Convergence to the maximal invariant measure for a zero-range process with random rates

Abstract

We consider a one-dimensional totally asymmetric nearest-neighbor zero-range process with site-dependent jump-rates -an environment. For each environment p we prove that the set of all invariant measures is the convex hull of a set of product measures with geometric marginals. As a consequence we show that for environments p satisfying certain asymptotic property, there are no invariant measures concentrating on configurations with density bigger than ρ * (p), a critical value. If ρ * (p) is finite we say that there is phase-transition on the density. In this case we prove that if the initial configuration has asymptotic density strictly above ρ * (p), then the process converges to the maximal invariant measure.

Escape of mass in zero-range processes with random rates

“…These measures are invariant for the process [1,13,22]. In some cases it is known that all invariant measures (concentrated on X ) are convex combinations of measures in {ν λ,v : 0 ≤ v ≤ c} (see [1,2]). …”

Quenched convergence and strong local equilibrium for asymmetric zero-range process with site disorder

“…The purpose of this paper is to introduce a new approach for the derivation of quenched strong local equilibrium in order to address this question, which was left open by the previous works [10,21]. In the case of supercritical hydrodynamic density ρ(t, x) > ρ c , we prove that the local equilibrium property fails, and that, locally around "typical points" of the environment, the distribution of the microscopic state is close to the critical measure with density ρ c : this can be viewed as a dynamic version of the loss of mass property studied in [4,7,8]. The dynamic loss of mass that we establish here allows us to remove the convexity assumption used in [7,8], but for a slightly less general class of initial configurations.…”

“…This corresponds to a critical density above which no product invariant measure exists. More generally, [4] showed that there were no invariant measures (whether product or not) of supercritical density, which can be interpreted as a phase transition. The model of M/M/1 queues in tandem has specific properties which make its analysis more tractable than that of the general model.…”

“…However, in this case such a measure does not exist. For the totally asymmetric M/M/1 model, it was shown in [4] that the system converges to the maximal invariant measure (thereby implying a loss of mass). This was established in [7,8] for the general nearest-neighbour model under a weak convexity assumption, and we showed that this could fail for non nearestneighbour jump kernels.…”

Hydrodynamics in a condensation regime: The disordered asymmetric zero-range process

“…≡ 1 and p(.) concentrated on the value 1, we obtain M/M/1 queues in tandem, for which [2] showed that there were no invariant measures of supercritical density.…”