Critical indices for singular diffusion.

Thus the systems studied by CCGS do seem to be examSandslide or avalanche models were introduced in the work of Bak, Tang, and Wiesenfeld [1]. They looked at the process of adding grains of sand arbitrarily slowly to a model sandpile, and studied the spectrum of avalanches produced by the individual addition events. They suggested that the pile would adjust its slope to a critical value in which there would be a very wide range of event sizes. They suggested that such qualitatively similar behavior, which they called self organized -criticality, would occur in many systems. Carlson, Chayes, Grannon, and Swindle (CCGS) [2] studied two sandslide models [2,3] in some detail and found support for such behavior. In particular, they saw that a coarse-grained variable in their models —generically described as a slope S—approached a critical value as the linear size of the system L became infinite. For larger L, they found that the modal slope S minus the critical slope S, obeys a simple scaling law: S,—S-L P . ples of self-organized criticality. This paper is intended to give scaling arguments for the divergence of Eq. (1.2) and scaling estimates of P and P. Two models are discussed here. (a) Two state (2S) model [2] of CCGS. Here CCGS construct an analytical solution, involving a state of local equilibrium, which gives a rigorous proof of diffusion and the index values (b= 3 and p= —, '. (b) The local limited model [3] (L ) with My=2. In their simulations CCGS see diffusion and estimate, /=4 and P= —, '.