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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics