Renormalization-group study of the Nagel-Schreckenberg model

We study the phase transition from free flow to congested phases in the Nagel-Schreckenberg (NS) model by using the dynamically driven renormalization group (DDRG). The breaking probability p that governs the driving strategy is investigated. For the deterministic case p = 0, the dynamics remain invariant in each renormalization-group (RG) transformation. Two fully attractive fixed points, rho(c)* = 0 and 1, and one unstable fixed point, rho(c)* = 1/(nu(max) + 1), are obtained. The critical exponent. which is related to the correlation length is calculated for various v(max). The critical exponent appears to decrease weakly with v(max) from nu = 1.62 to the asymptotical value of 1.00. For the random case p > 0, the transition rules in the coarse-grained scale are found to be different from the NS specification. To have a qualitative understanding of the effect of stochasticity, the case p -> 0 is studied with simulation, and the RG flow in the rho - p plane is obtained. The fixed points p = 0 and 1 that govern the driving strategy of the NS model are found. A short discussion on the extension of the DDRG method to the NS model with the open-boundary condition is outlined.