Cette thèse est consacrée à l'étude du modèle de Potts p-adique à q états sur les arbres de Cayley. Plus précisément, nous étudions les mesures de Gibbs p-adiques du modèle de Potts sur les arbres de Cayley d’ordres 3 et 4 et leurs systèmes dynamiques p-adiques associés.Dans la première partie, nous décrivons les mesures de Gibbs p-adiques invariantes par translations pour le modèle de Potts sur l’arbre de Cayley d'ordre 4. L’existence de mesures de Gibbs p-adiques invariantes par translations est équivalente à l’existence de points fixes d’une fonction rationnelle appelée fonction de Potts--Bethe. Cette fonction de Potts--Bethe est obtenue à partir de l'équation récurrente d'une fonction à valeur dans Q_p^q rencontrée lors de la construction des mesures de Gibbs p-adiques du modèle de Potts sur les arbres de Cayley. Afin de décrire ces mesures de Gibbs p-adiques invariantes par translations, nous trouvons les solutions d'une équation quartique dans certains domaines E_p de Q_p. En général, nous trouvons aussi des conditions de solvabilité pour les équations quartiques dépressées sur Q_p.Dans la deuxième partie, nous étudions la dynamique des fonctions de Potts--Bethe dans le cas d’arbres de Cayley d'ordres 3 et 4. Premièrement, nous décrivons la fonction de Potts--Bethe ayant une bonne réduction. Pour une fonction de Potts--Bethe ayant une bonne réduction, la droite projective P^1(Q_p) peut être décomposée en composants minimaux et leur bassins attractifs. Cependant, les fonctions de Potts--Bethe associées au modèle de Potts sur les arbres de Cayley d'ordres 3 et 4 ont une mauvaise réduction : pour de nombreux nombres premiers p, ces fonctions correspondantes sont en effet chaotiques. En fait, pour ces nombres premiers p, nous prouvons que, restreintes à leurs ensembles de Julia, les fonctions de Potts--Bethe sont topologiquement conjuguées à une dynamique de décalage. Pour des autres nombres premiers p, l'ensemble de Julia correspondant peut être vide. La propriété chaotique de la fonction de Potts-Bethe implique l'immensité de l'ensemble des mesures de Gibbs p-adiques et une transition de phase. Comme application, nous obtenons que pour de nombreux nombres premiers p, les modèles de Potts p-adiques sur les arbres de Cayley d'ordres 3 et 4 ont une transition de phase. Nous remarquons également que l'affirmation que la transition de phase implique le chaos n'est pas vraie. / This thesis is devoted to the study of the q-state p-adic Potts model on Cayley trees. Specifically, we investigate the p-adic Gibbs measures of the Potts model on the Cayley trees of orders 3 and 4 and their related p-adic dynamical systems.In the first part, we describe the existence of the translation-invariant p-adic Gibbs measures of the Potts model on the Cayley tree of order 4. The existence of translation-invariant p-adic Gibbs measures is equivalent to the existence of fixed points of a rational map called Potts–Bethe mapping. The Potts–Bethe mapping is derived from the recurrent equation of a Q_p^q-valued function in the construction of the p-adic Gibbs measures of the Potts model on Cayley trees. In order to describe the existence of these translation-invariant p-adic Gibbs measures, we find the solutions of some quartic equation in some domains E_p of Q_p. In general, we also provide some solvability conditions for the depressed quartic equation on Q_p.In the second part, we study the dynamics of the Potts–Bethe mapping of degrees 3 and 4. First, we describe the Potts–Bethe mapping having good reduction. For a Potts– Bethe mapping with good reduction, the projective line P^1(Qp) can be decomposed into minimal components and their attracting basins. However, the Potts–Bethe mapping associated to the Potts model on the Cayley trees of orders 3 and 4 have bad reduction. For many prime numbers p, such Potts–Bethe mappings are chaotic. In fact, for these primes p, we prove that restricted to their Julia sets, the Potts–Bethe mappings are topologically conjugate to the full shift dynamics. For other primes p, the corresponding Julia set might be empty. The chaotic property of the Potts-Bethe mapping implies the vastness of the set of the p-adic Gibbs measures, and hence implies the phase transition. As application, for many prime numbers p, the Potts models over Q_p on the Cayley trees of orders 3 and 4 have phase transition. We also remark the statement that phase transition implies chaos is not true.

Kollektive Migration und Schwarmverhalten sind Beispiele für Selbstorganisation und können in verschiedenen biologischen Systemen beobachtet werden, beispielsweise in Vogel-und Fischschwärmen oder Bakterienpopulationen. Im Zentrum dieser Arbeit steht ein räumlich diskretes und zeitlich stetiges Model, welches das kollektive Migrieren von Individuen mittels eines stochastischen Vielteilchensystems (VTS) beschreibt und analysierbar macht. Das konstruierte Modell ist in keiner Klasse gut untersuchter Vielteilchensysteme enthalten, sodass der größte Teil der Arbeit der Entwicklung von Methoden zur Untersuchung des Langzeitverhaltens bestimmter VTS gewidmet ist. Eine entscheidende Rolle spielen hier Gibbs-Maße, die zu zeitlich invarianten Maßen in Beziehung gesetzt werden. Durch eine Simulationsstudie und die Analyse des Einflusses der Parameter Migrationsgeschwindigkeit, Sensitivität der Individuen und (räumliche) Dichte der Anfangsverteilung können Eigenschaften kollektiver Migration erklärt und Hypothesen für weitere Analysen aufgestellt werden. / Collective migration and swarming behavior are examples of self-organization and can be observed in various biological systems, such as in flocks of birds, schools of fish or populations of bacteria. In the center of this thesis lies a stochastic interacting particle system (IPS), which is a spatially discrete model with a continuous time scale that describes collective migration and which can be treated using analytical methods. The constructed model is not contained in any class of well-understood IPS’s. The largest part of this work is used to develop methods that can be used to study the long-term behavior of certain IPS’s. Thereby Gibbs-Measures play an important role and are related to temporally invariant measures. One can explain the properties of collective migration and propose a hypothesis for further analyses by a simulation study and by analysing the parameters migration velocity, sensitivity of individuals and (spatial) density of the initial distribution.

Avec l'arrivée de l'environnement comme enjeu mondial, le secteur de l'efficacité énergétique prend une place de plus en plus importante pour les entreprises autant au niveau économique que pour l'image de la compagnie. Par le fait même, le domaine des technologies de l'énergie est un créneau de recherche dont les projets en cours se multiplient. D'ailleurs, un des problèmes qui peut survenir fréquemment dans certaines entreprises est d'aller mesurer la composition des matériaux dans des conditions difficiles d'accès. C'est le cas par exemple de l'électrolyse de l'aluminium qui se réalise à des températures très élevées. Pour pallier à ce problème, il faut créer et valider des modèles mathématiques qui vont calculer la composition et les propriétés à l'équilibre du système chimique. Ainsi, l'objectif global du projet de recherche est de développer un outil de calcul d'équilibres chimiques complexes (plusieurs réactions et plusieurs phases) et l'adapter aux problèmes électrochimiques et d'équilibres contraints. Plus spécifiquement, la plateforme de calcul doit tenir compte de la variation de température due à un gain ou une perte en énergie du système. Elle doit aussi considérer la limitation de l'équilibre due à un taux de réaction et enfin, résoudre les problèmes d'équilibres électrochimiques. Pour y parvenir, les propriétés thermodynamiques telles que l'énergie libre de Gibbs, la fugacité et l'activité sont tout d'abord étudiées pour mieux comprendre les interactions moléculaires qui régissent les équilibres chimiques. Ensuite, un bilan énergétique est inséré à la plateforme de calcul, ce qui permet de calculer la température à laquelle le système est le plus stable en fonction d'une température initiale et d'une quantité d'énergie échangée. Puis, une contrainte cinétique est ajoutée au système afin de calculer les équilibres pseudo-stationnaires en évolution dans le temps. De plus, la contrainte d'un champ de potentiel électrique est considérée pour l'évaluation des équilibres électrochimiques par des techniques classiques de résolution et fera l'objet de travaux futurs via une technique d'optimisation. Enfin, les résultats obtenus sont comparés avec ceux présents dans la littérature scientifique pour valider le modèle. À terme, le modèle développé devient un bon moyen de prédire des résultats en éliminant beaucoup de coût en recherche et développement. Les résultats ainsi obtenus sont applicables dans une grande variété de domaines tels que la chimie et l'électrochimie industrielle ainsi que la métallurgie et les matériaux. Ces applications permettraient de réduire la production de gaz à effet de serre en optimisant les procédés et en ayant une meilleure efficacité énergétique.

It is commonly known that water plays a crucial role in many natural and industrial processes. One of these processes is the formation of water trees, and the subsequent breakdown of polyethylene used for high voltage cable insulation purposes. It has been shown that the mechanism for water molecules diffusing through amorphous polyethylene includes the formation of small water clusters. Gibbs Ensemble Monte Carlo molecular simulations has been performed to study the clustering of vapour phase water under vapour - liquid equilibrium conditions at temperatures ranging from 300 K to 600 K. The increase in vapour density with increasing temperature leads to a radical increase in the fraction of molecules belonging to clusters with two or more water molecules. It is also seen that the size of the clusters increases with temperature. The topologies of the smaller clusters, up to pentamers, have also been studied. A structural transition is observed from a large percentage with cyclic topology, which is the minimum energy configuration, at lower temperatures to predominantly linear clusters, favoured by entropic effects, at higher temperatures. Similar water properties have been obseved in simulations where the vapour phase has been replaced with a hydrocarbon rich phase ( n-alkanes and polyethylene ). Application of an external electric field to the polymer system reduces the water solubility and affects the water structure. A dramatic increase in water solubility in the hydrocarbon phase is observed when two oppositely charged ions are introduced in the hydrocarbon. The structure of the water have changed from several small clusters to a single large cluster with a rod-like shape. The cluster is extremely stable during the simulation. Application of an external electric field may enhance or reduce the effect of the ions depending on the direction of the field. Based on these observations is an alternative mechanism for water tree propagation proposed.

This thesis suggests a Bayesian vector autoregressive (VAR) model which allows for explicit parametrization of the unconditional mean for data measured at different frequencies, without the need to aggregate data to the lowest common frequency. Using a normal prior for the steady-state and a normal-inverse Wishart prior for the dynamics and error covariance, a Gibbs sampler is proposed to sample the posterior distribution. A forecast study is performed using monthly and quarterly data for the US macroeconomy between 1964 and 2008. The proposed model is compared to a steady-state Bayesian VAR model estimated on data aggregated to quarterly frequency and a quarterly least squares VAR with standard parametrization. Forecasts are evaluated using root mean squared errors and the log-determinant of the forecast error covariance matrix. The results indicate that the inclusion of monthly data improves the accuracy of quarterly forecasts of monthly variables for horizons up to a year. For quarterly variables the one and two quarter forecasts are improved when using monthly data.

Perfect Gibbs sampling is a method to turn Markov Chain Monte Carlo (MCMC) samplers into exact generators for independent random vectors. We show that a perfect Gibbs sampling algorithm suggested in the literature is not always generating from the correct distribution. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

The first project in this work used classical molecular dynamics to study the ice recrystallization inhibition potential of a series of carbohydrates and alcochols, using the hydration index, partial molar volumes and isothermal compressibilities as parameters for measuring their cryogenic efficacy. Unfortunately, after 8 months of testing, this work demonstrates that the accuracy and precision of the density extracted from simulations is not sufficient in providing accurate partial molar volumes. As a result, this work clearly demonstrates that current classical molecular dynamics technology cannot probe the volumetric properties of interest with sufficient accuracy to aid in the research and development of novel cryoprotectants.The second project in this work used molecular dynamics simulations to evaluate the Gibbs free energy change of substituting CO2 in sI clathrate hydrates by N2,CH4, SO2 and H2S flue gas impurities under conditions proposed for CO2 capture (273 K, 10 bar). Our results demonstrate that CO2 substitutions by N2 in the small sI cages were thermodynamically favored. This substitution is problematic in terms of efficient CO2 capture, since the small cages make up 25% of the sI clathrate cages, therefore a significant amount of energy could be spent on removing N2 from the flue gas rather than CO2. The thermodynamics of CO2 substitution by CH4, SO2 and H2S in sI clathrate hydrates was also examined. The substitution of CO2 by these gases in both the small and large cages were determined to be favorable. This suggests that these gases may also disrupt the CO2 capture by sI clathrate hydrates if they are present in large concentrations in the combustion flue stream. Similar substitution thermodynamics at 200 K and 10 bar were also studied. With one exception, we found that the substitution free energies do not significantly change and do not alter the sign of thermodynamics. Thus, using a lower capture temperature does not significantly change the substitution free energies and their implications for CO2 capture by sI clathrate hydrates.

This work is concerned with sampling and computation of rare events in molecular systems. In particular, we present new methods for sampling the canonical ensemble corresponding to the Boltzmann-Gibbs probability measure. We combine an equation for controlling the kinetic energy of the system with a random noise to derive a highly degenerate diffusion (i.e. a diffusion equation where diffusion happens only along one or few degrees of freedom of the system). Next the concept of hypoellipticity is used to show that the corresponding Fokker-Planck equation of the highly degenerate diffusion is well-posed, hence we prove that the solution of the highly degenerate diffusion is ergodic with respect to the Boltzmann-Gibbs measure. We find that the new method is more efficient for computation of dynamical averages such as autocorrelation functions than the commonly used Langevin dynamics, especially in systems with many degrees of freedom. Finally we study the computation of free energy using an adaptive method which is based on the adaptive biasing force technique.

Student Number : 0002483R - MSc dissertation - School of Molecular and Cell Biology - Faculty of Science / Isothermal titration calorimetry was used to investigate the forces that drive ligand binding to the active site of rGST M1-1. In an attempt to gain insight into the recognition of non-substrate ligands by GSTs, this study also investigates interactions between rGST M1-1 and ANS, a non-substrate ligand. At 25 °C, complex formation between rGST M1-1 and GSH, GSO3 -, and S-hexylglutathione is characterised by a monophasic binding isotherm with Kd values of 38.5 mM, 2.1 mM and 0.2 mM, respectively. One molecule of each ligand is bound per monomer of rGST M1-1. Binding of these ligands is enthalpically favourable and entropically unfavourable with a resultant favourable Gibbs free energy, overall. The effects of temperature and buffer ionisation on the energetics of binding were studied. The enthalpic and entropic contributions for all three ligands exhibited temperature dependence over the temperature range investigated (5-30 °C). The Gibbs free energy showed negligible changes with increasing temperature due to enthalpy-entropy compensation. The temperature dependence of the binding enthalpy yielded heat capacity changes of – 2.69 kJ/mol/K and –3.68 kJ/mol/K at 25 °C for GSH and S-hexylglutathione binding and –1.86 kJ/mol/K overall for GSO3 -. The linear dependence of DH on temperature for GSO3 - binding to rGST M1-1 suggests the formation of a more constrained complex which limits the fluctuations in conformations of the mu-loop at the active site. The non-linear dependence of DH on temperature for GSH and Shexylglutathione binding to the enzyme suggests the formation of a complex that samples different bound conformations due to the mobility of the mu-loop even after ligand is bound. Calorimetric binding experiments in various buffer systems with different ionisation enthalpies suggest that the binding of GSH to rGST M1-1 is coupled to the deprotonation of the thiol of GSH while GSO3 - binding to rGST M1-1 is independent of the buffer ionisation. At 25 °C, the rGST M1-1#1;ANS association is represented by a monophasic binding isotherm with one molecule of ANS bound per monomer of rGST M1-1. The interaction is both enthalpically and entropically driven with a Kd value of 27.2 mM representing moderate affinity. The effect of temperature on the interaction was investigated over the temperature range of 5-30 °C. The linear dependence of the binding enthalpy on temperature indicates that no significant structural changes occur upon binding of ANS to the enzyme (DCp = -0.34 kJ/mol/K). The change in heat capacity associated with the interaction can be attributed to the burial of the polar sulphonate group of ANS and the exposure of the anilino and naphthyl rings to solvent as well as the possibility of weak electrostatic interactions between ANS and residues at the active site. The effect of ethacrynic acid, GSH, GSO3 - and S-hexylglutathione on the fluorescence of ANS was investigated in order to obtain some idea as to the location of the ANS binding site on rGST M1-1. ANS was displaced by GSO3 -, S-hexylglutathione and ethacrynic acid, while no displacement occurred upon binding of GSH to the active site of rGST M1-1. Displacement studies and molecular docking simulations indicate that ANS binds to the H-site of rGST M1-1 and the possibility of a second binding site for the molecule cannot be ruled out.

A study of how the matrix environment impacts the structure and IR spectra of N-acetylglycine conformers. The conformational composition of this compound is determined according to an analysis of the FTIR spectra of N-acetylglycine isolated in low temperature argon matrices. Bands of three N-acetylglycine conformers are identified based on the spectra: one major and two minor. The structure of all observed conformers is stabilized by different intramolecular hydrogen bonds. The Gibbs free energies of the conformers were calculated (CCSD(T)/CBS method), and these energy values were used to calculate conformer population at a temperature of 360 K, of which 85.3% belonged to the main conformer, and 9.6% and 5.1% to the minor conformers. We also determined the size and shape of the cavities that form when the N-acetylglycine conformers are embedded in the argon crystal during matrix deposition. It is established that the most energetically favorable cavity for the planar main conformer is the cavity that forms when 7 argon atoms are replaced. At the same time, bulky minor conformers were embedded into cavities that correspond to 8 removed argon atoms. We calculated the complexation energy between argon clusters and conformers, and the deformation energy of the argon crystal and the N-acetylglycine conformers. The matrix-induced shifts to the conformer oscillation frequency are calculated. Published by AIP Publishing.