Structure and dynamics in the low-density phase of a two-dimensional cellular automaton model of traffic flow

We analyze the structure and dynamics in the low-density phase of the deterministic two-dimensional cellular automaton model of traffic flow introduced in [O. Biham, A.A. Middleton, and D. Levine, Phys. Rev. A46, R6124(R) (2026) 10.1103/PhysRevA.46.R6124]. The model consists of horizontally oriented (H) cars that move to the right and vertically oriented (V) cars that move downward, on a square lattice of size L with periodic boundary conditions. Starting from a random initial state of density p, which is equally divided between the H- and V-cars, the model exhibits a phase transition at a critical density pc. For p<pc, it evolves toward a free-flowing periodic (FFP) state, while for p>pc, it evolves toward a fully jammed state or to an intermediate state of congested traffic. In the FFP states, the H- and V-cars segregate into homogeneous diagonal bands, in which they move freely without obstruction. To analyze the convergence toward the FFP states, we introduce a configuration-space distance measure D(t)=D∥(t)+D⊥(t) between the state of the system at time t and the set of FFP states. The D∥(t) term accounts for the interactions between homotypic pairs of H- (or V-) cars, while D⊥(t) accounts for the interactions between heterotypic pairs of H- and V-cars. We show that in the FFP states D(t)=0, while in all the other states D(t)>0. As the system evolves toward the FFP states, there is a separation of timescales, where D∥(t) decays very fast, while D⊥(t) decays much more slowly. Moreover, the time dependence of D⊥(t) is well fitted by an exponentially truncated power-law decay of the form D⊥(t)∼t−γexp(−t/τ⊥), where τ⊥ depends on L and p. The power-law decay suggests avalanche-like dynamics with no characteristic scale, while the exponential cutoff is imposed by the finite lattice size.