ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Loskutov A. J., Rjabov A. B. Billiard type systems and Fermi acceleration. Izvestiya VUZ. Applied Nonlinear Dynamics, 2008, vol. 16, iss. 5, pp. 83-98. DOI: 10.18500/0869-6632-2008-16-5-83-98

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 106)
Article type: 

Billiard type systems and Fermi acceleration

Loskutov Aleksandr Jurevich, Lomonosov Moscow State University
Rjabov Aleksej Borisovich, Lomonosov Moscow State University

Systems of billiard types with perturbed boundaries are described. A generalized dispersing billiard – the Lorentz gas with the open horizon – and a focusing billiard in the form of stadium are considered. It is analytically and numerically shown that, if the billiard possesses the property of the developed chaos, the consequence of the boundary perturbation is the Fermi acceleration. However, the perturbation of the nearly integrable billiard system leads to a new interesting phenomenon – the separation of the billiard particles in their velocities. This consists of the fact that if the initial particle velocities exceed some critical value (specific for the given billiard geometry) then the racing of the particle ensemble is observed. If the initial value is below the critical value, then the billiard particles are not accelerated. The dependence of the separation effect on the characteristic billiard parameters and the frequency of the boundary oscillations is described. 

Key words: 
  1. Zaslavsky GM. Chaos of Dynamic Systems. Routledge; 1985. 370 p.
  2. Chen RH, editor. Fundamental Problems in Statistical Mechanics. Vol. 3. Elsevier, Amsterdam; 1975. 303 p.
  3. Cornfeld IP, Sinai YG, Fomin SV. Ergodic Theory. NY: Springer; 1982. 486 p. DOI: 10.1007/978-1-4615-6927-5.
  4. Bunimovich LA. Systems of hyperbolic type with singularities. In: Dynamical Systems. Vol. 2. Moscow: VINITI; 1985. P. 173–204 (in Russian).
  5. Bunimovich LA, Sinai YG. Statistical properties of lorentz gas with periodic configuration of scatterers. Commun. Math. Phys. 1981;78(4):479–497. DOI: 10.1007/BF02046760.
  6. Baldwin PR. The billiard algorithm and KS entropy. J. Phys. A. 1991;24(16):L941–L947. DOI: 10.1088/0305-4470/24/16/010.
  7. Chernov N. Entropy, Lyapunov exponents, and mean free path for billiards. J. Stat. Phys. 1997;88(1–2):1–29. DOI: 10.1007/BF02508462.
  8. Garrido PL. Kolmogorov–Sinai entropy, Lyapunov exponents, and mean free time in billiard systems. J. Stat. Phys. 1997;88(3–4):807–824. DOI: 10.1023/B:JOSS.0000015173.74708.2a.
  9. Bunimovich LA. Conditions of stochasticity of two-dimensional billiards. Chaos. 1991;1(2):187–193. DOI: 10.1063/1.165827.
  10. Fermi E. On the origin of the cosmic radiation. Phys. Rev. 1949;75(8):1169–1174. DOI: 10.1103/PhysRev.75.1169.
  11. Ulam S.M. On some statistical properties of dynamical systems. In: Proc. 4th Berkeley Sympos. Math. Statist. and Prob. Vol. 3. California Univ. Press; 1961. P. 315–320.
  12. Lichtenberg AJ, Lieberman MA. Regular and Chaotic Dynamics. New York: Springer; 1992. 692 p. DOI: 10.1007/978-1-4757-2184-3.
  13. Brahic A. Numerical study of a simple dynamical system. I. The associated plane area-preserving mapping. Astron. Astrophys. 1971;12:98–110.
  14. Zaslavsky GM. Stochastic Irreversibility in Nonlinear Systems. Moscow: Nauka; 1970. 144 p. (in Russian).
  15. Lichtenberg AJ, Lieberman MA, Cohen RH. Fermi acceleration revisited. Physica D. 1980;1(3):291–305. DOI: 10.1016/0167-2789(80)90027-5.
  16. Pustyl’nikov LD. About the Fermi-Ulam model. Proc. Acad. Sci. USSR. 1987;292(3):549–553 (in Russian).
  17. Pustyl’nikov LD. Existence of invariant curves for maps close to degenerate maps, and a solution of the Fermi-Ulam problem. Russian Academy of Sciences. Sbornik Mathematics. 1995;82(1):231–241. DOI: 10.1070/SM1995v082n01ABEH003561.
  18. Krüger T, Pustyl’nikov LD, Troubetzkoy SE. Acceleration of bouncing balls in external fields. Nonlinearity. 1995;8(3):397–410. DOI: 10.1088/0951-7715/8/3/006.
  19. Pustyl'nikov LD. Poincaré models, rigorous justification of the second element of thermodynamics on the basis of mechanics, and the Fermi acceleration mechanism. Russian Mathematical Surveys. 1995;50(1):145–189. DOI: 10.1070/RM1995v050n01ABEH001663.
  20. Koiller J, Markarian R, Kamphorst SQ, de Carvalho SP. Time-dependent billiards. Nonlinearity. 1995;8(6):983–1003. DOI: 10.1088/0951-7715/8/6/006.
  21. Koiller J, Markarian R, Oliffson S, Pintos S. Static and time-dependent perturbations of the classical elliptical billiard. J. Stat. Phys. 1996;83(1–2):127–143. DOI: 10.1007/BF02183642.
  22. Kamphorst SQ, de Carvalho SP. Bounded gain of energy on the breathing circle billiard. Nonlinearity. 1999;12(5):1363–1373. DOI: 10.1088/0951-7715/12/5/310.
  23. Tsang KJ, Ngai KL. Dynamics of relaxing systems subjected to nonlinear interactions. Phys. Rev. E. 1997;56(1):R17–R20. DOI: 10.1103/PhysRevE.56.R17.
  24. Tsang KJ, Lieberman MA. Analytical calculation of invariant distributions on strange attractors. Physica D. 1984;11(1–2):147–166. DOI: 10.1016/0167-2789(84)90440-8.
  25. Tsang KJ, Lieberman MA. Invariant distribution on strange attractors in highly dissipative systems. Phys. Lett. A. 1984;103(4):175–181. DOI: 10.1016/0375-9601(84)90245-7.
  26. Loskutov AY, Ryabov AB, Akinshin LG. Mechanism of Fermi acceleration in dispersing billiards with time-dependent boundaries. J. Exp. Theor. Phys. 1999;89(5):966–974. DOI: 10.1134/1.558939.
  27. Loskutov A, Ryabov AB, Akinshin LG. Properties of some chaotic billiards with time-dependent boundaries. J. Phys. A. 2000;33(44):7973–7987. DOI: 10.1088/0305-4470/33/44/309.
  28. de Carvalho R.E., Souza F.C., Leonel E.D. Fermi acceleration on the annular billiard: A simplified version J. Phys. A. 2006;39(14):3561–3575. DOI: 10.1088/0305-4470/39/14/005.
  29. de Carvalho RE, Souza FC, Leonel ED. Fermi acceleration on the annular billiard. Phys. Rev. E. 2006;73(6):066229. DOI: 10.1103/PhysRevE.73.066229.
  30. Karlis AK, Papachristou PK, Diakonos FK, et al. Hyperacceleration in a stochastic Fermi-Ulam model. Phys. Rev. Lett. 2006;97(19):194102. DOI: 10.1103/physrevlett.97.194102.
  31. Leonel ED. Breaking down the Fermi acceleration with inelastic collisions. J. Phys. A. 2007;40(50):F1077. DOI: 10.1088/1751-8113/40/50/F02.
  32. Loskutov A, Chichigina O, Ryabov A. Thermodynamics of dispersing billiards with time-dependent boundaries. Int. J. Bifurc. Chaos. 2008;18(9):2863–2869. DOI: 10.1142/S0218127408022123.
  33. Loskutov A, Ryabov A. Particle dynamics in time-dependent stadium-like billiards. J. Stat. Phys. 2002;108(5):995–1014. DOI: 10.1023/A:1019735313330.
  34. Loskutov A, Ryabov AB. (Sent to press).
Short text (in English):
(downloads: 71)