| Summary: | 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.
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