Sand piles: From physics to cellular automata models

We analyze the dynamical behavior of the usual one dimensional sand pile model which actually describes the physical situation in which the pile is submitted to the uniform blow of a unidirectional wind. In the first step the Lagrangian formalism is investigated, showing that the stationary action principle does not select in a unique way the path which satisfies either the minimal or the maximal action principle. This drawback is solved making use of the information (Shannon) entropy which enables one to determine the unique path in which at any time step the entropy variation is minimal (adiabatic) or maximal (anti-adiabatic). A cellular automata (CA) model describing this sand pile behavior is introduced, and the consequent deterministic dynamic is compared with the entropy results, showing that also in this case there are some drawbacks. Moreover, it is shown that our CA local rule is a particular case of some standard CA sand pile models present in literature.