Before avalanche-size distributions from several system sizes can be collapsed to…
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
Before avalanche-size distributions from several system sizes can be collapsed to estimate τ and D, the far tail must be resolved well above its Monte Carlo noise.
((a)) The numbers in this problem are synthetic, made up for practice for a model pile; they are not the Bak-Tang-Wiesenfeld pile's values. A simulation of the model at L=64 is expected to give P(S≥2×104)≈8×10−5, far out beyond the cutoff. Treating each simulated avalanche as an independent Bernoulli trial for the rare event S≥2×104, and using the standard Np counting-error result, find how many avalanches Nav must be simulated for the resulting estimate of this probability to have a relative standard error of 10%.
((b)) With that many avalanches recorded, the cumulative distribution is found to follow a slope of −(τ−1)=−0.30 over the two decades 3≤s≤300, well below the cutoff. State τ.
((c)) The cutoff size (where the cumulative distribution turns over) is measured as sc=3000 grains at L=64 and sc=21700 grains at L=128. Estimate D from sc∝LD.
((d)) Check the pair (τ,D) from parts (b) and (c) against the exact relation D(2−τ)=2.