A generalized model which describes a family of antiferromagnetic Heisenberg magnets on a three-dimensional stacked triangular lattice is introduced. The model contains a constraint parameter which changes the details of the interactions but not the symmetry of the model. We investigate the question of whether a first or second order phase transition occurs in these systems using a short time dynamics method. This method does not suffer from the problem of critical slowing down which occurs in the usual equilibrium Monte Carlo simulations. The effective critical exponents are determined as a function of the constraint parameter. Our results provide strong evidence that the phase transition is first order. In addition, for a particular value of the constraint parameter, the model corresponds to an antiferromagnet on a stacked Kagome lattice. In this case, our results are not inconsistent with the existence of a finite temperature first order phase transition. / October 2007

A generalized model which describes a family of antiferromagnetic Heisenberg magnets on a three-dimensional stacked triangular lattice is introduced. The model contains a constraint parameter which changes the details of the interactions but not the symmetry of the model. We investigate the question of whether a first or second order phase transition occurs in these systems using a short time dynamics method. This method does not suffer from the problem of critical slowing down which occurs in the usual equilibrium Monte Carlo simulations. The effective critical exponents are determined as a function of the constraint parameter. Our results provide strong evidence that the phase transition is first order. In addition, for a particular value of the constraint parameter, the model corresponds to an antiferromagnet on a stacked Kagome lattice. In this case, our results are not inconsistent with the existence of a finite temperature first order phase transition.

A generalized model which describes a family of antiferromagnetic Heisenberg magnets on a three-dimensional stacked triangular lattice is introduced. The model contains a constraint parameter which changes the details of the interactions but not the symmetry of the model. We investigate the question of whether a first or second order phase transition occurs in these systems using a short time dynamics method. This method does not suffer from the problem of critical slowing down which occurs in the usual equilibrium Monte Carlo simulations. The effective critical exponents are determined as a function of the constraint parameter. Our results provide strong evidence that the phase transition is first order. In addition, for a particular value of the constraint parameter, the model corresponds to an antiferromagnet on a stacked Kagome lattice. In this case, our results are not inconsistent with the existence of a finite temperature first order phase transition.

Thesis (Ph. D.)--Physics, Georgia Institute of Technology, 2006. / Schatz, Michael, Committee Chair ; Cvitanovic, Predrag, Committee Member ; Uzer, Turgay, Committee Member ; Grigoriev, Roman, Committee Member ; Mischaikow, Konstantin, Committee Member.

Yes / In this article we briefly review the Boxed Molecular Dynamics (BXD) method, which allows analysis of thermodynamics and kinetics in complicated molecular systems. BXD is a multiscale technique, in which thermodynamics and long-time dynamics are recovered from a set of short-time simulations. In this article, we review previous applications of BXD to peptide cyclization, diamond etching, solution-phase organic reaction dynamics, and desorption of ions from self-assembled monolayers (SAMs). We also report preliminary results of simulations of diamond etching mechanisms and protein unfolding in AFM experiments. The latter demonstrate a correlation between the protein’s structural motifs and its potential of mean force (PMF). Simulations of these processes by standard molecular dynamics (MD) is typically not possible, since the experimental timescales are very long. However, BXD yields well-converged and physically meaningful results. Compared to other methods of accelerated MD, our BXD approach is very simple; it is easy to implement, and it provides an integrated approach for simultaneously obtaining both thermodynamics and kinetics. It also provides a strategy for obtaining statistically meaningful dynamical results in regions of configuration space that standard MD approaches would visit only very rarely. / DRG is grateful for funding from a Royal Society Research Fellowship. JB and DVS acknowledge the support of EPSRC (Grant No EP/E009824/1). E.M.-N. and S.A.V. are grateful to the “Centro de Supercomputación de Galicia (CESGA)” for the use of its computational resources, as well as to “Ministerio de Economía y Competitividad” (Grant No. CTQ2009-12588) for financial support. DS and E.M.-N. acknowledge the Leverhulme Trust for funding the E.M.-N. visit to Leeds by the grant “Accelerated classical and quantum molecular dynamics and its applications” (Grant No. VP1-2012-013).

The differences between refolding mechanisms of sperm whale apomyoglobin subsequent to three different unfolding conditions have been examined by atomistic level computer simulations. The three unfolding conditions used in this work are high-temperature, low temperature and low pH. The folding of this protein has been extensively studied experimentally, providing a large data base of folding parameters which can be probed using simulations. The crystal structure of sperm whale myoglobin was taken from Protein Data Bank, followed by the removal of the heme unit and a subsequent energy minimization was performed in order to generate the native apomyoblogin form. Thus, the native conformation of apomyoglobin utilized is the same in all the three different refolding simulations done in the present work. The differences are the way the initial unfolded conformations were obtained. The refolding trajectories were obtained at room temperature using the Stochastic Difference Equation in Length algorithm. The results reveal differences between the three refolding routes. In contrast to previous molecular simulations that modeled low pH denaturation, an extended intermediate with large helical content was not observed in the refolding simulations from the high-temperature unfolded state. Otherwise, a structural collapse occurs without formation of helices or native contacts. Once the protein structure is more compact (radius of gyration less than 18 angstroms) secondary and tertiary structures appear. The low pH simulations show some agreement with the low pH experimental data and previous molecular dynamics simulations, like formation of a conformation having radius of gyration around 20 angstroms and large helical content. And the refolding simulations after the low temperature unfolding present differences in the properties of apomyoglobin folding route, comparing to the other two previous conditions. The collapse of the protein during folding occurs later in the simulation when compared with high-temperature denaturing state, but earlier when compared to low pH simulations. These differences strongly suggest that a protein can follow different folding routes, depending on the nature and the structure of the unfolded state.

Nous étudions la recombinaison radiative des porteurs de charges photogénérés dans les puits quantiques InGaN/GaN étroits (2 nm). Nous caractérisons le comportement de la photoluminescence face aux différentes conditions expérimentales telles la température, l'énergie et la puissance de l'excitation et la tension électrique appliquée. Ces mesures montrent que l'émission provient d'états localisés. De plus, les champs électriques, présents nativement dans ces matériaux, n'ont pas une influence dominante sur la recombinaison des porteurs. Nous avons montré que le spectre d'émission se modifie significativement et subitement lorsque la puissance de l'excitation passe sous un certain seuil. L'émission possède donc deux ``phases'' dont nous avons déterminé le diagramme. La phase adoptée dépend à la fois de la puissance, de la température et de la tension électrique appliquée. Nous proposons que la phase à basse puissance soit associée à un état électriquement chargé dans le matériau. Ensuite, nous avons caractérisé la dynamique temporelle de notre échantillon. Le taux de répétition de l'excitation a une influence importante sur la dynamique mesurée. Nous concluons qu'elle ne suit pas une exponentielle étirée comme on le pensait précédemment. Elle est exponentielle à court temps et suit une loi de puissance à grand temps. Ces deux régimes sont lié à un seul et même mécanisme de recombinaison. Nous avons développé un modèle de recombinaison à trois niveaux afin d'expliquer le comportement temporel de la luminescence. Ce modèle suppose l'existence de centres de localisation où les porteurs peuvent se piéger, indépendamment ou non. L'électron peut donc se trouver sur un même centre que le trou ou sur n'importe quel autre centre. En supposant le transfert des porteurs entre centres par saut tunnel on détermine, en fonction de la distribution spatiale des centres, la dynamique de recombinaison. Ce modèle indique que la recombinaison dans les puits InGaN/GaN minces est liée à des agglomérats de centre de localisation. / We study the radiative recombination of optically generated charges in thin (2 nm) InGaN quantum wells. We characterise the behaviour of the photoluminescence with varying experimental conditions such as temperature, energy and power of the excitation and externally applied voltage. These measurements show that emission comes from localised states. We also show that electric fields, natively present in these materials, do not have a dominating effect on charge carrier dynamics. We have shown that the emission spectrum changes significantly and rapidly when the excitation power drops below a certain level. The emission has two phases of which we have measured the diagram. The phase of the emission depends on the power of the excitation, the temperature and the electric field. We propose that the low power phase is associated with an electrically charged state in the material. Decay dynamics was then characterised. We find that the excitation repetition rate has an influence on the measured dynamics. We conclude that the dynamics are not stretched-exponential as it was originally thought. The dynamics are exponential at short time and follow a power law at long time. This byphasic character results from a single recombination process. We have developped a three-level recombination model to describe experimental dynamics. It supposes the existence of localisation states where carriers can localise, independently or not. This means that the electron can be localised on the same state as the hole or on any other state. If we suppose that inter-state transitions occurs by a tunnel effect, one can determine the decay dynamics as a function of the localisation states' spatial distribution. Henceforth, we then show that radiative recombination in thin InGaN/GaN quantum wells is dominated by localisation and charge separation.

Dans cette thèse, on s'intéresse à l'étude qualitative des solutions d'équations aux dérivées partielles hamiltoniennes par le biais de la théorie des mesures invariantes. L'existence d'une telle mesure pour une EDP fournit, en effet, des informations sur sa dynamique en temps long. Nous étudierons deux situations quelque peu "extrémales". Dans une première partie, nous nous intéressons aux équations ayant une infinité de lois de conservation et dans une seconde, aux équations dont on ne connaît qu'une seule loi de conservation non triviale.Nous étudions les premières équations par le biais de l'équation de Benjamin-Ono. Il s'agit d'un modèle de description des ondes internes dans un fluide de grande profondeur.Nous nous intéressons à la dynamique de cette équation sur l'espace C^infty(T) en lui construisant une mesure invariante sur cet espace. Par conséquent, une propriété de récurrence presque sûre (par rapport à cette mesure) est établie pour les solutions infiniment lisses de cette équation. Nous prouvons, ensuite, des propriétés de non-dégénérescence pour cette mesure. En effet, nous montrons que, via cette mesure, une infinité de fonctionnelles indépendantes ont des distributions absolument continues par rapport à la mesure de Lebesgue sur R. Enfin, nous montrons que cette mesure est de nature au moins $2$-dimensionnelle. Dans ce travail, nous avons utilisé l'approche Fluctuation-Dissipation-Limite (FDL) introduite par Kuksin-Shirikyan. Notons qu'une propriété de récurrence presque sûre a été établie pour les solutions de régularité Sobolev de l'équation de Benjamin-Ono, dans les travaux de Deng, Tzvetkov et Visciglia.Dans l'autre partie de la thèse, nous abordons l'équation de Klein-Gordon à non-linéarité cubique, c'est un exemple d'EDPs hamiltoniennes pour lesquelles il n'est connu qu'une seule loi de conservation non triviale. Cette équation modélise l'évolution d'une particule massive relativiste. Ici, nous considérons les cas où l'équation est posée sur le tore tri-dimensionnel ou sur un domaine borné de R^3 à bord assez régulier. Nous lui construisons une mesure invariante concentrée sur l'espace de Sobolev H^2, en utilisant toujours l'approche FDL. Un autre aspect de ce travail est d'étendre le cadre de cette approche au contexte des EDPs à une seule loi de conservation, en effet, dans les travaux antérieurs, l'approche FDL avait nécessité deux lois de conservation pour fonctionner. Puis nous établissons une propriété de non-dégénérescence pour la mesure construite. Par conséquent, une propriété de récurrence presque sûre, par rapport à la mesure construite, est prouvée. Notons que des travaux antérieurs dus à Burq-Tzvetkov, de Suzzoni, Bourgain-Bulut et Xu ont traité la question de mesure de Gibbs invariante pour des équations des ondes dans un contexte radial. / In this thesis, we are concerned with the qualitative study of solutions of Hamiltonian partial differential equations by the way of the invariant measures theory. Indeed, existence of such a measure provides some informations concerning the large time dynamics of the PDE in question. In this thesis we treat two "extremal" situations. In the first part, we consider equations with infinitely many conservation laws, and in the second, we study equations for which we know only one non-trivial conservation law.We study the first equations by considering the Benjamin-Ono equation. The latter is a model describing internal waves in a fluide of great depth.We are concerned with the dynamics of that equation on the space C^infty(T) by constructing for it an invariant measure on that space. Accordingly, an almost sure (w.r.t. this measure) recurrence property is established for infinitely smooth solutions of that equation. Then, we prove qualitative properties for the constructed measure by showing that there are infinitely many independent observables whose distributions via this measure are absolutely continuous w.r.t. the Lebesgue measure on R. Moreover, we establish that the measure is of at least 2-dimensional nature. In this work, we used the Fluctuation-Dissipation-Limit (FDL) approach introduced by Kuksin and Shirikyan. Notice that an almost sure recurrence property for the Benjamin-Ono equation was established on Sobolev spaces by Deng, Tzvetkov and Visciglia.In the second part of the thesis, we consider the cubic Klein-Gordon equation, which is an example of Hamiltonian PDEs for which we know only one conservation law. This equation models the evolution of a massive relativistic particle. Here, we consider both the case of the tri-dimensional periodic solutions and those defined on a bounded domain of R^3. In both settings, we construct an invariant measure concentrated on the Sobolev space H^2xH^1, again with use of the FDL approach. Another aspect of this work is to extend the FDL approach to the context of PDEs having only one conservation law; indeed, in previous works, this approach required two conservation laws. Qualitative properties for the measure and almost sure (w.r.t. this measure) recurrence for H^2-solutions are proven. Notice that previous works by Burq-Tzvetkov, de Suzzoni, Bourgain-Bulut and Xu have treated the invariant Gibbs measure problem in the radial symmetry context for waves equations.

Nous étudions la recombinaison radiative des porteurs de charges photogénérés dans les puits quantiques InGaN/GaN étroits (2 nm). Nous caractérisons le comportement de la photoluminescence face aux différentes conditions expérimentales telles la température, l'énergie et la puissance de l'excitation et la tension électrique appliquée. Ces mesures montrent que l'émission provient d'états localisés. De plus, les champs électriques, présents nativement dans ces matériaux, n'ont pas une influence dominante sur la recombinaison des porteurs. Nous avons montré que le spectre d'émission se modifie significativement et subitement lorsque la puissance de l'excitation passe sous un certain seuil. L'émission possède donc deux ``phases'' dont nous avons déterminé le diagramme. La phase adoptée dépend à la fois de la puissance, de la température et de la tension électrique appliquée. Nous proposons que la phase à basse puissance soit associée à un état électriquement chargé dans le matériau. Ensuite, nous avons caractérisé la dynamique temporelle de notre échantillon. Le taux de répétition de l'excitation a une influence importante sur la dynamique mesurée. Nous concluons qu'elle ne suit pas une exponentielle étirée comme on le pensait précédemment. Elle est exponentielle à court temps et suit une loi de puissance à grand temps. Ces deux régimes sont lié à un seul et même mécanisme de recombinaison. Nous avons développé un modèle de recombinaison à trois niveaux afin d'expliquer le comportement temporel de la luminescence. Ce modèle suppose l'existence de centres de localisation où les porteurs peuvent se piéger, indépendamment ou non. L'électron peut donc se trouver sur un même centre que le trou ou sur n'importe quel autre centre. En supposant le transfert des porteurs entre centres par saut tunnel on détermine, en fonction de la distribution spatiale des centres, la dynamique de recombinaison. Ce modèle indique que la recombinaison dans les puits InGaN/GaN minces est liée à des agglomérats de centre de localisation. / We study the radiative recombination of optically generated charges in thin (2 nm) InGaN quantum wells. We characterise the behaviour of the photoluminescence with varying experimental conditions such as temperature, energy and power of the excitation and externally applied voltage. These measurements show that emission comes from localised states. We also show that electric fields, natively present in these materials, do not have a dominating effect on charge carrier dynamics. We have shown that the emission spectrum changes significantly and rapidly when the excitation power drops below a certain level. The emission has two phases of which we have measured the diagram. The phase of the emission depends on the power of the excitation, the temperature and the electric field. We propose that the low power phase is associated with an electrically charged state in the material. Decay dynamics was then characterised. We find that the excitation repetition rate has an influence on the measured dynamics. We conclude that the dynamics are not stretched-exponential as it was originally thought. The dynamics are exponential at short time and follow a power law at long time. This byphasic character results from a single recombination process. We have developped a three-level recombination model to describe experimental dynamics. It supposes the existence of localisation states where carriers can localise, independently or not. This means that the electron can be localised on the same state as the hole or on any other state. If we suppose that inter-state transitions occurs by a tunnel effect, one can determine the decay dynamics as a function of the localisation states' spatial distribution. Henceforth, we then show that radiative recombination in thin InGaN/GaN quantum wells is dominated by localisation and charge separation.

The aim of this thesis is to introduce a general framework for what is informally referred to as finite-time dynamics. Within this framework, we study hyperbolicity of reference trajectories, existence of invariant manifolds as well as normal hyperbolicity of invariant manifolds called Lagrangian Coherent Structures. We focus on a simple derivation of analytical results. At the same time, our approach together with the analytical results has strong impact on the numerical implementation by providing calculable expressions for known functions and continuity results that ensure robust computation. The main results of the thesis are robustness of finite-time hyperbolicity in a very general setting, finite-time analogues to classical linearization theorems, an approach to the computation of so-called growth rates and the generalization of the variational approach to Lagrangian Coherent Structures.