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.