The text is written in December 2008. I review the final chapters 6 and 7 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

## Chapter 6: Thermodynamic Systems with Linear Energy Exchange

In this chapter the authors consider the case from Chapter 3 but they assume now that the heat exchange between different compartments is linear, that is, the heat conductivity is constant and there is no radiation. In this case they come to a system matrix that seems to resemble one after the finite volume discretization. In my view, it would be interesting to conduct such a comparison between finite volumes for a thermal problem and the theory in the book.

## Chapter 7: Continuum Thermodynamics

In this chapter the authors generalize the results from Chapter 3 for infinite-dimensional systems. They show that the results remain basically very similar.

## Conclusion

In general I like the book. It shows indeed that one can use a dynamical system as an object to model a thermodynamic system. I was impressed by the power of mathematical analysis achieved in the book.

On the other hand, it should be also clear that the general theory of thermodynamics has not been developed in the book and this has to be done yet. The main problem with the book in my view was that the authors has started with a toy problem in Chapter 3 and have decided that they can describe the direction of the heat flow based on the energy value for all the systems without introducing temperature. This statement was repeated several times in the book. Let me cite from the Conclusion

“Energy flows from more energetic subsystems to less energetic subsystems”

Such a statement spoils the good impression made by the book. What is occasionally valid for a toy problem (T_i = E_i) cannot be generalized. It should be clear that the values of energy cannot replace temperature in the general case. Thermodynamics is impossible without clear separation between extensive and intensive properties.

Nevertheless, the book seems to show that the Zeroth, First and Second Laws are enough to build a strict mathematical construction. What is left after Chapter 4 is to repeat the procedure after introducing the caloric equation of state in the general form. Then when it comes to work (Chapter 5) one must consider the equation of state in general to relate pressure, volume and temperature (thermodynamics of the ideal gas is actually silly). After that one needs an extra axiom to deal with the Third Law – we cannot reach zero Kelvin and the dynamic system must obey this law as well. Finally there should be multicomponent systems and other types of work, for example electrochemistry. Hence the book leaves some job for others who are not happy with the current status of classical thermodynamics.