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 (ξ t , ζ t ) t≥0 satisfies:(A) if ξ 0 ≤ ζ 0 (coordinate-wise), then for all t ≥ 0, ξ t ≤ ζ 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 ≥ 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.

Constructive Euler Hydrodynamics for One-Dimensional Attractive Particle Systems

“…Part 1. It is an extension to the joint particle-disorder process of a classical scheme in a non-disordered setting due to [35], which is also the basis for similar results in [1,19,29,27]. It relies on couplings.…”

Invariant measures of mass migration processes

“…In this section, we take advantage of the work done in [14] on multiple-jump conservative dynamics of misanthrope type, a class of models including MMP, to analyze attractiveness properties of MMP. In Subsection 5.1, we derive the extremal translation invariant and invariant probability measures for MM-ZRP and MM-TP.…”

“…In Subsection 5.1, we derive the extremal translation invariant and invariant probability measures for MM-ZRP and MM-TP. In Lemma 5.1 and in Subsection 5.2 we give necessary and sufficient conditions for attractiveness of MMP, MM-ZRP and MM-TP based on the conditions established in [14].…”

“…The fact that attractiveness is a very strong property of conservative particle systems is illustrated by the characterization of the set of all translation-invariant and invariant measures of MMP. The latter follows from [14,Theorem 5.13], which requires attractiveness and irreducibility conditions for the coupling transition rates, denoted by (IC). Under Conditions (IC), if an initial coupled configuration (ξ 0 , ζ 0 ) contains a pair of discrepancies of opposite signs (that is, on two sites x, y, (ξ 0 (x) − ζ 0 (x))(ξ 0 (y) − ζ 0 (y)) < 0), there is a positive probability (for the coupled evolution) that it evolves into a locally ordered coupled configuration (that is, for some t ≥ 0, (ξ t (x) − ζ t (x))(ξ t (y) − ζ t (y)) ≥ 0; we refer to [14, Section 2.3] for a detailed analysis of the evolution of discrepancies).…”

“…Without being exhaustive, we quote some of them that will illustrate our results: The totally asymmetric stick process has a constant jump rate (independent of particles' numbers); it was studied in [25] in the context of Ulam's problem, and a generalization with nearest neighbour jumps was considered in [14]; it possesses a one parameter family of product geometric invariant measures. The MMP is a particular case of the multiple particle jump model studied in [14] in the context of attractiveness properties; under some conditions on the rates, the latter possesses a one parameter family of translation invariant and invariant measures, that are not always explicit. Dynamics with zero range interaction and multiple jumps were studied in a finite volume setup in the context of condensation properties: they were called mass transport models in [10,11,12], and Date: August 28, 2018. generalized zero range processes in [16] (where the number k of particles that may jump together was bounded).…”