scholarly journals Inertia triggers nonergodicity of fractional Brownian motion

2021 ◽  
Author(s):  
Andrey Cherstvy ◽  
Wei Wang ◽  
Ralf Metzler ◽  
Igor Sokolov
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Particle Tracking ◽  
Single Particle ◽  
Single Particle Tracking ◽  
Langevin Equations ◽  
Ergodic Properties ◽  
Massive Particles ◽  
Breaking Parameter ◽  
Ergodicity Breaking

How related are the ergodic properties of the over- and underdamped Langevin equations driven by fractional Gaussian noise? We here find that for massive particles performing fractional Brownian motion (FBM) inertial effects not only destroy the stylized fact of the equivalence of the ensemble-averaged mean-squared displacement (MSD) to the time-averaged MSD (TAMSD) of overdamped or massless FBM, but also concurrently dramatically alter the values of the ergodicity breaking parameter (EB). Our theoretical results for the behavior of EB for underdamped ot massive FBM for varying particle mass m, Hurst exponent H, and trace length T are in excellent agreement with the findings of extensive stochastic computer simulations. The current results can be of interest for the experimental community employing various single-particle-tracking techniques and aiming at assessing the degree of nonergodicity for the recorded time series (studying e.g. the behavior of EB versus lag time). To infer FBM as a realizable model of anomalous diffusion for a set single-particle-tracking data when massive particles are being tracked, the EBs from the data should be compared to EBs of massive (rather than massless) FBM.

scholarly journals Time-averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

2021 ◽  
Author(s):  
Wei Wang ◽  
Andrey G. Cherstvy ◽  
Holger Kantz ◽  
Ralf Metzler ◽  
Igor M. Sokolov
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Diffusion Processes ◽  
Stochastic Simulations ◽  
Time Averaging ◽  
Weak Ergodicity ◽  
Experimental Systems ◽  
Long Time ◽  
Breaking Parameter ◽  
Ergodicity Breaking

How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does the process of stochastic resetting impact nonergodicity? These are the main questions addressed in this study. Specifically, we examine, both analytically and by stochastic simulations, the implications of resetting on the MSD-and TAMSD-based spreading dynamics of fractional Brownian motion (FBM) with a long-time memory, of heterogeneous diffusion processes (HDPs) with a power-law-like space-dependent diffusivity D(x) = D0 |x| γ, and of their “combined” process of HDP-FBM. We find, i.a., that the resetting dynamics of originally ergodic FBM for superdiffusive choices of the Hurst exponent develops distinct disparities in the scaling behavior and magnitudes of the MSDs and mean TAMSDs, indicating so-called weak ergodicity breaking (WEB). For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD, and additionally observe a new trimodal form of the probability density function (PDF) of particle’ displacements. For all three reset processes (FBM, HDPs, and HDP-FBM) we compute analytically and verify by stochastic computer simulations the short-time (normal and anomalous) MSD and TAMSD asymptotes (making conclusions about WEB) as well as the long-time MSD and TAMSD plateaus, reminiscent of those for “confined” processes. We show that certain characteristics of the reset processes studied are functionally similar, despite the very different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicity breaking parameter EB as a function of the resetting rate r. For all the reset processes studied, we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB ∼ (1/r)-decay at large r values. Together with the emerging MSD-versus-TAMSD disparity, this pronounced r-dependence of the EB parameter can be an experimentally testable prediction. We conclude via discussing some implications of our results to experimental systems featuring resetting dynamics.

scholarly journals A new Paradigm in Single-Particle Tracking in Live Cells: Onset of Ergodicity Breaking

2011 ◽  
Vol 100 (3) ◽  
pp. 251a
Author(s):  
Diego Krapf ◽  
Aubrey V. Weigel ◽  
Blair Simon ◽  
Michael M. Tamkun
Keyword(s):  
Particle Tracking ◽  
Single Particle ◽  
Single Particle Tracking ◽  
Live Cells ◽  
New Paradigm ◽  
Ergodicity Breaking

scholarly journals Information-Efficient, Off-Center Sampling Results in Improved Precision in 3D Single-Particle Tracking Microscopy

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 498
Author(s):  
Chen Zhang ◽  
Kevin Welsher
Keyword(s):  
Fisher Information ◽  
Particle Tracking ◽  
Single Particle ◽  
Biological Samples ◽  
Tracking System ◽  
Single Particle Tracking ◽  
Uniform Sampling ◽  
Single Dimension ◽  
3D Localization ◽  
Particle Localization

In this work, we present a 3D single-particle tracking system that can apply tailored sampling patterns to selectively extract photons that yield the most information for particle localization. We demonstrate that off-center sampling at locations predicted by Fisher information utilizes photons most efficiently. When performing localization in a single dimension, optimized off-center sampling patterns gave doubled precision compared to uniform sampling. A ~20% increase in precision compared to uniform sampling can be achieved when a similar off-center pattern is used in 3D localization. Here, we systematically investigated the photon efficiency of different emission patterns in a diffraction-limited system and achieved higher precision than uniform sampling. The ability to maximize information from the limited number of photons demonstrated here is critical for particle tracking applications in biological samples, where photons may be limited.

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