  We investigate stellar metallicity distribution functions (MDFs), including
Fe and ${\alpha}$-element abundances, in dwarf galaxies from the Feedback in
Realistic Environments (FIRE) project. We examine both isolated dwarf galaxies
and those that are satellites of a Milky Way-mass galaxy. In particular, we
study the effects of including a sub-grid turbulent model for the diffusion of
metals in gas. Simulations that include diffusion have narrower MDFs and
abundance ratio distributions, because diffusion drives individual gas and star
particles toward the average metallicity. This effect provides significantly
better agreement with observed abundance distributions of dwarf galaxies in the
Local Group, including the small intrinsic scatter in [${\alpha}$/Fe] vs.
[Fe/H] (less than 0.1 dex). This small intrinsic scatter arises in our
simulations because the interstellar medium (ISM) in dwarf galaxies is
well-mixed at nearly all cosmic times, such that stars that form at a given
time have similar abundances to within 0.1 dex. Thus, most of the scatter in
abundances at z = 0 arises from redshift evolution and not from instantaneous
scatter in the ISM. We find similar MDF widths and intrinsic scatter for
satellite and isolated dwarf galaxies, which suggests that environmental
effects play a minor role compared with internal chemical evolution in our
simulations. Overall, with the inclusion of metal diffusion, our simulations
reproduce abundance distribution widths of observed low-mass galaxies, enabling
detailed studies of chemical evolution in galaxy formation.
