Abstract
We consider the Boltzmann-Grad limit for the Lorentz, or wind-tree, model. We prove that if ω is a fixed configuration of scatterer centers belonging to a set of full measure with respect to the Poisson distribution with parameter λ>0, then the evolution of an initial a.c. particle density tends in the Boltzmann-Grad limit to the solution of the Boltzmann equation for the model. As an intermediate step we prove that the process of the free path lengths and impact parameters induced by the Lebesgue measure on a small region tends to a limiting independent process.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Grad, Principles of the Kinetic Theory of Gases, inHandbunch der Physik, Vol. 12, S. Flügge, ed. (Springer, Berlin, 1958).
H. Spohn,Rev. Mod. Phys. 52:569–615 (1980).
G. Gallavotti, Rigorous Theory of the Boltzmann equation in the Lorentz Gas, Nota interna n∘ 358, Istituto di Fisica, Università di Roma, 1972.
G. Gallavotti,Phys. Rev. 185:308 (1969).
Ya. G. Sinai,Russ. Math. Surv. 25:132 (1970).
G. Gallavotti, Lectures on the Billiard, inDynamical Systems, Theory and Application, J. Moser, ed. (Springer, Berlin, 1975).
L. A. Bunimovich and Ya. G. Sinai,Commun. Math. Phys. 78:247–280 (1980).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Boldrighini, C., Bunimovich, L.A. & Sinai, Y.G. On the Boltzmann equation for the Lorentz gas. J Stat Phys 32, 477–501 (1983). https://doi.org/10.1007/BF01008951
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01008951