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

Abstract

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 + 〈– β_ord + α_nord〉 = 1 + 〈– p_i (β) + p_i (α)〉, где – β_ord ≡ 1/nΣ_iⁿβi и α_nopd ≡ 1/nΣⁿ_iα_i  are relative average characteristics (p_i – 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 α_nord = 0 g → 0, for β_ord = 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: g_th = 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) N_d and the number of elements in the structure of the object in the perfect condition, having the property of multi-scale and self-similarity, N_D, where d and D are the fractal and Euclidean dimensions. As a thermodynamic characteristic g_th is defined by the ratio of thermodynamic functions, functionals, for example, ΔG_i*/ΔG_i, where ΔG_i* = –RTlnа_i is real and, ΔG_i = –RTlnN_i is ideal state; the number of moles of n* − real state of matter to n − ideal state of matter; relative entropies of the system ΔS_real/ΔS_id (ΔS_id − Boltzmann entropy). New expressions of the information and thermodynamic entropies with a fractional (0:1) moment of order and with the entropic g_S and g_th non-ideality factors are obtained for the analysis of self-organized quasi-equilibrium structures in the Renyi formalism S_gS^M–Rn(p) = R/(1 – g_S)lnΣ^N_ip^gS_i; S_gth^M–Rn = R/(g_th)ln(Σ^N_i=1p_i^{gth – 1}); in the Tsallis formalism S_gS^M–TS (p) = R(1 – Σ_i^N(ε)p_i^gS)/(g_S – 1); S_gth^M–TS (p) = R(1 – Σ_i^N(ε)p_i^1–gth)/g_th with an application for studying the topological properties of high-molecular compounds by hydrodynamic methods, as well as the thermodynamics of polymer solutions.
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