Decay of classical chaotic systems: The case of the Bunimovich stadium

Abstract

The escape of an ensemble of particles from the Bunimovich stadium via a small hole has been studied numerically. The decay probability starts out exponentially but has an algebraic tail. The weight of the algebraic decay tends to zero for vanishing hole size. This behaviour is explained by the slow transport of the particles close to the marginally stable bouncing ball orbits. It is contrasted with the decay function of the corresponding quantum system.

Poincaré recurrences and transient chaos in systems with leaks

“…[44]). The above interpretation of p(n) given by (25) expresses the view, first suggested in [32], that the nonhyperbolic component of the saddle is approached through the hyperbolic part [32,50,52,54], i.e., the fraction of The unstable manifold of the hyperbolic part of the saddle ( black dots), obtained by using the method of [26] choosing an escape time ns = 80 < nc = 307. The support of ρr is fully outside the unstable manifold.…”

On Chaotic Dynamics in Rational Polygonal Billiards

“…In chaotic closed and weakly open classical systems (including Hamiltonian systems), exemplified by BB [23,24,25], infinite-horizon SB [20,26], and by corresponding low-density LG [14,27], the overall algebraic decay was found numerically through the geometry-dependent exponents δ ≥ 1. The channel of algebraic-type relaxation with δ = 1, common for both chaotic [20,24,26] and non-chaotic [10,22] billiards, seems to be generic for all incompletely hyperbolic systems with smooth convex boundaries.…”

“…A temporal behavior of N (t) non-escaped orbits (particles) is commonly scaled by the mean escape time [7,10,22,24] τ e = τ c P ∆ with ∆ P,…”

Particle–wall collision statistics in the open circular billiard

“…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.…”

“…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.br…”