Advertisement

Communications in Mathematical Physics

, Volume 372, Issue 2, pp 679–711 | Cite as

A Lower Bound for the Number of Elastic Collisions

  • Krzysztof Burdzy
  • Mauricio DuarteEmail author
Article

Abstract

We prove by example that the number of elastic collisions of n balls of equal mass and equal size ind-dimensional space can be greater than n3/27 for \({n \geq 3}\) and \({d \geq 2}\). The previously known lower bound was of order n2.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We are grateful to Jayadev Athreya and Jaime San Martin for very helpful advice. We thank the referees for very helpful suggestions.

References

  1. ABD18.
    Athreya, J.S., Burdzy, K., Duarte, M.: On pinned billiard balls and foldings. arXiv e-prints arXiv:1807.08320 (July 2018)
  2. BFK98a.
    Burago D., Ferleger S., Kononenko A.: A geometric approach to semi-dispersing billiards. Ergodic Theory Dyn. Syst.18(2), 303–319 (1998)MathSciNetCrossRefGoogle Scholar
  3. BFK98b.
    Burago D., Ferleger S., Kononenko A.: Unfoldings and global bounds on the number of collisions for generalized semi-dispersing billiards. Asian J. Math.2(1), 141–152 (1998)MathSciNetCrossRefGoogle Scholar
  4. BFK98c.
    Burago D., Ferleger S., Kononenko A.: Uniform estimates on the number of collisions in semi-dispersing billiards. Ann. of Math. (2)147(3), 695–708 (1998)MathSciNetCrossRefGoogle Scholar
  5. BFK00.
    Burago, D., Ferleger, S., Kononenko, A.: A geometric approach to semi-dispersing billiards. In: Hard Ball Systems and the Lorentz Gas, Volume 101 of Encyclopaedia Math. Sci., pp. 9–27. Springer, Berlin (2000)CrossRefGoogle Scholar
  6. BFK02.
    Burago, D., Ferleger, S., Kononenko, A.: Collisions in semi-dispersing billiard on Riemannian manifold. In: Proceedings of the International Conference on Topology and its Applications (Yokohama, 1999), vol. 122, pp. 87–103 (2002)MathSciNetCrossRefGoogle Scholar
  7. BI18.
    Burago, D., Ivanov, S.: Examples of exponentially many collisions in a hard ball system. arXiv e-prints, pp. arXiv:1809.02800 (September 2018)
  8. Bil68.
    Billingsley Patrick: Convergence of Probability Measures. Wiley, New York (1968)zbMATHGoogle Scholar
  9. CI04.
    Chen Xinfu, Illner Reinhard: Finite-range repulsive systems of finitely many particles. Arch. Ration. Mech. Anal.173(1), 1–24 (2004)MathSciNetCrossRefGoogle Scholar
  10. Ill89.
    Illner Reinhard: On the number of collisions in a hard sphere particle system in all space. Transp. Theory Stat. Phys.18(1), 71–86 (1989)ADSMathSciNetCrossRefGoogle Scholar
  11. Ill90.
    Illner Reinhard: Finiteness of the number of collisions in a hard sphere particle system in all space. II. Arbitrary diameters and masses. Transp. Theory Stat. Phys.19(6), 573–579 (1990)ADSMathSciNetCrossRefGoogle Scholar
  12. KT91.
    Kozlov, V.V., Treshchëv, D.V.: Billiards, volume 89 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, (1991). A genetic introduction to the dynamics of systems with impacts, Translated from the Russian by J. R. Schulenberger.Google Scholar
  13. MC93.
    Murphy T. J., Cohen E. G. D.: Maximum number of collisions among identical hard spheres. J. Stat. Phys.71(5-6), 1063–1080 (1993)ADSMathSciNetCrossRefGoogle Scholar
  14. MC00.
    Murphy, T.J., Cohen, E.G.D.: On the sequences of collisions among hard spheres in infinite space. In: Hard Ball Systems and the Lorentz Gas, volume 101 of Encyclopaedia Math. Sci., pp. 29–49. Springer, Berlin (2000)Google Scholar
  15. Vas79.
    Vaserstein L. N.: On systems of particles with finite-range and/or repulsive interactions. Commun. Math. Phys.69(1), 31–56 (1979)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Departamento de MatematicasUniversidad Andres BelloSantiagoChile

Personalised recommendations