Quite early, too, there was a realization of the fact that entropy had somehow a statistical character, that it had to do with mean values only. This was well brought out by the long known, and much quoted, "demon" experiment suggested by Maxwell, in which a being of superhuman power separated, without doing any work, the colder and hotter particles of a gas, thus effecting an apparent violation of the Second Law. This, to be sure, was getting close to the crux of the whole matter, but still lacked much to give entropy a precise physical meaning. Nevertheless, we see here a notable approach to the fundamental requirement that entropy must be tied down to the condition of "elementary chaos" (elementare-unordnung).
We have already dwelt somewhat fully on this hypothesis of "elementary chaos."
"It follows from this presentation that the concepts of entropy and temperature in their essence are tied to the condition of "elementare Unordnung." Thus a purely periodic absolute plane wave possesses neither entropy nor temperature because it contains nothing whatever in the way of uncheckable, non-measurable magnitudes, and therefore cannot be "elementar-ungeordnet," just as little as can be the case with the motion of a single rigid atom. When there is [an irregular co-operation of many partial oscillations of different periods, which independently of each other propagate themselves in the different directions of space, or] an irregular, confused, whirring intermingling of many atoms, then (and not till then) is there furnished the preliminary condition for the validity of the hypothesis of "elementare Unordnung and consequently for the existence of entropy and of temperature."
"Now what mechanical or electro-dynamic magnitude represents the entropy of a state? Evidently this magnitude depends in some way on the "Probability" of the state. For because "elementare Unordnung" and the lack of every individual check (or measurement) is of the essence of entropy it follows that only combination or probability considerations can furnish the necessary foothold for the computation of this magnitude. Even the hypothesis of "elementare Unordnung" by itself is essentially a proposition in Probability, for, out of a vast number of equally possible cases, it selects a definite number and declares they do not exist in Nature."
Now since the idea of entropy, and likewise the content of Second Law, is a universal one, and since, moreover, the theorems of probability possess no less universal significance, we may conjecture (surmise) that the connection between Entropy and Probability will be a very close one. We therefore place at the head (forefront) of our further presentation the following proposition: "The Entropy of a physical system in a definite condition depends solely on the probability of this state." The permissibility and fruitfulness of this proposition will become manifest later in different cases. A general and rigorous proof of this proposition will not be attempted at this place. Indeed, such an attempt would have no sense here because without a numerical statement of the probability of a state it could not be tested numerically.
[17]This relation is not a valid one, unless the external work performed by a gas during its change is equal to
.
(2) Planck's Formula for the Relation between Entropy and the Number of Complexions
Now we have already seen, from the permutation considerations presented on [p. 27], that the Theory of Probabilities leads very directly to the theorem, "The number of complexions included in a given state constitutes the probability W of that state." The next step (omitted here) is to identify the thermodynamically found expression for entropy of any state with the logarithm of its number of complexions.