Modelling the Galactic disc: perturbed distribution functions in the
  presence of spiral arms
  Starting from an axisymmetric equilibrium distribution function (DF) in
action space, representing a Milky Way thin disc stellar population, we use the
linearized Boltzmann equation to explicitly compute the response to a
three-dimensional spiral potential in terms of the perturbed DF. This DF, valid
away from the main resonances, allows us to investigate a snapshot of the
velocity distribution at any given point in three-dimensional configuration
space. Moreover, the first order moments of the DF give rise to non-zero radial
and vertical bulk flows -- namely breathing modes -- qualitatively similar to
those recently observed in the extended Solar neighbourhood. We show that these
analytically predicted mean stellar motions are in agreement with the outcome
of test-particle simulations. Moreover, we estimate for the first time the
reduction factor for the vertical bulk motions of a stellar population compared
to the case of a cold fluid. Such an explicit expression for the full perturbed
DF of a thin disc stellar population in the presence of spiral arms will be
helpful in order to dynamically interpret the detailed information on the Milky
Way disc stellar kinematics that will be provided by upcoming large astrometric
and spectroscopic surveys of the Galaxy.
