CHOOSEY HOT SAND: REFLECTION OF GRAIN SENSITIVITY ON PATTERN MORPHOLOGY

Abstract

We build and investigate a nonstandard model of pattern formation in a system of discrete entities evolving in discrete space and time. We chose a sandpile paradigm to fit our ideas in the frame of current research. In our model sand is hot because a grain can topple against gradient, i.e., the grain can walk to another node even when a number of grains in its current node is less than a number of neighboring nodes. Sand is choosey because behavior of the grains is not determined by any global parameter or any threshold of a number of neighboring grains (called here a grain sensitivity) but depends on the exact number of grains in the neighboring nodes. Namely, we assume that a grain being at a node x goes to one of the eight neighboring nodes, chosen at random, if there is another grain at the node x or if the number of grains in eight neighboring nodes lies in some set of 2{1,…,8}. These 256 rules of sensitivity are investigated. The classification of the rules if offered, based on the morphology of the patterns generated by each rule. Eight morphological classes are found. Fine structure of every class is investigated and transient phenomena are analyzed. Three kinds of description of class rules by Boolean expressions are offered. Evolution of the classes governed by several one-dimensional parameters is considered.