Asymmetric exclusion process with long-range interactions

Abstract

We consider asymmetric simple exclusion processes with $N$ particles on the one-dimensional discrete torus with $L$ sites with following properties: (i) nearest-neighbor jumps on the torus, (ii) the jump rates depend only on the distance to the next particle in the direction of the jump, (iii) the jump rates are independent of $N$ and $L$. For measures with a long-range two-body interaction potential that depends only on the distance between neighboring particles we prove a relation between the interaction potential and particle jump rates that is necessary and sufficient for the measure to be invariant for the process. The normalization of the measure and the stationary current are computed both for finite $L$ and $N$ and in the thermodynamic limit. For a finitely many particles that evolve on $\mathbb{Z}$ with totally asymmetric jumps it is proved, using reverse duality, that a certain family of nonstationary measures with a microscopic shock and antishock evolves into a convex combination of such measures with weights given by random walk transition probabilities. On macroscopic scale this domain random walk is a travelling wave phenomenon tantamount to phase separation with a stable shock and stable antishock. Various potential applications of this result and open questions are outlined.