THERMODYNAMIC MODELING, ENERGY EQUIPARTITION, AND NONCONSERVATION OF ENTROPY FOR DISCRETE-TIME DYNAMICAL SYSTEMS WASSIM M. HADDAD, QING HUI, SERGEY G. NERSESOV, AND VIJAYSEKHAR CHELLABOINA Received 19 November 2004 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 MODELING ENERGY EQUIPARTITION AND NONCONSERVATION OF ENTROPY FOR DISCRETE-TIME DYNAMICAL SYSTEMS WASSIM M. HADDAD QING HUI SERGEY G. NERSESOV AND VIJAYSEKHAR CHELLABOINA Received 19 November 2004 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 . mass and energy between coupled subsystems 5 6 8 11 20 23 24 . In particular conservation laws . 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
