Skip to main content
Log in

Time evolution for infinitely many hard spheres

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript


We construct the time evolution for infinitely many particles in ℝv interacting by the hard-sphere potential

$$\Phi (x) = \left\{ {\begin{array}{*{20}c} { + \infty } \\ 0 \\ \end{array} } \right. \begin{array}{*{20}c} {|x|< a} \\ {|x| \geqq a} \\ \end{array}$$

. Because there are abundant examples of hard-sphere configurations with more than one solution to the Newtonian equations of motion, we introduce the concept of aregular solution, in which the growth of velocities and crowding of particles at infinity are limited. We prove that (1) regular solutions exist with probability one in every equilibrium state, and (2) any configuration of the infinite system is the initial point of at most one regular solution. Equilibrium states are invariant under the time-evolution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Alexander, R.: The infinite hard-sphere system. Thesis, University of California, Berkeley, 1975

    Google Scholar 

  2. Bourbaki, N.: General topology, Part 2. Paris: Hermann 1966

    Google Scholar 

  3. Gowrisankaran, N. K.: Measurability of functions in product spaces. Proc. Amer. Math. Soc.31, 485–488 (1972)

    Google Scholar 

  4. Lanford, O. E. III: One-dimensional systems of infinitely many Particles I: An existence theorem. Commun. math. Phys.9, 169–181 (1968)

    Google Scholar 

  5. Lanford, O. E., III: One-dimensional systems of infinitely many particles II: Kinetic Theory. Commun. math. Phys.11, 257–292 (1969)

    Google Scholar 

  6. Lanford, O. E., III: Time evolution of large classical systems. In: Dynamical systems, theory and applications. Lecture Notes in Physics, Vol. 38, (ed. J. Moser). Berlin-Heidelberg-New York: Springer 1975

    Google Scholar 

  7. Marchioro, C., Pellegrinotti, A., Presutti, E.: Existence of time evolution forv-dimensional statistical mechanics. Commun. math. Phys.40, 175–186 (1975)

    Google Scholar 

  8. Ruelle, D.: Statistical mechanics. New York: Benjamin 1969

    Google Scholar 

  9. Schwartz, L.: Radon measures on arbitrary topological spaces and cylindrical measures. Oxford University Press, Tata Institute for Fundamental Research, 1973

  10. Sinai, Ya. G.: Construction of the dynamics for one-dimensional systems of statistical mechanics. Teoret. Mat. Fiz.11, 248–258 (1972). English translation: Theoret. Math. Phys. 487–494 (1973)

    Google Scholar 

  11. Sinai, Ya. G.: The construction of cluster dynamics for dynamical systems of statistical mechanics. Vestn. Mosk. Univ. Ser. I: Mat.-Meh.29, 152–159 (1974). English translation: Mosc. Univ. Math. Bull.29, 124–130 (1974)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Additional information

Communicated by J. L. Lebowitz

Part of this work forms part of the author's doctoral dissertation written at the University of California, Berkeley, under the direction of O. E. Lanford III

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alexander, R. Time evolution for infinitely many hard spheres. Commun.Math. Phys. 49, 217–232 (1976).

Download citation

  • Received:

  • Issue Date:

  • DOI: