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Kinetic description of collision avoidance in pedestrian crowds by sidestepping

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  • In this paper we study a kinetic model for pedestrians, who are assumed to adapt their motion towards a desired direction while avoiding collisions with others by stepping aside. These minimal microscopic interaction rules lead to complex emergent macroscopic phenomena, such as velocity alignment in unidirectional flows and lane or stripe formation in bidirectional flows. We start by discussing collision avoidance mechanisms at the microscopic scale, then we study the corresponding Boltzmann-type kinetic description and its hydrodynamic mean-field approximation in the grazing collision limit. In the spatially homogeneous case we prove directional alignment under specific conditions on the sidestepping rules for both the collisional and the mean-field model. In the spatially inhomogeneous case we illustrate, by means of various numerical experiments, the rich dynamics that the proposed model is able to reproduce.

    Mathematics Subject Classification: Primary: 35Q20, 35Q70; Secondary: 90B20, 91F99.


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  • Figure 1.  Calculation of the time to collision.

    Figure 2.  The state variables describing pedestrians moving with constant speed with $\alpha_d = 0$.

    Figure 3.  Comparison between (5) (blue curves computed with $\gamma = 0.8$, $\tau = 1$) and (6) (red curves). Left: $\theta_j = \pi$; right: $\theta_j = -\frac{\pi}{4}$ (see the text for details).

    Figure 4.  The set of pairs $(\rho,\,|\alpha_c|)\in [0,\,1]\times [0,\,\pi]$ (shaded area) for which Theorem 4.1 guarantees alignment in the case study of Section 4.2.

    Figure 5.  Alignment to the desired direction $\alpha_d = 0$ for the case study of Section 4.2 with $\rho = \frac{1}{2}$, $\alpha_c = \frac{\pi}{5}$. Left: $t = 10^{-2}$, right: $t = 10$.

    Figure 6.  The mapping $t\mapsto\bar{\theta}(t)$ computed numerically (blue solid curve) and estimated theoretically by (19) (red dashed curve). Left: $(\rho,\,\alpha_c) = \left(\frac{1}{2},\,\frac{\pi}{5}\right)$; right: $(\rho,\,\alpha_c) = \left(\frac{1}{3},\,\frac{3}{5}\pi\right)$.

    Figure 7.  The mapping $t\mapsto{{\mathit{\bar{\theta }}}_{\rm{MC}}}(t)$ for: (left) $\rho = \frac{1}{2}$ and different choices of $\alpha_c\in (0,\,\pi]$; (centre) $\alpha_c = \frac{\pi}{5}$ and different choices of $\rho\in (0,\,1]$; (right) $\rho = 1$, $\alpha_c = \frac{2}{5}\pi$ and different values of $\bar{\theta}_0$ obtained by varying the parameter $\hat{\theta}$ in (22). Dashed curves indicate the cases in which alignment fails.

    Figure 8.  Conditions (14) (blue) with $\bar{\theta}_0 = \frac{\pi}{2}$ and (31) (red) for two different choices of $a(\rho)$. Purple areas correspond to the pairs $(\rho,\,|\alpha_c|)$ satisfying both conditions. The values $\rho_1$, $\rho_2$ in the right graph are $\rho_{1,2} = \frac{21\mp\sqrt{105}}{42}$, i.e. $\rho_1\approx 0.256$, $\rho_2\approx 0.744$.

    Figure 9.  Top left: snapshots in time of the solution to (27) with initial condition (20). Top right: the same solution plotted in the $\theta t$-plane. Bottom: snapshots in time of the solution to (27) with initial condition (22) for two different choices of $\hat{\theta}$, $\sigma^2$.

    Figure 10.  The mapping $t\mapsto{{\mathit{\bar{\theta }}}_{\rm{FEM}}}(t)$ for: (left) $\rho = \frac{1}{2}$ and different choices of $\alpha_c\in (0,\,\pi]$; (right) $\alpha_c = \frac{\pi}{5}$ and different choices of $\rho\in (0,\,1]$. Alignment is always observed.

    Figure 11.  Test 1 -Alignment in the spatially inhomogeneous case and time trend of ${{\mathit{\bar{\theta }}}_{\rm{MC}}}$ over the whole space domain $\Omega$.

    Figure 12.  Test 2 -Lane formation in pedestrian counterflow. For pictorial purposes, in order to represent the two crowds in the same picture, the blue density is plotted on a negative scale. In white areas where the velocity is non-zero the two densities take nearly the same values.

    Figure 13.  Test 3 -Crossing flows and stripe formation.

    Table 1.  Parameters used in the numerical simulations of Section 5.

    $L$ $\alpha_d$ $\alpha_d^1$ $\alpha_d^2$ $\alpha_c$ $\tau$ $\gamma$ $\eta$
    Test 1 10 $\pi$ - - $\frac{\pi}{4}$ 1 $\frac{1}{2}$ -
    Test 2 10 - 0 $-\pi$ $\frac{\pi}{4}$ 1 $\frac{1}{2}$ $\frac{1}{5}$
    Test 3 10 - 0 $\frac{\pi}{2}$ $\frac{\pi}{4}$ 10 $\frac{2}{5}$ $\frac{1}{5}$
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