##
Self-Organized Criticality: An Investigation of Energy Dissipation and Bead Pile Dynamics

###
Andrew Kindschuh

Self-organized criticality (SOC) is a theory first proposed by Bak, Tang, and Wiesenfeld
to describe systems displaying 1/f noise and fractals. It has since been suggested to describe
a vast array of additional systems that are not well described by simple causality. These complex, dynamical systems self-organize to critical states characterized by scale invariance (both spatial and temporal) and power law behavior. Bak, Tang, and Wiesenfeld first modeled SOC in a numerical simulation of a sand pile, and the sand pile has since become the quintessential example of an SOC system, being extensively studied both in the literature and at The College of Wooster. Sand pile experiments at The College of Wooster have consisted of a uniform, spherical, glass or steel beads dropped singly from a constant height onto the apex of a conical pile of beads. Different sizes, shapes, and surfaces of bases have been investigated, as well as beads glued to the base in different patterns, and different drop heights. In this experiment two investigations are undertaken. Firstly the effects of drop location are investigated by dropping stainless steel beads from a constant height of 2.0 cm onto the pile at three locations, horizontally offset 0, 1, or 2 cm from the center of the pile. Each location consisted of multiple runs lasting 54 hours each. It was found that while small variations (~10% the base diameter) in horizontal drop location h ad negligible effects on the distribution of avalanches coming off the pile, larger horizontal displacements displayed finite size effects that mimicked the effects of a smaller base. The second investigation further explored the effects of bead material and density on avalanche distribution by using zirconium beads. Zirconium beads were dropped onto the apex of the pile from three different heights: 0.7 cm, 1.6, cm, and 5.7 cm. Zirconium has a density approximately halfway between those of glass and stainless steel, and thus a zirconium bead of similar size will impart a quantity of energy approximately halfway between the densities of glass and stainless steel upon impact. The energy of this impact was previously suspected to govern the cut-off large size avalanches, but a dependence on bead density was not found.