One-sided convergence in the Boltzmann-Grad limit

We review various contributions on the fundamental work of Lanford [21] de- riving the Boltzmann equation from (reversible) hard-sphere dynamics in the low density limit. We focus especially on the assumptions made on the initial data and on how they encode irreversibility. The impossibility to reverse time in the Boltzmann equation (expressed for instance by Boltzmann’s H-theorem) is related to the lack of convergence of higher order marginals on some singular sets. Explicit counterexamples single out the sets with vanishing measure where the initial data should converge in order to produce the Boltzmann dynamics.