## 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) ~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^{−βj} 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) ~ nα, 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, the associated boundary conditions at slow sites change dynamically depending on the masses on the slow sites.

Original language | English (US) |
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Pages (from-to) | 1048-1092 |

Number of pages | 45 |

Journal | Annals of Probability |

Volume | 52 |

Issue number | 3 |

DOIs | |

State | Published - 2024 |

## Keywords

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

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty