The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems

Abstract

The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the Boltzmann-Grad limit, where the radius of each scatterer tends to zero, and prove the existence of a limiting distribution for the free path length. We also discuss related problems, such as the statistical distribution of directions of lattice points that are visible from a fixed position.

Spherical averages in the space of marked lattices

“…Proof The proof follows from Corollary 6.5 in the case of one-dimensional distributions (n = 1) by the same arguments as in [17]. This proves the existence of the limits with Note that the limit process is independent of the choice of ξ when ξ / ∈ Q d , and only depends on the denominator of ξ when ξ ∈ Q d \ Z d ; cf.…”

“…The distribution of the free path length is well understood for random [2], periodic [1,5,17] and quasiperiodic [19,20] scatterer configurations. We will here consider the periodic Lorentz gas with random defects introduced in the previous section, where the scatterers are placed at the defect lattice Note that the papers [4,23] discuss the convergence of a defect periodic Lorentz gas to a random flight process governed by the linear Boltzmann equation in the limit when the removal probability of a scatterer tends to one.…”

“…As in [17], we will consider more general initial conditions, which for instance permit us to launch a particle from the boundary of a scatterer (which is moving as r → 0). Proof The proof follows from Corollary 6.5 in the case of one-dimensional distributions (n = 1) by the same arguments as in [17].…”

“…Spherical averages were used in [17] and [19] to establish the limit distribution for the free path length in crystals and quasicrystals, respectively. The plan for the remainder of this paper is to explain how spherical averages on marked lattices can be exploited to yield the path length distribution for crystals with random defects.…”

“…If the space X is homogeneous, then the weak convergence of the spherical average P t can be proved for all x 0 , with a complete classification of all limit measures, by means of measure rigidity techniques that are based on Ratner's measure classification theorem for subgroups generated by unipotent elements [22]. There is now a large body of literature on this topic, see for instance [7,8,16,17,19,24]. In some settings, spherical equidistribution may also be deduced directly from the mixing property of R [7].…”