Equilibration of energy in slow–fast systems

Kushal Shah, Dmitry Turaev, Vassili Gelfreich, Vered Rom-Kedar

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


Ergodicity is a fundamental requirement for a dynamical system to reach a state of statistical equilibrium. However, in systems with several characteristic timescales, the ergodicity of the fast subsystem impedes the equilibration of the whole system because of the presence of an adiabatic invariant. In this paper, we show that violation of ergodicity in the fast dynamics can drive the whole system to equilibrium. To show this principle, we investigate the dynamics of springy billiards, which are mechanical systems composed of a small particle bouncing elastically in a bounded domain, where one of the boundary walls has finite mass and is attached to a linear spring. Numerical simulations show that the springy billiard systems approach equilibrium at an exponential rate. However, in the limit of vanishing particle-to-wall mass ratio, the equilibration rates remain strictly positive only when the fast particle dynamics reveal two or more ergodic components for a range of wall positions. For this case, we show that the slow dynamics of the moving wall can be modeled by a random process. Numerical simulations of the corresponding springy billiards and their random models show equilibration with similar positive rates.

Original languageEnglish
Pages (from-to)E10514-E10523
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number49
StatePublished - 5 Dec 2017
Externally publishedYes


  • Chaos
  • Dynamical billiards
  • Fermi acceleration
  • Hamiltonian systems
  • Mixed phase space

ASJC Scopus subject areas

  • General


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