Write a program for the two-dimensional Bak-Tang-Wiesenfeld sandpile with open boundaries

PHYSICS · Subjective · 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

Write a program for the two-dimensional Bak-Tang-Wiesenfeld sandpile with open boundaries, using a stack of unstable sites as in the chapter and a seeded random number generator for the drop sites. (a) For L=16L = 16, 3232 and 6464, start from an empty grid, discard a transient of 5L25L^2 dropped grains, then record the number of topplings caused by each of the next 10510^5 grains, zero-toppling drops included. Estimate each mean sˉ(L)\bar s(L) with a standard error from 2020 consecutive block averages, and say why the formula σs/105\sigma_s/\sqrt{10^5}, which treats the avalanches as independent, is not safe here. (b) Compute the exact means ⟨s⟩=N−1∑ij(Δ−1)ij\langle s\rangle = N^{-1}\sum_{ij}(\boldsymbol\Delta^{-1})_{ij} by solving Δx=1\boldsymbol\Delta\mathbf x = \mathbf 1 for each LL, and check your simulated means against them. (c) Give the exact ratios ⟨s⟩(32)/⟨s⟩(16)\langle s\rangle(32)/\langle s\rangle(16) and ⟨s⟩(64)/⟨s⟩(32)\langle s\rangle(64)/\langle s\rangle(32), and say why they fall short of the factor 44 that ⟨s⟩∝L2\langle s\rangle \propto L^2 predicts. (d) Under the scaling form P(s;L)=s−τF(s/LD)P(s; L) = s^{-\tau}F(s/L^D), the ratio ⟨s2⟩/⟨s⟩\langle s^2\rangle/\langle s\rangle grows as LDL^D. Use your L=32L = 32 and L=64L = 64 runs to estimate DD, obtain τ=2−2/D\tau = 2 - 2/D, and plot sτP(s)s^\tau P(s) against s/LDs/L^D for all three sizes to see whether the curves meet at their cutoffs. Say why the pair you find is an effective value.

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