Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Ergodic Theory: Recurrence

  • Nikos Frantzikinakis
  • Randall McCutcheon
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_184

Definition of the Subject

The basic principle that lies behind several recurrence phenomena is that the typical trajectory of a system with finite volume comes backinfinitely often to any neighborhood of its initial point. This principle was first exploited by Poincaré in his 1890 King Oscar prize‐winningmemoir that studied planetary motion. Using the prototype of an ergodic‐theoretic argument, he showed that in any system of point masses having fixedtotal energy that restricts its dynamics to bounded subsets of its phase space, the typical state of motion (characterized by configurations andvelocities) must recur to an arbitrary degree of approximation.

Among the recurrence principle's more spectacularly counterintuitive ramifications is that isolated ideal gas systems that do not lose energy willreturn arbitrarily closely to their initial states, even when such a return entails a decrease in entropy from equilibrium, in apparentcontradiction to the second law of thermodynamics . Such...

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Nikos Frantzikinakis
    • 1
  • Randall McCutcheon
    • 1
  1. 1.Department of MathematicsUniversity of MemphisMemphisUSA