Facilitated Asymmetric Exclusion

We introduce a class of facilitated asymmetric exclusion processes in which particles are pushed by neighbors from behind. For the simplest version in which a particle can hop to its vacant right neighbor only if its left neighbor is occupied, we determine the steady state current and the distribution of cluster sizes on a ring. We show that an initial density downstep develops into a rarefaction wave that can have a jump discontinuity at the leading edge, while an upstep results in a shock wave. This unexpected rarefaction wave discontinuity occurs generally for facilitated exclusion processes.

💡 Research Summary

The paper introduces a new class of driven lattice gases called facilitated asymmetric exclusion processes (FAEP), in which particle motion to the right is only allowed when the particle’s left neighbor is present. The authors focus on the simplest incarnation of this rule—“a particle may hop to its empty right site only if its left site is occupied”—and study the model on a one‑dimensional periodic lattice (a ring).

First, they construct the exact stationary measure. Because a solitary particle cannot move, the system organizes into clusters of at least two consecutive particles. By solving the master equation for the Markov chain, they obtain a closed‑form expression for the cluster‑size distribution: the probability of finding a cluster of length k (k ≥ 2) is proportional to ρ^k (1 − ρ), where ρ = N/L is the global particle density. This non‑trivial stationary state differs markedly from the product‑measure stationary state of the ordinary asymmetric simple exclusion process (ASEP).

From the cluster statistics they derive the exact steady‑state current. Each cluster advances as a whole, and the average particle flux is J = ρ(1 − ρ²). The current vanishes both at ρ → 0 (as in any exclusion model) and at ρ → 1 (because the lattice is fully packed), but it also displays a pronounced suppression relative to the ASEP current J_AS = ρ(1 − ρ). The maximum current occurs at ρ = 1/√2, illustrating how facilitation reduces transport efficiency at high densities.

The dynamical evolution from step‑like initial conditions is then examined using the macroscopic continuity equation ∂_t ρ + ∂_x J(ρ) = 0. For a downstep (ρ_L > ρ_R) the characteristic curves fan out, producing a rarefaction wave. Uniquely for FAEP, the leading edge of this wave can develop a discontinuous jump in density—a “rarefaction discontinuity.” This phenomenon does not appear in the standard ASEP, where rarefaction profiles are smooth. For an upstep (ρ_L < ρ_R) the characteristics intersect, generating a shock wave whose speed obeys the Rankine‑Hugoniot condition v_s =