Controlling self-organized criticality in sandpile models

We introduce an external control to reduce the size of avalanches in some sandpile models exhibiting self-organized criticality. This rather intuitive approach seems to be missing in the vast literature on such systems. The control action, which amounts to triggering avalanches in sites that are near to become critical, reduces the probability of very large events, so that energy dissipation occurs most locally. The control is applied to a directed Abelian sandpile model driven by both uncorrelated and correlated depositions. The latter is essential to design an efficient and simple control heuristic, but has only small influence in the uncontrolled avalanche probability distribution. The proposed control seeks a trade-off between control cost and large event risk. Preliminary results hint that the proposed control works also for an undirected sandpile model.

Figure 1

Probability distribution of avalanche sizes $p\left(s\right)$ in region $X$ when $N_R=128$. Points were obtained by logarithmic size bins over the whole range of $s$. In the inset, curves are for $N_R=64$ (diamonds) and 32 (triangles). Solid and hollow symbols denote uncontrolled and controlled systems, respectively. Both straight lines result from least-square fitting to the solid symbols in the main panel. $a_c=4$ for all three cases.

Figure 2

Complementary probability $Q\left(s^{\prime }<s\right)$ when $N_R=64$ and $a_c=4$. Solid (hollow) symbols refer to uncontrolled (controlled) system: triangles, diamonds, and squares indicate internal, external, and control generated avalanches.

Figure 3

Ratio $f$ between total number of avalanches in the controlled and uncontrolled simulations. The number of time steps (equal for both simulations) depend on the system size. Line types indicate the following values of $\left(N_R,a_c\right)$: (64,4), solid curve; (64,8), dashed curve; (32,4), dotted curve; and (128,4), dotted-dashed curve. Decrease in $f$ in the large size region indicates control success. The increase in the value of $f$ for very large avalanche sizes, observed for some curves, is due to finite-size effects. The logarithmic scale in vertical axis shows that, in the inset, the performance of random control (dashed curve) is much worse that that provided by targeted control (solid curve). The inset also displays results for the BTW model, with $\left(N_R,a_c\right)=\left(32,4\right)$ (dotted curve), showing that the control works also very efficiently.