A Cellular Automata Study of Constraints (Dissolvence) in a Percolating Many-Particle System

Abstract

In a recent study by Kier, Cheng and Testa, simulations were carried out to monitor and quantify the emergence of a collective phenomenon, namely percolation, in a many-particle system modeled by cellular automata (CA). In the present study, the same setup was used to monitor the counterpart to collective behavior, namely the behavior of individual particles, as modeled by occupied cells in the CA simulations. As in the previous study, the input variables were the concentration of occupied cells and their joining and breaking probabilities. The first monitored attribute was the valence configuration (state) of the occupied cells, namely the percent of occupied cells in configuration Fi (%Fi), where i = number of occupied cells joined to that cell. The second monitored attribute was a functional one, namely the probability (in %) of a occupied cell in configuration Fi to move during one iteration (%Mi). First, this study succeeded in quantifying the expected, strong direct influences of the initial conditions on the configuration and movement of occupied cells. Statistical analyses unveiled correlations between initial conditions and cell configurations and movements. In particular, the distribution of configurations (%Fi) varied with concentration with a kinematic-like regularity amenable to mathematical modeling. However, another result also emerged from the work, such that the joining, breaking and concentration factors not only influenced the movement of occupied cells, they also modified each other's influence (Figure 1). These indirect influences have been demonstrated quite clearly, and some partial statistical descriptions were established. Thus, constraints at the level of ingredients (dissolvence) have been characterized as a counterpart to the emergence of a collective behavior (percolation) in very simple CA simulations.