  We consider the symmetric simple exclusion process evolving on the interval
of length $n-1$ in contact with reservoirs of density $\rho \in (0,1)$ at the
boundary. We use Yau's relative entropy method to show that if the initial
measure is associated with a profile $u_0:[0,1] \to (0,1)$, then at explicit
times $t^n(b)$ that depend on $u_0$, the distance to equilibrium, in total
variation distance, converges, as $n \to \infty$, to a profile $\mathcal
G(\gamma e^{-b})$. The parameter $\gamma$ also depends on the initial profile
$u_0$ and $\mathcal G(m)$ stands for the total variation distance $\|\mathcal N
(m,1) - \mathcal N(0,1)\|_{\mathrm{TV}}$.
