Irreversibility: from Rarefied Gases to the Expanding Universe

We review an approach to macroscopic irreversibility from microscopic time reversal invariant dynamics, that can be traced back to Boltzmann; an approach that rests on the special initial conditions of the universe, and on its expansion. We discuss the notion of entropy as a measure of disorder, and how the (irreversible) Boltzmann equation can be harmonized with the (reversible) dynamics of classical particles. This way we illustrate Boltzmann's reply to the macroscopically pardoxical but dynamically justified observations on his theory, due to Loschmidt and Zermelo. We finally discuss possible advances with respect to this approach, highlighting points in which it may be reviewed, from both a mathematical and a physical standpoint, drawing from recent results on time reversibility in presence of magnetic fields, and on consequences of the so-called fluctuation theorems.