Critical behavior of number-conserving cellular automata with nonlinear fundamental diagrams

Abstract

We investigate critical properties of a class of number-conserving cellular automata (CA) which can be interpreted as deterministic models of traffic flow with anticipatory driving. These rules are among the only known CA rules for which the shape of the fundamental diagram has been rigorously derived. In addition, their fundamental diagrams contain nonlinear segments, as opposed to majority of number-conserving CA which exhibit piecewise-linear diagrams. We found that the nature of singularities in the fundamental diagram of these rules is the same as for rules with piecewise-linear diagrams. The current converges toward its equilibrium value like $t^{-1/2}$, and the critical exponent $\beta$ is equal to 1. This supports the conjecture of universal behavior at singularities in number-conserving rules. We discuss properties of phase transitions occurring at singularities as well as properties of the intermediate phase.