One-sided convergence in the Boltzmann-Grad limit
  We review various contributions on the fundamental work of Lanford deriving the Boltzmann equation from 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 microscopic sets where the initial data should converge in order to produce the Boltzmann dynamics. 
Thierry Bodineau (CMAP), Isabelle Gallagher (IMJ), Laure Saint-Raymond   (DMA), Sergio Simonella
math.PR math-ph math.AP math.MP