  We consider the symmetric simple exclusion processes with a slow site in the
discrete torus with $n$ sites. In this model, particles perform
nearest-neighbor symmetric random walks with jump rates everywhere equal to
one, except at one particular site, \textit{the slow site}, where the jump rate
of entering that site is equal to one, but the jump rate of leaving that site
is given by a parameter $g(n)$. Two cases are treated, namely $g(n)=1+o(1)$,
and $g(n)=\alpha n^{-\beta}$ with $\beta>1$, $\alpha>0$. In the former, both
the hydrodynamic behavior and equilibrium fluctuations are driven by the heat
equation (with periodic boundary conditions when in finite volume). In the
latter, they are driven by the heat equation with Neumann boundary conditions.
We therefore establish the existence of a dynamical phase transition. The
critical behavior remains open.
