The power of a single trajectory

Schnellbächer, Nikolas D.; Schwarz, Ulrich S.

doi:10.1088/1367-2630/aab038

Cited by 7 publications

(7 citation statements)

References 9 publications

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“…Namely γ assumes three different values in the limit of large frequency depending on whether we have sub-, normal or superdiffusion, but independent of the precise value of α. In the SBM case, recalling the asymptotic results for the mean and variance in equations (19) and (21) respectively, we obtain at fixed observation time T (see figure 5 for the behaviour for arbitrary f ) In this limit the moment-generating function simplifies and the PDF is the gamma distribution with scale 2μ ( f=0, T) and shape parameter 1/2. In figure 5 analytical and numerical results for the coefficient of variation γ are shown.…”

Section: Variance and The Coefficient Of Variationmentioning

confidence: 64%

“…Only when we have sufficiently precise data over a large frequency window, we could use the α-dependent subleading term to identify the anomalous diffusion exponent α. The only explicit α-dependence in the leading order of expression (19) is in the ageing behaviour encoded by the dependence on T α−1 in the prefactor, which therefore becomes a relevant behaviour to check. The dependence on α of the ageing factor leads to the convergence of the limit T  ¥ in the subdiffusive case and to a divergence in the superdiffusive case.…”

Section: Ensemble-averaged Psdmentioning

confidence: 99%

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Single-trajectory spectral analysis of scaled Brownian motion

2019

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A standard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T  ¥. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T  ¥ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion. We demonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent. We also compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.

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Section: Variance and The Coefficient Of Variationmentioning

confidence: 64%

Section: Ensemble-averaged Psdmentioning

confidence: 99%

Single-trajectory spectral analysis of scaled Brownian motion

2019

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“…However, to a large extent, the analysis of a single trajectory of a single TP could indeed provide rich information regarding the particle's physical properties and its interactions with the environment. Methods like single trajectory analysis [17,18] and power spectral analysis [19,20] can be utilized to extract useful information of the particle, e.g., diffusion coefficient [21] and mean squared displacement [22]. Most recently there is exciting new model based on information theory to infer the external force field from a stochastic trajectory [23].…”

Section: Introductionmentioning

confidence: 99%

Mean Trajectories of Multiple Tracking Points on A Brownian Rigid Body: Convergence, Alignment and Twist

Xu¹

2021

Preprint

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We consider mean trajectories of multiple tracking points on a rigid body that conducts Brownian motion in the absence and presence of an external force field. Based on a naïve representation of rigid body -polygon and polyhedron where hydrodynamic interactions are neglected, we study the Langevin dynamics of these Brownian polygons and polyhedra. Constant force, harmonic force and an exponentially decaying force are investigated as examples. In two dimensional space, depending on the magnitude and form of the external force and the isotropy and anisotropy of the body, mean trajectories of these tracking points can exhibit three regimes of interactions: convergence, where the mean trajectories converge to either a point or a single trajectory; alignment, where the mean trajectories juxtapose in parallel; twist, where the mean trajectories twist and intertwine, forming a plait structure. Moreover, we have shown that in general a rigid body can sample from these regimes and transit between them. And its Brownian behavior could be modified during such transition. Notably, from a polygon in two dimensional space to a polyhedron in three dimensional space, the alignment and twist regimes disappear and there is only the convergence regime survived, due to the two more rotational degrees of freedom in three dimensional space.

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“…As a result, the spectral properties have been investigated at length, both experimentally and theoretically. In particular, the scale-free power spectrum -1/f α form with the exponent α lying between 1 and 2 -is termed 1/f noise [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Such a noise shows the existence of long time-correlation, a sign of complexity observed in nature, and one can find the exemplary instances in diverse contexts.…”

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confidence: 99%

“…Here, a = 0.1 and T = (2L − 1) 2 , with L = 2 4 , 25 , and 2 6 . (b) The input is a Gaussian process with ∼ 1/f 1+b for b = 1 and T = 2 10 , 2 12 , 2 14 , and 216 .…”

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confidence: 99%

Scaling theory for the $1/f$ noise

Yadav¹,

Kumar²

2021

Preprint

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We show that in a broad class of processes that show a 1/f α spectrum, the power also explicitly depends on the characteristic time scale. Despite an enormous amount of work, this generic behavior remains so far overlooked and poorly understood. An intriguing example is how the power spectrum of a simple random walk on a ring with L sites shows 1/f 3/2 not 1/f 2 behavior in the frequency range 1/L 2 f 1/2. We address the fundamental issue by a scaling method and discuss a class of solvable processes covering physically relevant applications.

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The power of a single trajectory

Cited by 7 publications

References 9 publications

Single-trajectory spectral analysis of scaled Brownian motion

Single-trajectory spectral analysis of scaled Brownian motion

Mean Trajectories of Multiple Tracking Points on A Brownian Rigid Body: Convergence, Alignment and Twist

Scaling theory for the $1/f$ noise

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