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), whic h 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. Ramaswa my, Phys. Rev. Lett. 63, 1659 (1989)]. For 0 < p < 1 metastable column s of sand may be formed, which are relaxed when one of the local slope s exceeds a critical value sigma(c). By varying the probability of top pling p we find that a continuous phase transition occurs at the criti cal probability p(c), at which the steady states with zero average slo pe (above p(c)) are replaced by states characterized by a finite avera ge slope (below p(c)). We study this phase transition in detail by int roducing an appropriate order parameter and the order-parameter suscep tibility chi. In a certain range of p < 1 we find the self-organized c ritical behavior that is characterized by nonuniversal p-dependent sca ling exponents for the probability distributions of size and length of avalanches. We also calculate the anisotropy exponent zeta and the fr actal dimension df of relaxation clusters in the entire range of value s 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 isotro pic and compact.