Max-min-plus expressions for one-dimensional particle cellular automata obtained from a fundamental diagram
This work provides a new mathematical framework for analyzing particle cellular automata, but it is incremental as it extends existing max-min-plus methods to a specific class of CA.
The authors derive max-min-plus expressions for one-dimensional neighborhood-five conservative cellular automata from a fundamental diagram, enabling asymptotic analysis of general solutions. They also present Lagrange representation equations and discuss generalization to neighborhood-n CA.
We study one-dimensional neighborhood-five conservative cellular automata (CA), referred to as particle cellular automata five (particle CA5). We show that evolution equations for particle CA5s that belong to certain types can be obtained in the form of max-min-plus expressions from a fundamental diagram. The obtained equations are transformed into other max-min-plus expressions by ultradiscrete Cole-Hopf transformation, which enable us to analyze the asymptotic behaviors of general solutions. The equations in the Lagrange representation, which describe particle motion, are also presented, which too can be obtained from a fundamental diagram. Finally, we discuss the generalization to a one-dimensional conservative neighborhood-$n$ CA, i.e., particle CA$n$.