published_or_final_version / abstract / toc / Physics / Master / Master of Philosophy

This thesis studies deformation quantization and its application to contact interactions and systems with dissipation. We consider the subtleties related to quantization when contact interactions and boundaries are present. We exploit the idea that discontinuous potentials are idealizations that should be realized as limits of smooth potentials. The Wigner functions are found for the Morse potential and in the proper limit they reduce to the Wigner functions for the infinite wall, for the most general (Robin) boundary conditions. This is possible for a very limited subset of the values of the parameters -- so-called fine tuning is necessary. It explains why Dirichlet boundary conditions are used predominantly. Secondly, we consider deformation quantization in relation to dissipative phenomena. For the damped harmonic oscillator we study a method using a modified noncommutative star product. Within this framework we resolve the non-reality problem with the Wigner function and correct the classical limit. / iii, 188 leaves ; 29 cm

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This thesis numerically solves the tensor order parameter continuum theory equations for nematic liquid crystals to investigate liquid crystal texturing mechanisms during an isotropic to nematic phase transition in a bulk unstable isotropic phase and next to solid surfaces. The Time Dependent Ginsburg Landau equation with a Landau de Gennes Helmholtz free energy density description is used to predict the shapes, textures and defect mechanisms that occur in the expanding droplets and films of a 4'-pentyl-4-cyanobiphenyl (5CB) nematic phase immediately after their nucleation from an unstable isotropic phase, due to a temperature quench. To create a robust simulation method able to tackle high curvature, defect nucleation, heterogeneous substrates and phase ordering interfaces, particular attention was paid to adapting the mathematical model and computational methods to what was previously known about the nucleation and growth events that occur experimentally during a bulk 5CB isotropic to nematic phase transition and next to decorated solid surfaces. The numerical simulations provide detailed predictions about (i) growth rates for different temperature quenches, (ii) structure of the isotropic-nematic interface, (iii) shapes of expanding nano and submicron nematic droplets, (iv) texturing within growing nano and submicron nematic droplets, (v) a new defect formation mechanism called "interfacial defect shedding", and (vi) the effect of contact angle and interface curvature next to a solid surface with anchoring switches. The main contributions of this thesis are its detailed predictions that emerge from the liquid crystal simulation results, the careful adaptation of the mathematical model and numerical method to what is currently known about early stage growth in a nematic liquid crystal phase, and the validation of new theory by the simulation results.

Percolation theory enters in various areas of research including critical phenomena and phase transitions. Bond-site percolation is a generalization of pure percolation motivated by the fact that bond-site is close to many physical realities. This work relies on a numerical study of percolation in lattices. A lattice is a regular pattern of sites also known as nodes or vertices connected by bonds also known as links or edges. Sites may be occupied or unoccupied, where the concentration ps is the fraction of occupied sites. The quantity pb is the fraction of open bonds. A cluster is a set of occupied sites connected by opened bonds. The bond-site percolation problem is formulated as follows: we consider an infinite lattice whose sites and bonds are at random or correlated and either allowed or forbidden with probabilities ps and pb that any site and any bond are occupied and open respectively. If those probabilities are small, there appears a sprinkling of isolated clusters each consisting of occupied sites connected by open bonds surrounded by numerous unoccupied sites. As the probabilities increase, reaching critical values above which there is an infinitely large cluster, then percolation is taking place. This means that one can cross the entire lattice by going successively from one occupied site connected by a opened bond to a neighbouring occupied site. The sudden onset of a spanning cluster happens at particular values of ps and pb, called the critical concentrations. Quantities related to cluster configuration (mean cluster and correlation length) and individual cluster structure (size and gyration radius of clusters ) are determined and compared for different models. In our studies, the Monte Carlo approach is applied while some authors used series expansion and renormalization group methods. The contribution of this work is the application of models in which the probability of opening a bond depends on the occupancy of sites. Compared with models in which probabilities of opening bonds are uncorrelated with the occupancy of sites, in the suppressed bond-site percolation, the higher site occupancy is needed to reach percolation. The approach of suppressed bond-site percolation is extended by considering direction of percolation along bonds (directed suppressed bond-site percolation). Fundamental results for models of suppressed bond-site percolation and directed suppressed bond-site percolation are the numerical determination of phase boundary between the percolating and non-percolating regions. Also, it appears that the spanning cluster around critical concentration is independent on models. This is an intrinsic property of a system. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2007.

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The study of non-equilibrium physics is an area of interest since, unlike for their equilibrium counterparts, there exists no general framework for solving such systems. In this thesis I investigate the emergence of structure and front propagation in driven systems, a special type of system within the area of non-equilibrium physics. In particular I focus on three particular one-dimensional models each of which illustrate this in a different way. The Driven Asymmetric Contact Process (DACP) describes a system where activity is continuously generated at one end of a one-dimensional lattice and where this activity is allowed to spread in one direction along the lattice. In the DACP one observes a propagating wave of activity which appears to abruptly vanish as the system undergoes a phase transition. Using a modified Fisher equation to model the system reveals the continued existence of the propagating wave, now contained within a decaying envelope. Furthermore this establishes relations between properties of the travelling wave and Directed Percolation critical exponents. The Zero-Range Process (ZRP) is a much studied system exhibiting a condensation transition. In the ZRP individual particles hop along a lattice at rates which depend only on the occupancy of the departure site. Here I investigate a modi cation of the ZRP where instead the majority of the particles at a site depart during a single hopping event. For this, the Chipping model, a condensate which propagates along the lattice is observed. It is found that this condensation transition is present even for hop rates which fall foul of the condensation requirements of the normal ZRP. Further it is observed that, unlike for normal ZRP, condensation occurs even in the low-density limit. As a result I suggest a condensation mechanism which depends only on the hop rates of low occupancy sites. The Host-Solute-Vacancy model (HSV) is a three-species system designed to model electromigration in a circuit. As the parameter space is navigated the system undergoes what appear to be two separate phase transitions from a randomly distributed state to a condensed state with either of two structures. To investigate the model new measures for determining condensation are developed. These show that, again, condensation occurs in the low-density limit. By a reduction to a ZRP an effective hop rate of the system is measured. This effective hop rate is found to beta function of the occupancy of a site as a fraction of the total system size. To explain this behaviour I invoke a description whereby there is a step in the hop rate as a function of occupancy. Through these three examples I illustrate how minor modi cations to the dynamics of known systems can result in a new and rich phenomenology. I draw particular attention to the effect of asymmetry in the dynamics.

This thesis describes a diversity of scattering experiments on a number of different systems. Using time-of-flight neutron scattering, a study of polycrystalline sodium in the highmomentum limit known as the impulse approximation has been performed. The purpose of this study was to look for anharmonic effects in the neutron recoil scattering of sodium as the temperature was increased from 30K to 300K. No such effects were detected and the results agreed with an isotropic harmonic solid to an accuracy of about 4%. Two experiments were carried out on antiferromagnetic systems using triple-axis neutron scattering techniques to measure the spin-wave dispersion relations. The first was on CuO to verify its description as a spin 1/2 one-dimensional antiferromagnet. The dispersion relation was measured along the chain direction up to an energy transfer of 8OmeV. This was done above and below the Néel temperature (T_N =240K). However, no evidence was seen to justify the description of CuO as a one-dimensional antiferromagnet, with the spin waves behaving like those in a classical three-dimensional system. The other spin-wave study examined the two-dimensional antiferromagnet KFeF₄ . The measurement of the spin-wave dispersion relation at two temperatures (50K and 100K) below the Néel temperature (T_N =136.75±0.25K), confirmed the description of KFeF₄ as a two-dimensional Heisenberg antiferromagnet with small Ising anisotropy. Studies of the magnetic phase transition in KFeF₄ revealed that below the Néel temperature, the critical behaviour is described by two-dimensional Ising models, and above a crossover to Heisenberg behaviour is seen. This crossover was detected by measuring the order parameter below T_N, and the static and dynamic susceptibilities above T_N using neutron scattering techniques. The results were compared to power-law behaviour and also to theories for the classical Heisenberg antiferromagnet and the more recent quantum Heisenberg antiferromagnetic model. The final study of KFeF₄ involved an x-ray experiment on the structural phase transition around 400K. It has been suggested that there is a second-order transition at 410K to an incommensurate phase, which then undergoes a first-order lock-in transition at 400K to the low-temperature structure. This single crystal x-ray scattering study confirms the existence of the first-order phase transition, but shows no evidence for a higher temperature second-order transition or for the incommensurate phase.

The final Chapter concerns a problem of partitioning graphs subject to certain restrictions. We prove that several subproblems are NP-complete.