Probability Distributions in Complex Systems

Sornette, Didier

doi:10.1007/978-0-387-30440-3_418

Cited by 40 publications

(41 citation statements)

References 64 publications

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“…To grasp the impact of our finding, consider, for example, the important debate on the inference of power laws in relaxing systems [18][19][20][21] or in the long tails of probability distributions, which suggests the existence of universal laws underlying many physical phenomena generally associated with complex systems and networks [22][23][24][25]. The basis for these inferences is the comparison of partial, i.e., nonequilibrium, distributions observed up to an experimentally accessible time t to model equilibrium distributions.…”

Section: Discussionmentioning

confidence: 99%

Intrinsic irreversibility of Markovian chains

2014

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We show that for a large class of stationary Markov processes the total variation distance between the final equilibrium distribution and that at a given time is a strongly monotonic vanishing function. We illustrate this for basic paradigmatic processes and discuss how, for systems susceptible to a canonical description, this can be interpreted as a statistical arrow of time that exists besides the standard decrease of free energy.

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Section: Discussionmentioning

confidence: 99%

Intrinsic irreversibility of Markovian chains

2014

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“…It seems that perhaps a combination of dynamics and statistics is necessary to describe systems with complicated dynamics (Cohen, 2005). Sornette (2007) discusses the ubiquity of observed power law distributions in complex systems as follows. The extension of Boltzmann's distribution to out-of-equilibrium systems is the subject of intense scrutiny.…”

Section: General Systems Theory and Classical Statistical Physicsmentioning

confidence: 99%

“…Power law distributions incarnate the notion that extreme events are not exceptional. Instead, extreme events should be considered as rather frequent and part of the same organization as the other events (Sornette, 2007).…”

Section: General Systems Theory and Classical Statistical Physicsmentioning

confidence: 99%

Universal Inverse Power-Law Distribution for Fractal Fluctuations in Dynamical Systems: Applications for Predictability of Inter-Annual Variability of Indian and USA Region Rainfall

Selvam

2016

Pure Appl. Geophys.

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Dynamical systems in nature exhibit self-similar fractal space-time fluctuations on all scales indicating long-range correlations and therefore the statistical normal distribution with implicit assumption of independence, fixed mean and standard deviation cannot be used for description and quantification of fractal data sets. The author has developed a general systems theory based on classical statistical physics for fractal fluctuations which predicts the following. (i) The fractal fluctuations signify an underlying eddy continuum, the larger eddies being the integrated mean of enclosed smaller-scale fluctuations. (ii) The probability distribution of eddy amplitudes and the variance (square of eddy amplitude) spectrum of fractal fluctuations follow the universal Boltzmann inverse power law expressed as a function of the golden mean. (iii) Fractal fluctuations are signatures of quantum-like chaos since the additive amplitudes of eddies when squared represent probability densities analogous to the subatomic dynamics of quantum systems such as the photon or electron. (iv) The model predicted distribution is very close to statistical normal distribution for moderate events within two standard deviations from the mean but exhibits a fat long tail that are associated with hazardous extreme events. Continuous periodogram power spectral analyses of available GHCN annual total rainfall time series for the period 1900 to 2008 for Indian and USA stations show that the power spectra and the corresponding probability distributions follow model predicted universal inverse power law form signifying an eddy continuum structure underlying the observed inter-annual variability of rainfall. Global warming related atmospheric energy input will result in intensification of fluctuations of all scales and can be seen immediately in high frequency (shortterm) fluctuations such as devastating floods/droughts resulting from excess/deficit annual, quasi-biennial and other shorter period (years) rainfall cycles.

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“…Current usage of these terms regards the idea that heavy-tailed distributions are those that have power-law frequency-size distributions, and where not all moments are finite; these distributions arise from scale-invariant processes (Ghil et al, 2011). What is interesting is that distributions with a wild character, such as the power-law distributions, in contrast to those with mild ones, such as Gaussian distributions, are ubiquitous in many natural phenomena characterised by some criticality process (Sornette, 2006(Sornette, , 2009. This can be explained by the presence of their heavy tail that emerges from the non-linear coupling properties, when the probability of values far from the statistical mode (such as the occurrence of disasters with respect to normal situations) has still some chance of occurring.…”

Section: Introductionmentioning

confidence: 99%

Power-law frequency distribution of H/V spectral ratio of seismic signals: Evidence for a critical crust

Signanini

Santis

2012

Earth Planet Sp

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We analyse records from several seismic stations in Central Italy (including some with multiple recording sites) for the statistical properties of spectra of both noise and earthquake signals. The main result is that the power-law scaling of most of the spectral H/V ratio statistical distributions leads to a so-called heavy tail. This is interpreted as a statistical distribution with a fractal dimension of about 2, which is almost certainly caused by a porous percolating medium beneath the stations due to some universal property of the crustal rocks crossed by the elastic waves propagation suggesting that the uppermost crust is in a critical state.

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Probability Distributions in Complex Systems

Cited by 40 publications

References 64 publications

Intrinsic irreversibility of Markovian chains

Intrinsic irreversibility of Markovian chains

Universal Inverse Power-Law Distribution for Fractal Fluctuations in Dynamical Systems: Applications for Predictability of Inter-Annual Variability of Indian and USA Region Rainfall

Power-law frequency distribution of H/V spectral ratio of seismic signals: Evidence for a critical crust

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