Kinetic transport in the two-dimensional periodic Lorentz gas

Abstract

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.One of the central challenges in kinetic theory is the derivation of macroscopic evolution equations-describing for example the dynamics of an electron gas-from the underlying fundamental microscopic laws of classical or quantum mechanics. An iconic mathematical model in this research area is the Lorentz gas [15], which describes an ensemble of non-interacting point particles in an infinite array of spherical scatterers. In the case of a disordered scatterer configuration, well known results by Gallavotti [12], Spohn [22], and Boldrighini, Bunimovich and Sinai [5] show that the time evolution of a macroscopic particle cloud is governed, in the limit of small scatterer density (Boltzmann-Grad limit), by the linear Boltzmann equation. We have recently proved an analogous statement for a periodic configuration of scatterers [16], [17]. In this case the linear Boltzmann equation fails (cf. also Golse [13]), and the random flight process that emerges in the Boltzmann-Grad limit is substantially more complicated.In the present paper we focus on the two-dimensional case, and derive explicit formulae for the collision kernels of the limiting random flight process. These include information not only of the velocity before and after the collision (as in the case of the linear Boltzmann equation), but also the path length until the next hit and the velocity thereafter. Our formulae thus generalize those for the limiting distributions of the free path length found by Dahlqvist [10] Our results on the Boltzmann-Grad limit complement classical studies in ergodic theory, where the scatterer size remains fixed. Bunimovich and Sinai [7] showed that the dynamics of the two-dimensional periodic Lorentz gas is diffusive in the limit of large times, and satisfies a central limit theorem. They assumed that the periodic Lorentz gas has a finite horizon, i.e., the scatterers are configured in such a way that the path length between consecutive collisions is bounded. The corresponding result for infinite horizon has recently been established by Szasz and Varju [21] following initial work by Bleher [2]. See also the recent papers by Dolgopyat, Szasz and Varju [11], and Melbourne and Nicol [19], [20] for related studies of statistical properties of the two-dimensional periodic Lorentz gas. The case of higher dimensions is still open, even for models with finite horizon, cf. Chernov [9], and Balint and Toth [1]. Scaling limits that are intermediate between the ...

Free Path Lengths in Quasicrystals

“…The distribution of free path lengths in the limit of small scatterer density (Boltzmann-Grad limit) has been studied extensively when P is a fixed realisation of a random point process (such as a spatial Poisson process) [5,13,26,35] and when P is a Euclidean lattice [1,2,8,9,11,12,14,19,24,26]. In the Boltzmann-Grad limit, the Lorentz process in fact converges to a random flight process, see [13,35,5] for the case of random P and [10,20,21,22] for periodic P.…”

Superdiffusion in the Periodic Lorentz Gas

“…This formula has been derived, independently and with different methods, by Marklof and Strömbergsson [19], Caglioti and Golse [11,12] and by Bykovskii and Ustinov [10].…”

“…The upper bound (6.16) follows from (5.7) and (5.9): for ϕ ∈ [0, 19) and for ϕ ∈ [ π 2 , π], we have…”

Diameters of random circulant graphs

“…The function F 0,0 (0, σ) is in turn related to the free path length Φ 0 (ξ) of the two-dimensional periodic Lorentz gas via formula (4.3) in [30]. This implies for the density ofP 2,scl (R): The explicit formula for Φ 0 in [5] (denoted there by h; the formula can also be obtained from [29,Eqs. (15) and (34)] or from [37, Prop.…”