Aging 1/f^β Noise and Non-stationary Power Spectra

The power spectrum of a stationary process is calculated using the Wiener-Khinchin theorem which gives the connectionbetweenthepowerspectrumandthecorrelationfunctionoftheobservedprocess[1,2]. Inmanyexperiments the spectrum exhibits 1/f β noise, i.e. the spectral density is observed as S(ω) ∼ ω−β where 0 < β < 2 at low frequencies. This power-law behavior seems unphysical since the total energy diverges, thus the noise cannot be described with the known stationary Wiener-Khinchin theorem. We have found that the 1/f β noise possess non- stationarity, i.e. the spectrum is time-dependent [3–5]. The time dependence of the 1/f β noise resolves the so called“1/f paradox”.

The nonstationary 1/f β noise is backed by two experimental evidences. The spectrum of intermittent quantum dots was measured showing that the 1/f β noise ages as the measurement time is increased, indicating a nonstationary behavior [6]. Furthermore, this aging 1/f β behavior was measured in the interface fluctuations in the (1+1)- dimensional Kardar-Parisi-Zhang universality class [7]. However, in many other processes, the 1/f β noise does not exhibit experimental evidences of non-stationarity and the famous paradox apparently remains open. The tension between the requirement of time-dependent 1/f β noise and the experimental evidences which support the stationarity is reduced in three levels: (i) we have shown that an unbounded process may present an appearance of time- independent 1/f β noise, while for bounded process the 1/f β noise ages [3–5], (ii) in macroscopic measurements the spectrum appears stationary while in the single-particle measurements the aging 1/f β is recovered [8], and (iii) if the fixed waiting time is much longer than the measurement time the 1/f β does not depend on the measurement time [9].

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Lett. 115, 080603 (2015).
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1/f β Noise, Phys. Rev. E, 94, 052130 (2016).
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New J. Phys. 16, 113054 (2014).
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50, 264006 (2017).
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Phys,Rev. E 96, 032132 (2017).
[9] N. Leibovich and E. Barkai, 1/f β noise for scale-invariant processes: How long you wait matters, Eur. Phys.

J. B 90 229 (2017).