An investigation into jamming percolation using renormalization group methods
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2010
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Swarthmore College. Dept. of Physics & Astronomy
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Thesis (B.A.)
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Abstract
Many systems in nature fail to satisfy the ergodic hypothesis and feature strong nonequilibrium
behavior. These systems have proven remarkably difficult to study. In recent years theorists
have studied a phase transition known as the jamming phase transition. The jamming phase
transition unifies several far from equilibrium phase transitions including: the glass transition,
the solidification of granular media in response to shearing forces, and the stiffening of colloids in
response to increases in pressure. It has recently been shown that certain correlated percolation
models can exhibit behavior analogous to the jamming phase transition. In particular this has been
shown for two models, coined the spiral model and the force-balance model.
The primary goal of this study was to determine whether or not the spiral model and the
force-balance model reside in the same universality class. To investigate this issue we used the
renormalization group to perform a numerical investigation of the spiral model and force balance
model. There is evidence to suggest that both models feature exponential, as opposed to power-law
scaling of the correlation function. It was therefore difficult to investigate, in a computationally
feasible way, large enough systems so as to reduce the finite size effects. To do this we came up
with a variety of algorithms including: a Monte-Carlo method, a binary search, and a linear time
culling algorithm.
The combination of these three strategies allowed us to investigate significantly larger systems
than had been investigated in the past. Specifically the two preceding studies on which we based
our work had investigated systems up to sizes 1600 and 3000. We managed to investigate systems
up to 15000 in size. This gave us promising results that agree with exact results found in work by
Fisher et al. and Schwarz et al. Although we do not yet have conclusive results, we believe that
determining whether these models lie in the same universality class should now be within reach.