Self-organized branching process for a one-dimensional ricepile model

A self-organized branching process is introduced to describe one-dimensional ricepile model with stochastic topplings. Although the branching processes are generally supposed to describe well the systems in high dimension, our modification grasps some of the peculiarities present in one dimension. We find analytically the crossover behavior from the trivial one-dimensional BTW behavior to self-organized criticality characterized by power-law distribution of avalanches. The finite-size effects, which are crucial in the crossover, are calculated.