Aging Wiener-Khinchin Theorem

The Wiener-Khinchin theorem shows how the power spectrum of a stationary random signal I(t) is related to its correlation function (t)I(t+τ). We consider nonstationary processes with the widely observed aging correlation function I(t)I(t+τ)∼tγφEA(τ/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 φEA(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.