Thermodynamics: A Dynamical Systems Approach – Work, Heat, and the Carno Cycle

The text is written in December 2008. I review the fifth chapter, Work, Heat, and the Carno Cycle 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

In this chapter the authors extend the formalism of Chapter 3 for a system with deformable wall, that is, a system that can make work. They show how the entropy can be defined in this case, prove theorems on its uniqueness and demonstrate that the system does not possess Poincaré recurrence. It is really interesting to see the possibilities of the system approach.

From my viewpoint there were several things in the chapter that could be improved.

1) Authors again using energy as a criterion of thermal equilibrium, that is, T_i = E_i. I have already got used to this notation and presumably was able to distinguish places where authors by E_i meant temperature from those where this stays for energy. Nevertheless I find it a bad practice, as it leads to wrong conclusions.

2) The authors have limited themselves to the ideal gas (Eq. 5.2). I have nothing against that one starts with a simple case and the ideal gas is a usual starting point in thermodynamics. However, this was not explicitly mentioned and no doubt in the general case one must introduce the equation of state.

3) The authors have not considered that the compartments can make work against each other. I have not understood why. This had a funny consequence. When they consider an equilibrium state of an isolated system (for example Proposition 5.3, p. 122), they come to the conclusion that equilibrium volume is an arbitrary point in R^q_plus. This is unphysical as implies that at equilibrium the pressures in different compartments could be different.

4) The authors claim at p. 125 that their system complies with the Third Law but this is obviously wrong. It seems that the authors have missed the fact that the entropy is not only zero at 0 K, but its derivatives are also zero. In other words, according to the Third Law we cannot reach the state of zero entropy (0 K) but in the system design by the authors it is possible.

5) The meaning of the axes in Fig. 5.2 is hard to understand. At the beginning I thought that this is equivalent to Fig. 3.2, that is, the energy of the two compartments. Yet, from the text under Fig. 5.2 it follows that this is not the case.

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Overview
Chapter 1: Introduction
Chapter 2: Dynamical system theory
Chapter 3: A System Foundation for Thermodynamics
Chapter 4: Temperature Equipartition and the Kinetic Theory of Gases

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Chapter 6 and 7


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