Aging Wiener-Khinchin Theorem

TL;DR

This paper extends the Wiener-Khinchin theorem to non-stationary aging processes, establishing relations between power spectra and correlation functions, and demonstrating the emergence of 1/f spectra in aging systems.

Contribution

It formulates two aging Wiener-Khinchin theorems for non-stationary processes with aging correlations, linking power spectra to time and ensemble averages, and applies these to various physical systems.

Findings

Aging 1/f spectra arise from non-analytical correlation functions.

Theorems are validated with quantum dots, diffusion, and logarithmic potential models.

Approach applies broadly to physical mechanisms with aging correlations.

Abstract

The Wiener-Khinchin theorem shows how the power spectrum of a stationary random signal $I(t)$ is related to its correlation function $\left\langle I(t)I(t+\tau)\right\rangle$. We consider non-stationary processes with the widely observed aging correlation function $\langle I(t) I(t+\tau) \rangle \sim t^\gamma \phi_{\rm EN}(\tau/t)$ and relate it to the sample spectrum. We formulate two aging Wiener-Khinchin theorems relating the power spectrum to the time and ensemble averaged correlation functions, discussing briefly the advantages of each. When the scaling function $\phi_{\rm EN}(x)$ exhibits a non-analytical behavior in the vicinity of its small argument we obtain aging $1/f$ type of spectrum. We demonstrate our results with three examples: blinking quantum dots, single file diffusion and Brownian motion in a logarithmic potential, showing that our approach is valid for a wide range of physical mechanisms.