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

We study the effect of reaction times on the kinetics of relaxation to stationary states and on congestion transitions in heterogeneous traffic using simulations of Newell's model on a ring. Heterogeneity is modeled as quenched disorders in the parameters of Newell's model and in the reaction time 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 of the platoon. 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 starts interacting 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 point is approached from above than 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, i.e., when the vehicles react instantaneously to a perturbation ahead (e.g., automated driving). Overall, we conclude that the nonzero reaction times of drivers in heterogeneous traffic significantly change the behavior of the free flow to congestion transition while it doesn't alter the kinetics of relaxation to stationary state.

Figure 1

A schematic of the speed $V\left(s\right)$ versus gap $s$ profile of a driver as per Newell's model. A typical profile is shown by black line. Below jam gap $S_{\mathrm{j}}$, the vehicle is at rest. If the gap $s$ is between $S_{\mathrm{j}}$ and $S_{\mathrm{c}}$, the speed increases at a rate of $w_{\mathrm{b}}{S}_{\mathrm{j}}^{-1}$ and if the gap $s$ is above $S_{\mathrm{c}}$, the speed is maximum and constant at $v_{\mathrm{f}}$. The filled blue area encompasses various possible driver profiles in the system as all the parameters of the model are random in our present study.

Figure 2

Top: Position ($x$) versus time ($t$) plot illustrating the formation of a stop-go wave. The vehicles move from left to right as depicted in the cartoon. The curves corresponds to the cars with matching colors. Bottom: Speed ($v$) versus time ($t$) plot for the vehicles in the top figure. Dotted arrows are guides to the eye. A small trough in the speed of vehicle 1 induces a larger trough in vehicle 2 and so on. As this reaches vehicle 7, the perturbation develops into a stop-go wave. The damped oscillations occur as the vehicle equalizes its speed with that of its leader.

Figure 3

A plot of typical positions of the $N$ vehicles versus time after the stationary state is reached when reaction time is included. Each line corresponds to a vehicle. The vehicles at the front of the platoon move smoothly without any stop-go waves. Some small oscillations generated in the vehicles at the front develop into stop-go waves as they move upstream. The stop-go waves (dark patterns moving upstream in the figure) get dissipated at the end of the platoon.

Figure 4

Gap size as a function of time. Scaled gap length versus scaled time is shown in the inset. Collapse of curves upon the scaling implies $L_{\mathrm{g}}\sim t^{2/3}$.

Figure 5

Average speed as a function of time for various track lengths. Scaled average relative speed is shown in the inset. Collapse of curves with scaling implies that $\langle \overline{v}\rangle \sim t^{-1/3}$.

Figure 6

Stationary state probability of gap $p\left(s\right)$; one part of it is the gap distribution behind the leader of the platoon $p_{\mathrm{p}}$ (the left curve) and the other part is the gap distribution ahead of the leader $p_{\mathrm{g}}$ (the right curve).

Figure 7

Positions of the $N$ vehicles versus time on a 10-km track. For this case, ${\rho }_{\mathrm{c}}$ lies between 49 and 50 veh/km. Top plot is for $\rho =49$ veh/km, which is below ${\rho }_{\mathrm{c}}$. Bottom plot is for $\rho =50$ veh/km, which is above ${\rho }_{\mathrm{c}}$.

Figure 8

Flow versus average density when reaction time is included

Figure 9

Maximum gap versus density when the reaction time is included. Inset: Plot depicting the scaling $\langle s^{max}\rangle \sim (\overline{\rho }{\rho }_{\mathrm{c}}^{-1}-1{)}^{-\gamma }$ with $\gamma =3/4$.

Figure 10

Variance of the gap distribution as function of density. Inset: Plot depicting the scaling ${\mathrm{\Delta }}^2\sim {(\overline{\rho }{\rho }_{\mathrm{c}}^{-1}-1)}^{-\eta }$ with $\eta =1$.

Figure 11

Scaled gap distributions near ${\rho }_{\mathrm{c}}$ for various $\{L,N\}$ pairs.

Figure 12

Gap ahead of the follower ($\varphi$) as a function of time for a typical case calculated using Eq. (A13). For small $\tau$, the relaxation to stationary gap is monotonic. As $\tau$ approaches $S_{\mathrm{j},\varphi }{w}_{\mathrm{b},\varphi }^{-1}$, $s_{\varphi }\left(t\right)$ becomes nonmonotone.