$\rho=\frac12$ (this assumption, which aims at removing a transport phenomenon inside the system, will be explained in detail in Remark \[rem:rho\]):

-   if $\varepsilon_n = E/\sqrt n$ (for some $E>0$), the sequence of processes ${{\mathcal Y}}^n$ converges, in a suitable space of test functions, towards the unique energy solution ${{\mathcal Y}}$ to with Dirichlet boundary conditions, with parameters $A=1$, $B=E$ and $C=\frac12$;

-   whenever $\sqrt n\varepsilon_n \to 0$ as $n\to \infty$, the sequence ${{\mathcal Y}}^n$ converges towards the unique Ornstein-Uhlenbeck (OU) process with Dirichlet boundary conditions, which is equivalent to the unique energy solution to with parameters $A=1$, $B=0$ and $C=\frac12$.

### The second order Boltzmann-Gibbs principle

The main ingredient that we use at the microscopic level is the BGP, that we have to reprove completely in our particular setting. Indeed, this is the first time that the BGP is established in a space with boundaries.

This tool, which was first introduced in [@gj2014], permits to investigate the space-time fluctuations of the microscopic density fluctuation field, and to prove that, when properly recentered, the latter is close in the macroscopic limit to a quadratic functional of the conservative field. It was originally proved for general weakly asymmetric simple exclusion processes 