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+τ)⟩. We consider nonstationary processes with the widely observed aging correlation function ⟨I(t)I(t+τ)⟩∼t(γ)ϕ(EA)(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 ϕ(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.

Conditional 1/fα noise: From single molecules to macroscopic measurement

“…Models of such nonstationary behavior are found in the theory of glasses [17,18], blinking quantum dots, analytically and experimentally [8,14], nanoscale electrodes [19], and interface fluctuations in the (1+1)-dimensional KPZ class, both experimentally and numerically, using liquid-crystal turbulence [20]. Thus one school of thought supports the idea that the sample spectrum exhibits nonstationary features of a particular kind [16,19,[21][22][23]. However, the others argue that while Mandelbrot's nonstationarity scenario is theoretically elegant, it is not a universal explanation since it is backed only by several experiments [8,19,20], and the spectrum is stationary [2,4,5].…”

1/fαpower spectrum in the Kardar–Parisi–Zhang universality class

“…In this work, taking the example of scale-invariant fluctuations of growing interfaces, we characterize the power spectrum by a set of recently proposed 'critical exponents' [10,[20][21][22]. This analysis has the advantage that one does not make any a priori assumption about the stationarity; instead, with the obtained exponents, one can judge whether the system is stationary or not, and, if there is aging, determine its time dependence.…”

On the continuing relevance of Mandelbrot’s non-ergodic fractional renewal models of 1963 to 1967

“…An important step [37] has been to distinguish between one which relates the spectrum and the ensemble averaged correlation function, and a second relating the spectrum to the time averaged correlation function. The importance of this distinction can be seen by considering Fourier inverting the power spectrum, i.e.…”

“…-experimental confirmation of the time-dependent spectrum [8,32], the absence of which may have contributed to Mandelbrot's relative lack of subsequent emphasis on his fractional renewal models; -a modern theory [37,42] using scale invariant autocorrelation functions of the form < I(t)I(t + τ >= t g φ(τ /t), implying a wider range of models and systems beyond renewal theory; -extension of the Wiener-Khinchine theorem to this class of processes [37,42]; -explicit calculation of the effect of conditional stationarity on non-ergodicity [7]; -the emphasisis of Bouchaud et al [43] on the effect on the power spectrum of the waiting time t w between the onset of a nonstationary process and the beginning of a measurement of duration T in the interval t w , t w + T , as distinct from the previously noted dependence of the spectrum on the measurement interval T . While Mandelbrot considered the case t w = 0, the opposite case t w T can be physically important.…”