Appendix I

On Deductions from Stirling’s Formula

The formula is

lim_{n = \infty} \frac{n!}{n^{n} e^{- n} \sqrt{2 π n}} = 1,

(a)

or, to an approximation quite suﬃcient for all practical purposes, provided that $n$ is larger than $7$

n! = {(\frac{n}{e})}^{n} \sqrt{2 π n} .

(b)

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.

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