  We study turbulent gas diffusion affects on $r$-process abundances in Milky
Way stars, by a combination of an analytical approach and a Monte Carlo
simulation. Higher $r$-process event rates and faster diffusion, lead to more
efficient mixing corresponding to a reduced scatter of $r$-process abundances
and causing $r$-process enriched stars to start appearing at lower
metallicities. We use three independent observations to constrain the model
parameters: (i) the scatter of radioactively stable $r$-process element
abundances, (ii) the largest $r$-process enrichment values observed in any
solar neighborhood stars and (iii) the isotope abundance ratios of different
radioactive $r$-process elements ($^{244}$Pu/$^{238}$U and
$^{247}$Cm/$^{238}$U) at the early solar system as compared to their formation.
Our results indicate that the Galactic $r$-process rate and the diffusion
coefficient are respectively $r<4\times 10^{-5}\mbox{ yr}^{-1}, D>0.1 \mbox{
kpc}^2\mbox{Gyr}^{-1}$ ($r<4\times 10^{-6}\mbox{ yr}^{-1}, D>0.5 \mbox{
kpc}^2\mbox{Gyr}^{-1}$ for collapsars or similarly prolific $r$-process
sources) with allowed values satisfying an approximate anti-correlation such
that $D\approx r^{-2/3}$, implying that the time between two $r$-process events
that enrich the same location in the Galaxy, is $\tau_{\rm mix}\approx
100-200\mbox{ Myr}$. This suggests that a fraction of $\sim 0.8$ ($\sim 0.5$)
of the observed $^{247}$Cm ($^{244}$Pu) abundance is dominated by one
$r$-process event in the early solar system. Radioactively stable element
abundances are dominated by contributions from $\sim 10$ different events in
the early solar system. For metal poor stars (with [Fe/H]$\lesssim -2$), their
$r$-process abundances are dominated by either a single or several events,
depending on the star formation history.
