Sand piles: from physics to cellular automata models
information and thermodynamical entropies
The article deals with the sand pile model: finitely many piles of finitely many grains are juxtaposed and grains of a pile sequentially fall down to the right-neighboring pile if the latter does not get higher than the former.NEWLINENEWLINEThis system is intrinsically nondeterministic: sometimes several piles are simultaneously in the situation where they can be fired (they have at least 2 grains more than their respective right neighbors), but it is (conjectured in the article, but actually) known (from [\textit{E. Formenti} and \textit{B. Masson}, ``On computing fixed points for generalized sandpiles'', Int. J. Unconv. Comput. 2, 51--72 (2006)]) that all possible evolutions eventually lead to the same equilibrium configuration.NEWLINENEWLINEThe authors here propose several ways to chose between these possible evolution paths. First, one can select the path that maximizes or minimizes locally the (discrete) integral (called associated action) of the Lagrangian, which is basically (a simple affine map of) the potential energy (depending on the height of every grain). Nevertheless, this does not determine a unique path.NEWLINENEWLINEInstead, one can do the same with the (Lagrangian computed from the) information entropy (average uncertainty of the pile distribution seen as an integer partition) instead of the potential energy. This does determine a path uniquely: actually, one for local minimization, called adiabatic evolution, and one for local maximization, called anti-adiabatic evolution. However, these do not coincide with the global maximizing or minimizing paths.NEWLINENEWLINEOne can do the same with thermodynamical entropy (logarithm of the thermodynamics probability) which, again, does not determine unique maximizing or minimizing paths.NEWLINENEWLINEA simple cellular automaton model is also presented, that corresponds to the parallel evolution: each pile that can be fired is fired synchronously. Unfortunately, this evolution may not involve the same intermediary configurations than any of the distinguished evolution paths presented before.
scientific article; zbMATH DE number 6049406
scientific article