The Nineteenth Israeli Mini-Workshop in Applied and Computational Mathematics
Eli Barkai (Bar-Ilan University)
1/f noise and the low frequency cutoff paradox
Starting with the work of Bernamont (1937) on resistance fluctuations, noisy signals of
a vast number of natural processes exhibit 1/f power-spectrum. Such spectra are found in
weather data, brain activity, currents of ion-channels and certain chaotic systems to name
a few. The wide applicability of this spectrum resulted in conflicting theories distributed
among many disciplines.
A unifying feature is that 1/f power spectrum is non-integrable at low frequencies
implying that the total energy in the system is infinite, i.e. the spectral desnity is not
normalizable. As pointed out by Mandelbrot (1950's) this infrared catastrophe suggests
that one should abandon the stationary mind set and hence go beyond the widely applicable
Wiener-Khinchin theorem for the power spectrum. Recent theoretical and experimental
advances renewed the discussion on this old paradox, for example in the context of
blinking quantum dots [1,2]. Importantly the removal of ensemble averaging in nano-scale
measurement revealed time dependent spectrum, at least for nano-crystals.
In this talk ageing, intermittency, weak ergodicity breaking, and critical exponents of
the sample power spectrum are discussed within a theoretical framework which hopefully
provides new insights on the 1/f enigma [1,3]. A general theoretical framework based
on non stationary but scale invariant correlation functions leads to an ageing Wiener-Khinchin
theorem which replaces the standard spectral theory [3]. The non-integrable
spectral density is reminiscent of the infinite invariant measure found in infinite ergodic
theory.
References
-
M. Niemann, H. Kantz, E. Barkai, Fluctuations of 1/f noise and the low frequency
cutoff paradox, Phys. Rev. Lett. 110, 140603 (2013). M. Niemann, E. Barkai,
and H. Kantz, Renewal theory for a system with internal states, Mathematical
Modelling of Natural Phenomena (special issue on anomalous diffusion) 11
3 (2016) 191-239.
- S. Sadegh, E. Barkai, and D. Krapf, 1/f noise for intermittent quantum dots
exhibits non-stationarity and critical exponents, New. J. of Physics 16 113054
(2014).
- N. Leibovich and E. Barkai, Aging Wiener-Khinchin Theorem, Phys. Rev. Lett.
115, 080602 (2015). N. Leibovich, A. Dechant, E. Lutz, and E. Barkai, Aging
Wiener-Khinchin theorem and critical exponents of 1/fβ noise, Phys. Rev. E.
94, 052130 (2016).
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