Unraveling the puzzling intermediate states in the Biham-Middleton-Levine traffic model

The Biham-Middleton-Levine (BML) traffic model, a cellular automaton with eastbound and northbound cars moving by turns on a square lattice, has been an underpinning model in the study of collective behavior by cars, pedestrians, and even internet packages. Contrary to initial beliefs that the model exhibits a sharp phase transition from freely flowing to fully jammed, it has been reported that it shows intermediate stable phases, where jams and freely flowing traffic coexist, but there is no clear understanding of their origin. Here, we analyze the model as an anisotropic system with a preferred fluid direction (northeast) and find that it exhibits two differentiated phase transitions: the system is either longer in the flow direction (longitudinal) or perpendicular to it (transversal). The critical densities where these transitions occur enclose the density interval of intermediate states and can be approximated by mean-field analysis, all derived from the anisotropic exponent relating the longitudinal and transversal correlation lengths. Thus, we arrive at the interesting result that the puzzling intermediate states in the original model are just a superposition of these two different behaviors of the phase transition, solving by the way most mysteries behind the BML model, which turns out to be a paradigmatic example of such anisotropic critical systems.

Figure 1

(a) Average velocity $\langle v\rangle$ vs density $\rho$ (circles) for a BML model on a ${45}^{\circ }$-rotated $256\times 256$ lattice [as depicted in (b), with eastbound cars in yellow (light gray) and northbound cars in black]. The mean-field approach for longitudinal systems (dashed red line) and the numerical prediction for an infinite transverse system (blue solid line) are also included for comparison. Typical configurations for (c) free flow, (d) intermediate states, (e) one global jam, and (f) random jams are also included.

Figure 2

Longitudinal ${\xi }_{\parallel }$ and transversal ${\xi }_{\perp }$ correlation lengths from final configurations at densities $\rho$ in the range [0.49–0.54] for square lattices of different sizes. The power-law fit gives $\theta =1.64\left(3\right)$ for the anisotropy exponent.

Figure 3

Finite-size scaling analysis for the phase transition of both longitudinal and transversal systems. (a) Transition curves for several longitudinal systems, including the mean-field prediction (magenta solid line) and a typical configuration for the jammed phase (inset). (b) Transition curves for several transversal systems [sizes as in (a), but with $L_{\parallel }$ and $L_{\perp }$ interchanged], including the asymptotic limit (magenta solid line) obtained from the FSS analysis. (c) Scaling of the transition width $\mathrm{\Delta }\left(L_{\parallel }\right)$ for both cases, giving ${\nu }_{\parallel }=3.0\left(3\right)$ and ${\nu }_{\perp }=1.8\left(2\right)$. (d) Scaling of the finite critical density gives ${\rho }_{c_{\parallel }}=0.521\left(2\right)$ and ${\rho }_{c_{\perp }}=0.283\left(2\right)$. All systems were simulated for at least $1.5\times {10}^6$ time steps or until convergence $(v=0$ or $v=1)$.

Figure 4

Time evolution of (a) the fraction of stopped cars $p_{\mathrm{stop}}$ and (b) of the fraction of crossed $\left(c_{\to \uparrow }\right)$ and queued $\left(c_{\to \to }\right)$ stopped cars in a $1438\times 82$ longitudinal system. (c) The value of $p_{\mathrm{stop}}$ after one time step for several densities is also shown. (d) Schematic representation of the jamming process for transversal systems.