Luigi Palatella

Power-law time distribution of large earthquakes.

We study the statistical properties of time distribution of seismicity 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 mu=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.

ASYMMETRIC ANOMALOUS DIFFUSION: AN EFFICIENT WAY TO DETECT MEMORY IN TIME SERIES

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 exponent δ of the time in 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 (CLT), as well as adopting a numerical treatment. We show that rendering the diffusion process to be asymmetric 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

Compression and Diffusion: A Joint Approach to Detect Complexity

Abstract The adoption of the Kolmogorov–Sinai entropy is becoming a popular research tool among physicists, especially when applied to a dynamical system fitting the conditions of validity of the Pesin theorem. The study of time series that are a manifestation of system dynamics whose rules are either unknown or too complex for a mathematical treatment, is still a challenge since the KS entropy is not computable, in general, in that case. Here we present a plan of action based on the joint action of two procedures, both related to the KS entropy, but compatible with computer implementation through fast and efficient programs. The former procedure, called compression algorithm sensitive to regularity (CASToRE), establishes the amount of order by the numerical evaluation of algorithmic compressibility. The latter, called complex analysis of sequences via scaling and randomness assessment (CASSANDRA), establishes the complexity degree through the numerical evaluation of the strength of an anomalous effect. This is the departure, of the diffusion process generated by the observed fluctuations, from ordinary Brownian motion. The CASSANDRA algorithm shares with CASToRE a connection with the Kolmogorov complexity. This makes both algorithms especially suitable to study the transition from dynamics to thermodynamics, and the case of non-stationary time series as well. The benefit of the joint action of these two methods is proven by the analysis of artificial sequences with the same main properties as the real time series to which the joint use of these two methods will be applied in future research work.

Intermittency and scale-free networks: a dynamical model for human language complexity

Article discussing intermittency, scale-free networks, and a dynamical model for human language complexity.

A non extensive approach to the entropy of symbolic sequences

Symbolic sequences with long-range correlations are expected to result in a slow regression to a steady state of entropy increase. However, we prove that also in this case a fast transition to a constant rate of entropy increase can be obtained, provided that the extensive entropy of Tsallis with entropic index q is adopted, thereby resulting in a new form of entropy that we shall refer to as Kolmogorov–Sinai–Tsallis (KST) entropy. We assume that the same symbols, either 1 or −1, are repeated in strings of length l, with the probability distribution p(l)∝1/lμ. The numerical evaluation of the KST entropy suggests that at the value μ=2 a sort of abrupt transition might occur. For the values of μ in the range 1 3). It is argued that this phase-transition-like property signals the onset of the thermodynamical regime at μ=2.

Memory beyond memory in heart beating, a sign of a healthy physiological condition.

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.

Correlation function and generalized master equation of arbitrary age

We study a two-state statistical process with a non-Poisson distribution of sojourn times. In accordance with earlier work, we find that this process is characterized by aging and we study three different ways to define the correlation function of arbitrary age of the corresponding dichotomous fluctuation. These three methods yield exact expressions, thus coinciding with the recent result by Godrèche and Luck [J. Stat. Phys. 104, 489 (2001)]. Actually, non-Poisson statistics yields infinite memory at the probability level, thereby breaking any form of Markovian approximation, including the one adopted herein, to find an approximated analytical formula. For this reason, we check the accuracy of this approximated formula by comparing it with the numerical treatment of the second of the three exact expressions. We find that, although not exact, a simple analytical expression for the correlation function of arbitrary age is very accurate. We establish a connection between the correlation function and a generalized master equation of the same age. Thus this formalism, related to models used in glassy materials, allows us to illustrate an approach to the statistical treatment of blinking quantum dots, bypassing the limitations of the conventional Liouville treatment.

Lyapunov vectors and assimilation in the unstable subspace: theory and applications

Based on a limited number of noisy observations, estimation algorithms provide a complete description of the state of a system at current time. Estimation algorithms that go under the name of assimilation in the unstable subspace (AUS) exploit the nonlinear stability properties of the forecasting model in their formulation. Errors that grow due to sensitivity to initial conditions are efficiently removed by confining the analysis solution in the unstable and neutral subspace of the system, the subspace spanned by Lyapunov vectors with positive and zero exponents, while the observational noise does not disturb the system along the stable directions. The formulation of the AUS approach in the context of four-dimensional variational assimilation (4DVar-AUS) and the extended Kalman filter (EKF-AUS) and its application to chaotic models is reviewed. In both instances, the AUS algorithms are at least as efficient but simpler to implement and computationally less demanding than their original counterparts. As predicted by the theory when error dynamics is linear, the optimal subspace dimension for 4DVar-AUS is given by the number of positive and null Lyapunov exponents, while the EKF-AUS algorithm, using the same unstable and neutral subspace, recovers the solution of the full EKF algorithm, but dealing with error covariance matrices of a much smaller dimension and significantly reducing the computational burden. Examples of the application to a simplified model of the atmospheric circulation and to the optimal velocity model for traffic dynamics are given.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’.

Absorption and Emission in the Non-Poissonian Case

This Letter addresses the challenging problems posed to the Kubo-Anderson (KA) theory by the discovery of intermittent resonant fluorescence with a nonexponential distribution of waiting times. We show how to extend the KA theory from aged to aging systems, aging for a very extended time period or even forever, being a crucial consequence of non-Poisson statistics.

Non-Poisson Dichotomous Noise: Higher-Order Correlation Functions and Aging

We study a two-state symmetric noise, with a given waiting time distribution psi (tau) , and focus our attention on the connection between the four-time and two-time correlation functions. The transition of psi (tau) from the exponential to the nonexponential condition yields the breakdown of the usual factorization condition of high-order correlation functions, as well as the birth of aging effects. We discuss the subtle connections between these two properties and establish the condition that the Liouville-like approach has to satisfy in order to produce a correct description of the resulting diffusion process.