Caos e termalização na teoria de Yang-Mills-Higgs em uma rede espacial

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Multi-q pattern analysis: A case study in image classification

This paper compares the effectiveness of the Tsallis entropy over the classic Boltzmann-Gibbs-Shannon entropy for general pattern recognition, and proposes a multi-q approach to improve pattern analysis using entropy. A series of experiments were carried out for the problem of classifying image patterns. Given a dataset of 40 pattern classes, the goal of our image case study is to assess how well the different entropies can be used to determine the class of a newly given image sample. Our experiments show that the Tsallis entropy using the proposed multi-q approach has great advantages over the Boltzmann-Gibbs-Shannon entropy for pattern classification, boosting image recognition rates by a factor of 3. We discuss the reasons behind this success, shedding light on the usefulness of the Tsallis entropy and the multi-q approach. (C) 2012 Elsevier B.V. All rights reserved.

Measurement of event background fluctuations for charged particle jet reconstruction in Pb-Pb collisions at root s(NN)=2.76 TeV

The effect of event background fluctuations on charged particle jet reconstruction in Pb-Pb collisions at root s(NN) = 2.76 TeV has been measured with the ALICE experiment. The main sources of non-statistical fluctuations are characterized based purely on experimental data with an unbiased method, as well as by using single high p(t) particles and simulated jets embedded into real Pb-Pb events and reconstructed with the anti-k(t) jet finder. The influence of a low transverse momentum cut-off on particles used in the jet reconstruction is quantified by varying the minimum track p(t) between 0.15 GeV/c and 2 GeV/c. For embedded jets reconstructed from charged particles with p(t) > 0.15 GeV/c, the uncertainty in the reconstructed jet transverse momentum due to the heavy-ion background is measured to be 11.3 GeV/c (standard deviation) for the 10% most central Pb-Pb collisions, slightly larger than the value of 11.0 GeV/c measured using the unbiased method. For a higher particle transverse momentum threshold of 2 GeV/c, which will generate a stronger bias towards hard fragmentation in the jet finding process, the standard deviation of the fluctuations in the reconstructed jet transverse momentum is reduced to 4.8-5.0 GeV/c for the 10% most central events. A non-Gaussian tail of the momentum uncertainty is observed and its impact on the reconstructed jet spectrum is evaluated for varying particle momentum thresholds, by folding the measured fluctuations with steeply falling spectra.

Exact correlation functions in particle-reaction models with immobile particles

Exact results on particle densities as well as correlators in two models of immobile particles, containing either a single species or else two distinct species, are derived. The models evolve following a descent dynamics through pair annihilation where each particle interacts once at most throughout its entire history. The resulting large number of stationary states leads to a non-vanishing configurational entropy. Our results are established for arbitrary initial conditions and are derived via a generating function method. The single-species model is the dual of the 1D zero-temperature kinetic Ising model with Kimball-Deker-Haake dynamics. In this way, both in finite and semi-infinite chains and also the Bethe lattice can be analysed. The relationship with the random sequential adsorption of dimers and weakly tapped granular materials is discussed.

Nonlocal asymmetric exclusion process on a ring and conformal invariance

We present a one-dimensional nonlocal hopping model with exclusion on a ring. The model is related to the Raise and Peel growth model. A nonnegative parameter u controls the ratio of the local backwards and nonlocal forwards hopping rates. The phase diagram, and consequently the values of the current, depend on u and the density of particles. In the special case of half-lling and u = 1 the system is conformal invariant and an exact value of the current for any size L of the system is conjectured and checked for large lattice sizes in Monte Carlo simulations. For u > 1 the current has a non-analytic dependence on the density when the latter approaches the half-lling value.

Probabilistic approaches in civil engineering: generation of random fields and structural identification with genetic algorithms

The inherent stochastic character of most of the physical quantities involved in engineering models has led to an always increasing interest for probabilistic analysis. Many approaches to stochastic analysis have been proposed. However, it is widely acknowledged that the only universal method available to solve accurately any kind of stochastic mechanics problem is Monte Carlo Simulation. One of the key parts in the implementation of this technique is the accurate and efficient generation of samples of the random processes and fields involved in the problem at hand. In the present thesis an original method for the simulation of homogeneous, multi-dimensional, multi-variate, non-Gaussian random fields is proposed. The algorithm has proved to be very accurate in matching both the target spectrum and the marginal probability. The computational efficiency and robustness are very good too, even when dealing with strongly non-Gaussian distributions. What is more, the resulting samples posses all the relevant, welldefined and desired properties of “translation fields”, including crossing rates and distributions of extremes. The topic of the second part of the thesis lies in the field of non-destructive parametric structural identification. Its objective is to evaluate the mechanical characteristics of constituent bars in existing truss structures, using static loads and strain measurements. In the cases of missing data and of damages that interest only a small portion of the bar, Genetic Algorithm have proved to be an effective tool to solve the problem.

Thermal relaxation for particle systems in interaction with several bosonic heat reservoirs

The present thesis is concerned with the study of a quantum physical system composed of a small particle system (such as a spin chain) and several quantized massless boson fields (as photon gasses or phonon fields) at positive temperature. The setup serves as a simplified model for matter in interaction with thermal "radiation" from different sources. Hereby, questions concerning the dynamical and thermodynamic properties of particle-boson configurations far from thermal equilibrium are in the center of interest. We study a specific situation where the particle system is brought in contact with the boson systems (occasionally referred to as heat reservoirs) where the reservoirs are prepared close to thermal equilibrium states, each at a different temperature. We analyze the interacting time evolution of such an initial configuration and we show thermal relaxation of the system into a stationary state, i.e., we prove the existence of a time invariant state which is the unique limit state of the considered initial configurations evolving in time. As long as the reservoirs have been prepared at different temperatures, this stationary state features thermodynamic characteristics as stationary energy fluxes and a positive entropy production rate which distinguishes it from being a thermal equilibrium at any temperature. Therefore, we refer to it as non-equilibrium stationary state or simply NESS. The physical setup is phrased mathematically in the language of C*-algebras. The thesis gives an extended review of the application of operator algebraic theories to quantum statistical mechanics and introduces in detail the mathematical objects to describe matter in interaction with radiation. The C*-theory is adapted to the concrete setup. The algebraic description of the system is lifted into a Hilbert space framework. The appropriate Hilbert space representation is given by a bosonic Fock space over a suitable L2-space. The first part of the present work is concluded by the derivation of a spectral theory which connects the dynamical and thermodynamic features with spectral properties of a suitable generator, say K, of the time evolution in this Hilbert space setting. That way, the question about thermal relaxation becomes a spectral problem. The operator K is of Pauli-Fierz type. The spectral analysis of the generator K follows. This task is the core part of the work and it employs various kinds of functional analytic techniques. The operator K results from a perturbation of an operator L0 which describes the non-interacting particle-boson system. All spectral considerations are done in a perturbative regime, i.e., we assume that the strength of the coupling is sufficiently small. The extraction of dynamical features of the system from properties of K requires, in particular, the knowledge about the spectrum of K in the nearest vicinity of eigenvalues of the unperturbed operator L0. Since convergent Neumann series expansions only qualify to study the perturbed spectrum in the neighborhood of the unperturbed one on a scale of order of the coupling strength we need to apply a more refined tool, the Feshbach map. This technique allows the analysis of the spectrum on a smaller scale by transferring the analysis to a spectral subspace. The need of spectral information on arbitrary scales requires an iteration of the Feshbach map. This procedure leads to an operator-theoretic renormalization group. The reader is introduced to the Feshbach technique and the renormalization procedure based on it is discussed in full detail. Further, it is explained how the spectral information is extracted from the renormalization group flow. The present dissertation is an extension of two kinds of a recent research contribution by Jakšić and Pillet to a similar physical setup. Firstly, we consider the more delicate situation of bosonic heat reservoirs instead of fermionic ones, and secondly, the system can be studied uniformly for small reservoir temperatures. The adaption of the Feshbach map-based renormalization procedure by Bach, Chen, Fröhlich, and Sigal to concrete spectral problems in quantum statistical mechanics is a further novelty of this work.

Quantifying and modeling financial market fluctuations

Die Entstehung eines Marktpreises für einen Vermögenswert kann als Superposition der einzelnen Aktionen der Marktteilnehmer aufgefasst werden, die damit kumulativ Angebot und Nachfrage erzeugen. Dies ist in der statistischen Physik mit der Entstehung makroskopischer Eigenschaften vergleichbar, die von mikroskopischen Wechselwirkungen zwischen den beteiligten Systemkomponenten hervorgerufen werden. Die Verteilung der Preisänderungen an Finanzmärkten unterscheidet sich deutlich von einer Gaußverteilung. Dies führt zu empirischen Besonderheiten des Preisprozesses, zu denen neben dem Skalierungsverhalten nicht-triviale Korrelationsfunktionen und zeitlich gehäufte Volatilität zählen. In der vorliegenden Arbeit liegt der Fokus auf der Analyse von Finanzmarktzeitreihen und den darin enthaltenen Korrelationen. Es wird ein neues Verfahren zur Quantifizierung von Muster-basierten komplexen Korrelationen einer Zeitreihe entwickelt. Mit dieser Methodik werden signifikante Anzeichen dafür gefunden, dass sich typische Verhaltensmuster von Finanzmarktteilnehmern auf kurzen Zeitskalen manifestieren, dass also die Reaktion auf einen gegebenen Preisverlauf nicht rein zufällig ist, sondern vielmehr ähnliche Preisverläufe auch ähnliche Reaktionen hervorrufen. Ausgehend von der Untersuchung der komplexen Korrelationen in Finanzmarktzeitreihen wird die Frage behandelt, welche Eigenschaften sich beim Wechsel von einem positiven Trend zu einem negativen Trend verändern. Eine empirische Quantifizierung mittels Reskalierung liefert das Resultat, dass unabhängig von der betrachteten Zeitskala neue Preisextrema mit einem Anstieg des Transaktionsvolumens und einer Reduktion der Zeitintervalle zwischen Transaktionen einhergehen. Diese Abhängigkeiten weisen Charakteristika auf, die man auch in anderen komplexen Systemen in der Natur und speziell in physikalischen Systemen vorfindet. Über 9 Größenordnungen in der Zeit sind diese Eigenschaften auch unabhängig vom analysierten Markt - Trends, die nur für Sekunden bestehen, zeigen die gleiche Charakteristik wie Trends auf Zeitskalen von Monaten. Dies eröffnet die Möglichkeit, mehr über Finanzmarktblasen und deren Zusammenbrüche zu lernen, da Trends auf kleinen Zeitskalen viel häufiger auftreten. Zusätzlich wird eine Monte Carlo-basierte Simulation des Finanzmarktes analysiert und erweitert, um die empirischen Eigenschaften zu reproduzieren und Einblicke in deren Ursachen zu erhalten, die zum einen in der Finanzmarktmikrostruktur und andererseits in der Risikoaversion der Handelsteilnehmer zu suchen sind. Für die rechenzeitintensiven Verfahren kann mittels Parallelisierung auf einer Graphikkartenarchitektur eine deutliche Rechenzeitreduktion erreicht werden. Um das weite Spektrum an Einsatzbereichen von Graphikkarten zu aufzuzeigen, wird auch ein Standardmodell der statistischen Physik - das Ising-Modell - auf die Graphikkarte mit signifikanten Laufzeitvorteilen portiert. Teilresultate der Arbeit sind publiziert in [PGPS07, PPS08, Pre11, PVPS09b, PVPS09a, PS09, PS10a, SBF+10, BVP10, Pre10, PS10b, PSS10, SBF+11, PB10].

Rigorous results in Spin Glasses and Monomer-Dimer systems

In this work I reported recent results in the field of Statistical Mechanics of Equilibrium, and in particular in Spin Glass models and Monomer Dimer models . We start giving the mathematical background and the general formalism for Spin (Disordered) Models with some of their applications to physical and mathematical problems. Next we move on general aspects of the theory of spin glasses, in particular to the Sherrington-Kirkpatrick model which is of fundamental interest for the work. In Chapter 3, we introduce the Multi-species Sherrington-Kirkpatrick model (MSK), we prove the existence of the thermodynamical limit and the Guerra's Bound for the quenched pressure together with a detailed analysis of the annealed and the replica symmetric regime. The result is a multidimensional generalization of the Parisi's theory. Finally we brie y illustrate the strategy of the Panchenko's proof of the lower bound. In Chapter 4 we discuss the Aizenmann-Contucci and the Ghirlanda-Guerra identities for a wide class of Spin Glass models. As an example of application, we discuss the role of these identities in the proof of the lower bound. In Chapter 5 we introduce the basic mathematical formalism of Monomer Dimer models. We introduce a Gaussian representation of the partition function that will be fundamental in the rest of the work. In Chapter 6, we introduce an interacting Monomer-Dimer model. Its exact solution is derived and a detailed study of its analytical properties and related physical quantities is performed. In Chapter 7, we introduce a quenched randomness in the Monomer Dimer model and show that, under suitable conditions the pressure is a self averaging quantity. The main result is that, if we consider randomness only in the monomer activity, the model is exactly solvable.

Optische Mikroskopie an elektrostatisch stabilisierten kolloidalen Suspensionen unter dem Einfluss geladener Wände - ausgewählte Gleichgewichts- und Nichtgleichgewichtsphänomene

Das Verhalten kolloidaler Suspensionen unter räumlich beschränkter Geometrie ist von großer Bedeutung für die statistische Physik wie auch für die Technologie. Von speziellem Interesse sind Modellsysteme geladener kolloidaler Sphären aufgrund ihrer langreichweitigen und veränderbaren Wechselwirkungen. In dieser Arbeit wurde ein experimenteller Aufbau für die optische mikroskopische Untersuchung solcher, zwischen ebenen Wänden beschränkter Systeme realisiert. Anhand von Piezo-Aktuatoren kann die Zellgeometrie flexibel und präzise eingestellt werden. Unter Verwendung eines Pumpkreislaufs mit einer Ionentauschersäule können kolloidale Suspensionen unter stark entsalzten Bedingungen effizient präpariert werden. Anhand dieses Aufbaus wurde zunächst das Gleichgewichtsphasendiagramm monodisperser geladener kolloidaler Sphären zwischen parallelen Wänden untersucht. Es wurden quantitative Resultate für den Grenzfall starker Entsalzung erzielt, welche mit theoretischen Grundzustandsvorhersagen übereinstimmen. In Doppellagensystemen konnte die Existenz transienter kolloidaler Moiré-Rotationsmuster demonstriert werden, welche besondere zweidimensionale Kristallstrukturen mit komplexer Basis darstellen. Es wurden ferner Nichtgleichgewichtsphänomene untersucht, welche durch Gradienten von lokal freigesetzten Elektrolyten verursacht werden. Durch hauptsächlich diffusioosmotischen Partikeltransport entlang einer einzelnen geladenen Substratoberfläche konnten die Bildung kristalliner Ordnung sowie komplexe, selbstorganisierte Bewegungszustände in einem verdünnten kolloidalen Monolagenfluid bei kleinen Reynolds-Zahlen induziert werden. Interessante Perspektiven für die zukünftige Verwendung des experimentellen Aufbaus ergeben sich aus Beobachtungen verschiedener weiterer Phänomene.

Aplicación de la física estadística a los medios granulares

Esta tesis se centra en el estudio de medios granulares blandos y atascados mediante la aplicación de la física estadística. Esta aproximación se sitúa entre los tradicionales enfoques macro y micromecánicos: trata de establecer cuáles son las propiedades macroscópicas esperables de un sistema granular en base a un análisis de las propiedades de las partículas y las interacciones que se producen entre ellas y a una consideración de las restricciones macroscópicas del sistema. Para ello se utiliza la teoría estadística junto con algunos principios, conceptos y definiciones de la teoría de los medios continuos (campo de tensiones y deformaciones, energía potencial elástica, etc) y algunas técnicas de homogeneización. La interacción entre las partículas es analizada mediante las aportaciones de la teoría del contacto y de las fuerzas capilares (producidas por eventuales meniscos de líquido cuando el medio está húmedo). La idea básica de la mecánica estadística es que entre todas soluciones de un problema físico (como puede ser el ensamblaje en equilibrio estático de partículas de un medio granular) existe un conjunto que es compatible con el conocimiento macroscópico que tenemos del sistema (por ejemplo, su volumen, la tensión a la que está sometido, la energía potencial elástica que almacena, etc.). Este conjunto todavía contiene un número enorme de soluciones. Pues bien, si no hay ninguna información adicional es razonable pensar que no existe ningún motivo para que alguna de estas soluciones sea más probable que las demás. Entonces parece natural asignarles a todas ellas el mismo peso estadístico y construir una función matemática compatible. Actuando de este modo se obtiene cuál es la función de distribución más probable de algunas cantidades asociadas a las soluciones, para lo cual es muy importante asegurarse de que todas ellas son igualmente accesibles por el procedimiento de ensamblaje o protocolo. Este enfoque se desarrolló en sus orígenes para el estudio de los gases ideales pero se puede extender para sistemas no térmicos como los analizados en esta tesis. En este sentido el primer intento se produjo hace poco más de veinte años y es la colectividad de volumen. Desde entonces esta ha sido empleada y mejorada por muchos investigadores en todo el mundo, mientras que han surgido otras, como la de la energía o la del fuerza-momento (tensión multiplicada por volumen). Cada colectividad describe, en definitiva, conjuntos de soluciones caracterizados por diferentes restricciones macroscópicas, pero de todos ellos resultan distribuciones estadísticas de tipo Maxwell-Boltzmann y controladas por dichas restricciones. En base a estos trabajos previos, en esta tesis se ha adaptado el enfoque clásico de la física estadística para el caso de medios granulares blandos. Se ha propuesto un marco general para estudiar estas colectividades que se basa en la comparación de todas las posibles soluciones en un espacio matemático definido por las componentes del fuerza-momento y en unas funciones de densidad de estados. Este desarrollo teórico se complementa con resultados obtenidos mediante simulación de la compresión cíclica de sistemas granulares bidimensionales. Se utilizó para ello un método de dinámica molecular, MD (o DEM). Las simulaciones consideran una interacción mecánica elástica, lineal y amortiguada a la que se ha añadido, en algunos casos, la fuerza cohesiva producida por meniscos de agua. Se realizaron cálculos en serie y en paralelo. Los resultados no solo prueban que las funciones de distribución de las componentes de fuerza-momento del sistema sometido a un protocolo específico parecen ser universales, sino que también revelan que existen muchos aspectos computacionales que pueden determinar cuáles son las soluciones accesibles. This thesis focuses on the application of statistical mechanics for the study of static and jammed packings of soft granular media. Such approach lies between micro and macromechanics: it tries to establish what the expected macroscopic properties of a granular system are, by starting from a micromechanical analysis of the features of the particles, and the interactions between them, and by considering the macroscopic constraints of the system. To do that, statistics together with some principles, concepts and definitions of continuum mechanics (e.g. stress and strain fields, elastic potential energy, etc.) as well as some homogenization techniques are used. The interaction between the particles of a granular system is examined too and theories on contact and capillary forces (when the media are wet) are revisited. The basic idea of statistical mechanics is that among the solutions of a physical problem (e.g. the static arrangement of particles in mechanical equilibrium) there is a class that is compatible with our macroscopic knowledge of the system (volume, stress, elastic potential energy,...). This class still contains an enormous number of solutions. In the absence of further information there is not any a priori reason for favoring one of these more than any other. Hence we shall naturally construct the equilibrium function by assigning equal statistical weights to all the functions compatible with our requirements. This procedure leads to the most probable statistical distribution of some quantities, but it is necessary to guarantee that all the solutions are likely accessed. This approach was originally set up for the study of ideal gases, but it can be extended to non-thermal systems too. In this connection, the first attempt for granular systems was the volume ensemble, developed about 20 years ago. Since then, this model has been followed and improved upon by many researchers around the world, while other two approaches have also been set up: energy and force-moment (i.e. stress multiplied by volume) ensembles. Each ensemble is described by different macroscopic constraints but all of them result on a Maxwell-Boltzmann statistical distribution, which is precisely controlled by the respective constraints. According to this previous work, in this thesis the classical statistical mechanics approach is introduced and adapted to the case of soft granular media. A general framework, which includes these three ensembles and uses a force-moment phase space and a density of states function, is proposed. This theoretical development is complemented by molecular dynamics (or DEM) simulations of the cyclic compression of 2D granular systems. Simulations were carried out by considering spring-dashpot mechanical interactions and attractive capillary forces in some cases. They were run on single and parallel processors. Results not only prove that the statistical distributions of the force-moment components obtained with a specific protocol seem to be universal, but also that there are many computational issues that can determine what the attained packings or solutions are.

Statistical distributions obtained from the compression of monodisperse, soft and frictionless particles

The cyclic compression of several granular systems has been simulated with a molecular dynamics code. All the samples consisted of bidimensional, soft, frictionless and equal-sized particles that were initially arranged according to a squared lattice and were compressed by randomly generated irregular walls. The compression protocols can be described by some control variables (volume or external force acting on the walls) and by some dimensionless factors, that relate stiffness, density, diameter, damping ratio and water surface tension to the external forces, displacements and periods. Each protocol, that is associated to a dynamic process, results in an arrangement with its own macroscopic features: volume (or packing ratio), coordination number, and stress; and the differences between packings can be highly significant. The statistical distribution of the force-moment state of the particles (i.e. the equivalent average stress multiplied by the volume) is analyzed. In spite of the lack of a theoretical framework based on statistical mechanics specific for these protocols, it is shown how the obtained distributions of mean and relative deviatoric force-moment are. Then it is discussed on the nature of these distributions and on their relation to specific protocols.

Análisis de la actividad neuronal y cerebral mediante técnicas de redes complejas

El cerebro humano es probablemente uno de los sistemas más complejos a los que nos enfrentamos en la actualidad, si bien es también uno de los más fascinantes. Sin embargo, la compresión de cómo el cerebro organiza su actividad para llevar a cabo tareas complejas es un problema plagado de restos y obstáculos. En sus inicios la neuroimagen y la electrofisiología tenían como objetivo la identificación de regiones asociadas a activaciones relacionadas con tareas especificas, o con patrones locales que variaban en el tiempo dada cierta actividad. Sin embargo, actualmente existe un consenso acerca de que la actividad cerebral tiene un carácter temporal multiescala y espacialmente extendido, lo que lleva a considerar el cerebro como una gran red de áreas cerebrales coordinadas, cuyas conexiones funcionales son continuamente creadas y destruidas. Hasta hace poco, el énfasis de los estudios de la actividad cerebral funcional se han centrado en la identidad de los nodos particulares que forman estas redes, y en la caracterización de métricas de conectividad entre ellos: la hipótesis subyacente es que cada nodo, que es una representación mas bien aproximada de una región cerebral dada, ofrece a una única contribución al total de la red. Por tanto, la neuroimagen funcional integra los dos ingredientes básicos de la neuropsicología: la localización de la función cognitiva en módulos cerebrales especializados y el rol de las fibras de conexión en la integración de dichos módulos. Sin embargo, recientemente, la estructura y la función cerebral han empezado a ser investigadas mediante la Ciencia de la Redes, una interpretación mecánico-estadística de una antigua rama de las matemáticas: La teoría de grafos. La Ciencia de las Redes permite dotar a las redes funcionales de una gran cantidad de propiedades cuantitativas (robustez, centralidad, eficiencia, ...), y así enriquecer el conjunto de elementos que describen objetivamente la estructura y la función cerebral a disposición de los neurocientíficos. La conexión entre la Ciencia de las Redes y la Neurociencia ha aportado nuevos puntos de vista en la comprensión de la intrincada anatomía del cerebro, y de cómo las patrones de actividad cerebral se pueden sincronizar para generar las denominadas redes funcionales cerebrales, el principal objeto de estudio de esta Tesis Doctoral. Dentro de este contexto, la complejidad emerge como el puente entre las propiedades topológicas y dinámicas de los sistemas biológicos y, específicamente, en la relación entre la organización y la dinámica de las redes funcionales cerebrales. Esta Tesis Doctoral es, en términos generales, un estudio de cómo la actividad cerebral puede ser entendida como el resultado de una red de un sistema dinámico íntimamente relacionado con los procesos que ocurren en el cerebro. Con este fin, he realizado cinco estudios que tienen en cuenta ambos aspectos de dichas redes funcionales: el topológico y el dinámico. De esta manera, la Tesis está dividida en tres grandes partes: Introducción, Resultados y Discusión. En la primera parte, que comprende los Capítulos 1, 2 y 3, se hace un resumen de los conceptos más importantes de la Ciencia de las Redes relacionados al análisis de imágenes cerebrales. Concretamente, el Capitulo 1 está dedicado a introducir al lector en el mundo de la complejidad, en especial, a la complejidad topológica y dinámica de sistemas acoplados en red. El Capítulo 2 tiene como objetivo desarrollar los fundamentos biológicos, estructurales y funcionales del cerebro, cuando éste es interpretado como una red compleja. En el Capítulo 3, se resumen los objetivos esenciales y tareas que serán desarrolladas a lo largo de la segunda parte de la Tesis. La segunda parte es el núcleo de la Tesis, ya que contiene los resultados obtenidos a lo largo de los últimos cuatro años. Esta parte está dividida en cinco Capítulos, que contienen una versión detallada de las publicaciones llevadas a cabo durante esta Tesis. El Capítulo 4 está relacionado con la topología de las redes funcionales y, específicamente, con la detección y cuantificación de los nodos mas importantes: aquellos denominados “hubs” de la red. En el Capítulo 5 se muestra como las redes funcionales cerebrales pueden ser vistas no como una única red, sino más bien como una red-de-redes donde sus componentes tienen que coexistir en una situación de balance funcional. De esta forma, se investiga cómo los hemisferios cerebrales compiten para adquirir centralidad en la red-de-redes, y cómo esta interacción se mantiene (o no) cuando se introducen fallos deliberadamente en la red funcional. El Capítulo 6 va un paso mas allá al considerar las redes funcionales como sistemas vivos. En este Capítulo se muestra cómo al analizar la evolución de la topología de las redes, en vez de tratarlas como si estas fueran un sistema estático, podemos caracterizar mejor su estructura. Este hecho es especialmente relevante cuando se quiere tratar de encontrar diferencias entre grupos que desempeñan una tarea de memoria, en la que las redes funcionales tienen fuertes fluctuaciones. En el Capítulo 7 defino cómo crear redes parenclíticas a partir de bases de datos de actividad cerebral. Este nuevo tipo de redes, recientemente introducido para estudiar las anormalidades entre grupos de control y grupos anómalos, no ha sido implementado nunca en datos cerebrales y, en este Capítulo explico cómo hacerlo cuando se quiere evaluar la consistencia de la dinámica cerebral. Para concluir esta parte de la Tesis, el Capítulo 8 se centra en la relación entre las propiedades topológicas de los nodos dentro de una red y sus características dinámicas. Como mostraré más adelante, existe una relación entre ellas que revela que la posición de un nodo dentro una red está íntimamente correlacionada con sus propiedades dinámicas. Finalmente, la última parte de esta Tesis Doctoral está compuesta únicamente por el Capítulo 9, el cual contiene las conclusiones y perspectivas futuras que pueden surgir de los trabajos expuestos. En vista de todo lo anterior, espero que esta Tesis aporte una perspectiva complementaria sobre uno de los más extraordinarios sistemas complejos frente a los que nos encontramos: El cerebro humano. ABSTRACT The human brain is probably one of the most complex systems we are facing, thus being a timely and fascinating object of study. Characterizing how the brain organizes its activity to carry out complex tasks is highly non-trivial. While early neuroimaging and electrophysiological studies typically aimed at identifying patches of task-specific activations or local time-varying patterns of activity, there has now been consensus that task-related brain activity has a temporally multiscale, spatially extended character, as networks of coordinated brain areas are continuously formed and destroyed. Up until recently, though, the emphasis of functional brain activity studies has been on the identity of the particular nodes forming these networks, and on the characterization of connectivity metrics between them, the underlying covert hypothesis being that each node, constituting a coarse-grained representation of a given brain region, provides a unique contribution to the whole. Thus, functional neuroimaging initially integrated the two basic ingredients of early neuropsychology: localization of cognitive function into specialized brain modules and the role of connection fibres in the integration of various modules. Lately, brain structure and function have started being investigated using Network Science, a statistical mechanics understanding of an old branch of pure mathematics: graph theory. Network Science allows endowing networks with a great number of quantitative properties, thus vastly enriching the set of objective descriptors of brain structure and function at neuroscientists’ disposal. The link between Network Science and Neuroscience has shed light about how the entangled anatomy of the brain is, and how cortical activations may synchronize to generate the so-called functional brain networks, the principal object under study along this PhD Thesis. Within this context, complexity appears to be the bridge between the topological and dynamical properties of biological systems and, more specifically, the interplay between the organization and dynamics of functional brain networks. This PhD Thesis is, in general terms, a study of how cortical activations can be understood as the output of a network of dynamical systems that are intimately related with the processes occurring in the brain. In order to do that, I performed five studies that encompass both the topological and the dynamical aspects of such functional brain networks. In this way, the Thesis is divided into three major parts: Introduction, Results and Discussion. In the first part, comprising Chapters 1, 2 and 3, I make an overview of the main concepts of Network Science related to the analysis of brain imaging. More specifically, Chapter 1 is devoted to introducing the reader to the world of complexity, specially to the topological and dynamical complexity of networked systems. Chapter 2 aims to develop the biological, topological and functional fundamentals of the brain when it is seen as a complex network. Next, Chapter 3 summarizes the main objectives and tasks that will be developed along the forthcoming Chapters. The second part of the Thesis is, in turn, its core, since it contains the results obtained along these last four years. This part is divided into five Chapters, containing a detailed version of the publications carried out during the Thesis. Chapter 4 is related to the topology of functional networks and, more specifically, to the detection and quantification of the leading nodes of the network: the hubs. In Chapter 5 I will show that functional brain networks can be viewed not as a single network, but as a network-of-networks, where its components have to co-exist in a trade-off situation. In this way, I investigate how the brain hemispheres compete for acquiring the centrality of the network-of-networks and how this interplay is maintained (or not) when failures are introduced in the functional network. Chapter 6 goes one step beyond by considering functional networks as living systems. In this Chapter I show how analyzing the evolution of the network topology instead of treating it as a static system allows to better characterize functional networks. This fact is especially relevant when trying to find differences between groups performing certain memory tasks, where functional networks have strong fluctuations. In Chapter 7 I define how to create parenclitic networks from brain imaging datasets. This new kind of networks, recently introduced to study abnormalities between control and anomalous groups, have not been implemented with brain datasets and I explain in this Chapter how to do it when evaluating the consistency of brain dynamics. To conclude with this part of the Thesis, Chapter 8 is devoted to the interplay between the topological properties of the nodes within a network and their dynamical features. As I will show, there is an interplay between them which reveals that the position of a node in a network is intimately related with its dynamical properties. Finally, the last part of this PhD Thesis is composed only by Chapter 9, which contains the conclusions and future perspectives that may arise from the exposed results. In view of all, I hope that reading this Thesis will give a complementary perspective of one of the most extraordinary complex systems: The human brain.

Aspects of Markov Chains and Particle Systems

Thesis (Ph.D.)--University of Washington, 2016-06

The dynamics of a genetic algorithm under stabilizing selection

A formalism recently introduced by Prugel-Bennett and Shapiro uses the methods of statistical mechanics to model the dynamics of genetic algorithms. To be of more general interest than the test cases they consider. In this paper, the technique is applied to the subset sum problem, which is a combinatorial optimization problem with a strongly non-linear energy (fitness) function and many local minima under single spin flip dynamics. It is a problem which exhibits an interesting dynamics, reminiscent of stabilizing selection in population biology. The dynamics are solved under certain simplifying assumptions and are reduced to a set of difference equations for a small number of relevant quantities. The quantities used are the population's cumulants, which describe its shape, and the mean correlation within the population, which measures the microscopic similarity of population members. Including the mean correlation allows a better description of the population than the cumulants alone would provide and represents a new and important extension of the technique. The formalism includes finite population effects and describes problems of realistic size. The theory is shown to agree closely to simulations of a real genetic algorithm and the mean best energy is accurately predicted.