Stirling's approximation: Difference between revisions

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'''Stirling's approximation''' was invented by the Scottish mathematician James Stirling (1692-1770).
'''Stirling's approximation''' is named after the Scottish mathematician James Stirling (1692-1770).


:<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k .</math>
:<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k .</math>
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:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>
:<math>\ln N! \sim \int_1^N \ln x\,dx \sim N \ln N -N .</math>


after some further manipulation one arrives at  
after some further manipulation one arrives at (apparently  Stirling's contribution was the prefactor of <math>\sqrt{2 \pi}</math>)


:<math>N! = \sqrt{2 \pi N} \; N^{N} e^{-N} e^{\lambda_N}</math>
:<math>N! = \sqrt{2 \pi N} \; N^{N} e^{-N} e^{\lambda_N}</math>

Revision as of 17:28, 5 November 2008

Stirling's approximation is named after the Scottish mathematician James Stirling (1692-1770).

using Euler-MacLaurin formula one has

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,

after some further manipulation one arrives at (apparently Stirling's contribution was the prefactor of )

where

For example:

N N! (exact) N! (Stirling) Error (%)
2 2 1.91900435 4.22
3 6 5.83620959 2.81
4 24 23.5061751 2.10
5 120 118.019168 1.67
6 720 710.078185 1.40
7 5040 4980.39583 1.20
8 40320 39902.3955 1.05
9 362880 359536.873 0.93
10 3628800 3598695.62 0.84

When one is dealing with numbers of the order of the Avogadro constant () this formula is essentially exact. In computer simulations the number of atoms or molecules (N) is invariably greater than 100; for N=100 the percentage error is approximately 0.083%.

Applications in statistical mechanics