Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials

Dechant, Andreas; Lutz, Eric; Kessler, David A.; Barkai, Eli

doi:10.1103/physreve.85.051124

Cited by 31 publications

(50 citation statements)

References 48 publications

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“…As was mentioned, this scaling condition is valid in many physical systems, e.g. see [11,15,[27][28][29][30][31]. Here we assume that this scale invariance is valid for all τ and t. In reality this is an approximation which we discuss elsewhere [33], and briefly below.…”

Section: Aging Wiener-khinchin Theorem With the Time-averaged Correlamentioning

confidence: 90%

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

Leibovich

Barkai

2017

Eur. Phys. J. B

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We study the power spectrum which is estimated from a nonstationary signal. In particular we examine the case when the signal is observed in a measurement time window [t w , t w +t m ], namely the observation started after a waiting time t w , and t m is the measurement duration. We introduce a generalized aging Wiener-Khinchin theorem which relates between the spectrum and the time-and ensemble-averaged correlation function for arbitrary t m and t w . Furthermore we provide a general relation between the non-analytical behavior of the scale-invariant correlation function and the aging 1/f β noise. We illustrate our general results with two-state renewal models with sojourn times' distributions having a broad tail.

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Section: Aging Wiener-khinchin Theorem With the Time-averaged Correlamentioning

confidence: 90%

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

Leibovich

Barkai

2017

Eur. Phys. J. B

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“…A Brownian particle diffusing in an asymptotically logarithmic potential of the form U (x) U 0 ln |x/a|, for x a, with free diffusion coefficient D * exhibits subdiffusive behavior for 1 < U 0 /D * < 3, with a diffusion exponent α = 3/2 − U 0 /2D * [30]. The autocorrelation function is also of the scaling form (11) with a scaling function [31],…”

Section: (T)/t Of the Process X(t)mentioning

confidence: 99%

Wiener-Khinchin Theorem for Nonstationary Scale-Invariant Processes

Dechant

Lutz

2015

Phys. Rev. Lett.

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We derive a generalization of the Wiener-Khinchin theorem for nonstationary processes by introducing a time-dependent spectral density that is related to the time-averaged power. We use the nonstationary theorem to investigate aging processes with asymptotically scale-invariant correlation functions.As an application, we analyze the power spectrum of three paradigmatic models of anomalous diffusion: scaled Brownian motion, fractional Brownian motion and diffusion in a logarithmic potential. We moreover elucidate how the nonstationarity of generic subdiffusive processes is related to the infrared catastrophe of 1/f -noise. The Wiener-Khinchin theorem is a fundamental result of the theory of stochastic processes. In its simplest form, it states that the autocorrelation function, C(τ ) = x(t + τ )x(t) , of a stationary random signal x(t) is equal to the Fourier transform of the power spectral density,. Herex(ω) denotes the Fourier transform of x(t), T the total measurement time and the ensemble average ... is taken over many realizations of the signal. The Wiener-Khinchin theorem provides a simple relationship between time and frequency representations of a fluctuating process. Its great practical importance comes from the fact that the autocorrelation function may be directly determined from the measured power spectral density, and vice versa. Over the years, it has become an indispensable tool in signal and communication theory The Wiener-Khinchin theorem is only applicable to weakly stationary processes that are characterized by a time-independent mean x(t) and a correlation function C(τ ) that solely depends on the difference τ of its time arguments [8]. It fails for nonstationary processes with an autocorrelation function, C(τ, t) = x(t+τ )x(t) , that depends explicitly on time t and hence exhibits aging. Further, the power spectral density, S(ω, t)-now an explicit function of both frequency and time-is not uniquely defined for nonstationary noisy signals [8]. A commonly used generalization, the Wigner-Ville function [9,10], given in Eq. (4) below, is related to the instantaneous power; it has the disadvantage of not being a true spectral density as it can take on negative values [8]. We here address these critical issues by first introducing a spectral density that is related to the timeaveraged power. Contrary to the Wigner-Ville function, it is strictly positive and therefore a proper spectral density. We employ this quantity to derive an extension of the Wiener-Khinchin theorem that is valid for finitetime, nonstationary signals. We apply this generalization to aging processes that are characterized by a correlation function of the form, C(τ, t) Ct α φ(τ /t), where φ is an arbitrary scaling function. Correlation functions of this type describe scale-invariant dynamics that do not possess a distinctive time scale, contrary to exponentially correlated processes. They occur in a wide range of physical [11], chemical [12] and biological [13] systems and have also found applications in finance [14] and the soc...

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“…The corresponding velocity correlation function was derived in Refs. [15,48] and has the asymptotic behavior (29) is precisely of the type (5), and we can apply our scaling Green-Kubo relation (9) and immediately obtain, for the diffusion exponent,…”

Section: (28)mentioning

confidence: 99%

Scaling Green-Kubo Relation and Application to Three Aging Systems

et al. 2014

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The Green-Kubo formula relates the spatial diffusion coefficient to the stationary velocity autocorrelation function. We derive a generalization of the Green-Kubo formula that is valid for systems with longrange or nonstationary correlations for which the standard approach is no longer valid. For the systems under consideration, the velocity autocorrelation function hvðt þ τÞvðtÞi asymptotically exhibits a certain scaling behavior and the diffusion is anomalous, hx 2 ðtÞi ≃ 2D ν t ν . We show how both the anomalous diffusion coefficient D ν and the exponent ν can be extracted from this scaling form. Our scaling GreenKubo relation thus extends an important relation between transport properties and correlation functions to generic systems with scale-invariant dynamics. This includes stationary systems with slowly decaying power-law correlations, as well as aging systems, systems whose properties depend on the age of the system. Even for systems that are stationary in the long-time limit, we find that the long-time diffusive behavior can strongly depend on the initial preparation of the system. In these cases, the diffusivity D ν is not unique, and we determine its values, respectively, for a stationary or nonstationary initial state. We discuss three applications of the scaling Green-Kubo relation: free diffusion with nonlinear friction corresponding to cold atoms diffusing in optical lattices, the fractional Langevin equation with external noise recently suggested to model active transport in cells, and the Lévy walk with numerous applications, in particular, blinking quantum dots. These examples underline the wide applicability of our approach, which is able to treat very different mechanisms of anomalous diffusion.

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Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials

Cited by 31 publications

References 48 publications

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

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

Wiener-Khinchin Theorem for Nonstationary Scale-Invariant Processes

Scaling Green-Kubo Relation and Application to Three Aging Systems

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