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Graphic representation of the evolutionary processes of the compositions of multicomponent objects of any nature


A method for the graphic representation of evolutionary processes of compositions is described. Preparation of materials includes descending ranking of contents of components and standardization of the length of the obtained sequences while discarding excess contents. The following three parameters are calculated for the persistent distribution of contents: (1) information entropy H = −Σp ilogp i as a measure of complexity of the system’s composition, (2) anentropy A = −Σlogp i as a measure of the composition purity, and (3) tolerance T = log[(Σ1/p i)/n] as a measure of special purity. In order to represent the process of compositional change, paired diagrams with the axes En, An, and T are used. The obtained entropy diagrams describe separation and mixing processes that occur in nature, technology, and society in the most adequate manner.

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  1. Petrov, T.G., Reason of General Classification Variant of Geochemical Systems, Vestnik Leningr. Gos. Univ., Ser. 7. Geolog. Geogr. 1971, vol. 18, no. 3, pp. 30–38.

    Google Scholar 

  2. Petrov, T.G., Information Language for Description of Multicomponent Object Compositions, Nauchn. Tekhn. Inform., Ser. 2, 2001, no. 3, pp. 8–18.

  3. Hartley, R.V.L., Transmission of Information, Bell System Techn. J., 1928, vol. 7, no. 3, pp. 535–563.

    Google Scholar 

  4. Petrov, T.G., Range-Entropy Approach to the Description of the Geological Object Compositions and Their Alterations (on the Geological Coenology), Obshch. Prikladn. Tsoenolog., 2007, no. 5, pp. 27–33.

  5. Spenser, H., Principles of Biology in two volumes. Vol. I. Part I: The Data of Biology; Part II: The Inductions of Biology; Part III: The Evolution of Life; Appendices. 1864; Vol. II. Part IV: Morphological Development; Part V: Physiological Development; Part VI: Laws of Multiplication; Appendices. 1867.

  6. Collins, F.H., Epitome of the Synthetic Philosophy of Herbert Spencer, Williams and Norgate, 1901

  7. Petrov, T.G., RHA Method as Systematization Problem Solution for Analytical Data about Substantial Composition of Geological Objects, Otech. Geologiya, 2008, no. 4, pp. 98–105.

  8. Petrov, T.G., Hierarchical Periodic System of Chemical Compositions and Its Connection with Periodical System of Elements, Vestn. St. Petersb. Gos. Univ., Ser. 7, 2009, no. 2, pp. 21–28.

  9. Velikoslavinskii, D.A., Eliseev, E.N., and Kratts, K.O., Variatsionnyi analiz magmaticheskikh sistem (Variation Analysis of Magmatic Systems), Leningrad: Nauka, 1984.

    Google Scholar 

  10. Petrov, T.G. and Farafonova, O.I., Informatsionnokomponentnyi analiz. Metod RHA (Information-Component Analysis. RHA Method), St. Petersburg: St. Petersb. Gos. Univ., 2005.

    Google Scholar 

  11. Petrov, T.G., Farafonova, O.I., and Sokolov, P.B., Information-Entropy Characteristics in Composition of Minerals and Rocks as a Reflection of Their Crystallization Process, Proc. All-Russ. Mineral. Soc., 2003, Part 132, no. 2, pp. 33–40.

  12. Devyatykh, G.G., Karpov, Yu.A., Kovalev, I.D., et al., Additive Composition of Samples from the Exhibition-Collection of Substances of Special Purity. III. Simple Solid Substances from 3th and 4th Periods of Periodical Table of Elements, Vysokochistye Veshchestva, 1991, no. 2, pp. 22–32.

  13. Petrov, T.G., On the Failing of Ability to Determine the Succession of Crystallization by Individual Characteristics of Mineral, Proc. All-Union Mineral. Soc., 1977, Part 106, no. 4, pp. 499–502.

  14. Petrov, T.G., Theory of Information and Problems of Crystallogenesis, in sb. statei “protsessy rosta kristallov i plenok poluprovodnikov” (Coll. Papers’ Processes of Growth of Crystals and Films of Semiconductors’), Aleksandrov, L.N. and Borisova, L.A., Eds., Novosibirsk: 1970.

  15. Petrov, T.G., About Complexity of Geochemical Systems from the Position of Theory of Information, Dokl. Akad. Nauk SSSR, 1970, vol. 191, no. 4, pp. 1094–1096.

    Google Scholar 

  16. Petrov, T.G. and Krasnova, N.I., R-slovar’-katalog khimicheskikh sostavov mineralov (R-Dictionary-Catalog of Chemical Compositions of Minerals), St. Petersburg: Nauka, 2010.

    Google Scholar 

  17. Shannon, C.E., A Mathematical Theory of Communication, Bell System Techn. J., 1948, vol. 27, pp. 623–656.

    MathSciNet  Google Scholar 

  18. Moelwyn-Hudges, E.A., Physical Chemistry, London: Pergamon Press, 1961

    Google Scholar 

  19. Fano, R., Transmission of Information; A Statistical Theory of Communications, Cambridge: MIT Press, 1961.

    Google Scholar 

  20. Yaglom, A.M. and Yaglom, I.M., Probability and Information, Berlin: Springer Verlag, 1983.

    MATH  Google Scholar 

  21. Jaynes, E.T., Information Theory and Statistical Mechanics, Phys. Rev., 1957, vol. 106, no. 4, pp. 620–630; vol. 108, no. 2, pp. 171–190.

    MathSciNet  Article  Google Scholar 

  22. Traibus, M., Thermostatics and Thermodynamics, London, 1967.

  23. Pelto, C.R., Mapping of Multicomponent Systems, J. Geol., 1954, vol. 62, no. 5, pp. 501–511.

    Article  Google Scholar 

  24. Vistelius, A.B., Problems of Geochemistry and Information Measures, Sov. Geologiya, 1964, no. 12, pp. 5–26.

  25. Buryakovskii, L.A., Entropy as Measure of Heterogeneity of Mountain Rocks, Sov. Geologiya, 1968, no. 3, pp. 135–138.

  26. Logvinenko, N.V., Petrografiya osadochnykh porod (Petrography of Sedimentary Rocks), Moscow: Vysshaya Shkola, 1984.

    Google Scholar 

  27. Bekman, I.N., Informatika. Kurs lektsii (Informatics. Course of Lectures), Moscow: Mos. Gos. Univ., 2009.

    Google Scholar 

  28. Gordienko, V.V. and Petrov, T.G., Analysis of Processes of Formation of Rare Metal Granit Pegmatites with the Help of RHA Information Language, Proc. All-Union Mineral. Soc., 1981, Part 110, no. 5, pp. 546–558.

  29. Petrov, T.G., Problems of Separation and Mixing in Inorganic Systems, Sb. statei “Geologiya”, (Coll. Papers “Geologiya”), Trofimov, V.T., Ed., 1995, vol. 2, pp. 181–186.

  30. Gresens, R.L., Composition-Volume Relationships of Metasomatism, Chem. Geol., 1967, vol. 2, no. 1, pp. 47–65.

    Article  Google Scholar 

  31. Petrov, T.G., The Way of Equal Brevities for Revealing Passive Components and Character of Behavior of Active Components in Geochemical Processes, Proc. All-Union Mineral. Soc., 1983, Part 112, no. 6, pp. 641–650.

  32. Shurubor, Yu.V., About One Property of Geochemical System Complexity Measure, Dokl. Akad. Nauk SSSR, 1972, vol. 205, no. 2, pp. 453–456.

    Google Scholar 

  33. Le Maitre, R.W., The Chemical Variability of Some Common Igneous Rocks, J. Petrolog. 1976, vol. 17, pp. 589–637.

    Google Scholar 

  34. Spiridonov, E.M., Baksheev, I.A., Seredkin, M.V., et al., Gumbeities of Ural and Conjugated Rock Mineralization, Geolog. Rudn. Mestorozhd., 1998, vol. 40, no. 2, pp. 174–190.

    Google Scholar 

  35. Sobolev, A.P., Mezozoiskie granitoidy Severo-Vostoka SSSR i problemy ikh rudonosnosti (Mesozoic Granitoids of North-East of USSR and Problems of Their Ore-Bearingness), Moscow: Nauka, 1989.

    Google Scholar 

  36. Demograficheskii entsiklopedicheskii slovar’ (Demographic Encyclopaedic Dictionary), Moscow: Sov. Entsiklopediya, 1985.

  37. Shvanov, V.N., Petrografiya peschanykh porod (Petrography of Sandy Rocks), Leningrad: Nedra, 1987.

    Google Scholar 

  38. Rossiya v tsifrakh. 2008 (Russia in Numbers-2008), Moscow, 2008.

  39. Gini, C., I Fattori Demografici Dell’Evolucione Delle Nazioni, Torino, 1912.

  40. Petrov, T.G., State of Information about Substance Composition of Geological Objects, Sov. Geologiya, 1976, no. 7, pp. 108–118.

  41. Mazur, M., Kachestvennaya teoriya informatsii (A Quality Theory of Information), Warsaw: Widavnictvo Naukovo-Techniczne, 1970.

    Google Scholar 

  42. Efremova, S.V. and Stafeev, K.G., Petrokhimicheskie metody issledovaniya gornykh porod (Petrochemical Methods of Study of Mountain Rocks), Moscow: Nedra, 1985.

    Google Scholar 

  43. Moshkin, S.V., Shelemotov, A.S., Petrov, T.G., and Krasnova, N.I., “RETROS-2”—Program Complex for Treatment of Petrochemical Data, in Geokhimiya magmaticheskikh porod, (Geochemistry of Magmatic Rocks), Apatity: Karel. Nauchn. Tsentr Ross. Akad. Nauk, 2003.

    Google Scholar 

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Correspondence to T. G. Petrov.

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Original Russian Text © T.G. Petrov, 2012, published in Nauchno-Tekhnicheskaya Informatsiya, Seriya 2, 2012, No. 3, pp. 21–31.

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Petrov, T.G. Graphic representation of the evolutionary processes of the compositions of multicomponent objects of any nature. Autom. Doc. Math. Linguist. 46, 79–93 (2012).

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  • evolution of compositions
  • information entropy
  • complexity of composition
  • entropy of mixing
  • anentropy
  • entropy of separation
  • purity of composition
  • entropy diagram
  • tolerance
  • Shurubor theorem
  • rock
  • ethnic composition
  • Gini coefficient