Power spectrum of rare events in a two-dimensional BTW model

One of the primary aims of the two-dimensional BTW model had been to explain the 1/f ^α 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 (τ(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 τ(n) which are finite size effects. Two characteristic time scales δT _s and δT _τ emerge in our analysis, and the power spectrum of s(T) and τ(n) behave like ${(b_{s,\tau }{\left(L\right)}^2+{\omega }^2)}^{-1}$, in which b _s and b _τ are some L-dependent parameters and ω is the angular frequency. The 1/f ² noise is therefore obtained in the limit $\omega \gg b_{s,\tau }$. b _s and b _τ 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. (paper)