Two-component self-gravitating isothermal slab models
  We revisit the problem of the isothermal slab (in standard Cartesian
coordinates, density distributions and mean gravitational potential are
considered to be independent of $x$ and $y$ and to be a function of $z$,
symmetric with respect to the $z = 0$ plane) in the context of the general
issues related to the role of weak collisionality in inhomogeneous
self-gravitating stellar systems. We thus consider the two-component case, that
is a system of heavy and light stars with assigned mass ratio ($\mu$) and
assigned global relative abundance ($\alpha$; the ratio of the total mass of
the heavy and light stars). The system is imagined to start from an initial
condition in which the two species are well mixed and have identical spatial
and velocity distributions and to evolve into a final configuration in which
collisions have generated equipartition and mass segregation. Initial and final
distribution functions are assumed to be Maxwellian. Application of mass and
energy conservation allows us to derive the properties of the final state from
the assumed initial conditions. In general, the derivation of these properties
requires a simple numerical integration of the Poisson equation. Curiously, the
case in which the heavy stars are exactly twice as massive as the light stars
($\mu = 2$) turns out to admit a relatively simple analytic solution. Although
the general framework of this investigation is relatively straightforward, some
non-trivial issues related to energy conservation and the possible use of a
virial constraint are noted and clarified. The formulation and the results of
this paper prepare the way to future studies in which the evolution induced by
weak collisionality will be followed either by considering the action of
standard collision operators or by means of dedicated numerical simulations.
