Multiscale Analysis of a Kinetic Model of Confined Suspensions of Self-Propelled Rods

Abstract

The behavior of active matter under confinement poses significant challenges due to the intricate coupling between dynamics near boundaries and those in the bulk. A defining feature of active matter systems is that a substantial portion of their dynamics takes place near confining boundaries. In our previous work, we developed a kinetic framework that enables direct computation of the probability distribution functions for both the position and orientation of active rods. A distinguishing aspect of this approach is its explicit treatment of wall accumulation through the use of two coupled probability distribution functions: one describing the bulk population and the other representing rods accumulated at the boundary. Another novel feature is the structure of the governing equation, which is degenerate: it is second-order in one non-temporal variable and first-order in another. The main focus of this paper is to rigorously justify this model via multi-scale analysis. We first establish well-posedness of the system and then employ two distinct multi-scale derivations to obtain the model as a singular limit of a more classical kinetic system in the regime of vanishing translational diffusion. For analytical clarity, we consider the case in which active rods, once accumulated at the wall, remain permanently confined there. This work provides a rigorous mathematical foundation for reduced kinetic models of confined active matter, bridging microscopic dynamics and macroscopic accumulation phenomena.