125.

By the preceding developments the calculation of the entropy of a system of $N$ molecules in a given thermodynamic state is, in general, reduced to the single problem of finding the magnitude $G$ of the region elements in the state space.

That such a definite finite quantity really exists is a characteristic feature of the theory we are developing, as contrasted with that due to Boltzmann, and forms the content of the so-called hypothesis of quanta.

As is readily seen, this is an immediate consequence of the proposition of [sect:120.] Sec. 120 that the entropy $S$ has an absolute, not merely a relative, value; for this, according to [eqn:(164)] (164), necessitates also an absolute value for the magnitude of the thermodynamic probability $W$, which, in turn, according to [sect:123.] Sec. 123, is dependent on the number of complexions, and hence also on the number and size of the region elements which are used.

Since all different complexions contribute uniformly to the value of the probability $W$, the region elements of the state space represent also regions of equal probability. If this were not so, the complexions would not be all equally probable.

However, not only the magnitude, but also the shape and position of the region elements must be perfectly definite.

For since, in general, the distribution density $w$ is apt to vary appreciably from one region element to another, a change in the shape of a region element, the magnitude remaining unchanged, would, in general, lead to a change in the value of $w$ and hence to a change in $S$.

We shall see that only in special cases, namely, when the distribution densities $w$ are very small, may the absolute magnitude of the region elements become physically unimportant, inasmuch as it enters into the entropy only through an additive constant. This happens, e.g., at high temperatures, large volumes, slow vibrations (state of an ideal gas, [sect:132.] Sec. 132, Rayleigh's radiation law, [sect:159.] Sec. 159).

Hence it is permissible for such limiting cases to assume, without appreciable error, that $G$ is infinitely small in the macroscopic sense, as has hitherto been the practice in statistical mechanics. As soon, however, as the distribution densities $w$ assume appreciable values, the classical statistical mechanics fail.