The Two Species Totally Asymmetric Simple Exclusion Process

Abstract

The two species totally asymmetric simple exclusion process ^1,2, or two species TASEP, is an interacting particle system³ in which two types of particles, first class and second class, live on the sites of a one dimensional lattice, hopping at random times to the adjacent site to their right; the model is called totally asymmetric because particles can jump only to the right, while more general ASEP models permit jumps in both directions, with a preference for one or the other. Here we study steady states—time invariant measures on the space of all configurations—for this model, and in particular discuss a family of explicitly computable translation invariant steady states on the infinite lattice; we show that these are precisely the extremal members of the set of all translation invariant steady states for the system. The family is parameterized by the densities ρ ₁ and ρ ₂ of the two species, and includes states with all densities lying within the triangle ρ ₁,≤ ρ ₂ 0, ρ ₁ + ρ ₂ ≤ 1. On the boundary of this triangle the model reduces to the the standard one species TASEP³ and the states constructed here reduce to the product states invariant for this simpler model.