Max-min-plus expressions for one-dimensional particle cellular automata obtained from a fundamental diagram

TL;DR

This paper derives max-min-plus algebraic expressions for one-dimensional particle cellular automata from fundamental diagrams, enabling analysis of their asymptotic behavior and generalization to higher neighborhood sizes.

Contribution

It introduces a method to obtain evolution equations of particle CA5s in max-min-plus form from fundamental diagrams, and extends the approach to neighborhood-n CA.

Findings

Max-min-plus expressions derived for particle CA5s

Ultradiscrete Cole-Hopf transformation used for analysis

Framework generalized to neighborhood-n CA

Abstract

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$n$.