Scaling theory for the $1/f$ noise

TL;DR

This paper introduces a scaling theory explaining how the power spectrum of processes with $1/f^{eta}$ noise depends on characteristic time scales, revealing overlooked behaviors in simple stochastic models.

Contribution

It develops a scaling framework for understanding $1/f^{eta}$ noise spectra, including explicit dependence on characteristic time scales, and analyzes solvable models like random walks.

Findings

Power spectrum depends explicitly on characteristic time scales.

Random walk on a ring exhibits $1/f^{3/2}$ behavior, not $1/f^2$.

Scaling method applies to physically relevant processes.

Abstract

We show that in a broad class of processes that show a $1/f^{\alpha}$ spectrum, the power also explicitly depends on the characteristic time scale. Despite an enormous amount of work, this generic behavior remains so far overlooked and poorly understood. An intriguing example is how the power spectrum of a simple random walk on a ring with $L$ sites shows $1/f^{3/2}$ not $1/f^2$ behavior in the frequency range $1/L^2 \ll f \ll 1/2$. We address the fundamental issue by a scaling method and discuss a class of solvable processes covering physically relevant applications.