Statistics, distillation, and ordering emergence in a two-dimensional stochastic model of particles in counterflowing streams

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.

Mobile-to-clogging transition in a Fermi-like model of counterflowing particles

“…The patterns arising from counterflowing stream of particles can be studied considering a wide range of apparently very different systems such as pedestrian dynamics [5], and charged colloids motion [6,7], which suggest more similarities between the micro and macro systems that we can anticipate in this kind of modelling. For this reason the straight formation of lanes, distillation, originated from the complex emergent process of self-propelled and/or field-directed objects/particles have raised a lot of interesting questions in the context of statistical mechanics and the physics of stochastic process modelled by Monte Carlo (MC) simulations or Partial Differential Equations (PDE) [8,9].…”

Numerical study of condensation in a Fermi-like model of counterflowing particles via Gini coefficient

“…The formation of patterns is a general property of such systems and different models have been proposed to explain the emergence of structures such as lanes, clogging and jamming. [3,7,8].…”

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

“…( 1)-( 3) for species A as well as it can be done for species B, similarly to what was done in Ref. [5].…”

“…The statistical mechanics of self-driven diffusive systems has great importance in the context of nonequilibrium phenomena [1]. In particular, counterflowing streams of particles [2][3][4][5][6] can lead to interesting patterns, such as those presented by studies with charged colloids [7][8][9][10][11], for example. Exactly as pedestrians organize their motion to make that system flow [12,13], particles without a "natural intelligence," self-propelled or simply oriented by a field, sometimes can organize their motion by lanes (see, for example, Refs.…”

“…For example, a lattice model for oppositely driven binary particles with purely repulsive interactions was investigated on a square lattice presenting such an effect [22]. On the other hand, two-dimensional directed random walks in counterflow can also present such phenomena [5].…”