Self-organization, ergodicity breaking, phase transition and synchronization in two-dimensional traffic-flow model

Analytical investigation is made on the two-dimensional traffic-flow model with alternative movement and exclude-volume effect between right and up arrows [Phys. Rev. {\bf A} 46 R6124 (1992)]. Several exact results are obtained, including the upper critical density above which there are only jamming configurations, and the lower critical density below which there are only moving configurations. The observed jamming transition takes place at another critical density $p_{c}(N)$, which is in the intermidiate region between the lower and upper critical densities. It is derived that $p_{c}(N)\,=\,CN^{\alpha}$, where $C$ and $\alpha$ are determined to be respectively $0.76$ and $-0.14$ from previous numerical simulation. This transition is suggested to be a second-order phase transition, the order parameter is found. The nature of self-organization, ergodicity breaking and synchronization are discussed, Comparison with the sandpile model is made.