Three-dimensional Fundamental Diagrams of Five-neighbor Particle Cellular Automata

Abstract

We analyze five-neighbor particle cellular automata whose conventional two-dimensional fundamental diagrams are multivalued, but whose mean flow is uniquely determined by introducing a second density. We first consider binary rules for which the second density is conserved, and then examine rules for which the second density is not conserved but converges asymptotically. These examples give three-dimensional fundamental diagrams in which the mean flow is determined by the particle density and the second density. We then investigate whether this single-valued structure is preserved under real-valued max-plus extensions. There are some rules where two different max-plus extensions are introduced, and numerical simulations show that both extensions preserve the same single-valued three-dimensional fundamental diagram. These observations imply that, in constructing real-valued max-plus extensions, it is important to choose the flux function and the second density consistently.