caa a22 4500 467911312 CHVBK 20180406152850.0 cr unu---uuuuu 170328e20061101xx s 000 0 eng 10.1007/s11232-006-0138-x doi (NATIONALLICENCE)springer-10.1007/s11232-006-0138-x Bashkirov A. aut Renyi entropy as a statistical entropy for complex systems [Elektronische Daten] [A. Bashkirov] To describe a complex system, we propose using the Renyi entropy depending on the parameter q (0 < q ≤ 1) and passing into the Gibbs-Shannon entropy at q = 1. The maximum principle for the Renyi entropy yields a Renyi distribution that passes into the Gibbs canonical distribution at q = 1. The thermodynamic entropy of the complex system is defined as the Renyi entropy for the Renyi distribution. In contrast to the usual entropy based on the Gibbs-Shannon entropy, the Renyi entropy increases as the distribution deviates from the Gibbs distribution (the deviation is estimated by the parameter η = 1 − q) and reaches its maximum at the maximum possible value ηmax. As this occurs, the Renyi distribution becomes a power-law distribution. The parameter η can be regarded as an order parameter. At η = 0, the derivative of the thermodynamic entropy with respect to η exhibits a jump, which indicates a kind of phase transition into a more ordered state. The evolution of the system toward further order in this phase state is accompanied by an entropy gain. This means that in accordance with the second law of thermodynamics, a natural evolution in the direction of self-organization is preferable. Springer Science+Business Media, Inc., 2006 Renyi entropy nationallicence complex system nationallicence self-organization nationallicence phase transition nationallicence power-law Zipf-Pareto distribution nationallicence second law of thermodynamics nationallicence direction of evolution nationallicence Theoretical and Mathematical Physics Kluwer Academic Publishers-Consultants Bureau 149/2(2006-11-01), 1559-1573 0040-5779 149:2<1559 2006 149 11232 https://doi.org/10.1007/s11232-006-0138-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11232-006-0138-x text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Bashkirov A. aut NATIONALLICENCE 773 0- Theoretical and Mathematical Physics Kluwer Academic Publishers-Consultants Bureau 149/2(2006-11-01), 1559-1573 0040-5779 149:2<1559 2006 149 11232 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer