The Wiener-Khinchin theorem shows how the power spectrum of a stationary random signal $I(t)$ is related to its correlation function $⟨I(t)I(t+\tau )⟩$. We consider nonstationary processes with the widely observed aging correlation function $⟨I(t)I(t+\tau )⟩\sim t^{\gamma }{\varphi }_{\mathrm{EA}}(\tau /t)$ and relate it to the sample spectrum. We formulate two aging Wiener-Khinchin theorems relating the power spectrum to the time- and ensemble-averaged correlation functions, discussing briefly the advantages of each. When the scaling function ${\varphi }_{\mathrm{EA}}(x)$ exhibits a nonanalytical behavior in the vicinity of its small argument we obtain the aging $1/f$-type of spectrum. We demonstrate our results with three examples: blinking quantum dots, single-file diffusion, and Brownian motion in a logarithmic potential, showing that our approach is valid for a wide range of physical mechanisms.

• Received 6 May 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.080602