Aging Wiener-Khinchin Theorem and Critical Exponents of $1/f$ Noise

Abstract

The power spectrum of a stationary process may be calculated in terms of the autocorrelation function using the Wiener-Khinchin theorem. We here generalize the Wiener-Khinchin theorem for nonstationary processes and introduce a time-dependent power spectrum $⟨S_{t_m}(\omega )⟩$ where $t_m$ is the measurement time. For processes with an aging correlation function of the form $⟨I(t)I(t+\tau )⟩=t^{{\rm Y}}{\varphi }_{\mathrm{EA}}(\tau /t)$, where ${\varphi }_{\mathrm{EA}}(x)$ is a nonanalytic function when $x$ is small, we find aging $1/f$ noise. Aging $1/f$ noise is characterized by five critical exponents. We derive the relations between the scaled correlation function and these exponents. We show that our definition of the time-dependent spectrum retains its interpretation as a density of Fourier modes and discuss the relation to the apparent infrared divergence of $1/f$ noise. We illustrate our results for blinking quantum dot models, single-file diffusion and Brownian motion in logarithmic potential.

References in corpus (44)

• Energy flow, partial equilibration and effective temperatures in systems with slow dynamics
• Universal emission intermittency in quantum dots, nanorods, and nanowires
• Models of traps and glass phenomenology
• Violation of the fluctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence
• Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes
• Aging on Parisi's tree
• Non-ergodic Intensity Correlation Functions for Blinking Nano Crystals
• Response of non-equilibrium systems at criticality: Ferromagnetic models in dimension two and above
• Off equilibrium dynamics and aging in unfrustrated systems
• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
• Exact Solution for the Influence of Spectral Diffusion on Single-Molecule Photon-Statistics
• Large Deviations in Single File Diffusion
• Characterizing Quantum-Dot Blinking Using Noise Power Spectra
• Experimental realization of directed percolation criticality in turbulent liquid crystals
• Aging Correlation Functions for Blinking Nano-Crystals, and Other On - Off Stochastic Processes
• Fluctuations of $1/f$ noise and the low frequency cutoff paradox
• Random Walks in Logarithmic and Power-Law Potentials, Nonuniversal Persistence, and Vortex Dynamics in the Two-Dimensional XY Model
• DNA bubble dynamics as a quantum Coulomb problem
• Generalized Elastic Model yields Fractional Langevin Equation
• Nonergodisity of a time series obeying Lévy statistics
• Persistence of Kardar-Parisi-Zhang Interfaces
• The Lasting Effect of Initial Conditions on Single File Diffusion
• Experimental study of the Fluctuation-Dissipation-Relation during an aging process
• Nonergodic dynamics of force-free granular gases
• Aging Wiener-Khinchin Theorem
• Beyond the Death of Linear Response: 1/f optimal information transport
• Universal Large Deviations for the Tagged Particle in Single File Motion
• Modeling scaled processes and 1/f^b noise by the nonlinear stochastic differential equations
• 1/f noise for intermittent quantum dots exhibits non-stationarity and critical exponents
• Fluctuations of time averages for Langevin dynamics in a binding force field
• Wiener-Khinchin theorem for nonstationary scale-invariant processes
• Aging classification in glassy dynamics
• Scaling Green-Kubo relation and application to three aging systems
• Dynamical ultrametricity in the critical trap model
• Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials
• Self-organised criticality and 1/f noise in single-channel current of voltage-dependent anion channel
• Power spectrum of mass and activity fluctuations in a sandpile
• Sample-to-sample fluctuations of power spectrum of a random motion in a periodic Sinai model
• Ageing in the contact process: Scaling behavior and universal features
• Statistics of the occupation time of renewal processes
• Dephasing by a nonstationary classical intermittent noise
• Single-file diffusion with non-thermal initial conditions
• Temporal Correlations and Persistence in the Kinetic Ising Model: the Role of Temperature
• Deconstructing $1/f$ noise and its universal crossover to non-$1/f$ behavior

Cited by in corpus (27)

• 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
• Diffusion with resetting in a logarithmic potential
• Spectral content of fractional Brownian motion with stochastic reset
• Universal spectral features of different classes of random diffusivity processes
• Aging power spectrum of membrane protein transport and other subordinated random walks
• Single-trajectory spectral analysis of scaled Brownian motion
• Scale invariant Green-Kubo relation for time averaged diffusivity
• Langevin equations from experimental data: the case of rotational diffusion in granular media
• The Heat Distribution in a Logarithm Potential
• Spectral density of individual trajectories of an active Brownian particle
• $1/f^{\alpha }$ power spectrum in the Kardar-Parisi-Zhang universality class
• Power-law scaling of correlations in statistically polarised nano-NMR
• Shannon Entropy for Time-Varying Persistence of Cell Migration
• $1/f^{\beta }$ noise for scale-invariant processes: How long you wait matters
• Conditional $1/f^{\alpha }$ noise: from single molecules to macroscopic measurements
• Passive advection of fractional Brownian motion by random layered flows
• $1/f$ noise from the sequence of nonoverlapping rectangular pulses
• Understanding the nature of the long-range memory phenomenon in socioeconomic systems
• $1/f$ noise in semiconductors arising from the heterogeneous detrapping process of individual charge carriers
• Frequency-frequency correlations of single-trajectory spectral densities of Gaussian processes
• Temporal scaling theory for bursty time series with clusters of arbitrarily many events
• Noise-to-signal ratio of single-trajectory spectral densities in centered Gaussian processes
• Effects of the kinetic energy in heat for overdamped systems
• Heat fluctuations in the logarithm-harmonic potential
• Long-range correlations with finite-size effects from a superposition of uncorrelated pulses with power-law distributed durations
• Classification of anomalous diffusion in animal movement data using power spectral analysis