PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems
Many Parts, One Behaviour, Ch Self-organised criticality and avalanches
Problem
The numbers below are synthetic, made up for practice: they describe a model pile that shares the Bak-Tang-Wiesenfeld pile's large-L law ⟨s⟩∝L2 but not its measured size distributions, which are the ones plotted in the chapter. At L=256 the cumulative distribution of avalanche sizes, with standard errors below 5% of the quoted values, is P(S≥20)=0.100 and P(S≥2000)=0.0398. Separately, the cutoff size (the value of s at which the cumulative plot turns down from its straight line) is sc=1000 at L=64 and sc=5660 at L=128.
((a)) Use the two cumulative values at L=256 (two decades apart in s) to estimate the slope −(τ−1) of the cumulative plot, and hence τ.
((b)) Use the two cutoff sizes to estimate D from sc∝LD.
((c)) Check the pair (τ,D) against the exact relation D(2−τ)=2, and comment on whether this level of agreement should be expected for the real two-dimensional pile.