On the phase-mixed eccentricity and inclination distributions of wide
  binaries in the Galaxy
  Modern observational surveys allow us to probe the phase space distribution
function (DF) of wide binaries in the Solar neighbourhood. This DF exhibits
non-trivial features, in particular a superthermal distribution of
eccentricities for semimajor axes $a\gtrsim 10^3$AU. To interpret such features
we must first understand how the binary DF is affected by dynamical
perturbations, which typically fall into two classes: (i) stochastic kicks from
passing stars, molecular clouds, etc. and (ii) secular torques from the
Galactic tide. Here we isolate effect (ii) and calculate the time-asymptotic,
phase-mixed DF for an ensemble of wide binaries under quadrupole-order tides.
For binaries wide enough that the phase-mixing assumption is valid, none of our
results depend explicitly on semimajor axes, masses, etc. We show that unless
the initial DF is both isotropic in binary orientation and thermal in
eccentricity, then the final phase-mixed DF is always both anisotropic and
non-thermal. However, the only way to produce a superthermal DF under phase
mixing is for the initial DF to itself be superthermal.
