1/f^αpower spectrum in the Kardar–Parisi–Zhang universality class

Abstract

The power spectrum of interface fluctuations in the (1 + 1)-dimensional Kardar-Parisi-Zhang (KPZ) universality class is studied both experimentally and numerically. The 1/f α -type spectrum is found and characterized through a set of "critical exponents" for the power spectrum. The recently formulated "aging Wiener-Khinchin theorem" accounts for the observed exponents. Interestingly, the 1/f α spectrum in the KPZ class turns out to contain information on a universal distribution function characterizing the asymptotic state of the KPZ interfaces, namely the Baik-Rains universal variance. It is indeed observed in the presented data, both experimental and numerical, and for both circular and flat interfaces, in the long time limit.

An appetizer to modern developments on the Kardar–Parisi–Zhang universality class

“…While the liquid-crystal experiment has successfully identified a number of universal features of the circular and flat subclasses, the stationary subclass remains to be directly identified. However, signatures of the stationary subclass, in particular the Baik-Rains distribution, were found by crossover analysis [112] and by power spectrum of height fluctuations δh(x, t) [113]. The crossover toward the Baik-Rains distribution was also identified numerically [112,114] and theoretically [106,107].…”

1∕ f β noise for scale-invariant processes: how long you wait matters

“…The nonstationary route to 1/f β started with the work of Mandelbrot [17][18][19]. Recently, the aged 1/f β spectrum was found experimentally in the growing interface fluctuations in the (1 + 1)-dimensional KPZ class, using liquid-crystal turbulence [24]. This together with theoretical models, and the mentioned blinking quantum dots, motivate us to investigate the subject in further depth.…”

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

“…-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.…”

“…Direct experimental evidence for the predicted time dependent prefactor in the power spectrum has only recently been available from experiments on blinking quantum dots [8]. Another pioneering measurement [32] [23], the multilevel switching process studied by Mandelbrot where the switching times are drawn from a probability distribution. The paper considered both the finite-mean stable distributions as shown in Figure 1 and their infinite-mean counterpart.…”