Asymmetric exclusion process with long-range interactions

Source abstract

We introduce the headway exclusion process which is an exclusion process with N particles on the one-dimensional discrete torus with L sites with jump rates that depend only on the distance to the next particle in the direction of the jump but not on 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 finitely many particles that evolve on Z unidirectionally it is proved by reverse duality that a certain family of non-stationary 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 traveling wave phenomenon tantamount to phase separation with a stable shock and a stable antishock. Various potential applications of this result and open questions are outlined.