ABSTRACT. We consider a classical system of point particles interacting by means of a short range potential. We prove that, in the low-density (Boltzmann-Grad) limit, the system behaves, for short times, as predicted by the associated Boltzmann equation. This is a revisitation and an extension of the thesis of King [9] (appeared after the well known result of Lanford [10] for hard spheres) and of a recent paper by Gallagher et al [5]. Our analysis applies to any stable and smooth potential. In the case of repulsive potentials (with no attractive parts), we estimate explicitly the rate of convergence.

We present a quantitative analysis of the Boltzmann-Grad (low-density) limit of a hard sphere system. We introduce and study a set of functions (correlation errors) measuring the deviations in time from the statistical independence of particles (propagation of chaos). In the context of the BBGKY hierarchy, a correlation error of order k measures the event where k particles are connected by a chain of interactions preventing the factorization. We show that, provided k < ε −α , such an error flows to zero with the average density ε, for short times, as ε γk , for some positive α, γ ∈ (0, 1). This provides an information on the size of chaos, namely, j different particles behave as dictated by the Boltzmann equation even when j diverges as a negative power of ε. The result requires a rearrangement of Lanford perturbative series into a cumulant type expansion, and an analysis of many-recollision events.

We review various contributions on the fundamental work of Lanford [20] 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.

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