## Abstract

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 language | English |
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Article number | 052130 |

Journal | Physical Review E |

Volume | 94 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2016 Nov 17 |

## ASJC Scopus subject areas

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