Analytic approach to the critical density in cellular automata for traffic flow

The jamming transition in the stochastic traffic cellular automaton of Nagel and Schreckenberg [J. Phys. I 2, 2221 (1992)] is examined. We argue that most features of the transition found in the deterministic limit do pot persist in the presence of noise, and suggest instead to define the transition to take place at that critical density rho(c) at which a large initial jam just fails to dissolve. We show that rho(c) = upsilon(J)l(upsilon(J) + upsilon(F)), where upsilon(F) is the velocity of noninteracting vehicles and upsilon(J) is the speed of the dissolution wave moving into the jam. An approximate analytic calculation of upsilon(J) in the framework of a simple renormalization scheme is presented, which explicitly displays the effect of the interaction between vehicles during the acceleration stage of the Nagel-Schreckenberg rules with maximum velocity upsilon(max) > 1. The analytic prediction is compared to numerical simulations. We find a remarkable correspondence between the analytic expression for upsilon(J) and a phase diagram obtained numerically by Lubeck et al. [S1063-651X(99)05907-3].