Rényi entropies and observables

Lesche, B.

doi:10.1103/physreve.70.017102

Cited by 62 publications

(50 citation statements)

References 3 publications

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“…It is dynamic entropy 17,18 and exhibits 19 concavity, positivity and Lesche 20,21 stability. Note that S should not be confused with Cov[χ, ln χ] ≡ χ ln χ − χ ln χ since in general ln χ = ln χ .…”

Section: Introductionmentioning

confidence: 99%

Fluctuations, under time reversal, of the natural time and the entropy distinguish similar looking electric signals of different dynamics

Varotsos

Sarlis

Skordas

et al. 2008

Journal of Applied Physics

111

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We show that the scale dependence of the fluctuations of the natural time itself under time reversal provides a useful tool for the discrimination of seismic electric signals (critical dynamics) from noises emitted from manmade sources as well as for the determination of the scaling exponent. We present recent data of electric signals detected at the Earth's surface, which confirm that the value of the entropy in natural time as well as its value under time reversal are smaller than that of the entropy of a "uniform" distribution.PACS numbers: 91.25. Qi, 91.30.Px, 05.45.Tp

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Section: Introductionmentioning

confidence: 99%

Fluctuations, under time reversal, of the natural time and the entropy distinguish similar looking electric signals of different dynamics

Varotsos

Sarlis

Skordas

et al. 2008

Journal of Applied Physics

111

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“…It is a dynamic entropy depending on the sequential order of pulses [5,6] and exhibits [9] concavity, positivity and Lesche [12,13] stability. When the system enters the critical stage (infinitely ranged long range temporal correlations [3,4]), the S-value is smaller [8,10] than the value S u (= 1/2 ln 2 − 1/4 ≈ 0.0966) of a "uniform" distribution (defined in Refs.…”

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confidence: 99%

Entropy of seismic electric signals: Analysis in natural time under time reversal

et al. 2006

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Electric signals have been recently recorded at the Earth's surface with amplitudes appreciably larger than those hitherto reported. Their entropy in natural time is smaller than that, S u , of a "uniform" distribution. The same holds for their entropy upon time-reversal. This behavior, as supported by numerical simulations in fBm time series and in an on-off intermittency model, stems from infinitely ranged long range temporal correlations and hence these signals are probably Seismic Electric Signals (critical dynamics). The entropy fluctuations are found to increase upon approaching bursting, which reminds the behavior identifying sudden cardiac death individuals when analysing their electrocardiograms.

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confidence: 99%

“…Despite this apparent progress in the field, however, there have been some criticisms regarding its applicability and scope. Among such criticisms, one can particularly cite the ones related to the Bayesian updating procedure [14], Lesche stability [15][16][17], and the methodology of the entropy maximization [18].Recently, Abe pinpointed that the nonadditive q-entropy is inherently limited to the finite discrete systems, since its continuum expression has not been obtained yet [19] (see also Refs. [20,21]).…”

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Route from discreteness to the continuum for the Tsallis q -entropy

Oikonomou¹,

Bagci²

2018

Phys. Rev. E

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The existence and exact form of the continuum expression of the discrete nonlogarithmic q-entropy is an important open problem in generalized thermostatistics, since its possible lack implies that nonlogarithmic qentropy is irrelevant for the continuous classical systems. In this work, we show how the discrete nonlogarithmic q-entropy in fact converges in the continuous limit and the negative of the q-entropy with continuous variables is demonstrated to lead to the (Csiszár type) q-relative entropy just as the relation between the continuous Boltzmann-Gibbs expression and the Kullback-Leibler relative entropy. As a result, we conclude that there is no obstacle for the applicability of the q-entropy to the continuous classical physical systems.Since its advent, the nonadditive q-entropy [1, 2] has found numerous fields of application in many diverse fields [3][4][5][6][7][8][9][10][11][12][13]. Despite this apparent progress in the field, however, there have been some criticisms regarding its applicability and scope. Among such criticisms, one can particularly cite the ones related to the Bayesian updating procedure [14], Lesche stability [15][16][17], and the methodology of the entropy maximization [18].Recently, Abe pinpointed that the nonadditive q-entropy is inherently limited to the finite discrete systems, since its continuum expression has not been obtained yet [19] (see also Refs. [20,21]). In this work, we show that one can indeed obtain the concomitant continuum expressions of the nonadditive entropy and therefore point out that the nonadditive q-entropy can also be used for continuous physical systems.Before proceeding further with the nonadditive case, one should be convinced why taking the route from discreteness to a continuum is essential concerning any entropy measure in general. Setting the Boltzmann constant to unity, the finite discrete Boltzmann-Gibbs (BG) entropy readswhere p i denotes the probability of the ith event. Let us now consider its continuous counterpart to be the following expressionwhere ρ(x) is a probability density function satisfying the normalization condition in the interval [a, b]. Although the continuous expression above seems reasonable at first sight, it has three serious drawbacks. First, the continuous version in Eq. (2) has an overall unit of log(length) whereas the discrete entropy in Eq. (1) is dimensionless [22]. Second, the probability density S(ρ) is not invariant with respect to coordinate transformations [22]. Last but not the least, the discrete BG entropy S({p}) in the n → ∞ limit and S(ρ) yield different results [22]: To see this more explicitly, consider a uniform distribution ρ(x) in the interval [a, b] as 1/(b − a) so that its discrete counterpart p(x i ) is given by 1/n obtained through dividing the same interval [a, b] into n equal subintervals where the index i runs from 1 to n. Then, the continuous entropy S(ρ) for this uniform distribution yields ln(b − a) while the discrete expression S({p}) attains infinity in the n → ∞ limit. In other words, the ...

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Rényi entropies and observables

Abstract: Evidence is given that Rényi entropies of macroscopic thermodynamic systems defined on the bases of probabilities of microstates cannot be related to observables. The notion of observable is clarified.

Cited by 62 publications

References 3 publications

Fluctuations, under time reversal, of the natural time and the entropy distinguish similar looking electric signals of different dynamics

Fluctuations, under time reversal, of the natural time and the entropy distinguish similar looking electric signals of different dynamics

Entropy of seismic electric signals: Analysis in natural time under time reversal

Route from discreteness to the continuum for the Tsallis q -entropy

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