Quasistatic dynamical systems
Abstract
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. Timeevolution 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 timeevolution is specified by strongly chaotic maps on the circle, we obtain a description of the statistical behaviour as a stochastic diffusion process, under surprisingly mild conditions on the initial distribution, by solving a wellposed 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2015
 arXiv:
 arXiv:1504.01926
 Bibcode:
 2015arXiv150401926D
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematical Physics;
 Mathematics  Probability;
 37C60;
 60G44;
 60H10
 EPrint:
 40 pages