Kinetic models - mathematical models of everything? : Comment on "Collective learning modelling based on the kinetic theory of active particles" by D. Burini et al.

Abstract

Since the emergence of systematic science it has been recognized that a natural phenomenon can be described by different models that vary in their complexity and their ability to capture the details of the features relevant at the required level of the resolution. It has been tacitly assumed that whenever two such models are applicable at the same level, they must provide equivalent descriptions of the phenomenon. One of the earliest and most celebrated examples of this type is offered by gas flow which can be described either by the Boltzmann equation at a suitably understood molecular level or by the Euler or Navier-Stokes equations at the level of continuum. More precisely, the flow of a gas as a continuous medium, or, in other words, at the macro level, can be explained in more detail by analysing elementary collisions between pairs of molecules. Thus, the Boltzmann equation is often recognized as a more detailed equation of gas at the so-called mesoscopic, or kinetic, level from which macroscopic properties of gas, such as density, momentum or temperature, can be derived. It should be noted that one can model gas at an even more fundamental, or micro, level by tracing the motion of individual molecules by solving the system of the Newton equations that describe their interactions.