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.

No description available.

Experimental data on the structural phase changes in zirconia are summarized. The computational techniques of molecular dynamics are reviewed and equations of motion are formulated which allow the study of phase changes as a function of temperature and pressure. The molecular dynamics program NPT which was written for this purpose is described. This program performs numerical integration of the classical equations of motions in atomistic simulations which allow a varying cell size and shape. The simulations produce time averages which are related to thermodynamic ensemble averages. Routines used to calculated the interatomic forces are implemented for potentials which vary as the inverse power of the separation distance between atoms. Calculation of Coulomb forces is done with the Ewald method and with a multipole method. The two methods are shown to be analytically equivalent and the precision and speed of the two routines are compared. Results generated by the program NPT are presented for energy minimization of crystal structures and for dynamic simulations. A number of different minimum-energy structures for soft-sphere potentials are found. Simulations are performed for several soft-sphere structures and dynamic properties are established. Structural phases changes are observed in two cases. A potential derived from ab initio calculations for monoclinic zirconia is tested. / Graduation date: 1994

Thesis (M.S.)--University of Texas at El Paso, 2007. / Title from title screen. Vita. CD-ROM. Includes bibliographical references. Also available online.

Thesis (Ph.D.)--Tufts University, 1999. / Adviser: Alexander Vilenkin. Submitted to the Dept. of Physics. Includes bibliographical references (leaves 119-122). Access restricted to members of the Tufts University community. Also available via the World Wide Web;