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

We investigate critical properties of a class of number-conserving cellular automata (CAs) 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 the majority of number-conserving CAs 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 as t−1/2, and the critical exponent β is equal to unity. This supports the conjecture of universal behaviour at singularities in number-conserving rules. We discuss properties of phase transitions occurring at singularities as well as properties of the intermediate phase.