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On the extended Kolmogorov–Nagumo information-entropy theory, the q→1/q duality and its possible implications for a non-extensive two-dimensional Ising model

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  • Masi, Marco

Abstract

The aim of this paper is to investigate the q→1/q duality in an information-entropy theory of all q-generalized entropy functionals (Tsallis, Renyi and Sharma–Mittal measures) in the light of a representation based on generalized exponential and logarithm functions subjected to Kolmogorov's and Nagumo's averaging. We show that it is precisely in this representation that the form invariance of all entropy functionals is maintained under the action of this duality. The generalized partition function also results to be a scalar invariant under the q→1/q transformation which can be interpreted as a non-extensive two-dimensional Ising model duality between systems governed by two different power law long-range interactions and temperatures. This does not hold only for Tsallis statistics, but is a characteristic feature of all stationary distributions described by q-exponential Boltzmann factors.

Suggested Citation

  • Masi, Marco, 2007. "On the extended Kolmogorov–Nagumo information-entropy theory, the q→1/q duality and its possible implications for a non-extensive two-dimensional Ising model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 377(1), pages 67-78.
  • Handle: RePEc:eee:phsmap:v:377:y:2007:i:1:p:67-78
    DOI: 10.1016/j.physa.2006.11.019
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    References listed on IDEAS

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    1. Touchette, Hugo, 2002. "When is a quantity additive, and when is it extensive?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(1), pages 84-88.
    2. Salazar, R. & Toral, R., 2001. "Thermostatistics of extensive and non-extensive systems using generalized entropies," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 290(1), pages 159-191.
    3. Almeida, M.P., 2001. "Generalized entropies from first principles," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 300(3), pages 424-432.
    4. Andrade, R.F.S., 1994. "Remarks on the behavior of the Ising chain in the generalized statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 203(3), pages 486-494.
    5. Tsallis, Constantino & Mendes, RenioS. & Plastino, A.R., 1998. "The role of constraints within generalized nonextensive statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 261(3), pages 534-554.
    6. Frank, T.D. & Daffertshofer, A., 2000. "Exact time-dependent solutions of the Renyi Fokker–Planck equation and the Fokker–Planck equations related to the entropies proposed by Sharma and Mittal," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 285(3), pages 351-366.
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    Cited by:

    1. Asgarani, Somayeh & Mirza, Behrouz, 2015. "Two-parameter entropies, Sk,r, and their dualities," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 417(C), pages 185-192.

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