The role of the Principle of Inertia in KTAP models

The mathematical models for describing the evolution of complex systems in the framework of the Kinetic Theory for Active Particles (KTAPs), based on suitable generalizations of Boltzmann’s kinetic theory of gases, nowadays seem to be the most versatile and effective ones to formulate in sufficiently precise and accurate terms, and possibly to solve, the most important and difficult problems not only of our everyday’s individual life but also of the life of human collectivities in the long term. In this connection, a simple examination of the application of KTAP to practical problems in which the state variables are not the usual position-velocity variables (and the kinetic state can, indeed, be completely disregarded), such as the prediction of the evolution of an epidemic in a given population in view of possible contain- ment measures, suggests that the scheme is not always so general as it would be desirable for the simple reason that it is conditioned by the implicit (and natural) assumption of the principle of inertia, which in non-mechanical contexts should be rejected. Starting from this remark, this paper has a threefold aim: (1) to point out that KTAP equations make a tacit use of the principle of inertia; (2) to recall the reasons why the principle of inertia is naturally connected to the origin of KTAP models; and (3) to suggest a suitable modification of the KTAP general system, enabling the user to choose to adopt or reject the principle of inertia.