Traffic Model

In the previous discussion, we assumed that all drivers are identical for simplicity. However, it is impossible due to humanity. For instance, an experienced driver might stop smoother and slower than a novice driver under the same inter-vehicle distance. Therefore, let's modify the classic Bando's model to introduce individual differences in the perception of spatial distance. Reminding that the optimal velocity $v_{\text{opt}}(\Delta x)$ in Bando's model describes how drivers change their velocity when inter-vehicle distance changes. Thus, when we control the changing rate of optimal velocities for all drivers at a different rate, they behave differently even when the inter-vehicle distances are the same. That is, $$ v_{\text{opt}} = v_{\text{opt}}(w_i\cdot \Delta x_i), $$ where $\Delta x_i$ is the inter-vehicle distance between vehicle $i$ and $i+1$, and $w_i$ illustrates the behavioral property of $i$-th driver. Fig. 1 shows the profile of optimal velocity for different w. Apparantly, for a small value of w, tthe optimal velocity changes more smoothly, illustrating the behavior of an experienced driver. While for a large value of w,the optimal velocity changes more aggressively, similar to the behavior of the novice driver.

In our study, we assume the set of driver properties ${w_i}$ forms a Gaussian distribution since it is a common assumption to illustrate humanity, and such an assumption will simplify our calculations. Though the assumption has been applied, the physical interpretation of the final results implies that the introduction of individual differences will suppress the jamming transition, which is independent of the distribution style. Fig 2 shows one example of the Gaussian distribution in our simulations.

With these modifications, we can introduce the individual difference into our simulations by normalizing the mean value of the distributions to a unit $\bar{w}=1$. Since the properties of drivers have been a Gaussian distribution recently, one hundred realizations are carried out for all simulations with different standard deviations of the distribution $\sigma$. The final results shown in Fig. 3 represent phases when the standard deviations and reaction times of drivers are changed. To understand this figure, we can focus on a fixed standard deviation. As we discussed before, for drivers with a large reaction time, drivers have no time to react to the inter-vehicle change, and the system starts to jam together. Thus, the bottom region of the figure, which is shown in green squares, represents the jammed state of the system. In contrast, drivers with short reaction time can react quickly enough to the inter-vehicle change, and the system will remain in a steady flow. Therefore, the upper region of the figure, which is shown in blue triangles, represents the steady flow state. Now, if we fix the reaction time and increase the standard deviation, we can find that the system transforms from a jamming state into a steady flow state. Apparently, the introduction of individual differences suppresses the jamming transition.

The simplest way to uncover the physics of such individual difference-induced jamming suppression is by calculating the average inter-vehicle distance change when there are only two vehicles. The equilibrium state will occur under the condition $$ w_1 \Delta x_1 = w_2 \Delta x_2 \\ \Delta x_1 + \Delta x_2 = L \\ $$ Thus, $$ w_1 \Delta x_1 = \frac{w_1w_2}{w_1 + w_2}L. $$ By assuming a symmetric condition $w_1 = \bar{w}+\Delta w$ and $w_2 = \bar{w}-\Delta w$, we get $$ w_1 \Delta x_1 = \frac{\bar{w}^2 - \Delta^2}{2N\bar{w}}L < \frac{\bar{w}}{2N}. $$ This implies that the average inter-vehicle perspective distance is shorter than the condition when all drivers are the same. Since drivers drive less aggressively in the high-density region (the side in the hyperbolic tangent diagram near the origin), drivers have enough time to react to the inter-vehicle distance change. Hence, the jamming transition is suppressed when the individual difference is introduced to the system in a high-density region.

In this section, I will briefly show you how I derive analytical solutions for phase boundary. The linearized equation of the perturbation to the steady state in Fourier mode is $$ \ddot{\tilde{y}}_k + \frac{1}{\tau}\dot{\tilde{y}}_k = \frac{f_1}{\tau}\sum_l \tilde{y}_l \tilde{w}_{k-l}[\exp (i\alpha_l)-1]. $$ For the case of classic Bando's model, all drivers are identical, and the only non-zero Fourier coefficient of $\tilde{w}_i$ is $\tilde{w}_0$.herefore, all Fourier modes of perturbations are decoupled in this case. However, since drivers are different in our research, different Fourier modes of perturbations couple with each other, which turns the solution nontrivial. Since the distribution of drivers properties is a Gaussian distribution, the Fourier modes of the Gaussian random field have been proven as $$ \langle \tilde{w}_0 \rangle= \bar{w}, \quad \langle |\tilde{w}_{k\neq 0}|^2\rangle = \frac{\sigma^2}{N}. $$ Thus, by rewriting the summation into two different parts and using matrix form, the equation becomes $$ \frac{d^2}{dt^2}\tilde{\textbf{y}} + \frac{1}{\tau}\frac{d}{dt}\tilde{\textbf{y}} = \frac{f_1}{\tau}(H^0+H^1)\tilde{\textbf{y}}, $$ where $\tilde{\textbf{y}} = (\tilde{y}_{-N/2+1},\tilde{y}_{-N/2+2},\dots, \tilde{y}_{N/2})$, and $$ H^0_{kl} = \tilde{w}_0[\exp (i\alpha_l)-1]\delta_{kl} \\[1.5em] H^1_{kl} = \tilde{w}_{k-l}[\exp (i\alpha_l)-1](1-\delta_{kl}) $$ The first matrix $H^0$ is a diagonal matrix, which represents the conditions when all vehicles are identical. The second matrix $H^1$ is the off-diagonal matrix, which represents the influence of the introduction of individual drivers. Since for a Gaussian distribution, $\langle |\tilde{w}_{k\neq 0}|^2\rangle = \frac{\sigma^2}{N}$, which tends to infinitesimal value when the number of drivers tends to be infinity. Thus, we can view the second matrix as a perturbation of the first matrix. Because the first matrix is diagonal and the second matrix is off-diagonal, the first order correction $\langle \phi_1|H^1|\phi_1\rangle = 0$, where $\phi_1$ represent the eigenfunction of the first matrix with the largest eigenvalue. Thus, to obtain the influence of the individual difference, the second-order correction $\lambda^{(2)}_1$ must be calculated. That is $$ \lambda^{(2)}_1 = \sum_{k\neq 1}\frac{|\tilde{w}_{1-k}|^2(e^{i\alpha_k}-1)(e^{i\alpha_1}-1)}{\tilde{w}_0(e^{i\alpha_1}-e^{i\alpha_k})} \simeq - \frac{i\sigma^2}{\tilde{w}_0}\sin \alpha_1. $$ Finally, up to the second-order correction, the equation of motion of the perturbation in Fourier mode will turn out to be $$ \ddot{\tilde{y}}_k + \frac{1}{\tau}\dot{\tilde{y}}_k = \frac{f_1}{\tau}\left[ e^{i\alpha_k} -1 - i\frac{\sigma^2}{\tilde{w}_0^2}\sin \alpha_1 \right], $$ and the instability condition is modified accordingly $$ f_1 > \frac{1}{2\tau \tilde{w}_0(1-\sigma^2/\tilde{w}_0^2)^2\cos^2(\alpha_k/2)}. $$ For thermodynamic limit $N\gg 0$ with a fixed density, $\cos^2(\alpha_k/2)\simeq 1$, the phase boundary will be predicted when the left-hand side equals the right-hand side and is plotted as the solid black line in Fig. 3. Detail calculation can be found in my master thesis. See the links below.