Residence time of symmetric random walkers in a strip with large reflective obstacles

We study the effect of a large obstacle on the so called residence time, i.e., the time that a particle performing a symmetric random walk in a rectangular (2D) domain needs to cross the strip. We observe a complex behavior, that is we find out that 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 explain the residence time behavior by developing a 1D analog of the 2D model where the role of the obstacle is played by two defect sites having a smaller probability to be crossed with respect to all the other regular sites. The 1D and 2D models behave similarly, but in the 1D case we are able to compute exactly the residence time finding a perfect match with the Monte Carlo simulations

Numerical evidence for the approximation of dissipative systems by gyroscopically coupled oscillator chains

Inspired by the equations of motion of the gyroscope, where a Lagrangian and conservative system may appear to mimic a dissipative one when focusing on a single degree of freedom for finite time intervals, we introduce a gyroscopic-type coupling between harmonic oscillators. The aim is to propose a tentative scheme of solution for the problem of finding a higher-dimensional Lagrangian system approximating a lower-dimensional dissipative one. Specifically, we consider a certain family of Lagrangian systems, for which the time evolution of the first Lagrangian parameter is conjectured to be a good approximation for the evolution of a one-dimensional linear dissipative system in finite time intervals, up to a fixed precision. The behavior of the selected family of gyroscopic couplings is compared with a given dissipative system, properly optimizing a family of parameters according to a described scheme. Numerical calculations are reported, suggesting the validity of the proposed conjecture

On the linear Boltzmann transport equation: a Monte Carlo algorithm for stationary solutions and residence times in presence of obstacles.

We consider the linear Boltzmann transport equation (LBTE) in a 2D strip. We present a Monte Carlo algorithm to directly simulate the motion of single particles following the dynamics prescribed by this equation. We construct the stationary solution of the LBTE in presence of large reflective obstacles in the strip. We compare the simulation steady state with that one obtained as stationary profile of the diffusion equation with mixed boundary conditions on the strip. We present the results of some numerical tests and we show the closeness of the two stationary solutions in the diffusive limit and the efficacy of our algorithm. We introduce the concept of residence time of particles, that is the typical time spent by particles to cross the strip. We show by a numerical simulation that in presence of obstacles the residence time is not monotonic with respect to the obstacles sizes

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

Conditional expectation of the duration of the classical gambler problem with defects

The effect of space inhomogeneities on a diffusing particle is studied in the framework of the 1D random walk. The typical time needed by a particle to cross a one-dimensional finite lane, the so-called residence time, is computed possibly in presence of a drift. A local inhomogeneity is introduced as a single defect site with jumping probabilities differing from those at all the other regular sites of the system. We find complex behaviors in the sense that the residence time is not monotonic as a function of some parameters of the model, such as the position of the defect site. In particular we show that introducing at suitable positions a defect opposing to the motion of the particles decreases the residence time, i.e., favors the flow of faster particles. The problem we study in this paper is strictly connected to the classical gambler’s ruin problem, indeed, it can be thought as that problem in which the rules of the game are changed when the gambler’s fortune reaches a particular a priori fixed value. The problem is approached both numerically, via Monte Carlo simulations, and analytically with two different techniques yielding different representations of the exact result

Propagation of chaos for a stochastic particle system modelling epidemics

International audienc

Conditional expectation of the duration of the classical gambler problem with defects

The effect of space inhomogeneities on a diffusing particle is studied in the framework of the 1D random walk. The typical time needed by a particle to cross a one-dimensional finite lane, the so-called residence time, is computed possibly in presence of a drift. A local inhomogeneity is introduced as a single defect site with jumping probabilities differing from those at all the other regular sites of the system. We find complex behaviors in the sense that the residence time is not monotonic as a function of some parameters of the model, such as the position of the defect site. In particular we show that introducing at suitable positions a defect opposing to the motion of the particles decreases the residence time, i.e., favors the flow of faster particles. The problem we study in this paper is strictly connected to the classical gambler’s ruin problem, indeed, it can be thought as that problem in which the rules of the game are changed when the gambler’s fortune reaches a particular a priori fixed value. The problem is approached both numerically, via Monte Carlo simulations, and analytically with two different techniques yielding different representations of the exact result

Inhomogeneities in Boltzmann-SIR models

International audienc

Nonlinear waves in pantographic beams induced by transverse impulses

We present and discuss the results of some numerical simulations dealing with wave motion in one-dimensional pantographic media, also known as pantographic beams. Specifically, the analysis is carried out in large displacements regime and pantographic beams are modeled by using a completely discrete approach. Nonlinear vibrations induced by a transversal impulse are studied. A simple yet efficient time-stepping scheme was used to reconstruct the motion and, consequently, to explore the propagation of strain waves in the pantographic medium. The numerical study concludes that the main features of the observed strain wave propagation are extremely useful toward the understanding of the dynamic behavior of the pantographic micro-structure and its continuum counterparts