Kinetic transport in the two-dimensional periodic Lorentz gas

Abstract

The periodic Lorentz gas describes an ensemble of non-interacting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (Boltzmann-Grad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic transport equation is not the linear Boltzmann equation--in contrast to the Lorentz gas with a disordered scatterer configuration. The present paper focuses on the two-dimensional set-up, and reports an explicit, elementary formula for the collision kernel of the transport equation.

arXiv comment10 pages, 5 figures

References in corpus (4)

• Almost Sure Invariance Principle For Nonuniformly Hyperbolic Systems
• The Lyapunov exponent in the Sinai billiard in the small scatterer limit
• Density-dependent diffusion in the periodic Lorentz gas
• The statistics of the trajectory in a certain billiard in a flat two-torus

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• On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas
• Free path lengths in quasicrystals
• Recent Results on the Periodic Lorentz Gas
• Power-law distributions for the free path length in Lorentz gases
• Free path lengths in quasi crystals
• Locally Perturbed Random Walks with Unbounded Jumps
• An analytic approximation to the Diffusion Coefficient for the periodic Lorentz Gas
• Volumes and distributions for random unimodular complex and quaternion lattices
• Efficient algorithms for general periodic Lorentz gases in two and three dimensions
• The Lorentz Process with a Nearly Periodic Distribution of Scatterers
• Quantum Transport in a Low-Density Periodic Potential: Homogenisation via Homogeneous Flows
• Convergence to the equilibrium for the kinetic transport equation in the two-dimensional periodic Lorentz Gas