We demonstrate that dual entropy expressions of the Tsallis type apply naturally to statistical–mechanical systems that experience an exceptional contraction of their configuration space. The entropic index α>1 describes the contraction process, while the dual index α′=2−α<1 defines the contraction dimension at which extensivity is restored. We study this circumstance along the three routes to chaos in low-dimensional nonlinear maps where the attractors at the transitions, between regular and chaotic behavior, drive phase-space contraction for ensembles of trajectories. We illustrate this circumstance for properties of systems that find descriptions in terms of nonlinear maps. These are size-rank functions, urbanization and similar processes, and settings where frequency locking takes place.

We show that size-rank distributions with power-law decay (often only over a limited extent) observed in a vast number of instances in a widespread family of systems obey Tsallis statistics. The theoretical framework for these distributions is analogous to that of a nonlinear iterated map near a tangent bifurcation for which the Lyapunov exponent is negligible or vanishes. The relevant statistical-mechanical expressions associated with these distributions are derived from a maximum entropy principle with the use of two different constraints, and the resulting duality of entropy indexes is seen to portray physically relevant information. Whereas the value of the index α fixes the distribution's power-law exponent, that for the dual index 2 − α ensures the extensivity of the deformed entropy.rank-ordered data | generalized entropies Z ipf's law refers to the (approximate) power law obeyed by sets of data when these are sorted out and displayed by rank in relation to magnitude or rate of recurrence (1). The sets of data originate from many different fields: astrophysical, geophysical, ecological, biological, technological, financial, urban, social, etc., suggesting some kind of universality. Over the years this circumstance has attracted much attention and the rationalization of this empirical law has become a common endeavor in the study of complex systems (2, 3). Here we pursue further the view (4, 5) that an understanding of the omnipresence of this type of rank distribution hints at an underlying structure similar to that which confers systems with many degrees of freedom the familiar macroscopic properties described by thermodynamics. That is, the quantities used in describing this empirical law obey expressions derived from principles akin to a statistical-mechanical formalism (4, 5). The most salient result presented here is that the reproduction of the data via a maximum entropy principle indicates that access to its configurational space is severely hindered to a point that the allowed configurational space has a vanishing measure. This feature appears to be responsible for the entropy expression not to be of the Boltzmann-Gibbs or Shannon type but instead to take that of the Tsallis form (6), while the extensivity of entropy is preserved. It is perhaps worth clarifying that our study is set in discrete space and it does not consider any formal Hamiltonian system.In Fig. 1 we show three examples of ranked data that appear to display power-law behavior along a considerably large interval of rank values. Fig. 1 (Top) shows data for the wealth of billionaires in the United States (7), Fig. 1 (Middle) shows data for the energy released by earthquakes in California (8), and Fig. 1 (Bottom) shows data for the intensity of solar flares (9). In Fig. 1 (Left), logarithmic scales are used for both size and rank, whereas Fig. 1 (Right) shows the same data in log-linear scales. Fig. 1 (Left) indicates approximate power-law decay for large rank and a clear deviation from this for small to moderate rank. As we shall sho...

Cosmic ray energy spectra exhibit power law distributions over many orders of magnitude that are very well described by the predictions of q-generalized statistical mechanics, based on a q-generalized Hagedorn theory for transverse momentum spectra and hard QCD scattering processes. QCD at largest center of mass energies predicts the entropic index to be q = 11. Here we show that the escort duality of the nonextensive thermodynamic formalism predicts an energy split of effective temperature given byMeV, where T H is the Hagedorn temperature. We carefully analyse the measured data of the AMS-02 collaboration and provide evidence that the predicted temperature split is indeed observed, leading to a different energy dependence of the e + and e − spectral indices. We also observe a distinguished energy scale E * ≈ 50 GeV where the e + and e − spectral indices differ the most. Linear combinations of the escort and non-escort q-generalized canonical distributions yield excellent agreement with the measured AMS-02 data in the entire energy range.Statistical mechanics is a universal formalism based on the maximization of the Boltzmann-Gibbs-Shannon entropy subject to suitable constraints. Despite its universal validity and success for short-range equilibrium systems, the applicability of the Boltzmann-Gibbs formalism has severe restrictions: It is not valid for nonequilibrium systems, it is not valid for systems with long-range interactions (such as gravity), and it is not valid for systems with a very small volume and fluctuating temperature (as probed in scattering processes of cosmic ray particles at very high energies). For these types of complex systems it is useful to generalize the formalism to a more general setting, based on the maximization of more general entropy measures which contain the Shannon entropy as a special case. Probably the most popular one of these generalizations is based on q-entropy (or Tsallis entropy), which leads to power law distributions (the so-called q-exponentials), but other generalized entropic approaches are possible as well [1][2][3][4][5][6][7][8] . In high energy physics, a recent success of the q-generalized approach is that excellent fits of measured transverse momentum spectra in high energy scattering experiments have been obtained 9 , based on an extension of Hagedorn's theory 10-12 to a q-generalized version and a generalized thermodynamic theory [13][14][15][16][17][18][19] . This includes recent experiments in the TeV region for pp and pp collisions 9,20-25 but there is also early work on cosmic ray spectra 26,27 and e + e − annihilation 15,28,29 . In this paper we systematically investigate the relevant degrees of freedom of the q-generalized statistical mechanics formalism at highest center of mass energies and develop an effective theory of energy spectra of cosmic rays, which are produced by scattering processes at extremely high energies (e.g. supernovae explosions). Some remarkable predictions come out of the formalism in its full generality. First of all, the para...

We analyse the probability densities of daily rainfall amounts at a variety of locations on the Earth. The observed distributions of the amount of rainfall fit well to a q-exponential distribution with exponent q close to q ≈ 1.3. We discuss possible reasons for the emergence of this power law. On the contrary, the waiting time distribution between rainy days is observed to follow a near-exponential distribution. A careful investigation shows that a q-exponential with q ≈ 1.05 yields actually the best fit of the data. A Poisson process where the rate fluctuates slightly in a superstatistical way is discussed as a possible model for this. We discuss the extreme value statistics for extreme daily rainfall, which can potentially lead to flooding. This is described by Fréchet distributions as the corresponding distributions of the amount of daily rainfall decay with a power law. On the other hand, looking at extreme event statistics of waiting times between rainy days (leading to droughts for very long dry periods) we obtain from the observed near-exponential decay of waiting times an extreme event statistics close to Gumbel distributions. We discuss superstatistical dynamical systems as simple models in this context.

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