1/f^beta noise in a model for weak ergodicity breaking

TL;DR

This paper investigates the Rebenshtok-Barkai model exhibiting 1/f^beta noise in systems with weak ergodicity breaking, revealing that the power spectrum remains a random variable even over infinite time, but the spectral exponent can still be reliably estimated.

Contribution

It provides analytical and numerical analysis of the power spectrum in a weak ergodicity breaking model, highlighting the non-convergence of the spectrum and the robustness of beta estimation.

Findings

Power spectrum does not converge to a fixed value over infinite time.

The spectral exponent beta can be reliably estimated despite non-convergence.

The spectrum's value is a finite-variance random variable due to strong correlations.

Abstract

In systems with weak ergodicity breaking, the equivalence of time averages and ensemble averages is known to be broken. We study here the computation of the power spectrum from realizations of a specific process exhibiting 1/f^beta noise, the Rebenshtok-Barkai model. We show that even the binned power spectrum does not converge in the limit of infinite time, but that instead the resulting value is a random variable stemming from a distribution with finite variance. However, due to the strong correlations in neighboring frequency bins of the spectrum, the exponent beta can be safely estimated by time averages of this type. Analytical calculations are illustrated by numerical simulations.