Aging Wiener-Khinchin theorem and critical exponents of 1/fβ noise

N. Leibovich, A. Dechant, E. Lutz, E. Barkai

Research output: Contribution to journalArticlepeer-review

24 Citations (Scopus)


The power spectrum of a stationary process may be calculated in terms of the autocorrelation function using the Wiener-Khinchin theorem. We here generalize the Wiener-Khinchin theorem for nonstationary processes and introduce a time-dependent power spectrum Stm(ω) where tm is the measurement time. For processes with an aging autocorrelation function of the form I(t)I(t+τ)=tϒφEA(τ/t), where φEA(x) is a nonanalytic function when x is small, we find aging 1/fβ noise. Aging 1/fβ noise is characterized by five critical exponents. We derive the relations between the scaled autocorrelation function and these exponents. We show that our definition of the time-dependent spectrum retains its interpretation as a density of Fourier modes and discuss the relation to the apparent infrared divergence of 1/fβ noise. We illustrate our results for blinking-quantum-dot models, single-file diffusion, and Brownian motion in a logarithmic potential.

Original languageEnglish
Article number052130
JournalPhysical Review E
Issue number5
Publication statusPublished - 2016 Nov 17

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics


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