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

We consider the space-time scaling limit of the particle mass in zero-range particle systems on a $1$D discrete torus $\mathbb{Z}/N\mathbb{Z}$ with a finite number of defects. We focus on two classes of increasing jump rates $g$, when $g(n)\sim n^\alpha$, for $0<\alpha\leq 1$, 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 $\lambda_j^{-1}N^{-\beta_j}g(n)$ when $g(n)\sim n^\alpha$, and to $\lambda_j^{-1}g(n)$ 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 $x_j = \lim_{N\uparrow \infty} k_{j, N}/N$, for all defect strengths. For rates $g(n)\sim n^\alpha$, at critical or super-critical slow sites ($\beta_j=\alpha$ or $\beta_j>\alpha$), 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 ($\lambda_j>1$), 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.