Solving and averaging over every drop site gives the exact stationary mean avalanche…

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

Solving Δx=1\boldsymbol\Delta\mathbf x = \mathbf 1 and averaging x\mathbf x over every drop site gives the exact stationary mean avalanche size of the two-dimensional Bak-Tang-Wiesenfeld sandpile (zero-toppling drops included) as ⟨s⟩=40.6\langle s\rangle = 40.6 topplings on an L=32L=32 grid and 337.5337.5 topplings on an L=96L=96 grid. ((a)) Using only the L=32L=32 value and the large-LL law ⟨s⟩∝L2\langle s\rangle \propto L^2, predict the mean avalanche size for L=96L = 96. ((b)) By what percentage does this prediction overshoot the exact value at L=96L = 96, and why does scaling up from the smaller grid overshoot rather than undershoot?

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