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Boltzmann's equation provides a microscopic model for the evolution of diluteclassical gases. A fundamental problem in mathematical physics is to rigorouslyderive Boltzmann's equation starting from Newton's laws. In the 1970s, OscarLanford provided such a derivation, for the hard sphere interaction, on a smalltime interval. One of the subtleties of Lanford's original proof was that thestrength of convergence proven at positive times was much weaker than thatwhich had to be assumed at the initial time, which is at odds with the idea ofpropagation of chaos. Several authors have addressed this situation withvarious notions of strong one-sided chaos, which is the true property which ispropagated by the dynamics. We provide a new approach to the problem based onduality and the evolution of observables; the observables encode the detailedinteraction and allow us to define a new notion of strong one-sided chaos.

Observables and Strong One-Sided Chaos in the Boltzmann-Grad Limit