Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T06:01:21.222Z Has data issue: false hasContentIssue false
  • English
  • Français

Examples of exponentially many collisions in a hard ball system

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  23 October 2020

DMITRI BURAGO  and
SERGEI IVANOV
DMITRI BURAGO
Affiliation:
The Pennsylvania State University, Department of Mathematics, University Park, PA16802, USA (e-mail: burago@math.psu.edu)
SERGEI IVANOV
Affiliation:
St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia (e-mail: svivanov@pdmi.ras.ru)
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider the system of n identical hard balls in ${\mathbb {R}}^3$ moving freely and colliding elastically. We show that there exist initial conditions such that the number of collisions is exponential in n.

Keywords

hard ball gas modelnumber of collisionsexamples with many collisions

MSC classification

Primary: 37D50: Hyperbolic systems with singularities (billiards, etc.)
Secondary: 70F35: Collision of rigid or pseudo-rigid bodies
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 9 , September 2021 , pp. 2754 - 2769
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Burago, D.. Hard ball gas and Alexandrov spaces of curvature bounded above. Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. Extra Vol. II (1998), 289–298.Google Scholar
Burago, D., Ferleger, S. and Kononenko, A.. Uniform estimates on the number of collisions in semi-dispersing billiards. Ann. of Math. (2) 147 (3) (1998), 695–708.CrossRefGoogle Scholar
Burago, D., Ferleger, S. and Kononenko, A.. Topological entropy of semi-dispersing billiards. Ergod. Th. & Dynam. Sys. 18 (4) (1998), 791–805.CrossRefGoogle Scholar
Burdzy, K. and Duarte, M.. A lower bound for the number of collisions. Comm. Math. Phys. 372 (2019), 679–711.CrossRefGoogle Scholar
Galperin, G. A.. Systems of locally interacting and repelling particles that are moving in space. Tr. Mosk. Mat. Obs. 43 (1981), 142–196 (in Russian).Google Scholar
Murphy, T. J. and Cohen, E. G. D.. Maximum number of collisions among identical hard sphere. J. Statist. Phys. 71 ( 5–6) (1993), 1063–1080.CrossRefGoogle Scholar
Murphy, T. J. and Cohen, E. G. D.. On the sequences of collisions among hard spheres in infinite space. Hard Ball Systems and the Lorentz Gas (Encyclopaedia of Mathematical Science, 101). Springer, Berlin, 2000, pp. 29–49.CrossRefGoogle Scholar
Serre, D.. Estimating the number and the strength of collisions in molecular dynamics. Preprint, 2019, arxiv:1903.05866 v1.Google Scholar
Sinai, Ja. G.. Billiard trajectories in a polyhedral angle (Russian). Uspehi Mat. Nauk 33 1(199) (1978), 229–230.Google Scholar
Sinai, Ya. G.. Hyperbolic billiards. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990). Mathematical Society of Japan, Tokyo, 1991, pp. 249–260.Google Scholar
Thurston, W. and Sandri, G.. Classical hard sphere 3-body problem. Bull. Amer. Phys. Soc. 9 (1964), 386.Google Scholar
Vaserstein, L. N.. On systems of particles with finite-range and/or repulsive interactions. Comm. Math. Phys. 69 ( 1) (1979), 31–56.CrossRefGoogle Scholar