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On the Periodic Lorentz Gas and the Lorentz Kinetic Equation

François Golse[1]

  • [1] École polytechnique, CMLS, F91128 Palaiseau cedex & Laboratoire Jacques-Louis Lions Boîte courrier 187, F75252 Paris cedex 05

Annales de la faculté des sciences de Toulouse Mathématiques (2008)

  • Volume: 17, Issue: 4, page 735-749
  • ISSN: 0240-2963

Abstract

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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)].

How to cite

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Golse, François. "On the Periodic Lorentz Gas and the Lorentz Kinetic Equation." Annales de la faculté des sciences de Toulouse Mathématiques 17.4 (2008): 735-749. <http://eudml.org/doc/10103>.

@article{Golse2008,
abstract = {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)].},
affiliation = {École polytechnique, CMLS, F91128 Palaiseau cedex & Laboratoire Jacques-Louis Lions Boîte courrier 187, F75252 Paris cedex 05},
author = {Golse, François},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {6},
number = {4},
pages = {735-749},
publisher = {Université Paul Sabatier, Toulouse},
title = {On the Periodic Lorentz Gas and the Lorentz Kinetic Equation},
url = {http://eudml.org/doc/10103},
volume = {17},
year = {2008},
}

TY - JOUR
AU - Golse, François
TI - On the Periodic Lorentz Gas and the Lorentz Kinetic Equation
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2008/6//
PB - Université Paul Sabatier, Toulouse
VL - 17
IS - 4
SP - 735
EP - 749
AB - 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)].
LA - eng
UR - http://eudml.org/doc/10103
ER -

References

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  3. Boca (F.), Zaharescu (A.).— The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit. Commun. Math. Phys. 269, no. 2, 425-471, (2007). Zbl1143.37002MR2274553
  4. Boldrighini (C.), Bunimovich (L.A.), Sinai (Ya.G.).— On the Boltzmann equation for the Lorentz gas. J. Statist. Phys.32, p; 477-501, (1983). Zbl0583.76092MR725107
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  6. Bunimovich (L.), Chernov (N.), Sinai (Ya.G.).— Statistical properties of two-dimensional hyperbolic billiards. Russian Math. Surveys 46, p. 47-106, (1991). Zbl0780.58029MR1138952
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  8. Bourgain (J.), Golse (F.), Wennberg (B.).— On the distribution of free path lengths for the periodic Lorentz gas. Commun. Math. Phys.190, p. 491-508 (1998). Zbl0910.60082MR1600299
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  14. Golse (F.), Wennberg (B.).— On the distribution of free path lengths for the periodic Lorentz gas II. M2AN Modél. Math. et Anal. Numér.34, p. 1151-1163 (2000). Zbl1006.82025MR1812731
  15. Lorentz (H.).— Le mouvement des électrons dans les métaux. Arch. Néerl.10, p. 336-371 (1905). Zbl36.0922.02
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  17. Ricci (L.), Wennberg (B.).— On the derivation of a linear Boltzmann equation from a periodic lattice gas. Stochastic Process. Appl.111, p. 281-315 (2004). Zbl1094.82013MR2056540
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