Stirling's approximation

James Stirling (1692-1770, Scotland)

$\ln N!=\ln 1+\ln 2+\ln 3+...+\ln N=\sum _{k=1}^N\ln k.$

Because of Euler-MacLaurin formula

$\sum _{k=1}^N\ln k=\underset{1}{\overset{N}{\int }}\ln xdx+\sum _{k=1}^p\frac{B_{2k}}{2k(2k-1)}(\frac{1}{n^{2k-1}}-1)+R,$

where B₁ = −1/2, B₂ = 1/6, B₃ = 0, B₄ = −1/30, B₅ = 0, B₆ = 1/42, B₇ = 0, B₈ = −1/30, ... are the Bernoulli numbers, and R is an error term which is normally small for suitable values of p.

Then, for large N,

$\ln N!\sim \underset{1}{\overset{N}{\int }}\ln xdx\sim N\ln N-N.$

after some further manipulation one arrives at

$N!=\sqrt{2\pi N}{N}^N{e}^{-N}{e}^{{\lambda }_N}$

where

$\frac{1}{12N+1}<{\lambda }_N<\frac{1}{12N}.$

For example:

N	N! (exact)	N! (Stirling)	Error (%)
5	120	118.019168	1.016
6	720	710.078185	1.014
7	5040	4980.39583	1.012
8	40320	39902.3955	1.010
9	362880	359536.873	1.009
10	3628800	3598695.62	1.008

As one usually deals with number of the order of the Avogadro constant ($10^{23}$) this formula is essentially exact.

Applications in statistical mechanics

• Ideal gas Helmholtz energy function