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

Florin P. Boca[1]; Radu N. Gologan[2]

  • [1] University of Illinois at Urbana-Champaign Department of Mathematics 1409 W. Green St. Urbana, IL 61801 (USA)
  • [2] Institute of Mathematics of the Romanian Academy P.O.Box 1-764 Bucharest 014700 (Romania)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 3, page 1043-1075
  • ISSN: 0373-0956

Abstract

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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 → 0 + . For every integer number ℓ ≥ 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 → 0 + centered at the corners and trajectory starting at the center in a regular hexagon, respectively in a square.

How to cite

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Boca, Florin P., and Gologan, Radu N.. "On the distribution of the free path length of the linear flow in a honeycomb." Annales de l’institut Fourier 59.3 (2009): 1043-1075. <http://eudml.org/doc/10416>.

@article{Boca2009,
abstract = {Consider the region obtained by removing from $\{\mathbb\{R\}\}^2$ the discs of radius $\varepsilon $, centered at the points of integer coordinates $(a,b)$ with $b\nequiva \hspace\{4.44443pt\}(\@mod \; \ell )$. We are interested in the distribution of the free path length (exit time) $\tau _\{\ell ,\varepsilon \} (\omega )$ of a point particle, moving from $(0,0)$ along a linear trajectory of direction $\omega $, as $\varepsilon \rightarrow 0^+$. For every integer number $\ell \ge 2$, we prove the weak convergence of the probability measures associated with the random variables $\varepsilon \tau _\{\ell ,\varepsilon \}$, explicitly computing the limiting distribution. For $\ell =3$, respectively $\ell =2$, this result leads to asymptotic formulas for the exit time of a billiard with pockets of radius $\varepsilon \rightarrow 0^+$ centered at the corners and trajectory starting at the center in a regular hexagon, respectively in a square.},
affiliation = {University of Illinois at Urbana-Champaign Department of Mathematics 1409 W. Green St. Urbana, IL 61801 (USA); Institute of Mathematics of the Romanian Academy P.O.Box 1-764 Bucharest 014700 (Romania)},
author = {Boca, Florin P., Gologan, Radu N.},
journal = {Annales de l’institut Fourier},
keywords = {Periodic Lorentz gas; linear flow; Farey fractions; honeycomb lattice; periodic Lorentz gas},
language = {eng},
number = {3},
pages = {1043-1075},
publisher = {Association des Annales de l’institut Fourier},
title = {On the distribution of the free path length of the linear flow in a honeycomb},
url = {http://eudml.org/doc/10416},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Boca, Florin P.
AU - Gologan, Radu N.
TI - On the distribution of the free path length of the linear flow in a honeycomb
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 3
SP - 1043
EP - 1075
AB - Consider the region obtained by removing from ${\mathbb{R}}^2$ the discs of radius $\varepsilon $, centered at the points of integer coordinates $(a,b)$ with $b\nequiva \hspace{4.44443pt}(\@mod \; \ell )$. We are interested in the distribution of the free path length (exit time) $\tau _{\ell ,\varepsilon } (\omega )$ of a point particle, moving from $(0,0)$ along a linear trajectory of direction $\omega $, as $\varepsilon \rightarrow 0^+$. For every integer number $\ell \ge 2$, we prove the weak convergence of the probability measures associated with the random variables $\varepsilon \tau _{\ell ,\varepsilon }$, explicitly computing the limiting distribution. For $\ell =3$, respectively $\ell =2$, this result leads to asymptotic formulas for the exit time of a billiard with pockets of radius $\varepsilon \rightarrow 0^+$ centered at the corners and trajectory starting at the center in a regular hexagon, respectively in a square.
LA - eng
KW - Periodic Lorentz gas; linear flow; Farey fractions; honeycomb lattice; periodic Lorentz gas
UR - http://eudml.org/doc/10416
ER -

References

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  1. F. P. Boca, C. Cobeli, A. Zaharescu, Distribution of lattice points visible from the origin, Comm. Math. Phys. 213 (2000), 433-470 Zbl0989.11049MR1785463
  2. F. P. Boca, R. N. Gologan, A. Zaharescu, The average length of a trajectory in a certain billiard in a flat two-torus, New York J. Math. (electronic) 9 (2003), 303-330 Zbl1066.37021MR2028172
  3. F. P. Boca, R. N. Gologan, A. Zaharescu, The statistics of the trajectory of a certain billiard in a flat two-torus, Comm. Math. Phys. 240 (2003), 53-73 Zbl1078.37006MR2004979
  4. F. P. Boca, A. Zaharescu, On the correlations of directions in the Euclidean plane, Trans. Amer. Math. Soc. 358 (2006), 1797-1825 Zbl1154.11022MR2186997
  5. F. P. Boca, A. Zaharescu, The distribution of the free path lengths in the periodic two-dimensional Lorentz gas in the small-scatterer limit, Comm. Math. Phys. 269 (2007), 425-471 Zbl1143.37002MR2274553
  6. E. Caglioti, F. Golse, On the distribution of free path lengths for the periodic Lorentz gas. III., Comm. Math. Phys. 236 (2003), 199-221 Zbl1041.82016MR1981109
  7. E. Caglioti, F. Golse, The Boltzman-Grad limit of the periodic Lorentz gas in two space dimensions, C. R. Math. Acad. Sci. Paris 346 (2008), 477-482 Zbl1145.82019MR2417573
  8. P. Dahlqvist, The Lyapunov exponent in the Sinai billiard in the small scatterer limit, Nonlinearity 10 (1997), 159-173 Zbl0907.58038MR1430746
  9. F. Golse, The periodic Lorentz gas in the Boltzman-Grad limit, International Congress of Mathematicians III (2006), 183-201, Eur. Math. Soc., Zürich Zbl1109.82020MR2275676
  10. J. Marklof, A. Strömbergsson, The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems Zbl1211.82011

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