The text is written in December 2008. I review the fourth chapter, Temperature Equipartition and the Kinetic Theory of Gases of the book Thermodynamics: A Dynamical Systems Approach by Wassim M. Haddad, Vijay Sekhar Chellaboina, & Sergey G. Nersesov
In this chapter the authors extend the theory of Chapter 3 from the case when T_i = E_i to the case T_i = beta_i E_i. In other words, they consider the case when different compartments possess different (but constant) heat capacity. As a result, the equations become much more readable. Now it is much clearer: if one sees beta_i E_i, then it is a temperature. Only now I have understood that by ectropy the authors mean TdQ.
In my view, it was a mistake to start the Chapter 3 with the statement, that one can define the Zeroth and Second Law without introducing temperature (it is only possible with the assumption T_i = E_i that is very unphysical) and then in the next chapter to generalize this. It would be much better to start with a more general case and then say that now for simplicity we consider a simpler case but we will return to the general case in the next chapter.
Actually the case of constant heat capacity is also not the general one. For the general case one should consider a caloric equation of state, that is, a temperature dependent heat capacity (in the general form) plus heats of the phase transitions. And no doubts the authors still have to harmonize the theory with the Third Law.
The second section in this Chapter “Boltzmann Thermodynamics” I have not understood. The model considered by the authors has nothing to do with Bolzmann’s kinetic theory of gases, as a compartment has already the temperature from the start by definition. Proposition 4.5 looks very strange:
“For every state reversible adiabatic process performed on a system consisting of q ideal gases connected by diathermal walls, the total entropy and total ectropy of the system remain constant”.
However, so far we have considered only a thermal system. That is, if the heat exchange with the environment is absent, the adiabatic system is actually an isolated system. What is meant here by a reversible adiabatic process is a puzzle for me.
A small note. The heat capacity Cv of a monoatomic gas at temperatures not close to zero is
Cv = (3/2) n R = 1.5 k n_i
where n is the number of mole and n_i is the number of molecules. That is, beta_i = 1/(1.5 k n_i) and not k/n_i as written in the book. Well, mathematicians took k=1 for simplicity and this does not influence the results. Still, it is good to know physics.