An avalanche in the sandpile's self-organised critical state is modelled
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
An avalanche in the sandpile's self-organised critical state is modelled, in mean field, as a critical branching process: each toppling triggers a number of further topplings drawn independently from a fixed offspring distribution with mean equal to the branching ratio σ. Take the offspring number to be Poisson-distributed with mean λ, and set λ=1 so that σ=1 (grain conservation).
((a)) State the offspring variance v of this distribution at σ=1.
((b)) Using the large-generation result for a critical branching process, that the survival probability to generation T falls off as 1−qT∼2/(vT), give the numerical prediction for the probability that an avalanche's chain of topplings survives at least T=100 generations, and at least T=400 generations.
((c)) Check that the ratio of these two survival probabilities is consistent with a duration density P(T)∝T−2 (equivalently a survival probability ∝T−1), and state the duration exponent this confirms.