Many phenomena of physical interest are described by systems of many interacting particles. As the number of particles is typically very big, this models are usually very difficult to study.The underlying idea of kinetic theory is that what matters is the collective behaviour of the system and not the motion of each single component. Such a collective behaviour arises on space and time scales which are much larger than the ones characterising the microscopic dynamics. At macroscopic scales, the systems can be described by partial differential equations. The core of this project addresses the validity of such a reduction from microscopic to macroscopic scale in the framework of classical and quantum gases.
The Boltzmann equation was introduced at the end of the XIX century by Boltzmann and Maxwell in the attempt of modelling the evolution in time of a rarefied gas at a macroscopic scale starting from the fundamental laws of classical mechanics, thus providing for the first time a justification to the second principle of thermodynamics.
The validity of such a description is nowadays endorsed by the scientific community, as manifested by its large use in applications. Despite that, a fully rigorous mathematical comprehension is still missing. This project aims to deal with some of the major open problems in the field, such as modelling long range interactions, boundaries and quantum effects. Special relevance is given to quantitative methods which allow for explicit bounds of the errors produced when considering Boltzmann-like equations at the macroscopic scale instead of interacting particles at a microscopic scale.