Max-plus analysis on some binary particle systems

TL;DR

This paper applies max-plus algebra to analyze certain binary particle cellular automata, deriving exact solutions and fundamental diagrams, and studying their asymptotic behaviors.

Contribution

It introduces max-plus algebra techniques to exactly solve and analyze specific particle cellular automata models, providing explicit solutions and fundamental diagrams.

Findings

Exact solutions for PCA4-1, 4-2, 4-3 derived

Asymptotic behaviors of solutions analyzed

Fundamental diagrams proved exactly

Abstract

We concern with a special class of binary cellular automata, i.e., the so-called particle cellular automata (PCA) in the present paper. We first propose max-plus expressions to PCA of 4 neighbors. Then, by utilizing basic operations of the max-plus algebra and appropriate transformations, PCA4-1, 4-2 and 4-3 are solved exactly and their general solutions are found in terms of max-plus expressions. Finally, we analyze the asymptotic behaviors of general solutions and prove the fundamental diagrams exactly.