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

A one-dimensional sand-pile model (Manna model), which has astochastic redistribution process, is studied both in discreteand continuous manners. The system evolves into a critical stateafter a transient period. A detailed analysis of the probabilitydistribution of the avalanche size and duration is numericallyinvestigated. Interestingly, contrary to the deterministicone-dimensional sand-pile model, where multifractal analysisworks well, the analysis based on simple finite-size scaling issuited to fitting the data on the distribution of the avalanchesize and duration. The exponents characterizing these probabilitydistributions are measured. Scaling relations of these scalingexponents and their universality class are discussed.