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### Particle-wall collision statistics in the open circular billiard

In the open circular billiard particles are placed initially with a uniform
distribution in their positions inside a planar circular vesicle. They all have
velocities of the same magnitude, whose initial directions are also uniformly
distributed. No particle-particle interactions are included, only specular
elastic collisions of the particles with the wall of the vesicle. The particles
may escape through an aperture with an angle $2\delta$. The collisions of the
particles with the wall are characterized by the angular position and the angle
of incidence. We study the evolution of the system considering the probability
distributions of these variables at successive times $n$ the particle reaches
the border of the vesicle. These distributions are calculated analytically and
measured in numerical simulations. For finite apertures $\delta<\pi/2$, a
particular set of initial conditions exists for which the particles are in
periodic orbits and never escape the vesicle. This set is of zero measure, but
the selection of angular momenta close to these orbits is observed after some
collisions, and thus the distributions of probability have a structure formed
by peaks. We calculate the marginal distributions up to $n=4$, but for
$\delta>\pi/2$ a solution is found for arbitrary $n$. The escape probability as
a function of $n^{-1}$ decays with an exponent 4 for $\delta>\pi/2$ and
evidences for a power law decay are found for lower apertures as well.Comment: 11 pages, 14 figures. Typos corrected and two new figures added,
figure captions changed and additional discussions added. Version accepted
for publication in Physica

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