One-sided convergence in the Boltzmann–Grad limit

Bodineau, Thierry; Gallagher, Isabelle; Saint-Raymond, Laure; Simonella, Sergio

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 5, pp. 985-1022.

Ce papier présente diverses contributions basées sur le travail fondamental de Lanford [21] qui a permis d’obtenir l’équation de Boltzmann à partir de la dynamique (réversible) des sphères dures dans la limite de densité faible.

On s’intéresse en particulier aux hypothèses sur la donnée initiale et sur la façon dont elles codent l’irréversibilité. On montre que l’impossibilité de renverser le sens du temps dans l’équation de Boltzmann (qui est exprimée notamment dans le théorème H) est liée à l’absence de convergence des marginales d’ordre supérieur sur des ensembles singuliers. Un contre exemple explicite permet de caractériser les ensembles, de mesure asymptotiquement nulle, où la donnée initiale doit converger pour obtenir la dynamique de Boltzmann.

We review various contributions on the fundamental work of Lanford [21] deriving 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.

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