Statistical mechanical approach to Fukui-Ishibashi traffic flow models

We consider cellular automaton models for one-dimensional traffic flow problems. Starting with a microscopic relation for the updating rule describing the occupancy on each site of the road, a macroscopic evolution relation for the average speed of cars can be obtained by carrying out statistical averages. Mean field equations are obtained by considering the asymptotic form of the evolution relation. This gives the average car speed in the long time limit as a function of the car density. The evolution relation is a nonlinear mapping between the average speeds at two consecutive time steps. The mean field results can be obtained by studying the attractors of the mapping. The approach is applied to study the model recently proposed by Fukui and Ishibashi. Our calculations show that for models in which the maximum speed of each car is M, a decoupling scheme retaining correlations up to M+1 sites can be applied to the calculation of spatial correlations involving more than M+1 sites. Exact results are obtained using our approach for models without random delay. For models with random delay, results are in good agreement with simulation results.