Diffusion Scaling in Event-Driven Random Walks: An Application to Turbulence
Introduction
Complex systems with many degrees of freedom and strong nonlinear interactions are ubiquitous in nature and in many human activities. In critical phenomena [1] and self-organized criticality [2, 3], the system typically moves towards a critical point, corresponding to a phase transition from an uncorrelated to a correlated condition. This transition is characterized by means of the critical value of a parameter characterizing the cooperation, i.e. the nonlinear coupling, among many individual units. A complex system at the critical point develops power-law long-range correlation in both time and space and it also displays a typical intermittent behaviour, given by a birth-death process of cooperation. In fact, the dynamical evolution is characterized by the alternance of metastable self-organized states and short-time events marking the transition among two successive metastable states. In self-organized systems, an inverse power-law decay typically emerges in the distribution of inter-event times or Waiting Times (WTs): ϕ(τ) ∼ 1/τμ. There are several examples of complex systems displaying power-law behaviour: ecological systems [4], neural dynamics [5, 6], blinking quantum dots [7, 8, 9], social dynamics [10], brain information processing [11, 12, 13, 14, 15, 16], atmospheric turbulence [17], earthquakes [18].
In recent years, the measurement of a single molecule tracking in cellular dynamics is allowing the analysis of single trajectories [19]. Many authors found a strong inhomogeneity in the statistical ensemble of trajectories and, thus, a strong evidence of ergodicity violation. Lubelski et al. [20] showed that this ergodicity violation is due to the presence of critical events with slow power-law decay in the WT distribution, a condition sometimes denoted as fractal intermittency. In fact, these authors compared time and ensemble averages in a transport model known as Continuous Time Random Walk (CTRW) [21, 22, 23], which is based on the presence of critical events in the particle dynamics.
The power-law exponent μ is then a basic parameter measuring, in some sense, the level of complexity of the system, where by complexity we mean the ability of the system to develop self-organized, metastable structures, which can survive for relatively long random times, thus introducing long-range time correlations. These metastable structures affect also the space correlations, as they are usually associated with a coherent motion in the configuration space, such as vortex motion in turbulence. The interpretation of μ is made clear by considering that the limit of large μ is associated with a weak coupling in the nonlinear interactions and, consequently, with a low level of self-organization. On the contrary, as μ decreases, the coupling becomes stronger and the level of self-organization increases. For this reason, μ will be denoted as complexity index.
An important feature of critical events in complex systems is often given by a fast decay of the system's memory in correspondence with event occurrence. In this case, a straightforward approach to the description of fractal intermittency is given by the theoretical framework of renewal processes [24]. At variance with other approaches, such as Fractional Brownian Motion (FBM) [25], in renewal modeling critical events, which are assumed to be statistically independent, play a central role and the WT probability distribution ψ (τ) is the main feature of the model. FBM is more focused on long-range memory, as a memory kernel is explicitly considered in the dynamical equation. In FBM, the memory kernel is the main modeling ingredient whose inverse power-law tail is related to the long-range auto-correlation function. FBM is a model for anomalous diffusion, thus generalizing the standard Brownian motion for normal diffusion. Similarly, the standard Langevin equation can be generalized to the fractional Langevin equation, where a convolution operator is introduced also in the dissipation term with a memory kernel associated with the noise auto-correlation. It is interesting to note that long-range correlations emerge also in renewal models. In fact, even if the transition events determine a fast drop of system's memory, this same memory remains strong when the considered time lag is entirely included into the time interval between two successive events. In this case, the power-law decay in the correlation is related to the probability of-long WTs among two events. Long WTs are but, due to the inverse power-law tail in the probability distribution, they occur often enough to determine an inverse power-law tail in the auto-correlation function [13]. Then, at variance with FBM, no explicit memory function is given in the dynamical equation, and the long-range correlation is associated with the fractal time intermittency, i.e. the WT distribution ψ(τ). Other modeling approaches to complex systems, which were also applied to turbulence and intermittency, are given by nonlinear Fokker–Planck equations [26], superstatistics [27] and multifractality [28].
Regarding renewal approach to intermittency, when dealing with experimental observations the renewal assumption must be Verified. Further, the direct estimation of the complexity index from the statistical distribution not always a reliable approach, as spurious events can be mixed with the genuine critical events. Spurious events have been shown to deeply affect the shape of the time distribution in such a way that the power-law decay could be hidden or unclear, with the apparent emergence of a stretched exponential decay [61]. In some cases, a power-law decay with a power exponent completely different from the real one could also appear, thus leading to misleading results. The presence of spurious events could be associated-with added noise in the time series or even with the method used to detect events, due to detection of false positives.
In order to minimize the effect of added noise and spurious events, we propose a method based on the scaling features of diffusion generated by three random walks. The main idea is that the random walks have different walking rules but are driven by the same critical events, that is, by the same underlying intermittent system. When added noise and/or spurious events are mixed with the genuine complex events, different regimes emerge in the diffusion scaling at different time ranges. Consequently, the diffusion method is able to extract the genuine scaling and the complexity of the intermittent system.
In Section 2 we give some definitions and results about renewal processes. In Section 3 the Continuous Time Random Walk (CTRW) model is briefly reviewed. The mathematical tools of CTRW allow to derive analytical results for the scaling exponents of the three random walks, given the assumption of a renewal process with inverse power-law tail in the WT distribution. In Section 4 we give some directions about the possibility of using the random walk approach to estimate not only the diffusion scaling, but also the complexity index and, finally, to evaluate the renewal assumption. A practical application to turbulence data is shown in order to make clear the basic concepts. Finally, in Section 5 some conclusions are drawn.
Section snippets
Renewal processes
A sequence of random events, randomly distributed in time, is described by a point process. This is formally defined as a stochastic counting process N(t), where δ(·) is the Dirac function and {tn}∞n=0 the sequence of event occurrence times. The times tn are obviously increasing with the index n: tn+1 > tn and t0 = 0 is the occurrence time of the first event. A renewal process is a point process with statistically independent events. In a renewal process the
Event-driven random walks
Renewal processes are the basic ingredient of transport models described with the Continuous Time Random Walk (CTRW) approach [21, 22, 23]. At variance with random walks with fixed time steps, in the CTRW the walker can make random jumps only in correspondence with an event occurrence. The standard random walk is associated with a Markov property of the probability distribution and, then, with a Markovian master equation. On the contrary, the CTRW is non-Markovian and, in particular, is a
Application to data analysis: the case of turbulence
Assuming the renewal hypothesis, the estimation of the scaling exponents δ and H allows to derive the complexity index μ by inverting the expressions given in Eqs. (18), (19), (20). This can be done for each walking rule and for both δ and H separately. When the computed values of μ are in agreement with each other, apart from experimental and statistical errors, this analysis not only gives a robust estimation of μ, but also a confirmation of the renewal assumption. For example, considering
Concluding remarks
We have reviewed some analytical results about the scaling laws of diffusion generated by three random walks driven by renewal non-Poisson events with inverse power-law tails in the WT-PDF. We have showed that the joint use of the random walks and of the two scaling analyzes (DE and DFA) can be useful in characterizing the complexity of cooperative systems, often displaying self-organized metastable structures, such as the turbulent coherent structures emerging in the dynamics of the
References (64)
- et al.
AIP Conf. Proc.
(2005) - et al.
Chem. Phys. Lett.
(2007) - O. C. Akin, P. Paradisi, P. Grigolini: J. Stat. Mech.: Theory and Experiments, P01013, January...
- et al.
Phys. Rev. Lett.
(1991) - et al.
Rev. Mod. Phys.
(1998) J. Stat. Phys.
(1974)- et al.
Phys. Rev. E
(1996) - et al.
Fractals
(2001) - et al.
J. Phys.: Conf. Series
(2011) Critical Phenomena in Natural Sciences
(2006)
Self-organized Criticality
Phys. Rev. Lett.
J. Phys. A
Self-organization in Complex Ecosystems
Spiking Neuron Models
Math. Biosc.
J. Chem. Phys.
Phys. Today
Phys. Rev. E
J. Neurosci.
Phys. Rev. E
Nature Physics
Front. Physiol.
Phys. Rev. E
Eur. Phys. J. Special Topics
Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Rev. Lett.
Proc. Nat. Acad. Sci.
Phys. Chem. Chem. Phys.
Phys. Rev. Lett.
Phys. Rev. Lett.
Random walks on lattices
Proc. Symp. Appl. Math., Am. Math. Soc.
J. Stat. Phys.
Adv. Chem: Phys.
Renewal Theory
Cited by (13)
Scaling law of diffusivity generated by a noisy telegraph signal with fractal intermittency
2015, Chaos, Solitons and FractalsCitation Excerpt :Even more, the added noise can also determine the emergence of an effective complexity index that is completely different from the real one [53]. On the contrary, it has been proved that the EDDiS approach is able to separate different regimes of both diffusion scalings δ and H, but only for the Asymmetric Jump (AJ) and Symmetric Jump (SJ) walking rules (see Appendix A) [9,15,17]. This is a crucial property when noise affects the recorded time series.
Sleep unconsciousness and breakdown of serial critical intermittency: New vistas on the global workspace
2013, Chaos, Solitons and FractalsCitation Excerpt :The renewal condition is generated by burstiness with fast memory decay, and it was found to well describe the intermittency features of several complex systems, from blinking quantum dots [33,34] to turbulence [35–37] and brain dynamics [38]. The renewal property seems to play a crucial role in the perturbation of complex systems [39–43] and it is a fundamental assumption in the derivation of a new Fluctuation–Dissipation Theorem (FDT) based on renewal events [44,45], whose main prediction is that two complex systems have a maximum interaction when they have similar complexities. The power-law relaxation foreseen by this new FDT was also experimentally validated in the weak turbulence regime of a liquid crystal [46].
Finite-energy Lévy-type motion through heterogeneous ensemble of Brownian particles
2019, Journal of Physics A: Mathematical and TheoreticalGaussian processes in complex media: New vistas on anomalous diffusion
2019, Frontiers in PhysicsIntermittency-driven complexity in signal processing
2017, Complexity and Nonlinearity in Cardiovascular Signals