“Correct entropy”in the analysis of complex systems: what is the consequence of rejecting the postulate of equal a priori probabilities?
DOI:
https://doi.org/10.15587/1729-4061.2015.47332Keywords:
entropic divergence, evolution of complex systems, power-law distribution, entropy production, synergeticsAbstract
Entropy can clearly reflect the probability of the macrostate of a system only in case of validity of the basic postulate of statistical mechanics—the postulate of equal a priori probabilities of microstates. We have proved that for most non-physical macrosystems it loses its power, and the role of entropy has to be performedby a more general property, which has been found out in this study and called entropic divergence. In accordance with the principle of continuity, it includes the Boltzmann entropy.
The properties of entropic divergence were considered through proving a number of theorems. This property has generally been found to have a minimal effect on the equilibrium state of the system.
The conditional minimum of entropic divergence was disclosed as an example of formalism through which we have derived exponential and marginal hyperbolic distributions that take into account unequal a priori probabilities. The multiplicative form of the combined distribution allows consideration of the process of interaction between two or more macrosystems as the realization of a complex macroexperience.
We have disclosed the possibility of using the research findings for analysing adaptive statistical interaction of aggregate macrosystems. The termsused in the studyfacilitate the development of quantitative methods of such analysis. We have given an example of the possibility of using this approach to compute the exponent in the power-law (hyperbolic) distributions.References
- Jaynes, E. T. (1957). Information Theory and Statistical Mechanics. Physical Review, 106 (4), 620–630. doi: 10.1103/physrev.106.620
- Plank, M. (1935). Vvedenie v teoreticheskuiu fiziku. Teoriia. Part 5. Moscow: Nauchno-tehnicheskoe izd-vo NKTP SSSR, 229.
- Frenkel', Ya. I. (1948). Statisticheskaia fizika. Moscow: Izd-vo akademii nauk SSSR, 760.
- Zommerfel'd, A. (1955). Termodinamika i statisticheskaia fizika. Moscow: Izd-vo inostr. lit., 481.
- Kittel', Ch. (1977). Statisticheskaia termodinamika. Moscow: Prosveshchenie, 336.
- Yakovenko, V. (2010). Statistical mechanics approach to the probability distribution of money. Department of Physics, University of Maryland, 11.
- Galkin, S. A., Elagin, O. I., Kozlov, A. A., Potapenko, V. A., Romanovskii, M. Yu. (2009). Eksponentsial'nye raspredeleniia individual'nyh dohodov i rashodov grazhdan: nabliudeniia i modeli. Trudy instituta obshchei fiziki im. A.M. Prohorova, RAN, 65, 29–49.
- Vil'son, A. J. (1978). Entropiinye metody modelirovaniia slozhnyh sistem. Moscow: Nauka, 248.
- Levich, A. P., Alekseev, V. L., Nikulin, V. A. (1994). Matematicheskie aspekty variatsionnogo modelirovaniia v ekologii soobshchestv. Matematicheskoe modelirovanie, 6 (5), 55–76.
- Prangishvili, I. V. (2003). Entropiinye i drugie sistemnye zakonomernosti: Voprosy upravleniia slozhnymi sistemami. Moscow: Nauka, 428.
- In: Prohorov, A. M. (1995). Fizicheskaia entsiklopediia. Vol. 4. Moscow: Sovetskaia entsiklopediia, 640.
- Delas, N., Kasianov, V. (2012). Extremely hyperbolic law of self-organized distribution systems. Eastern-European Journal of Enterprise Technologies, 4/4(58), 13–18. Available at: http://journals.uran.ua/eejet/article/view/4901/4543
- Delas, N. (2013). Evolution of complex systems with hyperbolic distribution. Eastern-European Journal of Enterprise Technologies, 3/4(63), 67–73. Available at: http://journals.uran.ua/eejet/article/view/14769/12571
- Kullback, S., Leibler, R. A. (1951). On Information and Sufficiency. The Annals of Mathematical Statistics, 22 (1), 79–86. doi: 10.1214/aoms/1177729694
- Trubnikov, B. A., Trubnikova, O. B. (2004). Piat' velikih raspredelenii veroiatnostei. Priroda, 11, 13–20.
- Gorban', I. I. (2014). Fenomen statisticheskoi ustoichivosti. Kiev: Naukova dumka, 444.
- Mandel'brot, B. (1973). Teoriia informatsii i psiholingvisticheskaia teoriia chastot slov. Moscow: Progress, 316–337.
- Botvina, L. R., Barenblatt, G. I. (1985). Avtomodel'nost' nakopleniia povrezhdaemosti. Problemy prochnosti, 12, 17–24.
- Golitsin, G. S., Mohov, I. I., Akperov, M. G., Bardin, M. Yu. (2007). Funktsii raspredeleniia veroiatnostei dlia tsiklonov i antitsiklonov. Doklady RAN, 413 (2), 254–256.
- Andersen, K. (2008). Dlinnyi hvost. Novaia model' vedeniia biznesa. Translated from English. Moscow: Vershina, 272.
- Haibullov, R. A. (2009). Rangovyi analiz kosmicheskih sistem. Izvestiia glavnoi astronomicheskoi observatorii v Pulkove, 3 (219), 95–104.
- Orlov, Yu. K. (1980). Nevidimaia garmoniia. Chislo i misl', 3, 70–105.
- Gelashvili, D. B., Yudin, D. I., Rozenberg, G. S., Yakimov, V. N. (2007). Stepennoi harakter nakopleniia vidovogo bogatstva kak proiavlenie fraktal'noi struktury biotsenoza. Zhurnal obshchei biologii, 68 (2), 115–124.
- Delas, N., Kasianov, V. (2012). Nongauss distribution as a property of complex systemes that are organized by type of cenoses. Eastern-European Journal of Enterprise Technologies, 3/4(57), 27–32. Available at: http://journals.uran.ua/eejet/article/view/4010/3677
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2015 Николай Иванович Делас
This work is licensed under a Creative Commons Attribution 4.0 International License.
The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.
A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with TECHNOLOGY CENTER PC, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.