I’m studying thermodynamics and statistical mechanics from the book by Kerson Huang. I’m having a conceptual difficulty with the notion of thermodynamic equilibrium.
At the beginning of the first chapter, the author states
A thermodynamic state is specified by a set of values of all the thermodynamic parameters necessary for the description of the system.
and
Thermodynamic equilibrium prevails when the thermodynamic state of the system does not change in time.
Then, after stating the laws of thermodynamics, a criterion for finding the equilibrium state is provided: for a closed system, the equilibrium state is the one that maximizes the entropy. This only makes sense if entropy is defined for non-equilibrium states, too. The same thing applies for the minimization of the free energy etc.
However, in the statistical mechanics part of the book, there is the following statement
The entropy in thermodynamics, just as $S$ here, is defined only for equilibrium situations.
It's not that this does not make sense: in fact, one talks about “equilibrium thermodynamics” because the subject is only concerned with equilibrium states. Thermodynamic parameters are average values over times longer than the relaxation time, and as such they require equilibrium situations in order to be defined. However, in my understanding, this makes the criterion of maximization of entropy meaningless.
I found this question, which is basically the same I'm asking, but I do not understand the answer. To be more specific, the accepted answer seems to boil down to the following example: one imagines that some “constraint” is placed on the system, that maintains the system in equilibrium. Then the constraint is relaxed, allowing the system to evolve, and then it is placed again. In this way, one can approximate the evolution of the system as a sequence of equilibrium states separated by “infinitesimal transformations”. The state that maximizes the entropy is then the one which does not require a constraint to remain in equilibrium.
I don't understand this explanation because it seems to me that the constraint invalidates the hypothesis that the system is closed. In other words, the presence of the constraint “forces” a state to be an equilibrium state, but then would no longer be described by the equilibrium parameters of the system alone.
I also have something else to add: in the book by Landau and Lifshitz (Course of theoretical physics V) the picture for the law of increase of entropy is that a non equilibrium system can be viewed as a composition of smaller system which are approximately in equilibrium, and for them an entropy can be defined. The total entropy is then the sum of the individual entropies. As the subsystem reach mutual equilibrium, the total entropy increases. This is to me a more logical picture. However, does it correspond to the thermodynamics one? i.e., can such a non equilibrium system be represented by a set of thermodynamic parameters? I guess not, but then the law of increase of entropy would have a different meaning.
In short, I'm terribly confused.
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Given everything that I said (if it is correct), what does the criterion of maximization of entropy actually mean?
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Does the law of increase of entropy require to consider non equilibrium systems?
Edit:
I now think that what I said about the hypothesis of closed system is nonsense, because my definition is wrong (see comments below) and also because the entropy increases when the system is thermally isolated, not when it is closed: from Clausius theorem,
$$
\Delta S \ge \int_A^B \frac{\delta Q}{T},
$$
and the right hand side is zero when $\delta Q = 0$. However, the questions don’t strictly depend on this.
Best Answer
Instead of saying that "The entropy in thermodynamics, just as S here, is defined only for equilibrium situations," it is better to say that "The entropy in thermo-statics, just as S here, is defined only for equilibrium situations." The entropy concept would be of very limited use if it were not applicable to dynamic situations. Below is a quote from Truesdell: Rational Thermodynamics,(1984, 2nd ed.) pp79-80 regarding this issue, and the whole book is about how to extend the concept of equilibrium entropy into non-equilibrium entropy. I am surprised that this is still controversial. Note too that the concept of a non-equilibrium entropy is not any more controversial than non-equilibrium temperature is.