京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19472号 / 理博第4132号 / 新制||理||1594(附属図書館) / 32508 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 佐々 真一, 准教授 藤 定義, 准教授 武末 真二 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

We study the fluctuation effects in the seminal cyclic predator-prey model in population dynamics due to Robert May and Warren Leonard both in the zero-dimensional and two-dimensional spatial version. We compute the mean time to extinction of a stable set of coexisting populations driven by large fluctuations. We see that the contribution of large fluctuations to extinction can be captured by a quasi-stationary approximation and the Wentzel–Kramers–Brillouin (WKB) eikonal ansatz. We see that near the Hopf bifurcation, extinctions are fast owing to the flat non-Gaussian distribution whereas away from the bifurcation, extinctions are dominated by large fluctuations of the fat tails of the distribution. We compare our results to Gillespie simulations and a single-species theoretical calculation. In addition, we study the spatio-temporal pattern formation of the stochastic May--Leonard model through the Doi-Peliti coherent state path integral formalism to obtain a coarse-grained Langevin description, i.e. the Complex Ginzburg Landau equation with stochastic noise in one complex field. We see that when one restricts the internal reaction noise to small amplitudes, one can obtain a simple form for the stochastic noise correlations that modify the Complex Ginzburg Landau equation. Finally, we study the effect of coupling a spatially extended May--Leonard model in two dimensions with symmetric predation rates to one with asymmetric rates that is prone to reach extinction. We show that the symmetric region induces otherwise unstable coexistence spiral patterns in the asymmetric May--Leonard lattice. We obtain the stability criterion for this pattern induction as we vary the strength of the extinction inducing asymmetry. This research was sponsored by the Army Research Office and was accomplished under Grant Number W911NF-17-1-0156. / Doctor of Philosophy / In the field of ecology, the cyclic predator-prey patterns in a food web are relevant yet independent to the hierarchical archetype. We study the paradigmatic cyclic May--Leonard model of three species, both analytically and numerically. First, we employ well--established techniques in large-deviation theory to study the extinction of populations induced by large but rare fluctuations. In the zero--dimensional version of the model, we compare the mean time to extinction computed from the theory to numerical simulations. Secondly, we study the stochastic spatial version of the May--Leonard model and show that for values close to the Hopf bifurcation, in the limit of small fluctuations, we can map the coarse-grained description of the model to the Complex Ginsburg Landau Equation, with stochastic noise corrections. Finally, we explore the induction of ecodiversity through spatio-temporal spirals in the asymmetric version of the May--Leonard model, which is otherwise inclined to reach an extinction state. This is accomplished by coupling to a symmetric May-Leonard counterpart on a two-dimensional lattice. The coupled system creates conditions for spiral formation in the asymmetric subsystem, thus precluding extinction.

One would like to evaluate and compare complex digital communication systems based upon their overall bit error rate. Unfortunately, analytical expressions for bit error rate for even simple communication systems are notoriously difficult to evaluate accurately. Therefore, communication engineers often resort to simulation techniques to evaluate these error probabilities. In this thesis importance sampling techniques (variations of standard Monte Carlo methods) are studied in relation to both linear and non-linear detectors. Quick simulation, an importance sampling method based upon the asymptotics of the error estimator, is studied in detail. The simulated error probabilities are compared to values obtained by numerically inverting Laplace Transform expressions for these quantities.

Cette thèse porte sur la dynamique des grandes échelles des écoulements géophysiques turbulents, en particulier sur leur organisation en écoulements parallèles orientés dans la direction est-ouest (jets zonaux). Ces structures ont la particularité d'évoluer sur des périodes beaucoup plus longues que la turbulence qui les entoure. D'autre part, on observe dans certains cas, sur ces échelles de temps longues, des transitions brutales entre différentes configurations des jets zonaux (multistabilité). L'approche proposée dans cette thèse consiste à moyenner l'effet des degrés de liberté turbulents rapides de manière à obtenir une description effective des grandes échelles spatiales de l'écoulement, en utilisant les outils de moyennisation stochastique et la théorie des grandes déviations. Ces outils permettent d'étudier à la fois les attracteurs, les fluctuations typiques et les fluctuations extrêmes de la dynamique des jets. Cela permet d'aller au-delà des approches antérieures, qui ne décrivent que le comportement moyen des jets.Le premier résultat est une équation effective pour la dynamique lente des jets, la validité de cette équation est étudiée d'un point de vue théorique, et les conséquences physiques sont discutées. De manière à décrire la statistique des évènements rares tels que les transitions brutales entre différentes configurations des jets, des outils issus de la théorie des grandes déviations sont employés. Des méthodes originales sont développées pour mettre en œuvre cette théorie, ces méthodes peuvent par exemple être appliquées à des situations de multistabilité. / This thesis deals with the dynamics of geophysical turbulent flows at large scales, more particularly their organization into east-west parallel flows (zonal jets). These structures have the particularity to evolve much slower than the surrounding turbulence. Besides, over long time scales, abrupt transitions between different configurations of zonal jets are observed in some cases (multistability). Our approach consists in averaging the effect of fast turbulent degrees of freedom in order to obtain an effective description of the large scales of the flow, using stochastic averaging and the theory of large deviations. These tools provide theattractors, the typical fluctuations and the large fluctuations of jet dynamics. This allows to go beyond previous studies, which only describe the average jet dynamics. Our first result is an effective equation for the slow dynamics of jets, the validityof this equation is studied from a theoretical point of view, and the physical consequences are discussed. In order to describe the statistics of rare events such as abrupt transitions between different jet configurations, tools from large deviation theory are employed. Original methods are developped in order to implement this theory, those methods can be applied for instance in situations of multistability.

In this thesis, I study fluctuations and rare events of time-additive observables of discrete-time Markov chains on finite state spaces. The observable of interest is the mean node connectivity visited by a random walk running on instances of an Erdős-Rényi (ER) random graph. I implement and analyze the Adaptive Power Method (APM) which converges to the driven process, a biased random walk defined through a control parameter that simulates trajectories corresponding to rare events of the observable in the original dynamics. The APM demonstrates good convergence and accurately produces the desired quantities from a single trajectory. Due to the bulk-dangling-chain structure in the ER graph, the driven process seems to undergo a dynamical phase transition (DPT) for infinitely large graphs, meaning the behavior of the trajectories changes abruptly as the control parameter is varied. Observations show that the random walk visits two distinct phases, being de-localized in the bulk or localized in the chain. Through two simpler models capturing the bulk-dangling-chain property of the ER graph I study how the DPT occurs as the graph size increases. I observe that the trajectories of the driven process near the transition show intermittent behavior between the two phases. The diverging time scale of the DPT is found to be the average time that the random walk spends in a phase before it transitions to the other one. On the ER graph the trajectories are also intermittent but the form of the time scaling remains open due to computational limits on the graph size.

Cette thèse porte sur l'étude numérique des fluctuations extrêmes de la force de traînée exercée par un écoulement turbulent sur un corps immergé.Ce type d'évènement, très rare, est difficile à caractériser par le biais d'un échantillonnage direct, puisqu'il est alors nécessaire de simuler l'écoulement sur des durées extrêmement longues. Cette thèse propose une approche différente, basée sur l'application d'algorithmes d'échantillonnage d'événements rares. L'objectif de ces algorithmes, issus de la physique statistique, est de modifier la statistique d'échantillonnage des trajectoires d'un système dynamique, de manière à favoriser l'occurrence d'événements rares. Si ces techniques ont été appliquées avec succès dans le cas de dynamiques relativement simples, l'intérêt de ces algorithmes n'est à ce jour pas clair pour des dynamiques déterministes extrêmement complexes, comme c'est le cas pour les écoulement turbulents.Cette thèse présente tout d'abord une étude de la dynamique et de la statistique associée aux fluctuations extrêmes de la force de traînée sur un obstacle carré fixe immergé dans un écoulement turbulent à deux dimensions. Ce cadre simplifié permet de simuler la dynamique sur des durées très longues, permettant d'échantillonner un grand nombre de fluctuations dont l'amplitude est assez élevée pour être qualifiée d'extrême.Dans un second temps, l'application de deux algorithmes d’échantillonnage est présentée et discutée.Dans un premier cas, il est illustré qu'une réduction significative du temps de calcul d'extrêmes peut être obtenue. En outre, des difficultés liées à la dynamique de l'écoulement sont mises en lumière, ouvrant la voie au développement de nouveaux algorithmes spécifiques aux écoulements turbulents. / This thesis discusses the numerical simulation of extreme fluctuations of the drag force acting on an object immersed in a turbulent medium.Because such fluctuations are rare events, they are particularly difficult to investigate by means of direct sampling. Indeed, such approach requires to simulate the dynamics over extremely long durations.In this work an alternative route is introduced, based on rare events algorithms.The underlying idea of such algorithms is to modify the sampling statistics so as to favour rare trajectories of the dynamical system of interest.These techniques recently led to impressive results for relatively simple dynamics. However, it is not clear yet if such algorithms are useful for complex deterministic dynamics, such as turbulent flows.This thesis focuses on the study of both the dynamics and statistics of extreme fluctuations of the drag experienced by a square cylinder mounted in a two-dimensional channel flow.This simple framework allows for very long simulations of the dynamics, thus leading to the sampling of a large number of events with an amplitude large enough so as they can be considered extreme.Subsequently, the application of two different rare events algorithms is presented and discussed.In the first case, a drastic reduction of the computational cost required to sample configurations resulting in extreme fluctuations is achieved.Furthermore, several difficulties related to the flow dynamics are highlighted, paving the way to novel approaches specifically designed to turbulent flows.