We introduce the mixed model of sandpile+anti-sandpile, which is called SA model. In the SA model, we are free to add or remove a particle from a chosen site. Because of the non-abelian property of...Show moreWe introduce the mixed model of sandpile+anti-sandpile, which is called SA model. In the SA model, we are free to add or remove a particle from a chosen site. Because of the non-abelian property of the toppling operators and anti-toppling operators, the SA model becomes subtle, and the group property existing in the pure sandpile model and anti-sandpile model does not hold in the SA model. Because of the non-local property of addition operators and anti-addition operators, the processes related to the sandpile, and anti-sandpile are not Feller. The traditional way of constructing the interacting particle processes in infinite volume, e.g., via Hille-Yoshida, does not work in these cases, other ways of construction are necessary. In the construction of sandpile+anti-sandpile process (SA process), we obtain the semigroup of the process for some special configurations and some special functions by series expansion and then using monotonicity of the process, we can extend it to the general case. The SA process shows a new transition phenomenon: it seems that the stationary measure is the result of a “competition” between the generators. The sandpile model, anti-sandpile model are “self-organized” critical systems. In recent years, this is challenged because the special nature of the dynamics can be considered as an implicit fine tuning that makes sure the system can reach criticality, therefore the “selforganized” critical behavior of these systems can be thought of as a more conventional phase transition between “stabilizable” and “non-stabilizable”. We discuss the conditions for a mixed system to reach a stable state both in finite volume and infinite volume.Show less
