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Non-Boltzmannian Entropies for Complex Classical Systems, Quantum Coherent States and Black Holes

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Chaos, Nonlinearity, Complexity

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 206))

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Abstract

The Renyi entropy is derived as a cumulant average of the Boltzmann entropy in the same way as the Helmholtz free energy can be obtained by cumulant averaging of a Hamiltonian. Such a form of the information entropy and the principle of entropy maximum (MEP) for it are justified by the Shore–Johnson theorem. The application of MEP to the Renyi entropy gives rise to the Renyi distribution. Thermodynamic entropy in the Renyi thermostatistics increases with system complexity (gain of an order parameter n = 1-q) and reaches its maximal value at q min. The Renyi distribution for such q becomes a pure power–law distribution. Because a power–law distribution is characteristic for self-organizing systems the Renyi entropy can be considered as a potential that drives the system to a self-organized state. The derivative of difference of entropies in the Renyi and Gibbs thermostatistics in respect to n exhibits a jump at n = 0. This permits us to consider the transfer to the Renyi thermostatistics as a peculiar kind of a phase transition into a more organized state.

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Authors and Affiliations

  1. Institute Dynamics of Geospheres, Russian Academy of Sciences, Moscow, 119334, RUSSIA

    A. G. Bashkirov

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  1. Department of Mechanical Engineering, Nuclear Engineering and Technology Programme Indian Institute of Technology Kanpur, Kanpur, 208016, India

    A. Sengupta Professor

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Bashkirov, A. (2006). Non-Boltzmannian Entropies for Complex Classical Systems, Quantum Coherent States and Black Holes. In: Sengupta, A. (eds) Chaos, Nonlinearity, Complexity. Studies in Fuzziness and Soft Computing, vol 206. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31757-0_4

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  • DOI: https://doi.org/10.1007/3-540-31757-0_4

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