Quasistatic dynamical systems

TL;DR

This paper introduces quasistatic dynamical systems, inspired by thermodynamics, describing slow, deterministic state evolution that can be viewed as stochastic diffusion processes under certain conditions.

Contribution

It formalizes quasistatic dynamical systems, linking them to stochastic diffusion processes, and explores their statistical behavior and centering methods.

Findings

States evolve as stochastic diffusion processes under mild conditions

The process can be described by solving a martingale problem

Certain centering methods may not produce expected diffusions

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. 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 behaviour 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.