Sandpile model with activity inhibition

Abstract

A sandpile model is studied in which bonds of the system are inhibited for activity after a certain number of transmission of grains. This condition impels an unstable sand column to distribute grains only to those neighbors that have toppled less than m times. In this non-Abelian model grains effectively move faster than the ordinary diffusion ͑superdiffusion͒. A system size dependent crossover from Abelian sandpile behavior to a new critical behavior is observed for all values of the parameter m. ͓S1063-651X͑97͒51111-1͔ PACS number͑s͒: 05.40.ϩj, 05.70.Jk, 05.70.Ln The concept of self-organized criticality ͑SOC͒ was introduced to describe how a system, starting from an arbitrary initial condition may evolve to a scale free critical state following some specific dynamical rules while under the action of repeated external perturbations ͓1͔. Naturally occurring physical phenomena like sandpiles ͓2͔, forest fires ͓3͔, river networks ͓4͔, earthquakes ͓5͔, etc. are argued as systems showing SOC. To demonstrate the idea of SOC a simple model known as the ''sandpile'' model was introduced in which a stochastically driven cellular automata evolves under a nonlinear, diffusive, self-organizing mechanism leading to a nonequilibrium critical state ͓1͔.At present many different versions of the sandpile model are available. However, precise classification of various models in different universality classes in terms of their critical exponents is not yet fully complete and still attracts much attention ͓6͔. Among the different models most widely studied is the Abelian sandpile model ͑ASM͒ in which many analytical ͓7͔ as well as numerical ͓8͔ results are known. Some efforts have also been given towards the analytical calculation of avalanche size exponents ͓9,10͔. Second, a two-state sandpile model with stochastic evolution rules was also studied ͓11͔ that was initially thought to belong to the same universality class as that of ASM ͓11,12͔ but later claimed to be different ͓13͔.We consider a situation in which an intermediate time scale is associated with every bond of the system. Each bond allows only a certain number of grains to cross from its one end to the other and after that it has a dead time and cannot support any further traffic until a new avalanche starts. This dead time is much greater than the time scale of avalanche propagations but much less than the input rate of grains. We call this model as the ''sandpile model with activity inhibition'' ͑SMAI͒.Similar to different sandpile models we also define our model on a regular lattice with open boundary. Non-negative integer numbers (h i ) assigned at the lattice sites represent the heights of the sand columns. Sand grains are added at randomly chosen sites by increasing the h values by unity: h i →h i ϩ1. The possibility of a sand column becoming unstable arises only when the height h i becomes greater than a threshold value h c . Such a column becomes unstable only if the number n i of nearest-neighbor sites that have toppled less than a preassigned cutoff numb...

Characteristics of deterministic and stochastic sandpile models in a rotational sandpile model

“…This inherent randomness makes the model not only non-abelian but also "quasideterministic". Due to the stochastic dynamical rules, MSM had already been found nonabelian [25]. It is also important to notice that the local correlation in the rotational toppling rule then can not propagate throughout the avalanche as in BTW because of the "internal stochasticity" in the model.…”

Autoregressive cascades on random networks

“…In case of the sandpile model, a toppled node is allowed to participate in further interactions, while in ARC model once a node fails it takes no further part in the process. There is also an inhibition sandpile model [17], where a toppled node is stopped temporarily or permanently from taking part in further interaction, however, only empirical results are available for the same. Another similar form of interaction is the bootstrap percolation, where the failure of a node depends on the failure of a fixed number of neighbors and not the weights at the failed neighboring nodes [18].…”