Geodesic
The Relationship between the Fermi--Dirac Distribution and Statistical Distributions in Languages
Matematičeskie zametki, Tome 101 (2017) no. 4, pp. 531-548.

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In this article, we study, from the mathematical point of view, the analogies between language and multi-particle systems in thermodynamics. We attempt to introduce an appropriate mathematical apparatus and the technical tools of statistical physics to descriptions of language. In particular, we apply the notions of number of degrees of freedom, Bose condensate, phase transition and others to linguistics objects. On the basis of a statistical analysis of dictionaries and statistical distributions in languages, we conjecture that the transition from the semiotic communication system of the higher primates to human language can be described as a phase transition of the first kind. We show that the number of words appearing with frequency 1 in a corpus of texts is equal to the number of ones in the corresponding Fermi–Dirac distribution, while the high frequency of stop-words corresponds to the large number of particles in the Bose condensate, when the number of degrees of freedom is less than two, provided there is a gap in the spectrum. The presented considerations are illustrated by examples from the Russian language. Some of the illustrative examples are untranslatable into English, and so they were replaced in translation by similar examples from the English language.
Keywords: number of degrees of freedom, frequency of occurrence, frequency dictionary, Zipf law, statistical language distribution, Bose–Einstein distribution, number theory, Van-der-Waals model, isotherm, tropical topology, tropical analysis, telegraphic style, stop-word.
Mots-clés : Fermi–Dirac distribution 
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V. P. Maslov. The Relationship between the Fermi--Dirac Distribution and Statistical Distributions in Languages. Matematičeskie zametki, Tome 101 (2017) no. 4, pp. 531-548. http://geodesic.mathdoc.fr/item/MZM_2017_101_4_a3/

[1] V. P. Maslov, “The Fermi–Dirac distribution as a model of a thermodynamically ideal liquid. Phase transition of the first kind for neutral gases (corresponding to nonpolarMolecules)”, Math. Notes, 101:1 (2017), 100–114 | DOI

[2] W.-S. Dai, M. Xie, “Gentile statistics with a large maximum occupation number”, Ann. Physics, 309:2 (2004), 295–305 | DOI | MR | Zbl

[3] A. S. Monin, A. M. Yaglom, Statisticheskaya gidromekhanika, Ch. 1, Nauka, M., 1965; Ч. 2, 1967

[4] V. P. Maslov, T. V. Maslova, “O zakone Tsipfa i rangovykh raspredeleniyakh v lingvistike i semiotike”, Matem. zametki, 80:5 (2006), 718–732 | DOI | MR | Zbl

[5] T. V. Maslova, “Ob utochnenii zakona Tsipfa dlya chastotnykh slovarei”, Nauchno-tekhnicheskaya informatsiya. Ser. 2, 11 (2006), 37–44

[6] M. V. Arapov, E. N. Efimova, Yu. A. Shreider, “O smysle rangovykh raspredelenii”, Nauchno-tekhnicheskaya informatsiya. Ser. 2, 1 (1975), 9–20 | Zbl

[7] O. Lyashevskaya, S. Sharov, Chastotnyi slovar sovremennogo russkogo yazyka (na materialakh Natsionalnogo korpusa russkogo yazyka), Azbukovnik, M., 2009

[8] P. M. Alekseev, Chastotnye slovari, Izd-vo SPbGU, SPb., 2001

[9] Lingvisticheskie raspredeleniya (elementy kolichestvennogo analiza teksta), LGPI im. A. I. Gertsena, L., 1985

[10] Yu. Tupdava, Problemy i metody kvantitativno-sistemnogo issledovaniya leksiki, Valgus, Tallin, 1987

[11] S. Burlak, Proiskhozhdenie yazyka. Fakty, issledovaniya, gipotezy, Astrel, M., 2011

[12] V. P. Maslov, “New approach to classical thermodynamics”, Math. Notes, 100:1 (2016), 154–185 | DOI | MR

[13] V. P. Maslov, “Large negative numbers in number theory, thermodynamics, information theory, and human thermodynamics”, Russ. J. Math. Phys., 23:4 (2016), 510–528 | DOI | MR

[14] V. P. Maslov, “Tropical topology”, Math. Notes, 100:6 (2016), 886–889 | DOI

[15] A. M. Vershik, “Statisticheskaya mekhanika kombinatornykh razbienii i ikh predelnye konfiguratsii”, Funkts. analiz i ego pril., 30:2 (1996), 19–39 | DOI | MR | Zbl

[16] A. M. Vershik, “Predelnoe raspredelenie energii kvantovogo idealnogo gaza s tochki zreniya teorii razbienii naturalnykh chisel”, UMN, 52:2 (314) (1997), 139–146 | DOI | MR | Zbl

[17] A. M. Vershik, G. A. Freiman, Yu. V. Yakubovich, “Lokalnaya predelnaya teorema dlya sluchainykh razbienii naturalnykh chisel”, Teoriya veroyatn. i ee primen., 44:3 (1999), 506–525 | DOI | MR | Zbl

[18] P. Erdős, “On some asymptotic formulas in the theory of partitions”, Bull. Amer. Math. Soc., 52 (1946), 185–188 | DOI | MR | Zbl

[19] O. S. Issers, Kommunikativnye strategii i taktiki russkoi rechi, URSS, M., 2006

[20] V. P. Maslov, Kvantovaya ekonomika, Nauka, M., 2006

[21] L. D. Landau, E. M. Lifshits, Kvantovaya mekhanika, Nauka, M., 1976

[22] Kh. Logen, Metod trigonometricheskikh summ i ego primenenie v teorii chisel, Mir, M., 1964 | Zbl

[23] R. Feinman, A. Khibs, Kvantovaya mekhanika i integraly po traektoriyam, Mir, M., 1968

[24] V. P. Maslov, “Dvukhflyuidnaya kartina nadkriticheskikh yavlenii”, TMF, 180:3 (2014), 394–432 | DOI | Zbl

[25] V. P. Maslov, “Distribution corresponding to classical thermodynamics”, Phys. Wave Phenom., 23:2 (2015), 81–95 | DOI

[26] V. P. Maslov, “Novye raspredeleniya v termodinamike”, Pisma v ZhTF, 42:18 (2016), 46–54