Thermodynamic modeling, energy equipartition, and nonconservation of entropy for discrete-time dynamical systems
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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. 1. Introduction 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]. In particular, conservation laws (e.g., mass and energy) are used to capture the exchange of material between coupled macroscopic subsystems known as compartments. Each compartment is assumed to be kinetically homogeneous; that is, any material entering the compartment is instantaneously mixed with the material in the compartment. These models are known as compartmental models and are widespread in engineering systems as well as in biological and ecological sciences [1, 7, 9, 16, 17, 22]. 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 actually occur; that is, they do not address the second law of thermodynamics involving entropy notions in the energy flow between subsystems. The goal of the present paper is directed towards developing nonlinear discrete-time compartmental models that are consistent with thermodynamic principles. Specifically, Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:3 (2005) 275–318 DOI: 10.1155/ADE.2005.275
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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. Furthermore, using graph-theoretic notions, we state three thermodynamic axioms consistent with the zeroth and second laws of thermodynamics that ensure that our large-scale dynamical system model gives rise to a thermodynamically consistent energy flow model. Specifically, using a large-scale dynamical systems theory perspective, we show that our compartmental dynamical system model leads to a precise for
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