Stability of the entropy for superstatistics

doi:10.1016/j.physleta.2003.10.025

Physics Letters A

Volume 319, Issues 3–4, 8 December 2003, Pages 273-278

https://doi.org/10.1016/j.physleta.2003.10.025 Get rights and content

Abstract

The Boltzmann–Gibbs celebrated entropy S_BG=−k∑_ip_ilnp_i is concave (with regard to all probability distributions {p_i}) and stable (under arbitrarily small deformations of any given probability distribution). It seems reasonable to consider these two properties as necessary for an entropic form to be a physical one in the thermostatistical sense. Most known entropic forms (e.g., Rényi entropy) violate these conditions, in contrast with the basis of nonextensive statistical mechanics, namely $S_{q} =k 1−∑_{i} p_{i}^{q} q−1$ $(q∈ R; S_{1} =S_{BG})$ , which satisfies both (∀q>0). We have recently generalized S_q (into S) in order to yield, through optimization, the Beck–Cohen superstatistics. We show here that S satisfies both conditions as well. The satisfaction of both concavity and stability appears to be very helpful to identify physically admissible entropic forms. More precisely, the use of this criterion raises up a sensible part of the uncomfortable degeneracy emerging from the fact that the distribution which optimizes some constrained entropic form, also optimizes (under the same constraints) any monotonic function of it. The simplifying criterion is based on the mathematical fact that neither concavity nor stability are invariant through monotonicity.

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Acknowledgements

Useful remarks from E. Brigatti are acknowledged, as well as partial support from PCI/MCT, CNPq, PRONEX/MCT and FAPERJ (Brazilian agencies).

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It seems reasonable to consider concavity (with regard to all probability distributions) and stability (under arbitrarily small deformations of any given probability distribution) as necessary for an entropic form to be a physical one in the thermostatistical sense. Most known entropic forms, e.g. Renyi entropy $S_{q}^{R} =(ln ∑_{i} p_{i}^{q})/(1−q)$ , violate one and/or the other of these conditions. In contrast, the Boltzmann–Gibbs entropy $S_{BG} =−∑_{i} p_{i} ln p_{i}$ and the nonextensive one $S_{q} =(1−∑_{i} p_{i}^{q})/(q−1) (q>1)$ satisfy both. S_BG and S_q belong in fact to a larger class of entropies S satisfying both, namely those which, through appropriate optimization, yield the Beck–Cohen superstatistics. We briefly review here the proof of stability of S, and illustrate for an important particular case, namely the log-normal superstatistical entropy. The satisfaction of both concavity and stability appears to be very helpful to identify physically admissible entropic forms.
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Stability of the entropy for superstatistics

Abstract

Section snippets

Acknowledgements

Physica A

Physica A

Physica A

Phys. Rev. Lett.

Phys. Rev. Lett.

Europhys. Lett.

Phys. Rev. E

Phys. Rev. Lett.

Phys. Rev. E

Phys. Rev. E

Phys. Rev. Lett.

Physica A

Phys. Rev. Lett.

Phys. Rev. Lett.

J. Phys. A

Phys. Rev. E

Physica A

Phys. Rev. E

Phys. Rev. E

Phys. Rev. E

Phys. Rev. E

Phys. Rev. Lett.

Nonextensive Statistical Mechanics and Thermodynamics

Braz. J. Phys.

Non Extensive Statistical Mechanics and Physical Applications

Physica A

J. Stat. Phys.