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I'm studying Thermodynamics on Callen's book and he introduces entropy through the entropy maximum postulate, namely:

There exists a function (called the entropy $S$) of the extensive parameters of any composite system, defined for all equilibrium states and having the following property: The values assumed by the extensive parameters in the absence of an internal constraint are those that maximize the entropy over the manifold of constrained equilibrium states.

So, if a system has some constraint and is at an equilibrium state $A$, if we modify some constraint new equilibrium states become available and the system will go to the equilibrium state $B$ that maximizes the entropy.

Now, when discussing reversibility the author states the following

A quasi-static locus can be approximated by a real process in a closed system only if the entropy is monotonically nondecreasing along the quasi-static locus.

The limiting case of a quasi-static process in which the increase in entropy becomes vanishingly small is called a reversible process. For such a process the final entropy is equal to the initial entropy, and the process can be traversed in either direction.

But this seems to contradict the entropy maximum postulate. The postulate states that the new equilibrium state a system goes after the change of some constraint is the one that maximizes the entropy. How can a system go to another state with the same entropy? This seems to be a contradiction. When he says that the entropy must be monotonically nondecreasing, I think in view of the postulate it should be monotonically increasing.

What's wrong with my reasoning? Why having the system going to a state with the same entropy does not violate the entropy maximum postulate?

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What she means (yes, Mrs. Callen is a lady) is the system is able to undergo a process that changes the state of the system, as long as both states (final and initial) have the same entropy. Hence the term quasi-static.

What is wrong with your reasoning is to think that any given system is only allowed one state of maximum entropy. There can be an infinite degree of degeneracy: think of the Mexican hat potential. That's an example of an infinite number of states where a system would be in equilibrium and they are all allowed to said system. Apply the same reasoning to the entropy function.

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