Power-law distributions for the free path length in Lorentz gases
arXiv:1310.0328 · doi:10.1007/s10955-014-0935-9
Abstract
It is well known that, in the Boltzmann-Grad limit, the distribution of the free path length in the Lorentz gas with disordered scatterer configuration has an exponential density. If, on the other hand, the scatterers are located at the vertices of a Euclidean lattice, the density has a power-law tail proportional to xi^{-3}. In the present paper we construct scatterer configurations whose free path lengths have a distribution with tail xi^{-N-2} for any positive integer N. We also discuss the properties of the random flight process that describes the Lorentz gas in the Boltzmann-Grad limit. The convergence of the distribution of the free path length follows from equidistribution of large spheres in products of certain homogeneous spaces, which in turn is a consequence of Ratner's measure classification theorem.
arXiv comment13 pages
References in corpus (8)
- The Lyapunov exponent in the Sinai billiard in the small scatterer limit
- On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas
- The statistics of the trajectory in a certain billiard in a flat two-torus
- On the distribution of free-path lengths for the periodic Lorentz gas III
- On the Periodic Lorentz Gas and the Lorentz Kinetic Equation
- Kinetic transport in the two-dimensional periodic Lorentz gas
- New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis
- Free path lengths in quasi crystals
Cited by in corpus (5)
- Free path lengths in quasicrystals
- A Spectral Approach for Solving the Nonclassical Transport Equation
- Horizons and free path distributions in quasiperiodic Lorentz gases
- Efficient algorithms for general periodic Lorentz gases in two and three dimensions
- Convergence to the equilibrium for the kinetic transport equation in the two-dimensional periodic Lorentz Gas