Particle-based modeling of dynamics in presence of obstacles

The aim of this dissertation is to study the effects of the presence of obstacles in the dynamics of some particle-based models. The interest is due to the existence of non-trivial phenomena observed in systems modeling different contexts, from biological scenarios to pedestrian dynamics. Indeed, obstacles can interfere with the motion of particles producing opposite effects, both “slowing down” and “speeding up” the dynamics, depending on the model and on some features of the obstacles, such as position, size, shape. In many different contexts it is required to pay great attention to this matter. In cells, for instance, it is possible to observed anomalous diffusion: 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. Different situations in grain and pedestrian dynamics are known where the presence of an obstacle favour the egress of the agents of the system. In the first part of this work we introduce the linear Boltzmann equation in our domain: a 2D rectangular strip with two open sides and two reflective sides with large reflective obstacles. We state our results concerning the characterization of the stationary solution. We construct a Monte Carlo algorithm to simulate the dynamics and we use it to construct numerically the stationary states and to evaluate the residence time in presence of obstacles, i.e. the time needed by particles to cross the strip. Finally, we prove the results on the stationary solution of the equation. 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 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. In the second part we consider a 2D finite rectangular lattice where a rectangular fixed obstacle is present. We consider particles performing a simple symmetric random walk and we study numerically the residence time behavior. The residence time does not depend monotonically on the geometric properties of the obstacle, such as its width, length, and position. In some cases, due to the presence of the obstacle, the mean residence time is shorter with respect to the one measured for the obstacle-free strip. We give a complete interpretation of the results on the residence times, even by constructing the reduced 1D picture of the lattice: a simple symmetric random walk on an interval with two singular sites mimicking the 2D case. We calculate the residence time for this 1D model via Monte Carlo simulations, finding good correspondence between the results of the 1D and the 2D model. Finally, we produce analytical calculation of the residence time for the 1D model. In the third part we introduce a probabilistic cellular automaton model to study the motion of pedestrians: a lattice model without exclusion based on a variation of the simple symmetric random walk on the square lattice. The model, despite the basic rules of its dynamics, captures some interesting features of the difficulty in obtaining an optimal strategy of evacuation in a very difficult situation (no visibility, no external lead to the exits). We introduce an interpersonal attraction parameter for the evacuation of confined and darkened spaces: a "buddying" threshold (of no-exclusion per site) mimicking the tendency of pedestrians to form groups (herding effect) and to cooperate. We examine how the dynamics of the crowd is influenced by the tendency to form big groups. The effect of group sizes on outgoing fluxes, evacuation times and wall effects is carefully studied with a Monte Carlo framework accounting also for the presence of an internal obstacle. A strong asymmetry emerges in the effect of the same obstacle placed in different position, opening the possibility to optimize the evacuation by adding a suitable barrier positioned appropriately. The last part of this thesis is devoted to the Lorentz model. Strategies to study the convergence of the particle density in the dynamics of the Lorentz model to the solution of the linear Boltzmann equation in a suitable low-density limit are discussed in the geometry of the rectangular strip with two open side and two reflective side in presence of obstacles.