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

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.