Abstract
We study a method for detecting the origins of anomalous diffusion, when it is observed in an ensemble of times-series, generated experimentally or numerically, without having knowledge about the exact underlying dynamics. The reasons for anomalous diffusive scaling of the mean-squared displacement are decomposed into three root causes: increment correlations are expressed by the ‘Joseph effect’ (Mandelbrot and Wallis 1968 Water Resour. Res. 4 909), fat-tails of the increment probability density lead to a ‘Noah effect’ (Mandelbrot and Wallis 1968 Water Resour. Res. 4 909), and non-stationarity, to the ‘Moses effect’ (Chen et al 2017 Phys. Rev. E 95 042141). After appropriate rescaling, based on the quantification of these effects, the increment distribution converges at increasing times to a time-invariant asymptotic shape. For different processes, this asymptotic limit can be an equilibrium state, an infinite-invariant, or an infinite-covariant density. We use numerical methods of time-series analysis to quantify the three effects in a model of a non-linearly coupled Lévy walk, compare our results to theoretical predictions, and discuss the generality of the method.
Highlights
Diffusive processes that scale anomalously with time, such that the Mean-Squared Displacement (MSD) of the expanding particle packet is x2(t) ∼ t2H, (1)and the Hurst exponent H = 1/2, are widely observed
We study a method for detecting the origins of anomalous diffusion, when it is observed in an ensemble of times-series, generated experimentally or numerically, without having knowledge about the exact underlying dynamics
The reasons for anomalous diffusive scaling of the mean-squared displacement are decomposed into three root causes: increment correlations are expressed by the “Joseph effect” [1], fat-tails of the increment probability density lead to a “Noah effect” [1], and nonstationarity, to the “Moses effect” [2]
Summary
Diffusive processes that scale anomalously with time, such that the Mean-Squared Displacement (MSD) of the expanding particle packet is x2(t) ∼ t2H ,. For processes with stationary increments, where the probability distribution of δxj is independent of time, anomalous diffusive scaling can occur because of long-time increment correlations This is called the Joseph effect [1, 2, 15]. A paradigmatic process that exhibits this effect is fractional Brownian motion [2, 16] Another cause of anomalous scaling may be that the increment distribution is fat-tailed, in the sense that its second moment is divergent. We explore the emergence of the three effects in different parameter regimes of the model using simulations and methods of time-series analysis of single Levy walk trajectories, and compare our findings with analytical results based on the well developed theory for this process This example shows that the analysis based on the three constitutive effects is a useful tool that can be applied to study other systems as well (see discussion).
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