Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT:In this thesis we discuss some of the anomalies present in systems with long-range interactions, for instance negative speci c heat and negative magnetic susceptibility, and show how they can be related to the convexity properties of the thermodynamic potentials and nonequivalence of ensembles. We also discuss the possibility of engineering long-range quantum spin systems with cold atoms in optical lattices to experimentally verify the existence of nonequivalence of ensembles. We then formulate an expression for the density of states when the energy and magnetisation correspond to a pair of non-commuting operators. Finally we analytically compute the entropy s( ;m) as a function of energy, , and magnetisation, m, for the anisotropic Heisenberg model with Curie-Weiss type interactions. The results show that the entropy is non-concave in terms of magnetisation under certain circumstances which in turn indicates that the microcanonical and canonical ensembles are not equivalent and that the magnetic susceptibility is negative. After making an appropriate change of variables we show that a second-order phase transition can be present at negative temperatures in the microcanonical ensemble which cannot be represented in the canonical ensemble. / AFRIKAANSE OPSOMMING: In hierdie tesis bespreek ons van die onverwagte eienskappe wat sisteme met lang afstand wisselwerkings kan openbaar, byvoorbeeld negatiewe spesi eke warmte en negatiewe magnetiese suseptibiliteit. Ons dui ook die ooreenkoms tussen hierdie gedrag en die konveksiteit van die termodinamiese potensiale en nie-ekwivalente ensembles aan. Hierna bespreek ons die moontlikheid om lang afstand kwantum spin sisteme te realiseer met koue atome in 'n optiese rooster. Daarna wys ons hoe dit moontlik is om 'n uitdrukking vir die digtheid van toestande te formuleer vir sisteme waar die energie en magnetisasie ooreenstem met operatore wat nie met mekaar kommuteer nie. Uiteindelik bepaal ons die entropie, s( ;m), in terme van die energie, , en magnetisasie, m, vir die anisotropiese Heisenberg model met Curie-Weiss tipe interaksies. Die resultate wys dat die entropie onder sekere omstandighede nie konkaaf in terme van magnetisasie is nie. Dit, op sy beurt, dui aan dat die mikrokanoniese en kanoniese ensembles nie ekwivalent is nie en dat die magnetiese suseptibiliteit negatief kan wees. Nadat ons 'n toepaslike transformasie van veranderlikes maak, wys ons dat 'n tweede orde fase-oorgang by negatiewe temperature kan plaasvind in die mikrokanoniese ensemble wat nie verteenwoordig kan word in die kanoniese ensemble nie.

We consider a gas of indistinguishable bosons, confined to a ring of radius R, and interacting via a pair-wise cosine potential. This may be thought of as the quantized Hamiltonian Mean Field (HMF) model for bosons originally introduced by Chavanis as a generalization of Antoni and Ruffo’s classical model. This thesis contains three parts: In part one, the dynamics of a Bose-condensate are considered by studying a generalized Gross-Pitaevskii equation (GGPE). Quantum effects due to the quantum pressure are found to substantially alter the system’s dynamics, and can serve to inhibit a pathological instability for repulsive interactions. The non-commutativity of the large-N , long-time, and classical limits is discussed. In part two, we consider the GGPE studied above and seek static solutions. Exact solutions are identified by solving a non-linear eigenvalue problem which is closely related to the Mathieu equation. Stationary solutions are identified as solitary waves (or solitons) due to their small spatial extent and the system’s underlying Galilean invariance. Asymptotic series are developed to give an analytic solution to the non- linear eigenvalue problem, and these are then used to study the stability of the solitary wave mentioned above. In part three, the exact solutions outlined above are used to study quantum fluctuations of gapless excitations in the HMF model’s symmetry broken phase. It is found that this phase is destroyed at zero temperature by large quantum fluctuations. This demonstrates that mean-field theory is not exact, and can in fact be qualitatively wrong, for long-range interacting quantum systems, in contrast to conventional wisdom. / Thesis / Doctor of Philosophy (PhD) / The Hamiltonian Mean Field (HMF) model was initially proposed as a simplified description of self-gravitating systems. Its simplicity shortens calculations and makes the underlying physics more transparent. This has made the HMF model a key tool in the study of systems with long-range interactions. In this thesis we study a quantum extension of the HMF model. The goal is to understand how quantum effects can modify the behaviour of a system with long-range interactions. We focus on how the model relaxes to equilibrium, the existence of special “solitary waves”, and whether quantum fluctuations can prevent a second order (quantum) phase transition from occurring at zero temperature.

One of the most important aspects for all life on this planet is the act to keep their biological processes in a state where they do not reach equilibrium. One part in the upholding of this imbalanced state is the barrier between the cells and their surroundings, created by the cell membrane. Additionally, terrestrial animal life often requires a barrier that protects the organism's body from external hazards and water loss. As an alternative to experiments, the investigation of the processes occurring at these barriers can be performed by using molecular dynamics simulations. Through this method we can obtain an atomistic description of the dynamics associated with events that are not accessible to experimental setups.  In this thesis the first paper presents an improved particle-mesh Ewald method for the calculation of long-range Lennard-Jones interactions in molecular dynamics simulations, which solves the historical performance problem of the method. The second paper demonstrate an improved implementation, with a higher accuracy, that only incurs a performance loss of roughly 15% compared to conventional simulations using the Gromacs simulation package. Furthermore, the third paper presents a study of cholesterol's effect on the permeation of six different solutes across a variety of lipid bilayers. A laterally inhomogeneous permeability in cholesterol-containing membranes is proposed as an explanation for the large differences between experimental permeabilities and calculated partition coefficients in simulations. The fourth paper contains a coarse-grained simulation study of a proposed structural transformation in ceramide bilayer structures, during the formation of the stratum corneum. The simulations show that glycosylceramides are able to stabilize a three-dimensionally folded bilayer structure, while simulations with ceramides collapse into a lamellar bilayer structure. / <p>QC 20160308</p>

Mon travail traite des propriétés géométriques de domaines présents dans un modèle magnétique simple, le modèle d'Ising. Par domaines nous entendons des régions où les spins prennent des valeurs similaires. En plus des propriétés telles que l'aimantation ou la susceptibilité magnétique, il est intéressant d'étudier la structure des domaines qu'il est naturel de décrire dans le cadre de la percolation. Dans cette thèse je me suis intéressé à plusieurs problèmes liés aux domaines de spins à l'équilibre et hors d'équilibre. J'ai ainsi étudié la dynamique des amas après des trempes critiques ou sous-critiques. Dans le cas de trempes cyritiques, le scaling dynamique a été étudié finement et l'influence des propriétés d'équilibre sur la dynamique mise en avant. Pour les trempes sous-critiques, nous avons considéré comme conditions initiales l'équilibre à une température critique ou infinie. Nous avons montré que dans le cas d'une température initiale critique la probabilité que le système finisse son évolution dans un état avec des bandes est exactement celle qu'un amas percole initialement. Concernant une température initiale infinie nous avons mis en évidence un régime transitoire conduisant le système vers le point critique de percolation. À l'équilibre à la température critique, nous avons obtenu une formule exacte pour la probabilité qu'un amas s'enroule autour d'un système avec des conditions aux bords périodiques. Nous avons aussi étudié le comportement critique du modèle d'Ising avec des interactions à longue portée en nous intéressant au passage des comportements longue portée à courte portée. / In this work I have considered the geometrical properties of the domains found in the Ising model. Those domains are regions where the spins have the same value. In addition to the properties such as magnetisation and magnetic susceptibility, it is interesting to study the domains' structure and this is done naturally within percolation theory. In this thesis, I considered several situations concerning spin domains be it in equilibrium or out of equilibrium. I studied the dynamics of domains after critical or sub-critical quenches. For critical quenches the dynamical scaling has been carefully checked and the influence of the equilibrium properties on the dynamics has been shown. For sub-critical quenches we have considered both critical and infinite temperature initial conditions. We have shown that for critical initial condition the probability that the system ends up in a stripe state is exactly the probability that a spin cluster percolates initially. For the infinite temperature initial condition, we have discovered a transient regime which brings very quickly the system to a state similar to critical percolation. In equilibrium at the critical temperature we obtained an exact formula for the wrapping probabilities of Ising spin clusters on a system with periodic boudary conditions. We have also studied the critical behaviour of the Ising model with long-range interactions with a special interest to the cross-over between the long-range and short-range regimes.

Les interactions à longue portée concernent de nombreux systèmes naturels. Un exemple notable est celui de la gravitation newtonienne qui est pertinent dans le cas de l'étude de systèmes d'étoiles ou d'amas de galaxies. Ces systèmes ont notamment la particularité de ne pas respecter l'additivité des potentiels thermodynamiques et présentent une dynamique dominée par les effets collectifs. Une caractéristique remarquable est qu'après une évolution très rapide, ces systèmes restent piégés dans des états quasi-stationnaires pendant un temps qui peut être extrêmement grand (divergeant avec la taille du système). C'est seulement sur des échelles de temps plus longue que les simulation montre que ces systèmes relaxent à l’équilibre thermodynamique.Les états quasi-stationnaire sont interprétés théoriquement comme les solution stationnaires de l'équation de Vlasov. Cette équation de champs moyen représente un très bonne approximation de la dynamique macroscopique des systèmes en interaction à longue portée dans la limite ou le nombre de particule tend vers l'infini. Dans une premier temps, nous nous attacherons à comprendre, en fonction de la portée de la force de paire et de sa régularisation à court distance, quel est le champs de validité de cette équation, et en particulier, dans quelle cas le phénomène d'état quasi-stationnaire est attendu.Dans une seconde partie, combinant les approches théoriques et numériques, nous étudions la modification de la dynamique des systèmes à longue portée soumis à différentes sortes de perturbation non-Hamiltonienne. La robustesse des états quasi-stationnaires en présence des différentes perturbation est analysée en détails. / Long range interactions concern numerous natural systems. A notable example is the one of the gravitation which is relevant in the case of the study of a stars system or galaxy clusters. In particular, these systems does not respect the additivity of thermodynamical potential and present a dynamics dominated by collective effects. One of the most remarkable feature is that, after a very rapid evolution, these systems remains trapped into quasi-stationary states up to a very long time (diverging with the system size). It is only on longer time scales, that simulations have shown that the system relaxes to thermal equilibrium.Quasi-stationary states are theoretically interpreted as solutions of the Vlasov equation. This mean filed equation represents a very good approximation of the dynamics of long range systems in the limit of a large number of particles. Firstly we give a limit on the validity of the Vlasov equation depending of the range of the pair force and on its short scales regularisation. In a second part, using theoretical an numerical approach, we study the modification of the dynamics of long range systems when subjected to different kinds of non-Hamiltonian perturbations. In particular, the robustness of quasi-stationary states, in presence of this different perturbations is analysed in details.

One of the most important aspects for all life on this planet is theact to keep their cellular processes in a state where they do notreach equilibrium. One part in the upholding of this imbalanced stateis the barrier between the cells and their surroundings, created bythe cell membrane. In addition to experiments, the investigation ofprocesses occuring in the cell membrane can be performed by usingmolecular dynamics simulations. Through this method we can obtain anatomistic description of the dynamics associated with events that arenot accessible to experimental setups. Molecular dynamics relies onthe integration of Newton's equations of motion in order to sample therelevant parts of phase-space for the system, and therefore it isdependent on a correct description of the interactions between all thesimulated particles. In this thesis I first present an improved methodfor the calculation of long-range interactions in molecular dynamicssimulations, followed by a study of cholesterol's impact on thepermeation of small solutes across a lipid bilayer. The first paper presents a previously derived modification to theparticle-mesh Ewald method, which makes it possible to apply thisto long-range Lennard-Jones interactions. Old implementations of themethod have been haunted by an extreme performance degradation andhere I propose a solution to this problem by applying a modifiedinteraction potential. I further show that the historical treatmentof long-range interactions in simulations of lipid bilayers hasnon-negligible effects on their structural properties.In the second paper, this modification is improved such that the smallerrors introduced by the modified interaction potential becomenegligible. Furthermore, I demonstrate that I have also improved theimplementation of the method so that it now only incurs a performanceloss of roughly 15% compared to conventional simulations using theGromacs simulation package.The third paper presents a simulation study of cholesterol's effect onthe permeation of six different solutes across a variety of lipidbilayers. I analyze the effect of different head groups, tail lengths,and tail saturation by performing simulations of the solutes in fourdifferent bilayers, with cholesterol contents between 0% and50%. Analysis of the simulations shows that the impact of the surfacearea per lipid on the partitioning of the solute could be lower thanpreviously thought. Furthermore, a model with a laterallyinhomogeneous permeability in cholesterol-containing membranes isproposed, which could explain the large differences betweenpermeabilities from experiments and calculated partition coefficientsin simulations. / <p>QC 20140609</p>

This thesis investigates how the low-energy magnetic configuration of a mesoscopic chain can be tuned by geometrical modifications. The magnetic arrays made by single-domain stadium shaped elements positioned side-by-side were fabricated by patterning into a sputtered ferromagnetic thin film. The thickness of the thin film was determined by X-ray reflectivity measurements while Scanning Electron Microscopy and Atomic Force Microscopy were used to characterize the surface morphology of the nanostructures. Magnetic Force Microscopy was used to image the magnetic configuration of mesoscopic chains after applying a thermal annealing protocol and a field demagnetization protocol. By gradually modifying the geometrical arrangement of the half of mesospins, the magnetic chain is found to exhibit a transition from antiferromagnetic to dimer antiferromagnetic configuration after the thermal annealing treatment. After the field demagnetization protocol, both antiferromagnetic and dimer antiferromagnetic domains are formed. Micromagnetic simulations were performed to investigate how the interaction between the mesospins is affected by the geometrical modifications and a qualitative method was invented to examine the theoretical low-energy state of the magnetic chains. It is found that the low-energy magnetic configuration of the mesoscopic arrays is formed after the competition and collaboration of different interactions and is the one observed after the thermal annealing treatment.

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No description available.

Nous étudions l’influence du champ magnétique aléatoire et des interactions à longue portée sur le comportement critique du modèle d’Ising ; notre approche est basée sur une version non perturbative et fonctionnelle du groupe de renormalisation. Les concepts du groupe de renormalisation non perturbatif sont tout d’abord introduits, puis illustrés dans le cadre simple d’une théorie classique d’un champ scalaire. Nous discutons ensuite les propriétés critiques de cette dernière en présence d’un champ magnétique aléatoire gelé qui traduit le désordre dans le système. Celui-ci est distribué comme un bruit blanc gaussien dans l’espace. Nous insistons principalement sur la propriété de réduction dimensionnelle qui prédit un comportement critique identique pour le modèle en champ aléatoire à d dimensions et le modèle pur (c’est à dire sans champ aléatoire) en dimension d − 2. Bien que cette propriété soit démontrée à tous les ordres par la théorie de perturba- tion, on montre que celle-ci est brisée en dessous d’une dimension critique dDR = 5.13. La réduction dimensionnelle et sa brisure sont alors reliées aux caractéristiques d’échelle des grandes avalanches intervenant dans le système à température nulle. Nous considérons, dans un second temps, une généralisation du modèle d’Ising dans laquelle l’interaction ferromagnétique décroit désormais à longue portée comme r^−(d+σ) avec σ > 0 (d désigne toujours la dimension de l’espace). Dans un tel système, il est possible de travailler en dimension fixée (incluant la dimension d = 1) et de varier l’exposant σ afin de parcourir une gamme de comportements critiques similaire à celle obtenue entre les dimensions critiques inférieure et supérieure de la version à courte portée du modèle. Nous avons caractérisé la transition de phase dans le plan (σ, d), et notamment calculé les exposants critiques en fonction du paramètre σ pour les dimensions physiquement intéressantes d = 1, 2 et 3. Finalement, on s’intéresse aussi à la théorie en présence d’un champ magnétique aléatoire dont les corrélations décroissent à grande distance comme r^−d+ρ avec ρ > −d. Dans le cas particulier où ρ = 2 − σ, on montre que la propriété de réduction dimensionnelle est vérifiée lorsque σ est suffisamment petit, mais brisée à grand σ (en dimension inférieure à dDR ). En particulier, concernant le modèle tridimensionnel, nos résultats prédisent une brisure de réduction dimensionnelle lorsque σ > σDR = 0.71 / We study the influence of the presence of a random magnetic field and of long-ranged interactions on the critical behavior of the Ising model. Our approach is based on a nonperturbative and functional version of the renormalization group. The bases of the nonperturbative renormalization group are introduced first and then illustrated in the simple case of the classical scalar field theory. We next discuss the critical properties of the latter in the presence of a random magnetic field, which is associated with frozen disorder in the system. The distribution of the random field in space is taken as that of a gaussian white noise. We focus on the property of dimensional reduction that predicts identical critical behavior for the random-field model in dimension $d$ and the pure model, \textit{i.e.} in the absence of random field, in dimension d-2. Although this property is found at all orders of the perturbation theory, it is violated below a critical dimension $d_{DR} \approx 5.13$. We show that the dimensional reduction and its breakdown are related to the large-scale properties of the avalanches that are present in the system at zero temperature. We next consider a generalization of the Ising model in which the ferromagnetic interaction varies at large distance like $r^{-(d+\sigma)}$ with $\sigma > 0$ ($d$ being the spatial dimension). In this system, it is possible to obtain a range of critical behavior similar to that encountered in the short-ranged version of the model between the lower and the upper critical dimensions by varying the exponent $\sigma$ while keeping the dimension $d$ fixed (including the case $d=1$).We have characterized the phase transition of this long-ranged model in the plane $(\sigma,d)$ and computed the critical exponents as a function of the parameter $\sigma$ for the physically interesting dimensions, $d=1,2$ and $3$. Finally, we have also studied the long-ranged random-field Ising model when the correlations of the random magnetic field decrease at large distance as $r^{-d+\rho}$ with $\rho > -d$. In the special case where $\rho=2-\sigma$, we have shown that the dimensional-reduction property is satisfied when $\sigma$ is small enough but breaks down above a critical value (when the spatial dimension $d$ is less than $d_{DR}$). In particular, for $d=3$, we predict a breakdown of dimensional reduction for $\sigma_{DR}\approx 0.71$.