Power laws and phase transitions in heterogenous car following with reaction times

Abstract

We study the effect of reaction times on the kinetics of relaxation to stationary states and on congestion transitions in heterogeneous traffic. Heterogeneity is modeled as quenched disorders in the parameters of the car following model and in the reaction times of the drivers. We observed that at low densities, the relaxation to stationary state from a homogeneous initial state is governed by the same power laws as derived by E. Ben-Naim et al., Kinetics of clustering in traffic flow, Phys. Rev. E 50, 822 (1994). The stationary state, at low densities, is a single giant platoon of vehicles with the slowest vehicle being the leader. We observed formation of spontaneous jams inside the giant platoon which move upstream as stop-go waves and dissipate at its tail. The transition happens when the head of the giant platoon interacts with its tail, stable stop-go waves form, which circulate in the ring without dissipating. We observed that the system behaves differently when the transition density is approached from above that it does when approached from below. When the transition density is approached from below, the gap distribution behind the leader has a double peak and is fat-tailed but has a bounded support and thus the maximum gap in the system and the variance of the gap distribution tend to size-independent values. When the transition density is approached from above, the gap distribution becomes a power law and, consequently, the maximum gap in the system and the variance in the gaps diverge as a power law, thereby creating a discontinuity at the transition. Thus, we observe a phase transition of unusual kind in which both a discontinuity and a power law are observed at the transition density. These unusual features vanish in the absence of reaction time (e.g., automated driving).