@article{58c5023ed4264cf58ad943cc4242334c,
title = "Equilibration of energy in slow–fast systems",
abstract = "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.",
keywords = "Chaos, Dynamical billiards, Fermi acceleration, Hamiltonian systems, Mixed phase space",
author = "Kushal Shah and Dmitry Turaev and Vassili Gelfreich and Vered Rom-Kedar",
note = "Funding Information: Part of this work was done while K.S. was at the Indian Institute of Technology Delhi. K.S. was supported by Science and Engineering Research Board, Government of India File SR/FTP/PS-108/2012. K.S., D.T., and V.G. acknowledge the financial support and hospitality of the Weizmann Institute of Science, where part of this work was done. D.T. was supported by Russian Science Foundation Grant 14-41-00044 for this research and Engineering and Physical Sciences Research Council (EPSRC) Grant EP/P026001/1 and thanks the Royal Society. The research of V.G. was supported by EPSRC Grant EP/J003948/1. V.R.-K. acknowledges support from Israel Science Foundation Grant 1208/16. Funding Information: ACKNOWLEDGMENTS. Part of this work was done while K.S. was at the Indian Institute of Technology Delhi. K.S. was supported by Science and Engineering Research Board, Government of India File SR/FTP/PS-108/2012. K.S., D.T., and V.G. acknowledge the financial support and hospitality of the Weizmann Institute of Science, where part of this work was done. D.T. was supported by Russian Science Foundation Grant 14-41-00044 for this research and Engineering and Physical Sciences Research Council (EPSRC) Grant EP/P026001/1 and thanks the Royal Society. The research of V.G. was supported by EPSRC Grant EP/J003948/1. V.R.-K. acknowledges support from Israel Science Foundation Grant 1208/16. Publisher Copyright: {\textcopyright} 2017, National Academy of Sciences. All rights reserved.",
year = "2017",
month = dec,
day = "5",
doi = "10.1073/pnas.1706341114",
language = "English",
volume = "114",
pages = "E10514--E10523",
journal = "Proceedings of the National Academy of Sciences of the United States of America",
issn = "0027-8424",
publisher = "National Academy of Sciences",
number = "49",
}