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Quasistatic dynamical systems

Published online by Cambridge University Press:  12 May 2016

NEIL DOBBS  and
MIKKO STENLUND
NEIL DOBBS
Affiliation:
Département de Physique Théorique, Université de Genève, Geneva 1211, Switzerland email neil.dobbs@gmail.com
MIKKO STENLUND
Affiliation:
Department of Mathematics and Statistics, P.O. Box 68, Fin-00014 University of Helsinki, Finland email mikko.stenlund@helsinki.fi
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Abstract

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We introduce the notion of a quasistatic dynamical system, which generalizes that of an ordinary dynamical system. Quasistatic dynamical systems are inspired by the namesake processes in thermodynamics, which are idealized processes where the observed system transforms (infinitesimally) slowly due to external influence, tracing out a continuous path of thermodynamic equilibria over an (infinitely) long time span. Time evolution of states under a quasistatic dynamical system is entirely deterministic, but choosing the initial state randomly renders the process a stochastic one. In the prototypical setting where the time evolution is specified by strongly chaotic maps on the circle, we obtain a description of the statistical behavior as a stochastic diffusion process, under surprisingly mild conditions on the initial distribution, by solving a well-posed martingale problem. We also consider various admissible ways of centering the process, with the curious conclusion that the ‘obvious’ centering suggested by the initial distribution sometimes fails to yield the expected diffusion.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 8 , December 2017 , pp. 2556 - 2596
Copyright
© Cambridge University Press, 2016 

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