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Equilibration of energy in slow–fast systems

  1. Vered Rom-Kedare,1
  1. aDepartment of Electrical Engineering and Computer Science, Indian Institute of Science Education and Research, Bhopal 462066, India;
  2. bDepartment of Mathematics, Imperial College, London SW7 2AZ, United Kingdom;
  3. cLobachevsky University of Nizhny, Novgorod 603950, Russia;
  4. dMathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom;
  5. eDepartment of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
  1. Edited by Jean-Pierre Eckmann, University of Geneva, Geneva, Switzerland, and accepted by Editorial Board Member Herbert Levine October 29, 2017 (received for review April 17, 2017)


Do partial energies in slow–fast Hamiltonian systems equilibrate? This is a long-standing problem related to the foundation of statistical mechanics. Altering the traditional ergodic assumption, we propose that nonergodicity in the fast subsystem leads to equilibration of the whole system. To show this principle, we introduce a set of mechanical toy models—the springy billiards—and describe stochastic processes corresponding to their adiabatic behavior. We expect that these models and this principle will play an important role in the quest to establish and study the underlying postulates of statistical mechanics, one of the long-standing scientific grails.


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.


  • ?1To whom correspondence should be addressed. Email: vered.rom-kedar{at}weizmann.ac.il.
  • Author contributions: K.S., D.T., V.G., and V.R.-K. designed research; K.S., D.T., V.G., and V.R.-K. performed research; K.S., D.T., V.G., and V.R.-K. contributed new reagents/analytic tools; K.S., D.T., V.G., and V.R.-K. analyzed data; K.S., D.T., V.G., and V.R.-K. conceived work ideas by discussions; K.S., V.G., and V.R.-K. performed simulations; and K.S., D.T., V.G., and V.R.-K. wrote the paper.

  • The authors declare no conflict of interest.

  • This article is a PNAS Direct Submission. J.-P.E. is a guest editor invited by the Editorial Board.

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