Abelian sandpile model

The BTW model was the first discovered example of a dynamical system displaying self-organized criticality.

In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile.

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.

As a direct consequence of this fact, it is shown that if two sand grains are added to the stable configuration in two different orders, e.g., first at site A and then at site B, and first at B and then at A, the final stable configuration of sand grains turns out to be exactly the same.

Every avalanche is guaranteed to eventually stop, i.e. after a finite number of topplings some stable configuration is reached such that the automaton is well defined.

To generalize the sandpile model from the rectangular grid of the standard square lattice to an arbitrary undirected finite multigraph

Given an arbitrary but fixed ordering of the non-sink vertices, multiple toppling operations, which can e.g. occur during the stabilization of an unstable configuration, can be efficiently encoded by using the graph Laplacian

, the set of recurrent configurations forms an abelian group isomorphic to the cokernel of the reduced graph Laplacian

More generally, the set of stable configurations (transient and recurrent) forms a commutative monoid under the operation

, which naturally appears in a cellular automaton closely related to the sandpile model, referred to as the chip firing or dollar game.

The model also displays[7] 1/ƒ noise, a feature common to many complex systems in nature.

For two dimensions, it has been hypothesized that the associated conformal field theory consists of symplectic fermions with a central charge c = −2.

The standard way of stabilizing the sandpile is to find a maximal legal sequence; i.e., by toppling so long as it is possible.

The animation shows the recurrent configuration corresponding to the identity of the sandpile group on different

Visually, the identities on larger grids seem to become more and more detailed and to "converge to a continuous image".

Indeed, existence of scaling limits of recurrent sandpile configurations has been proved by Wesley Pegden and Charles Smart.

[10][11] In further joint work with Lionel Levine, they use the scaling limit to explain the fractal structure of the sandpile on square grids.

[12] Another scaling limit, when the relaxations of a perturbation of the maximal stable state converge to a picture defined by tropical curves, is established in works of Nikita Kalinin and Mikhail Shkolnikov.

with is unstable and can topple (or fire), sending one of its chips to each of its four neighbors: Since the initial configuration is finite, the process is guaranteed to terminate, with the grains scattering outward.

A popular special case of this model is given when the initial configuration is zero for all vertices except the origin.

If the origin carries a huge number of grains of sand, the configuration after relaxation forms fractal patterns (see figure).

When letting the initial number of grains at the origin go to infinity, the rescaled stabilized configurations were shown to converge to a unique limit.

, then the stabilization operation on finite graphs is well-defined and the sandpile group can be written as before.

In contrast, vertices in the interior of the grid are still only allowed to carry integer numbers of grains.

[15] Of specific interest is the question how the recurrent configurations dynamically change along the continuous geodesics of this torus passing through the identity.

This question leads to the definition of the sandpile dynamics respectively induced by the integer-valued harmonic function

resemble the "smooth stretching" of the identity along the main diagonals visualized in the animation.

The configurations appearing in the dynamics induced by the same harmonic function on square grids of different sizes were furthermore conjectured to weak-* converge, meaning that there supposedly exist scaling limits for them.

on the larger grid to the corresponding configurations which appear at the same time in the sandpile dynamics induced by the restriction of

The Bak–Tang–Wiesenfeld sandpile was mentioned on the Numb3rs episode "Rampage," as mathematician Charlie Eppes explains to his colleagues a solution to a criminal investigation.