Mathematical tools of the kinetic theory of active particles with some reasoning on the modelling progression and heterogeneity

The mathematical approach proposed in this paper refers to the modelling, and related mathematical problems, of large systems of interacting entities whose microscopic state includes not only geometrical and mechanical variables (typically position and velocity), but also peculiar functions or specific activities. The number of the above entities is sufficiently large for describing the overall state of the system using a suitable probability distribution over the microscopic state. The first part of the paper is devoted to the derivation of suitable mathematical structures which can be properly used to model a variety of models in different fields of applied sciences. Then some research perspectives are analyzed, focussed on applications to biological systems.