No description available.

Orientador: Amir Ordacgi Caldeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-08-18T12:04:56Z (GMT). No. of bitstreams: 1 GarciaRodriguez_AlexisOmar_D.pdf: 2441638 bytes, checksum: 00dfa78fb7b0c9f69778a51704c587b7 (MD5) Previous issue date: 2011 / Resumo: Este trabalho foi motivado por várias dificuldades encontradas no estudo do artigo de M. Ueda, Transmission Spectrum of a Tunneling Particle Interacting with Dynamical Fields: Real- Time Functional Integral Approach, Phys. Rev. B 54, 8676 (1996). Nesse artigo, num formalismo de tempo real, é descrito o tunelamento de uma partícula através de uma barreira utilizando tempos não reais de travessia através dessa barreira. No presente trabalho é proposto um formalismo mais amplo de tempo real para uma introdução mais natural de valores complexos do tempo na descrição do tunelamento de uma partícula cm interação com o ambiente. Esta proposta está baseada no chamado método do tempo complexo utilizado no caso do tunelamento de uma partícula sem interação com o ambiente estudado nos trabalhos de D. W. McLaughlin, J. Math. Phys. 13, 1099 (1972) c B. R. Holstein c A. R. Swift, Am. J. Phys. 50, 833 (1982). Seguindo o trabalho citado de Ueda, o ambiente da partícula é representado através de um conjunto, ou banho térmico, de osciladores harmônicos caracterizados por uma função de densidade espectral J(w). Utilizando o método de Feynman de integrais de trajetória, integramos sobre as coordenadas dos osciladores do banho c obtemos uma expressão exata para o espectro de transmissão da partícula para uma temperatura do banho T > O. Limitando-nos então ao caso mais simples T = O, estudamos o tunelamento dissipativo da partícula através da barreira. Considerando h um parâmetro pequeno (limite semiclássico), aproximamos o espectro de transmissão da partícula através da contribuição das trajetórias clássicas c suas trajetórias vizinhas. Nesta aproximação consideramos a variação da ação efetiva da partícula para tempos dados de duração das trajetórias c deste modo substituímos o procedimento variacional seguido no trabalho indicado de Ueda onde não é considerada a variação nos tempos de travessia da partícula através da barreira. Num segundo problema variacional nos tempos de duração das trajetórias clássicas de acordo com o método do tempo complexo e considerando também a variação nas posições iniciais c finais dessas trajetórias, obtemos as equações de movimento das chamadas trajetórias clássicas especiais. Este tratamento das coordenadas iniciais c finais das trajetórias clássicas substitui o procedimento seguido no trabalho de Ueda onde é considerc1da uma aceleração nula durante todo o trajeto de movimento incluindo o trajeto na região da barreira. Diferentemente do artigo citado de Ueda, no presente trabalho utilizamos pacotes de ondas relativamente bem localizados para descrever os estados inicial e final da partícula. Em consequência, aproximamos o espectro de transmissão da partícula através de trajetórias clássicas especiais com coordenadas iniciais c finais iguais ao valor médio da coordenada para esses pacotes de ondas. O procedimento seguido neste trabalho, baseado no método do tempo complexo, permite obter o fator ele acoplamento apropriado entre as duas trajetórias que descrevem a ação efetiva ela partícula substituindo assim o procedimento de tipo ad hoc seguido com este fim no trabalho indicado de Ueda. O método do tempo complexo permite obter também o termo ela diferença entre a ação efetiva da partícula c o expoente ele tunelamento, sendo que estas grandezas são tratadas como iguais no trabalho citado de Ueda. Considerando termos até primeira ordem num campo elétrico externo c na interação da partícula com o banho de osciladores, obtemos expressões gerais para o expoente de tunelamento, o espectro de transmissão, a taxa total de tunelamento c o tempo de travessia da partícula através da barreira, válidas para um banho de osciladores com uma função de densidade espectral arbitrária. Assim temos que a interação da partícula com um banho de osciladores com uma função de densidade espectral arbitrária diminui a taxa total de tunelamento. Adicionalmente, obtemos que a interação da partícula com os osciladores do banho com frequências ?a = ?C ~ 1.9 T , onde T0 é o tempo característico de travessia através da barreira no caso cm que não há interação da partícula com o banho de osciladores nem campo elétrico, não afeta o tempo característico de travessia através da barreira. Por outro lado, a interação da partícula com os osciladores do banho que têm frequências ?a < ?C (?a > ?C) diminui (aumenta) o tempo característico de travessia através da barreira. No caso de um banho de osciladores com uma única frequência w c uma constante de acoplamento com a partícula dada por Ca = Ca (wT)a , são identificados cinco comportamentos diferentes em função de w para o expoente característico de tunelamento e o tempo característico de travessia através da barreira. Estes comportamentos correspondem aos valores de s < 1, s = 1, 1 < s < 2, s = 2 e s > 2. No trabalho de M. Ueda, Phys. Rev. B 54, 8676 (1996), foi considerado somente o expoente característico de tunelamento no caso s = 1. No caso de um banho ôhmico de osciladores a temperatura zero, assim corno no caso de um banho de osciladores com uma única frequência, obtemos que o espectro de transmissão da partícula é zero para urna energia final característica da partícula maior que a energia inicial característica. Este resultado corrige o resultado correspondente no trabalho citado de Ueda, o qual não é consistente do ponto de vista físico, permitindo também obter de um modo mais coerente a corrente de tunelamento entre dois metais separados por um material isolante a temperatura zero. Obtém-se também que a interação da partícula com um banho ôhmico de osciladora não afeta o tempo característico de travessia através da barreira até primeira ordem nessa interação / Abstract: This work was motivated by several difficulties found when studying the article by M . Ueda, Transmission Spectrum of a Tunneling Particle Interacting with Dynamical Fields: Real-Time Functional-Integral Approach, Phys. Rev. B 54, 8676 (1996). In that paper, using a real-time formalism, a tunneling particle is described by complex traversal times of tunneling. In the present work we propose a broader real-time formalism that allows for a more natural introduction of complex values of time in the description of a tunneling particle interacting with the environment. This proposal is based on the well-known complex time method used in the case of a tunneling particle with no interaction with the environment studied in the works of D. W. McLaughlin, J. Math. Phys. 13, 1099 (1972) and B. R. Holstein and A. R. Swift, Am. J. Phys. 50, 833 (1982). Following the cited work of Ueda, the environment of the particle is represented by a set, or heat bath, of harmonic oscillators which is characterized by a spectral density function J(w). Using the Feynman path integrals method, we integrate out the coordinates of the bath oscillators and obtain an exact expression for the transmission spectrum of the particle for a bath temperature T > O. Limiting ourselves to the simpler case T = O, we study the case of a dissipative tunneling of the particle. Considering h a small parameter (semiclassical limit) we approximate the transmission spectrum of the particle by the contribution of the classical trajectories and its neighboring paths. In this approach we consider the variation of the effective action of the particle for given duration times of the paths and replace the variation procedure followed in the cited work of Ueda where the variation in the traversal times of tunneling is not considered. In a second variation problem for the duration times of the classical paths, according to the complex time method and considering also the variation in the initial and final positions of these paths, we obtain the equations of motion for the so-called special classical paths. This treatment of the initial and final coordinates of the classical paths replaces the procedure followed in the cited work of Ueda where an acceleration equal to zero is considered during the entire path of motion including the region under the barrier. Unlike the cited article of Ueda, we use in the present work wave packets relatively well localized to describe the init.ial and final statics of the particle. Conscqncnt.ly, we approximate the transmission spectrum of the particle through special classical paths with initial and final coordinates equal to the average value of the coordinate for those wave packets. The procedure followed in this work, based on the complex time method, gives the appropriate coupling factor between the two paths describing the effective action of the particle and thus replaces the ad hoc procedure followed for this purpose in the cited work of Ueda. The complex time method also allows us to obtain the difference term between the effective action of the particle and the tunneling exponent. These quantities are treated as equal in Ueda\'s work. Considering terms up to first order in an external electric field and the interaction of the particle with the bath of oscillators, we obtain general expressions for the tunneling exponent, transmission spectrum, total tunneling rate and traversal time of tunneling, which are valid for a bath of oscillators with an arbitrary spectral lenity function. We find that the interaction of the particle with a bath of oscillators with an arbitrary spectral density function decreases the total tunneling rate. Also, we find that the interaction of the particle with the bath oscillators with frequencies ?a = ?C ~ 1.9 T , where To is the characteristic traversal time of tunneling when there is no interaction of the particle with the bath of oscillators nor electric field. , does not affect the characteristic traversal time of tunneling. On the other hand, the interaction of the particle with the bath oscillators having frequencies ?a< ?c (?a: > ?c decreases (increases) the characteristic traversal time of tunneling. In the case of a bath of oscillators with a single frequency w and a coupling constant with the particle given by Ca = Ca (wT)a we identify five different behaviors deepening on w for the characteristic tunneling exponent and the characteristic traversal time of tunneling. These behaviors correspond to the values of s < 1, s = 1, 1 < s < 2, s = 2 and s > 2. In the work of M. Ueda, Phys. Rev. B 54, 8676 (1996), it was only considered the characteristic tunneling exponent in the case s = 1. In the case of an ohmic bath of oscillators at zero temperature, as well as in the case of a bath of oscillators with a single frequency, we obtain that the transmission spectrum of the particle is ;1,cro for a final characteristic energy of the particle greater than the initial characteristic energy. This result corrects the corresponding result in Ueda work, which is not consistent from a physical point of view, allowing also for a more coherent derivation of the tunneling current between two metals separated by an insulating material at zero temperature. It is also obtained that the interaction of the particle with an ohmic bath of oscillators does not affect the characteristic traversal time of tunneling up to first order in that interaction / Doutorado / Física / Doutor em Ciências

In den vergangenen Jahrzehnten rückte das häufige Auftreten von multiplen Polylogarithmen und multiplen Zeta-Werten, in Feynman-Diagramm Rechnungen niedriger Ordnung, verstärkt in den wissenschaftlichen Fokus. Hierbei offenbarte sich eine Verbindung zu den mathematischen Theorien der Perioden und der iterierten Integrale von Chen. Eine ähnliche Allgegenwärtigkeit von multiplen Zeta-Werten wurde jüngst auch in der α'-Entwicklung von Genus-Null Stringtheorie Amplituden beobachtet. Davon inspiriert befasst sich diese Arbeit mit der Systematik der iterierten Integralen in den Streuamplituden der offenen Stringtheorie. Unser Fokus liegt insbesondere auf der Genus-Eins Amplitude, welche sich vollständig durch iterierte Integrale, die bezüglich einer punktierten elliptischen Kurve definiert sind, ausdrücken lässt. Wir führen den Begriff der getwisteten elliptischen multiplen Zeta-Werte ein. Dieser Begriff beschreibt eine Klasse von iterierten Integralen, die auf einer elliptischen Kurve definiert sind, bei welcher ein rationales Gitter entfernt wurde. Anschließend zeigen wir, dass die Entwicklung eines jeden getwisteten elliptischen multiplen Zeta-Wertes, bezüglich des modularen Parameters τ, durch ein Anfangswertproblem beschrieben wird. Weiterhin präsentieren wir ein Argument dafür, dass sich im Limes τ→i∞ jeder getwistete elliptische multiple Zeta-Wert durch zyklotomische multiple Zeta-Werte ausdrücken lässt. Schließlich beschreiben wir wie sich Genus-Eins Amplituden in offener Stringtheorie mithilfe von getwisteten elliptischen multiplen Zeta-Werten ausdrücken lassen und illustrieren dies für die Vier-Punkt Amplitude. Hierbei zeigt es sich, dass bis zu dritter Ordnung in α' alle Beiträge durch die Unterklasse der elliptischen multiplen Zeta-Werte ausgedrückt werden können, was wiederum äquivalent zu der Abwesenheit unphysikalischer Pole in Gliozzi-Scherk-Olive projizierter Superstringtheorie ist. / Over the last few decades the prevalence of multiple polylogarithms and multiple zeta values in low order Feynman diagram computations of quantum field theory has received increased attention, revealing a link to the mathematical theories of Chen’s iterated integrals and periods. More recently, a similar ubiquity of multiple zeta values was observed in the α'-expansion of genus-zero string theory amplitudes. Inspired by these developments, this work is concerned with the systematic appearance of iterated integrals in scattering amplitudes of open superstring theory. In particular, the focus will be on studying the genus-one amplitude, which requires the notion of iterated integrals defined on punctured elliptic curves. We introduce the notion of twisted elliptic multiple zeta values that are defined as a class of iterated integrals naturally associated to an elliptic curve with a rational lattice removed. Subsequently, we establish an initial value problem that determines the expansions of twisted elliptic multiple zeta values in terms of the modular parameter τ of the elliptic curve. Any twisted elliptic multiple zeta value degenerates to cyclotomic multiple zeta values at the cusp τ → i∞, with the corresponding limit serving as the initial condition of the initial value problem. Finally, we describe how to express genus-one open-string amplitudes in terms of twisted elliptic multiple zeta values and study the four-point genus-one open-string amplitude as an example. For this example we find that up to third order in α' all possible contributions in fact belong to the subclass formed by elliptic multiple zeta values, which is equivalent to the absence of unphysical poles in Gliozzi-Scherk-Olive projected superstring theory.

In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.

In dieser Dissertation erweitern wir die Ideen des Yangian-Bootstrap-Algorithmus auf Feynman-Diagramme mit massiven Teilchen. Ausgehend von der massiven dual-konformen Symmetrie der N = 4 Super-Yang-Mills Theorie auf dem Coulomb-Zweig konstruieren wir einen Satz von bilokalen Yangian Level-Eins Generatoren und zeigen, dass sie eine unendliche Anzahl von planaren ein- und zwei-Schleifen-Diagrammen vernichten. Wir beschreiben außerdem wie der dual-konforme Level-Eins Impuls-Operator auf eine massive Verallgemeinerung des gewöhnlichen spezial-konformen Generators im Impulsraum abgebildet wird. Als nächstes wenden wir den Yangian-Bootstrap-Algorithmus mit großem Erfolg auf eine Reihe von massiven Ein-Schleifen-Diagrammen mit verallgemeinerten Propagatorexponenten und in beliebiger Anzahl von Raumdimensionen an. Im Spezialfall der dual-konformen Integrale, deren Propagatorexponenten sich zur Raumdimension addieren, finden wir neue sehr einfache Darstellungen durch hypergeometrische Funktionen, die eine natürliche Verallgemeinerung für Diagramme mit beliebig vielen äußeren Punkten erlauben. Außerdem diskutieren wir Aspekte des Yangian-Bootstrap-Algorithmus in Minkowski-Raumzeit am Beispiel des masselosen Box-Integrals. Wir zeigen, dass dessen Yangian-Symmetrie gemeinsam mit seinen diskreten Permutationssymmetrien das Box-Integrals bis auf 12 unbestimmte Konstanten komplett festlegt. Schließlich schlagen wir vor, dass das Auftreten von Yangian-Symmetrie in massiven Fischnetz-Diagrammen mit deren Rolle als Ein-Spur-Streuamplituden in einer massiven Fischnetz-Theorie zusammenhängen könnte. In Analogie mit der masselosen Fischnetz-Theorie zeigen wir, wie diese Theorie als Deformation der N = 4 Super-Yang-Mills Theorie auf dem Coulomb-Zweig definiert werden kann. Wir diskutieren eine bestimmte Klasse von planaren Grenzfällen, in der die off-shell Streuamplituden der Theorie eine massive dual-konforme Symmetrie sowie Yangian-Symmetrie aufweisen. / In this dissertation, we extend the ideas of the Yangian bootstrap algorithm to massive Feynman diagrams. Based on the massive dual-conformal symmetry of Coulomb branch N = 4 super-Yang-Mills theory, we construct a set of bi-local Yangian level-one generators and show that they annihilate infinite classes of massive planar Feynman integrals at one and two loops. We also describe how the dual-conformal level-one momentum generator maps to a massive deformation of the ordinary momentum space special conformal generator. We then apply the Yangian bootstrap to a set of massive one-loop integrals with generalised propagator powers and in an arbitrary number of space dimensions to great success. In the special case of dual-conformal integrals, whose propagator powers sum to the space dimension, we find very simple novel hypergeometric structures, suggesting a natural generalisation to diagrams with an arbitrary number of external points. In the particular case of the massless box integral we also discuss elements of the Yangian bootstrap in Minkowski space. We show that its Yangian and discrete permutation symmetries constrain it up to 12 undetermined constants. We then derive the values of these constants via analytic continuation from the box integral in the Euclidean region. Finally, we provide evidence that the appearance of Yangian symmetry for massive fishnet diagrams is related to their role as colour-ordered scattering amplitudes in a massive fishnet theory. We show how to construct this theory from Coulomb branch N = 4 super-Yang-Mills theory, paralleling the original construction of the massless fishnet theory. We discuss how a particular class of planar limits leads to the emergence of massive dual-conformal symmetry as well as massive Yangian symmetry for the theory’s off-shell scattering amplitudes.

In dieser Dissertation untersuchen wir unterschiedliche Aspekte im Zusammenhang mit Korrelationsfunktionen in der Fischnetz-Theorie. Zunächst betrachten wir einen der einfachsten Korrelatoren der Fischnetz Theorie, das konforme Box-Integral, in Minkowski Signatur. Während dieses Integral in Euklidischer Signatur eine konforme Symmetrie aufweist, wird diese Symmetrie in Minkowski-Raumzeit subtil gebrochen. Wir beschreiben die Brechung der konformen Symmetrie quantitativ, indem wir die funktionale Form des Box-Integrals in allen kinematischen Regionen untersuchen. Ausserdem untersuchen wir das Ausmass zu dem das Box integral durch seine Yangian-Symmetrie festgelegt ist. Als nächstes widmen wir uns den Basso-Dixon-Graphen, die ebenfalls konforme Vier-Punkt-Integrale sind und Verallgemeinerungen des Box-Integrals zu höheren Schleifenordnungen darstellen. Wir leiten die Yangian-Ward-Identitäten ab, die diese Klasse von Integralen erfüllen. Die Ward-Identitäten sind einhomogene Erweiterungen der partiellen Differentialgleichungen, die im homogenen Fall durch Appell-Hypergeometrische Funktionen gelöst werden. Die Ward-Identitäten können natürlicherweise auf eine Ein-Parameter-Familie von D-dimensionalen Integralen erweitert werden, die Korrelatoren in der verallgemeinerten Fischnetz-Theorie von Kazakov und Olivucci darstellen. Schliesslich untersuchen wir den Dilatationsoperator in einem Drei-Skalar-Sektor der Fischnetztheorie, der auch als Eklektisches Modell bezeichnet wird. In diesem Sektor der Dilatationsoperator nimmt nicht--diagonalisierbare Form an. Das führt dazu, dass die Zwei-Punkt-Korrelationsfunktionen eine logarithmische Abhängigkeit von der Raumzeitseparierung der Operatoren annimmt. Unter Zuhilfenahme von kombinatorischen Argumenten führen wir eine generierende Funktion ein, die das Jordan-Block-Spektrum eines verwandten Modells, der hypereklektischen Spinkette, vollständig charakterisiert. / We study various aspects of correlation functions in fishnet theory. We begin with the study of the simplest correlator in theory theory, represented by the conformal box integral, in Minkowski space. While this integral is conformally invariant in Euclidean space, this symmetry is subtly broken in Minkowski space. We quantify the extent to which conformal symmetry is broken by analysing the functional form of the box in each kinematic region. We propose a new method to calculate the box integral directly in Minkowski space, by introducing a family of configurations with two points at infinity. Furthermore, we investigate the extent to which the box integral is constrained by Yangian symmetry. We constrain the functional form of the box integral in all kinematic regions up to twelve undetermined constants, which we fix by three separate analytic continuations from the Euclidean region. Next, we study the Basso-Dixon graphs, which represent higher-loop versions of the box integral. We derive and study Yangian Ward identities for this class of integrals. These take the form of inhomogeneous extensions of the partial differential equations defining the Appell hypergeometric functions. The Ward identities naturally generalise to a one-parameter family of D dimensional integrals representing correlators in a generalised fishnet theory. Finally, we study the dilatation operator in a particular three scalar sector of the fishnet theory, which has been dubbed the eclectic model. This dilatation operator is non-diagonalisable in this sector. This leads to logarithmic spacetime dependence in the corresponding two-point functions. Using combinatorial arguments, we introduce a generating function which fully characterises the Jordan block spectrum of a related model: the hypereclectic spin chain. This function is found by purely combinatorial means and can be expressed in terms of the q-binomial coefficient.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O presente trabalho tem como objetivo auxiliar professores de Educação Básica a abordar alguns conceitos básicos de Cálculo Diferencial e Integral no terceiro ano do ensino médio, utilizando os softwares wxMaxima e Geogebra. Nossa proposta visa resgatar o ensino do Cálculo no âmbito escolar, mas de uma maneira diferente da tradicional, utilizando a tecnologia em nosso favor, como uma ferramenta facilitadora no processo ensino-aprendizagem de conceitos de limites, derivadas e integrais. A ideia é que toda a parte algébrica e gráfica, que exige conhecimentos matemáticos específicos, seja feita pelos softwares e que os estudantes aprendam a interpretar as soluções que as ferramentas nos fornecem e a conhecerem um pouco mais sobre a matemática e suas aplicações. / The present work aims to help Basic Education teachers to approach some basic concepts of Differential and Integral Calculus in the third year of high school using the software wxMaxima and Geogebra. Our purpose is to recover the teaching of Calculus in the school context, but in a different way from the traditional one, using the technology in our favor, as a facilitating tool in the teaching-learning process of boundary, derivative and integral concepts. The idea is that all the algebraic and graphic part, which requires specific mathematical knowledge, is done by software and that students learn to interpret the solutions that the tools provide us and to know a little more about mathematics and its applications.

This work is devoted to investigating the algebraic and analytic integrability of first order polynomial partial differential equations via an understanding of the well-developed area of local and global integrability of polynomial vector fields. In the view of characteristics method, the search of first integrals of the first order partial differential equations P(x,y,z)∂z(x,y) ∂x +Q(x,y,z)∂z(x,y) ∂y = R(x,y,z), (1) is equivalent to the search of first integrals of the system of the ordinary differential equations dx/dt= P(x,y,z), dy/dt= Q(x,y,z), dz/dt= R(x,y,z). (2) The trajectories of (2) will be found by representing these trajectories as the intersection of level surfaces of first integrals of (1). We would like to investigate the integrability of the partial differential equation (1) around a singularity. This is a case where understanding of ordinary differential equations will help understanding of partial differential equations. Clearly, first integrals of the partial differential equation (1), are first integrals of the ordinary differential equations (2). So, if (2) has two first integrals φ1(x,y,z) =C1and φ2(x,y,z) =C2, where C1and C2 are constants, then the general solution of (1) is F(φ1,φ2) = 0, where F is an arbitrary function of φ1and φ2. We choose for our investigation a system with quadratic nonlinearities and such that the axes planes are invariant for the characteristics: this gives three dimensional Lotka– Volterra systems x' =dx/dt= P = x(λ +ax+by+cz), y' =dy/dt= Q = y(µ +dx+ey+ fz), z' =dz/dt= R = z(ν +gx+hy+kz), where λ,µ,ν 6= 0. v Several problems have been investigated in this work such as the study of local integrability and linearizability of three dimensional Lotka–Volterra equations with (λ:µ:ν)–resonance. More precisely, we give a complete set of necessary and sufficient conditions for both integrability and linearizability for three dimensional Lotka-Volterra systems for (1:−1:1), (2:−1:1) and (1:−2:1)–resonance. To prove their sufficiency, we mainly use the method of Darboux with the existence of inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable. Also, more general three dimensional system have been investigated and necessary and sufficient conditions are obtained. In another approach, we also consider the applicability of an entirely different method which based on the monodromy method to prove the sufficiency of integrability of these systems. These investigations, in fact, mean that we generalized the classical centre-focus problem in two dimensional vector fields to three dimensional vector fields. In three dimensions, the possible mechanisms underling integrability are more difficult and computationally much harder. We also give a generalization of Singer’s theorem about the existence of Liouvillian first integrals in codimension 1 foliations in Cnas well as to three dimensional vector fields. Finally, we characterize the centres of the quasi-homogeneous planar polynomial differential systems of degree three. We show that at most one limit cycle can bifurcate from the periodic orbits of a centre of a cubic homogeneous polynomial system using the averaging theory of first order.

Ce travail a pour but de proposer des outils visant `a comparer des résultats exp´erimentaux avec des modèles pour la dispersion de traceur en milieu poreux, dans le cadre de la dispersion anormale.Le “Mobile Immobile Model” (MIM) a été à l’origine d’importants progrès dans la description du transport en milieu poreux, surtout dans les milieux naturels. Ce modèle généralise l’quation d’advection-dispersion (ADE) e nsupposant que les particules de fluide, comme de solut´e, peuvent ˆetre immo-bilis´ees (en relation avec la matrice solide) puis relˆachées, le piégeage et le relargage suivant de plus une cin´etique d’ordre un. Récemment, une version stochastique de ce modèle a ´eté proposée. Malgré de nombreux succès pendant plus de trois décades, le MIM reste incapable de repr´esenter l’´evolutionde la concentration d’un traceur dans certains milieux poreux insaturés. Eneffet, on observe souvent que la concentration peut d´ecroˆıtre comme unepuissance du temps, en particulier aux grands temps. Ceci est incompatible avec la version originale du MIM. En supposant une cinétique de piégeage-relargage diff´erente, certains auteurs ont propos´e une version fractionnaire,le “fractal MIM” (fMIM). C’est une classe d’´equations aux d´eriv´ees par-tielles (e.d.p.) qui ont la particularit´e de contenir un op´erateur int´egral li´e`a la variable temps. Les solutions de cette classe d’e.d.p. se comportentasymptotiquement comme des puissances du temps, comme d’ailleurs cellesde l’´equation de Fokker-Planck fractionnaire (FFPE). Notre travail fait partie d’un projet incluant des exp´eriences de tra¸cageet de vélocimétrie par R´esistance Magn´etique Nucl´eaire (RMN) en milieuporeux insatur´e. Comme le MIM, le fMIM fait partie des mod`eles ser-vant `a interpréter de telles exp´eriences. Sa version “e.d.p.” est adapt´eeaux grandeurs mesur´ees lors d’exp´eriences de tra¸cage, mais est peu utile pour la vélocimétrie RMN. En effet, cette technique mesure la statistiquedes d´eplacements des mol´ecules excit´ees, entre deux instants fixés. Plus précisément, elle mesure la fonction caractéristique (transform´ee de Fourier) de ces d´eplacements. Notre travail propose un outil d’analyse pour ces expériences: il s’agit d’une expression exacte de la fonction caract´eristiquedes d´eplacements de la version stochastique du mod`ele fMIM, sans oublier les MIM et FFPE. Ces processus sont obtenus `a partir du mouvement Brown-ien (plus un terme convectif) par des changement de temps aléatoires. Ondit aussi que ces processus sont des mouvement Browniens, subordonnéspar des changements de temps qui sont eux-mˆeme les inverses de processusde L´evy non d´ecroissants (les subordinateurs). Les subordinateurs associés aux modèles fMIM et FFPE sont des processus stables, les subordinateursassoci´es au MIM sont des processus de Poisson composites. Des résultatsexp´erimenatux tr`es r´ecents on sugg´er´e d’´elargir ceci `a des vols de L´evy (plusg´en´eraux que le mouvement Brownien) subordonnés aussi.Le lien entre les e.d.p. fractionnaires et les mod`eles stochastiques pourla sous-diffusion a fait l’objet de nombreux travaux. Nous contribuons `ad´etailler ce lien en faisant apparaˆıtre les flux de solut´e, en insistant sur une situation peu ´etudiée: nous examinons le cas o`u la cinétique de piégeage-relargage n’est pas la mˆeme dans tout le milieu. En supposant deux cinétiques diff´erentes dans deux sous-domaines, nous obtenons une version du fMIMavec un opérateur intégro-diff´erentiel li´e au temps, mais dépendant de la position.Ces r´esultats sont obtenus au moyen de raisonnements, et sont illustrés par des simulations utilisant la discrétisation d’intégrales fractionnaires etd’e.d.p. ainsi que la méthode de Monte Carlo. Ces simulations sont en quelque sorte des preuves numériques. Les outils sur lesquels elles s’appuient sont présentés aussi. / We propose tools for to compare experimental data and models for anomalousdispersion in porous media.The “Mobile Immobile Model” (MIM) significantly improved the descrip-tion of mass transport in natural porous media. This model generalizes theadvection-dispersion equation (ADE) by assuming that fluid and solute parti-cles can be found in mobile on immobile states, exchanging matter accordingto first order kinetics. Moreover, it has a stochastic version. Nevertheless,the original MIM does not represent the power-law decrease of some break-through curves observed in some media, better described by a fractionalversion, the “fractal MIM” (fMIM) which assumes a different kinetics. Theacronym “fMIM” denotes partial differential equations (p.d.e.) involving afractional integral with respect to time, having solutions falling-off as powerof times, asymptotically. It keeps in similarity with the fractional Fokker-Planck equation (FFPE). As this equation, the fMIM describes the evolutionof the probability density function of stochastic processes, namely Brownianmotion sujected to a time change that is the hitting time of a stable sub-ordinator, strictly stable or not, according FFPE or fMIM is considered.Using probabilistic arguments and numerical simulation, we extend this re-sult to the case when the transport parameters and the time scales of thetime change vary in space. P.d.es are well suited for comparing with tracer tests data. Yet, they arenot very useful to discuss signals recorded by pulsed field gradient (PFG)nuclear magnetic resonance (NMR), a technique which measures the char-acteristic function (Fourier transform) of molecular displacements betweentwo fixed instants. For to process such data, we derive an expression of thecharacteristic function of the displacements of Brownian motions subordi-nated by the hitting times of stable subordinators, i.e. of processes whosedensity satisfies FFPE of fMIM. We also consider time changes that are hit-ting times of composite Poisson processes (CPP), which correspond to theoriginal version of the MIM.

Dans cette thèse, on étudie des processus stochastiques issus de la physique statistique. Le mouvement Brownien fractionnaire, objet central des premiers chapitres, généralise le mouvement Brownien aux cas où la mémoire est importante pour la dynamique. Ces effets de mémoire apparaissent par exemple dans les systèmes complexes et la diffusion anormale. L’absence de la propriété de Markov rend difficile l’étude probabiliste du processus. On développe une approche perturbative autour du mouvement Brownien pour obtenir de nouveaux résultats, sur des observables liées aux statistiques des extrêmes. En plus de leurs applications physiques, on explore les liens de ces résultats avec des objets mathématiques, comme les lois de Lévy et la constante de Pickands. / In this thesis, we study stochastic processes appearing in different areas of statistical physics: Firstly, fractional Brownian motion is a generalization of the well-known Brownian motion to include memory. Memory effects appear for example in complex systems and anomalous diffusion, and are difficult to treat analytically, due to the absence of the Markov property. We develop a perturbative expansion around standard Brownian motion to obtain new results for this case. We focus on observables related to extreme-value statistics, with links to mathematical objects: Levy’s arcsine laws and Pickands’ constant. Secondly, the model of elastic interfaces in disordered media is investigated. We consider the case of a Brownian random disorder force. We study avalanches, i.e. the response of the system to a kick, for which several distributions of observables are calculated analytically. To do so, the initial stochastic equation is solved using a deterministic non-linear instanton equation. Avalanche observables are characterized by power-law distributions at small-scale with universal exponents, for which we give new results.