$b_N\approx N^{-1}$ is the one observed for the hydrodynamic limit: It corresponds to a crossover regime where the limit is given by a viscous Burger’s equation [@Demasi89; @Gartner88; @KipLan] which interpolates between the heat and the inviscid Burgers’ equations. The one occurring for $b_N\approx N^{-1}\log N$ is however not observed in the macrospic profile and is specific to mixing times.

In the present work, we identify the full asymptotic of the mixing time (with the right constant) when the bias is either negligible compared to, or much larger than $\log N / N$ (or $\log k/N$ when the total number of particle is not of order $N$). This implies cutoff in these two regimes. Our result and its proof provide a better understanding of the effect of asymmetry on microscopic mixing: When $b_N\gg N^{-1}\log N$, the pattern of relaxation is identical to that of the fully asymmetric case and microscopic equilibrium is reached exactly when the macroscopic profile hits its equilibrium state. When $b_N\ll N^{-1}\log N$ the pattern of relaxation resembles that of the symmetric case, the mixing time corresponds to the time needed to equilibrate local fluctuations, in particular in the case $(B)$ described above (or more precisely when 