Temporal Power-Law Distribution

The fact that there are no characteristic length scales immediately implies a similar lack of any characteristic time scales for the fluctuations. Consider the effect of a single perturbation of a random site of a system in the critical state. The perturbation will spread to the neighbors of the site, to the next nearest neighbors, and so on, until, after a time r and a total of / sand slides, the effects will die out. The distribution of the life-times of the avalanches, D t), obeys the power law 

It is not hard to see that, given a power-law distribution of lifetimes (equation 8.113), the power spectrum S f), defined by 

Now consider the case where the system is perturbed randomly in space and time and F(t) represents a superposition of many avalanches (occurring simulta-neou.sly and independently). The total power spectrum is the (incoherent) sum of individual ( ontributions for single relaxation event due to single perturbations. 

Consider the relaxation due to a single perturbation in a given domain. The result- 

Wiesenfeld, et. al. [wiesen89] compare the simplicity of the independent relaxation time interpretation of l//-noise fluctuations in the saudpile CA model to other recent models yielding 1// noise  

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