Scaling law of diffusivity generated by a noisy telegraph signal with fractal intermittency

Paradisi P., Allegrini P.

Statistical and Nonlinear Physics  Noise  Time series analysis  Fractal intermittency  Signal processing  Scaling  Applied Mathematics  Complex systems  General Mathematics  General Physics and Astronomy 

In many complex systems the non-linear cooperative dynamics determine the emergence of self-organized, metastable, structures that are associated with a birth-death process of coop- eration. This is found to be described by a renewal point process, i.e., a sequence of crucial birth-death events corresponding to transitions among states that are faster than the typical long-life time of the metastable states. Metastable states are highly correlated, but the occur- rence of crucial events is typically associated with a fast memory drop, which is the reason for the renewal condition. Consequently, these complex systems display a power-law decay and, thus, a long-range or scale-free behavior, in both time correlations and distribution of inter-event times, i.e., fractal intermittency. The emergence of fractal intermittency is then a signature of complexity. However, the scaling features of complex systems are, in general, affected by the presence of added white or short-term noise. This has been found also for fractal intermittency. In this work, after a brief review on metastability and noise in complex systems, we discuss the emerging paradigm of Temporal Complexity. Then, we propose a model of noisy fractal in- termittency, where noise is interpreted as a renewal Poisson process with event rate r p . We show that the presence of Poisson noise causes the emergence of a normal diffusion scaling in the long-time range of diffusion generated by a telegraph signal driven by noisy fractal intermittency. We analytically derive the scaling law of the long-time normal diffusivity coef- ficient. We find the surprising result that this long-time normal diffusivity depends not only on the Poisson event rate, but also on the parameters of the complex component of the signal: the power exponent μ of the inter-event time distribution, denoted as complexity index, and the time scale T needed to reach the asymptotic power-law behavior marking the emergence of complexity. In particular, in the range μ < 3, we find the counter-intuitive result that normal diffusivity increases as the Poisson rate decreases. Starting from the diffusivity scaling law here derived, we propose a novel scaling analysis of complex signals being able to estimate both the complexity index μ and the Poisson noise rate r_p .

Source: Chaos, solitons and fractals 81 (2015): 451–462. doi:10.1016/j.chaos.2015.07.003

Publisher: Pergamon., Oxford, Regno Unito

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