Diffusion Scaling in Event-Driven Random Walks: An Application to Turbulence

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Scaling laws for the diffusion generated by three different random walk models are reviewed. The random walks, defined on a one-dimensional lattice, are driven by renewal intermittent events with non-Poisson statistics and inverse power-law tail in the distribution of the inter-event or waiting times, so that the event sequences are characterized by self-similarity. Intermittency is a ubiquitous phenomenon in many complex systems and the power exponent of the waiting time distribution, denoted as complexity index, is a crucial parameter characterizing the system's complexity. It is shown that different scaling exponents emerge from the different random walks, even if the self-similarity, i.e. the complexity index, of the underlying event sequence remains the same. The direct evaluation of the complexity index from the time distribution is affected by the presence of added noise and secondary or spurious events. It is possible to minimize the effect of spurious events by exploiting the scaling relationships of the random walk models. This allows to get a reliable estimation of the complexity index and, at the same time, a confirmation of the renewal assumption. An application to turbulence data is shown to explain the basic ideas of this approach.

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), N(t)=0tz(t)dt;z(t)=nδ(ttn),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

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