Domain wall theory (DWT) has proved to be a powerful tool for the analysis of one-dimensional transport processes. A simple version of it was found very accurate for the Totally Asymmetric Simple Exclusion Process (TASEP) with random sequential updat
e. However, a general implementation of DWT is still missing in the case of updates with less fluctuations, which are often more relevant for applications. Here we develop an exact DWT for TASEP with parallel update and deterministic (p=1) bulk motion. Remarkably, the dynamics of this system can be described by the motion of a domain wall not only on the coarse-grained level but also exactly on the microscopic scale for arbitrary system size. All properties of this TASEP, time-dependent and stationary, are shown to follow from the solution of a bivariate master equation whose variables are not only the position but also the velocity of the domain wall. In the continuum limit this exactly soluble model then allows us to perform a first principle derivation of a Fokker-Planck equation for the position of the wall. The diffusion constant appearing in this equation differs from the one obtained with the traditional `simple DWT.
We study two perpendicular intersecting flows of pedestrians. The latter are represented either by moving hard core particles of two types, eastbound ($symbp$) and northbound ($symbm$), or by two density fields, $rhop_t(brr)$ and $rhom_t(brr)$. Each
flow takes place on a lattice strip of width $M$ so that the intersection is an $Mtimes M$ square. We investigate the spontaneous formation, observed experimentally and in simulations, of a diagonal pattern of stripes in which alternatingly one of the two particle types dominates. By a linear stability analysis of the field equations we show how this pattern formation comes about. We focus on the observation, reported recently, that the striped pattern actually consists of chevrons rather than straight lines. We demonstrate that this `chevron effect occurs both in particle simulations with various different update schemes and in field simulations. We quantify the effect in terms of the chevron angle $Deltatheta_0$ and determine its dependency on the parameters governing the boundary conditions.
A one-way {em street} of width M is modeled as a set of M parallel one-dimensional TASEPs. The intersection of two perpendicular streets is a square lattice of size M times M. We consider hard core particles entering each street with an injection pro
bability alpha. On the intersection square the hard core exclusion creates a many-body problem of strongly interacting TASEPs and we study the collective dynamics that arises. We construct an efficient algorithm that allows for the simulation of streets of infinite length, which have sharply defined critical jamming points. The algorithm employs the `frozen shuffle update, in which the randomly arriving particles have fully deterministic bulk dynamics. High precision simulations for street widths up to M=24 show that when alpha increases, there occur jamming transitions at a sequence of M critical values alphaM,M < alphaM,M-1 < ... < alphaM,1. As M grows, the principal transition point alphaM,M decreases roughly as sim 1/(log M) in the range of M values studied. We show that a suitable order parameter is provided by a reflection coefficient associated with the particle current in each TASEP.
We introduce a new update algorithm for exclusion processes, more suitable for the modeling of pedestrian traffic. Pedestrians are modeled as hard-core particles hopping on a discrete lattice, and are updated in a fixed order, determined by a phase a
ttached to each pedestrian. While the case of periodic boundary conditions was studied in a companion paper, we consider here the case of open boundary conditions. The full phase diagram is predicted analytically and exhibits a transition between a free flow phase and a jammed phase. The density profile is predicted in the frame of a domain wall theory, and compared to Monte Carlo simulations, in particular in the vicinity of the transition.
First we consider a unidirectional flux omega_bar of vehicles each of which is characterized by its `natural velocity v drawn from a distribution P(v). The traffic flow is modeled as a collection of straight `world lines in the time-space plane, with
overtaking events represented by a fixed queuing time tau imposed on the overtaking vehicle. This geometrical model exhibits platoon formation and allows, among many other things, for the calculation of the effective average velocity w=phi(v) of a vehicle of natural velocity v. Secondly, we extend the model to two opposite lanes, A and B. We argue that the queuing time tau in one lane is determined by the traffic density in the opposite lane. On the basis of reasonable additional assumptions we establish a set of equations that couple the two lanes and can be solved numerically. It appears that above a critical value omega_bar_c of the control parameter omega_bar the symmetry between the lanes is spontaneously broken: there is a slow lane where long platoons form behind the slowest vehicles, and a fast lane where overtaking is easy due to the wide spacing between the platoons in the opposite direction. A variant of the model is studied in which the spatial vehicle density rho_bar rather than the flux omega_bar is the control parameter. Unequal fluxes omega_bar_A and omega_bar_B in the two lanes are also considered. The symmetry breaking phenomenon exhibited by this model, even though no doubt hard to observe in pure form in real-life traffic, nevertheless indicates a tendency of such traffic.
