Advances in Difference Equations
Volume 2005 (2005), Issue 3, Pages 275-318

Thermodynamic modeling, energy equipartition, and nonconservation of entropy for discrete-time dynamical systems

Wassim M. Haddad,1 Qing Hui,1 Sergey G. Nersesov,1 and Vijaysekhar Chellaboina2

1School of Aerospace Engineering, Georgia Institute of Technology, Atlanta 30332-0150, GA, USA
2Department of Mechanical, Aerospace and Biomedical Engineering, The University of Tennessee, Knoxville 37996-2210, TN, USA

Received 19 November 2004

Copyright © 2005 Wassim M. Haddad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.