali liquidi e vetri. Un liquido ´e un sistema caratterizzato da invarianza traslazionale a lungo raggio e da una struttura ordinata a corto raggio. Lo stato liquido, a differenza di quello cristallino, non gode di una periodicit`a strutturale e tutto ci`o che avevamo imparato dalla fisica dello stato solido (teorema di Block, fononi, zone di Brillouin, autostati di onde piane, etc.) deve essere completamente rivisto. Le propriet`a collettive della materia sono il risultato di interazioni a livello inter- e intra-molecolare identificate da caratteristiche scale temporali e spaziali. Uno strumento ormai affermato e utile per lo studio della natura collettiva dei processi microscopici attivi all’interno del sistema ´e la spettroscopia acustica. Mentre nel caso della fase cristallina si poteva sfruttare la natura periodica della struttura del sistema, limitando lo studio alla cosiddetta prima zona di Brillouin, per caratteriz- zare un sistema disordinato ´e necessario esplorare un intervallo spettrale il pi`u ampio possibile cosí da poter aver accesso alle scale spazio-temporali in cui le dinamiche avvengo. Questo tipo di approccio pu`o essere chiamato Spettroscopia Acustica ad Ampia Banda Spettrale . Il mio dottorato ´e stato dedicato allo sviluppo di nuovi metodi e tecniche speri- mentali per esplorare con continuit`a i processi dinamici la cui evoluzione avviene su scale di tempi tra le decine di ns e i ps. Tale Spettroscopia Acustica ad Ampia Banda Spettrale ´e stata testata su un campione prototipo di acetonitrile, il liquido con il pi´u alto momento di dipolo esistente, noto per le sue molteplici dinamiche di orgine inter- e intra-molecolare. Usando diversi approcci sperimentali e computazionali ´e stato possibile caratterizzare i principali processi di rilassamento per tale composto in tutta la sua fase liquida. Combinando poi tutte le informazioni acquisite, ´e stato possibile indentificare una mutua influenza tra i diversi processi di rilassamento il cui comportamento altrimenti sarebbe rimasto incompreso. Dopo una breve introduzione per contestualizzare la Spettroscopia Acustica ad Ampia Banda Spettrale nell’ambito dei sistemi disordinati, nel Capitolo 1 viene offerta una panoramica delle tecniche sperimentali usate per effettuare le misure nel corso di questo lavoro. Trova posto in questo capitolo la descrizione del mio primo originale contributo all’estensione dell’intervallo spettrale acustico. Grazie alla progettazione e realizzazione di particolari filtri spaziali ´e ora possibile effet- tuare esperimenti di diffusione di luce Brillouin con angolo di scattering variabile senza dover incorrere in fastidiosi problemi di contributi spuri. Nel capitolo vengono anche richiamati i principali elementi di fisica alla base di ogni tecnica, marcando in particolare la caratteristica di complementariet`a tra esperimenti risolti in tempo e in energia. Siccome tutte le informazioni a proposito delle dinamiche interne del campione sono mediate dall’interazione coi modi acustici, il Capitolo 2´e dedicato al formalismo delle fluttuazioni di densit`a, evidenziando le differenze tra il modello idrodinamico, valido per scale spaziali macroscopiche, e l’approccio delle funzioni memoria, nec- essario per descrivere la regione mesoscopica dove le dimensioni caratteristiche dei modi acustici diventano confrontabili con le distanze tra le particelle. Nel Capitolo 3 sono riportati i risultati sperimentali ottenuti mediante la Spet- troscopia Acustica ad Ampio Intervallo Spettrale nell’intero range di temperature in cui l’acetonitrile permane allo stato liquido. Per prima cosa, attraverso un tecnica fotonica di spettroscopia non lineare (Transient Grating) ´e stato possibile misurare alcune variabili termodinamiche, potendo fare chiarezza tra i vari contributi presenti in letteratura. Tra i risultati ottenuti, la dipendenza in temperatura del rapporto tra i calori specifici laddove usualmente viene considerata costante. Partendo da questi risultati e usando l’approccio di filtraggio introdotto nel Capitolo 1, ´e stato possibile studiare il rilassamento vibrazionale dell’acetonitrile scoprendo alcune ril- evanti discrepanze con i precedenti lavori riportati in letteratura. Abbiamo cosí dimostrato l’importanza di eliminare eventuali contributi spuri e di poter attingere a informazioni complementari da diversi domini spettrali. Verr`a inoltre mostrato il risultato dello studio del cosiddetto rilassamento strutturale effettuato nel range di alte energie grazie allo Scattering di raggi X. Una interessante correlazione si ´e po- tuta riscontrare tra i risultati di questo esperimento e quelli ottenuti con la tecnica Transient Grating nella sua risposta ultraveloce. Ancora una volta, per ottenere tale risultato ´e stato necessario combinare informazioni provenienti da molti inter- valli spettrali diversi, confermando la validit`a di un approccio multi-tecnica come quello della spettroscopia acustica a larga banda spettrale. L’andamento in tem- peratura dei suddetti processi di rilassamento suggerivano di valutare la presenza di un eventuale accoppiamento tra i due fenomeni. ´E esattamente ci`o che abbiamo osservato esplorando il dominio del profondo ultravioletto e per il quale abbiamo proposto un modello fenomenologico in grado di fornire una rappresentazione delle interazioni in gioco. Per poter approfondire tale fenomeno di accoppiamento sarebbe estremamente utile poter esplorare l’intervallo spettrale degli UV soffici, ma fino ad oggi nessuna tecnica era in grado accedere a tale range. Nel Capitolo 4 mostriamo la nostra innovativa soluzione al problema: un nuovo interferometro Farby-Perot “table-top”, concepito per lavorare con una sorgente UV laser e con sole ottiche riflettive. Gli ultimi capitoli della tesi descrivono il mio contributo al prossimo esperi- mento avente come sorgente di luce un laser ad elettroni liberi: il progetto TIMER, destinato ad aumentare ulteriormente il range spettrale sperimentalmente acces- sibile con la spettroscopia acustica. In particolare, mostrer`o i risultati ottenuti nell’esperimentopilotachiamato“mini-TIMER”graziealqualedimostratolapossibilit`a di effettuare un esperimento Transient Grating anche nel range spettrale dell’estremo UV (EUV). Quando TIMER sar`a operativo sar`a il primo setup sperimentale in grado di sondare la regione mesoscopica nell’EUV, zona di cruciale importanza per lo studio dei vetri e delle nanostrutture. Questo risultato apre inoltre la strada ad una nuova classe di interessanti esperimenti ad oggi oggetto solo di considerazioni teoriche: i cosiddetti esperimenti di “four wave mixing” combinati con la selettivit`a elementale che la radiazione EUV pu`o fornire.
XXVI Ciclo
1985

I obtain a number of results for the dynamics of several disordered spin systems, of successively greater complexity. I commence with the generalised Hopfield model trained with an intensive number of patterns, where in the thermodynamic limit macroscopic, deterministic equations of motion can be derived exactly for both the synchronous discrete time and asynchronous continuous time dynamics. I show that for symmetric embedding matrices Lyapunov functions exist at the macroscopic level of description in terms of pattern overlaps. I then show that for asymmetric embedding matrices several types of bifurcation phenomena to complex non-transient dynamics occur, even in this simplest model. Extending a recent result of Coolen and Sherrington, I show how the dynamics of the generalised Hopfield model trained with extensively many patterns and non-trivial embedding matrix can be described by the evolution of a small number of overlaps and the disordered contribution to the 'energy', upon calculation of a noise distribution by the replica method. The evaluation of the noise distribution requires two key assumptions: that the flow equations are self averaging, and that equipartitioning of probability occurs within the macroscopic sub-shells of the ensemble. This method is inexact on intermediate time scales, due to the microscopic information integrated out in order to derive a closed set of equations. I then show how this theory can be improved in a systematic manner by introducing an order parameter function - the joint distribution of spins and local alignment fields, which evolves in time deterministically, according to a driven diffusion type equation. I show how the coefficients in this equation can be evaluated for the generalised Sherrington-Kirkpatrick model, both within the replica symmetric ansatz, and using Parisi's ultrametric ansatz for the replica matrices, upon making once again the two key assumptions (self averaging and equipartitioning). Since the order parameter is now a continuous function, however, the assumption of equipartitioning within the macroscopic sub-shells is much less restricting.

Cette thèse présente plusieurs aspects de la physique des systèmes élastiques désordonnés et des méthodes analytiques utilisées pour les étudier. On s’intéressera d’une part aux propriétés universelles des processus d’avalanches statiques et dynamiques (à la transition de dépiégeage) d’interfaces élastiques de dimension arbitraire en milieu aléatoire à température nulle. Pour étudier ces questions nous utiliserons le groupe de renormalisation fonctionnel. Après une revue de ces aspects,nous présenterons plus particulièrement les résultats obtenus pendant la thèse sur (i) la structure spatiale des avalanches et (ii) les corrélations entre avalanches.On s’intéressera d’autre part aux propriétés statiques à température finie de polymères dirigés en dimension 1+1, et en particulier aux observables liées à la classe d’universalité KPZ. Dans ce contexte l’étude de modèles exactement solubles a récemment permis de grands progrès. Après une revue de ces aspects, nous nous intéresserons plus particulièrement aux modèles exactement solubles de polymère dirigé sur le réseau carré, et présenterons les résultats obtenus pendantla thèse dans cette voie: (i) classification des modèles à température finie sur le réseau carré exactement solubles par ansatz de Bethe; (ii) universalité KPZ pour les modèles Log-Gamma et Inverse-Beta; (iii) universalité et nonuniversalitéKPZ pour le modèle Beta; (iv) mesures stationnaires du modèle Inverse-Beta et des modèles à température nulle associés
This thesis presents several aspects of the physics of disordered elastic systems and of the analytical methods used for their study.On one hand we will be interested in universal properties of avalanche processes in the statics and dynamics (at the depinning transition) of elastic interfaces of arbitrary dimension in disordered media at zero temperature. To study these questions we will use the functional renormalization group. After a review of these aspects we will more particularly present the results obtained during the thesis on (i) the spatial structure of avalanches and (ii) the correlations between avalanches.On the other hand we will be interested in static properties of directed polymers in 1+1 dimension, and in particular in observables related to the KPZ universality class. In this context the study of exactly solvable models has recently led to important progress. After a review of these aspects we will be more particularly interested in exactly solvable models of directed polymer on the square lattice and present the results obtained during the thesis in this direction: (i) classification ofBethe ansatz exactly solvable models of directed polymer at finite temperature on the square lattice; (ii) KPZ universality for the Log-Gamma and Inverse-Beta models; (iii) KPZ universality and non-universality for the Beta model; (iv) stationary measures of the Inverse- Beta model and of related zero temperature models