Aging Wiener-Khinchin Theorem

Abstract

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 non-stationary processes with the widely observed aging correlation function $⟨I(t)I(t+\tau )⟩\sim t^{\gamma }{\varphi }_{\mathrm{EN}}(\tau /t)$ 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 ${\varphi }_{\mathrm{EN}}(x)$ exhibits a non-analytical behavior in the vicinity of its small argument we obtain 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.

References in corpus (5)

• Universal emission intermittency in quantum dots, nanorods, and nanowires
• Large Deviations in Single File Diffusion
• Generalized Elastic Model yields Fractional Langevin Equation
• Nonergodisity of a time series obeying Lévy statistics
• Wiener-Khinchin theorem for nonstationary scale-invariant processes

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