We prove that the Boltzmann-Grad limit of the Lorentz gas with periodic distribution of scatterers cannot be described with a linear Boltzmann equation. This is at variance with the case of a Poisson distribution of scatterers, for which the convergence to the linear Boltzmann equation was proved by Gallavotti [Phys. Rev. (2) 185, 308 (1969)]. The arguments presented here complete the analysis in [Golse-Wennberg, M2AN Modél. Math. et Anal. Numér. 34, 1151 (2000)], where the impossibility of a kinetic description was established only in the case of absorbing obstacles. The proof is based on estimates on the distribution of free-path lengths established in [Golse-Wennberg loc.cit.] and in [Bourgain-Golse-Wennberg, Commun. Math. Phys. 190, 491 (1998)], and on a classical result on the spectrum of the linear Boltzmann equation which can be found in [Ukai-Point-Ghidouche, J. Math. Pures Appl. (9) 57, 203 (1978)].
On démontre dans cet article que la limite de Boltzmann-Grad du gaz de Lorentz dans une configuration périodique d’obstacles ne peut être décrite par une équation de type Boltzmann linéaire. Rappelons qu’au contraire, dans le cas où la configuration des obstacles est aléatoire et suit une loi de Poisson, Gallavotti a démontré la convergence en moyenne de la densité de particules vers la solution d’une équation de type Boltzmann linéaire [Phys. Rev. (2) 185, 308 (1969)]. La démonstration présentée ici complète l’analyse faite dans [Golse-Wennberg, Modél. Math. et Anal. Numér. 34, 1151 (2000)], où l’impossibilité d’une description cinétique est établie dans le seul cas d’obstacles absorbants. Cette preuve est basée sur la distribution des temps de sortie démontrée dans [Golse-Wennberg, loc. cit.] et dans [Bourgain-Golse-Wennberg, Commun. Math. Phys. 190, 491 (1998)], ainsi que sur un résultat classique concernant le spectre de l’équation de Boltzmann linéaire — voir par exemple [Ukai-Point-Ghidouche, J. Math. Pures et Appl. (9) 57, 203 (1978)].
@article{AFST_2008_6_17_4_735_0,
author = {Fran\c{c}ois Golse},
title = {On the {Periodic} {Lorentz} {Gas} and the {Lorentz} {Kinetic} {Equation}},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {735--749},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 17},
number = {4},
year = {2008},
doi = {10.5802/afst.1200},
mrnumber = {2499853},
zbl = {1166.82304},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1200/}
}
TY - JOUR AU - François Golse TI - On the Periodic Lorentz Gas and the Lorentz Kinetic Equation JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2008 SP - 735 EP - 749 VL - 17 IS - 4 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1200/ DO - 10.5802/afst.1200 LA - en ID - AFST_2008_6_17_4_735_0 ER -
%0 Journal Article %A François Golse %T On the Periodic Lorentz Gas and the Lorentz Kinetic Equation %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2008 %P 735-749 %V 17 %N 4 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1200/ %R 10.5802/afst.1200 %G en %F AFST_2008_6_17_4_735_0
François Golse. On the Periodic Lorentz Gas and the Lorentz Kinetic Equation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 17 (2008) no. 4, pp. 735-749. doi : 10.5802/afst.1200. https://afst.centre-mersenne.org/articles/10.5802/afst.1200/
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