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

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. For any given environment satisfying suitable averaging properties, we establish a hydrodynamic limit given by a scalar conservation law including the domain above critical density, where the flux is shown to be constant.MSC 2010 subject classification: 60K35, 82C22.

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

“…Let us first recall the results of [9] with respect to the hydrodynamic limit of our process. We begin with the following standard definitions in hydrodynamic limit theory.…”

“…The connection alluded to above between creation of local equilibrium and convergence results can be understood by letting the system start initially from a uniform hydrodynamic profile with density ρ. The result of [9] implies that no evolution is seen at all on the hydrodynmic scale, because uniform profiles are stationary solutions of the hydrodynamic equation. If ρ < ρ c , one way to achieve this profile is by distributing the initial configuration according to the product stationary state of the dynamics with mean density ρ.…”

“…In the companion paper [9] and in previous works [7,8], we started developing robust approaches to study various aspects of this phase transition for the general model in dimension one with nearest-neighbour jumps (here by "general" we mean that we consider a general function g and not only the particular one g(n) = max(n, 1), but do not refer to the dimension or jump kernel). For instance, an interesting signature of the above phase transition is the mass escape phenomenon.…”

“…Phase transition also arises in the hydrodynamic limit. We showed in [9] that the hydrodynamic behavior of our process is given under hyperbolic time scaling by entropy solutions of a scalar conservation law ∂ t ρ(t, x) + ∂ x [f (ρ(t, x))] = 0 (1) where ρ(t, x) is the local particle density field, with a flux-density function ρ → f (ρ) that is increasing up to critical density and constant thereafter.…”

“…Next, the local equilibrium property is expectedly wrong at supercritical hydrodynamic densities ρ(t, x) > ρ c , since a corresponding equilibrum measure does not exist in this case. This already poses a problem at the level of the hydrodynamic limit ( [9]), since the usual heuristic for (1) is based on the idea that the macroscopic flux function f is the expectation of microscopic flux function under local equilibrium.…”