Coupling, 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 \((\xi_t,\zeta_t)_{t\geq 0}\) satisfies: (A) if \(\xi_0\leq\zeta_0\) (coordinate-wise), then for all \(t\geq 0\), \(\xi_t\leq\zeta_t\) a.s. In this paper, we consider generalized misanthrope models which are conservative particle systems on \(\Z^d\) such that, in each transition, \(k\) particles may jump from a site \(x\) to another site \(y\), with \(k\geq 1\). 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 \(k\le 2\)) 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.