Boltzmann provided a scenario to explain why individual macroscopic systems composed of a large number N of microscopic constituents are inevitably (i.e., with overwhelming probability) observed to approach a unique macroscopic state of thermodynamic equilibrium, and why after having done so, they are then observed to remain in that state, apparently forever. We will discuss rigourous results that mathematically prove the basic features of Boltzmann’s scenario for two simple classical models : a boundary-free model for the spatial homogenization of a non-interacting gas of point particles, and the well-known Kac ring model. (Joint work with P. E. Parris (Missouri))