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• W1969926935 abstract "We report numerical results of the investigation of coarse-grained properties of the classical orbits of the non-integrable stadium having a linear part of extension ε. More precisely, we investigate properties of orbits starting inside a small disk of radius r and returning to it for the first time after a distance L. The dependences of the orbit length L on the launching angle as well as the number of orbits N(L) are explored. Four different regimes are discussed: (1) For L≤〈h〉-1, where 〈h〉 is the Kolmogorov entropy, orbits do not feel the mixing character of the dynamics and correspond to the integrable case where the stadium reduces to a disk. For sufficiently small r, N(L) ∼ exp(-β0rL). This can be rationalized from ergodic properties of irrational numbers. (2) For h-1≤L and sufficiently small r, N(L) can be fitted by a pure exponential N(L) ∼ exp[-β(ε)rL] with β(ε) = (π/2+ε)-1. This regime comes from the strong chaotic character of the orbits. (3 ) For h-1≤L and larger r, N(L) is no more a pure exponential. The typical orbit length is too small to achieve a sufficient mixing. N(L) can be fitted by an exponential with power-law corrections or by a stretched exponential. (4) For very large L, N(L) should eventually cross over to an algebraic decay N(L ∼ L-2. This power-law tail is the result of th existence of nearly trapped vertical bouncing-ball trajectories, which are very rare but nevertheless dominant at sufficiently large L. We do not observe this tail because of our condition that orbits which end on the disk have also started inside it. Understanding the properties of coarse-grained trajectories is interesting for (1) connecting the dynamical properties of billiards (internal problem) to the (external) problem of chaotic scattering, (2) for defining tests of the general statistical laws of geometrical acoustics in auditoriums (room acoustics) and (3) for exploring one aspect of the correspondence between classical and quantum chaos, namely, the introduction of a finite de Broglie wavelength." @default.
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• W1969926935 date "1990-08-01" @default.
• W1969926935 modified "2023-10-18" @default.
• W1969926935 title "Coarse-grained properties of the chaotic trajectories in the stadium" @default.
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• W1969926935 doi "https://doi.org/10.1016/0167-2789(90)90057-v" @default.
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