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

The power spectral density (PSD) of any time-dependent stochastic process Xt is a meaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of Xt 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.

• Publication year

  2018

• Venue

  New Journal of Physics

• Publication date

  2018-01-09

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.1088/1367-2630/aaa67carXiv 1801.02986
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Passive advection of fractional Brownian motion by random layered flows

  2019cited by this paper
• Quantifying non-ergodicity of anomalous diffusion with higher order moments

  2017cited by this paper
• Conditional 1/f^{α} noise: From single molecules to macroscopic measurement.

  2017cited by this paper
• Manipulation and Motion of Organelles and Single Molecules in Living Cells.

  2017cited by this paper
• Time averaging, ageing and delay analysis of financial time series

  2017cited by this paper
• Aging Wiener-Khinchin theorem and critical exponents of 1/f^{β} noise.

  2016cited by this paper
• How to connect time-lapse recorded trajectories of motile microorganisms with dynamical models in continuous time.

  2016cited by this paper
• 1/fα power spectrum in the Kardar–Parisi–Zhang universality class

  2016cited by this paper
• Time averages in continuous-time random walks.

  2016cited by this paper
• Sample-to-sample fluctuations of power spectrum of a random motion in a periodic Sinai model.

  2016cited by this paper
• Long range correlations generated by phase separation. Exact results from field theory

  2016cited by this paper
• Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes.

  2016cited by this paper
• Temporal Correlations of the Running Maximum of a Brownian Trajectory.

  2016cited by this paper
• Diffusion in a Crowded, Rearranging Environment.

  2016cited by this paper
• Underdamped scaled Brownian motion: (non-)existence of the overdamped limit in anomalous diffusion

  2016cited by this paper
• Universal proximity effect in target search kinetics in the few-encounter limit

  2016cited by this paper
• Estimation of motility parameters from trajectory data

  2015cited by this paper
• First passage time distribution in heterogeneity controlled kinetics: going beyond the mean first passage time

  2015cited by this paper
• Aging Wiener-Khinchin Theorem.

  2015cited by this paper
• Superdiffusion dominates intracellular particle motion in the supercrowded cytoplasm of pathogenic Acanthamoeba castellanii

  2015cited by this paper
• The Power Spectrum of Ionic Nanopore Currents: The Role of Ion Correlations.

  2015cited by this paper
• Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion.

  2014cited by this paper
• Weak Ergodicity Breaking of Receptor Motion in Living Cells Stemming from Random Diffusivity

  2014cited by this paper
• Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking.

  2014cited by this paper
• Nonergodic subdiffusion from Brownian motion in an inhomogeneous medium.

  2014cited by this paper
• Diffusing diffusivity: a model for anomalous, yet Brownian, diffusion.

  2014cited by this paper
• Collective dynamics effect transient subdiffusion of inert tracers in flexible gel networks

  2014cited by this paper
• 1/f noise for intermittent quantum dots exhibits non-stationarity and critical exponents

  2013cited by this paper
• Single-particle tracking data reveal anticorrelated fractional Brownian motion in crowded fluids.

  2013cited by this paper
• Anomalous transport in the crowded world of biological cells

  2013cited by this paper
• Aging renewal theory and application to random walks

  2013cited by this paper
• Universality of the fluorescence intermittency in nanoscale systems: experiment and theory.

  2013cited by this paper
• Low-frequency 1/f noise in graphene devices.

  2013cited by this paper
• Trajectory to trajectory fluctuations in first-passage phenomena in bounded domains

  2013cited by this paper
• Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes

  2013cited by this paper
• Nonergodicity in nanoscale electrodes.

  2013cited by this paper
• Optimal estimation of diffusion coefficients from single-particle trajectories.

  2013cited by this paper
• Optimal least-squares estimators of the diffusion constant from a single Brownian trajectory

  2013cited by this paper
• Distribution of the least-squares estimators of a single Brownian trajectory diffusion coefficient

  2013cited by this paper
• A Comprehensive Treatment of q-Calculus

  2012cited by this paper
• Optimal estimates of the diffusion coefficient of a single Brownian trajectory.

  2012cited by this paper
• Fluctuations of 1/f noise and the low-frequency cutoff paradox.

  2012cited by this paper
• Time-averaged MSD of Brownian motion

  2012cited by this paper
• Strange kinetics of single molecules in living cells

  2012cited by this paper
• Models of anomalous diffusion in crowded environments

  2012cited by this paper
• First passages in bounded domains: when is the mean first passage time meaningful?

  2012cited by this paper
• Viscoelastic Subdiffusion: Generalized Langevin Equation Approach

  2012cited by this paper
• Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking

  2011cited by this paper
• Probability distribution of the time-averaged mean-square displacement of a Gaussian process.

  2011cited by this paper
• The Nature and Perception of Fluctuations in Human Musical Rhythms

  2011influential reference
• First passages for a search by a swarm of independent random searchers

  2011cited by this paper
• Time-averaged quadratic functionals of a Gaussian process.

  2011cited by this paper
• Quantitative analysis of random ameboid motion

  2010cited by this paper
• Variety in intracellular diffusion during the cell cycle

  2009cited by this paper
• Ergodic properties of fractional Brownian-Langevin motion.

  2008cited by this paper
• Random time-scale invariant diffusion and transport coefficients.

  2008cited by this paper
• Functional Analysis of Spontaneous Cell Movement under Different Physiological Conditions

  2008cited by this paper
• Power spectrum analysis for optical tweezers. II: Laser wavelength dependence of parasitic filtering, and how to achieve high bandwidth

  2006cited by this paper
• Cooperative behavior and pattern formation in mixtures of driven and nondriven colloidal assemblies.

  2006cited by this paper
• Characterizing quantum-dot blinking using noise power spectra

  2004cited by this paper
• Power spectrum analysis for optical tweezers

  2004cited by this paper
• Fundamentals of Noise and Vibration Analysis for Engineers: Introductory comments

  2003cited by this paper
• Random walk with shrinking steps

  2003cited by this paper
• Self-similar Gaussian processes for modeling anomalous diffusion.

  2002cited by this paper
• Magnetic tweezers: micromanipulation and force measurement at the molecular level.

  2002cited by this paper
• Spectra and correlations of climate data from days to decades

  2001cited by this paper
• Power spectrum and detrended fluctuation analysis: application to daily temperatures

  2000cited by this paper
• Single-particle tracking: applications to membrane dynamics.

  1997cited by this paper
• Ecology, evolution and 1 f -noise.

  1996cited by this paper
• 1/f noise in human cognition.

  1995cited by this paper
• Power spectrum of a driven bistable system

  1994cited by this paper
• 1/f noise in computer network traffic

  1994cited by this paper
• Single particle tracking. Analysis of diffusion and flow in two-dimensional systems.

  1991cited by this paper
• Self-Organized Criticality and Earthquakes

  1989cited by this paper
• Generic 1/f noise in chaotic Hamiltonian dynamics.

  1987cited by this paper
• On the interpretation of 1/f noise

  1983cited by this paper
• Random walk in a random environment and 1f noise

  1983cited by this paper
• On 1/f noise and other distributions with long tails.

  1982cited by this paper
• Experimental studies on 1/f noise

  1981cited by this paper
• Low-frequency fluctuations in solids: 1/f noise

  1981cited by this paper
• ‘1/fnoise’ in music and speech

  1975cited by this paper
• Some Probability Density Functions and Their Characteristic Functions

  1973cited by this paper
• Some long‐run properties of geophysical records

  1969cited by this paper
• A BESSEL FUNCTION DISTRIBUTION

  1932cited by this paper
• Versuche zur Messung der Avogadro-Loschmidtschen Zahl aus der Brownschen Bewegung einer Drehwaage

  1931cited by this paper
• The Schottky Effect in Low Frequency Circuits

  1925cited by this paper

• Inference of a time delay in stochastic systems

  2025cites this paper
• Statistical testing-based framework for differentiating anomalous diffusion models with constant and random parameters

  2025cites this paper
• Loewner evolution for critical invasion percolation tree

  2025cites this paper
• Microscopic theory of a precessing ferromagnet for ultrasensitive magnetometry

  2025cites this paper
• Identification of bridge modal frequencies using a vehicle with stochastic velocities and initial vibrations

  2025cites this paper
• Decoding Anomalous Diffusion Using Higher-Order Spectral Analysis and Multiple Signal Classification

  2025cites this paper
• Modeling and Predicting Noise-Induced Failure Rates in Ultra-Low-Voltage SRAM Bitcells Affected by Process Variations

  2025cites this paper
• Collective Hard Core Interactions Leave Multiscale Signatures in Number Fluctuation Spectra

  2025cites this paper
• Moment-based spectral modeling for progressive damage assessment in structures: a 3D spiral visualization approach

  2025cites this paper
• Deep learning for heterogeneous anomalous dynamics in cellular and molecular biology

  2025cites this paper
• Two-dimensional fractional Brownian motion: Analysis in time and frequency domains

  2025cites this paper
• Harmonic Fourier decomposition and its application in fault identification of rotating machinery system

  2025cites this paper
• Design Principles of DNA-Barcodes for Nanopore-FET Readout, Based on Molecular Dynamics and TCAD Simulations.

  2024cites this paper
• Two-dimensional Brownian motion with dependent components: Turning angle analysis.

  2024cites this paper
• Regular and anomalous diffusion: I. Foundations

  2024cites this paper
• Gramian Angular Fields for leveraging pretrained computer vision models with anomalous diffusion trajectories

  2023cites this paper
• Exact Response Theory for Time-Dependent and Stochastic Perturbations

  2023cites this paper
• Mean back relaxation for position and densities.

  2023cites this paper
• Feasibility of a trapped atom interferometer with accelerating optical traps

  2023cites this paper
• Extended-gate field-effect transistor chemo- and biosensors: State of the art and perspectives

  2023cites this paper
• Intelligent equipment maintenance and diagnosis method based on VS-Harmogram method

  2023cites this paper
• Accurate Estimation of Diffusion Coefficients and their Uncertainties from Computer Simulation

  2023cites this paper
• Frequency–frequency correlations of single-trajectory spectral densities of Gaussian processes

  2022cites this paper
• Fixational eye movements as active sensation for high visual acuity

  2022cites this paper
• Spectral fingerprints of nonequilibrium dynamics: The case of a Brownian gyrator.

  2022cites this paper
• Characteristics of intrinsic non-stationarity and its effect on eddy-covariance measurements of CO₂ fluxes

  2022cites this paper
• Point-kinetics neutron noise modeling and analysis via probabilistic finite state automata

  2022cites this paper
• Characteristics of intrinsic non-stationarity and its effect on eddy-covariance measurements of CO2 fluxes

  2022cites this paper
• Characteristics of intrinsic non-stationarity and its effect on eddy-covariance measurements of CO 2 fluxes

  2022cites this paper
• Classification of anomalous diffusion in animal movement data using power spectral analysis

  2022influential citation
• Extracting, quantifying, and comparing dynamical and biomechanical properties of living matter through single particle tracking.

  2022cites this paper
• Towards a robust criterion of anomalous diffusion

  2022cites this paper
• Chiral run-and-tumble walker: Transport and optimizing search

  2022cites this paper
• Spectral Properties of Water Hammer Wave

  2022cites this paper
• Diffusion coefficient and power spectrum of active particles with a microscopically reversible mechanism of self-propelling.

  2022cites this paper
• Noise-to-signal ratio of single-trajectory spectral densities in centered Gaussian processes

  2022cites this paper
• Characteristics of intrinsic non-stationarity and its effect on eddy-covariance measurements of CO2 fluxes

  2021cites this paper
• Characterisation of anomalous diffusion statistics powered by deep learning

  2021cites this paper
• Aging power spectrum of membrane protein transport and other subordinated random walks

  2021cites this paper
• Identifying heterogeneous diffusion states in the cytoplasm by a hidden Markov model

  2021cites this paper
• Feature extraction method based on adaptive and concise empirical wavelet transform and its applications in bearing fault diagnosis

  2021cites this paper
• Spectral Analysis of ATP-Dependent Mechanical Vibrations in T Cells

  2021cites this paper
• Characterization of anomalous diffusion classical statistics powered by deep learning (CONDOR)

  2021cites this paper
• Probing the metallic energy spectrum beyond the Thouless energy scale using singular value decomposition

  2021cites this paper
• Single-Particle Tracking Reveals Anti-Persistent Subdiffusion in Cell Extracts

  2021cites this paper
• Assessing the polymer coil-globule state from the very first spectral modes

  2021cites this paper
• Efficient recurrent neural network methods for anomalously diffusing single particle short and noisy trajectories

  2021cites this paper
• Spectral density of individual trajectories of an active Brownian particle

  2021influential citation
• Intrinsic non-stationarity of carbon dioxide turbulent fluxes in the urban boundary layer

  2021cites this paper
• A Story on the Wave Spectral Properties of Water Hammer

  2021influential citation
• Randomness of Möbius coefficients and Brownian motion: growth of the Mertens function and the Riemann hypothesis

  2021cites this paper
• Stochastic gravitational wave background due to gravitational wave memory

  2021cites this paper
• Characteristics of intrinsic non-stationarity and its effect on eddy-covariance measurements of CO2 fluxes

  2021cites this paper
• The diffusion of doxorubicin drug molecules in silica nanoslits is non-Gaussian, intermittent and anticorrelated.

  2020cites this paper
• Quantifying active diffusion in an agitated fluid.

  2020cites this paper
• Ultrahigh-Speed Imaging of Rotational Diffusion on a Lipid Bilayer

  2020cites this paper
• Exact first-passage time distributions for three random diffusivity models

  2020cites this paper
• Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories

  2020influential citation
• Superstatistics and non-Gaussian diffusion

  2020cites this paper
• Quantifying the non-equilibrium activity of an active colloid.

  2020cites this paper
• The anomalous diffusion challenge: single trajectory characterisation as a competition

  2020cites this paper
• Single trajectory characterization via machine learning

  2020cites this paper
• Measurements and characterization of the dynamics of tracer particles in an actin network.

  2020cites this paper
• Fluctuations in the Homogeneity of Cell Medium Distinguish Benign from Malignant Lymphocytes in a Cellular Model of Acute T Cells Leukemia

  2020cites this paper
• Brownian motion and beyond: first-passage, power spectrum, non-Gaussianity, and anomalous diffusion

  2019cites this paper
• Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems

  2019cites this paper
• Machine learning method for single trajectory characterization

  2019cites this paper
• Ensemble Empirical Mode Decomposition of Photoplethysmogram Signals in Biometric Recognition

  2019cites this paper
• Single-trajectory spectral analysis of scaled Brownian motion

  2019cites this paper
• Tracer diffusion on a crowded random Manhattan lattice

  2019cites this paper
• Spectral Content of a Single Non-Brownian Trajectory

  2019influential citation
• Codifference can detect ergodicity breaking and non-Gaussianity

  2019cites this paper
• Universal spectral features of different classes of random-diffusivity processes

  2019influential citation
• Codifference can detect ergodicity breaking and non-Gaussianity

  2019cites this paper
• Full distribution of first exit times in the narrow escape problem

  2019cites this paper
• Passive advection of fractional Brownian motion by random layered flows

  2019cites this paper
• Fraudulent White Noise: Flat power spectra belie arbitrarily complex processes

  2019cites this paper
• Reconstruction of substrate’s diffusion landscape by the wavelet analysis of single particle diffusion tracks

  2018cites this paper
• The power of a single trajectory

  2018cites this paper
• Universal spectral densities: white and flicker noises

  2018cites this paper
• Spectral content of fractional Brownian motion with stochastic reset

  2018cites this paper
• Generalized Riemann hypothesis, time series and normal distributions

  2018cites this paper
• Generalized Riemann hypothesis and stochastic time series

  2018cites this paper
• Viscoelastic subdiffusion in a random Gaussian environment.

  2017cites this paper
• Computational Statistics and Data Analysis

  year unknowncites this paper