Anomalous Self-Organization in Active Piles

Abstract

Inspired by recent observations on active self-organized critical (SOC) systems, we designed an active pile (or ant pile) model with two ingredients: beyond-threshold toppling and under-threshold active motions. By including the latter component, we were able to replace the typical power-law distribution for geometric observables with a stretched exponential fat-tailed distribution, where the exponent and decay rate are dependent on the activity's strength (ζ). This observation helped us to uncover a hidden connection between active SOC systems and α-stable Levy systems. We demonstrate that one can partially sweep α-stable Levy distributions by changing ζ. The system undergoes a crossover towards Bak-Tang-Weisenfeld (BTW) sandpiles with a power-law behavior (SOC fixed point) below a crossover point ζ<ζ*≈0.1.

Keywords: Levy-stable distribution; self-organized active pile; stretched exponential distribution.

Figure 1

The time evolution of $\overline{n}$ versus time T. The left inset shows $T_{\zeta }^{*}/L^2$ and ${〈\overline{n}〉}_T$ in terms of $1/L$, with $\zeta =1$. The small change in $T_{\zeta }^{*}/L^2$ versus L indicates that $T_{\zeta }^{*}\propto L^2$, similar to the ordinary BTW model [53]. The right inset shows $T_{\zeta }^{*}/L^2$ and ${〈\overline{n}〉}_T$ in terms of $\zeta$, with $L=256$. The fitting for $〈\overline{n}〉$ is according to Equation (1), with ${\gamma }_{{〈\overline{n}〉}_T}=0.53\pm 0.02$. Although $T_{\zeta }^{*}/L^2$ exhibits fluctuations, they can be attributed to the statistical uncertainty in its determination.

Figure 2

(a) The distribution function of the avalanche size ($p(s,\zeta )$) and mass ($p(m,\zeta )$) with parameters $\mu$ and $\sigma$ defined in Equation (2) in terms of $1/L$ (insets), with $\zeta =1$ and $L=256$. The insets demonstrate that $\mu$ and $\sigma$ show power-law behavior in terms of system size L. (b) ${\mu }_m$, ${\mu }_s$, ${\sigma }_m$, and ${\sigma }_s$ in terms of $\zeta$, with $L=256$.

Figure 3

(a) $\ln (-\ln \frac{p(y,\zeta )}{p_{0y}})$ in terms of $\ln y$, where $y=l$ (main), $y=r_g^2$ (upper inset), and $y=sm$ (lower inset), with $L=256$. (b) ${\alpha }_y$ in terms of $\zeta$, with the corresponding fits for $L=256$. (c) ${\alpha }_y$ and ${\lambda }_y$ (inset) in terms of $\zeta$ on semilog scale, with $L=256$. The light blue box shows the crossover region to the BTW fixed point.

Figure 4

Phase diagram of the ant pile model. FP shows a fixed point. There are two FPs: the BTW FP ($\zeta \to 0$) and the hyper-active ($\zeta \to \infty$) FP; there is a crossover point between them around ${\zeta }^{*}\approx 0.1$.

Figure 5

${\tilde{p}}_{\alpha }^{\left(l\right)}(s,\zeta )$ in terms of s for l. Inset shows peak point $s_{\max }$ in terms of $\zeta$ for all three cases. A crossover is observed around ${\zeta }^{*}\approx 0.1$ (the blue region).

Figure 6

The fractal structure of avalanches. The main figure is $〈\log l〉$ in terms of $〈\log r_g〉$, with slope $D_f$ for $L=256$. Inset shows $D_f$ in terms of $\zeta$.