Observations in the Sandpile Model

Thomas Worsch

University of Karlsruhe

In the sandpile model by Bak, Tang, and Wiesenfeld one alternately does two things:

• runs a specific two-dimensional CA until a stable configuration is reached
• changes the state of a randomly selected cell

The cells of the CA have eight possible states 0, 1, ..., 7 and the local rule can be described best by interpreting the state as a number of “grains of sand” currently contained in a cell. If it has four grains or more, then it will move one grain to each of the four nearest neighbors. Thus a configuration is stable iff each cell has a state 0, 1, 2, or 3.

The randomly selected cell is always changed by adding one grain of sand.

This process gives rise to a Markov chain (MC) whose states are the stable configurations of the CA.

The set of recurrent configurations of this MC is well studied. For example, it is known that they form a group under the operation of pointwise addition and subsequent relaxation to a stable state.

Less is known about the set of transient configurations of the MC. We will present some new results by Matthias Schulz about the structure of this set. We will also show the outcome of some experiments (which give rise to more questions than answers).

[presentation materials]