Abstract
We develop thermodynamic models for discrete-time large-scale dynamical systems. Specifically, using compartmental dynamical system theory, we develop energy flow models possessing energy conservation, energy equipartition, temperature equipartition, and entropy nonconservation principles for discrete-time, large-scale dynamical systems. Furthermore, we introduce a new and dual notion to entropy; namely, ectropy, as a measure of the tendency of a dynamical system to do useful work and grow more organized, and show that conservation of energy in an isolated thermodynamic system necessarily leads to nonconservation of ectropy and entropy. In addition, using the system ectropy as a Lyapunov function candidate, we show that our discrete-time, large-scale thermodynamic energy flow model has convergent trajectories to Lyapunov stable equilibria determined by the system initial subsystem energies.
Highlights
Thermodynamic principles have been repeatedly used in continuous-time dynamical system theory as well as in information theory for developing models that capture the exchange of nonnegative quantities between coupled subsystems [5, 6, 8, 11, 20, 23, 24]
Even though the compartmental models developed in the literature are based on the first law of thermodynamics involving conservation of energy principles, they do not tell us whether any particular process can occur; that is, they do not address the second law of thermodynamics involving entropy notions in the energy flow between subsystems
To develop discrete-time compartmental models that are consistent with thermodynamic principles, consider the discrete-time large-scale dynamical system Ᏻ shown in Figure 3.1 involving q interconnected subsystems
Summary
Thermodynamic principles have been repeatedly used in continuous-time dynamical system theory as well as in information theory for developing models that capture the exchange of nonnegative quantities (e.g., mass and energy) between coupled subsystems [5, 6, 8, 11, 20, 23, 24]. 276 Thermodynamic modeling for discrete-time systems since thermodynamic models are concerned with energy flow among subsystems, we develop a nonlinear compartmental dynamical system model that is characterized by energy conservation laws capturing the exchange of energy between coupled macroscopic subsystems. Using the system ectropy as a Lyapunov function candidate, we show that our thermodynamically consistent large-scale nonlinear dynamical system model possesses a continuum of equilibria and is semistable; that is, it has convergent subsystem energies to Lyapunov stable energy equilibria determined by the large-scale system initial subsystem energies.
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