Computer Simulations of Critical Phenomena in Systems with LongRange Interactions: a Study of Ising Dipoles and SelfOrganized Criticality in Earthquakes.
Abstract
This thesis discusses scaling and critical behavior of two different models. One model describes Ising dipoles, originates in condensed matter physics, and depicts equilibrium critical phenomena. The other model, taken from the earth sciences, describes faulting instabilities and the resulting earthquakes, and involves selforganized criticalitya nonequilibrium phenomenon. Both models are characterized by long range interactions, with a resulting sensitivity to boundary conditions. The ordering properties of Ising dipoles on lattices are studied in a mean field theory and by Monte Carlo simulations. The mean field theory is manifestly shape independent in zero external field. In the case of dipoles on a diluted lattice the mean field theory predicts a critical concentration above which the low temperature phase is ferroelectric (or antiferroelectric depending on the lattice structure). Extensive Monte Carlo simulation results are in agreement with those of mean field theory. We propose a finite size scaling form that includes logarithmic corrections for systems at the critical dimensionality. In the case of dipoles on a body centered tetragonal lattice we found that the finite scaling form significantly improved the data collapse over the scaling form with mean field exponents. With lattice parameters appropriate to the Ising ferromagnetic compound LiHoF_4, we obtain a ferromagnetic transition temperature T 1.51 K in excellent agreement with experiment. This indicates that the material LiHoF_4 is dominated by the dipoledipole interaction: since in the simulations we only include dipole dipole interactions. For dipoles on the simple cubic lattice, the ordered state is made up of antiferromagnetic rows. The critical exponents obtained by finite size scaling are beta~ 1/7, gamma ~ 8/7 and o~ 1/7. These results are in good agreement with those of high temperature series expansions. A model of selforganized ruptures in an elastic medium is developed; and applied to earthquakes. In the model the local ruptures are represented by double couples to be consistent with elastic theory. The explicit form of this double couple source is derived. The system is driven by slowly increasing the shear stress. The model evolves towards a selforganized critical state in which the earthquake distribution follows the GutenbergRichter law with an exponent in agreement with observational data. By modeling the local static fatigue for the rocks, we also obtained Omori's law for the rate of aftershocks. The effects of annealing are investigated.
 Publication:

Ph.D. Thesis
 Pub Date:
 1992
 Bibcode:
 1992PhDT........31X
 Keywords:

 Physics: General; Geophysics