Avalanches on a conical bead pile: scaling with tuning parameters
Uniform spherical beads were used to explore the scaling behavior of a granular system near its critical angle of repose on a conical 3D bead pile. We found two tuning parameters that could take the system to a critical point. The existence of those tuning parameters violates the fundamental assumption of self-organized criticality, which proposed that complex dynamical systems self-organize to a critical point without need for tuning. Our avalanche size distributions were well described by a simple power-law, as is characteristic of a critical point, with the power τ = 1.5 when dropping beads slowly onto the apex of a bead pile from a small height. However, we could also move the system from the critical point using either of two tuning parameters: the height from which the beads fell onto the top of the pile or the region over which the beads struck the pile. As the drop height increased, the system did not reach the critical point yet the resulting distributions were independent of the bead mass, coefficient of friction, or coefficient of restitution. All our apex-dropping distributions for any type of bead (glass, stainless steel, zirconium) showed universality by scaling onto a common curve with τ = 1.5 and σ = 1.0, where 1/ σ is the power of the tuning parameter. From independent calculations using the moments of the distribution, we find values for τ = 1.6 ± 0.1 and σ = 0.91 ± 0.15. When beads were dropped across the surface of the pile instead of solely on the apex, then the system also moved from the critical point and again the avalanche size distributions fell on a common curve when scaled similarly using the same values of τ and σ . We also observed that an hcp structure on the base of the pile caused an emergent structure in the pile that had six faces with some fcc or hcp structure; this structure did not affect the distribution of avalanche sizes.