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https://hdl.handle.net/20.500.14365/4710
Title: | Necessary Condition of Self-Organisation in Nonextensive Open Systems | Authors: | Afşar, Ozgur Tırnaklı, Uğur |
Keywords: | S-theorem q-renormalized entropy complexity measures logistic map Power-Law Sensitivity Quantitative-Analysis Renormalized Entropy Initial Conditions Complexity Origin |
Publisher: | Mdpi | Abstract: | In this paper, we focus on evolution from an equilibrium state in a power law form by means of q-exponentials to an arbitrary one. Introducing new q-Gibbsian equalities as the necessary condition of self-organization in nonextensive open systems, we theoretically show how to derive the connections between q-renormalized entropies (Delta(S) over tilde (q)) and q-relative entropies (KLq) in both Bregman and Csiszar forms after we clearly explain the connection between renormalized entropy by Klimantovich and relative entropy by Kullback-Leibler without using any predefined effective Hamiltonian. This function, in our treatment, spontaneously comes directly from the calculations. We also explain the difference between using ordinary and normalized q-expectations in mean energy calculations of the states. To verify the results numerically, we use a toy model of complexity, namely the logistic map defined as Xt +1 = 1 - aX(t)(2), where a is an element of [0, 2] is the map parameter. We measure the level of self-organization using two distinct forms of the q-renormalized entropy through period doublings and chaotic band mergings of the map as the number of periods/chaotic-bands increase/decrease. We associate the behaviour of the q-renormalized entropies with the emergence/disappearance of complex structures in the phase space as the control parameter of the map changes. Similar to Shiner-Davison-Landsberg (SDL) complexity, we categorize the tendencies of the q-renormalized entropies for the evaluation of the map for the whole control parameter space. Moreover, we show that any evolution between two states possesses a unique q = q* value (not a range for q values) for which the q-Gibbsian equalities hold and the values are the same for the Bregmann and Csiszar forms. Interestingly, if the evolution is from a = 0 to a = a(c) similar or equal to 1.4011, this unique q* value is found to be q* similar or equal to 0.2445, which is the same value of qsensitivity given in the literature. | URI: | https://doi.org/10.3390/e25030517 https://hdl.handle.net/20.500.14365/4710 |
ISSN: | 1099-4300 |
Appears in Collections: | PubMed İndeksli Yayınlar Koleksiyonu / PubMed Indexed Publications Collection Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection WoS İndeksli Yayınlar Koleksiyonu / WoS Indexed Publications Collection |
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