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# Linear Boltzmann dynamics in a strip with large reflective obstacles: Stationary state and residence time

• * Corresponding author: Alessandro Ciallella
• The presence of obstacles modifies the way in which particles diffuse. In cells, for instance, it is observed that, due to the presence of macromolecules playing the role of obstacles, the mean-square displacement of biomolecules scales as a power law with exponent smaller than one. On the other hand, different situations in grain and pedestrian dynamics in which the presence of an obstacle accelerates the dynamics are known. We focus on the time, called the residence time, needed by particles to cross a strip assuming that the dynamics inside the strip follows the linear Boltzmann dynamics. We find that the residence time is not monotonic with respect to the size and the location of the obstacles, since the obstacle can force those particles that eventually cross the strip to spend a smaller time in the strip itself. We focus on the case of a rectangular strip with two open sides and two reflective sides and we consider reflective obstacles into the strip. We prove that the stationary state of the linear Boltzmann dynamics, in the diffusive regime, converges to the solution of the Laplace equation with Dirichlet boundary conditions on the open sides and homogeneous Neumann boundary conditions on the other sides and on the obstacle boundaries.

Mathematics Subject Classification: Primary: 82B21, 76P05, 82B40; Secondary: 60J60.

 Citation: • • Figure 1.  Domain $\Omega$: strip with large fixed obstacles, where $\partial\Omega_L$ and $\partial\Omega_R$ are the vertical open boundaries and $\partial\Omega_E$ are reflective boundaries.

Figure 2.  Elastic collision with a scatterers: impact parameter $\delta$ and angle of incidence $\alpha$.

Figure 3.  Plot of the simulated solutions $h_{t_m}$ in a $3D$ plot and in a $2D$ plot constructed by averaging on the $x_2$ variable: in dark gray $t_m = 2\cdot 10^{-1}$, in gray $t_m = 10^{-1}$, in light gray $t_m = 2\cdot10^{-2}$. In black (grid and dashed line) the analytic solution $\rho$ of the associated Laplace problem.

Figure 4.  Simulation parameter $t_m = 10^{-2}$: relative error ${|h_{t_m}-\rho|}/{\rho}$.

Figure 5.  Simulation parameter $t_m = 10^{-2}$: on the left in gray the numerical solution $h_{t_m}$ and in black the solution $\rho$ of the associated Laplace problem; on the right the relative error ${|h_{t_m}-\rho|}/{\rho}$. Into the strip there is a square obstacle with side $8\cdot10^{-1}$.

Figure 6.  Simulation parameter $t_m = 10^{-2}$: on the left in gray the numerical solution $h_{t_m}$ and in black the solution $\rho$ of the associated Laplace problem; on the right the relative error ${|h_{t_m}-\rho|}/{\rho}$. In the strip is placed a very thin obstacle with height of $0.8$.

Figure 7.  Simulation parameter $t_m = 10^{-2}$: on the left in gray the numerical solution $h_{t_m}$ and in black the solution $\rho$ of the associated Laplace problem; on the right the relative error ${|h_{t_m}-\rho|}/{\rho}$. In the first line we show the case of two squared obstacles with side $6\cdot 10^{-1}$, in the second one a couple of rectangular obstacles, taller and thinner than the squares.

Figure 8.  Residence time vs. height of a centered rectangular obstacle with fixed width $4\cdot 10^{-2}$ (on the left) and $4 \cdot 10^{-1}$ (on the right). Simulation parameters: $L_1 = 4$, $L_2 = 1$, $t_m = 2\cdot 10^{-2}$, total number of inserted particles $10^8$, the total number of particles exiting through the right boundary varies from $5.3\cdot 10^{5}$ to $3.6\cdot 10^{5}$ (on the left) and from $5.3\cdot 10^{5}$ to $2.1\cdot 10^{5}$ (on the right) depending on the obstacle height. The solid lines represent the value of the residence time measured for the empty strip (no obstacle).

Figure 9.  Residence time vs. height of a centered rectangular obstacle with fixed width $8\cdot 10^{-1}$ (on the left) and $12 \cdot 10^{-1}$ (on the right). Simulation parameters: $L_1 = 4$, $L_2 = 1$, $t_m = 2\cdot 10^{-2}$, total number of inserted particles $10^8$, the total number of particles exiting through the right boundary varies from $5.2\cdot 10^{5}$ to $1.4\cdot 10^{5}$ (on the left) and from $5.2 \cdot 10^{5}$ to $1.1 \cdot 10^{5}$ (on the right) depending on the obstacle height. The solid lines represent the value of the residence time measured for the empty strip (no obstacle).

Figure 10.  Residence time vs. width of a centered rectangular obstacle with fixed height $0.8$ (on the left) and vs. the side length of a centered squared obstacle (on the right). Simulation parameters: $L_1 = 4$, $L_2 = 1$, $t_m = 2\cdot 10^{-2}$, total number of inserted particles $10^{8}$, the total number of particles exiting through the right boundary varies from $4.2 \cdot 10^{5}$ to $1.1\cdot 10^{5}$ (on the left) and from $5.3\cdot 10^5$ to $1.3\cdot 10^5$ (on the right) depending on the obstacle width. The solid lines represent the value of the residence time measured for the empty strip (no obstacle).

Figure 11.  Residence time vs. position of the center of the obstacle. The obstacle is a square of side length $0.8$ on the left and a rectangle of side lengths $0.04$ and $0.8$ on the right. Simulation parameters: $L_1 = 4$, $L_2 = 1$, $t_m = 2\cdot 10^{-2}$, total number of inserted particles $10^8$, the total number of particles exiting through the right boundary is stable at the order of $2.6\cdot 10^5$ (on the left) and of $4\cdot 10^5$ (on the right) not depending on the obstacle position. The solid lines represent the value of the residence time measured for the empty strip (no obstacle).

Figure 12.  As in the right panel in Figure 8. In the left panel the height of the obstacle is equal to $0.8$. Left panel: the mean time spent by particles crossing the strip in each point of the strip ($0.02\times0.02$ cells have been considered) for the empty strip case (black) and in presence of the obstacle (gray). Right panel: residence time in regions L (circles), C (squares), and R (triangles) in presence of the obstacle (gray) and for the empty strip case (black).

Figure 13.  As in Figure 12 for the geometry in the left panel in Figure 10. In the left panel the width of the obstacle is $2.28$.

Figure 14.  As in Figure 12 for the geometry in the left panel in Figure 11. In the left panel the position of the center of the obstacle is $0.8$.

Figure 15.  Domain $\Lambda$: infinite strip with big fixed obstacles: the whole boundaries of $\Lambda$ is a specular reflective boundary.

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