History Induced Critical Scaling in Disordered Media and Super Diffusive Growth in Highly Advective Random Environments
Carpenter, John Halsey
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https://hdl.handle.net/2142/32103
Description
Title
History Induced Critical Scaling in Disordered Media and Super Diffusive Growth in Highly Advective Random Environments
Author(s)
Carpenter, John Halsey
Issue Date
2004-10
Director of Research (if dissertation) or Advisor (if thesis)
Dahmen, Karin A.
Department of Study
Physics
Discipline
Physics
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Random Field Ising Model (RFIM)
demagnetization
population biology
simulation
hysteresis
Language
en
Abstract
The behavior of non-equilibrium systems in the presence of quenched random
disorder is studied. In noisy, hysteretic systems, the role of driving force history
is studied in the context of the non-equilibrium, zero-temperature Random Field
Ising Model (RFIM). The RFIM was originally developed as a simple model for
disordered magnets, but has applications far beyond magnetic systems. Previous work
examining history effects in models and experiments are briefly reviewed, followed by
a summary of the saturation loop behavior of the RFIM. A numerical scaling analysis
of the AC demagnetization curve of the RFIM is performed, examining the effect of
the underlying disorder on avalanche size distributions, correlation functions, and
spanning avalanches. Furthermore, a similar scaling analysis for nested, concentric,
symmetric subloops is performed via an analysis of history-induced disorder. Next the
effects of long range demagnetizing fields on the demagnetization curve and subloops
are studied. Finally, an analysis of corrections to scaling for subloops is presented,
along with a derivation of the exponent relations.
Disorder in population biology is studied for the case of a spreading cluster of
bacteria in a highly advective environment with inhomogeneous nutrient concentration.
A model reaction-diffusion equation with Fisher growth terms is introduced
with a brief discussion of previous work on similar equations and experiments. The
linear two-dimensional problem is mapped onto a simplified one-dimensional equation.
Numerical simulations of concentration profiles reveal anomalous growth and
super-diffusive spreading in the direction perpendicular to the convection velocity. A time characterizing the crossover from pure diffusion to this super-diffusive behavior
is perturbatively calculated. The crossover time’s dependence on the velocity and
disorder strength is then tested numerically. Two-dimensional simulations of the full
linear reaction-diffusion equation also show the onset of super-diffusive growth in
concentration contour maps. On the other hand, with nonlinear growth in two dimensions,
a symmetric wave front develops with a propagation velocity greater than
the minimum Fisher velocity. An expression is derived and tested for this velocity.
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