Critical Behaviors in a Stochastic One-Dimensional Sand-Pile Model

A one-dimensional sand-pile model (Manna model), which has a stochastic redistribution process, is studiedboth in discrete and continuous manners. The system evolves into a critical state after a transient period. A detailedanalysis of the probability distribution of the avalanche size and duration is numerically investigated. Interestingly,contrary to the deterministic one-dimensional sand-pile model, where multifractal analysis works well, the analysis basedon simple finite-size scaling is suited to fitting the data on the distribution of the avalanche size and duration. Theexponents characterizing these probability distributions are measured. Scaling relations of these scaling exponents andtheir universality class are discussed.