Particle–wall collision statistics in the open circular billiard

Abstract

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 2δ. 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 δ < π/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 δ > π/2 a solution is found for arbitrary n. The escape probability as a function of n −1 decays with an exponent 4 for δ > π/2 and evidences for a power law decay are found for lower apertures as well.PACS numbers: 45.50.-j,05.20.-y,02.70.-c I. INTRODUCTIONIn the theory of dynamical systems, the study of the decay of simple hamiltonian systems [1]- [5] is of much interest. These studies are an extension of earlier research concerning the closed version of such systems, where the main question is their ergodicity. The pioneering work in this area is centered on the system called Sinai billiard, a circular billiard with a smaller circular exclusion area in its interior [6], which was shown to be ergodic. Other two-dimensional ballistic billiards are known to be ergodic as well, such as the Bunimowitch stadium [4,7]. One main interest in the studies of open billiards is the decay dynamics for long times. Bauer and Bertsch [1] found an exponential decay in an chaotic dynamics and a power law decay for a system with regular dynamics. These first results concerning the integrable case where later questioned by Legrand and Sornette [2], but it became clear that the difficulty in settling this question using numerical experiments is related to the high sensitivity of the results to initial conditions [3]. In a more detailed simulational study for the chaotic two-dimensional Bunimovich stadium [4], algebraic tails were found at sufficiently long times, but the weight of the algebraic tail tends to zero in the limit where the size of the aperture vanishes.In the classical circular billiard the (non-interacting) particles undergo elastic specular collisions with the wall. * Electronic address: jstilck@if.uff.brSince two quantities are conserved (kinetic energy and angular momentum), the system is integrable and therefor...

Classical dynamics and particle transport in kicked billiards

“…The initial momenta of all particles are assumed to be equal p 0 . Classical dynamics of unperturbed open billiards has been extensively studied both for integrable and chaotic geometries [24]- [30]. In particular, it was found in the Refs.…”

“…We note that classical dynamics of unperturbed open billiards have been studied for integrable and chaotic geometries by many authors (see, e.g., Refs. [24]- [30]). In particular, it was found that the number of (non-interacting) particles in nonintegrable open billiard decreases exponentially, while in case of regular billiard it decreases according to power law [24]- [26].…”