Couplings, attractiveness and hydrodynamics for conservative particle systems

Abstract

Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in simple exclusion. The derived Markovian coupled process (ξt , ζt )t ≥ 0 satisfies: (A) if ξ0 ≤ ζ0 (coordinate-wise), then for all t ≥ ; 0, ξ t ≤ ζt a.s. In this paper, we consider generalized misanthrope models which are conservative particle systems on ℤd such that, in each transition, k particles may jump from a site x to another site y, with k ≥ ; 1. These models include simple exclusion for which k = 1, but, beyond that value, the basic coupling construction is not possible and a more refined one is required. We give necessary and sufficient conditions on the rates to insure attractiveness; we construct a Markovian coupled process which both satisfies (A) and makes discrepancies between its two marginals non-increasing. We determine the extremal invariant and translation invariant probability measures under general irreducibility conditions.We apply our results to examples including a two-species asymmetric exclusion process with charge conservation (for which k ≤ 2) which arises from a solid-on-solid interface dynamics, and a stick process (for which k is unbounded) in correspondence with a generalized discrete Hammersley-Aldous-Diaconis model.We derive the hydrodynamic limit of these two one-dimensional models.