  Motivated by recent studies on the dynamics of colloidal solutions in narrow
channels, we consider the steady state properties of an assembly of non
interacting particles subject to the action of a traveling potential moving at
a constant speed while the solvent is modeled by a heat bath at rest in the
laboratory frame. Since the description, we propose here, takes into account
the inertia of the colloidal particles it is necessary to consider the
evolution of both positions and momenta and study the governing equation for
the one-particle phase-space distribution. We first derive the asymptotic form
of its solutions as an expansion in Hermite polynomials and their generic
properties, such as the force and energy balance and then we particularize our
study to the case of an inverted parabolic potential barrier. We obtain
numerically the steady state density and temperature profile and show that the
expansion is rapidly convergent for large values of the friction constant and
small drifting velocities. The present results on the one hand confirm the
previous studies based on the dynamic density functional theory (DDFT) when the
friction constant is large, on the other hand display effects such as the
presence of a wake behind the barrier and a strong inhomogeneity in the
temperature field which are beyond the DDFT description.
