Areas and sizes of cascades in dissipative one-dimensional sandpile model

A prototypical self-organized sandpile mode is studied on a one-dimensional (1D) chain with periodic boundary conditions. A dissipation mechanism in which every grain being transferred between nodes has a probability ϵ to be taken out is needed, as the system has no boundary nodes for grains to fall off - a feature reminiscent of complex networks. Detailed numerical analysis revealed distributions of cascade areas D(a) and cascade sizes D(s) that are intrinsically different from other 1D sandpile models with an open end. Analyzing cascading processes on a chain, independent-site approximations to D(a) and D(s) are developed. The approximated distributions are given in terms of a single parameter ϕ0, which is the fraction of empty nodes when the system is stable. The approximations are shown to capture the key features of the distributions. The distribution of cascade sizes D(s) is shown to exhibit large fluctuations that cannot be suppressed by averaging over different realizations. Our approximation provides a physically transparent explanation of the intrinsic large fluctuations in terms of the number of ways that a cascade can proceed for achieving a certain size. To close the approximations, a semi-empirical formula for the parameter ϕ0 as a function of the dissipation probability ϵ is found. Our work thus reports non-trivial results on a seemingly simple model and sheds light on analyzing cascading processes in other complex networks with no boundary nodes.