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The distributions of a galaxy's gas and stars in chemical space encodes atremendous amount of information about that galaxy's physical properties andassembly history. However, present methods for extracting information fromchemical distributions are based either on coarse averages measured overgalactic scales (e.g., metallicity gradients) or on searching for clusters inchemical space that can be identified with individual star clusters or gasclouds on $\sim 1$ pc scales. These approaches discard most of the information,because in galaxies gas and young stars are observed to be distributedfractally, with correlations on all scales, and the same is likely to be trueof metals. In this paper we introduce a first theoretical model, based onstochastically-forced diffusion, capable of predicting the multi-scalestatistics of metal fields. We derive the variance, correlation function, andpower spectrum of the metal distribution from first principles, and determinehow these quantities depend on elements' astrophysical origin sites and on thelarge-scale properties of galaxies. Among other results, we explain for thefirst time why the typical abundance scatter observed in the interstellar mediaof nearby galaxies is $\approx 0.1$ dex, and we predict that this scatter willbe 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 futurechemical tagging studies.

Metallicity Fluctuation Statistics in the Interstellar Medium and Young Stars. I. Variance and Correlation