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.
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Figure 3.
Plot of the simulated solutions
Figure 7.
Simulation parameter
Figure 8.
Residence time vs. height of a centered rectangular obstacle with fixed width
Figure 9.
Residence time vs. height of a centered rectangular obstacle with fixed width
Figure 10.
Residence time vs. width of a centered rectangular obstacle with fixed height
Figure 11.
Residence time vs. position of the center of the obstacle. The obstacle is a square of side length
Figure 12.
As in the right panel in Figure 8. In the left panel the height of the obstacle is equal to
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