Address: Naberezhnaya Severnoy Dviny, 17, Arkhangelsk, 163002, Russian Federation, Northern (Arctic) Federal University named after M.V.Lomonosov, office 1425

Phone: +7 (8182) 21-61-18

Lesnoy Zhurnal

Non-Ideality Factor in Multifractal and Entropy-Based Analysis of Self-Organized Structures of Plant Polymers (Lignins)

Версия для печати

N.A. Makarevich

Complete text of the article:

Download article (pdf, 0.7MB )




An attempt has been made to introduce the generalized non-ideality factor of systems g (GNF) into information entropy equations that describe self-organized structures of essentially nonequilibrium systems with the use of studying the topological properties of high molecular weight compounds in solutions using wood lignins as an example. The factor g as a relative thermodynamic characteristic connects the ideal and real models of systems in which two competitive (opposite in sign and action) processes can be distinguished: order (−) ↔ chaos (+); attraction (−) ↔ repulsion (+); compression (−) ↔ extension (+); clustering (−) ↔ decay (+), etc. g = 1 + 〈– βord + αnord〉 = 1 + 〈– pi (β) + pi (α)〉, где – βord ≡ 1/nΣinβi и αnopd ≡ 1/nΣinαi are relative average characteristics (pi – probabilities) of oppositely occurring processes. The factor g varies in the interval 0≤ g ≥1 and depends on which of the competitive processes prevails. For αnord = 0 g → 0, for βord = 0 g→2, for g = 1 the behavior of the elements of the system will be ideal. The factor g is introduced into any classical equations suitable for studying ideal systems with the aim of using them to describe real systems (for example, the equations of Henry, Raoult, Van’t Hoff, general gas, etc.). Strictly mathematically, the factor g is defined through the values M – measure, ε – size (scale), and d – dimension as a ratio of logarithms of measures of real (М*) and ideal (М0) states of the object: gth = lnМ*/lnМ0 = d/D, where M* and M0 can be the number of elements in the structure of the fractal real (for example, cluster) or mathematical object (for example, Sierpiński triangle) Nd and the number of elements in the structure of the object in the perfect condition, having the property of multi-scale and self-similarity, ND, where d and D are the fractal and Euclidean dimensions. As a thermodynamic characteristic gth is defined by the ratio of thermodynamic functions, functionals, for example, ΔGi*/ΔGi, where ΔGi* = –RTlnаi is real and, ΔGi = –RTlnNi is ideal state; the number of moles of n* − real state of matter to n − ideal state of matter; relative entropies of the system ΔSreal/ΔSidSid − Boltzmann entropy). New expressions of the information and thermodynamic entropies with a fractional (0:1) moment of order and with the entropic gS and gth non-ideality factors are obtained for the analysis of self-organized quasi-equilibrium structures in the Renyi formalism SgSMRn(p) = R/(1 – gS)lnΣNipgSi; SgthM–Rn = R/(gth)ln(ΣNi=1pigth – 1); in the Tsallis formalism SgSM–TS (p) = R(1 – ΣiN(ε)pigS)/(gS – 1); SgthM–TS (p) = R(1 – ΣiN(ε)pi1–gth)/gth with an application for studying the topological properties of high-molecular compounds by hydrodynamic methods, as well as the thermodynamics of polymer solutions.


Nikolay A. Makarevich1,2, Doctor of Chemistry, Prof.;
ResearcherID: ABF-6367-2020, ORCID:


1Military Academy of the Republic of Belarus, prosp. Nezavisimosti, 220, Minsk, 220000, Republic of Belarus; e-mail:
2Northern (Arctic) Federal University named after M.V. Lomonosov, Naberezhnaya Severnoy Dviny, 17, Arkhangelsk, 163002, Russian Federation


non-ideality factor of systems, fractal structures, fractal dimensionality, information and thermodynamic entropy, polymer solutions, thermodynamics of polymer solutions, lignins

For citation

Makarevich N.A. Non-Ideality Factor in Multifractal and Entropy-Based Analysis of Self-Organized Structures of Plant Polymers (Lignins). Lesnoy Zhurnal [Russian Forestry Journal], 2021, no. 2, pp. 194–212. DOI: 10.37482/0536-1036-2021-2-194-212


  1. Afanas’yev N.I. Structure of Macromolecules in Solutions and at the Phase Boundaries, the Surface-Active Properties of Lignosulfonates: Dr. Chem. Sci. Diss. Saint Petersburg, LTA Publ., 1996. 302 p.
  2. Afanas’yev N.I., Korobova E.N., Ferofontova S.D. Intermolecular Interactions in Solutions of Lignosulfonates. Lesnoy Zhurnal [Russian Forestry Journal], 1996, no. 1-2, pp. 110–117. URL:
  3. Bashkirov A.G. Renyi Entropy as a Statistical Entropy for Complex Systems. Teoreticheskaya i Matematicheskaya Fizika [Theoretical and Mathematical Physics], 2006, vol. 149, no. 2, pp. 299–317. DOI: 10.4213/tmf4235
  4. Bogolitsyn K.G., Lunin V.V., Kosyakov D.S., Karmanov A.P., Skrebets T.E. et al. Physical Chemistry of Lignin. Moscow, Akademkniga Publ., 2010. 492 p.
  5. Brillouin L. Scientific Uncertainty and Information. Translated from English. Moscow, Mir Publ., 1966. 272 p. DOI: 10.1016/C2013-0-12512-3
  6. Budtov V.P. Physical Chemistry of Polymer Solutions. Saint Petersburg, Khimiya Publ., 1992. 384 p.
  7. Zaripov R.G. Discrimination Information and Disorder − Order Transitions. Kazan, KSTU Publ., 1999. 155 p.
  8. Zubarev D.N., Morozov V.G., Repke G. Statistical Mechanics of Nonequilibrium Processes. Moscow, Fizmatlit Publ., 2002. 432 p.
  9. Karmanov A.P. Self-Organization and Structural Organization of Lignin. Yekaterinburg, UB RAS Publ., 2004. 270 p.
  10. Karmanov A.P., Matveyev D.V., Monakov Yu.B. The Dynamic of the Polymerization of Monomeric Precursors of Guaicyl Lignins. Doklady Akademii Nauk, 2001, vol. 380, no. 5, pp. 635–638.
  11. Karmanov A.P., Kuz’min D.V., Shamshina I.N., Belyaev V.Yu., Kocheva L.S., Matveev D.V., Monakov Yu.B. Hydrodynamic and Conformational Properties of Lignins from Sorbus aucuparia and Robinia pseudoacacia Woody Plants. Vysokomolekulyarnyye soyedineniya. Seriya A: Fizika polimerov [Polymer Science. Series A - Polymer Physics], 2004, vol. 46, no. 6, pp. 997–1004.
  12. Klimontovich Yu.L. Statistical Theory of Open Systems. Vol. 1. Moscow, Yanus Publ., 1995. 624 p.
  13. Kolesnichenko A.V., Marov M.Ya. Turbulence and Self-Organization. Problems of Modeling Space and Natural Environments. Moscow, BINOM. Laboratoriya znaniy Publ., 2014. 632 p.
  14. Makarevich N.A. Non-Ideality Factor in the Classical Equations for Real Gases and Condensed Systems: Universal Associative-Ionized Multiplier in the Classical Equations for Solutions of Nonelectrolytes and Electrolytes. Doklady Nacional’noj akademii nauk Belarusi [The Doklady of the National Academy of Sciences of Belarus], 2016, vol. 60, no. 1, pp. 94–101.
  15. Makarevich N.A. Interface “Gas−Liquid− Solid”: Monograph. Arkhangelsk, NArFU Publ., 2018. 411 p.
  16. Polak L.S., Mikhaylov A.S. Self-Organization in Nonequilibrium Physical and Chemical Processes. Moscow, Nauka Publ., 1975. 351 p.
  17. Prigogine I., Kondepudi D. Modern Thermodynamics. From Heat Engines to Dissipative Structures. Translated from English. Moscow. Mir Publ., 2002. 461 p.
  18. Feder J. Fractals. Translated from English. Moscow, Mir Publ., 1991. 254 p. DOI: 10.1007/978-1-4899-2124-6
  19. svetkov V.N., Lavrenko P.N., Bushin S.V. A Hydrodynamic Invariant of Polymer Molecules. Uspekhi khimii [Russian Chemical Reviews], 1982, vol. 51, no. 10, pp. 1698–1732. DOI: 10.1070/RC1982v051n10ABEH002935
  20. Tsvetkov V.N., Eskin V.E., Frenkel’ S.Ya. Structure of Macromolecules in Solutions. Moscow, Nauka Publ., 1964. 720 p.
  21. BBI JU Strengthens Collaboration on Sustainable Bioeconomy with the European Forest Institute. Materials of the Bio-Based Industries Joint Undertaking Website. 2020. Available at: (accessed 23.12.19).
  22. Cohen E.G.D. Superstatistics. Physica D: Nonlinear Phenomena, 2004, vol. 193, iss. 1-4, pp. 35–52. DOI: 10.1016/j.physd.2004.01.007
  23. Efimova E.A., Ivanov A.O., Camp P.J. Thermodynamic Properties of Ferrofluids in Applied Magnetic Fields. Physical Review E, 2013, vol. 88, iss. 4, art. 042310. DOI: 10.1103/PhysRevE.88.042310
  24. Jaynes E.T. Information Theory and Statistical Mechanics. Brandeis University Summer Institute Lectures in Theoretical Physics. Vol. 3 – Statistical Physics. New York, W.A. Benjamin Inc., 1963, pp. 181–218.
  25. Joslin C., Goldman S. The Third Dielectric and Pressure Virial Coefficients of Dipolar Hard Sphere Fluids: II. Numerical Results. Molecular Physics, 1993, vol. 79, iss. 3, pp. 499–514. DOI:10.1080/00268979300101401
  26. Karmanov A.P., Monakov Yu.B. Hydrodynamic Properties and Structure of Lignin. International Journal of Polymeric Materials and Polymeric Biomaterials, 2000, vol. 48, iss. 2, pp. 151–175. DOI: 10.1080/00914030008050614
  27. Karmanov A.P., Poleschikov S.M., Kocheva L.S. Theoretical and Experimental Simulation of Lignin Biosynthesis. Butlerovskiye soobshcheniya [Butlerov Communications], 2015, vol. 41, no. 3, pp. 147–151. DOI: 10.37952/ROI-jbc-01/15-41-3-147
  28. Kullback S., Leibler R.A. On Information and Sufficiency. The Annals of Mathematical Statistics, 1951, vol. 22, iss. 1, pp. 79–86. DOI: 10.1214/aoms/1177729694
  29. Maher P. Explication of Inductive Probability. Journal of Philosophical Logic, 2010, vol. 39, iss. 6, pp. 593–616. DOI: 10.1007/s10992-010-9144-4
  30. Naimark O.B. Defect-Induced Transitions as Mechanisms of Plasticity and Failure in Multifield Continua. Advances in Multifield Theories of Continua with Substructure. Ed. by G. Capriz, P. Mariano. Boston, MA, Birkhäuser, 2004, pp. 75–114. DOI: 10.1007/978-0-8176-8158-6_4
  31. Pavlos G.P. Complexity Theory, Time Series Analysis and Tsallis q-Entropy Principle Part One: Theoretical Aspects. Journal of the Mechanical Behavior of Materials, 2017, vol. 26, no. 5-6, pp. 139–180. DOI: 10.1515/jmbm-2017-0023
  32. Rényi A. Probability Theory. North-Holland, Amsterdam, 1970. 670 p.
  33. Szilard L. Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. Zeitschrift für Physik, 1929, vol. 53, iss. 11-12, pp. 840–856. DOI: 10.1007/BF01341281
  34. Tsallis C. Possible Generalization of Boltzmann-Gibbs Statistics. Journal of Statistical Physics, 1988, vol. 52, pp. 479–487. DOI: 10.1007/BF01016429
  35. Wohlgemuth R., Twardowski T., Aguilar A. Bioeconomy Moving forward Step by Step – A Global Journey. New Biotechnology, 2021, vol. 61, pp. 22–28. DOI: 10.1016/j.nbt.2020.11.006

Non-Ideality Factor in Multifractal and Entropy-Based Analysis of Self-Organized Structures of Plant Polymers (Lignins)


Make a Submission


Lesnoy Zhurnal (Russian Forestry Journal) was awarded the "Seal of Recognition for Active Data Provider of the Year 2024"