Nonequilibrium phase transition and self-organized criticality in a sandpile model with stochastic dynamics

We introduce and study numerically a directed two-dimensional sandpile automaton with probabilistic toppling (probability parameter p), which provides a good laboratory to study both self-organized criticality and the far-from-equilibrium phase transition. In the limit p=1 our model reduces to the critical height model in which the self-organized critical behavior was found by exact solution [D. Dhar and R. Ramaswamy, Phys. Rev. Lett. 63, 1659 (1989)]. For 0p${\mathrm{\sigma }}_{\mathit{c}}$. By varying the probability of toppling p we find that a continuous phase transition occurs at the critical probability ${\mathit{p}}_{\mathit{c}}$, at which the steady states with zero average slope (above ${\mathit{p}}_{\mathit{c}}$) are replaced by states characterized by a finite average slope (below ${\mathit{p}}_{\mathit{c}}$). We study this phase transition in detail by introducing an appropriate order parameter and the order-parameter susceptibility χ. In a certain range of pp-dependent scaling exponents for the probability distributions of size and length of avalanches. We also calculate the anisotropy exponent ζ and the fractal dimension ${\mathit{d}}_{\mathit{f}}$ of relaxation clusters in the entire range of values of the toppling parameter p. We show that the relaxation clusters in our model are anisotropic and can be described as fractals for values of p above the transition point. Below the transition they are isotropic and compact. © 1996 The American Physical Society.