Generalized flux-weighted boundary walls in kinetic models

TL;DR

This paper introduces a new analytical framework for studying stationary states in collisionless systems with boundary reservoirs, revealing non-thermal profiles and boundary-induced temperature gradients.

Contribution

It generalizes boundary conditions in kinetic models and derives explicit stationary distributions linking microscopic boundary rules to macroscopic profiles.

Findings

Thermal equilibrium occurs only with standard flux-weighted Maxwellian injection.

Non-thermal stationary states with complex density and temperature profiles are observed.

Analytical predictions match particle-based numerical simulations.

Abstract

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.