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On the distribution of free path lengths for the periodic Lorentz gas. III. (English) Zbl 1041.82016

[For part II see F. Golse and B. Wennberg, Math. Model. Numer. Anal. 34, No. 6, 1151-1163 (2000; Zbl 1006.82025).]
Summary: For \(r\in (0,1)\), let \(Z_r = \{x\in \mathbb{R}^2 | \text{dist}(x,\mathbb{Z}^2)>r/2\}\) and define \[ \tau_r(x,v) = \inf \{t>0 | x+tv \in \partial Z_r\}. \] Let \(\Phi_r(t)\) be the probability that \(\tau_r(x,v) \geq t\) for \(x\) and \(v\) uniformly distributed in \(Z_r\) and \(\mathbf S^1\) respectively. We prove in this paper that \[ \begin{aligned} \limsup_{\varepsilon\to0^+} \frac1{|\ln\varepsilon|} \int_\varepsilon^{1/4} \Phi_r \left(\frac tr\right) \frac{dr}r &= \frac2{\pi^2t} + O\left(\frac1{t^2}\right),\\ \liminf_{\varepsilon\to0^+} \frac1{|\ln\varepsilon|} \int_\varepsilon^{1/4} \Phi_r \left(\frac tr\right) \frac{dr}r &= \frac2{\pi^2t} +O\left(\frac1{t^2}\right)\end{aligned} \] as \(t\to +\infty\). This result improves upon the bounds on \(\Phi_r\) in the paper of J. Bourgain, F. Golse and B. Wennberg [Commun. Math. Phys. 190, No. 3, 491–508 (1998; Zbl 0910.60082)]. We also discuss the applications of this result in the context of kinetic theory.

MSC:

82C40 Kinetic theory of gases in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
37A60 Dynamical aspects of statistical mechanics
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