  This paper introduces the idea that the general mixing inequality obeyed by
evolving stellar phase densities may place useful constraints on the possible
history of the over-all galaxy population. We construct simple models for the
full stellar phase space distributions of galaxies' disk and spheroidal
components, and reproduce the well-known result that the maximum phase density
of an elliptical galaxy is too high to be produced collisionlessly from a disk
system, although we also show that the inclusion of a bulge component in the
disk removes this evolutionary impediment. In order to draw more general
conclusions about the evolution of the galaxy population, we use the Millennium
Galaxy Catalogue to construct a model of the entire phase density distribution
of stars in a representative sample of the local Universe. In such a composite
population, we show that the mixing inequality rules out some evolutionary
paths that are not prohibited by consideration of the maximum phase density
alone, and thus show that the massive ellipticals in this population could not
have formed purely from collisionless mergers of a low mass galaxy population
like that found in the local Universe. Although the violation of the mixing
inequality is in this case quite minor, and hence avoidable with a modest
amount of non-collisionless star formation in the merger process, it does
confirm the potential of this approach. The future measurement of stellar phase
densities at higher redshift will allow this potential to be fully exploited,
offering a new way to look at the possible pathways for galaxy evolution, and
to learn about the environment of star formation through the way that this
phase space becomes populated over time.
