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

Abstract

We consider the space-time scaling limit of the particle mass in zero-range particle systems on a 1D discrete torus $\mathbb{Z}/\mathit{N}\mathbb{Z}$ with a finite number of defects. We focus on two classes of increasing jump rates g, when $\mathit{g}(\mathit{n})\sim {\mathit{n}}^{\mathit{\alpha }}$, for $0<\mathit{\alpha }\le 1$, 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 $\mathit{g}(\mathit{n})$, depending only on the number of particles n at k. At a defect site ${\mathit{k}}_{\mathit{j},\mathit{N}}$, however, the jump rate is slowed down to ${\mathit{\lambda }}_{\mathit{j}}^{-1}{\mathit{N}}^{-{\mathit{\beta }}_{\mathit{j}}}\mathit{g}(\mathit{n})$ when $\mathit{g}(\mathit{n})\sim {\mathit{n}}^{\mathit{\alpha }}$, and to ${\mathit{\lambda }}_{\mathit{j}}^{-1}\mathit{g}(\mathit{n})$ when g is bounded. Here N is a scaling parameter where the grid spacing is seen as $1/\mathit{N}$ and time is speeded up by ${\mathit{N}}^2$.

Starting from initial measures with $\mathit{O}(\mathit{N})$ relative entropy with respect to an invariant measure, we show the hydrodynamic limit and characterize boundary behaviors at the macroscopic defect sites ${\mathit{x}}_{\mathit{j}}={lim}_{\mathit{N}\uparrow \infty }{\mathit{k}}_{\mathit{j},\mathit{N}}/\mathit{N}$, for all defect strengths. For rates $\mathit{g}(\mathit{n})\sim {\mathit{n}}^{\mathit{\alpha }}$, at critical or super-critical slow sites (${\mathit{\beta }}_{\mathit{j}}=\mathit{\alpha }$ or ${\mathit{\beta }}_{\mathit{j}}>\mathit{\alpha }$), 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 (${\mathit{\lambda }}_{\mathit{j}}>1$), 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.