Non Uniform Cellular Automata Description of Signed Partition Versions of Ice and Sand Pile Models

This paper reviews the well-known formalisations for ice andsand piles, based on a finite sequence of non-negative integers and itsrecent extension to signed partitions, i.e. sequences of a non-negativeand a non-positive part of integers, both non increasing.The ice pile model can be interpreted as a discrete time dynamicalsystem under the action of a vertical and a horizontal evolution rule,whereas the sand pile model is characterized by the unique action of thevertical rule.The signed partition extension, besides these two dynamical evolutionrules, also takes into account an annihilation rule at the boundary regionbetween the non-negative and the non-positive regions. We provide anoriginal physical interpretation of this model as a p-n junction of twosemiconductors.Moreover, we show how the sand pile extension of the signed partitionenvironment can be formalized by mean of a non-uniform cellularautomaton (CA) since the vertical and the annihilation evolution ruleshave the formal description of two CA local rules. Finally, we provide asimilar construction for the ice pile extension.