Title: A rigourous demonstration of the validity of Boltzmann's scenario for the spatial homogenization of a freely expanding gas and the equilibration of the Kac ring,
Abstract: 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 provide here rigourous
new results that mathematically prove the basic features of Boltzmann's
scenario for two classical models: a simple boundary-free model for the spatial
homogenization of a non-interacting gas of point particles, and the well-known
Kac ring model. Our results, based on concentration inequalities that go back
to Hoeffding, and which focus on the typical behavior of individual macroscopic
systems, improve upon previous results by providing estimates, exponential in
$N$, of probabilities and time scales involved.