Three systems are each held slightly below a threshold at which a mean-field…

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

Three systems are each held slightly below a threshold at which a mean-field branching process would become critical: a sparse Erdos-Renyi random graph with mean degree

c = 0.95

, whose giant component appears once the mean number of further neighbours reached along an edge exceeds

1

; an SIR epidemic in a well-mixed population with basic reproduction number

R_0 = 0.95

; a Bak-Tang-Wiesenfeld sandpile in which a fraction of grains is lost at every toppling, giving a branching ratio

\sigma = 0.95

(each grain passed to a neighbour lost with probability

0.05

, as in the chapter's bulk-dissipation model). ((a)) Explain why all three reduce, in mean field, to the same branching-process criterion, a mean offspring

x

crossing

1

, and use the reused formula

\langle s\rangle = 1/(1-x)

to compute the common mean size (mean component size, mean outbreak size, mean avalanche size) and the common mean-field cutoff scale

s_{\rm c} \sim (1-x)^{-2}

for

x=0.95

. ((b)) The sandpile is a slowly driven threshold system: grains are added far more slowly than an avalanche relaxes, and grains lost at the boundary balance the drive. State, referring to this separation of time scales, why the sandpile's branching ratio can sit near

1

without anyone dialling it from outside, whereas the graph's

c

and the epidemic's

R_0

are parameters chosen or measured externally. Then state the counter-argument for why the sandpile's own approach to

\sigma = 1

also rests on a hidden tuning. ((c)) At

x=1

exactly, the survival probability of the underlying branching process to duration

T

falls as

T^{-1}

, so the probability density of durations falls as

T^{-2}

. Using the same mean-field scaling that gives the size cutoff

s_{\rm c}\sim(1-x)^{-2}

, together with

s \propto T^2

near threshold, estimate the duration

T_{\rm c}

beyond which these power laws are cut off for

x=0.95

, and describe in one sentence how survival behaves for

T \gg T_{\rm c}

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