Thermodynamics: A Dynamical Systems Approach

In 2008 I have read the book Thermodynamics: A Dynamical Systems Approach by Wassim M. Haddad, Vijay Sekhar Chellaboina, & Sergey G. Nersesov.

and then I have posted my comments to it to the thermodynamicslib group. Now I have added the comments to my blog. You will find below the links to my comments for each chapter of the book.

The book is an interesting mixture of thermodynamics and system theory. I have been working in chemical thermodynamics for awhile and now my work is partly related to system theory. I have found the information about the book at the homepages of Jan C. Willems

where there is also an interesting review of the book


In Introduction there is an enjoyable overview of thermodynamics, a short overview of system thermodynamics and then an outline of the book. I should say that during the first reading the section about system thermodynamics was hardly understandable. Now I am reading the book for the second time and I already can follow it. Yet, it does not make sense to make comments about it right now; I’d better do it while commenting on other chapters. Hence I limited my comments in this message to Section 1.1 An overview of thermodynamics.

I like this section. It is a concise summary and nice text. Actually I have decided to buy the book only after I have read it for free from Princeton Press. It was a very clever advertisement for the book.

1) Thermodynamics and Universe

I guess since Clausius (“The entropy of the Universe tends to a maximum”) there are some fascinating relationships between thermodynamics and the Universe. There is hardly a textbook where this has not been discussed. The authors have stuck to this tradition and mentioned this several times.

I am personally sceptic if the thermodynamics laws could be applied to the Universe. First, thermodynamics assumes that energy and entropy are additive functions. It seems not to be valid in the case of Universe. Second, I find it dangerous to extrapolate models too far away.

Recently I have found a fascinating site Are You Living In a Computer Simulation? I think that this is a good example that shows the danger of extrapolation. But if we speak about thermodynamics itself, then it actually does not matter if we live in a simulation or not. The concept of expanding universe could suffer, but the thermodynamics remains the same.

2) Thermodynamics is hard to teach

The authors make several good statements about this. I would agree. It would be nice to have more logic here. Recently I have read The Tragicomical History of Thermodynamics by Truesdell and this aspect of thermodynamics is nicely covered there.

The best characteristics of thermodynamics that I have seen is

“Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don’t understand it, but by that time you are so used to it, it doesn’t bother you any more.”

This is attributed to Anold Sommerfeld. I would say that this is the very true. When one starts using the thermodynamics in practice, then actually it is okay. I have seen no problem by using the thermodynamics laws in chemical thermodynamics. So it was my strategy in teaching, to bring students to the third level as soon as possible.

3) Equilibrium and nonequilibrium states

In order to shows inconsistency in conventional thermodynamics, the authors give a simple example on p. 8 with the conclusion “Hence, the entropy of the system can only increase if the system is not isolated!” The solution in chemical thermodynamics to this apparent paradox is quite simple – introducing so called a frozen state. Imagine a box with gases that could react with each other but the reaction rate is negligible. Then we have some equilibrium state that could be changed by exposing gases to a catalyst. This is a pretty standard example in chemical thermodynamics. I guess that it would make sense for mathematicians to start learning thermodynamics from chemical thermodynamics and only after that go to heat engines. Unfortunately along this way they first have to learn a bit of chemistry and this makes such a suggestion completely unfeasible.

4) Modern thermodynamics

At the end of this section the authors list modern works on the foundation of thermodynamics. To my shame I have completely missed them. Just recently I came across Rational Thermodynamics by Truessdel but have not read it yet. It is a tough reading for a practitioner.

There is a nice paragraph about rational thermodynamics in the section (p. 10) but unfortunately the authors have decided not to disclose what is wrong with it. And if they decided to write their own book, then presumably there should be something in rational thermodynamics with what they do not quite agree. However, the authors have concealed that reason. It is a pity. I have to read Rational Thermodynamics by myself.

Dynamical system theory

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.

A System Foundation for Thermodynamics

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?

Temperature Equipartition and the Kinetic Theory of Gases

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.

Work, Heat, and the Carno Cycle

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.

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.

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.


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.