Transport Processes from Mechanics: Minimal and Simplest Models

We review the current state of a fundamental problem of rigorous derivation of transport processes in classical statistical mechanics from classical mechanics. Such derivations for diffusion and momentum transport (viscosities) were obtained for minimal models of these processes involving one and two particles respectively. However, a minimal model which demonstrates heat conductivity contains three particles. Its rigorous analysis is currently out of reach for existing mathematical techniques. The gas of localized balls is widely accepted as a basis for a simplest model for derivation of Fourier’s law. We suggest a modification of the localized balls gas and argue that this gas of localized activated balls is a good candidate to rigorously prove Fourier’s law. In particular, hyperbolicity is derived for a reduced version of this model.