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Statistical Complexity pp 215–235Cite as

Rényi Entropy and Complexity

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Abstract

Several important properties of the Rényi entropy and Rényi entropy power are presented. Uncertainty relations for the Rényi entropy including uncertainty relations for single particle densities of many particle systems in position and momentum spaces are discussed. Connection between Fisher information and Rényi entropy is studied. The Fisher-Rényi information plane and entropic product are presented.

Position and momentum space Rényi entropies of order α are presented for ground-state neutral atoms with atomic numbers Z=1–103. It is emphasized that the values of α≤1 (α≥1) stress the shell structure for position-space (momentum-space) Rényi entropies. Position and momentum space relative Rényi entropies of order α are presented for ground-state neutral atoms with atomic numbers Z=1–103. Simple hydrogen-like model densities are used as the reference. A relationship with the atomic radius and quantum capacitance is also discussed.

The relationship between the statistical complexity and the Rényi entropy is studied. A recently introduced, one-parameter extension of the LMC complexity is presented.

The maximum Rényi entropy principle is used to generalize the Thomas-Fermi model. A simple relation between the dimension and the Rényi parameter is emphasized.

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Acknowledgements

E.R. acknowledges the Spanish project FQM-165/0207 (Junta de Andalucía) and No. FIS2008-01143. Á.N. acknowledges grant OTKA No. T67923. The work was also supported by the TAMOP 4.2.1/B-09/1/KONV-2010-0007 project. The project is co-financed by the European Union and the European Social Fund.

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Authors and Affiliations

  1. Department of Theoretical Physics, University of Debrecen, 4010, Debrecen, Hungary

    Á. Nagy

  2. Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, 18071, Granada, Spain

    E. Romera

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  1. Á. Nagy
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  2. E. Romera
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Correspondence to Á. Nagy .

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Editors and Affiliations

  1. School of Chemistry, University of Hyderabad, Hyderabad, 500046, India

    K.D. Sen

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Nagy, Á., Romera, E. (2011). Rényi Entropy and Complexity. In: Sen, K. (eds) Statistical Complexity. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3890-6_7

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