  The distributions of a galaxy's gas and stars in chemical space encodes a
tremendous amount of information about that galaxy's physical properties and
assembly history. However, present methods for extracting information from
chemical distributions are based either on coarse averages measured over
galactic scales (e.g., metallicity gradients) or on searching for clusters in
chemical space that can be identified with individual star clusters or gas
clouds on $\sim 1$ pc scales. These approaches discard most of the information,
because in galaxies gas and young stars are observed to be distributed
fractally, with correlations on all scales, and the same is likely to be true
of metals. In this paper we introduce a first theoretical model, based on
stochastically-forced diffusion, capable of predicting the multi-scale
statistics of metal fields. We derive the variance, correlation function, and
power spectrum of the metal distribution from first principles, and determine
how these quantities depend on elements' astrophysical origin sites and on the
large-scale properties of galaxies. Among other results, we explain for the
first time why the typical abundance scatter observed in the interstellar media
of nearby galaxies is $\approx 0.1$ dex, and we predict that this scatter will
be correlated on spatial scales of $\sim 0.5-1$ kpc, and over time scales of
$\sim 100-300$ Myr. We discuss the implications of our results for future
chemical tagging studies.
