Zero-Range Process in Random Environment

Bahadoran, Christophe; Mountford, Thomas; Ravishankar, K.; Saada, Ellen

doi:10.1007/978-3-030-69784-6_4

Cited by 4 publications

(1 citation statement)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [19], with respect to a 1D totally asymmetric zero-range process, with bounded, increasing rate function g(•), effects of a slow site and a slow particle are found when starting from a 'flat' initial measure. In [4] (see also [5]), with respect to a class of such 1D totally asymmetric zero-range systems, however with a nontrivial density of site disorders, hydrodynamics is shown with respect to an effective flux function, constant at supercritical densities; see also the conjectures in Section 3.3 in [4] about slow sites in asymmetric models.…”

Section: Introductionmentioning

confidence: 93%

Condensation, boundary conditions, and effects of slow sites in zero-range systems

Sethuraman¹,

Xue²

2022

Preprint

View full text Add to dashboard Cite

We consider the space-time scaling limit of the particle mass in zero-range particle systems on a 1D discrete torus Z/N Z with a finite number of defects. We focus on two classes of increasing jump rates g, when g(n) ∼ n α , for 0 < α ≤ 1, and when g is a bounded function. In such a model, a particle at a regular site k jumps equally likely to a neighbor with rate g(n), depending only on the number of particles n at k. At a defect site k j,N , however, the jump rate is slowed down to λ −1 j N −β j g(n) when g(n) ∼ n α , and to λ −1 j g(n) when g is bounded. Here, N is a scaling parameter where the grid spacing is seen as 1/N and time is speeded up by N 2 .Starting from initial measures with O(N ) relative entropy with respect to an invariant measure, we show the hydrodynamic limit and characterize boundary behaviors at the macroscopic defect sites x j = lim N↑∞ k j,N /N , for all defect strengths. For rates g(n) ∼ n α , at critical or super-critical slow sites (β j = α or β j > α), associated Dirichlet boundary conditions arise as a result of interactions with evolving atom masses or condensation at the defects. Differently, when g is bounded, at any slow site (λ j > 1), we find the hydrodynamic density must be bounded above by a threshold value reflecting the strength of the defect. Moreover, due to interactions with masses of atoms stored at the slow sites, the associated boundary conditions bounce between being periodic and Dirichlet.

show abstract

Section: Introductionmentioning

confidence: 93%

Condensation, boundary conditions, and effects of slow sites in zero-range systems

Sethuraman¹,

Xue²

2022

Preprint

View full text Add to dashboard Cite

show abstract

On a nonlinear SPDE derived from a hydrodynamic limit in a Sinai-type random environment

et al. 2023

View full text Add to dashboard Cite

Hydrodynamic limit of the zero range process on a randomly oriented graph

Balázs

Maxey-Hawkins

2022

Electron. J. Probab.

View full text Add to dashboard Cite

We prove the hydrodynamic limit of a totally asymmetric zero range process on a torus with two lanes and randomly oriented edges. The asymmetry implies that the model is non-reversible. The random orientation of the edges is constructed in a bistochastic fashion which keeps the usual product distribution stationary for the quenched zero range model. It is also arranged to have no overall drift along the Z direction, which suggests diffusive scaling despite the asymmetry present in the dynamics. Indeed, using the relative entropy method, we prove the quenched hydrodynamic limit to be the heat equation with a diffusion coefficient depending on ergodic properties of the orientation of the edges.The zero range process on this graph turns out to be non-gradient. Our main novelty is the introduction of a local equilibrium measure which decomposes the vertices of the graph into components of constant density. A clever choice of these components eliminates the non-gradient problems that normally arise during the hydrodynamic limit procedure.

show abstract

Zero-Range Process in Random Environment

Cited by 4 publications

References 29 publications

Condensation, boundary conditions, and effects of slow sites in zero-range systems

Condensation, boundary conditions, and effects of slow sites in zero-range systems

On a nonlinear SPDE derived from a hydrodynamic limit in a Sinai-type random environment

Hydrodynamic limit of the zero range process on a randomly oriented graph

Contact Info

Product

Resources

About