Power spectrum of rare events in two-dimensional BTW model, violation of $1/f$ noise hypothesis

TL;DR

This study investigates the power spectrum of activity, avalanche sizes, and rare event waiting times in the 2D BTW model, revealing finite-size effects and the violation of the $1/f$ noise hypothesis.

Contribution

It introduces the analysis of rare event waiting times and demonstrates the finite-size effects leading to the breakdown of $1/f$ noise in the model.

Findings

Power spectra follow a specific form involving system size-dependent parameters.

Finite-size effects cause deviations from ideal $1/f$ noise.

Power spectrum tends to a delta function as system size increases.

Abstract

One of the primitive aims of the two-dimensional BTW model had been to explain the $1/f^{\alpha}$ noise which is widely seen in the natural systems. In this paper we study some time signals, namely the activity inside an avalanche ($x(t)$), the avalanches sizes ($s(T)$) and the rare events waiting time (REWT) ($\tau(n)$ as a new type of noise). The latter is expected to be important in predicting the period and also the vastness of the upcoming large scale events in a sequence of self-organized natural events. Especially we report some exponential anti-correlation behaviors for $s(T)$ and $\tau(n)$ which are finite-size effects. Two characteristic time scales $\delta T_{s}$ and $\delta T_{\tau}$ emerge in our analysis. It is proposed that the power spectrum of $s(T)$ and $\tau(n)$ behave like $\left( b_{s,\tau}(L)^2+\omega^2\right)^{-\frac{1}{2}}$, in which $b_{s}$ and $b_{\tau}$ are some $L$-dependent parameters and $\omega$ is the angular frequency. The $1/f$ noise is therefore obtained in the limit $\omega\gg b_{s,\tau}$. $b_{s}$ and $b_{\tau}$ decrease also in a power-law fashion with the system size $L$, which signals the fact that in the thermodynamic limit the power spectrum tends to the Dirac delta function.