  The classical notion of a single-particle scalar distribution function or
phase space density can be generalized to a matrix in order to accommodate
superpositions of states of discrete quantum numbers, such as neutrino
mass/flavor. Such a `neutrino distribution matrix' is thus an appropriate
construct to describe a neutrino gas that may vary in space as well as time and
in which flavor mixing competes with collisions. The Liouville equations obeyed
by relativistic neutrino distribution matrices, including the spatial
derivative and vacuum flavor mixing terms, can be explicitly but elegantly
derived in two new ways: from a covariant version of the familiar simple model
of flavor mixing, and from the Klein-Gordon equations satisfied by a quantum
`density function' (mean value of paired quantum field operators). Associated
with the latter derivation is a case study in how the joint position/momentum
dependence of a classical gas (albeit with Fermi statistics) emerges from a
formalism built on quantum fields.
