Generalized flux-weighted boundary walls in kinetic models

We present a technique to investigate the stationary states of a system of a collisionless system confined by an external potential and coupled to boundary reservoirs through prescribed reinjection rules. We consider a family of boundary conditions parametrized by an integer $n$, corresponding to different velocity distributions imposed at the boundaries, generalizing the standard flux-weighted Maxwellian scheme. By combining Liouville's theorem with the boundary injection rule, we derive an explicit analytical expression for the stationary distribution function. This framework provides a direct link between microscopic boundary dynamics and macroscopic stationary profiles. We show that thermal equilibrium is recovered only for the standard flux-weighted injection method, while for all other cases the system relaxes to manifestly non-thermal stationary states. The resulting density and temperature profiles exhibit non-trivial spatial structures, including non-monotonic behaviour and temperature gradients induced by the boundary conditions alone. Analytical predictions for stationary moments are obtained in closed form for representative cases and are nicely reproduced by particle-based numerical simulations.