Real space renormalisation group scaling techniques are used to investigate the static critical behaviour of the pure and dilute, classical, anisotropic Heisenberg model. Transfer matrix methods are employed to obtain asymptotically exact expressions for the correlation lengths and susceptibilities of the one-dimensional system. The resulting scaling relationships are combined with an approximate bond moving scheme to treat pure and dilute models in higher dimensionalities. Detailed discussions are given for the dependence of correlation lengths and susceptibilities on temperature, anisotropy and concentration, and fcr the critical temperature on anisotropy and concentration. Particular emphasis is given to the weakly anisotropic system near percolation threshold and comparisons are made between the results of the present analysis and those of neutron-scattering experiments on dilute quasi-two- and three-dimensional systems.

In this thesis we study several different models which display glassy behaviour. Firstly, we investigate a simple, purely topological, cellular model for which the Hamiltonian is non-interacting but the dynamics are constrained. We find a non-thermodynamic transition to a glassy phase in which the energy fails to reach the equilibrium value below a characteristic temperature which is dependent on the cooling rate. This model involves activated processes and displays two-step relaxation in both the energy and the correlation functions; the latter also exhibit signs of aging. The relaxation time can be well-fitted at all temperatures by an offset Arrhenius law. Some predictions of Mode-coupling Theory are tested with some agreement found, but no convincing evidence that this description is the most fitting. By defining a suitable response function, we find that the equilibrium Fluctuation-Dissipation Theorem (FDT) is upheld for all but very short waiting-times, despite the fact that the system is not in equilibrium. This topological model is simplified to a hexagonally-based spin model, which also displays glassy behaviour, involves activated processes and exhibits two-step relaxation. This is a consequence of reaction-diffusion processes on two different time-scales, one temperature-independent and the other an exponential function of inverse temperature. We study two versions of this model, one with a single absorbing ground state, and the other with a highly degenerate ground state. These display qualitatively similar but quantitatively distinct macroscopic behaviour, and related but different microscopic behaviour. We extend this work to a square lattice, and find that the geometry of the lattice has a considerable impact on the behaviour, and to three dimensions, which provides support for the reaction-diffusion classification of the early behaviour. We find observable-dependent FDT plots; the observable can be chosen such that FDT is upheld for a region whilst the system is out of equilibrium — this observation is supported by some preliminary results for one-dimensional kinetically-constrained Ising chains.

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2002. / Includes bibliographical references (leaves 81-82). Also available in electronic version. Access restricted to campus users.

We conduct a computational statistical study of nonequilibrium processes with and without small-world interactions. We first investigate the motion of a passive random walker on growing nonequilibrium one-dimensional surfaces with or without small-world connections. The walker always moves to a higher connected site on the evolving surface. The surfaces examined are related to the evolution of parallel discrete-event simulations, with or without small-world connections. We have also examined the Kim-Kosterlitz surface growth model. In particular, we study the probability distribution function of the distance between the walker and the global maximum of the surface at saturation. We find that the availability of small-world connections for either the surface or the walker dramatically changes this probability distribution function. We next report of the lifetime of the metastable state of the square-lattice Ising model. We have used a macroscopic meanield dynamic using the density of states from a modified Wang-Landau sampling procedure. The Wang-Landau sampling, was used to give the density of states g, either as a function of two parameters, g(E,M), or as a function of only the magnetization, g(M). From the density of states the constrained free energy, F(m) , is calculated. Using a macroscopic meanield dynamic, constrained to having only single spin flips, we obtain the lifetime, tau, of the metastable state with and without small-world connections. From F(m) we obtain the exact first-passage time, tau Comparisons to recent predictions of the droplet theory of nucleation and growth will be made.

No description available.

This report presents the general Ising model and its basic assumptions. This study aims to, from diagonalization of the Transfer Matrix, obtain the Helmholtz free energy and the exclusion of a phase transition for the one-dimensional Ising model under an external magnetic field. Furthermore from establishing the commutation relations of the Transfer matrices and using the Kramers-Wannier duality one finds the free energy and the presence of a phase transition for the square-lattice Ising model.

No description available.

In many systems of interest, both physical and biological, disorder inhibits the organization and cooperative properties of the system. Disorder can originate from a variety of system defects and the degree of disorder also varies. Geometric frustration introduces disorder into a system in which all the preferred interactions between the elements of the system cannot be satisfied due to the topology of an underlying lattice that describes the position of these elements. Recently, geometric frustration has been recognized as an important organizing principle in a diverse range of systems from superconducting networks to neural computation. The correlation behavior of such systems is often complicated and poorly understood. The myosin lattice of higher vertebrate muscle is a geometrically frustrated system, and the presence of this kind of disorder has prevented a rigorous interpretation of X-ray diffraction patterns from muscle fibres for the purposes of studying muscle molecular structure. This thesis investigates the correlation behavior of two geometrically frustrated systems, the triangular Ising antiferromagnet (TIA) and the fully frustrated square Ising model (FFS), and its use to interpret X-ray fibre diffraction patterns. A combination of numerical evaluation of exact expressions and Monte Carlo simulation is used to study a number of aspects of the two-point correlation function of the TIA and FFS. In the case of the TIA, a simple functional expression is developed that allows accurate calculation of the correlation function. Theory is developed for calculating diffraction by polycrystalline fibres of helical molecules, in which the constituent crystallites contain correlated substitution disorder. The theory was used to study the characteristics of diffraction by fibres with TIA-type substitution disorder statistics. A quantitative model of the disorder in the myosin filament array is developed and the above theory is used to calculate X-ray fibre diffraction from low resolution models of the myosin filament array in higher vertebrate muscle. The calculated diffraction is compared to measured diffraction data, showing good agreement.

Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / Microfiche copy available in Archives and Science. / Bibliography: leaves 171-175. / Ph.D.

Neste trabalho apresentamos alguns resultados da invariância conforme e da teoria de escala para sistemas finitos. Estudamos, usando tais técnicas, dois modelos estatísticos (modelos 1 e 2). Para cada modelo obtivemos a anomalia conforme e as dimensões dos operadores energia e magnetização bem como seus respectivos descendentes / In this work we show some results of conformal invariance theory and finite-size scaling. We study by using these theories two statistical mechanics models (models 1 and 2). To each model we obtained the conformal anomaly, the dimensions of energy and magnetization operators as well their respective descendents