Statistical mechanical approach to cellular automaton models of highway traffic flow

Cellular automaton models for traffic flow problems in one dimension are considered. Starting with a microscopic relation for the updating rule describing the occupancy on each site of the road, a macroscopic time-evolution relation is obtained for the average speed of cars by carrying out statistical averages. Mean field equations are obtained as the asymptotic limit 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 can be regarded as a nonlinear mapping between the average speeds at two consecutive time steps. The mean field results are, thus, obtained by studying the attractors of the mapping. The approach is applied to the model recently proposed by Fukui and Ishibashi (FI) for one dimensional traffic flow. The calculation leads to the evaluation of spatial correlation functions involving a string of sites on the road. Our calculations show that for FI models in which the maximum speed of each car is M, a decoupling scheme retaining correlations upto M+1 sites can be applied to the calculation of spatial correlations involving more than M+1 sites. Details are given for the cases of M = 1 and M = 2 without and with random delay. Results are reported for the case of M = 3 without random delay. 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.