Fluctuations of $1/f$ noise and the low frequency cutoff paradox

TL;DR

This paper uncovers the universal statistical properties of $1/f$ noise, explaining its non-stationarity, resolving the low frequency cutoff paradox, and linking intermittency to observed noise behavior.

Contribution

It derives the universal distribution of the power spectrum for $1/f$ noise, addressing longstanding paradoxes and clarifying the absence of a physical low frequency cutoff.

Findings

Universal distribution of power spectrum for $1/f$ noise derived

Resolved the paradox of non-integrability and Parseval's theorem violation

Showed no physical low frequency cutoff exists in experiments

Abstract

Recent experiments on blinking quantum dots and weak turbulence in liquid crystals reveal the fundamental connection between $1/f$ noise and power law intermittency. The non-stationarity of the process implies that the power spectrum is random -- a manifestation of weak ergodicity breaking. Here we obtain the universal distribution of the power spectrum, which can be used to identify intermittency as the source of the noise. We solve an outstanding paradox on the non integrability of $1/f$ noise and the violation of Parseval's theorem. We explain why there is no physical low frequency cutoff and therefore cannot be found in experiments.