SANDPILE MODELS WITH AND WITHOUT AN UNDERLYING SPATIAL STRUCTURE

We present a simple mean-field model for the sandpile model introduced by Bak, Tang, and Wiesenfeld (BTW) [Phys. Rev. Lett. 59, 381 (1987)]. In the mean-field model we are able to pinpoint the process of self-organization as well as the emerging scale invariance displayed as a power-law distribution of avalanche sizes. We discuss the BTW sandpile model on a lattice and show that the dynamical behavior can be expressed as a transport problem. This implies that the average avalanche size scales with the system size, and additional heuristic arguments related to the transport properties more than indicate the origin of the power-law behavior. We review recent work in which scaling relations and additional constraints between the various critical exponents are addressed. We demonstrate that some of the proposed relations are inconsistent. We present a coherent ''theory'' in which the scaling relations along with additional constraints leave only one exponent unknown.