# Thermodynamics: A Dynamical Systems Approach – Dynamical system theory

The text is written in December 2008. I review the second chapter, Dynamical system theory of the book Thermodynamics: A Dynamical Systems Approach by Wassim M. Haddad, Vijay Sekhar Chellaboina, & Sergey G. Nersesov

http://blog.rudnyi.ru/2010/05/thermodynamics-dynamical-systems.html

I should confess that the level of mathematics is a bit high for me. It would be good to find some introductory text on these subjects and have some exercises. Still I hope that I was able to understand the main points in this chapter correctly.

Small digression. Should it be a dynamical system or dynamic system? Google gives 629 000 hits for “dynamic system” and 492 000 hits for “dynamical system”. English is not my native language and I just wonder. In this message I will follow the authors and use a dynamical system.

First the authors formally define a nonnegative dynamical system. I would agree. It seems that it is quite a general object and it would be interesting to define thermodynamics using it as a model object for a thermodynamic system.

I should say that I do not completely understand if we should limit a system to a nonnegative. On the other hand, I cannot say exactly why I am not satisfied with a nonnegative dynamical system. I will list two thoughts just to document them.

• 1) In chemical thermodynamics it is quite usual to work with negative energies.
• 2) The minimal energy is achievable at zero Kelvin. Yet, according to the Third Law we cannot achieve zero Kelvin.

In Section 2.2 Stability Theory for Nonnegative Dynamical Systems the stability is defined and the theorems in this respect are proved. The Lyapunov functions are used as the background framework.

In Section 2.3 the authors introduce and define reversibility, irreversibility, recoverability and irrecoverability. This was the hardest section for me. I have to learn the Banach spaces yet. Nevertheless intuitively it was actually understandable.

Finally in Section 2.4 Reversible Dynamical System, Volume-Preserving Flows, und Poincaré Recurrence the authors consider the circumstances when a dynamical system will have cyclic behavior. They again define this formally and find sufficient conditions when a dynamical system will not have Poincaré recurrence.

In general, I like the approach of the authors. Define a mathematical object formally, define properties and show what is necessary for the object to possess or not possess these properties.

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