May 2024 Condensation, boundary conditions, and effects of slow sites in zero-range systems
Sunder Sethuraman, Jianfei Xue
Author Affiliations +
Ann. Probab. 52(3): 1048-1092 (May 2024). DOI: 10.1214/24-AOP1684


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<α1, 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 λj1Nβjg(n) when g(n)nα, and to λj1g(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=limNkj,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.

Funding Statement

S.S. was supported by ARO-W911NF-18-1-0311


Download Citation

Sunder Sethuraman. Jianfei Xue. "Condensation, boundary conditions, and effects of slow sites in zero-range systems." Ann. Probab. 52 (3) 1048 - 1092, May 2024.


Received: 1 May 2022; Revised: 1 January 2024; Published: May 2024
First available in Project Euclid: 23 April 2024

Digital Object Identifier: 10.1214/24-AOP1684

Primary: 60K35

Keywords: boundary condition , Condensation , defect , hydrodynamic , inhomogeneity , Interacting particle system , Zero-range

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 3 • May 2024
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