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Extended Entropies And Disorder

Author

Listed:
  • MATT DAVISON

    (Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 5B7, Canada)

  • J. S. SHINER

    (The Shiner Group, Bergacher 3, CH-3325 Hettiswil, Switzerland;
    Abteilung Mathematik und Ingenieurwissenschaften, Fernfachhochschule Schweiz, CH-3900 Brig, Switzerland;
    Institut für mathematische Statistik und Versicherungslehre der Universität Bern, CH-3012 Bern, Switzerland)

Abstract

Landsberg's notion of disorder, entropy normalized to maximum entropy, was originally proposed for the Shannon information-theoretic entropy to overcome extensivity-based deficiencies of entropy as a measure of disorder. We generalize Landsberg's concept to three classes of extended entropies: Rényi, Tsallis and Landsberg–Vedral. We show an intimate connection between the Rényi disorders and the spectrum of dimensions known as multifractals. Three examples are treated, including one for power law distributions and one based on the logistic map. On the basis of the three examples, it is demonstrated that all three classes of extended disorder are required to fully characterize the corresponding properties of a system. We conjecture, and sketch a proof to support, that all three extended disorders are also sufficient to completely determine a dimension spectrum.

Suggested Citation

  • Matt Davison & J. S. Shiner, 2005. "Extended Entropies And Disorder," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 8(01), pages 125-158.
    • Handle: RePEc:wsi:acsxxx:v:08:y:2005:i:01:n:s0219525905000373
    DOI: 10.1142/S0219525905000373
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