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On the distribution of free path lengths for the periodic Lorentz gas II

François Golse, Bernt Wennberg (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Consider the domain Z ϵ = { x n ; d i s t ( x , ϵ n ) > ϵ γ } and let the free path length be defined as τ ϵ ( x , v ) = inf { t > 0 ; x - t v Z ϵ } . In the Boltzmann-Grad scaling corresponding to γ = n n - 1 , it is shown that the limiting distribution φ ϵ of τ ϵ is bounded from below by an expression of the form C/t, for some C> 0. A numerical study seems to indicate that asymptotically for large t, φ ϵ C / t . This is an extension of a previous work [J. Bourgain et al., Comm. Math. Phys.190 (1998) 491-508]. As a consequence, it is proved that the linear Boltzmann type transport equation is inappropriate...

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

Florin P. Boca, Radu N. Gologan (2009)

Annales de l’institut Fourier

Consider the region obtained by removing from 2 the discs of radius ε , centered at the points of integer coordinates ( a , b ) with b a ( mod ) . We are interested in the distribution of the free path length (exit time) τ , ε ( ω ) of a point particle, moving from ( 0 , 0 ) along a linear trajectory of direction ω , as ε 0 + . For every integer number 2 , we prove the weak convergence of the probability measures associated with the random variables ε τ , ε , explicitly computing the limiting distribution. For = 3 , respectively = 2 , this result leads...

On the finite blocking property

Thierry Monteil (2005)

Annales de l’institut Fourier

A planar polygonal billiard 𝒫 is said to have the finite blocking property if for every pair ( O , A ) of points in 𝒫 there exists a finite number of “blocking” points B 1 , , B n such that every billiard trajectory from O to A meets one of the B i ’s. Generalizing our construction of a counter-example to a theorem of Hiemer and Snurnikov, we show that the only regular polygons that have the finite blocking property are the square, the equilateral triangle and the hexagon. Then we extend this result to translation surfaces....

On the Hausdorff dimension of piecewise hyperbolic attractors

Tomas Persson (2010)

Fundamenta Mathematicae

We study non-invertible piecewise hyperbolic maps in the plane. The Hausdorff dimension of the attractor is calculated in terms of the Lyapunov exponents, provided that the map satisfies a transversality condition. Explicit examples of maps for which this condition holds are given.

Periodic billiard orbits in right triangles

Serge Troubetzkoy (2005)

Annales de l’institut Fourier

There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic, and (iii) the perpendicular periodic orbits fill the corresponding invariant surface.

Planar Lorentz process in a random scenery

Françoise Pène (2009)

Annales de l'I.H.P. Probabilités et statistiques

We consider the periodic planar Lorentz process with convex obstacles (and with finite horizon). In this model, a point particle moves freely with elastic reflection at the fixed convex obstacles. The random scenery is given by a sequence of independent, identically distributed, centered random variables with finite and non-null variance. To each obstacle, we associate one of these random variables. We suppose that each time the particle hits an obstacle, it wins the amount given by the random variable...

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