Recent Results on the Periodic Lorentz Gas

Abstract

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) vol. 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.

62 pages. Course at the conference "Topics in PDEs and applications 2008" held in Granada, April 7-11 2008; figure 13 and a misprint in Theorem 4.6 corrected in the new version

References in corpus (6)

• The Lyapunov exponent in the Sinai billiard in the small scatterer limit
• On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas
• On the Periodic Lorentz Gas and the Lorentz Kinetic Equation
• On the distribution of free-path lengths for the periodic Lorentz gas III
• The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions
• Kinetic transport in the two-dimensional periodic Lorentz gas

Cited by in corpus (13)

• Non-classical transport with angular-dependent path-length distributions. 1: Theory
• A reciprocal formulation of non-exponential radiative transfer. 1: Sketch and motivation
• A Spectral Approach for Solving the Nonclassical Transport Equation
• A Reciprocal Formulation of Nonexponential Radiative Transfer. 2: Monte Carlo Estimation and Diffusion Approximation
• The Nonclassical Diffusion Approximation to the Nonclassical Linear Boltzmann Equation
• Nonclassical Particle Transport in 1-D Random Periodic Media
• Distribution of the linear flow length in a honeycomb in the small-scatterer limit
• Efficient algorithms for general periodic Lorentz gases in two and three dimensions
• On the Accuracy of the Non-Classical Transport Equation in 1-D Random Periodic Media
• Asymptotic Derivation of the Simplified $P_N$ Equations for Nonclassical Transport with Anisotropic Scattering
• Statistical properties of Lorentz gases on aperiodic tilings, Part 1
• Efficient algorithm for simulating particles in real quasiperiodic environments
• Convergence to the equilibrium for the kinetic transport equation in the two-dimensional periodic Lorentz Gas