On the Periodic Lorentz Gas and the Lorentz Kinetic Equation
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چکیده
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)]. Key-words: Lorentz gas; Sinai billiards; Kinetic theory; BoltzmannGrad limit. MSC: 82C40 (37A60,37D50)
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تاریخ انتشار 2007