DOI: https://doi.org/10.15587/1729-4061.2018.134062

### Exponential and hyperbolic types of distribution in macrosystems: their combined symmetry and finite properties

Nikolaj Delas

#### Abstract

This paper proposes the extended entropy method that identifies certain new relations in the organization of macro systems and which sheds light on several existing theoretical issues. It is shown that the type of distribution within the macrosystem is determined by the ratio of the kinetic properties of its agents ‒ "carriers" and "resources". If the relaxation time is shorter for the "carriers", there forms the exponential type of distribution; if it is shorter for the "resources" ‒ there forms the extreme hyperbolic distribution with a heavy tail. Analytical expressions were derived for them and their spectra. A convenient technique to parametrically record them via modal characteristics was devised.

Author discovered the existence of the combined symmetry of these two types of distributions. They can be regarded as alternative statistical interpretations of a single state of the macrosystem.

Distributions of real macro systems possess finite properties; given the natural constraints, they form the right bounds. The proposed method makes it possible to determine their coordinates based on the extreme principle, considering the right bounds of finite distributions as a product of self-organization of macro systems. Strict ratios were constructed, taking into consideration the finite features of distributions. Parametrically, they depend on the specific volume of "resources", and the magnitude of a form-parameter ‒ ratio between modal and boundary coordinates.

The value of the obtained results is in that they shed light on a number of problematic issues in the statistical theory of macro systems, as well as include a set of convenient tools in order to analyze two types of distributions with finite properties.

#### Keywords

macro system; entropy; entropy modeling; finite distributions; hyperbolic distributions; distributions with a heavy tail

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#### References

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Golicin, G. S., Mohov, I. I., Akperov, M. G., Bardin, M. Yu. (2007). Funkcii raspredeleniya veroyatnostey dlya ciklonov i anticiklonov. Doklady RAN, 413 (2), 254–256.

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Delas, N. I. (2015). “Correct entropy”in the analysis of complex systems: what is the consequence of rejecting the postulate of equal a priori probabilities? Eastern-European Journal of Enterprise Technologies, 4 (4 (74)), 4–14. doi: 10.15587/1729-4061.2015.47332

Feigenbaum, M. J. (1979). The universal metric properties of nonlinear transformations. Journal of Statistical Physics, 21 (6), 669–706. doi: 10.1007/bf01107909

Delas, N. I. (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

#### GOST Style Citations

Vil'son A. Dzh. Entropiynye metody modelirovaniya slozhnyh sistem. Moscow: Nauka, 1978. 248 p.

Kas'yanov V. A. Sub'ektivniy analiz. Kyiv: NAU, 2007. 512 p.

Chernavskiy D. S., Nikitin A. P., Chernavskaya O. D. O mekhanizmah vozniknoveniya raspredeleniya Pareto v slozhnyh sistemah. Moscow: FIAN, 2007. 17 p.

Yablonskiy A. I. Matematicheskie modeli v issledovanii nauki. Moscow: Nauka, 1986. 395 p.

Feller V. Vvedenie v teoriyu veroyatnostey i ee prilozheniya. Vol. 2. Moscow: Mir, 1967. 765 p.

Trubnikov B. A., Trubnikova O. B. Pyat' velikih raspredeleniy veroyatnostey // Priroda. 2004. Issue 11. P. 13–20.

Buhovec A. G. Sistemnyy podhod i rangovye raspredeleniya v zadachah klassifikacii // Vestnik VGU. Seriya: Ekonomika i upravlenie. 2005. Issue 1. P. 130–142.

Sibatov R. T., Uchaikin V. V. Fractional differential approach to dispersive transport in semiconductors // Uspekhi Fizicheskih Nauk. 2009. Vol. 179, Issue 10. P. 1079–1104. doi: 10.3367/ufnr.0179.200910c.1079

Salat H., Murcio R., Arcaute E. Multifractal methodology // Physica A: Statistical Mechanics and its Applications. 2017. Vol. 473. P. 467–487. doi: 10.1016/j.physa.2017.01.041

Newman M. Power laws, Pareto distributions and Zipf's law // Contemporary Physics. 2005. Vol. 46, Issue 5. P. 323–351. doi: 10.1080/00107510500052444

Takagi K. An Analytical Model of the Power Law Distributions in the Complex Network // World Journal of Mechanics. 2012. Vol. 02, Issue 04. P. 224–227. doi: 10.4236/wjm.2012.24027

Gualandi S., Toscani G. Pareto tails in socioeconomic phenomena: a kinetic description // Economics Discussion Papers. 2017. URL: http://www.economics-ejournal.org/economics/discussionpapers/2017-111/file

Lisicin D. V., Gavrilov K. V. Ocenivanie parametrov finitnoy modeli, ustoychivoe k narusheniyu finitnosti // Sib. zhurn. industr. matem. 2013. Vol. 16, Issue 2. P. 109–121.

Delas N. I., Kas'yanov V. A. Extremely hyperbolic law of self-organized distribution systems // Eastern-European Journal of Enterprise Technologies. 2012. Vol. 4, Issue 4 (58). P. 13–18. URL: http://journals.uran.ua/eejet/article/view/4901/4543

Frenkel' Ya. I. Statisticheskaya fizika. Moscow: Izd-vo akademii nauk SSSR, 1948. 760 p.

Relaksaciya. Fizicheskaya enciklopediya. Vol. 3 / Zubarev D. N., Alekseev D. M., Baldin A. M. et. al.; A. M. Prohorov (Ed.). Moscow: Sovetskaya enciklopediya, 1992. 672 p.

Krylov N. S. Raboty po obosnovaniyu statisticheskoy fiziki. Moscow: «Editorial URSS», 2003. 207 p.

Endryus G. Teoriya razbieniy. Moscow: Nauka. Glavnaya redakciya fiziko-matematicheskoy literatury, 1982. 256 p.

Olver F. Vvedenie v asimptoticheskie metody i special'nye funkcii / A. P. Prudnikov (Ed.). Moscow: Nauka, 1978. 375 p.

Botvina L. R., Barenblatt G. I. Avtomodel'nost' nakopleniya povrezhdaemosti // Problemy prochnosti. 1985. Issue 12. P. 17–24.

Ivlev L. S., Dovgalyuk Yu. A. Fizika atmosfernyh aerozol'nyh sistem. Sankt-Peterburg: NIIH SPbGU, 1999. 194 p.

Fizika vzryva / Andreev S. G., Babkin A. V., Baum F. A. et. al.; L. P. Orlenko (Ed.). 3e izd. Moscow: Fizmatlit, 2004. 656 p.

Funkcii raspredeleniya veroyatnostey dlya ciklonov i anticiklonov / Golicin G. S., Mohov I. I., Akperov M. G., Bardin M. Yu. // Doklady RAN. 2007. Vol. 413, Issue 2. P. 254–256.

Yakovenko V. Statistical mechanics approach to the probability distribution of money. URL: https://arxiv.org/pdf/1007.5074.pdf

Delas N. I. “Correct entropy”in the analysis of complex systems: what is the consequence of rejecting the postulate of equal a priori probabilities? // Eastern-European Journal of Enterprise Technologies. 2015. Vol. 4, Issue 4 (74). P. 4–14. doi: 10.15587/1729-4061.2015.47332

Feigenbaum M. J. The universal metric properties of nonlinear transformations // Journal of Statistical Physics. 1979. Vol. 21, Issue 6. P. 669–706. doi: 10.1007/bf01107909

Delas N. I. Evolution of complex systems with hyperbolic distribution // Eastern-European Journal of Enterprise Technologies. 2013. Vol. 3, Issue 4 (63). P. 67–73. URL: http://journals.uran.ua/eejet/article/view/14769/12571