In this paper, we describe the long-time behavior of the non-cutoff Boltzmann
equation with soft potentials near a global Maxwellian background on the whole
space in the weakly collisional limit (i.e. infinite Knudsen number $1/\nu\to
\infty$). Specifically, we prove that for initial data sufficiently small
(independent of the Knudsen number), the solution displays several dynamics
caused by the phase mixing/dispersive effects of the transport operator $v
\cdot \nabla_x$ and its interplay with the singular collision operator. For
$x$-wavenumbers $k$ with $|k|\gg\nu$, one sees an enhanced dissipation effect
wherein the characteristic decay time-scale is accelerated to
$O(1/\nu^{\frac{1}{1+2s}} |k|^{\frac{2s}{1+2s}})$, where $s \in (0,1]$ is the
singularity of the kernel ($s=1$ being the Landau collision operator, which is
also included in our analysis); for $|k|\ll \nu$, one sees Taylor dispersion,
wherein the decay is accelerated to $O(\nu/|k|^2)$. Additionally, we prove
almost-uniform phase mixing estimates. For macroscopic quantities as the
density $\rho$, these bounds imply almost-uniform-in-$\nu$ decay of
$(t\nabla_x)^\beta \rho$ in $L^\infty_x$ due to Landau damping and dispersive
decay.
