Discrete model of information evolution of a complex system near a phase transition point

Abstract

Landau's theory of phase transitions and Frenkel's theory of heterophase fluctuations, due to their simplicity and efficiency, have acquired the status of universal theories that have numerous applications in various branches of science and technology. One of the important scientific directions in the development of the theory is the study of changes in information in complex systems. To date, an urgent task is the development of multilevel mathematical models of the dynamics of the transition of a macroscopic system to a new phase state, based on the analysis of changes in Shannon information. The paper proposes a discrete model of the phase transition at the microscopic level. The discrete phase transition model differs from the models used in the Landau and Frenkel theory by a smaller scale, which leads to a greater dependence on random factors. The model is based on the theory of Markov processes and the principle of maximum information or information entropy. The main mathematical apparatus is the Kolmogorov equations. For clarity, Kolmogorov graphs are used. Work results. A stochastic model of the phase transition at the microlevel is constructed. The model remains adequate both near the phase transition point and far from it. An estimate of the intensity of the evolution of the entropy of the system is given. The dynamics of phase change during transitions of the first and second order is obtained. Conditions for phase equilibrium are obtained for phase transitions of the first and second kind. A stochastic interpretation of the phenomenon of hysteresis at the microscopic level is given. The results obtained are in accordance with the principles of Landau and Frenkel's theory. The results of the work have prospects for application in forecasting crises of complex systems based on the dynamics of information changes.