1/fβ noise in a model for weak ergodicity breaking

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/fβ 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 β can be safely estimated by time averages of this type. Analytical calculations are illustrated by numerical simulations.