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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.

  • Given everything that I said (if it is correct), what does the criterion of maximization of entropy actually mean?

  • 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.

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  • $\begingroup$ "I don't understand this explanation because it seems to me that the constraint invalidates the hypothesis that the system is closed." The definition of a closed system is that it cannot exchange mass with its surroundings. That doesn't preclude the possibility system constraints. $\endgroup$
    – Bob D
    Jul 20, 2023 at 19:37
  • $\begingroup$ I think I got confused with a closed system in the mechanical sense? $\endgroup$ Jul 20, 2023 at 19:40
  • $\begingroup$ what do you mean by"mechanical sense"? $\endgroup$
    – Bob D
    Jul 20, 2023 at 19:41
  • $\begingroup$ “In nonrelativistic classical mechanics, a closed system is a physical system that doesn't exchange any matter with its surroundings, and isn't subject to any net force whose source is external to the system” by wikipedia. $\endgroup$ Jul 20, 2023 at 19:42
  • $\begingroup$ can you give link? Never heard about no net force. A closed system in thermodynamics allows for the possibility of both heat and work energy transfer. $\endgroup$
    – Bob D
    Jul 20, 2023 at 19:45

1 Answer 1

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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.

For the slightly disequilibrated professional I will make only two remarks in passing. First, in mechanics the concept of force originated in statics and was carried over bodily, if with much delay and discussion, to motions. If the restoring force exerted by a spring is proportional to the increase of its length in a static experiment, will it still be so when a ball is attached to the end and set into oscillation, especially if the experiment is performed in a spaceship in orbit around the moon? Indeed, does it make sense to talk about forces at all in a moving system? The forces, it seems, might be affected by the motions, yet we are supposed to know the forces first in order to determine what the motion will be. These questions, and far subtler ones of the same kind, were asked in the seventeenth century; today the freshman is trained specifically not to ask them. Perhaps he ought to; but if he does, he is more likely to take up philosophy than science; and if NEWTON had insisted that forces be used only when they can be measured by an operationalist who works and thinks very, very slowly, it is unlikely anyone would be designing spaceships today. The professionals who moan that temperature is a concept for equilibrium alone are not solving any problems themselves; they are merely pronouncing our problems insoluble and sneering at us for trying to solve them.

Second, the professional theorists of thermodynamics show a lack of respect for experiment I find hard to admire. Experiments are being done today in all sorts of extreme conditions. My friends and colleagues tell me they measure the temperatures in polymer solutions undergoing strong normal-stress effects, they control the temperatures in explosion fronts, they infer the temperatures on the skins of artificial satellites. It would be presumptuous on my part to question the details of the work of these men, and I see no reason to deny that they know their business. The temperatures occurring as values of the temperature function $\theta(\cdot)$ in modern theories of materials are intended to represent the numbers these experimenters report and call "temperatures", although their systems bear little likeness to those the old books on thermodynamics describe as being "in equilibrium ", and their processes seem to have little in common with whatever it is the thermodynamicists mean by "quasi-static". I do not claim that we yet know whether or not a particular theory be borne out by any particular experiment, but I do not see why the professionals should forbid us from trying to explain the experiments by a rational theory, approached in that very spirit everyone today regards as the right one for mechanics and electromagnetism.

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  • $\begingroup$ Would you agree, then, that in order for the criterion of maximization of entropy to make sense, it is necessary to consider non equilibrium states? $\endgroup$ Jul 20, 2023 at 20:52
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    $\begingroup$ I personally would accept it. In this framework, even with variations in the system, the entropy per unit mass is considered to be close to thermodynamic equilibrium at the local parameter values, and the total entropy of the system is then consider the integral over the mass of the system. $\endgroup$ Jul 20, 2023 at 21:09
  • $\begingroup$ Conventionally, entropy maximization can be done along the lines of Gibbs, Tisza, Callen, over purely equilibrium states. That is done by saying that, for example, for 2 variables the postulate is that $S=S(U,V)$ be a concave function of the variables. That is $S(U+\delta U, V+\delta V) + S(U-\delta U, V-\delta V) \le 2S(U, V)$ for arbitrary small deviations $\delta U, \delta V)$. Understood this way there is no need for a process, just the existence of equilibrium states. If the claim is that nothing else can be done then there is Truesdell's answer to it. $\endgroup$
    – hyportnex
    Jul 20, 2023 at 21:26
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    $\begingroup$ Alternatively and most commonly there is the so-called local equilibrium hypothesis to which @ChetMIller is alluding above, that is for a small enough piece of matter most material response can be taken in to be equilibrium, that is one can freely use the thermal and caloric equations of state obtained in equilibrium and use those locally. This is the most common generalization but not the only one, and Truesdell is thinking of something more general along the lines of Coleman, Bridgman or Eckart, but I think in practice it does not really get much beyond the local equilibrium hypothesis. $\endgroup$
    – hyportnex
    Jul 20, 2023 at 21:30
  • $\begingroup$ I see. It seems like the local equilibrium hypothesis is the thing employed by Landau in the textbook? And so the entropy of the system as a whole in equilibrium is greater than the integral over the equilibrium subsystems when they are not in mutual equilibrium, correct? $\endgroup$ Jul 20, 2023 at 22:02

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