We study time series concerning rare events. The occurrence of a rare event is depicted as a jump of constant intensity always occurring in the same direction, thereby generating an asymmetric diffusion process. We consider the case where the waiting time distribution is an inverse power law with index µ. We focus our attention on µ < 3, and we evaluate the scaling δ of the resulting diffusion process. We prove that δ gets its maximum value, δ = 1, corresponding to the ballistic motion, at µ = 2. We study the resulting diffusion process by means of joint use of the continuous time random walk and of the generalized central limit theorem, as well as adopting a numerical treatment. We show that rendering asymmetric the diffusion process yields the significant benefit of enhancing the value of the scaling parameter δ. Furthermore, this scaling parameter becomes sensitive to the power index µ in the whole region 1 < µ < 3. Finally, we show our method in action on real data concerning human heartbeat sequences.

We study the statistical properties of time distribution of seimicity in California by means of a new method of analysis, the Diffusion Entropy. We find that the distribution of time intervals between a large earthquake (the main shock of a given seismic sequence) and the next one does not obey Poisson statistics, as assumed by the current models. We prove that this distribution is an inverse power law with an exponent µ = 2.06 ± 0.01. We propose the Long-Range model, reproducing the main properties of the diffusion entropy and describing the seismic triggering mechanisms induced by large earthquakes. The search for correlation in the space-time distribution of earthquakes is a major goal in geophysics. At the short-time and the short-space scale the existence of correlation is well established. Recent geophysical observations indicate that main fracture episodes can trigger long-range as well as short-range seismic effects [1,2,3,4]. However, a clear evidence in support of these geophysical indications has not yet been provided. This is probably the reason why one of the models adopted to describe the time distribution of earthquakes is still the Generalized Poisson (GP) model [5,6,7,8,9]. Basically the GP model assumes that the earthquakes are grouped into temporal clusters of events and these clusters are uncorrelated: in fact the clusters are distributed at random in time and therefore the time intervals between one cluster and the next one follow a Poisson distribution. On the other hand, the intra-cluster earthquakes are correlated in time as it is expressed by the Omori's law [10,11], an empirical law stating that the main shock, i.e. the highest magnitude earthquake of the cluster, occurring at time t 0 is followed by a swarm of correlated earthquakes (after shocks) whose number (or frequency) n(t) decays in time as a power law, n(t) ∝ (t − t 0 ) −p , with the exponent p being very close to 1. The Omori's law implies [12] that the distribution of the time intervals between one earthquake and the next, denoted by τ , is a power law ψ(τ ) ∝ τ −p . This property has been recently studied by the authors of Ref.[12] by means of a unified scaling law for ψ L,M (τ ), the probability of having a time interval τ between two seismic events with a magnitude * Corresponding author: vito.latora@ct.infn.it larger than M and occurring within a spatial distance L. This has the effect of taking into account also space and extending the correlation within a finite time range τ * , beyond which the authors of Ref.[12] recover Poisson statistics.In this letter, we provide evidence of inter-clusters correlation by studing a catalog of seismic events in California with a new technique of analysis called Diffusion Entropy (DE) [13,14]. This technique, scarcely sensitive to predictable events such as the Omori cascade of aftershocks, is instead very sensitive when the deviation from Poisson statistics generates Lévy diffusion [14,15]. This deviation, on the other hand, implies that the geophysical process generating clusters ha...

We describe two types of memory and illustrate each using artificial and actual heartbeat data sets. The first type of memory, yielding anomalous diffusion, implies the inverse power-law nature of the waiting time distribution and the second the correlation among distinct times, and consequently also the occurrence of many pseudoevents, namely, not genuinely random events. Using the method of diffusion entropy analysis, we establish the scaling that would be determined by the real events alone. We prove that the heart beating of healthy patients reveals the existence of many more pseudoevents than in the patients with congestive heart failure. The analysis of time series of physiological significance is currently done by many research groups using the paradigm of anomalous scaling ͓1͔. This means that a time series is converted into a diffusion process described by the probability distribution p(x,t) of the diffusing variable x, which is expected to fit the scaling propertywith the ''degree of anomaly'' being measured by the distance of the scaling parameter ␦ from the standard value 0.5.It is straightforward to prove that the Shannon entropyof a process fitting the scaling condition of Eq. ͑1͒ yieldswhere A is a constant, whose explicit form is not relevant for the ensuing discussion. This result is immediately obtained by plugging Eq. ͑1͒ into Eq. ͑2͒. We thus find a method to evaluate the scaling parameter ␦, more efficient than the calculation of the second moment of the probability distribution. Note that when the distribution density under study departs from the ordinary Gaussian case and the function F(y) has slow tails with an inverse power-law nature ͓2,3͔ the second moment is a divergent quantity. This diverging quantity is made finite by the unavoidable statistical limitation. In this case, the second moment analysis would yield misleading results, determined by the statisticaly inaccuracy, while the method based on Eq. ͑3͒ yields correct results ͓2,3͔. This method is denoted as diffusion entropy ͑DE͒ method. The aim of this paper is to show that the entropy of a diffusion process generated by a physiological time series according to the prescriptions of Refs. ͓2,3͔ yields a scaling exponent that depends only on genuinely random events. The time distances 's between nearest-neighbor events can be evaluated numerically and can be associated to a density distribution (). In the case of physiological processes, the waiting time distribution is expected to be an inverse power law, with index . According to the theory of Ref. ͓3͔ there exists a simple relation between ␦ and . Thus, the experimental determination of () should yield the same information as the DE method. This is true when the events are genuinely random events. If the events are not genuinely random, and a memory, or time correlation exists, the DE method and the direct evaluation of () do not yield equivalent results, and the conflict betwen them is an important information on the physiological process under study.Prior to the physiological...

We discuss the problem of the equivalence between continuous-time random walk ͑CTRW͒ and generalized master equation ͑GME͒. The walker, making instantaneous jumps from one site of the lattice to another, resides in each site for extended times. The sojourn times have a distribution density (t) that is assumed to be an inverse power law with the power index . We assume that the Onsager principle is fulfilled, and we use this assumption to establish a complete equivalence between GME and the Montroll-Weiss CTRW. We prove that this equivalence is confined to the case where (t) is an exponential. We argue that is so because the Montroll-Weiss CTRW, as recently proved by Barkai ͓E. Barkai, Phys. Rev. Lett. 90, 104101 ͑2003͔͒, is nonstationary, thereby implying aging, while the Onsager principle is valid only in the case of fully aged systems. The case of a Poisson distribution of sojourn times is the only one with no aging associated to it, and consequently with no need to establish special initial conditions to fulfill the Onsager principle. We consider the case of a dichotomous fluctuation, and we prove that the Onsager principle is fulfilled for any form of regression to equilibrium provided that the stationary condition holds true. We set the stationary condition on both the CTRW and the GME, thereby creating a condition of total equivalence, regardless of the nature of the waiting-time distribution. As a consequence of this procedure we create a GME that is a bona fide master equation, in spite of being non-Markov. We note that the memory kernel of the GME affords information on the interaction between system of interest and its bath. The Poisson case yields a bath with infinitely fast fluctuations. We argue that departing from the Poisson form has the effect of creating a condition of infinite memory and that these results might be useful to shed light on the problem of how to unravel non-Markov quantum master equations.

The multi-scale and nonlinear nature of the ocean dynamics dramatically affects the spreading of matter, like pollutants, marine litter, etc., of physical and chemical seawater properties, and the biological connectivity inside and among different basins. Based on the Finite-Scale Lyapunov Exponent analysis of the largest available near-surface Lagrangian data set from the Global Drifter Program, our results show that, despite the large variety of flow features, relative dispersion can ultimately be described by a few parameters common to all ocean sub-basins, at least in terms of order of magnitude. This provides valuable information to undertake Lagrangian dispersion studies by means of models and/or of observational data. Moreover, our results show that the relative dispersion rates measured at submesoscale are significantly higher than for large-scale dynamics. Auxiliary analysis of high resolution GPS-tracked drifter hourly data as well as of the drogued/undrogued status of the buoys is provided in support of our conclusions. A possible application of our study, concerning reverse drifter motion and error growth analysis, is proposed relatively to the case of the missing Malaysia Airlines MH370 aircraft.