Time-dependent dynamics

    It is impossible to completely shield any physical system from the environment, so the latter must be modelled, by either deterministic or stochastic forcing, or thermostats. Stochastic systems can be understood using periodic orbits of the unperturbed dynamics. Depending on the parameters, the forcing can destabilised or, remarkably, stabilise, the motion. In the context of billiards we can consider time-dependent (eg sinusoidally vibrating) walls. In each collision the particle normally changes its speed. Often, many such collisions occur with effectively random changes in speed, well approximated by a normal diffusion process. If the speed is unbounded, this leads to the phenomenon of Fermi acceleration, originally proposed to understand the origin of high energy cosmic rays.

  1. Stochastic dynamics of relativistic turbulence. C. P. Dettmann and N. E. Frankel, Phys. Rev. E 53, 5502-5505 (1996) pdf
  2. Trace formulas for stochastic evolution operators: Weak noise perturbation theory P. Cvitanovic', C. P. Dettmann, R. Mainieri, and G. Vattay, J. Stat. Phys. 93, 981-999 (1998) ps.gz (1.2M when uncompressed) arxiv
  3. Traces and determinants of strongly stochastic operators C. P. Dettmann, Phys. Rev. E 59, 5231-5234 (1999) pdf ps html arxiv
  4. Trace formulas for stochastic evolution operators: Smooth conjugation method P. Cvitanovic', C. P. Dettmann, R. Mainieri, and G. Vattay, Nonlinearity 12, 939-953 (1999) ps arxiv
  5. Spectrum of stochastic evolution operators: Local matrix representation approach P. Cvitanovic', N. Sondergaard, G. Palla, G. Vattay, and C. P. Dettmann, Phys. Rev. E 60, 3936-3941 (1999) pdf ps Two distinct arxiv versions:arxiv arxiv
  6. Noise corrections to stochastic trace formulas G. Palla, G. Vattay, A. Voros, N. Sondergaard, C. P. Dettmann, Found. Phys. 31, 641-657 (2001). arxiv
  7. Fractal asymptotics, C. P. Dettmann, Physica D 187, 214-222 (2004). ps arxiv
  8. Stochastic stabilization of cosmological photons. C. P. Dettmann, J. P. Keating, and S. D. Prado, J. Phys. A.: Math. Theor.37 L377-L383 (2004) pdf. arxiv animation (0.3M) [Featured in the June 2010 issue of Mathematics Today, pdf available here.]
  9. Stochastic stabilization of chaos and the cosmic microwave background. C. P. Dettmann, J. P. Keating and S. D. Prado, Int. J. Mod. Phys. D 13 1461-1468 (2004). pdf
  10. Thermostats for "slow" configurational modes A. A. Samoletov, C. P. Dettmann and M. A. J. Chaplain, J. Stat. Phys. 128 1321-1336 (2007) [Origin of the "Nose-Hoover-Langevin" thermostat] pdf arxiv
  11. Notes on configurational thermostat schemes, A. A. Samoletov, C. P. Dettmann and M. A. J. Chaplain, J. Chem. Phys. 132 246101 (2010) pdf arxiv
  12. Escape of particles in a time-dependent potential well, D. R. Costa, C. P. Dettmann and E. D. Leonel, Phys. Rev. E, 83 066211 (2011). pdf
  13. Scaling invariance for the escape of particles from a periodically corrugated waveguide, E. D. Leonel, D. R. Costa and C. P. Dettmann, Phys. Lett. A 376 421-425 (2012). pdf
  14. Escape and transport for an open bouncer: Stretched exponential decays, C. P. Dettmann and E. D. Leonel, Physica D 241 403-408 (2012). pdf arxiv
  15. Recurrence of particles in static and time varying oval billiards, E. D. Leonel and C. P. Dettmann, Phys. Lett. A 376 1669-1674 (2012). pdf arxiv
  16. Stickiness in a bouncer model: A slowing mechanism for Fermi acceleration, A. L. P. Livorati, T. Kroetz, C. P. Dettmann, I. L. Caldas, E. D. Leonel, Phys. Rev. E 86 036203 (2012). pdf arxiv
  17. Scaling invariance of the diffusion coefficient in a family of two-dimensional Hamiltonian mappings, J. A. de Oliveira, C. P. Dettmann, D. R. Costa and E. D. Leonel, Phys. Rev. E 87 062904 (2013). pdf
  18. Periodic compression of an adiabatic gas: Intermittency enhanced Fermi acceleration, C. P. Dettmann and E. D. Leonel, EPL 103 40003 (2013). pdf
  19. Escape through a time-dependent hole in the doubling map, A. L. P. Livorati, O. Georgiou, C. P. Dettmann and E. D. Leonel, Phys. Rev. E, 89 052913 (2014). arxiv pdf
  20. Separation of particles leading to decay and unlimited growth of energy in a driven stadium-like billiard, A. L. P. Livorati, M. S. Palmero, C. P. Dettmann, I. L. Caldas, E. D. Leonel, J. Phys. A.: Math. Theor., 47 365101 (2014). arxiv
  21. Transport and dynamical properties for a bouncing ball model with regular and stochastic perturbations, D. R. Costa, C. P. Dettmann and E. D. Leonel, Commun. Nonlin. Sci. Numer. Sim. 20 871-881 (2015). pdf
  22. Circular, elliptic and oval billiards in a gravitational field, D. R. Costa, C. P. Dettmann and E. D. Leonel, Commun. Nonlin. Sci. Numer. Sim., 22 731-746 (2015). pdf
  23. On the statistical and transport properties of a non-dissipative Fermi-Ulam model, A. L. P. Livorati, C. P. Dettmann, I. L. Caldas, E. D. Leonel, Chaos 25 103107 (2015). pdf.
  24. Crises in a dissipative bouncing ball model, A. L. P. Livorati, I. L. Caldas, C. P. Dettmann and E. D. Leonel, Phys. Lett. A 379 2830-2838 (2015). arxiv.
  25. Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse, C. P. Dettmann, V. Fain and D. Turaev, Nonlinearity 31 667-700 (2018). arxiv.
  26. Conference paper: Stochastic stabilization of the cosmic microwave background radiation, C. P. Dettmann, J. P. Keating and S. D. Prado, Proceedings of the Tenth Marcel Grossmann Meeting on General Relativity, ed M. Novello, S. Perez-Bergliaffa and R. Ruffini, World Scientific, Singapore (2006) ps pdf.

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