Thermodynamics: A Dynamical Systems Approach – A System Foundation for Thermodynamics

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

In this chapter the authors have considered a simple thermal system and showed that a resultant dynamic system, obeying the two axioms i) and ii) resembling the Zeroth and Second Laws, possesses entropy and appropriate behavior. I personally was impressed. The formal mathematical approach seems to function quite well in this case. The authors have defined the entropy in a uniform way for all system states, proved that the entropy is a unique function increasing in the adiabatically isolated system, and that there cannot be Poincaré recurrence in such a system. The system in question was simple but I also like to start with a simple example, to check that everything functions there and only then go further to more complex systems.

There are two things in this chapter that have disturbed me a bit. First the authors have defined the both axioms i) and ii) (p. 56-57) in terms of energy. They thought that this way they could make the next statement (footnote 3, p. 57).

“It is important to note that our formulation of the second law of thermodynamics as given by Axiom ii) does not require the mentioning of temperature nor the more primitive subjective notions of hotness or coldness. As we will see later, temperature is defined in terms of the system entropy after we establish the existence of a unique, continuously differentiable entropy function for G.”

I would say that this is self-delusion. This has worked in this chapter because the system in question was as follows. There was a number of compartments possessing the same constant heat capacity. That is, E_i = Cv T_i and as the heat capacity was assumed to be one, E_i = T_i. I bet that this cannot be generalized. This also makes the reading of the chapter particularly hard, as symbol E_i at some places is actually temperature but in other places means energy indeed. Mixing energy and temperature is a bad idea, as energy is an extensive property and temperature is an intensive temperature. At the same time, axioms i) and ii) must be expressed in terms of intensive property. Anyway it is possible to forgive this to mathematicians.

A system with a constant heat capacity does not obey the Third Law and there is a practical problem to integrate from 0 K: S(T) – S(0) = Cv ln(T/0). The authors have found a funny solution to remove this obstacle: in their world the energy is zero at some positive temperature, that is, they use some temperature scale that has some positive value at 0 K. Mathematicians are special people.

Second, the authors have introduced two terms in Fig. 3.1 (p. 47), S_i and sigma_ii to describe an energy flux going to the i-th compartment and energy dissipation from the i-th compartment to the environment. (Note that S here has nothing to do with entropy. Understanding the notation in the chapter is a good exercise for the brain.) The two terms were necessary in order to define inputs and outputs for the dynamic system but it is hard to say what the exact meaning of S_i and sigma_ii is. Worthy of noting is that they both could have positive and negative values. Jan C. Willems in his review of the book

even says that the input-output treatment may not be appropriate in the case of thermodynamics.

Finally there are some small comments.

1) p. 46

“The absence of a state space formalism in classical thermodynamics, and physics in general, is quite disturbing and in our view largely responsible for the monomeric state of classical thermodynamics.”

Classical thermodynamics does not have time by design. On the other hand, in nonequilibrium thermodynamics the kinetic equations look quite similar to Eq (2.1). Hard to understand what the authors have meant here.

2) p. 46, the paragraph at the bottom.

I was not able to understand this paragraph. It would be good to have some examples.

3) Theorem 3.4. p. 68

Could we tell after this theorem that Sr = Sa = S in Proposition 3.3?


Chapter 1: Introduction
Chapter 2: Dynamical system theory


Chapter 4: Temperature Equipartition and the Kinetic Theory of Gases
Chapter 5: Work, Heat, and the Carno Cycle
Chapter 6 and 7