Non-Abelian sandpile automata with height restrictions

We have studied the properties of a sandpile automata under the constraint of height restriction of sand columns. In this sandpile, an active site transfers a grain to a neighboring site if and only if the height of the sand column at the destination site is less than a preassigned value nc. This sandpile was studied by Dickman et al. [Phys. Rev. E 66, 016111 (2002)] in a conserved system with a fixed number of sand grains. In contrast, we have studied the avalanche dynamics of the driven sandpile under the open boundary conditions. The deterministic dynamics of the Bak, Tang, and Wiesenfeld (BTW) sandpile under the height restriction is found to be non-Abelian. Using numerical results, we argue that the steady states of the sandpile are exactly the recurrent states of the BTW sandpile, but occur with nonuniform probabilities. A detailed analysis of the cluster size distributions indicates that the associated exponent values are likely to be different from those of the BTW sandpile. The other differences include that the drop number distribution decays as a power law, and the largest avalanche size grows as the fourth power of the system size.

Automated summary

We studied a sandpile automata with height restriction of sand columns. In this sandpile, an active site transfers a grain to a neighboring site only if the height of the sand column at the destination site is less than a preassigned value nc. We analyzed the avalanche dynamics of the driven sandpile under open boundary conditions and found that the deterministic dynamics of the BTW sandpile under the height restriction are non-Abelian. Results from numerical simulations suggest that the steady states of the sandpile are recurrent states of the BTW sandpile, but with different probabilities and exponent values.