Erratum: Aging Wiener-Khinchin theorem and critical exponents of 
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 noise [Phys. Rev. E 
<b>94</b>
, 052130 (2016)]

Leibovich, N.; Dechant, Andreas; Lutz, Eric; Barkai, Eli

doi:10.1103/physreve.96.059902

Cited by 11 publications

(39 citation statements)

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“…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].…”

Section: Introductionmentioning

confidence: 75%

“…Mandelbrot suggested that 1/f α power spectrum ages which means that S t (ω) ∝ ω −2+β t −1+β , so α = 2−β [13]. Importantly, here the spectrum depends on the measurement time t (see details below), and the total power remains finite ∞ 1/t S t (ω)dω = const [14,16]. The time dependent amplitude of S t (ω) provides a normalizable spectral density, therefore it should naturally appear in a bounded process.…”

Section: Introductionmentioning

confidence: 99%

“…The time dependent amplitude of S t (ω) provides a normalizable spectral density, therefore it should naturally appear in a bounded process. In this scenario the spectrum is a density, as it should be, in the sense that S t (ω) is normalizable [16]. 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].…”

Section: Introductionmentioning

confidence: 99%

“…One way to resolve this low-frequency paradox is to assume that the underlying process is nonstationary [13][14][15][16]. Mandelbrot suggested that 1/f α power spectrum ages which means that S t (ω) ∝ ω −2+β t −1+β , so α = 2−β [13].…”

Section: Introductionmentioning

confidence: 99%

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

Leibovich

Barkai

2017

Phys. Rev. E

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We demonstrate that the measurement of 1/f α noise at the single molecule or nano-object limit is remarkably distinct from the macroscopic measurement over a large sample. The single particle measurements yield a conditional time-dependent spectrum. However, the number of units fluctuating on the time scale of the experiment is increasing in such a way that the macroscopic measurements appear perfectly stationary. The single particle power spectrum is a conditional spectrum, in the sense that we must make a distinction between idler and non-idler units on the time scale of the experiment. We demonstrate our results based on stochastic and deterministic models, in particular the well known superposition of Lorentzians approach, the blinking quantum dot model, and deterministic dynamics generated by non-linear mapping. Our results show that the 1/f α spectrum is inherently nonstationary even if the macroscopic measurement completely obscures the underlying time dependence of the phenomena.

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Section: Introductionmentioning

confidence: 75%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Leibovich

Barkai

2017

Phys. Rev. E

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“…Since this subject is related to 1/f noise, it still has aroused widespread interest. By using CTRW formalism one can focus on the origin of 1/f noise, something that was known for a while [42,43] (see also some remarks below).…”

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confidence: 99%

The continuous time random walk, still trendy: fifty-year history, state of art and outlook

2017

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Abstract.In this article we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism, the numerous modifications permitted by its flexibility, its various applications, and the promising perspectives in the various fields of knowledge. A short review of significant achievements and possibilities is given. However, this review is still far from completeness. We focused on a pivotal role of CTRWs mainly in anomalous stochastic processes discovered in physics and beyond. This article plays the role of an extended announcement of the Eur. Inspiring properties and achievementsIn their pioneering work published in year 1965 [1], physicists Eliott W. Montroll and George H. Weiss introduced the concept of continuous-time random walk (CTRW) as a way to make the interevent-time continuous and fluctuating. It is characterized by some distribution associated with a stochastic process, giving an insight into the process activity. This distribution, called pausing-or waiting-time one (WTD), permitted the description of both Debye (exponential) and, what is most significant, non-Debye (slowly-decaying) relaxations as well as normal and anomalous transport and diffusion [2,3] -thus the model involves fundamental aspects of the stochastic world -a real, complex world. Notably, ancestors of this concept are presented by Michael Shlesinger in [4].Let us incidentally comment that term "walk" in the name "continuous-time random walk" is commonly used in the generic sense comprising two concepts: namely, both the walk (associated with finite displacement velocity of the process) and flight (associated with an instantaneous displacement of the process). Thus we have to specify in a detailed way with what kind of process we are considering.The CTRW formalism was most conveniently developed by physicists Scher and Lax in terms of recursion relations [5][6][7][8][9][10]. In this context the distinction between Contribution to the Topical Issue "Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook", edited by Ryszard Kutner and Jaume Masoliver. a e-mail: ryszard.kutner@fuw.edu.pl discrete and continuous times [11] and also between separable and non-separable WTDs were introduced [12]. A thorough analysis of the latter, called also the nonindependent CTRW, was performed a decade ago [13] although, in the context of the concentrated lattice gases, it was performed much earlier [14,15]. These analyses took into account dependences over many correlated consecutive particle displacements and waiting (or interevent) times. The Scher and Lax formulation of the CTRW formalism is particularly convenient to study as well anomalous transport and diffusion as the non-Debye relaxation and their anomalous scaling properties (e.g., the nonlinear time growth of the process variance). Examples are the span of the walk, the first-passage times, survival probabilities, the number of distinct sites visited and, of course, mean and mean-square displacement if they exist. It is very interesting...

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Power spectral density of a single Brownian trajectory: what one can and cannot learn from it

et al. 2018

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The power spectral density (PSD) of any time-dependent stochastic process X t is a meaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of X t over an infinitely large observation time T, that is, it is defined as an ensemble-averaged property taken in the limit  ¥ T . A legitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation time T. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is a fluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail. Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discrete-time lattice random walk. We highlight some differences in the behavior of a single-trajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories.

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Erratum: Aging Wiener-Khinchin theorem and critical exponents of 1/fβ noise [Phys. Rev. E 94 , 052130 (2016)]

Abstract: This corrects the article DOI: 10.1103/PhysRevE.94.052130.

Cited by 11 publications

References 1 publication

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

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

The continuous time random walk, still trendy: fifty-year history, state of art and outlook

Power spectral density of a single Brownian trajectory: what one can and cannot learn from it

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