The Boltzmann-Grad limit of the periodic Lorentz gas

Abstract

Abstract. We study the dynamics of a point particle in a periodic array of spherical scatterers, and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (Boltzmann-Grad limit). A generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a Markov process with memory two.

Diffusion in the Lorentz Gas

“…[110][111][112]. The tail of the (closely related) free flight function in all dimensions agrees with the leading term of Eq.…”

Superdiffusion in the Periodic Lorentz Gas

“…We consider a test particle that moves along straight lines with unit speed until it hits a sphere, where it is scattered elastically. The above scaling of scattering radius vs. lattice spacing ensures that the mean free path length (i.e., the average distance between consecutive collisions) has the limit ξ = 1/v d−1 as r → 0, where In the case of the classic Lorentz gas the scattering mechanism is given by specular reflection, but as in [21] we will here also allow more general spherically symmetric scattering maps. The precise conditions will be stated in Sect.…”

“…The starting point of our analysis is the paper [21], which proves that, for every fixed t > 0, the limit r → 0 in (1.2) (resp. (1.6)) exists and is given by a continuous-time (resp.…”

“…The central limit theorem is stated as Theorem 3.2 in Sect. 3 after a brief survey of the relevant results from [21]. The subsequent sections of the paper are devoted to the proof of Theorem 3.2.…”

Ergodic theory and the duality principle on homogeneous spaces

“…• The quantitative solution of the lattice point counting problem which appears in [GN2] can be viewed as a quantitative duality argument, applied to the case where the subgroup H is in fact equal to G. • The Boltzmann-Grad limit for periodic Lorentz gas has been studied by Marklof and Strömbergsson [MS1,MS2]. Using periodicity, one can reduce the original problem to analysing distribution on the space of lattices…”