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On the validity of the Boltzmann equation for short range potentials. (English) Zbl 1296.82051

Summary: 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 F. King [BBGKY hierarchy for positive potentials, PhD Thesis. Department of Mathematics, Univ. California, Berkeley (1975)] (that appeared after the well-known result of O. E. Lanford III [Lect. Notes Phys. 38, 1–111 (1975; Zbl 0329.70011)] for hard spheres) and of a recent paper by I. Gallagher et al. [From Newton to Boltzmann: hard spheres and short-range potentials. Zürich: European Mathematical Society (EMS) (2013; Zbl 1315.82001)]. 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.

MSC:

82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
35Q20 Boltzmann equations
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