We derive the diffusion process generated by a correlated dichotomous fluctuating variable y starting from a Liouville-like equation by means of a projection procedure. This approach makes it possible to derive all statistical properties of the diffusion process from the correlation function of the dichotomous fluctuating variable ⌽ y (t). Of special interest is that the distribution of the times of sojourn in the two states of the fluctuating process is proportional to d 2 ⌽ y (t)/dt 2 . Furthermore, in the special case where ⌽ y (t) has an inverse power law, with the index ␤ ranging from 0 to 1, thus making it nonintegrable, we show analytically that the statistics of the diffusing variable approximate in the long-time limit the ␣-stable Lévy distributions. The departure of the diffusion process of dynamical origin from the ideal condition of the Lévy statistics is established by means of a simple analytical expression. We note, first of all, that the characteristic function of a genuine Lévy process should be an exponential in time. We evaluate the correction to this exponential and show it to be expressed by a harmonic time oscillation modulated by the correlation function ⌽ y (t). Since the characteristic function can be given a spectroscopic significance, we also discuss the relevance of our results within this context. ͓S1063-651X͑96͒08611-4͔

We study the electroencephalogram ͑EEG͒ of 30 closed-eye awake subjects with a technique of analysis recently proposed to detect punctual events signaling rapid transitions between different metastable states. After single-EEG-channel event detection, we study global properties of events simultaneously occurring among two or more electrodes termed coincidences. We convert the coincidences into a diffusion process with three distinct rules that can yield the same only in the case where the coincidences are driven by a renewal process. We establish that the time interval between two consecutive renewal events driving the coincidences has a waiting-time distribution with inverse power-law index Ϸ 2 corresponding to ideal 1 / f noise. We argue that this discovery, shared by all subjects of our study, supports the conviction that 1 / f noise is an optimal communication channel for complex networks as in art or language and may therefore be the channel through which the brain influences complex processes and is influenced by them.

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...

This exploratory study provides evidence that, despite some redundancies in the informative content of nonlinear indices and strong differences in their prognostic power, quantification of nonlinear properties of HRV provides independent information in risk stratification of CHF patients.

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.

We address the problem of DNA sequences, developing a "dynamical" method based on the assumption that the statistical properties of DNA paths are determined by the joint action of two processes, one deterministic with long-range correlations and the other random and b-function correlated. The generator of the deterministic evolution is a nonlinear map belonging to a class of maps recently tailored to mimic the processes of weak chaos responsible for the birth of anomalous diffusion. It is assumed that the deterministic process corresponds to unknown biological rules that determine the DNA path, whereas the noise mimics the inQuence of an infinite-dimensional environment on the biological process under study. We prove that the resulting diffusion process, if the effect of the random process is neglected, is an o.-stable Levy process with 1 ( o. ( 2. We also show that, if the diffusion process is determined by the joint action of the deterministic and the random process, the correlation effects of the "deterministic dynamics" are canceled on the shortrange scale, but show up in the long-range one. We denote our prescription to generate statistical sequences as the copying mistake map (CMM). We carry out our analysis of several DNA sequences and their CMM realizations with a variety of techniques and we especially focus on a method of regression to equilibrium, which we call the Onsager analysis. With these techniques we establish the statistical equivalence of the real DNA sequences with their CMM realizations. We show that long-range correlations are present in exons as well as in introns, but are diKcult to detect, since the exon "dynamics" is shown to be determined by the entanglement of three distinct and independent CMM's.

We study a system whose dynamics are driven by non-Poisson, renewal, and nonergodic events. We show that external perturbations influencing the times at which these events occur violate the standard fluctuation-dissipation prescription due to renewal aging. The fluctuation-dissipation relation of this Letter is shown to be the linear response limit of an exact expression that has been recently proposed to account for the luminescence decay in a Gibbs ensemble of semiconductor nanocrystals, with intermittent fluorescence. The main purpose of this Letter is to go beyond the ergodic condition and to propose a generalization of FDT compatible with the anomalous properties of intermittent fluorescence, denoted as blinking quantum dots (BQD) [5], including the important renewal property [6]. We note that Verberk et al. [7], using only the renewal and nonexponential distribution revealed by experimental observation, predicted an inverse power-law luminescence decay fitting the experiment results. They did not, however, make the connection between their prediction and the breakdown of FDT. To establish a connection between the theoretical prediction of Verberk et al. and the FDT breakdown, we write the most general form of the response of a physical system to a time-dependent perturbation:where A is a system variable, whose mean value in the absence of perturbation is assumed to vanish. Another system variable B is coupled to the time-dependent perturbation P t of strength and AB t; t 0 is called the response function. In the recent condensed matter literature [1] the assumption is made that the external perturbation corresponds to adding to the unperturbed Hamiltonian the interaction term H pert B P t , and the adoption of the Kubo approach [3] is shown to generate the response functionwhere the brackets denote an average on either the classical or the quantum equilibrium probability distribution. This simple formula does not imply that the correlation function depends on t ÿ t 0 , and is consequently a generalization of the LRT of Kubo [3] obtained using Liouville or Liouvillelike equations that determine the time evolution of the variable probability. Herein we shift the focus from variables to events, an event being a collision that may produce an abrupt change in the value of a variable. We assume these events to be the renewal, non-Poisson, and nonergodic events studied by Barkai and co-workers [8], that A B S and that the variable S has only two distinct values, S 1. Finally, we assume that the external perturbation changes the prescription that determines the times at which events occur, namely, the times at which the variable S may, or may not, change its sign. This final assumption reveals the lack of a Hamiltonian formalism, a condition shared by Refs. [5,7].We define the nonstationary autocorrelation function where t; t 0 is the waiting-time distribution density of age t 0 [8,9]. Note that Eq. (4), as well as Eq. (2), is a generalization of the Kubo FDT [3]. When t;t 0 Þ t ÿ t 0 , d=dt t; t 0