Escape and transport for an open bouncer: Stretched exponential decays

Abstract We consider time-dependence of dynamical transport, following a recent study of the stadium billiard in which classical transmission and reflection probabilities were shown to exhibit exponential or algebraic decays depending on the choice of the lead or “hole”, raising the question of whether this feature is due to special properties of the stadium. The system considered here is much more general, having a generic mixed phase space structure, time-dependence of the dynamics, and Fermi acceleration (trajectories with unbounded velocity). We propose an efficient numerical scheme for this model, observe escape and transport effects including similar asymmetry, and also clear stretched exponential decays.

[1]  W. Haas,et al.  Untersuchungen Über den Verlauf des Eindringens Eines Transversalen Magnetfeldes in Einen Supraleiter , 1934 .

[2]  Alexander Loskutov,et al.  Properties of Some Chaotic Billiards with Time-Dependent Boundaries , 2000 .

[3]  A. Loskutov,et al.  Mechanism of Fermi acceleration in dispersing billiards with time-dependent boundaries , 1999 .

[4]  M. Zworski,et al.  Fractal Weyl laws for chaotic open systems. , 2002, Physical review letters.

[5]  Stephen Wiggins,et al.  Escape from planetary neighbourhoods , 2005, Monthly Notices of the Royal Astronomical Society.

[6]  Enrico Fermi,et al.  On the Origin of the Cosmic Radiation , 1949 .

[7]  B. Dietz,et al.  Avoided-level-crossing statistics in open chaotic billiards. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  O. Piro,et al.  Opening up fractal structures of three-dimensional flows via leaking , 2004 .

[9]  Matthew Wright,et al.  New Directions in Linear Acoustics and Vibration: Index , 2010 .

[10]  G. Cristadoro,et al.  Universality of algebraic decays in Hamiltonian systems. , 2008, Physical review letters.

[11]  E. Altmann,et al.  Poincaré recurrences and transient chaos in systems with leaks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Dmitry Turaev,et al.  Fermi acceleration in non-autonomous billiards , 2008 .

[13]  L. Bunimovich,et al.  Open circular billiards and the Riemann hypothesis. , 2004, Physical review letters.

[14]  Edson D. Leonel,et al.  A hybrid Fermi–Ulam-bouncer model , 2005 .

[15]  Luna-Acosta.  Regular and chaotic dynamics of the damped Fermi accelerator. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[16]  Mark F. Demers,et al.  Escape rates and conditionally invariant measures , 2006 .

[17]  G. Keller,et al.  Rare Events, Escape Rates and Quasistationarity: Some Exact Formulae , 2008, 0810.2229.

[18]  Takahisa Harayama,et al.  Quantum Chaos and Quantum Dots , 2004 .

[19]  H. Buljan,et al.  Many-hole interactions and the average lifetimes of chaotic transients that precede controlled periodic motion. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  J. Sjöstrand.  Geometric bounds on the density of resonances for semiclassical problems , 1990 .

[21]  Orestis Georgiou,et al.  Transmission and reflection in the stadium billiard: time-dependent asymmetric transport. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  R. H. Cohen,et al.  Fermi acceleration revisited , 1980 .

[23]  Edson D. Leonel,et al.  Describing Fermi acceleration with a scaling approach: The Bouncer model revisited , 2008 .

[24]  H. Kantz,et al.  Stickiness in Hamiltonian systems: from sharply divided to hierarchical phase space. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  C. Dettmann,et al.  Peeping at chaos: Nondestructive monitoring of chaotic systems by measuring long-time escape rates , 2006, nlin/0610013.

[26]  L. Bunimovich,et al.  Where to place a hole to achieve a maximal escape rate , 2008, 0811.4438.

[27]  Didier Sornette,et al.  Transient chaos in room acoustics. , 1993, Chaos.

[28]  F. Saif.  Classical and quantum chaos in atom optics , 2005 .

[29]  Stephen Wiggins,et al.  Microcanonical rates, gap times, and phase space dividing surfaces. , 2009, The Journal of chemical physics.

[30]  E. Leonel,et al.  Scaling properties of the regular dynamics for a dissipative bouncing ball model , 2007 .

[31]  Orestis Georgiou,et al.  Survival probability for the stadium billiard , 2008, 0812.3095.

[32]  中村 勝弘,et al.  Quantum chaos and quantum dots , 2004 .

[33]  B. Chirikov.  A universal instability of many-dimensional oscillator systems , 1979 .

[34]  L. Bunimovich,et al.  Suppressing Fermi acceleration in a driven elliptical billiard. , 2010, Physical review letters.

[35]  Hazime Mori,et al.  Anomalous Diffusion Due to Accelerator Modes in the Standard Map , 1991 .

[36]  Focusing Components in Typical Chaotic Billiards Should be Absolutely Focusing , 2009 .

[37]  J. Wiersig,et al.  Fractal Weyl law for chaotic microcavities: Fresnel's laws imply multifractal scattering. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.