State of maximum entropy

In thermodynamics, state of maximum entropy, aka “maximum entropy state”, is the “state” of a system in which the equivalence-values of all uncompensated transformations N have reached a maximum value; a maximum value (see: maximum entropy) of entropy transformation, in a completed cycle. [1]

Overview
In 1854, Rudolf Clausius introduced the concept of “equivalence-values”, or Q/T values, as an upgrade to the caloric particle model, of heat.

Clausius also introduced the logic that equivalence-values or “transformation equivalents”, as Maxwell latter called them, could “compensate” each other, in the mechanical equivalent of heat sense of the matter, when going into or out of a system, in the sense of heat transforming into work, or work transforming into heat, in respect to a system transforming back to its “original” or “initial state”, following a series of transformations. In “idea” systems, these transformation equivalents, theoretically, would cancel each other out. in “real” systems, however, there would remain “uncompensated transformations” in each cycle.

This is what, after 1865, Clausius began to refer to as “entropy increase”. Hence, when any given system reaches the final stage of its series of transformation cycles, the sum of the equivalence values of all the “uncompensated transformations” would have a “maximum” positive numerical value, symbol N, defined as follows:

equivalence-value of all uncompensated transformations

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Confusions | Errors
In c.1875 to c.1895, what is called "confused thermodynamics", what are called "entropy as disorder confusions" began to solidify, resulting in completely incorrect renditions of what exactly "maximum entropy" means in respect to systems and order and disorder assignments.

In 1933, James Jeans, who sparked off the Jeans, Donnen, Guggenheim debate (1834), gave the following incorrect view of what constitutes the state of a system at maximal entropy, namely that it is one that "avoids concentration" and is one whose energy is "uniformly diffused", both of which are incorrect:

“These instances have shown us two final states in which the entropy is a maximum. They illustrate a very wide and very general principle — the final state of maximum entropy avoids concentration, whether of special substances (as with the ink) or of energy (as with the heat of the fire). The ‘commonest state’ is one in which both substance and energy are uniformly diffused, just as the commonest state in which we find a concert audience is that in which tall people and short, dark and fair, and so on, are uniformly diffused.”
— James Jeans (1933), The New Background of Science (pg. 267) (Ѻ)

The following is an over-typical confused example:

Entropy can be defined as a measure of disorder in a system. Thus, a system that has reached maximum entropy is in a state of complete disor­der. Maximum entropy can be compared to system death [see: heat death]. A system in a state of maximum entropy is completely lack­ing in structure or organization, and is basically in a random state of disarray. or maximum decay. A concrete physical system in maximum entropy has in effect expended all of its energy resources. Since it is closed, there is little hope of reversal, and it will essentially remain in a state of maxi­mum entropy, unless its boundaries arc somehow opened so that new energy (and information) can be used to renew the system (assuming that it is not beyond repair by this time).”
— George Ritzer (2004), Encyclopedia of Social Theory (§: General System Theory, pg. 310)

In 1975, Norman Dolloff, in his Heat Death and the Phoenix, was the first to give the generally correct view of things.

Quotes
The following are related quotes:

“With this [commonness] definition [of entropy] we find that, just because the numerical factors involved are so immense, conditions of ‘maximum’ entropy are not only more common, but incomparably more common, than those whose entropy is less, and so it is all down the ladder. Because of this, it is practically certain that each state of the universe will be succeeded by a state of higher entropy than itself, so that the universe will ‘evolve’ through a succession of states of ever-increasing entropy, until it finally reaches a state of maximum entropy. Beyond this it cannot go; it must come to rest — not in the sense that every atom in it will have come to rest (for maximum entropy does not involve this), but rather in the sense that its general characteristics cannot change any more. Yet if someone asserts that this will not happen, and that the universe will move to a state of lower entropy than the present, we cannot prove him wrong. He is entitled to his opinion, either as a speculation or as a pious hope. All we can say is that the odds against his dream coming true involve a very high power of 10^79 his disfavor.”
— James Jeans (1933), The New Background of Science (pg. 265) (Ѻ)

See also
Heat death

References
1. Clausius, Rudolf. (1865). The Mechanical Theory of Heat: with its Applications to the Steam Engine and to Physical Properties of Bodies (pg. 141). John van Voorst.

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