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. A 46, R6124(R) (1992)]. 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 $p_c$. For $p<p_c$, it evolves toward a free-flowing periodic (FFP) state, while for $p>p_c$, 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 $D\left(t\right)=D_{\parallel }\left(t\right)+D_{\perp }\left(t\right)$ between the state of the system at time $t$ and the set of FFP states. The $D_{\parallel }\left(t\right)$ term accounts for the interactions between homotypic pairs of H- (or V-) cars, while $D_{\perp }\left(t\right)$ accounts for the interactions between heterotypic pairs of H- and V-cars. We show that in the FFP states $D\left(t\right)=0$, while in all the other states $D\left(t\right)>0$. As the system evolves toward the FFP states, there is a separation of timescales, where $D_{\parallel }\left(t\right)$ decays very fast, while $D_{\perp }\left(t\right)$ decays much more slowly. Moreover, the time dependence of $D_{\perp }\left(t\right)$ is well fitted by an exponentially truncated power-law decay of the form $D_{\perp }\left(t\right)\sim t^{-\gamma }exp(-t/{\tau }_{\perp })$, where ${\tau }_{\perp }$ 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.

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• Dynamical phase transitions
• Nonequilibrium statistical mechanics
• Phase separation
• Self-organized systems