CONDENSATION, BOUNDARY CONDITIONS, AND EFFECTS OF SLOW SITES IN ZERO-RANGE SYSTEMS

Sunder Sethuraman, Jianfei Xue

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the space-time scaling limit of the particle mass in zero-range particle systems on a 1D discrete torus Z/NZ with a finite number of defects. We focus on two classes of increasing jump rates g, when g(n) ~nα, 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 kj,N, however, the jump rate is slowed down to λ−1 j N−βj g(n) when g(n) ∼ nα, 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 N2. 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 xj = limN↑∞ kj,N /N, for all defect strengths. For rates g(n) ~ nα, at critical or super-critical slow sites (βj = α or βj?> 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 (λj?> 1), we find the hydrodynamic density must be bounded above by a threshold value reflecting the strength of the defect. Moreover, the associated boundary conditions at slow sites change dynamically depending on the masses on the slow sites.

Original languageEnglish (US)
Pages (from-to)1048-1092
Number of pages45
JournalAnnals of Probability
Volume52
Issue number3
DOIs
StatePublished - 2024

Keywords

  • boundary condition
  • condensation
  • defect
  • hydrodynamic
  • inhomogeneity
  • Interacting particle system
  • zero-range

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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