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

TL;DR

The paper analyzes the low-density regime of the deterministic two-dimensional BML traffic model on a torus, introducing a configuration-space distance $D(t)=D_{\nparallel}(t)+D_{\nperp}(t)$ to quantify convergence to free-flowing periodic (FFP) states. It uncovers a separation of time scales with $D_{\nparallel}(t)$ decaying rapidly while $D_{\nperp}(t)$ decays as $D_{\nperp}(t)\sim t^{-\gamma}\exp(-t/\tau_{\nperp})$, where $\gamma\approx1.1$ and $\tau_{\nperp}$ depends on system size $L$ and density $p$, signaling avalanche-like relaxation limited by finite size. The study delineates the structure of FFP states—diagonal banding and per-diagonal segregation—and demonstrates how the distance measures capture convergence dynamics, including finite-size effects and avalanche phenomena near $p\approx0.25$. It discusses the implications for coarse-grained theories, potential Lyapunov function formulations, and extensions to stochastic variants or open-boundary setups, highlighting the broader relevance of deterministic 2D CA traffic models for phase-transition and non-equilibrium dynamics.

Abstract

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.

Figures (4)

• Figure 1: (Color online) Illustration of a free-flowing periodic state of the BML traffic flow model, obtained from a computer simulation of the model starting from a random initial state, for a lattice of size $L=32$ and density $p=0.25$. In this snapshot the time $t$ is even, such that in the next time step it will be the turn of the H-cars to move. One can thus observe that the cells in front of all the H-cars are empty, while the cells in front of some of the V-cars are occupied by H-cars. In this configuration and in subsequent time steps, all the cars move freely when their turn arrives, with no obstructions.
• Figure 2: (Color online) Simulation results for the distance measure $D_{\parallel}(t)$, given by Eq. (\ref{['eq:D_parallel_n']}), as a function of the time $t/(2L)$ (in units of cycles) on a lattice of size $L=1,024$ (left column) and $L=4,096$ (right column) and densities (a,d) $p=0.05$; (b,e) $p=0.15$ and (c,f) $p=0.25$. It is found that starting from a random initial state, $D_{\parallel}(t)$ sharply decays (by more than an order of magnitude) within a few time steps and then continues to decay as time evolves. It eventually converges toward $D_{\parallel}(t) = 0$. The results for the two lattice sizes are very similar, indicating that the dependence on the system size is weak. Data points are shown for all the even time steps. Each data point represents an average over 100 random initial conditions. The noise in the tail is due to the fact that at these long times the signal is very small and the fluctuations become significant. The noise level tends to decrease as the lattice size is increased.
• Figure 3: (Color online) Simulation results (circles) for the distance measure $D_{\perp}(t)$, given by Eq. (\ref{['eq:D_perp_n']}), as a function of the time $t/(2L)$ (in units of cycles) for lattices of size $L=1,024$ (left column) and $L=4,096$ (right column) and densities (a,d) $p=0.05$; (b,e) $p=0.15$ and (c,f) $p=0.25$. Each data point represents an average over 100 random initial conditions. The results are well fitted by exponentially truncated power-law decay functions (dashed lines) of the form $D_{\perp}(t) \sim t^{-\gamma} \exp ( - t/\tau_{\perp} )$, with $\gamma = 1.1$ in all the cases and the relaxation times (in units of cycles) $\tau_{\perp}/(2L)$ take the values (a) 1.1; (b) 3.1; (c) 7.2; (d) 1.1; (e) 3.6 and (f) 8.0.
• Figure 4: The relaxation time $\tau_{\perp}/(2L)$ (in units of cycles) of the distance measure $D_{\perp}(t)$ as a function of the density $p$, for lattices of size $L=1,024$ ($\times$) and $L=4,096$ ($\circ$). In both cases $\tau_{\perp}$ increases as $p$ is increased and then starts to saturate at $p=0.25$ (and even slightly decreases for $L=4,096$). The results indicate that for $p > 0.125$ the relaxation time tends to increase as the system size is increased. For $L=1,024$ each data point represent an average over $1,000$ random initial conditions, while for $L=4,096$ each data point represents an average over $100$ random instances.