Hilbert method toward a multiscale analysis from kinetic to macroscopic models for active particles

This paper develops a Hilbert type method to derive models at the macroscopic scale for large systems of several interacting living entities whose statistical dynamics at the microscopic scale is delivered by kinetic theory methods. The presentation is in three steps, where the first one presents the structures of the kinetic theory approach used toward the aforementioned analysis; the second step presents the mathematical method; while the third step provides a number of specific applications. The approach is focused on a simple system and with a binary mixture, where different time-space scalings are used. Namely, parabolic, hyperbolic, and mixed in the case of a mixture.