Renewal, modulation, and superstatistics in times series

Allegrini, Paolo; Barbi, Francesco; Grigolini, Paolo; Paradisi, Paolo

doi:10.1103/physreve.73.046136

Cited by 44 publications

(71 citation statements)

References 40 publications

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“…[22]. Restricting ourselves to brief comments we would only like to emphasize that, as the analysis of above simple variant shows, the general method developed in our work enables one to represent the results obtained in this paper in very compact and general form.…”

Section: Fluctuations Of Waiting Time Pdf Matricesmentioning

confidence: 99%

“…In some CTRW-variants the non-homogeneity of the process, consisting in the dependence of the jump-like fluctuations on the fluctuation number, have been taken into account [20,21]. In other variants the modification of simple CTRW time sequences of renewals is proposed [22]. These modifications are very interesting and essentially clarify specific features of CTRW-like processes.…”

Section: Introductionmentioning

confidence: 99%

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Non-Markovian stochastic Liouville equation and its Markovian representation: Extensions of the continuous-time random-walk approach

Shushin

2008

Phys. Rev. E

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Some specific features and extensions of the continuous time random walk (CTRW) approach are analyzed in detail within the Markovian representation (MR) and CTRW-based non-Markovian stochastic Liouville equation (SLE). In the MR CTRW processes are represented by multidimensional Markovian ones. In this representation the probability distribution function (PDF) W (t) of fluctuation renewals is associated with that of reoccurrences in a certain jump state of some Markovian controlling process. Within the MR the non-Markovian SLE, which describes the effect of CTRW-like noise on relaxation of dynamic and stochastic systems, is generalized to take into account the influence of relaxing systems on statistical properties of noise. The generalized nonMarkovian SLE is applied to study two modifications of the CTRW approach. One of them considers the cascaded CTRWs in which the controlling process is actually CTRW-like one controlled by another CTRW process, controlled in turn by the third one, etc. Within the MR simple expression for the PDF W (t) of total controlling process is obtained in terms of Markovian variants of controlling PDFs in the cascade. The expression is shown to be especially simple and instructive in the case of anomalous processes determined by long time tailed W (t). The cascaded CTRWs can model the effect of complexity of a system on relaxation kinetics (in glasses, fractals, branching media, ultrametric structures, etc.). Another CTRW-modification describes the kinetics of processes governed by fluctuating W (t). Within the MR the problem is analyzed in a general form without restrictive assumptions on correlations of PDFs of consecutive renewals. The analysis shows that W (t) can strongly affect the kinetics of the process. Possible manifestations of this effect are discussed.

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Section: Fluctuations Of Waiting Time Pdf Matricesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non-Markovian stochastic Liouville equation and its Markovian representation: Extensions of the continuous-time random-walk approach

Shushin

2008

Phys. Rev. E

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“…The presence of crucial events [24,26] makes the probability distribution density px; l satisfy the scaling condition…”

Section: -2mentioning

confidence: 99%

“…The importance of these events has been stressed in [26], showing that for S < 3 the long-time statistical properties of the network S are determined by these events, in spite of camouflage action exerted by the cloud of secondary events, within which these events are imbedded. Following the authors of Ref.…”

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confidence: 99%

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Sun-Climate Complexity Linking

West

Grigolini

2008

Phys. Rev. Lett.

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It is known that Earth's short-term temperature anomalies share the same complexity index as solar flares. We show that this property is not accidental and is a consequence of the phenomenon of information transfer based on the crucial role of non-Poisson renewal events in complex networks. DOI: 10.1103/PhysRevLett.100.088501 PACS numbers: 92.60.Ry, 05.20.ÿy, 05.40.Fb A cornerstone of statistical physics is the fluctuationdissipation theorem (FDT) of the first kind and the linear response theory (LRT) of Kubo [1] on which it rests. Recently, with the increasing importance of phenomena whose statistics do not satisfy the central limit theorem, a special version of the FDT was developed to study the response of renewal non-Poisson networks to external perturbations. This theory is a kind of generalization of the well-known theory of stochastic resonance (SR) proposed in 1981 to explain the periodic recurrence of ice ages [2]. The current generalization is called the event dominated fluctuation-dissipation theorem (EDFDT) of the first kind [3,4] and leads to the remarkable result that under certain conditions a complex network being perturbed by a second complex network takes on the statistical properties of the perturbation. This is the complexity matching (CM) phenomenon recently proposed by the authors of Ref. [5] involving the transfer of information, rather than energy, and can be realized with a perturbative linking of extremely weak intensity [4]. CM is expected to apply to all processes dominated by crucial renewal events, as has been shown in the case of earthquakes [6], blinking quantum dots [3], and brain dynamics [7]. Herein we apply this theory to the linking of Earth's climate to total solar irradiance (TSI). The average global temperature is a consequence of the TSI being absorbed and redistributed by Earth's atmosphere and oceans by means of nonlinear hydrothermal dynamic processes [8]. In the past two decades this phenomenon has been framed as a large-scale numerical simulation incorporating all identified physical or chemical mechanisms in an attempt to recreate and understand the variability in Earth's temperature time series [9]. What is not addressed in these simulations, over and above the temperature increase of the past 30 years, are the statistics of the average global temperature and the reasons why the statistics of the temperature anomalies are not Poisson, Gaussian, or any of the other elementary statistical distributions [10,11].Earth's surface temperature responds to variations in the TSI, which can be partitioned into low-frequency quasicycles and the more high-frequency random fluctuations. The low amplitude, approximately 11-year solar cycle influence on climate is well studied, leading to, for example, the decadal oscillation in the stratosphere and troposphere, as well as the resonant excitation of La Nina [12]. On the other hand, on the basis of large-scale computer simulations (see, for example, [13]), the highfrequency random fluctuations in solar activity have been found ...

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