Nesta dissertação estudamos uma probabilidade obtida a partir de conceitos da Mecânica Estatística Quântica do ponto de vista da Teoria Ergódica. A probabilidade é obtida a partir de um estado KMS sobre um lattice unidimensional de spins quânticos. Mostramos que esta probabilidade é mixing para o shift. Além disso, mostramos que vale um princípio dos grandes desvios para uma certa classe de funções e exploramos algumas propriedades do Jacobiano. Iremos considerar o estado KMS associado a um certo Hamiltoniano específico agindo sobre o lattice de spins quânticos. Nas seções iniciais vamos apresentar alguns conceitos e prerequisitos básicos (como operadores densidade, produto tensorial, C*-algebras e estados KMS) para o entendimento do resultado principal / In this dissertation we study a probability derived from Quantum Statistical Mechanics through the viewpoint of Ergodic Theory. The probability is obtained from a KMS state acting on a one dimensional lattice of quantum spins. We show that this probability is mixing for the shift map. Moreover, we show that a large deviation principle is true for a certain class of functions and we explore some properties of the Jacobian. We will consider the KMS state associated to a certain specific Hamiltonian acting on the quantum spin lattice. In the initial sections we will present some concepts and prerequisites (such as density operators, tensor product, C*-algebras and KMS states) for the understanding of our main results.

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

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24386号 / 理博第4885号 / 新制||理||1699(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 平岡 裕章, 教授 COLLINS Benoit Vincent Pierre, 教授 坂上 貴之 / 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.

<p>This thesis is centered around three infinite dimensional diffusion processes:</p> <p>(i). the infinitely-many-neutral-alleles diffusion model [Ethier and Kurtz, 1981],</p> <p>(ii). the two-parameter infinite dimensional diffusion model [Petrov, 2009] and [Feng and Sun, 2010],</p> <p>(iii). the infinitely-many-alleles diffusion with symmetric dominance [Ethier and Kurtz, 1998].</p> <p>The partition structures, the ergodic inequalities and the asymptotic theory of these three models are discussed. In particular, the asymptotic theory turns out to be the major contribution of this thesis.</p> <p>In Chapter 2, a slightly altered version of Kingman's one-to-one correspondence theorem on partition structures is provided, which in turn becomes a handy tool for obtaining the asymptotic result on the partition structures associated with models (i) and (ii).</p> <p>In Chapter 3, the three diffusion models are briefly introduced. New representations of the transition densities of models (i) and (ii) are obtained simply by rearranging the previous representations obtained in [Ethier, 1992] and [Feng et al., 2011] respectively. These two new representations have their own advantages, by making use of which the corresponding ergodic inequalities easily follow. Furthermore, thanks to the functional inequalities in [Feng et al., 2011], the ergodic inequality for model (iii) becomes available as well.</p> <p>In Chapter 4, the asymptotic properties of models (i) and (ii) are thoroughly studied. Various asymptotic results are obtained, such as the weak limits of models (i) and (ii) at different time scales when the mutation rate approaches infinity, and the large deviation principle for models (i) and (ii) at a fixed time, and that of the transient partition structures of models (i) and (ii). Of all these results, the weak limit and the large deviation principle of the transient partition structures are of particular interest.</p> <p>In Chapter 5, the asymptotic results on the stationary distribution and the transient distribution of model (iii) are both obtained. The weak limit of the infinitely-many- alleles diffusion with symmetric overdominance at fixed time t serves as an alternative answer to Gillespie's conjecture [Gillespie, 1999]. The weak limit of the stationary distribution of the infinitely-many-alleles diffusion with symmetric overdominance provides a complete solution to the remaining problem in [Feng, 2009].</p> / Doctor of Philosophy (PhD)

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.

Nesta tese são estudados alguns tópicos sobre otimização ergódica e formalismo termodinâmico que generalizam resultados, de Contreras, Lopes e Thieullen (2006), Garibaldi e Lopes (2008) no primeiro caso e Baraviera, Lopes e Thieullen (2006), Bissacot, Mengue e Pérez (2006) no segundo, para contextos onde não existem medidas de Gibbs, ou, em outras palavras, não é satisfeita a propriedade BIP. É demonstrada a existência de subações calibradas para potenciais coercivos de variação finita em espaços shift transitivos de alfabeto enumerável. O método usado é a construção da barreira de Peierls nesse contexto. Provam-se algumas das propriedades da barreira de Peierls e, como consequência das construções, é mostrada uma classificação dos shifts que possuem subações calibradas e limitadas. Posteriormente é realizado um estudo do formalismo termodinâmico para potenciais somáveis de variação finita e pressão finita com medida maximizante única f em shifts topologicamente mixing. Fazendo uso dos resultados de Freire e Vargas (2015), são estudadas a famlia de estados de equilbrio correspondente com f e a famlia de funções 1/B log h_ B , onde h_B são auto vetores do operador de Ruelle para Bf . É demonstrado que os pontos de acumulação quando B vai para infinito são subações uniformemente contnuas. Finalmente é provada uma propriedade dos grandes desvios para a famlia de estados de equilbrio \\mu_B com hipóteses sobre a convergência de uma famlia de funções g_B que normaliza o operador de Ruelle para cada B> 1 (Veja seção 4.4) / In this thesis, the study of topics on ergodic optimization and thermodynamic formalism for countable Markov shifts is presented. It provides a generalization of the previous results, in Contreras, Lopes and Thieullen (2006), Garibaldi and Lopes (2008) for the first subject and Baraviera, Lopes and Thieullen (2006), Bissacot, Mengue e Pérez (2006) for the second one, to situations where there are no Gibbs measures, ie, the BIP property is not verified. The existence of calibrated subactions for coercive potentials with finite variation over transitive countable Markov shifts is proved. The method is based on the construction of the Peierls barrrier in this context. Some properties of the Peierls barrier are proved and, as consequence of the proof, a classification of the Markov shifts which support calibrated and limited subactions is shown. Subsequently, the thermodynamic formalism for topologically mixing Markov shift and summable potentials with finite variation, finite pressure and unique maximizing measure f is studied. Using results in Freire and Vargas (2015), the class of equilibrium states corresponding with f and the class of functions 1/ log h_B are studied where h_B are the eigenfunctions for the Ruelle operator. It is proved that its accumulation points, as goes to infinity, are uniformly continuous subactions. Finally, it is proved a large deviation principle for the equilibrium states family \\mu_B , assuming a hypothesis about the convergence in a family of functions that normalizes the Ruelle operator (See section 4.4 for more details).

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.

Nesta tese são estudados alguns tópicos sobre otimização ergódica e formalismo termodinâmico que generalizam resultados, de Contreras, Lopes e Thieullen (2006), Garibaldi e Lopes (2008) no primeiro caso e Baraviera, Lopes e Thieullen (2006), Bissacot, Mengue e Pérez (2006) no segundo, para contextos onde não existem medidas de Gibbs, ou, em outras palavras, não é satisfeita a propriedade BIP. É demonstrada a existência de subações calibradas para potenciais coercivos de variação finita em espaços shift transitivos de alfabeto enumerável. O método usado é a construção da barreira de Peierls nesse contexto. Provam-se algumas das propriedades da barreira de Peierls e, como consequência das construções, é mostrada uma classificação dos shifts que possuem subações calibradas e limitadas. Posteriormente é realizado um estudo do formalismo termodinâmico para potenciais somáveis de variação finita e pressão finita com medida maximizante única f em shifts topologicamente mixing. Fazendo uso dos resultados de Freire e Vargas (2015), são estudadas a famlia de estados de equilbrio correspondente com f e a famlia de funções 1/B log h_ B , onde h_B são auto vetores do operador de Ruelle para Bf . É demonstrado que os pontos de acumulação quando B vai para infinito são subações uniformemente contnuas. Finalmente é provada uma propriedade dos grandes desvios para a famlia de estados de equilbrio \\mu_B com hipóteses sobre a convergência de uma famlia de funções g_B que normaliza o operador de Ruelle para cada B> 1 (Veja seção 4.4) / In this thesis, the study of topics on ergodic optimization and thermodynamic formalism for countable Markov shifts is presented. It provides a generalization of the previous results, in Contreras, Lopes and Thieullen (2006), Garibaldi and Lopes (2008) for the first subject and Baraviera, Lopes and Thieullen (2006), Bissacot, Mengue e Pérez (2006) for the second one, to situations where there are no Gibbs measures, ie, the BIP property is not verified. The existence of calibrated subactions for coercive potentials with finite variation over transitive countable Markov shifts is proved. The method is based on the construction of the Peierls barrrier in this context. Some properties of the Peierls barrier are proved and, as consequence of the proof, a classification of the Markov shifts which support calibrated and limited subactions is shown. Subsequently, the thermodynamic formalism for topologically mixing Markov shift and summable potentials with finite variation, finite pressure and unique maximizing measure f is studied. Using results in Freire and Vargas (2015), the class of equilibrium states corresponding with f and the class of functions 1/ log h_B are studied where h_B are the eigenfunctions for the Ruelle operator. It is proved that its accumulation points, as goes to infinity, are uniformly continuous subactions. Finally, it is proved a large deviation principle for the equilibrium states family \\mu_B , assuming a hypothesis about the convergence in a family of functions that normalizes the Ruelle operator (See section 4.4 for more details).

Gegenstand der vorliegenden Dissertation ist das Verhalten sogenannter Selbstüberschneidungslokalzeiten $\\|\\ell_t\\|_p^p$ einer zeitstetigen Irrfahrt $(S_r)_r$ auf dem $d$-dimensionalen Gitter $\\Z^d$. Dabei ist für $p>1$ die Funktion $\\ell_t$ definiert durch $$ \\ell_t(z):=\\int_{0}^{t}\\1_{\\{S_r=z\\}}\\,\\d r\\nonumber $$ und bezeichnet die Aufenthaltsdauer der Irrfahrt bis zum Zeitpunkt $t\\in(0,\\infty)$ im Punkt $z\\in\\Z^d$. Ziel ist es, ein Prinzip großer Abweichungen zu entwickeln, d.h. das Hauptaugenmerk liegt auf dem asymptotischen Verhalten der Wahrscheinlichkeit, dass die Selbstüberschneidungslokalzeiten von ihrem Erwartungswert in erheblichem Maße nach oben abweichen. Mit anderen Worten; es soll das asymptotische Verhalten von $$ \\log\\P(\\|\\ell_t\\|_p^p\\geq r^p_t) $$ genau bestimmt werden, wobei $r_t^p\\in(0,\\infty)$ schneller als der Erwartungswert $\\E[\\|\\ell_t\\|_p^p]$ gegen unendlich streben soll. Dieses Verhalten kann dabei durch $t$, $r_t$ und eine gewisse Variationsformel beschrieben werden. Es wird sich herausstellen, dass es zwei Fälle zu betrachten gilt, in denen sich das probabilistisch beste Verhalten stark unterscheidet; die genaue Position des Phasenübergangs hängt dabei von den Parametern $p$ und $d$ ab. Im Vorgriff auf die Resultate kann man festhalten, dass die nötigen Selbstüberschneidungen in kleinen Dimensionen (im sogenannten subkritischen Fall) über einen großen Bereich erfolgen, aufgrund dessen bei der mathematischen Modellierung eine Reskalierung erforderlich ist. In hohen Dimensionen (dem sogenannten superkritischen Fall) ist dies nicht nötig, da die erforderlichen Selbstüberschneidungen innerhalb eines begrenzten Intervalles erfolgen. Das Interesse an der Untersuchung entstand unter anderem aus der Verbindung zu Modellen der statistischen Mechanik (parabolisches Anderson Modell) und zur Variationsanalysis. In der Vergangenheit wurde eine Vielzahl an Methoden benutzt, um dieses Problem zu lösen. In der vorliegenden Dissertation soll die sogenannte Momentenmethode bestmöglich ausgereizt werden und es wird gezeigt, welche Ergebnisse damit möglich sind.