La matière active, désignant les systèmes hors d’équilibre composés de particules étant capable d’utiliser l’énergie présente dans leur environnement afin de se déplacer de façon systématique, a suscité beaucoup d’attention auprès des communautés de mécanique statistique et matière molle ces dernières décennies. Les systèmes actifs couvrent en effet un large panel d’exemples allant de la biologie aux granulaires. Cette thèse se concentre sur l’étude de modèles minimaux de matière active sèche (ceux pour lesquels le fluide dans lequel les particles sont immergées est négligé), tel que le modèle de Vicsek qui considère des particules ponctuelles se déplaçant à vitesse constante tout en alignant leur direction de mouvement avec celles de leurs voisins localement en présence de bruit, et définit une classe d’universalité hors équilibre pour la transition vers le mouvement collectif. Quatre problèmes en suspens ont été abordés : La définition d’une classe d’universalité en matière active sèche qui décrit des systèmes de particles présentant un alignement polaire et un mouvement apolaire. Cette nouvelle classe exhibe une transition continue vers un quasi-ordre polaire doté d’exposants variant continument, et donc analogue au modèle XY à l’équilibre, mais n’appartenant pas à la classe d’universalité Kosterlitz-Thouless. Ensuite, l’étude de la validité des théories cinétiques décrivant les modèles de type Vicsek, qui sont confrontées aux résultats obtenus aux niveaux microscopique et hydrodynamique. Puis une évaluation quantitative de la théorie de Toner et Tu, permettant de mesurer les exposants caractérisant les fluctuations dans la phase ordonnée du modèle de Vicsek, à partir de simulations numériques à grande échelle du modèle microscopique. Enfin, la création d’un formalisme pour la dérivation d’équations hydrodynamiques à partir de modèles de matière active sèche à trois dimensions, ainsi que leur étude au niveau linéaire. / Active matter, i.e. nonequilibrium systems composed of many particles capable of exploiting the energy present in their environment in order to produce systematic motion, has attracted much attention from the statistical mechanics and soft matter communities in the past decades. Active systems indeed cover a large variety of examples that range from biological to granular. This Ph.D. focusses on the study of minimal models of dry active matter (when the fluid surrounding particles is neglected), such as the Vicsek model: point-like particles moving at constant speed and aligning their velocities with those of their neighbors locally in presence of noise, that defines a nonequilibrium universalilty class for the transition to collective motion. Four current issues have been addressed: The definition of a new universality class of dry active matter with polar alignment and apolar motion, showing a continuous transition to quasilong-range polar order with continuously varying exponents, analogous to the equilibrium XY model, but that does not belong to the Kosterlitz-Thouless universality class. Then, the study of the faithfulness of kinetic theories for simple Vicsek-style models and their comparison with results obtained at the microscopic and hydrodynamic levels. Follows a quantitative assessment of Toner and Tu theory, which has allowed to compute the exponents characterizing fluctuations in the flocking phase of the Vicsek model, from large scale numerical simulations of the microscopic dynamics. Finally, the establishment of a formalism allowing for the derivation of hydrodynamic field theories for dry active matter models in three dimensions, and their study at the linear level.

Computational models of collective motion have yielded many insights about the way that groups of animals or simulated particles may move together and self-organize. Recent literature has compared predictions of models with large datasets of detailed observations of animal behavior, and found that there are important discrepancies, leading researchers to reexamine some of the most widely used assumptions. We introduce FlockOpt, an optimization-based, variable-speed, self-propelled particle model of collective motion that addresses important shortcomings of earlier models. In our model, each particle adjusts its velocity by performing a constrained optimization of a locally-defined objective function, which is computed at each time step over the kinematics of the particle and the relative position of neighboring particles. Our model explains how ordered motion can arise in the absence of an explicitly prescribed alignment term and simulations performed with our model exhibit a wide variety of patterns of motion, including several not possible with popular constant-speed models. Our model predicts that variations in speed and heading of particles are coupled due to costs associated with changes in relative position. We have found that a similar coupling effect may also be present in the flight of groups of gregarious bats. The Mexican Free-tailed bat (Tadarida brasiliensis) is a gregarious bat that forms large maternity colonies, containing hundreds of thousands to millions of individuals, in the southwestern United States in the summer. We have developed a protocol for calibrating cameras used in stereo videography and developed guidelines for data collection. Our field protocol can be deployed in a single afternoon, requiring only short video segments of light, portable calibration objects. These protocols have allowed us to reconstruct the three-dimensional flight trajectories of hundreds of thousands of bats in order to use their flight as a biological study system for our model.

abstract: The main objective of mathematical modeling is to connect mathematics with other scientific fields. Developing predictable models help to understand the behavior of biological systems. By testing models, one can relate mathematics and real-world experiments. To validate predictions numerically, one has to compare them with experimental data sets. Mathematical modeling can be split into two groups: microscopic and macroscopic models. Microscopic models described the motion of so-called agents (e.g. cells, ants) that interact with their surrounding neighbors. The interactions among these agents form at a large scale some special structures such as flocking and swarming. One of the key questions is to relate the particular interactions among agents with the overall emerging structures. Macroscopic models are precisely designed to describe the evolution of such large structures. They are usually given as partial differential equations describing the time evolution of a density distribution (instead of tracking each individual agent). For instance, reaction-diffusion equations are used to model glioma cells and are being used to predict tumor growth. This dissertation aims at developing such a framework to better understand the complex behavior of foraging ants and glioma cells. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2019

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O aprendizado de máquina consiste em conceitos e técnicas que permitem aos computadores melhorar seu desempenho com a experiência, ou em outras palavras, aprender com dados. Duas de suas principais categorias são o aprendizado não-supervisionado e o semissupervisionado, que respectivamente consistem em inferir padrões em bases cujos dados não têm rótulo (classe) e classificar dados em bases parcialmente rotuladas. Embora muito estudado, trata-se de um campo repleto de desafios e com muitos tópicos abertos. Sistemas dinâmicos coletivos, por sua vez, são sistemas constituídos por muitos indivíduos, cada qual um sistema dinâmico por si só, de modo que todos eles agem coletivamente, ou seja, a ação de cada indivíduo é influenciada pela ação dos vizinhos. Uma característica notável desses sistemas é que padrões globais podem surgir espontaneamente das interações locais entre os indivíduos, fenômeno conhecido como emergência. Os desafios intrínsecos e a relevância do tema vêm motivando sua pesquisa em diversos ramos da ciência e da engenharia. Este trabalho de doutorado consiste no desenvolvimento e análise de modelos dinâmicos coletivos para o aprendizado de máquina, especificamente suas categorias não-supervisionada e semissupervisionada. As tarefas de segmentação de imagens e de detecção de comunidades em redes, que de certo modo podem ser entendidas como tarefas do aprendizado de máquina, são também abordadas. Em especial, desenvolvem-se modelos nos quais a movimentação dos objetos é determinada pela localização e velocidade de seus vizinhos. O sistema dinâmico assim modelado é então conduzido a um estado cujo padrão formado por seus indivíduos realça padrões subjacentes do conjunto de dados. Devido ao seu caráter auto-organizável, os modelos aqui desenvolvidos são robustos e as informações geradas durante o processo (valores das variáveis do sistema) são ricas e podem, por exemplo, revelar características para realizar soft labeling e determinar classes sobrepostas. / Machine learning consists of concepts and techniques that enable computers to improve their performance with experience, i.e., learn from data. Unsupervised and semi-supervised learning are important categories of machine learning, which respectively consists of inferring patterns in datasets whose data have no label (class) and classifying data in partially-labeled datasets. Although intensively studied, machine learning is still a field full of challenges and with many open topics. Collective dynamical systems, in turn, are systems made of a large group of individuals, each one a dynamical system by itself, such that all of them behave collectively, i.e., the action of each individual is influenced by the action of its neighbors. A remarkable feature of those systems is that global patterns may spontaneously emerge from the local interactions among individuals, a phenomenon known as emergence. Their relevance and intrinsic challenges motivate research in various branches of science and engineering. In this doctorate research, we develop and analyze collective dynamical models for their usage in machine-learning tasks, specifically unsupervised and semi-supervised ones. Image segmentation and network community detection are also addressed, as they are related to machine learning as well. In particular, we propose to work on models in which the objects motion is determined by the location and velocity of their neighbors. By doing so, the dynamical system reaches a configuration in which the patterns developed by the set of individuals highlight underlying patterns of the dataset. Due to their self-organizing nature, it is also expected that the models can be robust and the information generated during the process (values of the system variables) can be rich and reveal, for example, features to perform soft labeling and determine overlapping classes.

Les bancs de poissons sont des entités pouvant regrouper plusieurs milliers d'individus qui se déplacent de façon synchronisée, dans un environnement sujet à de multiples perturbations, qu'elles soient endogènes (e.g. le départ soudain d'un congénère) ou exogènes (e.g. l'attaque d'un prédateur). La coordination de ces bancs de poissons, décentralisée, n'est pas encore totalement comprise. Si les mécanismes sous-jacents aux interactions sociales proposés dans des travaux précédents reproduisent qualitativement les structures collectives observées dans la nature, la quantification de ces interactions et l'accord quantitatif entre ces mesures individuelles et les motifs collectifs sont encore rares dans les recherches récentes et forment l'objet principal de cette thèse. L'approche de ce travail repose sur une étroite combinaison entre les méthodes expérimentales et de modélisation dans l'objectif de découvrir les liens entre les comportements individuels et les structures observées à l'échelle collective. Nous avons caractérisé et quantifié les interactions et mécanismes à l'origine, d'abord, de la coordination des individus dans les bancs de poissons et, ensuite, de la propagation d'information, quand le groupe subit une perturbation endogène ou exogène. Ces travaux, tous réalisés en étudiant la même espèce de poisson d'eau douce, le nez-rouge (Hemigrammus rhodostomus), ont mobilisé une diversité de méthodes expérimentales, d'analyses statistique et de modélisation, à l'interface de l'éthologie, de la physique statistique et des sciences computationnelles. / Fish schools are systems in which thousands of individuals can move in a synchronised manner in a changing environment, with endogenous perturbations (e.g. when a congener leaves the group) or exogenous (e.g. the attack of a predator). The coordination of fish schools, decentralised, is not completely understood yet. If the mechanisms underlying social interactions discussed in previous studies qualitatively match the social patterns observed in nature, the quantification of these interactions and the quantitative match between individual measurements and collective patterns are still sparse in recent works and are the main focus of this thesis. This work combines closely experimental and modelling methods in order to investigate the links between the individual behaviours and the patterns observed at the collective scale. We have characterised and quantified the interactions and mechanisms at the origin of, first, the coordination of individuals in fish schools and, second, the propagation of information, when the group is under endogenous or exogenous perturbations. This thesis focuses on one freshwater fish species, the rummy-nose tetra (Hemigrammus rhodostomus), and is the result of a diversity of experimental methods, statistical analyses and modelling, at the interface of ethology, statistical physics and computational sciences.

This study explores principles of motion based communication in animal and human group behavior. It develops models of cooperative control that involve communication through actions aimed at a shared objective. Moreover, it aims at understanding the collective motion in multi-agent models towards a desired objective which requires interaction with the environment. In conducting a formal study of these problems, first we investigate the leader-follower interaction in a dance performance. Here, the prototype model is salsa. Salsa is of interest because it is a structured interaction between a leader (usually a male dancer) and a follower (usually a female dancer). Success in a salsa performance depends on how effectively the dance partners communicate with each other using hand, arm and body motion. We construct a mathematical framework in terms of a Dance Motion Description Language (DMDL). This provides a way to specify control protocols for dance moves and to represent every performance as sequences of letters and corresponding motion signals. An enhanced form of salsa (intermediate level) is discussed in which the constraints on the motion transitions are described by simple rules suggested by topological knot theory. It is shown that the proficiency hierarchy in dance is effectively captured by proposed complexity metrics. In order to investigate the group behavior of animals that are reacting to environmental features, we have analyzed a large data set derived from 3-d video recordings of groups of Myotis velifer emerging from a cave. A detailed statistical analysis of large numbers of trajectories indicates that within certain bounds of animal diversity, there appear to be common characteristics of the animals' reactions to features in a clearly defined flight corridor near the mouth of the cave. A set of vision-based motion control primitives is proposed and shown to be effective in synthesizing bat-like flight paths near groups of obstacles. A comparison of synthesized paths and actual bat motions culled from our data set suggests that motions are not based purely on reactions to environmental features. Spatial memory and reactions to the movement of other bats may also play a role. It is argued that most bats employ a hybrid navigation strategy that combines reactions to nearby obstacles and other visual features with some combination of spatial memory and reactions to the motions of other bats.

O aprendizado de máquina consiste em conceitos e técnicas que permitem aos computadores melhorar seu desempenho com a experiência, ou em outras palavras, aprender com dados. Duas de suas principais categorias são o aprendizado não-supervisionado e o semissupervisionado, que respectivamente consistem em inferir padrões em bases cujos dados não têm rótulo (classe) e classificar dados em bases parcialmente rotuladas. Embora muito estudado, trata-se de um campo repleto de desafios e com muitos tópicos abertos. Sistemas dinâmicos coletivos, por sua vez, são sistemas constituídos por muitos indivíduos, cada qual um sistema dinâmico por si só, de modo que todos eles agem coletivamente, ou seja, a ação de cada indivíduo é influenciada pela ação dos vizinhos. Uma característica notável desses sistemas é que padrões globais podem surgir espontaneamente das interações locais entre os indivíduos, fenômeno conhecido como emergência. Os desafios intrínsecos e a relevância do tema vêm motivando sua pesquisa em diversos ramos da ciência e da engenharia. Este trabalho de doutorado consiste no desenvolvimento e análise de modelos dinâmicos coletivos para o aprendizado de máquina, especificamente suas categorias não-supervisionada e semissupervisionada. As tarefas de segmentação de imagens e de detecção de comunidades em redes, que de certo modo podem ser entendidas como tarefas do aprendizado de máquina, são também abordadas. Em especial, desenvolvem-se modelos nos quais a movimentação dos objetos é determinada pela localização e velocidade de seus vizinhos. O sistema dinâmico assim modelado é então conduzido a um estado cujo padrão formado por seus indivíduos realça padrões subjacentes do conjunto de dados. Devido ao seu caráter auto-organizável, os modelos aqui desenvolvidos são robustos e as informações geradas durante o processo (valores das variáveis do sistema) são ricas e podem, por exemplo, revelar características para realizar soft labeling e determinar classes sobrepostas. / Machine learning consists of concepts and techniques that enable computers to improve their performance with experience, i.e., learn from data. Unsupervised and semi-supervised learning are important categories of machine learning, which respectively consists of inferring patterns in datasets whose data have no label (class) and classifying data in partially-labeled datasets. Although intensively studied, machine learning is still a field full of challenges and with many open topics. Collective dynamical systems, in turn, are systems made of a large group of individuals, each one a dynamical system by itself, such that all of them behave collectively, i.e., the action of each individual is influenced by the action of its neighbors. A remarkable feature of those systems is that global patterns may spontaneously emerge from the local interactions among individuals, a phenomenon known as emergence. Their relevance and intrinsic challenges motivate research in various branches of science and engineering. In this doctorate research, we develop and analyze collective dynamical models for their usage in machine-learning tasks, specifically unsupervised and semi-supervised ones. Image segmentation and network community detection are also addressed, as they are related to machine learning as well. In particular, we propose to work on models in which the objects motion is determined by the location and velocity of their neighbors. By doing so, the dynamical system reaches a configuration in which the patterns developed by the set of individuals highlight underlying patterns of the dataset. Due to their self-organizing nature, it is also expected that the models can be robust and the information generated during the process (values of the system variables) can be rich and reveal, for example, features to perform soft labeling and determine overlapping classes.

Une étude expérimentale et numérique traitant de l'influence de structures tourbillonnaires sur la sédimentation de nuage de particules sphériques sous l'effet de la gravité est présentée dans une première partie de ce manuscrit. L'écoulement est créé par électro-convection, ce qui permet de générer un réseau de vortex contrôlés en vitesse et de taille constante qui imite un écoulement tourbillonnaire. Des techniques de PIV (Particle image-velocimetry) et de suivi de particules sont utilisés pour étudier la sédimentation du nuage.Le nuage est modélisé comme un ensemble de particules ponctuelles pour lesquelles les forces d'interaction hydrodynamiques entre particules sont prépondérantes. Le comportement du nuage est comparé aux prédictions obtenues avec des modèles numériques. Dans une seconde partie est présentée une étude expérimentale et numérique concernant la sédimentation à faible nombre de Reynolds de fibres flexibles dans un fluide visqueux au repos. L'état d'équilibre atteint par la fibre flexible est étudié. Nous identifions trois régimes ayant des signatures différentes sur l'état stationnaire de la fibre flexible: un régime de faibles déformations dans lequel la force de traînée est proportionnelle à celle d'une fibre sédimentant horizontalement par rapport à la gravité; un régime de grandes déformations dans lequel la force de traînée est aussi proportionnelle à la vitesse de la fibre, mais avec un coefficient de traînée qui est celui d'une fibre chutant parallèlement à la gravité; et un régime de reconfiguration élastique où le filament se déforme avec une traînée plus faible qui n'est plus proportionnelle à sa vitesse, mais à la racine carrée de celle-ci / In the first part, a jointed experimental and numerical study examining the influence of vortical structures on the settling of a cloud of solid spherical particles under the action of gravity at low Stokes numbers is presented. We use electro-convection to generate a two-dimensional array of controlled vortices which mimics a simplified vortical flow. Particle image-velocimetry and tracking are used to examine the motion of the cloud within this vortical flow. The cloud is modeled as a set of point-particles for which the hydrodynamic interaction is preponderant. The cloud behavior (trajectory, velocity, aspect ratio, break-up time …) is compared to the predictions of a two-way-coupling numerical simulation. In the second part, a jointed experimentally and numerical study on the dynamics of slender flexible filaments settling in a viscous fluid at low Reynolds number is presented. The equilibrium state of a flexible fiber settling in a viscous fluid is examined using a combination of macroscopic experiments, numerical simulations and scaling arguments. We identify three regimes having different signatures on this equilibrium configuration of the elastic filament: a weak deformation regime wherein the drag is proportional to the fiber velocity settling perpendicular to the gravity; a large deformation regime wherein the drag is proportional to the fiber velocity settling parallel to the gravity and an intermediate elastic reconfiguration regime where the filament deforms to adopt a shape with a smaller drag which is no longer linearly proportional to the velocity but to the square root of the velocity

Les symétries géométriques en usage en physique nucléaire sont assez peu variées, essentiellement la symétrie de l’ellipsoïde triaxial. On propose donc une méthode rigoureuse permettant d’étudier l’évolution et la possibilité de l’existence de symétries nouvelles dont la symétrie tétraédrique. Le formalisme de l’équation de SCHRÖDINGER est replacé dans le cadre des espaces de RIEMANN. Ce formalisme est utilisé dans le contexte du noyau de l’atome où l’on applique la théorie du champ moyen alliée à l’approximation adiabatique. Le noyau est le siège de deux catégories de mouvements adiabatiquement séparés, le mouvement rapide des nucléons dans le champ moyen, et le mouvement collectif modifiant lentement le champ moyen. Le second est régi par une équation de SCHRÖDINGER collective qui prend place dans un espace dont la métrique est donnée par le tenseur de masse. L’étude de la géométrie du noyau est alors calculable à l’aide de deux grands programmes développés dans le cadre de la thèse. / The geometrical symmetries used in nuclear physics are not very diversified, essentially the symmetry of the triaxial ellipsoid. One proposes therefore a rigourous method allowing to study the temporal evolution and the possibility of the existence of new symmetries among them the tetrahedral symmetry. The formalism of SCHRÖDINGER equation is reformulated in the framework of RIEMANN’s spaces. This formalism is used in the context of the atomic nucleus where one applies the mean-field theory combined with the adiabatic approximation. The nucleus is the terrain of two types of motions adiabatically separated, the quick motion of the nucleons in the mean-field and the collective motion modifying slowly the meanfield. The second one is governed by a collective SCHRÖDINGER equation written down in a space whose metric is given by the mass tensor. The study of the nucleus geometry is then computable with the help of two big programs developped within the thesis.