Computer simulations of critical phenomena in systems with long range interaction: A study of ising dipoles and self-organized criticality in earthquakes

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 ear...

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Main Author: Xu, Huangjian
Language:English
Published: 2008
Online Access:http://hdl.handle.net/2429/3143
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spelling ndltd-LACETR-oai-collectionscanada.gc.ca-BVAU.2429-31432014-03-14T15:38:38Z Computer simulations of critical phenomena in systems with long range interaction: A study of ising dipoles and self-organized criticality in earthquakes Xu, Huangjian 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 earthqnakes, and involves self-organized criticality — a 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 anti-ferroelectric 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 LiHoF4,we obtain a ferromagnetic transition temperature Tc= 1.51K in excellent agreement with experiment. This indicates that the material LiHoF4 is dominated by the dipole-dipole 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 anti-ferromagnetic rows. The critical exponents obtained by finite size scaling are ß~1/7,y ~ 8/7 and a ~4/7. These results are in good agreement with those of high temperature series expansions. A model of self—organized 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 self-organized critical state in which the earthquake distribution follows the Gutenberg Richter 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. 2008-12-18T21:18:15Z 2008-12-18T21:18:15Z 1992 2008-12-18T21:18:15Z 1992-11 Electronic Thesis or Dissertation http://hdl.handle.net/2429/3143 eng UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/]
collection NDLTD
language English
sources NDLTD
description 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 earthqnakes, and involves self-organized criticality — a 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 anti-ferroelectric 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 LiHoF4,we obtain a ferromagnetic transition temperature Tc= 1.51K in excellent agreement with experiment. This indicates that the material LiHoF4 is dominated by the dipole-dipole 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 anti-ferromagnetic rows. The critical exponents obtained by finite size scaling are ß~1/7,y ~ 8/7 and a ~4/7. These results are in good agreement with those of high temperature series expansions. A model of self—organized 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 self-organized critical state in which the earthquake distribution follows the Gutenberg Richter 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.
author Xu, Huangjian
spellingShingle Xu, Huangjian
Computer simulations of critical phenomena in systems with long range interaction: A study of ising dipoles and self-organized criticality in earthquakes
author_facet Xu, Huangjian
author_sort Xu, Huangjian
title Computer simulations of critical phenomena in systems with long range interaction: A study of ising dipoles and self-organized criticality in earthquakes
title_short Computer simulations of critical phenomena in systems with long range interaction: A study of ising dipoles and self-organized criticality in earthquakes
title_full Computer simulations of critical phenomena in systems with long range interaction: A study of ising dipoles and self-organized criticality in earthquakes
title_fullStr Computer simulations of critical phenomena in systems with long range interaction: A study of ising dipoles and self-organized criticality in earthquakes
title_full_unstemmed Computer simulations of critical phenomena in systems with long range interaction: A study of ising dipoles and self-organized criticality in earthquakes
title_sort computer simulations of critical phenomena in systems with long range interaction: a study of ising dipoles and self-organized criticality in earthquakes
publishDate 2008
url http://hdl.handle.net/2429/3143
work_keys_str_mv AT xuhuangjian computersimulationsofcriticalphenomenainsystemswithlongrangeinteractionastudyofisingdipolesandselforganizedcriticalityinearthquakes
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