On the mathematical transport theory in microporous media: The billiard approach

This paper is an expository of the main dynamical properties of billiards, which depend on the shape of the walls of the container, and the recent developments like the introduction of an external field, which mimic the coupling with a thermostat.The class of dynamical system dealt with in this paper exhibits characteristics of hybrid systems as it links discrete and continuous, deterministic and stochastic dynamics.The contents are focused on applications. Specifically, transport dynamics in highly-confined regions has been of interest in the last few decades because of industrial and medical applications. Aspects of confined transport remain elusive, considering that in microporous membranes, whose size pores is about that of the molecules, the transport is sometimes ballistic, and sometimes diffusive. The classical kinetic and macroscopic approach can not be directly applied because collisions of particle fluid with walls prevail. The microscopic mathematical billiard theory can be applied as a mathematical tool since the interstices between obstacles can be considered as the pores of the membranes.