Towards a robust criterion of anomalous diffusion

Abstract

Anomalous-diffusion, the departure of the spreading dynamics of diffusing particles from the traditional law of Brownian-motion, is a signature feature of a large number of complex soft-matter and biological systems. Anomalous-diffusion emerges due to a variety of physical mechanisms, e.g., trapping interactions or the viscoelasticity of the environment. However, sometimes systems dynamics are erroneously claimed to be anomalous, despite the fact that the true motion is Brownian—or vice versa. This ambiguity in establishing whether the dynamics as normal or anomalous can have far-reaching consequences, e.g., in predictions for reaction- or relaxation-laws. Demonstrating that a system exhibits normal- or anomalous-diffusion is highly desirable for a vast host of applications. Here, we present a criterion for anomalous-diffusion based on the method of power-spectral analysis of single trajectories. The robustness of this criterion is studied for trajectories of fractional-Brownian-motion, a ubiquitous stochastic process for the description of anomalous-diffusion, in the presence of two types of measurement errors. In particular, we find that our criterion is very robust for subdiffusion. Various tests on surrogate data in absence or presence of additional positional noise demonstrate the efficacy of this method in practical contexts. Finally, we provide a proof-of-concept based on diverse experiments exhibiting both normal and anomalous-diffusion.

Stochastic processes in a confining harmonic potential in the presence of static and dynamic measurement noise

“…The task of identifying a specific stochastic process and the best estimates of its parameters is hampered by the fact that measured data are compromised by the presence of detection noise. For unconfined motion the effects of both static and dynamic noise (as defined above) have been well studied, both from a conceptual point of view [72][73][74][75][76][77]81] and with respect to objective data analyses [58][59][60][61][62][63][64][65][66][67][68][69][70]. Here we presented a first approach to understanding the effects of static and dynamic noise on different normal-and anomalous-diffusion processes.…”

“…The effects of static and dynamic noise have been studied and methods how to remedy these effects proposed [77][78][79][80]. In particular, it was shown that power spectral analyses of single trajectories can still provide meaningful information on whether a measured dynamics is anomalous-or normal-diffusive [81]. Moreover, it was demonstrated that machine learning methods can provide reliable results for the trajectory analysis even in the presence of noise [69,70].…”

Extracting, quantifying, and comparing dynamical and biomechanical properties of living matter through single particle tracking

“…154,210,213,214 Recently it was demonstrated that the coefficient of variation of the PSD is a robust measure for anomalous diffusion in the presence of static and dynamic errors. 215…”

Frequency–frequency correlations of single-trajectory spectral densities of Gaussian processes

“…Importantly, for fixed ω > 0 and T → ∞, the coefficient of variation approaches a constant ω-independent value, dependent only on the spread of the process. It was thus suggested [31] (see also [32]) that γ can also serve as a robust criterion of anomalous diffusion-the issue to which we will return at the end of this paper. Moreover, the PDF reached in the T → ∞ limit depends on S(ω) = lim T →∞ S(ω, T ) and on its average μ(ω) only in the combination b ω = S(ω) μ(ω) .…”

“…These questions have been addressed in particular for the time series associated with blinking events in quantum dots [3], in standard Brownian motion [30] and in several anomalous diffusion processes [31][32][33][34], and also for the trajectories of the BG [35]. For instance, in [3] (see also [1] for some other examples) the distribution of S(ω, T ) was obtained for the time series of blinking events in quantum dots, and shown to converge to the exponential function in the limit T → ∞.…”