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 $({\xi }_t,{\zeta }_t{)}_{t\ge 0}$ satisfies: (A) if ${\xi }_0\le {\zeta }_0$ (coordinate-wise), then for all $t\ge 0$, ${\xi }_t\le {\zeta }_t$ a.s. In this paper, we consider generalized misanthrope models which are conservative particle systems on ${\mathbb{Z}}^d$ such that, in each transition, $k$ particles may jump from a site $x$ to another site $y$, with $k\ge 1$. 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.

References in corpus (5)

• Shocks in the asymmetric exclusion process with internal degree of freedom
• Multiclass processes, dual points and M/M/1 queues
• Euler hydrodynamics of one-dimensional attractive particle systems
• Nonequilibrium field-induced phase separation in single-file diffusion
• Strong hydrodynamic limit for attractive particle systems on Z

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