Wiener-Khinchin theorem for nonstationary scale-invariant processes

TL;DR

This paper generalizes the Wiener-Khinchin theorem for nonstationary, scale-invariant processes, enabling analysis of aging and anomalous diffusion models, and clarifies the origin of 1/f-noise in nonstationary systems.

Contribution

It introduces a time-dependent spectral density for nonstationary processes and applies it to analyze aging, anomalous diffusion, and 1/f-noise phenomena.

Findings

Derived a nonstationary Wiener-Khinchin theorem

Analyzed power spectra of anomalous diffusion models

Linked nonstationarity to 1/f-noise infrared catastrophe

Abstract

We derive a generalization of the Wiener-Khinchin theorem for nonstationary processes by introducing a time-dependent spectral density that is related to the time-averaged power. We use the nonstationary theorem to investigate aging processes with asymptotically scale-invariant correlation functions. As an application, we analyze the power spectrum of three paradigmatic models of anomalous diffusion: scaled Brownian motion, fractional Brownian motion and diffusion in a logarithmic potential. We moreover elucidate how the nonstationarity of generic subdiffusive processes is related to the infrared catastrophe of 1/f-noise.