TY - CHAP

T1 - Description of the Structure Evolution Using Methods of Control Theory of Adaptive Systems

AU - Хантулева, Татьяна Александровна

N1 - Bibliographic Information • Book Title Mathematical Modeling of Shock-Wave Processes in Condensed Matter • Book Subtitle From Statistical Thermodynamics to Control Theory • Authors Tatiana Aleksandrovna Khantuleva • Series Title Shock Wave and High Pressure Phenomena • DOI https://doi.org/10.1007/978-981-19-2404-0 • Publisher Springer Singapore • eBook Packages Physics and Astronomy, Physics and Astronomy (R0) • Copyright Information The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 • Hardcover ISBN 978-981-19-2403-3 • eBook ISBN 978-981-19-2404-0 • Series ISSN 2197-9529 • Series E-ISSN 2197-9537 • Edition Number1 • Number of Pages XV, 336 • Number of Illustrations 19 b/w illustrations, 58 illustrations in colour • Topics Statistical Physics, Classical and Continuum Physics, Condensed Matter, Solid Mechanics

PY - 2022/7/19

Y1 - 2022/7/19

N2 - The new approach to the modeling of non-equilibrium processes developed on the basis of the methods of non-equilibrium statistical mechanics and presented in Chapter 5, contains a physically significant way to describe self-organization of new dynamic structures. However, when describing dynamic processes, there is usually not enough information to determine the sizes of all turbulent structures, as a result of which some of the degrees of freedom begin to relax in the direction of more stable states. Since all structural processes in the medium are interconnected, the entire system begins to evolve over time. In order to close the modeling dynamic processes, it becomes necessary to describe temporal evolution of the forming structures. To do this, first we need to determine the goal of the evolution in accordance with Maximum Entropy (MEP) principle by Jaynes that is described in Chapter 4. From the results obtained by Zubarev in non-equilibrium statistical mechanics we know that only total entropy production in the system determined in Chapter 4 can be chosen as a goal of the temporal evolution. To describe the way to the goal, we used one of the methods of cybernetic physics that branched off from the control theory of adaptive systems as an independent discipline [1-3]. This method is based on the Speed Gradient (SG) principle [4] that determines the fastest way to the goal under constraints imposed. The most simple and physical SG algorithm defines the speed of temporal evolution taking into account feedbacks between turbulent structures and macroscopic response of the system to an external impact. Although the methods of the control theory of adaptive systems are actively used in solving problems with internal and external feedbacks, in the field of non-equilibrium transport processes, the closure of models through feedback plays a key role and is an inherent element of modeling high-rate and short-duration processes.In this chapter we first show that SG algorithm can not describe temporal evolution of the system far from local equilibrium neither at macroscopic nor at microscopic level. Then we apply the SG method to describe evolution of dynamic structures that emerge on the mesoscale. For the model correlation function constructed in Chapter 5, the SG algorithm in finite form is reduced to the gradient descent on the surface of the integral entropy production of the system under constraints imposed in accordance with the general evolution principle by Prigogine–Glansdorf [5]. Such graphical interpretation of the SG method gives us a possibility to predict future states of the system taking into account the information contained in the constraints and feedbacks with macroscopic evolution of the system. The description of stationary states out of equilibrium is discussed in section 6.7. In section 6.9 a new look at the problem of stability is proposed in the framework of the SG principle.The proposed interdisciplinary approach [6] to describing the time evolution of the system far from local equilibrium allows us to construct flexible mathematical models that are able to reconstruct themselves over time in a self-consistent way along with the system's response to external influences due to the mechanism of internal control. It became clear that all attempts to use traditional macroscopic and microscopic models cannot provide an adequate description of high-rate processes. Without the introduction of internal control, such rigid models will not have predictive ability. Internal control arising at the mesoscopic levels together with the synergistic formation of new dynamic structures far from local equilibrium seems to be inherent not only to living systems, but, in a varying degree, to almost all material phenomena.

AB - The new approach to the modeling of non-equilibrium processes developed on the basis of the methods of non-equilibrium statistical mechanics and presented in Chapter 5, contains a physically significant way to describe self-organization of new dynamic structures. However, when describing dynamic processes, there is usually not enough information to determine the sizes of all turbulent structures, as a result of which some of the degrees of freedom begin to relax in the direction of more stable states. Since all structural processes in the medium are interconnected, the entire system begins to evolve over time. In order to close the modeling dynamic processes, it becomes necessary to describe temporal evolution of the forming structures. To do this, first we need to determine the goal of the evolution in accordance with Maximum Entropy (MEP) principle by Jaynes that is described in Chapter 4. From the results obtained by Zubarev in non-equilibrium statistical mechanics we know that only total entropy production in the system determined in Chapter 4 can be chosen as a goal of the temporal evolution. To describe the way to the goal, we used one of the methods of cybernetic physics that branched off from the control theory of adaptive systems as an independent discipline [1-3]. This method is based on the Speed Gradient (SG) principle [4] that determines the fastest way to the goal under constraints imposed. The most simple and physical SG algorithm defines the speed of temporal evolution taking into account feedbacks between turbulent structures and macroscopic response of the system to an external impact. Although the methods of the control theory of adaptive systems are actively used in solving problems with internal and external feedbacks, in the field of non-equilibrium transport processes, the closure of models through feedback plays a key role and is an inherent element of modeling high-rate and short-duration processes.In this chapter we first show that SG algorithm can not describe temporal evolution of the system far from local equilibrium neither at macroscopic nor at microscopic level. Then we apply the SG method to describe evolution of dynamic structures that emerge on the mesoscale. For the model correlation function constructed in Chapter 5, the SG algorithm in finite form is reduced to the gradient descent on the surface of the integral entropy production of the system under constraints imposed in accordance with the general evolution principle by Prigogine–Glansdorf [5]. Such graphical interpretation of the SG method gives us a possibility to predict future states of the system taking into account the information contained in the constraints and feedbacks with macroscopic evolution of the system. The description of stationary states out of equilibrium is discussed in section 6.7. In section 6.9 a new look at the problem of stability is proposed in the framework of the SG principle.The proposed interdisciplinary approach [6] to describing the time evolution of the system far from local equilibrium allows us to construct flexible mathematical models that are able to reconstruct themselves over time in a self-consistent way along with the system's response to external influences due to the mechanism of internal control. It became clear that all attempts to use traditional macroscopic and microscopic models cannot provide an adequate description of high-rate processes. Without the introduction of internal control, such rigid models will not have predictive ability. Internal control arising at the mesoscopic levels together with the synergistic formation of new dynamic structures far from local equilibrium seems to be inherent not only to living systems, but, in a varying degree, to almost all material phenomena.

KW - entropy production, temporal evolution, self-organization, mesoscale, interbal control, Speed Gradient principle

UR - https://www.mendeley.com/catalogue/e1b6fb5d-e158-3fa7-be60-8671d57c9065/

U2 - 10.1007/978-981-19-2404-0_6

DO - 10.1007/978-981-19-2404-0_6

M3 - Chapter

SN - 978-981-19-2403-3

T3 - Shock Wave and High Pressure Phenomena

SP - 175

EP - 208

BT - Mathematical Modeling of Shock-Wave Processes in Condensed Matter

PB - Springer Nature

CY - Singapore

ER -