An avalanche is modelled as a critical branching process

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

An avalanche is modelled as a critical branching process, as in the random-graph component analysis of Chapter 5: each toppling triggers a random number of further topplings drawn independently from a fixed offspring distribution with generating function g(x)g(x), mean offspring g′(1)=1g'(1) = 1 (the critical branching ratio σ=1\sigma = 1) and offspring variance v=g′′(1)v = g''(1). (a) Let qTq_T be the probability that the process (started from a single individual) has gone extinct by generation TT, and πT=1−qT\pi_T = 1 - q_T the survival probability. Using the extinction recursion qT+1=g(qT)q_{T+1} = g(q_T) from Chapter 5's branching-process theory, and expanding gg near x=1x = 1, show that for large TT πT≈2vT,\pi_T \approx \frac{2}{vT}, and hence that the duration distribution behaves as P(T)∝T−2P(T) \propto T^{-2}. (b) For an offspring distribution with variance v=3v = 3, estimate the survival probability to generation T=100T = 100. (c) Separately, the exact mean avalanche size of the two-dimensional Bak-Tang-Wiesenfeld sandpile on a grid of linear size L=40L = 40, from solving Δx=1\boldsymbol\Delta\mathbf x = \mathbf 1, is ⟨s⟩=61.9\langle s\rangle = 61.9 topplings. Using the large-LL law ⟨s⟩∝L2\langle s\rangle \propto L^2 from the random-walk exit-time argument, give the leading-order prediction of ⟨s⟩\langle s\rangle for L=120L = 120.

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