Appendix I

On Deductions from Stirling’s Formula

The formula is

(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 $7$

(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 (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 (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

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

or,

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

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 (173).

The statements of Sec. 143 may be proven in a similar manner. From (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 (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

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

This is the relation of Sec. 143.