A model of irreversible jam formation in dense traffic

Abstract

We study an one-dimensional stochastic model of vehicular traffic on open segments of a single-lane road of finite size L. The vehicles obey a stochastic discrete-time dynamics which is a limiting case of the generalized Totally Asymmetric Simple Exclusion Process. This dynamics has been previously used by Bunzarova and Pesheva [Phys. Rev. E 95, 052105 (2017)] for an one-dimensional model of irreversible aggregation. The model was shown to have three stationary phases: a many-particle one, MP, a phase with completely filled configuration, CF, and a boundary perturbed MP+CF phase, depending on the values of the particle injection (α), ejection (β) and hopping (p) probabilities.Here we extend the results for the stationary properties of the MP+CF phase, by deriving exact expressions for the local density at the first site of the chain and the probability P(1) of a completely jammed configuration. The unusual phase transition, characterized by jumps in both the bulk density and the current (in the thermodynamic limit), as α crosses the boundary α = p from the MP to the CF phase, is explained by the finite-size behavior of P(1).By using a random walk theory, we find that, when α approaches from below the boundary α = p, three different regimes appear, as the size L → ∞:

One-dimensional discrete aggregation-fragmentation model

“…As expected, at p m = 1 these expressions reduce to equalities (4) in Ref. [26]. As is readily seen, in the alternative case of several coexisting gaps, the above probabilities apply exactly to the rightmost one.…”

“…1(b) the many-particle phase MP contains a macroscopic number of particles or on the left-hand side of it have a critical type of evolution with mean lifetime of the order O(L 1/2 ), see Ref. [26]. The phase MP+CF is mixed in the sense that the completely filled configurations are perturbed by short living gaps entering the chain from the first site.…”

“…We note that the above generalized backward-ordered dynamics was suggested as exactly solvable one by Wölki [20], and studied on a ring in [21][22][23]. The limit case of p m = 1 corresponds to irreversible aggregation, or jam formation in the case of vehicles, suggested and studied by Bunzarova and Pesheva [24] and further elaborated in [25,26]. Here, we focus on the stationary properties of gTASEP when p < p m < 1, describing the generic case of attraction between particles hopping stochastically and unidirectionally in discrete time along finite one-dimensional chains with given boundary conditions at the ends.…”

“…One of the methods, developed in Refs. [25,26], and which is used also here, is based on the study of the time evolution of single gaps in different regions of the CF phase.…”

gTASEP with Attraction Interaction on Lattices with Open Boundaries

“…The model incorporates two extreme cases: the TASEP with parallel update (PU) when pm=0 is set (see, e.g., Refs [5,6]) and the case with all particles irreversibly merging (when pm=1) into a single cluster moving as an isolated particle. The latter case is that of the irreversible aggregation (IA), studied in Refs [7][8][9]. The gTASEP reduces to the extensively studied ordinary TASEP with backward ordered sequential update (BOSU) when pm=p (see, e.g., Refs [10,11]).…”