The mean avalanche size of the two-dimensional Bak-Tang-Wiesenfeld sandpile

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

The mean avalanche size of the two-dimensional Bak-Tang-Wiesenfeld sandpile, averaged over the site where the grain is dropped, is ⟨s⟩=1N∑i,j(Δ−1)ij\langle s\rangle=\frac1N\sum_{i,j}(\boldsymbol\Delta^{-1})_{ij}, with Δ=4I−A\boldsymbol\Delta=4\mathbf I-\mathbf A the toppling matrix of the L×LL\times L grid joined to an edge sink (A\mathbf A is the grid-plus-sink adjacency matrix with the sink's row and column deleted). (a) Expand Δ−1=14(I−A/4)−1\boldsymbol\Delta^{-1}=\frac14(\mathbf I-\mathbf A/4)^{-1} as a series in powers of A/4\mathbf A/4, and interpret (A/4)ijn(\mathbf A/4)^n_{ij} as the probability that a walker, stepping to a uniformly chosen neighbour at each move and absorbed at the sink, is at site jj after nn steps having started at ii. Show that ∑j(Δ−1)ij=14⟨Texit(i)⟩\sum_j(\boldsymbol\Delta^{-1})_{ij}=\tfrac14\langle T_{\rm exit}(i)\rangle, where Texit(i)T_{\rm exit}(i) is the number of steps such a walk takes to reach the sink from ii. (b) Using the fact that the mean exit time of a two-dimensional random walk from a point a distance of order LL from an absorbing boundary grows as L2L^2, deduce the scaling ⟨s⟩∝L2\langle s\rangle\propto L^2. (c) The exact mean avalanche size at L=32L=32, from solving Δx=1\boldsymbol\Delta\mathbf x=\mathbf 1, is 40.640.6 topplings. Use the scaling of part (b) to give the leading-order prediction at L=128L=128, and compare it with the exact value 593.9593.9 at that size.

Browse the problem library