Recent Results on the Periodic Lorentz Gas

François Golse (1, 2)

1 CMLS - Centre de Mathématiques Laurent Schwartz
2 LJLL - Laboratoire Jacques-Louis Lions

The Drude-Lorentz model for the motion of electrons in a solid is a classical model in statistical mechanics, where electrons are represented as point particles bouncing on a fixed system of obstacles (the atoms in the solid). Under some appropriate scaling assumption --- known as the Boltzmann-Grad scaling by analogy with the kinetic theory of rarefied gases --- this system can be described in some limit by a linear Boltzmann equation, assuming that the configuration of obstacles is random [G. Gallavotti, [Phys. Rev. (2) {\bf 185} (1969), 308]). The case of a periodic configuration of obstacles (like atoms in a crystal) leads to a completely different limiting dynamics. These lecture notes review several results on this problem obtained in the past decade as joint work with J. Bourgain, E. Caglioti and B. Wennberg.

Mots clés: Periodic Lorentz gas; Boltzmann-Grad limit; Linear Boltzmann equation; Mean free path; Distribution of free path lengths; Continued fractions; Farey fractions

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Systèmes dynamiques [math.DS]

François Golse. Recent Results on the Periodic Lorentz Gas. Xavier Cabré, Juan Soler. Nonlinear Partial Differential Equations, Birkhäuser, pp.39-99, 2012, Advanced Courses in Mathematics CRM Barcelona, 978-3-0348-0190-4. ⟨10.1007/978-3-0348-0191-1⟩. ⟨hal-00390895v2⟩