Editing Stirling's approximation
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James Stirling (1692-1770, Scotland) | |||
:<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> | ||
Because of [http://en.wikipedia.org/wiki/Euler-Maclaurin_formula Euler-MacLaurin formula] | |||
:<math>\sum_{k=1}^N \ln k=\int_1^N \ln x\,dx+\sum_{k=1}^p\frac{B_{2k}}{2k(2k-1)}\left(\frac{1}{n^{2k-1}}-1\right)+R ,</math> | :<math>\sum_{k=1}^N \ln k=\int_1^N \ln x\,dx+\sum_{k=1}^p\frac{B_{2k}}{2k(2k-1)}\left(\frac{1}{n^{2k-1}}-1\right)+R ,</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 | ||
:<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> | ||
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| N || N! (exact) || N! (Stirling) || Error (%) | | 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 | |||
|10 || 3628800 || 3598695.62 || | |||
|} | |} | ||
As one usually deals with number of the order of the [[Avogadro constant ]](<math>10^{23}</math>) this formula is essentially exact. | |||
==Applications in statistical mechanics== | ==Applications in statistical mechanics== | ||
*[[Ideal gas Helmholtz energy function]] | *[[Ideal gas Helmholtz energy function]] | ||
[[Category: Mathematics]] | [[Category: Mathematics]] |