ABSTRACT
In this work, we propose a two-dimensional extension of a previously defined one-dimensional version of a model of particles in counterflowing streams, which considers an adapted Fermi-Dirac distribution to describe the transition probabilities. In this modified and extended version of the model, we consider that only particles of different species can interact, and they hop through the cells of a two-dimensional rectangular lattice with probabilities taking into account diffusive and scattering aspects. We show that for a sufficiently low level of randomness (α≥10), the system can relax to a mobile self-organized steady state of counterflow (lane formation) or to an immobile state (clogging) if the system has an average density near a certain crossover value (ρ_{c}). We also show that in the case of imbalance between the species, we can simultaneously have three different situations for the same density value set: (i) an immobile phase, (ii) a mobile pattern organized by lanes, and (iii) a profile with mobility but without lane formation, which essentially is the coexistence of situations (i) and (ii). All of our results were obtained by performing Monte Carlo simulations.
ABSTRACT
In this paper, we propose a stochastic model which describes two species of particles moving in counterflow. The model generalizes the theoretical framework that describes the transport in random systems by taking into account two different scenarios: particles can work as mobile obstacles, whereas particles of one species move in the opposite direction to the particles of the other species, or particles of a given species work as fixed obstacles remaining in their places during the time evolution. We conduct a detailed study about the statistics concerning the crossing time of particles, as well as the effects of the lateral transitions on the time required to the system reaches a state of complete geographic separation of species. The spatial effects of jamming are also studied by looking into the deformation of the concentration of particles in the two-dimensional corridor. Finally, we observe in our study the formation of patterns of lanes which reach the steady state regardless of the initial conditions used for the evolution. A similar result is also observed in real experiments involving charged colloids motion and simulations of pedestrian dynamics based on Langevin equations, when periodic boundary conditions are considered (particles counterflow in a ring symmetry). The results obtained through Monte Carlo simulations and numerical integrations are in good agreement with each other. However, differently from previous studies, the dynamics considered in this work is not Newton-based, and therefore, even artificial situations of self-propelled objects should be studied in this first-principles modeling.
