Narrow Results

Pedestrian queues like those observed at ticket counters or supermarket checkouts are usually described by classical queueing theory. However, models like the M/M/1 queue neglect the internal structure (density profile) of the queue by focussing on the system length as the only dynamical variable. This is different in the Exclusive Queueing Process (EQP) in which the queue is considered on a microscopic level. It is equivalent to a Totally Asymmetric Exclusion Process (TASEP) of varying length. The EQP has a surprisingly rich phase diagram with respect to the arrival probability α and the service probability β. The behavior on the phase transition line is much more complex than for the TASEP with a fixed system length. It is nonuniversal and depends strongly on the update procedure used. In this paper, we review the main properties of the EQP and its applications to pedestrian dynamics, vehicular traffic and biological systems. We also mention extensions of the EQP and some related models.

In this document, we present the Special Issue's projects; these include reviews and articles about mathematical solutions and formulations of single-file dynamics (SFD), yet also its computational modeling, experimental evidence, and value in explaining real life occurrences. In particular, we introduce projects focusing on electron dynamics on liquid helium in channels with changing width, on the zig-zag configuration in files with longitudinal movement, on expanding files, on both heterogeneous and slow files, on files with external forces, and on the importance of the interaction potential shape on the particle dynamics along the file. Applications of SFD are of intrinsic value in life sciences, biophysics, physics, and materials science, since they can explain a large diversity of many-body systems, e.g., biological channels, biological motors, membranes, crowding, electron motion in proteins, etc. These systems are explained in all the projects that participate in this topical issue. This Special Issue can therefore intrigue, inspire and advance scientifically young people, yet also those scientists that actively work in this field.

In this paper, we introduce and study one-dimensional models for the behavior of pedestrians in a narrow street or corridor. We begin at the microscopic level by formulating a stochastic cellular automata model with explicit rules for pedestrians moving in two opposite directions. Coarse-grained mesoscopic and macroscopic analogs are derived leading to the coupled system of PDEs for the density of the pedestrian traffic. The obtained first-order system of conservation laws is only conditionally hyperbolic. We also derive higher-order nonlinear diffusive corrections resulting in a parabolic macroscopic PDE model. Numerical experiments comparing and contrasting the behavior of the microscopic stochastic model and the resulting coarse-grained PDEs for various parameter settings and initial conditions are performed. These numerical experiments demonstrate that the nonlinear diffusion is essential for reproducing the behavior of the stochastic system in the nonhyperbolic regime.

This paper studies asymmetric exclusion processes on a microtubule-like system with two species of particles. The model is motivated by the structure of microtubules and kinesins and dyneins moving along microtubules in opposite directions. The proposed model is similar to that in J. Phys. A40, 2275 (2007) in which two-channel TASEPs with narrow entrances and parallel update are studied. This paper extended the above-mentioned work to a multiple-channel hollow cylinder case. Thus, each channel has two nearest neighbors in our model. The corresponding rule for narrow entrances is that particles cannot enter the system if either of two nearest-neighbor sites on neighboring channels is occupied by the other species of particles. The phase diagram of the model is obtained from a mean-field approximation and verified by computer simulations. It is shown that the spontaneous symmetry breaking exists with two asymmetric phases: high/low density and low/low density. The flipping process of particles is observed. Bulk density and particle currents are computed. Monte Carlo simulation results deviate from the mean-field prediction when entrance rate α is high, which is due to neglecting correlations among particles in mean-field calculations. The results are also compared with that obtained from two-channel system with one neighbor narrow entrance in parallel update.

A family of boundary conditions corresponding to exclusion processes is introduced. This family is a generalization of the boundary conditions corresponding to the simple exclusion process, the drop-push model, and the one-parameter solvable family of pushing processes with certain rates on the continuum.^1–3 The conditional probabilities are calculated using the Bethe ansatz, and it is shown that at large times they behave like the corresponding conditional probabilities of the family of diffusion-pushing processes introduced in Refs. 1–3.