Difusão singular e percolação em modelos do tipo “Pilha de Areia” confinados

Abstract

We investigate the behavior of a two-state sandpile model subjected to a confining potential in one and two dimensions. From the microdynamical description of this simple model with its intrinsic exclusion mechanism, it is possible to derive a continuum nonlinear diffusion equation that displays singularities in both the diffusion and drift terms. The stationary-state solutions of this equation, which maximizes the Fermi-Dirac entropy, are in perfect agreement with the spatial profiles of time-averaged occupancy obtained from model numerical simulations in one as well as in two dimensions. Surprisingly, in the onedimensional case, our results show that the presence of a confining potential can lead to the emergence of a power-law tail in the distribution of avalanche sizes. The onset of this regime is signaled by a maximum in the fluctuation of energy. In the two-dimensional case, by studying the systems at different confining conditions, we observe two scale-invariant regimes. At a given confining potential strength, the cluster size distribution takes the form of a power law. This regime corresponds to the situation in which the density at the center of the system approaches the critical percolation threshold. The analysis of the fractal dimension of the largest cluster frontier provides evidence that this regime is reminiscent of gradient percolation. By increasing further the confining potential, most of the particles coalesce in a giant cluster, and we observe a regime where the jump size distribution takes the form of a power law. Likewise the one-dimensional case, the onset of this second regime is also signaled by a maximum in the fluctuation of energy.