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Aging Wiener-Khinchin Theorem and Critical Exponents of $1/f$ Noise

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 $\left\langle S_{t_m}(ω)\right\rangle$ where $t_m$ is the measurement time. For processes with an aging correlation function of the form $\left\langle I(t)I(t+τ)\right\rangle=t^Υϕ_{\rm EA}(τ/t)$, where $ϕ_{\rm EA}(x)$ is a nonanalytic function when $x$ is small, we find aging $1/f$ noise. Aging $1/f$ noise is characterized by five critical exponents. We derive the relations between the scaled correlation 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 $1/f$ noise. We illustrate our results for blinking quantum dot models, single-file diffusion and Brownian motion in logarithmic potential.