Perturbation theory for the Fokker-Planck operator in chaos

TL;DR

This paper demonstrates that perturbation theory applied to the Fokker-Planck operator can accurately estimate long-term averages in chaotic systems with added noise, despite the fractal nature of their stationary distributions.

Contribution

It introduces a perturbative approach to the Fokker-Planck operator that works effectively for chaotic systems with noise, where traditional methods fail.

Findings

Perturbation expansion provides accurate long-time averages.

Additive noise smooths fractal distributions, enabling perturbation methods.

The approach is effective even in highly chaotic regimes.

Abstract

The stationary distribution of a fully chaotic system typically exhibits a fractal structure, which dramatically changes if the dynamical equations are even slightly modified. Perturbative techniques are not expected to work in this situation. In contrast, the presence of additive noise smooths out the stationary distribution, and perturbation theory becomes applicable. We show that a perturbation expansion for the Fokker-Planck evolution operator yields surprisingly accurate estimates of long-time averages in an otherwise unlikely scenario.