  We analyze Chandra X-ray spectra of the M0 V+M0 V binary GJ 338. As
quantified by X-ray surface flux, these are the most inactive M dwarfs ever
observed with X-ray grating spectroscopy. We focus on measuring coronal
abundances, in particular searching for evidence of abundance anomalies related
to First Ionization Potential (FIP). In the solar corona and wind, low FIP
elements are overabundant, which is the so-called "FIP effect." For other
stars, particularly very active ones, an "inverse FIP effect" is often
observed, with low FIP elements being underabundant. For both members of the GJ
338 binary, we find evidence for a modest inverse FIP effect, consistent with
expectations from a previously reported correlation between spectral type and
FIP bias. This amounts to strong evidence that all M dwarfs should exhibit the
inverse FIP effect phenomenon, not just the active ones. We take the first step
towards modeling the inverse FIP phenomenon in M dwarfs, building on past work
that has demonstrated that MHD waves coursing through coronal loops can lead to
a ponderomotive force that fractionates elements in a manner consistent with
the FIP effect. We demonstrate that in certain circumstances this model can
also lead to an inverse FIP effect, pointing the way to more detailed modeling
of M dwarf coronal abundances in the future.
