any $E>0$ and $\gamma{\geqslant}\frac12$ we define ${{\mathcal L}}_n^{\textrm{bulk}}$ and ${{\mathcal L}}_n^{\mathrm{bnd}}$ acting on functions $f: \Omega_n \to {{\mathbb R}}$ as follows: first, $$\big({{\mathcal L}}_n^{\textrm{bulk}} f\big)(\eta) = 
\sum_{x=1}^{n-2} \bigg\{\Big(1+\frac{E}{n^\gamma} \Big) \eta(x)\big(1-\eta(x+1)\big)+\eta(x+1)\big(1-\eta(x)\big)\bigg\} \big(\nabla_{x,x+1}f\big)(\eta),$$ where $\big(\nabla_{x,x+1}f\big)(\eta)=f(\sigma^{x,x+1}\eta)
-f(\eta)$ and $\sigma^{x,x+1} \eta$ is the configuration obtained from $\eta$ after exchanging $\eta(x)$ with $\eta(x+1)$, namely $$(\sigma^{x,x+1} \eta)(y)=
\left\{
\begin{array}{ll}
\eta(x)\,,& \textrm{ if } y=x+1,\\
\eta(x+1)\,,& \textrm{ if } y=x,\\
\eta(y)\,,& \textrm{ if } y\notin \{x,x+1\}.\\
\end{array}
\right.$$ Second, the generator of the dynamics at the boundaries is given by $$\begin{split}
\big({{\mathcal L}}_n^{\mathrm{bnd}} f\big)(\eta) & =  
 \Big\{\Big(1+\frac{E}{n^\gamma} \Big) \rho(1-\eta(1))+\eta(1)\big(1-\rho\big)\Big\}\big(f(\sigma^{1}\eta)
-f(\eta)\big)\\
&\; +
 \Big\{\Big(1+\frac{E}{n^\gamma} \Big) \eta(n-1)(1-\rho)+\rho\big(1-\eta(n-1)\big)\Big\}\big(f(\sigma^{n-1}\eta)
-f(\eta)\big)\\
\end{split}$$ where $$(\sigma^{x} \eta)(y)=
\left\{
\begin{array}{ll}
1-\eta(y)\,,& \textrm{ if } y=x\,,\\
\eta(y)\,,&  \textrm{ if } y\neq x.\\
\end{array}
\right.$$ From now on to simplify notation we denote the rates that appear above by $r_{x,x+1}(\eta)$: precisely, at any $x\in\{0,\dots,n-1\}$, and assuming the convention $\eta(0)=\eta(n)=\rho$, they are given by $$\label{eq:rate}
r_{x,x+1}(\eta)=\Big(1+\frac{E}{n^\gamma} \Big) \eta(x)\big(1-\eta(x+1)\big)+\eta(x+1)\big(1-\eta(x)\big).$$ We refer to Figure \[fig1\] for an illustration of the dynamics.

(0.5,0.3)(10:170:0.45); (0.5,-0.3)(-10:-170:0.45); (4.5,-0.3)(-10:-170:0.45); (4.5,0.3)(10:170:0.45); (8.5,-0.3)(-10:-170:0.45); (8.5,0.3)(10:170:0.45); (1,0) – (8,0);

(1,0) circle (0.25); (3,0) circle (0.25); (5,0) circle (0.25); (6,0) circle (0.25); (8,0) circle (0.25); (2,0) circle (.25) (4,0) circle (.25) (7,0) circle (.25);

(1.3,-0.05) node\[anchor=north\] [$\bf 1$]{} (2.3,-0.05) node\[anchor=north\] 