Avalanche properties of sandpile models in terms of a microscopic parameter

Sandpile models are usually characterized by studying the "macroscopic" critical properties of its non-equilibrium steady state. However, most of the information of a sandpile avalanche remains stored in a "microscopic" parameter, the toppling number s_i of each sand column at a lattice site i. In terms of s_i, a number of physical quantities such as distribution of s_i, fluctuation and correlation in s_i, density distribution of s_i etc.; are defined and analyzed for a variety of sandpile models at their respective steady states. A set of new critical exponents is estimated and scaling relations among them are established. The values of the critical exponents are found to satisfy the scaling relations and able to characterize the system properties distinctly for different sandpile models. The analysis not only reveals the complex geometrical properties associated with the toppling surface but also provides deeper insight into the avalanche properties.