Studying Of Critical Behaviors' Dependence On Driving Mechanisms And Spatial Topologies In The Sand-Pile Models

The work in this paper generalizes the recent international progress on self-organized critical sand-pile models. We introduce the basic concept and principle of self-organized criticality and its special field--sandpile models. We summarize the formations, driving mechanisms, toppling rules and avalanche dynamics of some typical sandpile models (BTW, Manna, Zhang and Oslo ricepile models). On this base, we establish some sandpile models with different driving mechanisms and spatial topology, use numerical simulation, study in detail the critical scaling behaviors and avalanche dynamics of these models. And we compare and analyze the obtained results with the relative results on the international literature, and get some innovative and valuable general conclusions. These conclusions not only push the research on sandpile models for international academia, but also are helpful to the whole investigation on complex system and self-organized criticality. First, from a new point view, we use moment analysis techniques, perform large scale simulations of a one dimensional Oslo rice-pile model, obtain well defined avalanche exponents whose precision are five or ten times higher than the general literature and use data collapse method to verify the obtained results. This shows moment analysis techniques is very effective for clearly analyzing scaling behaviors in the Oslo ricepile models. Compared to direct measurement method, extrapolations method and local slope analysis method, it can be more clearly for describing scaling behaviors. Second, we establish one-dimensional ricepile models random driving at multi-sites. We numerically study the critical behaviors for four cases: adding grains to randomly chosen positions from 2 sites, from 32 sites, 512 sites, equally spaced, and from all sites of the network (the system size is L=2048). The emphasis lies in discussing the dependence of scaling behaviors on driving mechanism. Our numerical study shows that the random driving will result in the violation of the finite-size scaling, and the critical behaviors in this model are further away from the finite-size scaling along with the number of driving sites increasing. Third,we establish a one-dimensional sand-pile model, which has a stochastic redistribution process and whose energy input rules and toppling rules are discrete and continuous(namely, its input energy can be a non-integer), respectively. We numerically study the critical behavior in this model. The results show the critical behaviors fulfill the simple finite-size scaling in both the discrete model and the continuous model, but their avalanche exponents are different. By studying scaling relations of these avalanche exponents, we find the discrete model and continuous model are in different universality classes. At the same time, we find that although both the one-dimensional discrete sandpile model and Oslo ricepile model are discrete, the analysis on avalanche exponents shows the two models don't have the same critical behaviors, thus they are not in the same universality class. At last, we establish four two-dimensional rice-pile models for different toppling mechanisms and with different spatial topologies, namely, stochastic toppling rice-pile models on the triangular lattice and square lattice, and deterministic toppling rice-pile models on the triangular lattice and square lattice. We numerically study critical behaviors in these four models. The results show critical behaviors of the system have a very sensitive dependence on toppling mechanisms for the same spatial topology (the same grid).Whereas, critical behaviors of the system have no obvious dependence on spatial topologies for the same toppling mechanism. More exactly, the model with the same spatial topology but different toppling mechanisms(stochastic toppling and deterministic toppling) belong to different universality classes, but the model with the same toppling mechanism but different spatial topologies (square lattice and triangular lattice) remains in the same universality class. Our results show that the different spatial topologies don't alter the critical behavior of the system and the different toppling rules (stochastic toppling and deterministictoppling) alter the behavior of the system significantly and change the universality class of models. Our numerical results show that the two dimensional rice-pile models are in different universality classes with the BTW, Manna and Zhang models in literatures. In the BTW, Manna and Zhang models, the critical slope is a constant, whereas it is random in the 2D rice-pile models. The difference between the critical slope rules results in the completely different critical behaviors.