Structure and dynamics in the low-density phase of a two-dimensional cellular automaton model of traffic flow
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| Format: | Preprint |
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2026
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| _version_ | 1866908801567817728 |
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| author | Rabinovich, Gilad Hertzberg Biham, Ofer Katzav, Eytan |
| author_facet | Rabinovich, Gilad Hertzberg Biham, Ofer Katzav, Eytan |
| contents | 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 (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 $D(t)=D_{\parallel}(t)+D_{\perp}(t)$ between the state of the system at time $t$ and the set of FFP states. The $D_{\parallel}(t)$ term accounts for the interactions between homotypic pairs of H (or V) cars, while $D_{\perp}(t)$ 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 time scales, where $D_{\parallel}(t)$ decays very fast while $D_{\perp}(t)$ decays much more slowly. Moreover, the time dependence of $D_{\perp}(t)$ is well fitted by an exponentially truncated power-law decay of the form $D_{\perp}(t)\sim t^{-γ} \exp(-t/τ_{\perp})$, where $τ_{\perp}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01180 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Structure and dynamics in the low-density phase of a two-dimensional cellular automaton model of traffic flow Rabinovich, Gilad Hertzberg Biham, Ofer Katzav, Eytan Cellular Automata and Lattice Gases 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 (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 $D(t)=D_{\parallel}(t)+D_{\perp}(t)$ between the state of the system at time $t$ and the set of FFP states. The $D_{\parallel}(t)$ term accounts for the interactions between homotypic pairs of H (or V) cars, while $D_{\perp}(t)$ 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 time scales, where $D_{\parallel}(t)$ decays very fast while $D_{\perp}(t)$ decays much more slowly. Moreover, the time dependence of $D_{\perp}(t)$ is well fitted by an exponentially truncated power-law decay of the form $D_{\perp}(t)\sim t^{-γ} \exp(-t/τ_{\perp})$, where $τ_{\perp}$ 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. |
| title | Structure and dynamics in the low-density phase of a two-dimensional cellular automaton model of traffic flow |
| topic | Cellular Automata and Lattice Gases |
| url | https://arxiv.org/abs/2601.01180 |