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The periodic Lorentz gas in the Boltzmann-Grad limit: asymptotic estimates. (English) Zbl 1242.82036

According to the authors, the linear Boltzmann equation that describe the macroscopic dynamics of a dilute gas in matter was postulated by Lorentz in 1905 by considering a gas of non-interacting point particles moving in an infinite, fixed array of hard sphere scatterers. Moreover, crucially, Lorentz assumed that, in the limit of small scatterers (Boltzmann-Grad limit), consecutive collisions become independent of each other and are solely determined by the single scatterer cross-section. In studies of periodic scatterer configurations [Ann. Math. (2) 172, No. 3, 1949–2033 (2010; Zbl 1211.82011); Ann. Math. (2) 174, No. 1, 225–298 (2011; Zbl 1237.37014)], the authors showed that, in this case, the Boltzmann-Grad limit is governed by a transport equation which is distinctly different from the linear Boltzmann equation. The authors stated that one of the main features is that (contrary to Lorentz’ assumption) consecutive collisions are no longer independent, i.e., the collision kernel of the transport equation does not only depend on the particle velocity before and after each collision, but also on the flight time until the next collision and the velocity thereafter. The collision kernel is thus significantly more complicated than in the linear Boltzmann equation, and explicit formulas are so far only known in dimension \(d = 2\) for different approaches. The objective of the present paper is to derive asymptotic estimates for the collision kernel for small and large inter-collision times when dimension \(d \geq 3\). These estimates yield, in particular, precise asymptotics for the distribution of the free path length in the periodic Lorentz gas.

MSC:

82C40 Kinetic theory of gases in time-dependent statistical mechanics
37A60 Dynamical aspects of statistical mechanics
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
11H06 Lattices and convex bodies (number-theoretic aspects)
35Q20 Boltzmann equations
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