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Abelian sandpile model

The Abelian sandpile model, also known as the Bak–Tang–Wiesenfeld model, was the first discovered example of a dynamical system displaying self-organized criticality. It was introduced by Per Bak, Chao Tang and Kurt Wiesenfeld in a 1987 paper. The Abelian sandpile model, also known as the Bak–Tang–Wiesenfeld model, was the first discovered example of a dynamical system displaying self-organized criticality. It was introduced by Per Bak, Chao Tang and Kurt Wiesenfeld in a 1987 paper. The model is a cellular automaton. In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile. This slope builds up as 'grains of sand' (or 'chips') are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope. Bak, Tang, and Wiesenfeld considered process of successive random placement of sand grains on the grid; each such placement of sand at a particular site may have no effect, or it may cause a cascading reaction that will affect many sites. The model has since been studied on the infinite lattice, on other (non-square) lattices, and on arbitrary graphs (including directed multigraphs). The iteration rules for the model on the square lattice with sites ( x , y ) ∈ Z 2 {displaystyle (x,y)in mathbb {Z} ^{2}} can be defined as follows: Begin with some nonnegative configuration of values z ( x , y ) ∈ Z {displaystyle z(x,y)in mathbf {Z} } which is finite, meaning Any site ( x , y ) {displaystyle (x,y)} with is unstable and can topple (or fire), sending one of its chips to each of its four neighbors: The process is guaranteed to terminate, with the chips scattering outward, given that the initial configuration was finite. Moreover, although there will often be many possible choices for the order in which to topple vertices, the final configuration does not depend on the chosen order; this is one sense in which the sandpile is abelian. The number of times each vertex topples in this process is also independent of the choice of toppling order. Generalizing from the lattice to an arbitrary undirected simple graph, a special vertex called a sink is specified that is not allowed to topple. Again, a state of the model is a chip configuration counting the number of chips on each non-sink vertex. A non-sink vertex v {displaystyle v} with

[ "Quantum mechanics", "Combinatorics", "Discrete mathematics", "Criticality", "Control theory" ]
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