Solutions for certain number-conserving deterministic cellular automata.
We explain the unexpected behavior of the generalizations of cellular automation traffic models introduced in [H. Fukś and N. Boccara, Int. J. Mod. Phys. C 9, 1 (1998)]. We analyze the steady-state flow in R(m,k) as a function of the initial density; we show that these rules correspond to a system with an infinite number of different kinds of virtual particles interacting according to complex annihilation rules. From simple considerations, we are able to predict the unexpected cutoff of the average flow at unity observed by Fukś and Boccara. We present an efficient algorithm for determining the exact final flow from a given finite initial state. An analysis of this algorithm in the infinite limit using generating functions yields an exact polynomial equation between the flow and density for R(m,k), of maximally 2(m+k)th degree in both.