  We consider the simple exclusion process with $k$ particles on a segment of
length $N$ performing random walks with transition $p>1/2$ to the right and
$q=1-p$ to the left. We focus on the case where the asymmetry in the jump rates
$b=p-q>0$ vanishes in the limit when $N$ and $k$ tend to infinity, and obtain
sharp asymptotics for the mixing times of this sequence of Markov chains in the
two cases where the asymmetry is either much larger or much smaller than $(\log
k)/N$. We show that in the former case ($b \gg (\log k)/N$), the mixing time
corresponds to the time needed to reach macroscopic equilibrium, like for the
strongly asymmetric (i.e.\ constant $b$) case studied in [LL18], while the
latter case ($b \ll (\log k)/N$) macroscopic equilibrium is not sufficient for
mixing and one must wait till local fluctuations equilibrate, similarly to what
happens in the symmetric case worked out in [Lac16b]. In both cases,
convergence to equilibrium is abrupt: we have a cutoff phenomenon for the
total-variation distance. We present a conjecture for the remaining regime when
the asymmetry is of order $(\log k) / N$.
