the system. Moreover, attached to the extremities of $\Lambda_n$ there are two reservoirs of particles: one at site $0$, which can send a particle to site $1$ (if this site is empty), and the other one at site $n$, which can send a particle to site $n-1$ (if this site is empty). Another interpretation of the boundary dynamics could be given as follows: particles can either be created (resp. annihilated) at the sites $x=1$ or $x=n-1$ if the site is empty (resp. occupied).

All possible moves are endowed with random Poissonian clocks, independently of each other. Along the time evolution, we launch the jump whose clock rings first, and after the jump all clocks are reset. Namely, we are given a family of independent Poisson processes, indexed by all the possibles moves $x \curvearrowleft y $, with intensities $\lambda(x,y)$, on the time line $[0,\infty)$. The intensities $\lambda(x,y)$ depend on the occupation variables $\eta(x),\eta(y)$ and on some small parameter $\varepsilon_n >0$ as follows:

-   if $x \in \{2,\dots,n-2\}$, then we assume $$\begin{aligned}
    \lambda(x,x+1) & = (1+\varepsilon_n)\eta(x)\big(1-\eta(x+1)\big), \label{eq:jumpright}\\
    \lambda(x,x-1) & = \eta(x)\big(1-\eta(x-1)\big), \label{eq:jumpleft}\end{aligned}$$

-   while at the boundaries, for some fixed $\rho \in(0,1)$ we set $$\begin{aligned}
    \lambda(0,1) & = \rho(1+\varepsilon_n) \big(1-\eta(1)\big), \qquad 