Critical exponents of the sand pile models in two dimensions

Abstract We study three sand pile automaton models namely, the critical height, critical slope and the critical Laplacian models in two dimensions, in which the stability criterion of the sand columns depend on the zeroth, first and the second derivatives of the sand height function. We carried out simulations on system sizes up to 2048 × 2048 and up to 108 avalanches were generated. The exponents of the critical height model were calculated by taking into account the strong corrections to scaling. In order to determine the exponents of the critical Laplacian model accurately we introduced a height restriction in the toppling criterion that maintains universality but accelerates convergence to the steady state by orders of magnitude. We see clear scaling in the critical height and the critical Laplacian models and find that they belong to different universality classes. However, we do not find any scaling in the critical slope model.

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