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| Titel |
Functional background of the Tsallis entropy: "coarse-grained" systems and "kappa" distribution functions |
| VerfasserIn |
A. V. Milovanov, L. M. Zelenyi |
| Medientyp |
Artikel
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| Sprache |
Englisch
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| ISSN |
1023-5809
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| Digitales Dokument |
URL |
| Erschienen |
In: Nonlinear Processes in Geophysics ; 7, no. 3/4 ; Nr. 7, no. 3/4, S.211-221 |
| Datensatznummer |
250004259
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| Publikation (Nr.) |
 copernicus.org/npg-7-211-2000.pdf |
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| Zusammenfassung |
| The concept of the generalized entropy is
analyzed, with the particular attention to the definition postulated by Tsallis
[J. Stat. Phys. 52, 479 (1988)]. We show that the Tsallis entropy can be
rigorously obtained as the solution of a nonlinear functional equation; this
equation represents the entropy of a complex system via the partial entropies of
the subsystems involved, and includes two principal parts. The first part is
linear (additive) and leads to the conventional, Boltzmann, definition of
entropy as the logarithm of the statistical weight of the system. The second
part is multiplicative and contains all sorts of multilinear products of the
partial entropies; inclusion of the multiplicative terms is shown to reproduce
the generalized entropy exactly in the Tsallis sense. We speculate that the
physical background for considering the multiplicative terms is the role of the
long-range correlations supporting the "macroscopic" ordering
phenomena (e.g., formation of the "coarse-grained" correlated
patterns). We prove that the canonical distribution corresponding to the Tsallis
definition of entropy, coincides with the so-called "kappa"
redistribution which appears in many physical realizations. This has led us to
associate the origin of the "kappa" distributions with the
"macroscopic" ordering ("coarse-graining") of the system.
Our results indicate that an application of the formalism based on the Tsallis
notion of entropy might actually have sense only for the systems whose
statistical weights, Ω, are relatively small. (For the
"coarse-grained" systems, the weight \omega could be interpreted as
the number of the "grains".) For large Ω (i.e., Ω -> ∞),
the standard statistical mechanical formalism is advocated, which implies the
conventional, Boltzmann definition of entropy as ln Ω. |
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