Billiards

    A (mathematical) billiard is a model in which a point particle moves in a straight line except for mirror-like reflections at the boundary or at obstacles, physically modelling any system with particles or (short wavelength) waves in a homogeneous cavity; the many examples include visible light (lasers etc), microwaves (including for wireless communications), electrons in semiconductor microstructures, atoms confined by lasers, and also acoustics. Almost any dynamical property can be obtained by varying the shape of the billiard. A particular example, the Lorentz gas, consists of an infinite space containing circular obstacles placed periodically or randomly; random placement arises in statistical mechanics as a model for more complicated systems of many atoms, while periodic placement is equivalent to the simplest molecular dynamics model, 2 particles in periodic boundary conditions. I have used the Lorentz gas and more general billiards to study thermostats, integrable and intermittent dynamics, diffusion and open systems. See also relevant quantum/wave problems in microresonators and quantum dots.

  1. Conjugate Pairing in the three dimensional periodic Lorentz gas C. P. Dettmann, G. P. Morriss and L. Rondoni, Phys. Rev. E 52, R5746-R5748 (1995) pdf
  2. Hamiltonian formulation of the Gaussian isokinetic thermostat. C. P. Dettmann and G. P. Morriss, Phys. Rev. E, 54, 2495-2500 (1996) pdf ps
  3. The field dependence of Lyapunov exponents for nonequilibrium systems G. P. Morriss, C. P. Dettmann and D. J. Isbister, Phys. Rev. E, 54 4748-4654 (1996) pdf
  4. Crisis in the periodic Lorentz gas. C. P. Dettmann and G. P. Morriss, Phys. Rev. E 54, 4782-4790 (1996) pdf (4.9M)
  5. Stability ordering of cycle expansions C. P. Dettmann and G. P. Morriss, Phys. Rev. Lett. 78, 4201-4204 (1997) pdf ps arxiv
  6. Irreversibility, diffusion and multifractal measures in thermostatted systems, C. P. Dettmann, G. P. Morriss, and L. Rondoni, Chaos, Solitons and Fractals 8, 783-792 (1997)
  7. Recent results for the thermostatted Lorentz gas, G. P. Morriss, C. P. Dettmann and L. Rondoni, Physica A 240, 84-95 (1997)
  8. Microscopic chaos from Brownian motion? C. P. Dettmann, E. G. D. Cohen and H. van Beijeren, Nature 401, 875-875 (1999) ps.gz (1.0M when uncompressed) arxiv
  9. The existence of Burnett coefficients in the periodic Lorentz gas, N. I. Chernov and C. P. Dettmann, Physica A 279, 37-44 (2000) ps arxiv
  10. Microscopic chaos and diffusion C. P. Dettmann and E. G. D. Cohen, J. Stat. Phys. 101, 775-817 (2000) ps.gz (28 pages; 2.1M when uncompressed) arxiv
  11. Note on chaos and diffusion C. P. Dettmann and E. G. D. Cohen, J. Stat. Phys. 103, 589-599 (2001) ps arxiv
  12. The Burnett expansion of the periodic Lorentz gas, C. P. Dettmann, Ergod. Th. Dyn. Sys. 23, 481-491 (2003) ps arxiv
  13. Open circular billiards and the Riemann hypothesis, L. A. Bunimovich and C. P. Dettmann, Phys. Rev. Lett. 94 100201 (2005) ps pdf arxivRH day slides
  14. Peeping at chaos: Nondestructive monitoring of chaotic systems by measuring long-time escape rates L. A. Bunimovich and C. P. Dettmann, EPL, 80 40001 (2007). pdf arxivanimation (6.1M)
  15. Survival probability for the stadium billiard, C. P. Dettmann and O. Georgiou, Physica D, 238, 2395-2403 (2009). pdf arxiv
  16. Transmission and reflection in the stadium billiard: Time-dependent asymmetric transport, C. P. Dettmann and O. Georgiou, Phys. Rev. E 83 036212 (2011). [Selected to appear in the PRE "Kaleidoscope"] pdf arxiv poster
  17. Open mushrooms: Stickiness revisited, C. P. Dettmann and O. Georgiou, J. Phys. A.: Math. Theor. 44 195102 (2011). [Highlighted in a JPA Insights article.] pdf arxiv poster
  18. 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
  19. 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
  20. New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis, C. P. Dettmann, J. Stat. Phys. 146 181-204 (2012). pdf arxiv animation (4.8M)
  21. 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
  22. 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
  23. Quantifying intermittency in the open drivebelt billiard, C. P. Dettmann and O. Georgiou, Chaos 22 026113 (2012). pdf arxiv
  24. 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
  25. Escape from a circle and Riemann hypotheses, L. A. Bunimovich and C. P. Dettmann, submitted ps pdf
  26. 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, submitted pdf
  27. Keyhole and Reflection Effects in Network Connectivity Analysis, M. Z. Bocus, C. P. Dettmann, J. P. Coon, M. R. Rahman, submitted arxiv.
  28. Periodic volume fluctuations with infinite horizon: Intermittency enhanced Fermi acceleration, C. P. Dettmann and E. D. Leonel, submitted pdf
  29. Book chapter: The Lorentz gas as a paradigm for nonequilibrium stationary states, C. P. Dettmann, pp 315-365 in Hard ball systems and the Lorentz gas (edited by D. Szasz), Encyclopaedia of Mathematical Sciences Vol 101 (Springer, 2000). Full size version, 50 pages ps. Environmental microscopic version, 25 pages ps.
  30. Book chapter: Recent advances in open billiards with some open problems, C. P. Dettmann, in Frontiers in the study of chaotic dynamical systems with open problems (Ed. Z. Elhadj and J. C. Sprott, World Scientific, 2011) arxiv

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