189.

The formula is 218 $\Label [e q n] (a) \upshape (a) {lim}_{n = \infty} \frac{n!}{n^{n} e^{- n} \sqrt{2 π n}} = 1,$ or, to an approximation quite sufficient for all practical purposes, provided that $n$ is larger than $7$ $\Label [e q n] (b) \upshape (b) n! = {(\frac{n}{e})}^{n} \sqrt{2 π n} .$

For a proof of this relation and a discussion of its limits of accuracy a treatise on probability must be consulted.

On substitution in [eqn:(170)] (170) this gives $W = \frac{{(\frac{N}{e})}^{N}}{{(\frac{N_{1}}{e})}^{N_{1}} \cdot {(\frac{N_{2}}{e})}^{N_{2}} \dots} \cdot \frac{\sqrt{2 π N}}{\sqrt{2 π N_{1}} \cdot \sqrt{2 π N_{2}} \dots} .$ On account of [eqn:(165)] (165) this reduces at once to $\frac{N_{N}}{N_{1}^{N_{1}} N_{2}^{N_{2}} \dots} \cdot \frac{\sqrt{2 π N}}{\sqrt{2 π N_{1}} \cdot \sqrt{2 π N_{2}} \dots} .$ Passing now to the logarithmic expression we get

$S = k \log W = k [N \log N - N_{1} \log N_{1} - N_{2} \log N_{2} - \dots + \log \sqrt{2 π N} - \log \sqrt{2 π N_{1}} - \log \sqrt{2 π N_{2}} - \dots],$

$S = k \log W = k [(N \log N - \log \sqrt{2 π N}) + (N_{1} \log N_{1} - \log \sqrt{2 π N_{1}}) + (N_{2} \log N_{2} - \log \sqrt{2 π N_{2}}) + \dots] . S = k \log W$

$= k [(N \log N + \log \sqrt{2 π N}) - (N_{1} \log N_{1} + \log \sqrt{2 π N_{1}}) - (N_{2} \log N_{2} + \log \sqrt{2 π N_{2}}) - \dots] .}$

plus 0.75em minus 0.25em Now, for a large value of $N_{i}$ , the term $N_{i} \log N_{i}$ is very much larger than $\log \sqrt{2 π N_{i}}$ , as is seen by writing the latter in the form $\frac{1}{2} \log 2 π + \frac{1}{2} \log N_{i}$ . Hence the last expression will, with a fair approximation, reduce to $S = k \log W = k [N \log N - N_{1} \log N_{1} - N_{2} \log N_{2} - \dots] .$

Introducing now the values of the densities of distribution $w$ by means of the relation $N_{i} = w_{i} N$ we obtain $S = k \log W = k N [\log N - w_{1} \log N_{1} - w_{2} \log N_{2} - \dots],$ or, since $w_{1} + w_{2} + w_{3} + \dots = 1,$ and hence $(w_{1} + w_{2} + w_{3} + \dots) \log N = \log N,$ and $\log N - \log N_{1} = \log \frac{N}{N_{1}} = \log \frac{1}{w_{1}} = - \log w_{1},$ we obtain by substitution, after one or two simple transformations $S = k \log W = - k N \sum w_{1} \log w_{1},$ a relation which is identical with [eqn:(173)] (173).

The statements of [sect:143.] Sec. 143 may be proven in a similar manner. From [eqn:(232)] (232) we get at once $S = k \log W_{m} = k \log \frac{(N + P - 1)!}{(N - 1)! P!}$ Now $\log (N - 1)! = \log N! - \log N,$ and, for large values of $N$ , $\log N$ is negligible compared with $\log N!$ . Applying the same reasoning to the numerator we may without appreciable error write $S = k \log W_{m} = k \log \frac{(N + P)!}{N! P!} .$ Substituting now for $(N + P)!$ , $N!$ , and $P!$ their values from [eqn:(b)] (b) and omitting, as was previously shown to be approximately correct, the terms arising from the $\sqrt{2 π (N + P)}$ etc., we get, since the terms containing $e$ cancel out

$S & = k [(N + P) \log (N + P) - N \log N - P \log P] & = k [(N + P) \log \frac{N + P}{N} + P \log N - P \log P] & = k N [(\frac{P}{N} + 1) \log (\frac{P}{N} + 1) - \frac{P}{N} \log \frac{P}{N}] .$

This is the relation of [sect:143.] Sec. 143.

IIReferences

Among general papers treating of the application of the theory of quanta to different parts of physics are:

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