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

TYPEStatistical & Bio Seminar
Speaker:Nava Reinitz
Time:14:30 - 15:30
Location:Lidow Nathan Rosen (300)

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∼ ωβ 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 [35]. 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 [35], (ii) in macroscopic measurements the spectrum appears stationary while in the single-particle measurements the aging 1/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].

[1] R.Kubo, M.Toda, and H. Hashitsume, Statistical Physics II- Nonequilibrium Statistical Mechanics, Springer (1995).

[2] M. B. Priestley, Spectral Analysis and Time Series, Academic Press, London (1981).
[3] N. Leibovich and E. Barkai, 
Aging Wiener-Khinchin Theorem, Phys. Rev. Lett. 115, 080602 (2015).
[4] A. Dechant and E. Lutz, 
Wiener-Khinchin theorem for nonstationary scale-invariant processes, Phys. Rev.

Lett. 115, 080603 (2015).
[5] N. Leibovich, A. Dechant, E. Lutz and E. Barkai, 
Aging Wiener-Khinchin Theorem and Critical Exponents of

1/f β Noise, Phys. Rev. E, 94, 052130 (2016).
[6] S.Sadegh,E.BarkaiandD.Krapf,
Five critical exponents describing 1/f noise for intermittent quantum dots,

New J. Phys. 16, 113054 (2014).
[7] K.A.Takeuchi, 
1/f α power spectrum in the KardarParisi Zhang universality class, J. Phys. A:Math. Theor.

50, 264006 (2017).
[8] N.LeibovichandE.Barkai,Conditional 1/f α noise: From single molecules to macroscopic measurement,

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).