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Local violation of conservation in the abelian sandpile model through fractal patterns of non-conservative sites. (English) Zbl 1479.82037

Summary: In this paper we investigate the two-dimensional abelian sandpile model in which the conservation of sand grains is locally violated using non-conservative lattice sites. We use spatially correlated arbitrary fractal patterns to mark the non-conservative sites. We have observed a “crossover” in the scaling behavior of the distribution functions of both the avalanche areas and avalanche sizes. The pre-crossover scaling exponents are known already and are related to the embedding dimension of the model. In addition to these, we have found that new “post-crossover” scaling exponents result from fractality. In fact we have found a spectrum of values for these exponents across the fractal dimensions \(1\leq d_{\mathrm{f}}\leq 2\) with a dominantly declining “linear” trend.

MSC:

82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82C27 Dynamic critical phenomena in statistical mechanics
28A80 Fractals

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