Coexistence and crossover phenomena in a Fermi-like model of particles in counterflowing streams

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.