The Statistics of the Trajectory of a Certain Billiard in a Flat Two-Torus

Abstract

Abstract. We consider a billiard in the punctured torus obtained by removing a small disk of radius ε > 0 from the flat torus T 2 , with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity ω only. Letτε(ω), and respectivelyRε(ω), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when T 2 is being identified with [0, 1) 2 . We prove that the probability measures on [0, ∞) associated with the random variables ετε and εRε are weakly convergent as ε → 0 + and explicitly compute the densities of the limits.

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

“…The free path in a square billiard with trajectory starting at the center Theorem 1.3 should be compared with the situation where the initial position is at one of the four vertices, and where the limiting distribution has compact support [2,3]. It is not clear whether these methods would directly extend to other concrete initial positions, such as (…”

“…As a result any trajectory with slope tan ω ∈ (γ, γ ) will intersect, as in the case of the square lattice [2,3], one of the scatterers (q, a) + V ε or (q , a ) + V ε .…”

“…with repartition function G given by Part (ii) of Theorem 1.1 can now be deduced from part (i) by a standard approximation argument [1,2,3,6] based on τ ,ε (ω) ≈ q ,ε (ω) cos ω for tan ω ∈ I with I ⊆ [0, 1] interval of length |I| ε c . We skip the detailed proof, which can be easily reconstructed from some of the arguments in the next section.…”

“…Recent progress led to a better understanding of the statistics of the free path length of the periodic Lorentz gas in the small scatterer limit [2,3,5,6,7,8,9,10]. The aim of this paper is to study situations where periodicity conditions are altered by imposing certain congruence conditions on the integer lattice points where scatterers are placed.…”

“…The case of the honeycomb lattice arises as a particular example in this context. In this paper we only consider the situation where motion originates at the origin, extending the results from [2] and [3]. The case where the initial position is randomly chosen is more intricate and will not be treated here.…”