Mean field theory for ordinary and hot sandpiles

A mean field theory is discussed for a sandpile model, a cellular automaton prototype of systems showing self-organized criticality. The previous formulation of the mean field does not take into account the dissipation effects that take place on boundaries. This gives rise to some inconsistencies that are eliminated by carefully considering the boundaries effects, as it is shown in this paper. We present here a revised version of the MF equations. The main result is that criticality arises in the thermodynamic limit for sandpile systems, confirming numerical observations on the behavior of the order parameter. The mean field approach is also generalized by applying it to the more general case of sandpiles in thermal equilibrium where a temperature-like parameter T is introduced. In this case we show that criticality is not destroyed at T > 0. © 1998 Published by Elsevier Science B.V. All rights reserved.