On the distribution of the free path length of the linear flow in a honeycomb

Boca, Florin P.; Gologan, Radu N.

On the distribution of the free path length of the linear flow in a honeycomb
[Sur la distribution du temps de sortie pour le flot linéaire dans un réseau hexagonal]

Boca, Florin P. ; Gologan, Radu N.

Annales de l'Institut Fourier, Tome 59 (2009), p. 1043-1075 / Harvested from Numdam

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Résumé
Abstract

Nous considérons la région obtenue en enlevant de $ℝ^{2}$ les disques de rayon $ε$ , centrés aux points de coordonnées entières $(a, b)$ avec $b ≢ a (mod ℓ)$ . Nous étudions la répartition de la longueur du libre parcours (temps de sortie) $τ_{ℓ, ε} (ω)$ d’une particule ponctuelle, partant de $(0, 0)$ sur une trajectoire rectiligne de direction $ω$ quand $ε \to 0^{+}$ . Pour tout nombre entier $ℓ \geq 2$ , on montre la convergence faible des mesures de probabilité attachées aux variables aléatoires $ε τ_{ℓ, ε}$ , en calculant la distribution limite d’une manière explicite. Pour $ℓ = 3$ , respectivement $ℓ = 2$ , ce résultat mène à des formules asymptotiques pour le temps de sortie d’un billard avec des poches de rayon $ε \to 0^{+}$ centrés aux coins dans un hexagone régulier, respectivement dans un carré.

Consider the region obtained by removing from $ℝ^{2}$ the discs of radius $ε$ , centered at the points of integer coordinates $(a, b)$ with $b ≢ a (mod ℓ)$ . We are interested in the distribution of the free path length (exit time) $τ_{ℓ, ε} (ω)$ of a point particle, moving from $(0, 0)$ along a linear trajectory of direction $ω$ , as $ε \to 0^{+}$ . For every integer number $ℓ \geq 2$ , we prove the weak convergence of the probability measures associated with the random variables $ε τ_{ℓ, ε}$ , explicitly computing the limiting distribution. For $ℓ = 3$ , respectively $ℓ = 2$ , this result leads to asymptotic formulas for the exit time of a billiard with pockets of radius $ε \to 0^{+}$ centered at the corners and trajectory starting at the center in a regular hexagon, respectively in a square.

Publié le : 2009-01-01

MR 2543662 | Zbl 1173.37036

DOI : https://doi.org/10.5802/aif.2457
Classification: 11P21, 37D50, 82C40
Mots clés: Gaz de Lorentz périodique, flot linéaire, suite de Farey, réseau hexagonal

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     author = {Boca, Florin P. and Gologan, Radu N.},
     title = {On the distribution of the free path length of the linear flow in a honeycomb},
     journal = {Annales de l'Institut Fourier},
     volume = {59},
     year = {2009},
     pages = {1043-1075},
     doi = {10.5802/aif.2457},
     zbl = {1173.37036},
     mrnumber = {2543662},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2009__59_3_1043_0}
}

Boca, Florin P.; Gologan, Radu N. On the distribution of the free path length of the linear flow in a honeycomb. Annales de l'Institut Fourier, Tome 59 (2009) pp. 1043-1075. doi : 10.5802/aif.2457. http://gdmltest.u-ga.fr/item/AIF_2009__59_3_1043_0/

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