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

Abstract

We study asymmetric zero-range processes on Z with nearestneighbour jumps and site disorder. The jump rate of particles is an arbitrary but bounded nondecreasing function of the number of particles. We prove quenched strong local equilibrium at subcritical and critical hydrodynamic densities, and dynamic local loss of mass at supercritical hydrodynamic densities. Our results do not assume starting from local Gibbs states. As byproducts of these results, we prove convergence of the process from given initial configurations with an asymptotic density of particles to the left of the origin. In particular, we relax the weak convexity assumption of [7,8] for the escape of mass property.MSC 2010 subject classification: 60K35, 82C22.

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

“…Proof of (ii). This boils down to proving that for any x ∈ Z, for any ρ ∈ [0, ρ c ), (12) and (24), Strassen's Theorem (see e.g. [30]) yields a coupling measureμ(dη, dζ) of µ α,ρ and µ α,ρ ′ under which η ≤ ζ holds a.s.. Then, setting…”

“…This can be used for instance to infer local equilibrium besides hydrodynamic limit. However, the problem of local equilibrium and loss of local equilibrium in our setting is deferred to [12], where it is investigated in depth. To obtain stationary processes, one should replace (114) with…”

“…that are not stationary (they can never be if ρ > ρ c ) are what we called "pseudo-equilibria" at the beginning of this section, because they are time-invariant on the macroscopic scale, where they correspond to a flat density profile with uniform density ρ at all times. However, this property does not hold on a smaller scale for supercritical densities, due to the mass escape at slow sites (see [2,20,11,12]).…”

“…Also partly conveyed by the interface process is the local equilibrium property, that is the natural question following the derivation of the hydrodynamic limit. This property is studied in depth in the companion paper [12]. Note that the situation is more delicate than usual in that the "freezing" of supercritical areas in the hydrodynamic scaling does not have local implications.…”

Zero-range process in random environment

“…In this review paper, we consider the one-dimensional attractive nearest neighbour process, that is d = 1, p(1) + p(−1) = 1 and g nondecreasing. We summarize the papers [13,14,15,16], by giving their results and the main ideas of their proofs. In these papers, we developed robust approaches to study various aspects of the phase transition mentioned above.…”